Subgroups — C₆₇×#61108 #30740359

Order 17152.

2592 conjugacy classes of subgroups.

The subgroup lattice has 912 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3586	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3584	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#578×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3293	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3290	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3289	C₂	1	8	yes	maximal, complemented
7	C₆₇×#3156	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3069	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3067	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#473×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3022	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3020	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2957	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2955	C₂	1	1	yes	maximal, characteristic, complemented
15	#585×C₆₇	C₂²	1	2	yes	—
16	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	complemented
17	#578×C₆₇	—	2	4	no	—
18	#578×C₆₇	—	2	2	no	—
19	#578×C₆₇	—	2	2	no	—
20	#578×C₆₇	—	2	1	no	—
21	#578×C₆₇	C₂²	1	4	yes	complemented
22	#578×C₆₇	C₂²	1	8	yes	complemented
23	#578×C₆₇	C₂²	1	4	yes	complemented
24	#578×C₆₇	C₂²	1	2	yes	complemented
25	#577×C₆₇	—	2	4	no	—
26	#577×C₆₇	—	2	2	no	—
27	#577×C₆₇	C₂²	1	4	yes	complemented
28	#577×C₆₇	C₂²	1	8	yes	complemented
29	#576×C₆₇	—	2	1	no	—
30	#576×C₆₇	C₂²	1	2	yes	complemented
31	#575×C₆₇	—	2	4	no	—
32	#575×C₆₇	—	2	2	no	—
33	#575×C₆₇	C₂²	1	2	yes	—
34	#575×C₆₇	C₂²	1	1	yes	characteristic
35	#575×C₆₇	C₂²	1	2	yes	complemented
36	C₂×#137×C₆₇	—	2	4	no	—
37	C₂×#137×C₆₇	—	2	2	no	—
38	C₂×#137×C₆₇	C₂²	1	2	yes	—
39	C₂×#137×C₆₇	C₂²	1	2	yes	complemented
40	C₂×#137×C₆₇	C₂²	1	2	yes	complemented
41	C₂×#135×C₆₇	—	2	2	no	—
42	C₂×#135×C₆₇	—	2	1	no	—
43	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
45	C₂²×Q₁₆×C₆₇	—	2	1	no	—
46	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
47	#568×C₆₇	C₂²	1	2	yes	complemented
48	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic, complemented
50	#536×C₆₇	C₂²	1	2	yes	complemented
51	#497×C₆₇	C₂²	1	4	yes	complemented
52	#495×C₆₇	C₂²	1	2	yes	complemented
53	#494×C₆₇	C₂²	1	2	yes	complemented
54	#473×C₆₇	C₂²	1	4	yes	complemented
55	#473×C₆₇	C₂²	1	8	yes	complemented
56	#473×C₆₇	C₂²	1	4	yes	complemented
57	#473×C₆₇	C₂²	1	2	yes	complemented
58	#472×C₆₇	C₂²	1	2	yes	complemented
59	#471×C₆₇	C₂²	1	4	yes	complemented
60	#471×C₆₇	C₂²	1	8	yes	complemented
61	#456×C₆₇	C₂²	1	8	yes	complemented
62	#456×C₆₇	C₂²	1	8	yes	complemented
63	#455×C₆₇	C₂²	1	8	yes	complemented
64	#454×C₆₇	C₂²	1	8	yes	complemented
65	#444×C₆₇	C₂²	1	4	yes	complemented
66	#444×C₆₇	C₂²	1	4	yes	complemented
67	#443×C₆₇	C₂²	1	4	yes	complemented
68	#443×C₆₇	C₂²	1	4	yes	complemented
69	#430×C₆₇	C₂²	1	1	yes	characteristic, complemented
70	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic, complemented
71	#421×C₆₇	C₂²	1	1	yes	characteristic
72	#421×C₆₇	C₂²	1	1	yes	characteristic, complemented
73	#421×C₆₇	C₂²	1	2	yes	complemented
74	#413×C₆₇	C₂²	1	2	yes	complemented
75	C₂×#101×C₆₇	C₂²	1	2	yes	complemented
76	#405×C₆₇	C₂²	1	1	yes	characteristic, complemented
77	2₋¹⁺⁴×C₆₇	—	2	4	no	—
78	2₋¹⁺⁴×C₆₇	—	2	4	no	—
79	2₋¹⁺⁴×C₆₇	—	2	2	no	—
80	2₋¹⁺⁴×C₆₇	—	2	2	no	—
81	2₋¹⁺⁴×C₆₇	—	4	2	no	—
82	2₋¹⁺⁴×C₆₇	C₂³	1	8	yes	—
83	2₋¹⁺⁴×C₆₇	C₂³	1	4	yes	—
84	H(5,𝔽₂)×C₆₇	—	2	2	no	—
85	H(5,𝔽₂)×C₆₇	—	2	2	no	—
86	H(5,𝔽₂)×C₆₇	—	4	2	no	—
87	H(5,𝔽₂)×C₆₇	C₂³	1	4	yes	—
88	C₂×#41×C₆₇	—	2	4	no	—
89	C₂×#41×C₆₇	—	2	4	no	—
90	C₂×#41×C₆₇	—	2	2	no	—
91	C₂×#41×C₆₇	—	2	2	no	—
92	C₂×#41×C₆₇	—	4	2	no	—
93	C₂×#41×C₆₇	—	4	2	no	—
94	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×#41×C₆₇	C₂³	1	2	yes	—
97	C₂×#41×C₆₇	C₂³	1	2	yes	—
98	C₂²×Q₈×C₆₇	—	2	2	no	—
99	C₂²×Q₈×C₆₇	—	2	2	no	—
100	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
101	#137×C₆₇	—	2	4	no	—
102	#137×C₆₇	—	2	4	no	—
103	#137×C₆₇	—	2	2	no	—
