Subgroups — C₆₇×#23885 #30703136

Order 17152.

1662 conjugacy classes of subgroups.

The subgroup lattice has 720 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	1	yes	maximal, characteristic, complemented
2	C₆₇×#3486	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3025	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3022	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#3004	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2010	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2009	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1909	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1905	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1891	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1796	C₂	1	1	yes	maximal, characteristic, complemented
12	#585×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
15	#578×C₆₇	—	2	4	no	—
16	#577×C₆₇	—	2	4	no	—
17	#575×C₆₇	—	2	2	no	—
18	#575×C₆₇	C₂²	1	2	yes	—
19	C₂×#137×C₆₇	—	2	2	no	—
20	C₂×#137×C₆₇	C₂²	1	2	yes	—
21	C₂×#137×C₆₇	C₂²	1	2	yes	—
22	#568×C₆₇	C₂²	1	1	yes	characteristic
23	D₄×Q₈×C₆₇	—	2	2	no	—
24	D₄×Q₈×C₆₇	—	2	2	no	—
25	#548×C₆₇	—	2	2	no	—
26	#548×C₆₇	—	2	2	no	—
27	#536×C₆₇	—	2	2	no	—
28	#536×C₆₇	C₂²	1	1	yes	characteristic
29	#536×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	#525×C₆₇	—	2	2	no	—
31	#525×C₆₇	C₂²	1	1	yes	characteristic
32	#525×C₆₇	C₂²	1	2	yes	—
33	#525×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	#519×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	#458×C₆₇	—	2	2	no	—
36	B(4,𝔽₂)×C₆₇	—	2	2	no	—
37	#456×C₆₇	—	2	2	no	—
38	#455×C₆₇	—	2	2	no	—
39	#451×C₆₇	—	2	4	no	—
40	#448×C₆₇	—	2	4	no	—
41	#421×C₆₇	C₂²	1	2	yes	—
42	#421×C₆₇	C₂²	1	2	yes	complemented
43	#419×C₆₇	C₂²	1	2	yes	complemented
44	#419×C₆₇	C₂²	1	2	yes	complemented
45	#413×C₆₇	C₂²	1	1	yes	characteristic
46	#410×C₆₇	C₂²	1	2	yes	—
47	#410×C₆₇	C₂²	1	1	yes	characteristic
48	#407×C₆₇	C₂²	1	2	yes	complemented
49	#344×C₆₇	C₂²	1	2	yes	complemented
50	H(3,ℤ/4)×C₆₇	C₂²	1	2	yes	complemented
51	2₋¹⁺⁴×C₆₇	—	2	4	no	—
52	2₋¹⁺⁴×C₆₇	—	2	2	no	—
53	2₋¹⁺⁴×C₆₇	—	2	1	no	—
54	2₋¹⁺⁴×C₆₇	—	4	2	no	—
55	H(5,𝔽₂)×C₆₇	—	2	2	no	—
56	H(5,𝔽₂)×C₆₇	—	2	2	no	—
57	H(5,𝔽₂)×C₆₇	—	2	1	no	—
58	C₂×#41×C₆₇	—	2	2	no	—
59	C₂×#41×C₆₇	—	2	1	no	—
60	C₂×#41×C₆₇	—	2	2	no	—
61	C₂×#41×C₆₇	—	4	2	no	—
62	C₂×#41×C₆₇	D₄	1	2	yes	complemented
63	C₂×#41×C₆₇	C₂³	1	2	yes	—
64	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
66	C₂²×Q₈×C₆₇	—	2	2	no	—
67	C₂²×Q₈×C₆₇	—	2	2	no	—
68	C₂²×Q₈×C₆₇	—	2	1	no	—
69	C₂²×Q₈×C₆₇	—	2	1	no	—
70	C₂²×Q₈×C₆₇	—	2	4	no	—
71	C₂²×Q₈×C₆₇	—	4	2	no	—
72	C₂²×Q₈×C₆₇	D₄	1	2	yes	complemented
73	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₇	—	2	1	no	—
76	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
77	#137×C₆₇	—	2	8	no	—
78	#137×C₆₇	—	2	4	no	—
79	#137×C₆₇	—	2	2	no	—
80	#137×C₆₇	—	2	4	no	—
81	#137×C₆₇	—	2	4	no	—
82	#137×C₆₇	—	2	2	no	—
83	#137×C₆₇	—	2	4	no	—
84	#137×C₆₇	—	4	2	no	—
85	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
86	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
87	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
88	#135×C₆₇	—	2	8	no	—
89	#135×C₆₇	—	2	4	no	—
90	C₂×Q₁₆×C₆₇	—	2	4	no	—
91	C₂×Q₁₆×C₆₇	—	2	2	no	—
92	C₂×Q₁₆×C₆₇	D₄	1	4	yes	complemented
93	C₂×SD₁₆×C₆₇	—	2	4	no	—
94	C₂×SD₁₆×C₆₇	—	2	2	no	—
95	C₂×SD₁₆×C₆₇	D₄	1	4	yes	complemented
96	#131×C₆₇	—	2	4	no	—
97	C₂×M₁₆×C₆₇	—	2	2	no	—
98	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
99	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
100	C₄⋊Q₈×C₆₇	—	2	4	no	—
101	C₄⋊Q₈×C₆₇	—	2	1	no	—
102	C₄⋊Q₈×C₆₇	D₄	1	2	yes	complemented
103	#124×C₆₇	—	2	4	no	—
104	#124×C₆₇	—	2	1	no	—
105	#124×C₆₇	D₄	1	2	yes	complemented
106	#123×C₆₇	—	2	2	no	—
107	#123×C₆₇	—	2	1	no	—
108	#123×C₆₇	—	2	2	no	—
109	#123×C₆₇	—	2	2	no	—
110	#123×C₆₇	D₄	1	2	yes	—
111	C₂²⋊Q₈×C₆₇	—	2	4	no	—
112	C₂²⋊Q₈×C₆₇	—	2	4	no	—
113	C₂²⋊Q₈×C₆₇	—	2	4	no	—
114	C₂²⋊Q₈×C₆₇	—	2	1	no	—
115	C₂²⋊Q₈×C₆₇	—	2	1	no	—
116	C₂²⋊Q₈×C₆₇	—	2	2	no	—
117	C₂²⋊Q₈×C₆₇	—	2	2	no	—
118	C₂²⋊Q₈×C₆₇	—	2	2	no	—
119	#121×C₆₇	—	2	4	no	—
120	#121×C₆₇	—	2	1	no	—
121	#121×C₆₇	—	2	1	no	—
122	#121×C₆₇	—	2	2	no	—
123	#120×C₆₇	—	2	1	no	—
124	#120×C₆₇	—	2	2	no	—
125	#120×C₆₇	D₄	1	2	yes	—
126	C₄×Q₈×C₆₇	—	2	2	no	—
127	C₄×Q₈×C₆₇	—	2	2	no	—
128	C₄×D₄×C₆₇	—	2	4	no	—
129	C₄×D₄×C₆₇	—	2	4	no	—
130	C₄×D₄×C₆₇	—	2	2	no	—
131	C₄×D₄×C₆₇	—	2	2	no	—
132	C₄×D₄×C₆₇	—	4	2	no	—
133	#117×C₆₇	—	2	2	no	—
134	#117×C₆₇	C₂³	1	2	yes	—
135	#117×C₆₇	C₂³	1	1	yes	characteristic
136	#117×C₆₇	C₂³	1	1	yes	characteristic
137	#104×C₆₇	—	2	4	no	—
138	#104×C₆₇	—	4	2	no	—
139	#103×C₆₇	—	2	4	no	—
140	#103×C₆₇	—	2	4	no	—
141	#101×C₆₇	—	2	1	no	—