104	#137×C₆₇	—	2	2	no	—
105	#137×C₆₇	—	2	8	no	—
106	#137×C₆₇	—	2	4	no	—
107	#137×C₆₇	—	2	8	no	—
108	#137×C₆₇	—	2	8	no	—
109	#137×C₆₇	—	2	8	no	—
110	#137×C₆₇	—	2	4	no	—
111	#137×C₆₇	—	2	8	no	—
112	#137×C₆₇	—	2	4	no	—
113	#137×C₆₇	—	2	4	no	—
114	#137×C₆₇	D₄	1	2	yes	—
115	#137×C₆₇	D₄	1	2	yes	—
116	#137×C₆₇	C₂³	1	8	yes	—
117	#137×C₆₇	C₂³	1	8	yes	—
118	#137×C₆₇	C₂³	1	4	yes	—
119	#137×C₆₇	C₂³	1	4	yes	—
120	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
121	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
122	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
123	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
124	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
125	AGL(1,ℤ/8)×C₆₇	D₄	1	2	yes	—
126	AGL(1,ℤ/8)×C₆₇	D₄	1	2	yes	—
127	AGL(1,ℤ/8)×C₆₇	C₂³	1	4	yes	—
128	AGL(1,ℤ/8)×C₆₇	C₂³	1	4	yes	—
129	#135×C₆₇	—	2	4	no	—
130	#135×C₆₇	—	2	4	no	—
131	#135×C₆₇	—	2	2	no	—
132	#135×C₆₇	—	2	4	no	—
133	#135×C₆₇	—	2	8	no	—
134	#135×C₆₇	—	2	2	no	—
135	#135×C₆₇	—	2	4	no	—
136	#135×C₆₇	—	2	4	no	—
137	#135×C₆₇	—	2	8	no	—
138	#135×C₆₇	—	2	1	no	—
139	#135×C₆₇	—	2	1	no	—
140	#135×C₆₇	—	2	1	no	—
141	#135×C₆₇	—	2	1	no	—
142	#135×C₆₇	D₄	1	2	yes	—
143	#135×C₆₇	D₄	1	2	yes	—
144	#135×C₆₇	C₂³	1	2	yes	—
145	#135×C₆₇	C₂³	1	2	yes	—
146	#135×C₆₇	C₂³	1	4	yes	—
147	C₂×Q₁₆×C₆₇	—	2	2	no	—
148	C₂×Q₁₆×C₆₇	—	2	2	no	—
149	C₂×Q₁₆×C₆₇	—	2	4	no	—
150	C₂×Q₁₆×C₆₇	—	2	4	no	—
151	C₂×Q₁₆×C₆₇	—	2	2	no	—
152	C₂×Q₁₆×C₆₇	—	2	8	no	—
153	C₂×Q₁₆×C₆₇	—	2	1	no	—
154	C₂×Q₁₆×C₆₇	—	2	1	no	—
155	C₂×Q₁₆×C₆₇	—	2	1	no	—
156	C₂×Q₁₆×C₆₇	—	2	1	no	—
157	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
158	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
159	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
160	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
161	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
162	C₂×SD₁₆×C₆₇	—	2	2	no	—
163	C₂×SD₁₆×C₆₇	—	2	2	no	—
164	C₂×SD₁₆×C₆₇	—	2	4	no	—
165	C₂×SD₁₆×C₆₇	—	2	8	no	—
166	C₂×SD₁₆×C₆₇	D₄	1	2	yes	—
167	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
168	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
169	C₂×D₈×C₆₇	—	2	2	no	—
170	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
171	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
172	#131×C₆₇	—	2	4	no	—
173	#131×C₆₇	—	2	1	no	—
174	#131×C₆₇	—	2	1	no	—
175	#131×C₆₇	—	2	1	no	—
176	#131×C₆₇	—	2	1	no	—
177	#131×C₆₇	C₂³	1	2	yes	—
178	#131×C₆₇	C₂³	1	8	yes	—
179	#131×C₆₇	C₂³	1	2	yes	—
180	#131×C₆₇	C₂³	1	2	yes	—
181	#131×C₆₇	C₂³	1	2	yes	—
182	C₂×M₁₆×C₆₇	—	2	4	no	—
183	C₂×M₁₆×C₆₇	—	2	1	no	—
184	C₂×M₁₆×C₆₇	—	2	1	no	—
185	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
186	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
187	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
188	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
189	C₂²×C₈×C₆₇	—	2	1	no	—
190	C₂²×C₈×C₆₇	—	2	1	no	—
191	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
192	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
193	#124×C₆₇	C₂³	1	4	yes	—
194	#123×C₆₇	—	4	2	no	—
195	C₂²⋊Q₈×C₆₇	—	4	2	no	—
196	#117×C₆₇	C₂³	1	1	yes	characteristic
197	#108×C₆₇	C₂³	1	2	yes	—
198	#108×C₆₇	C₂³	1	4	yes	—
199	#108×C₆₇	C₂³	1	2	yes	—
200	#104×C₆₇	C₂³	1	4	yes	—
201	#104×C₆₇	C₂³	1	8	yes	—
202	#104×C₆₇	C₂³	1	4	yes	—
203	#101×C₆₇	C₂³	1	8	yes	—
204	#101×C₆₇	C₂³	1	4	yes	—
205	#100×C₆₇	C₂³	1	4	yes	—
206	#97×C₆₇	C₂³	1	2	yes	—
207	C₄×C₈×C₆₇	C₂³	1	2	yes	—
208	#41×C₆₇	—	2	1	no	—
209	#41×C₆₇	—	2	2	no	—
210	#41×C₆₇	—	2	1	no	—
211	#41×C₆₇	—	2	1	no	—
212	#41×C₆₇	—	2	2	no	—
213	#41×C₆₇	—	2	1	no	—
214	#41×C₆₇	—	2	4	no	—
215	#41×C₆₇	—	2	4	no	—
216	#41×C₆₇	—	2	4	no	—
217	#41×C₆₇	—	2	8	no	—
218	#41×C₆₇	—	2	4	no	—
219	#41×C₆₇	—	2	8	no	—
220	#41×C₆₇	—	2	2	no	—
221	#41×C₆₇	—	2	2	no	—
222	#41×C₆₇	—	2	2	no	—
223	#41×C₆₇	—	2	2	no	—
224	#41×C₆₇	—	4	4	no	—
225	#41×C₆₇	—	4	4	no	—
226	#41×C₆₇	—	4	8	no	—
227	#41×C₆₇	—	4	4	no	—
228	#41×C₆₇	—	4	4	no	—
229	#41×C₆₇	—	4	4	no	—
230	#41×C₆₇	—	4	4	no	—
231	#41×C₆₇	—	4	8	no	—