142	#100×C₆₇	—	2	1	no	—
143	#99×C₆₇	—	2	2	no	—
144	#99×C₆₇	—	2	4	no	—
145	C₂²⋊C₈×C₆₇	—	2	4	no	—
146	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
147	#41×C₆₇	—	2	2	no	—
148	#41×C₆₇	—	2	2	no	—
149	#41×C₆₇	—	2	4	no	—
150	#41×C₆₇	—	2	4	no	—
151	#41×C₆₇	—	2	2	no	—
152	#41×C₆₇	—	2	2	no	—
153	#41×C₆₇	—	2	2	no	—
154	#41×C₆₇	—	2	2	no	—
155	#41×C₆₇	—	2	2	no	—
156	#41×C₆₇	—	2	2	no	—
157	#41×C₆₇	—	4	2	no	—
158	#41×C₆₇	—	4	8	no	—
159	#41×C₆₇	—	4	4	no	—
160	#41×C₆₇	—	4	2	no	—
161	#41×C₆₇	—	4	4	no	—
162	#41×C₆₇	—	4	2	no	—
163	#41×C₆₇	—	4	2	no	—
164	#41×C₆₇	—	4	2	no	—
165	C₂×Q₈×C₆₇	—	2	4	no	—
166	C₂×Q₈×C₆₇	—	2	2	no	—
167	C₂×Q₈×C₆₇	—	2	1	no	—
168	C₂×Q₈×C₆₇	—	2	1	no	—
169	C₂×Q₈×C₆₇	—	2	2	no	—
170	C₂×Q₈×C₆₇	—	2	2	no	—
171	C₂×Q₈×C₆₇	—	2	4	no	—
172	C₂×Q₈×C₆₇	—	2	2	no	—
173	C₂×Q₈×C₆₇	—	2	2	no	—
174	C₂×Q₈×C₆₇	—	4	4	no	—
175	C₂×Q₈×C₆₇	—	4	2	no	—
176	C₂×Q₈×C₆₇	—	4	8	no	—
177	C₂×Q₈×C₆₇	—	4	4	no	—
178	C₂×Q₈×C₆₇	—	4	2	no	—
179	C₂×Q₈×C₆₇	—	4	4	no	—
180	C₂×Q₈×C₆₇	—	4	4	no	—
181	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
182	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
183	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
184	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
185	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
186	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₂×D₄×C₆₇	—	2	2	no	—
188	C₂×D₄×C₆₇	—	2	2	no	—
189	C₂×D₄×C₆₇	—	2	2	no	—
190	C₂×D₄×C₆₇	—	2	2	no	—
191	C₂×D₄×C₆₇	—	2	1	no	—
192	C₂×D₄×C₆₇	—	2	2	no	—
193	C₂×D₄×C₆₇	—	4	2	no	—
194	C₂×D₄×C₆₇	—	4	4	no	—
195	C₂×D₄×C₆₇	—	4	2	no	—
196	C₂×D₄×C₆₇	—	4	2	no	—
197	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
198	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
199	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
200	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	—	2	1	no	—
202	C₂²×C₄×C₆₇	—	2	4	no	—
203	C₂²×C₄×C₆₇	—	2	2	no	—
204	C₂²×C₄×C₆₇	—	2	1	no	—
205	C₂²×C₄×C₆₇	—	2	2	no	—
206	C₂²×C₄×C₆₇	—	2	1	no	—
207	C₂²×C₄×C₆₇	—	2	1	no	—
208	C₂²×C₄×C₆₇	—	2	1	no	—
209	C₂²×C₄×C₆₇	—	2	2	no	—
210	C₂²×C₄×C₆₇	—	2	2	no	—
211	C₂²×C₄×C₆₇	—	2	2	no	—
212	C₂²×C₄×C₆₇	—	4	4	no	—
213	C₂²×C₄×C₆₇	—	4	2	no	—
214	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
215	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
216	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
217	Q₁₆×C₆₇	—	2	4	no	—
218	Q₁₆×C₆₇	—	2	8	no	—
219	Q₁₆×C₆₇	—	4	4	no	—
220	SD₁₆×C₆₇	—	2	4	no	—
221	SD₁₆×C₆₇	—	2	4	no	—
222	SD₁₆×C₆₇	—	2	8	no	—
223	SD₁₆×C₆₇	—	4	4	no	—
224	D₈×C₆₇	—	2	4	no	—
225	M₁₆×C₆₇	—	2	4	no	—
226	M₁₆×C₆₇	—	2	2	no	—
227	M₁₆×C₆₇	—	2	2	no	—
228	M₁₆×C₆₇	—	2	2	no	—
229	M₁₆×C₆₇	—	2	2	no	—
230	C₂×C₈×C₆₇	—	2	4	no	—
231	C₂×C₈×C₆₇	—	2	2	no	—
232	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
233	C₄⋊C₄×C₆₇	—	2	4	no	—
234	C₄⋊C₄×C₆₇	—	2	4	no	—
235	C₄⋊C₄×C₆₇	—	2	2	no	—
236	C₄⋊C₄×C₆₇	—	2	4	no	—
237	C₄⋊C₄×C₆₇	—	2	2	no	—
238	C₄⋊C₄×C₆₇	—	2	1	no	—
239	C₄⋊C₄×C₆₇	—	2	1	no	—
240	C₄⋊C₄×C₆₇	—	2	2	no	—
241	C₄⋊C₄×C₆₇	—	4	2	no	—
242	C₂²⋊C₄×C₆₇	—	2	2	no	—
243	C₂²⋊C₄×C₆₇	—	2	4	no	—
244	C₂²⋊C₄×C₆₇	—	2	2	no	—
245	C₂²⋊C₄×C₆₇	—	2	4	no	—
246	C₂²⋊C₄×C₆₇	—	2	2	no	—
247	C₂²⋊C₄×C₆₇	—	2	1	no	—
248	C₂²⋊C₄×C₆₇	—	2	2	no	—
249	C₂²⋊C₄×C₆₇	—	2	2	no	—
250	C₂²⋊C₄×C₆₇	—	4	2	no	—
251	C₂²⋊C₄×C₆₇	—	4	2	no	—
252	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
253	C₄²×C₆₇	—	2	4	no	—
254	C₄²×C₆₇	—	2	1	no	—
255	C₄²×C₆₇	—	2	2	no	—
256	C₄²×C₆₇	C₂×D₄	1	2	yes	—
257	C₂³×C₆₇	—	2	1	no	—
258	C₂³×C₆₇	—	2	2	no	—
259	C₂³×C₆₇	—	2	1	no	—
260	C₂³×C₆₇	—	2	2	no	—
261	C₂³×C₆₇	—	4	2	no	—
262	C₂³×C₆₇	—	4	2	no	—
263	C₂³×C₆₇	#120	1	2	yes	—
264	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
265	Q₈×C₆₇	—	2	2	no	—
266	Q₈×C₆₇	—	2	4	no	—
267	Q₈×C₆₇	—	2	2	no	—
268	Q₈×C₆₇	—	2	1	no	—
269	Q₈×C₆₇	—	2	2	no	—
270	Q₈×C₆₇	—	2	1	no	—
271	Q₈×C₆₇	—	2	4	no	—
272	Q₈×C₆₇	—	2	2	no	—
273	Q₈×C₆₇	—	4	4	no	—
274	Q₈×C₆₇	—	4	8	no	—
275	Q₈×C₆₇	—	4	2	no	—
276	Q₈×C₆₇	—	4	1	no	—
277	Q₈×C₆₇	—	4	1	no	—
278	Q₈×C₆₇	#120	1	4	yes	—
279	D₄×C₆₇	—	2	4	no	—
280	D₄×C₆₇	—	2	1	no	—
281	D₄×C₆₇	—	2	2	no	—
282	D₄×C₆₇	—	2	1	no	—
283	D₄×C₆₇	—	2	2	no	—
284	D₄×C₆₇	—	2	2	no	—
285	D₄×C₆₇	—	4	4	no	—
286	D₄×C₆₇	—	4	2	no	—
287	D₄×C₆₇	—	4	4	no	—
288	D₄×C₆₇	—	4	4	no	—
289	D₄×C₆₇	—	4	4	no	—
290	D₄×C₆₇	—	4	2	no	—
291	D₄×C₆₇	—	4	2	no	—
292	D₄×C₆₇	—	4	2	no	—
293	D₄×C₆₇	—	8	4	no	—
294	C₂×C₄×C₆₇	—	2	2	no	—
295	C₂×C₄×C₆₇	—	2	4	no	—
296	C₂×C₄×C₆₇	—	2	2	no	—
297	C₂×C₄×C₆₇	—	2	2	no	—
298	C₂×C₄×C₆₇	—	2	2	no	—
299	C₂×C₄×C₆₇	—	2	2	no	—
300	C₂×C₄×C₆₇	—	2	2	no	—
301	C₂×C₄×C₆₇	—	2	1	no	—
302	C₂×C₄×C₆₇	—	2	1	no	—
303	C₂×C₄×C₆₇	—	2	2	no	—
304	C₂×C₄×C₆₇	—	2	2	no	—
305	C₂×C₄×C₆₇	—	4	4	no	—
306	C₂×C₄×C₆₇	—	4	4	no	—
307	C₂×C₄×C₆₇	—	4	8	no	—