232	#41×C₆₇	—	4	2	no	—
233	#41×C₆₇	—	4	2	no	—
234	#41×C₆₇	C₂×D₄	1	2	yes	—
235	#41×C₆₇	C₂×D₄	1	4	yes	—
236	#41×C₆₇	C₂×D₄	1	2	yes	—
237	#41×C₆₇	C₂⁴	1	2	yes	—
238	#41×C₆₇	C₂⁴	1	4	yes	—
239	#41×C₆₇	C₂⁴	1	2	yes	—
240	C₂×Q₈×C₆₇	—	2	2	no	—
241	C₂×Q₈×C₆₇	—	2	8	no	—
242	C₂×Q₈×C₆₇	—	2	4	no	—
243	C₂×Q₈×C₆₇	—	2	8	no	—
244	C₂×Q₈×C₆₇	—	2	4	no	—
245	C₂×Q₈×C₆₇	—	2	2	no	—
246	C₂×Q₈×C₆₇	—	2	2	no	—
247	C₂×Q₈×C₆₇	—	2	2	no	—
248	C₂×Q₈×C₆₇	—	2	2	no	—
249	C₂×Q₈×C₆₇	—	4	2	no	—
250	C₂×Q₈×C₆₇	—	4	4	no	—
251	C₂×Q₈×C₆₇	—	4	4	no	—
252	C₂×Q₈×C₆₇	—	4	2	no	—
253	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
254	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
255	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
256	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
257	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
258	C₂×D₄×C₆₇	—	2	1	no	—
259	C₂×D₄×C₆₇	—	2	2	no	—
260	C₂×D₄×C₆₇	—	2	1	no	—
261	C₂×D₄×C₆₇	—	2	4	no	—
262	C₂×D₄×C₆₇	—	2	4	no	—
263	C₂×D₄×C₆₇	—	2	2	no	—
264	C₂×D₄×C₆₇	—	4	4	no	—
265	C₂×D₄×C₆₇	—	4	2	no	—
266	C₂×D₄×C₆₇	—	4	4	no	—
267	C₂×D₄×C₆₇	—	4	4	no	—
268	C₂×D₄×C₆₇	—	4	2	no	—
269	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
270	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
271	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
272	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
273	C₂²×C₄×C₆₇	—	2	1	no	—
274	C₂²×C₄×C₆₇	—	2	1	no	—
275	C₂²×C₄×C₆₇	—	2	2	no	—
276	C₂²×C₄×C₆₇	—	2	2	no	—
277	C₂²×C₄×C₆₇	—	2	2	no	—
278	C₂²×C₄×C₆₇	—	4	4	no	—
279	C₂²×C₄×C₆₇	—	4	2	no	—
280	C₂²×C₄×C₆₇	—	4	2	no	—
281	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
282	Q₁₆×C₆₇	—	2	8	no	—
283	Q₁₆×C₆₇	—	2	8	no	—
284	Q₁₆×C₆₇	—	2	4	no	—
285	Q₁₆×C₆₇	—	2	4	no	—
286	Q₁₆×C₆₇	—	2	2	no	—
287	Q₁₆×C₆₇	—	2	2	no	—
288	Q₁₆×C₆₇	—	2	1	no	—
289	Q₁₆×C₆₇	—	2	2	no	—
290	Q₁₆×C₆₇	—	2	2	no	—
291	Q₁₆×C₆₇	—	2	8	no	—
292	Q₁₆×C₆₇	—	2	1	no	—
293	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
294	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
295	Q₁₆×C₆₇	C₂×D₄	1	4	yes	—
296	SD₁₆×C₆₇	—	2	4	no	—
297	SD₁₆×C₆₇	—	2	4	no	—
298	SD₁₆×C₆₇	—	2	8	no	—
299	SD₁₆×C₆₇	—	2	4	no	—
300	SD₁₆×C₆₇	—	2	8	no	—
301	SD₁₆×C₆₇	—	2	4	no	—
302	SD₁₆×C₆₇	—	2	2	no	—
303	SD₁₆×C₆₇	—	2	8	no	—
304	SD₁₆×C₆₇	—	2	2	no	—
305	SD₁₆×C₆₇	C₂×D₄	1	4	yes	—
306	SD₁₆×C₆₇	C₂×D₄	1	4	yes	—
307	SD₁₆×C₆₇	C₂×D₄	1	4	yes	—
308	SD₁₆×C₆₇	C₂×D₄	1	4	yes	—
309	D₈×C₆₇	—	2	4	no	—
310	D₈×C₆₇	—	2	4	no	—
311	D₈×C₆₇	—	2	1	no	—
312	D₈×C₆₇	—	2	1	no	—
313	D₈×C₆₇	C₂×D₄	1	2	yes	—
314	D₈×C₆₇	C₂×D₄	1	2	yes	—
315	D₈×C₆₇	C₂×D₄	1	4	yes	—
316	M₁₆×C₆₇	—	2	2	no	—
317	M₁₆×C₆₇	—	2	1	no	—
318	M₁₆×C₆₇	—	2	1	no	—
319	M₁₆×C₆₇	—	2	2	no	—
320	M₁₆×C₆₇	—	2	2	no	—
321	M₁₆×C₆₇	—	2	8	no	—
322	M₁₆×C₆₇	—	2	2	no	—
323	M₁₆×C₆₇	—	2	2	no	—
324	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
325	M₁₆×C₆₇	C₂×D₄	1	2	yes	—
326	M₁₆×C₆₇	C₂⁴	1	2	yes	—
327	M₁₆×C₆₇	C₂⁴	1	4	yes	—
328	M₁₆×C₆₇	C₂⁴	1	2	yes	—
329	C₂×C₈×C₆₇	—	2	1	no	—
330	C₂×C₈×C₆₇	—	2	1	no	—
331	C₂×C₈×C₆₇	—	2	2	no	—
332	C₂×C₈×C₆₇	—	2	2	no	—
333	C₂×C₈×C₆₇	—	2	2	no	—
334	C₂×C₈×C₆₇	—	2	2	no	—
335	C₂×C₈×C₆₇	—	2	2	no	—
336	C₂×C₈×C₆₇	—	2	8	no	—
337	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
338	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
339	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
340	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
341	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
342	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
343	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
344	C₄⋊C₄×C₆₇	—	2	1	no	—
345	C₄⋊C₄×C₆₇	—	4	4	no	—
346	C₂²⋊C₄×C₆₇	—	2	1	no	—
347	C₂²⋊C₄×C₆₇	—	4	4	no	—
348	C₄²×C₆₇	C₂⁴	1	2	yes	—
349	C₂³×C₆₇	—	2	1	no	—
350	C₂³×C₆₇	—	2	2	no	—
351	C₂³×C₆₇	—	2	1	no	—
352	C₂³×C₆₇	—	2	2	no	—
353	C₂³×C₆₇	—	4	4	no	—
354	C₂³×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
355	Q₈×C₆₇	—	2	1	no	—
356	Q₈×C₆₇	—	2	2	no	—
357	Q₈×C₆₇	—	2	1	no	—
358	Q₈×C₆₇	—	2	4	no	—
359	Q₈×C₆₇	—	2	2	no	—
360	Q₈×C₆₇	—	2	4	no	—
361	Q₈×C₆₇	—	2	2	no	—