308	C₂×C₄×C₆₇	—	4	4	no	—
309	C₂×C₄×C₆₇	—	4	2	no	—
310	C₂×C₄×C₆₇	—	4	2	no	—
311	C₂×C₄×C₆₇	—	4	1	no	—
312	C₂×C₄×C₆₇	—	4	1	no	—
313	C₂×C₄×C₆₇	—	4	2	no	—
314	C₂×C₄×C₆₇	—	4	2	no	—
315	C₂×C₄×C₆₇	—	4	2	no	—
316	C₂×C₄×C₆₇	—	4	2	no	—
317	C₂×C₄×C₆₇	—	4	2	no	—
318	C₂×C₄×C₆₇	#120	1	2	yes	—
319	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
321	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
322	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
323	C₈×C₆₇	—	2	4	no	—
324	C₈×C₆₇	—	4	2	no	—
325	C₂²×C₆₇	—	2	2	no	—
326	C₂²×C₆₇	—	2	4	no	—
327	C₂²×C₆₇	—	2	2	no	—
328	C₂²×C₆₇	—	4	1	no	—
329	C₂²×C₆₇	—	4	2	no	—
330	C₂²×C₆₇	—	4	1	no	—
331	C₂²×C₆₇	—	4	2	no	—
332	C₂²×C₆₇	—	4	2	no	—
333	C₂²×C₆₇	—	4	1	no	—
334	C₂²×C₆₇	—	4	1	no	—
335	C₂²×C₆₇	—	4	2	no	—
336	C₂²×C₆₇	—	8	2	no	—
337	C₂²×C₆₇	—	16	2	no	—
338	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
339	C₂²×C₆₇	D₄²	1	2	yes	—
340	C₄×C₆₇	—	2	4	no	—
341	C₄×C₆₇	—	2	2	no	—
342	C₄×C₆₇	—	4	2	no	—
343	C₄×C₆₇	—	4	2	no	—
344	C₄×C₆₇	—	4	1	no	—
345	C₄×C₆₇	—	4	2	no	—
346	C₄×C₆₇	—	4	1	no	—
347	C₄×C₆₇	—	4	2	no	—
348	C₄×C₆₇	—	8	2	no	—
349	C₄×C₆₇	C₂×#120	1	1	yes	characteristic
350	C₄×C₆₇	#534	1	1	yes	characteristic
351	C₄×C₆₇	D₄²	1	2	yes	—
352	#23885	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
353	C₂×C₆₇	—	2	1	no	—
354	C₂×C₆₇	—	2	2	no	—
355	C₂×C₆₇	—	4	2	no	—
356	C₂×C₆₇	—	4	2	no	—
357	C₂×C₆₇	—	8	1	no	—
358	C₂×C₆₇	—	8	2	no	—
359	C₂×C₆₇	#2403	1	1	yes	characteristic
360	#3586	C₂×C₆₇	1	1	yes	characteristic, complemented
361	#3486	C₂×C₆₇	1	1	yes	characteristic, complemented
362	#3025	C₂×C₆₇	1	1	yes	characteristic, complemented
363	#3022	C₂×C₆₇	1	1	yes	characteristic, complemented
364	#3004	C₂×C₆₇	1	2	yes	complemented
365	#2010	C₂×C₆₇	1	2	yes	complemented
366	#2009	C₂×C₆₇	1	2	yes	complemented
367	#1909	C₂×C₆₇	1	1	yes	characteristic, complemented
368	#1905	C₂×C₆₇	1	2	yes	complemented
369	#1891	C₂×C₆₇	1	1	yes	characteristic, complemented
370	#1796	C₂×C₆₇	1	1	yes	characteristic, complemented
371	C₆₇	#23885	1	1	yes	characteristic, direct product factor, Sylow
372	#585	C₂²×C₆₇	1	1	yes	characteristic
373	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	1	yes	characteristic
374	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
375	#578	—	2	4	no	—
376	#577	—	2	4	no	—
377	#575	—	2	2	no	—
378	#575	C₂²×C₆₇	1	2	yes	—