362	Q₈×C₆₇	—	4	8	no	—
363	Q₈×C₆₇	—	4	4	no	—
364	Q₈×C₆₇	—	4	2	no	—
365	Q₈×C₆₇	—	4	2	no	—
366	Q₈×C₆₇	C₂²×D₄	1	2	yes	—
367	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
368	Q₈×C₆₇	C₂²×D₄	1	2	yes	—
369	D₄×C₆₇	—	2	2	no	—
370	D₄×C₆₇	—	2	4	no	—
371	D₄×C₆₇	—	2	2	no	—
372	D₄×C₆₇	—	2	2	no	—
373	D₄×C₆₇	—	2	4	no	—
374	D₄×C₆₇	—	2	2	no	—
375	D₄×C₆₇	—	2	1	no	—
376	D₄×C₆₇	—	2	2	no	—
377	D₄×C₆₇	—	2	1	no	—
378	D₄×C₆₇	—	2	2	no	—
379	D₄×C₆₇	—	2	2	no	—
380	D₄×C₆₇	—	4	4	no	—
381	D₄×C₆₇	—	4	4	no	—
382	D₄×C₆₇	—	4	4	no	—
383	D₄×C₆₇	—	4	8	no	—
384	D₄×C₆₇	—	4	2	no	—
385	D₄×C₆₇	—	4	2	no	—
386	D₄×C₆₇	C₂²×D₄	1	2	yes	—
387	D₄×C₆₇	C₂²×D₄	1	4	yes	—
388	D₄×C₆₇	C₂²×D₄	1	2	yes	—
389	C₂×C₄×C₆₇	—	2	1	no	—
390	C₂×C₄×C₆₇	—	2	2	no	—
391	C₂×C₄×C₆₇	—	2	1	no	—
392	C₂×C₄×C₆₇	—	2	2	no	—
393	C₂×C₄×C₆₇	—	2	4	no	—
394	C₂×C₄×C₆₇	—	2	2	no	—
395	C₂×C₄×C₆₇	—	2	2	no	—
396	C₂×C₄×C₆₇	—	2	4	no	—
397	C₂×C₄×C₆₇	—	2	2	no	—
398	C₂×C₄×C₆₇	—	2	2	no	—
399	C₂×C₄×C₆₇	—	2	4	no	—
400	C₂×C₄×C₆₇	—	2	2	no	—
401	C₂×C₄×C₆₇	—	4	4	no	—
402	C₂×C₄×C₆₇	—	4	4	no	—
403	C₂×C₄×C₆₇	—	4	4	no	—
404	C₂×C₄×C₆₇	—	4	8	no	—
405	C₂×C₄×C₆₇	—	4	8	no	—
406	C₂×C₄×C₆₇	—	4	2	no	—
407	C₂×C₄×C₆₇	—	4	2	no	—
408	C₂×C₄×C₆₇	—	4	2	no	—
409	C₂×C₄×C₆₇	—	4	2	no	—
410	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
411	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
412	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
413	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
414	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
415	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
416	C₈×C₆₇	—	2	2	no	—
417	C₈×C₆₇	—	2	4	no	—
418	C₈×C₆₇	—	2	2	no	—
419	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
420	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
421	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
422	C₈×C₆₇	C₂²×D₄	1	4	yes	—
423	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
424	C₂²×C₆₇	—	2	2	no	—
425	C₂²×C₆₇	—	2	4	no	—
426	C₂²×C₆₇	—	2	2	no	—
427	C₂²×C₆₇	—	4	1	no	—
428	C₂²×C₆₇	—	4	2	no	—
429	C₂²×C₆₇	—	4	2	no	—
430	C₂²×C₆₇	—	8	2	no	—
431	C₂²×C₆₇	—	8	1	no	—
432	C₂²×C₆₇	—	8	2	no	—
433	C₂²×C₆₇	—	16	4	no	—
434	C₂²×C₆₇	D₄²	1	1	yes	characteristic
435	C₂²×C₆₇	D₄²	1	1	yes	characteristic
436	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
437	C₄×C₆₇	—	2	2	no	—
438	C₄×C₆₇	—	2	4	no	—
439	C₄×C₆₇	—	2	2	no	—
440	C₄×C₆₇	—	4	2	no	—
441	C₄×C₆₇	—	4	2	no	—
442	C₄×C₆₇	—	4	4	no	—
443	C₄×C₆₇	D₄²	1	1	yes	characteristic
444	C₄×C₆₇	D₄²	1	1	yes	characteristic
445	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
446	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
447	#61108	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
448	C₂×C₆₇	—	2	1	no	—
449	C₂×C₆₇	—	2	1	no	—
450	C₂×C₆₇	—	2	1	no	—
451	C₂×C₆₇	—	4	2	no	—
452	C₂×C₆₇	—	4	4	no	—
453	C₂×C₆₇	—	4	2	no	—
454	C₂×C₆₇	—	8	2	no	—
455	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
456	#3586	C₂×C₆₇	1	2	yes	complemented
457	#3584	C₂×C₆₇	1	1	yes	characteristic, complemented
458	C₂×#578	C₂×C₆₇	1	1	yes	characteristic, complemented
459	#3293	C₂×C₆₇	1	4	yes	complemented
460	#3290	C₂×C₆₇	1	4	yes	complemented
461	#3289	C₂×C₆₇	1	8	yes	complemented
462	#3156	C₂×C₆₇	1	1	yes	characteristic, complemented
463	#3069	C₂×C₆₇	1	2	yes	complemented
464	#3067	C₂×C₆₇	1	1	yes	characteristic, complemented
465	C₂×#473	C₂×C₆₇	1	1	yes	characteristic, complemented
466	#3022	C₂×C₆₇	1	2	yes	complemented
467	#3020	C₂×C₆₇	1	2	yes	complemented
468	#2957	C₂×C₆₇	1	1	yes	characteristic, complemented
469	#2955	C₂×C₆₇	1	1	yes	characteristic, complemented
470	C₆₇	#61108	1	1	yes	characteristic, direct product factor, Sylow
471	#585	C₂²×C₆₇	1	2	yes	—
472	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	complemented
473	#578	—	2	4	no	—
474	#578	—	2	2	no	—
475	#578	—	2	2	no	—
476	#578	—	2	1	no	—
477	#578	C₂²×C₆₇	1	2	yes	complemented
478	#578	C₂²×C₆₇	1	8	yes	complemented
479	#578	C₂²×C₆₇	1	4	yes	complemented
480	#578	C₂²×C₆₇	1	4	yes	complemented
481	#577	—	2	4	no	—
482	#577	—	2	2	no	—
483	#577	C₂²×C₆₇	1	4	yes	complemented
484	#577	C₂²×C₆₇	1	8	yes	complemented
485	#576	—	2	1	no	—
486	#576	C₂²×C₆₇	1	2	yes	complemented
487	#575	—	2	4	no	—
488	#575	—	2	2	no	—
489	#575	C₂²×C₆₇	1	2	yes	—
490	#575	C₂²×C₆₇	1	1	yes	characteristic
491	#575	C₂²×C₆₇	1	2	yes	complemented
492	C₂×#137	—	2	4	no	—
493	C₂×#137	—	2	2	no	—
494	C₂×#137	C₂²×C₆₇	1	2	yes	—
495	C₂×#137	C₂²×C₆₇	1	2	yes	complemented
496	C₂×#137	C₂²×C₆₇	1	2	yes	complemented
497	C₂×#135	—	2	2	no	—
498	C₂×#135	—	2	1	no	—
499	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
500	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
501	C₂²×Q₁₆	—	2	1	no	—
502	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
503	#568	C₂²×C₆₇	1	2	yes	complemented
504	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
505	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic, complemented
506	#536	C₂²×C₆₇	1	2	yes	complemented
507	#497	C₂²×C₆₇	1	4	yes	complemented
508	#495	C₂²×C₆₇	1	2	yes	complemented
509	#494	C₂²×C₆₇	1	2	yes	complemented
510	#473	C₂²×C₆₇	1	4	yes	complemented
511	#473	C₂²×C₆₇	1	8	yes	complemented
512	#473	C₂²×C₆₇	1	4	yes	complemented
513	#473	C₂²×C₆₇	1	2	yes	complemented
514	#472	C₂²×C₆₇	1	2	yes	complemented
515	#471	C₂²×C₆₇	1	4	yes	complemented
516	#471	C₂²×C₆₇	1	8	yes	complemented
517	#456	C₂²×C₆₇	1	8	yes	complemented
518	#456	C₂²×C₆₇	1	8	yes	complemented
519	#455	C₂²×C₆₇	1	8	yes	complemented
520	#454	C₂²×C₆₇	1	8	yes	complemented
521	#444	C₂²×C₆₇	1	4	yes	complemented
522	#444	C₂²×C₆₇	1	4	yes	complemented
523	#443	C₂²×C₆₇	1	4	yes	complemented
524	#443	C₂²×C₆₇	1	4	yes	complemented
525	#430	C₂²×C₆₇	1	1	yes	characteristic, complemented
526	C₂×#108	C₂²×C₆₇	1	1	yes	characteristic, complemented
527	#421	C₂²×C₆₇	1	1	yes	characteristic
528	#421	C₂²×C₆₇	1	1	yes	characteristic, complemented
529	#421	C₂²×C₆₇	1	2	yes	complemented
530	#413	C₂²×C₆₇	1	2	yes	complemented
531	C₂×#101	C₂²×C₆₇	1	2	yes	complemented
532	#405	C₂²×C₆₇	1	1	yes	characteristic, complemented
533	2₋¹⁺⁴	—	2	4	no	—
534	2₋¹⁺⁴	—	2	4	no	—
535	2₋¹⁺⁴	—	2	2	no	—
536	2₋¹⁺⁴	—	2	2	no	—
537	2₋¹⁺⁴	—	4	2	no	—
538	2₋¹⁺⁴	C₂³×C₆₇	1	8	yes	—
539	2₋¹⁺⁴	C₂³×C₆₇	1	4	yes	—
540	H(5,𝔽₂)	—	2	2	no	—
541	H(5,𝔽₂)	—	2	2	no	—
542	H(5,𝔽₂)	—	4	2	no	—
543	H(5,𝔽₂)	C₂³×C₆₇	1	4	yes	—
544	C₂×#41	—	2	4	no	—
545	C₂×#41	—	2	4	no	—
546	C₂×#41	—	2	2	no	—
547	C₂×#41	—	2	2	no	—
548	C₂×#41	—	4	2	no	—
549	C₂×#41	—	4	2	no	—
550	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
551	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
552	C₂×#41	C₂³×C₆₇	1	2	yes	—
553	C₂×#41	C₂³×C₆₇	1	2	yes	—
554	C₂²×Q₈	—	2	2	no	—
555	C₂²×Q₈	—	2	2	no	—
556	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
557	#137	—	2	4	no	—
558	#137	—	2	4	no	—
559	#137	—	2	2	no	—
560	#137	—	2	2	no	—
561	#137	—	2	8	no	—
562	#137	—	2	4	no	—
563	#137	—	2	8	no	—
564	#137	—	2	8	no	—
565	#137	—	2	8	no	—
566	#137	—	2	4	no	—
567	#137	—	2	8	no	—
568	#137	—	2	4	no	—
569	#137	—	2	4	no	—
570	#137	D₄×C₆₇	1	2	yes	—
571	#137	D₄×C₆₇	1	2	yes	—
572	#137	C₂³×C₆₇	1	8	yes	—
573	#137	C₂³×C₆₇	1	8	yes	—
574	#137	C₂³×C₆₇	1	4	yes	—
575	#137	C₂³×C₆₇	1	4	yes	—
576	AGL(1,ℤ/8)	—	2	2	no	—
577	AGL(1,ℤ/8)	—	2	2	no	—
578	AGL(1,ℤ/8)	—	2	4	no	—
579	AGL(1,ℤ/8)	—	2	4	no	—
580	AGL(1,ℤ/8)	—	2	8	no	—
581	AGL(1,ℤ/8)	D₄×C₆₇	1	2	yes	—
582	AGL(1,ℤ/8)	D₄×C₆₇	1	2	yes	—
583	AGL(1,ℤ/8)	C₂³×C₆₇	1	4	yes	—
584	AGL(1,ℤ/8)	C₂³×C₆₇	1	4	yes	—
585	#135	—	2	4	no	—
586	#135	—	2	4	no	—
587	#135	—	2	2	no	—
588	#135	—	2	4	no	—
589	#135	—	2	8	no	—
590	#135	—	2	2	no	—
591	#135	—	2	4	no	—
592	#135	—	2	4	no	—
593	#135	—	2	8	no	—
594	#135	—	2	1	no	—
595	#135	—	2	1	no	—
596	#135	—	2	1	no	—
597	#135	—	2	1	no	—
598	#135	D₄×C₆₇	1	2	yes	—
599	#135	D₄×C₆₇	1	2	yes	—
600	#135	C₂³×C₆₇	1	2	yes	—
601	#135	C₂³×C₆₇	1	2	yes	—
602	#135	C₂³×C₆₇	1	4	yes	—
603	C₂×Q₁₆	—	2	2	no	—
604	C₂×Q₁₆	—	2	2	no	—
605	C₂×Q₁₆	—	2	4	no	—
606	C₂×Q₁₆	—	2	4	no	—
607	C₂×Q₁₆	—	2	2	no	—
608	C₂×Q₁₆	—	2	8	no	—
609	C₂×Q₁₆	—	2	1	no	—
610	C₂×Q₁₆	—	2	1	no	—
611	C₂×Q₁₆	—	2	1	no	—
612	C₂×Q₁₆	—	2	1	no	—