379	C₂×#137	—	2	2	no	—
380	C₂×#137	C₂²×C₆₇	1	2	yes	—
381	C₂×#137	C₂²×C₆₇	1	2	yes	—
382	#568	C₂²×C₆₇	1	1	yes	characteristic
383	D₄×Q₈	—	2	2	no	—
384	D₄×Q₈	—	2	2	no	—
385	#548	—	2	2	no	—
386	#548	—	2	2	no	—
387	#536	—	2	2	no	—
388	#536	C₂²×C₆₇	1	1	yes	characteristic
389	#536	C₂²×C₆₇	1	1	yes	characteristic, complemented
390	#525	—	2	2	no	—
391	#525	C₂²×C₆₇	1	1	yes	characteristic
392	#525	C₂²×C₆₇	1	2	yes	—
393	#525	C₂²×C₆₇	1	1	yes	characteristic, complemented
394	#519	C₂²×C₆₇	1	1	yes	characteristic, complemented
395	#458	—	2	2	no	—
396	B(4,𝔽₂)	—	2	2	no	—
397	#456	—	2	2	no	—
398	#455	—	2	2	no	—
399	#451	—	2	4	no	—
400	#448	—	2	4	no	—
401	#421	C₂²×C₆₇	1	2	yes	—
402	#421	C₂²×C₆₇	1	2	yes	complemented
403	#419	C₂²×C₆₇	1	2	yes	complemented
404	#419	C₂²×C₆₇	1	2	yes	complemented
405	#413	C₂²×C₆₇	1	1	yes	characteristic
406	#410	C₂²×C₆₇	1	1	yes	characteristic
407	#410	C₂²×C₆₇	1	2	yes	—
408	#407	C₂²×C₆₇	1	2	yes	complemented
409	#344	C₂²×C₆₇	1	2	yes	complemented
410	H(3,ℤ/4)	C₂²×C₆₇	1	2	yes	complemented
411	2₋¹⁺⁴	—	2	4	no	—
412	2₋¹⁺⁴	—	2	2	no	—
413	2₋¹⁺⁴	—	2	1	no	—
414	2₋¹⁺⁴	—	4	2	no	—
415	H(5,𝔽₂)	—	2	2	no	—
416	H(5,𝔽₂)	—	2	2	no	—
417	H(5,𝔽₂)	—	2	1	no	—
418	C₂×#41	—	2	2	no	—
419	C₂×#41	—	2	1	no	—
420	C₂×#41	—	2	2	no	—
421	C₂×#41	—	4	2	no	—
422	C₂×#41	D₄×C₆₇	1	2	yes	complemented
423	C₂×#41	C₂³×C₆₇	1	2	yes	—
424	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
425	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
426	C₂²×Q₈	—	2	2	no	—
427	C₂²×Q₈	—	2	2	no	—
428	C₂²×Q₈	—	2	1	no	—
429	C₂²×Q₈	—	2	1	no	—
430	C₂²×Q₈	—	2	4	no	—
431	C₂²×Q₈	—	4	2	no	—
432	C₂²×Q₈	D₄×C₆₇	1	2	yes	complemented
433	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
434	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
435	C₂³×C₄	—	2	1	no	—
436	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
437	#137	—	2	8	no	—
438	#137	—	2	4	no	—
439	#137	—	2	2	no	—
440	#137	—	2	4	no	—
441	#137	—	2	4	no	—
442	#137	—	2	2	no	—
443	#137	—	2	4	no	—
444	#137	—	4	2	no	—
445	AGL(1,ℤ/8)	—	2	4	no	—
446	AGL(1,ℤ/8)	—	2	2	no	—
447	AGL(1,ℤ/8)	—	2	2	no	—
448	#135	—	2	8	no	—
449	#135	—	2	4	no	—
450	C₂×Q₁₆	—	2	4	no	—
451	C₂×Q₁₆	—	2	2	no	—
452	C₂×Q₁₆	D₄×C₆₇	1	4	yes	complemented
453	C₂×SD₁₆	—	2	4	no	—
454	C₂×SD₁₆	—	2	2	no	—
455	C₂×SD₁₆	D₄×C₆₇	1	4	yes	complemented
456	#131	—	2	4	no	—
457	C₂×M₁₆	—	2	2	no	—