613	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
614	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
615	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
616	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
617	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
618	C₂×SD₁₆	—	2	2	no	—
619	C₂×SD₁₆	—	2	2	no	—
620	C₂×SD₁₆	—	2	4	no	—
621	C₂×SD₁₆	—	2	8	no	—
622	C₂×SD₁₆	D₄×C₆₇	1	2	yes	—
623	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
624	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
625	C₂×D₈	—	2	2	no	—
626	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
627	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
628	#131	—	2	4	no	—
629	#131	—	2	1	no	—
630	#131	—	2	1	no	—
631	#131	—	2	1	no	—
632	#131	—	2	1	no	—
633	#131	C₂³×C₆₇	1	2	yes	—
634	#131	C₂³×C₆₇	1	2	yes	—
635	#131	C₂³×C₆₇	1	2	yes	—
636	#131	C₂³×C₆₇	1	2	yes	—
637	#131	C₂³×C₆₇	1	8	yes	—
638	C₂×M₁₆	—	2	4	no	—
639	C₂×M₁₆	—	2	1	no	—
640	C₂×M₁₆	—	2	1	no	—
641	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
642	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
643	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
644	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
645	C₂²×C₈	—	2	1	no	—
646	C₂²×C₈	—	2	1	no	—
647	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
648	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
649	#124	C₂³×C₆₇	1	4	yes	—
650	#123	—	4	2	no	—
651	C₂²⋊Q₈	—	4	2	no	—
652	#117	C₂³×C₆₇	1	1	yes	characteristic
653	#108	C₂³×C₆₇	1	2	yes	—
654	#108	C₂³×C₆₇	1	2	yes	—
655	#108	C₂³×C₆₇	1	4	yes	—
656	#104	C₂³×C₆₇	1	4	yes	—
657	#104	C₂³×C₆₇	1	4	yes	—
658	#104	C₂³×C₆₇	1	8	yes	—
659	#101	C₂³×C₆₇	1	8	yes	—
660	#101	C₂³×C₆₇	1	4	yes	—
661	#100	C₂³×C₆₇	1	4	yes	—
662	#97	C₂³×C₆₇	1	2	yes	—
663	C₄×C₈	C₂³×C₆₇	1	2	yes	—
664	#41	—	2	1	no	—
665	#41	—	2	2	no	—
666	#41	—	2	1	no	—
667	#41	—	2	1	no	—
668	#41	—	2	2	no	—
669	#41	—	2	1	no	—
670	#41	—	2	4	no	—
671	#41	—	2	4	no	—
672	#41	—	2	4	no	—
673	#41	—	2	8	no	—
674	#41	—	2	4	no	—
675	#41	—	2	8	no	—
676	#41	—	2	2	no	—
677	#41	—	2	2	no	—
678	#41	—	2	2	no	—
679	#41	—	2	2	no	—
680	#41	—	4	4	no	—
681	#41	—	4	4	no	—
682	#41	—	4	8	no	—
683	#41	—	4	4	no	—
684	#41	—	4	4	no	—
685	#41	—	4	4	no	—
686	#41	—	4	4	no	—
687	#41	—	4	8	no	—
688	#41	—	4	2	no	—
689	#41	—	4	2	no	—
690	#41	C₂×D₄×C₆₇	1	2	yes	—
691	#41	C₂×D₄×C₆₇	1	4	yes	—
692	#41	C₂×D₄×C₆₇	1	2	yes	—
693	#41	C₂⁴×C₆₇	1	2	yes	—
694	#41	C₂⁴×C₆₇	1	4	yes	—
695	#41	C₂⁴×C₆₇	1	2	yes	—
696	C₂×Q₈	—	2	2	no	—
697	C₂×Q₈	—	2	8	no	—
698	C₂×Q₈	—	2	4	no	—
699	C₂×Q₈	—	2	8	no	—
700	C₂×Q₈	—	2	4	no	—
701	C₂×Q₈	—	2	2	no	—
702	C₂×Q₈	—	2	2	no	—
703	C₂×Q₈	—	2	2	no	—
704	C₂×Q₈	—	2	2	no	—
705	C₂×Q₈	—	4	2	no	—
706	C₂×Q₈	—	4	4	no	—
707	C₂×Q₈	—	4	4	no	—
708	C₂×Q₈	—	4	2	no	—
709	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
710	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
711	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
712	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
713	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
714	C₂×D₄	—	2	1	no	—
715	C₂×D₄	—	2	2	no	—
716	C₂×D₄	—	2	1	no	—
717	C₂×D₄	—	2	4	no	—
718	C₂×D₄	—	2	4	no	—
719	C₂×D₄	—	2	2	no	—
720	C₂×D₄	—	4	4	no	—
721	C₂×D₄	—	4	2	no	—
722	C₂×D₄	—	4	4	no	—
723	C₂×D₄	—	4	4	no	—
724	C₂×D₄	—	4	2	no	—
725	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
726	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
727	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
728	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
729	C₂²×C₄	—	2	1	no	—
730	C₂²×C₄	—	2	1	no	—
731	C₂²×C₄	—	2	2	no	—
732	C₂²×C₄	—	2	2	no	—
733	C₂²×C₄	—	2	2	no	—
734	C₂²×C₄	—	4	4	no	—
735	C₂²×C₄	—	4	2	no	—
736	C₂²×C₄	—	4	2	no	—
737	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
738	Q₁₆	—	2	8	no	—
739	Q₁₆	—	2	8	no	—
740	Q₁₆	—	2	4	no	—
741	Q₁₆	—	2	4	no	—
742	Q₁₆	—	2	2	no	—
743	Q₁₆	—	2	2	no	—
744	Q₁₆	—	2	1	no	—
745	Q₁₆	—	2	2	no	—
746	Q₁₆	—	2	2	no	—
747	Q₁₆	—	2	8	no	—
748	Q₁₆	—	2	1	no	—
749	Q₁₆	C₂×D₄×C₆₇	1	2	yes	—
750	Q₁₆	C₂×D₄×C₆₇	1	2	yes	—
751	Q₁₆	C₂×D₄×C₆₇	1	4	yes	—
752	SD₁₆	—	2	4	no	—
753	SD₁₆	—	2	4	no	—
754	SD₁₆	—	2	8	no	—
755	SD₁₆	—	2	4	no	—
756	SD₁₆	—	2	8	no	—
757	SD₁₆	—	2	4	no	—
758	SD₁₆	—	2	2	no	—
759	SD₁₆	—	2	8	no	—
760	SD₁₆	—	2	2	no	—
761	SD₁₆	C₂×D₄×C₆₇	1	4	yes	—
762	SD₁₆	C₂×D₄×C₆₇	1	4	yes	—
763	SD₁₆	C₂×D₄×C₆₇	1	4	yes	—
764	SD₁₆	C₂×D₄×C₆₇	1	4	yes	—
765	D₈	—	2	4	no	—
766	D₈	—	2	4	no	—
767	D₈	—	2	1	no	—
768	D₈	—	2	1	no	—
769	D₈	C₂×D₄×C₆₇	1	2	yes	—
770	D₈	C₂×D₄×C₆₇	1	2	yes	—
771	D₈	C₂×D₄×C₆₇	1	4	yes	—
772	M₁₆	—	2	2	no	—
773	M₁₆	—	2	1	no	—
774	M₁₆	—	2	1	no	—
775	M₁₆	—	2	2	no	—
776	M₁₆	—	2	2	no	—
777	M₁₆	—	2	8	no	—
778	M₁₆	—	2	2	no	—
779	M₁₆	—	2	2	no	—
780	M₁₆	C₂×D₄×C₆₇	1	2	yes	—
781	M₁₆	C₂×D₄×C₆₇	1	2	yes	—
782	M₁₆	C₂⁴×C₆₇	1	2	yes	—
783	M₁₆	C₂⁴×C₆₇	1	4	yes	—
784	M₁₆	C₂⁴×C₆₇	1	2	yes	—
785	C₂×C₈	—	2	1	no	—
786	C₂×C₈	—	2	1	no	—
787	C₂×C₈	—	2	2	no	—
788	C₂×C₈	—	2	2	no	—
789	C₂×C₈	—	2	2	no	—
790	C₂×C₈	—	2	2	no	—
791	C₂×C₈	—	2	2	no	—
792	C₂×C₈	—	2	8	no	—
793	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
794	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
795	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
796	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
797	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
798	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
799	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
800	C₄⋊C₄	—	2	1	no	—
801	C₄⋊C₄	—	4	4	no	—
802	C₂²⋊C₄	—	2	1	no	—
803	C₂²⋊C₄	—	4	4	no	—
804	C₄²	C₂⁴×C₆₇	1	2	yes	—
805	C₂³	—	2	1	no	—
806	C₂³	—	2	2	no	—
807	C₂³	—	2	1	no	—
808	C₂³	—	2	2	no	—
809	C₂³	—	4	4	no	—
810	C₂³	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
811	Q₈	—	2	1	no	—
812	Q₈	—	2	2	no	—
813	Q₈	—	2	1	no	—
814	Q₈	—	2	4	no	—
815	Q₈	—	2	2	no	—
816	Q₈	—	2	4	no	—
817	Q₈	—	2	2	no	—
818	Q₈	—	4	8	no	—
819	Q₈	—	4	4	no	—
820	Q₈	—	4	2	no	—
821	Q₈	—	4	2	no	—
822	Q₈	C₂²×D₄×C₆₇	1	2	yes	—
823	Q₈	C₂²×D₄×C₆₇	1	4	yes	—
824	Q₈	C₂²×D₄×C₆₇	1	2	yes	—
825	D₄	—	2	2	no	—
826	D₄	—	2	4	no	—
827	D₄	—	2	2	no	—
828	D₄	—	2	2	no	—
829	D₄	—	2	4	no	—
830	D₄	—	2	2	no	—
831	D₄	—	2	1	no	—
832	D₄	—	2	2	no	—
833	D₄	—	2	1	no	—
834	D₄	—	2	2	no	—
835	D₄	—	2	2	no	—
836	D₄	—	4	4	no	—
837	D₄	—	4	4	no	—
838	D₄	—	4	4	no	—
839	D₄	—	4	8	no	—
840	D₄	—	4	2	no	—
841	D₄	—	4	2	no	—
842	D₄	C₂²×D₄×C₆₇	1	2	yes	—
843	D₄	C₂²×D₄×C₆₇	1	4	yes	—
844	D₄	C₂²×D₄×C₆₇	1	2	yes	—
845	C₂×C₄	—	2	1	no	—
846	C₂×C₄	—	2	2	no	—
847	C₂×C₄	—	2	1	no	—
848	C₂×C₄	—	2	2	no	—
849	C₂×C₄	—	2	4	no	—
850	C₂×C₄	—	2	2	no	—
851	C₂×C₄	—	2	2	no	—
852	C₂×C₄	—	2	4	no	—
853	C₂×C₄	—	2	2	no	—
854	C₂×C₄	—	2	2	no	—
855	C₂×C₄	—	2	4	no	—
856	C₂×C₄	—	2	2	no	—
857	C₂×C₄	—	4	4	no	—
858	C₂×C₄	—	4	4	no	—
859	C₂×C₄	—	4	4	no	—
860	C₂×C₄	—	4	8	no	—
861	C₂×C₄	—	4	8	no	—
862	C₂×C₄	—	4	2	no	—
863	C₂×C₄	—	4	2	no	—
864	C₂×C₄	—	4	2	no	—
865	C₂×C₄	—	4	2	no	—
866	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
867	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
868	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
869	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
870	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
871	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
872	C₈	—	2	2	no	—
873	C₈	—	2	4	no	—
874	C₈	—	2	2	no	—
875	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
876	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
877	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
878	C₈	C₂²×D₄×C₆₇	1	4	yes	—
879	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
880	C₂²	—	2	2	no	—
881	C₂²	—	2	4	no	—
882	C₂²	—	2	2	no	—
883	C₂²	—	4	1	no	—
884	C₂²	—	4	2	no	—
885	C₂²	—	4	2	no	—
886	C₂²	—	8	2	no	—
887	C₂²	—	8	1	no	—
888	C₂²	—	8	2	no	—
889	C₂²	—	16	4	no	—
890	C₂²	D₄²×C₆₇	1	1	yes	characteristic
891	C₂²	D₄²×C₆₇	1	1	yes	characteristic
892	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
893	C₄	—	2	2	no	—
894	C₄	—	2	4	no	—
895	C₄	—	2	2	no	—
896	C₄	—	4	2	no	—
897	C₄	—	4	2	no	—
898	C₄	—	4	4	no	—
899	C₄	D₄²×C₆₇	1	1	yes	characteristic
900	C₄	D₄²×C₆₇	1	1	yes	characteristic
901	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
902	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
903	C₂	—	2	1	no	—
904	C₂	—	2	1	no	—
905	C₂	—	2	1	no	—
906	C₂	—	4	2	no	—
907	C₂	—	4	2	no	—
908	C₂	—	4	4	no	—
909	C₂	—	8	2	no	—
910	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	C₂×D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₆₇×#3586	C₂	semidirect
2. C₆₇×#3584	C₂	semidirect
3. C₂×#578×C₆₇	C₂	semidirect
4. C₆₇×#3293	C₂	semidirect
5. C₆₇×#3290	C₂	semidirect
6. C₆₇×#3289	C₂	semidirect
7. C₆₇×#3156	C₂	semidirect
8. C₆₇×#3069	C₂	semidirect
9. C₆₇×#3067	C₂	semidirect
10. C₂×#473×C₆₇	C₂	semidirect
11. C₆₇×#3022	C₂	semidirect
12. C₆₇×#3020	C₂	semidirect
13. C₆₇×#2957	C₂	semidirect
14. C₆₇×#2955	C₂	semidirect
16. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
21. #578×C₆₇	C₂²	semidirect
22. #578×C₆₇	C₂²	semidirect
23. #578×C₆₇	C₂²	semidirect
24. #578×C₆₇	C₂²	semidirect
27. #577×C₆₇	C₂²	semidirect
28. #577×C₆₇	C₂²	semidirect
30. #576×C₆₇	C₂²	semidirect
35. #575×C₆₇	C₂²	semidirect
39. C₂×#137×C₆₇	C₂²	semidirect
40. C₂×#137×C₆₇	C₂²	semidirect
46. C₂²×Q₁₆×C₆₇	C₂²	semidirect
47. #568×C₆₇	C₂²	semidirect
49. C₂×#131×C₆₇	C₂²	semidirect
50. #536×C₆₇	C₂²	semidirect
51. #497×C₆₇	C₂²	semidirect
52. #495×C₆₇	C₂²	semidirect
53. #494×C₆₇	C₂²	semidirect
54. #473×C₆₇	C₂²	semidirect
55. #473×C₆₇	C₂²	semidirect
56. #473×C₆₇	C₂²	semidirect
57. #473×C₆₇	C₂²	semidirect
58. #472×C₆₇	C₂²	semidirect
59. #471×C₆₇	C₂²	semidirect
60. #471×C₆₇	C₂²	semidirect
61. #456×C₆₇	C₂²	semidirect
62. #456×C₆₇	C₂²	semidirect
63. #455×C₆₇	C₂²	semidirect
64. #454×C₆₇	C₂²	semidirect
65. #444×C₆₇	C₂²	semidirect
66. #444×C₆₇	C₂²	semidirect
67. #443×C₆₇	C₂²	semidirect
68. #443×C₆₇	C₂²	semidirect
69. #430×C₆₇	C₂²	semidirect
70. C₂×#108×C₆₇	C₂²	semidirect
72. #421×C₆₇	C₂²	semidirect
73. #421×C₆₇	C₂²	semidirect
74. #413×C₆₇	C₂²	semidirect
75. C₂×#101×C₆₇	C₂²	semidirect
76. #405×C₆₇	C₂²	semidirect
447. #61108	C₆₇	direct (N × H)
456. #3586	C₂×C₆₇	semidirect
457. #3584	C₂×C₆₇	semidirect
458. C₂×#578	C₂×C₆₇	semidirect
459. #3293	C₂×C₆₇	semidirect
460. #3290	C₂×C₆₇	semidirect
461. #3289	C₂×C₆₇	semidirect
462. #3156	C₂×C₆₇	semidirect
463. #3069	C₂×C₆₇	semidirect
464. #3067	C₂×C₆₇	semidirect
465. C₂×#473	C₂×C₆₇	semidirect
466. #3022	C₂×C₆₇	semidirect
467. #3020	C₂×C₆₇	semidirect
468. #2957	C₂×C₆₇	semidirect
469. #2955	C₂×C₆₇	semidirect
470. C₆₇	#61108	direct (N × H)
472. C₂×2₋¹⁺⁴	C₂²×C₆₇	semidirect
477. #578	C₂²×C₆₇	semidirect
478. #578	C₂²×C₆₇	semidirect
479. #578	C₂²×C₆₇	semidirect
480. #578	C₂²×C₆₇	semidirect
483. #577	C₂²×C₆₇	semidirect
484. #577	C₂²×C₆₇	semidirect
486. #576	C₂²×C₆₇	semidirect
491. #575	C₂²×C₆₇	semidirect
495. C₂×#137	C₂²×C₆₇	semidirect
496. C₂×#137	C₂²×C₆₇	semidirect
502. C₂²×Q₁₆	C₂²×C₆₇	semidirect
503. #568	C₂²×C₆₇	semidirect
505. C₂×#131	C₂²×C₆₇	semidirect
506. #536	C₂²×C₆₇	semidirect
507. #497	C₂²×C₆₇	semidirect
508. #495	C₂²×C₆₇	semidirect
509. #494	C₂²×C₆₇	semidirect
510. #473	C₂²×C₆₇	semidirect
511. #473	C₂²×C₆₇	semidirect
512. #473	C₂²×C₆₇	semidirect
513. #473	C₂²×C₆₇	semidirect
514. #472	C₂²×C₆₇	semidirect
515. #471	C₂²×C₆₇	semidirect
516. #471	C₂²×C₆₇	semidirect
517. #456	C₂²×C₆₇	semidirect
518. #456	C₂²×C₆₇	semidirect
519. #455	C₂²×C₆₇	semidirect
520. #454	C₂²×C₆₇	semidirect
521. #444	C₂²×C₆₇	semidirect
522. #444	C₂²×C₆₇	semidirect
523. #443	C₂²×C₆₇	semidirect
524. #443	C₂²×C₆₇	semidirect
525. #430	C₂²×C₆₇	semidirect
526. C₂×#108	C₂²×C₆₇	semidirect
528. #421	C₂²×C₆₇	semidirect
529. #421	C₂²×C₆₇	semidirect
530. #413	C₂²×C₆₇	semidirect
531. C₂×#101	C₂²×C₆₇	semidirect
532. #405	C₂²×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#3586	2
C₆₇×#3584	2
C₂×#578×C₆₇	2
C₆₇×#3293	2
C₆₇×#3290	2
C₆₇×#3289	2
C₆₇×#3156	2
C₆₇×#3069	2
C₆₇×#3067	2
C₂×#473×C₆₇	2
C₆₇×#3022	2
C₆₇×#3020	2
C₆₇×#2957	2
C₆₇×#2955	2
#61108	67

Subgroups — C67×#61108 #30740359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61108 #30740359