458	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
459	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
460	C₄⋊Q₈	—	2	4	no	—
461	C₄⋊Q₈	—	2	1	no	—
462	C₄⋊Q₈	D₄×C₆₇	1	2	yes	complemented
463	#124	—	2	4	no	—
464	#124	—	2	1	no	—
465	#124	D₄×C₆₇	1	2	yes	complemented
466	#123	—	2	2	no	—
467	#123	—	2	1	no	—
468	#123	—	2	2	no	—
469	#123	—	2	2	no	—
470	#123	D₄×C₆₇	1	2	yes	—
471	C₂²⋊Q₈	—	2	4	no	—
472	C₂²⋊Q₈	—	2	4	no	—
473	C₂²⋊Q₈	—	2	4	no	—
474	C₂²⋊Q₈	—	2	1	no	—
475	C₂²⋊Q₈	—	2	1	no	—
476	C₂²⋊Q₈	—	2	2	no	—
477	C₂²⋊Q₈	—	2	2	no	—
478	C₂²⋊Q₈	—	2	2	no	—
479	#121	—	2	4	no	—
480	#121	—	2	1	no	—
481	#121	—	2	1	no	—
482	#121	—	2	2	no	—
483	#120	—	2	1	no	—
484	#120	—	2	2	no	—
485	#120	D₄×C₆₇	1	2	yes	—
486	C₄×Q₈	—	2	2	no	—
487	C₄×Q₈	—	2	2	no	—
488	C₄×D₄	—	2	4	no	—
489	C₄×D₄	—	2	4	no	—
490	C₄×D₄	—	2	2	no	—
491	C₄×D₄	—	2	2	no	—
492	C₄×D₄	—	4	2	no	—
493	#117	—	2	2	no	—
494	#117	C₂³×C₆₇	1	2	yes	—
495	#117	C₂³×C₆₇	1	1	yes	characteristic
496	#117	C₂³×C₆₇	1	1	yes	characteristic
497	#104	—	2	4	no	—
498	#104	—	4	2	no	—
499	#103	—	2	4	no	—
500	#103	—	2	4	no	—
501	#101	—	2	1	no	—
502	#100	—	2	1	no	—
503	#99	—	2	2	no	—
504	#99	—	2	4	no	—
505	C₂²⋊C₈	—	2	4	no	—
506	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
507	#41	—	2	2	no	—
508	#41	—	2	2	no	—
509	#41	—	2	4	no	—
510	#41	—	2	4	no	—
511	#41	—	2	2	no	—
512	#41	—	2	2	no	—
513	#41	—	2	2	no	—
514	#41	—	2	2	no	—
515	#41	—	2	2	no	—
516	#41	—	2	2	no	—
517	#41	—	4	2	no	—
518	#41	—	4	8	no	—
519	#41	—	4	4	no	—
520	#41	—	4	2	no	—
521	#41	—	4	4	no	—
522	#41	—	4	2	no	—
523	#41	—	4	2	no	—
524	#41	—	4	2	no	—
525	C₂×Q₈	—	2	4	no	—
526	C₂×Q₈	—	2	2	no	—
527	C₂×Q₈	—	2	1	no	—
528	C₂×Q₈	—	2	1	no	—
529	C₂×Q₈	—	2	2	no	—
530	C₂×Q₈	—	2	2	no	—
531	C₂×Q₈	—	2	4	no	—
532	C₂×Q₈	—	2	2	no	—
533	C₂×Q₈	—	2	2	no	—
534	C₂×Q₈	—	4	4	no	—
535	C₂×Q₈	—	4	2	no	—
536	C₂×Q₈	—	4	8	no	—
537	C₂×Q₈	—	4	4	no	—
538	C₂×Q₈	—	4	2	no	—
539	C₂×Q₈	—	4	4	no	—
540	C₂×Q₈	—	4	4	no	—
541	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
542	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
543	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
544	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
545	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
546	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
547	C₂×D₄	—	2	2	no	—
548	C₂×D₄	—	2	2	no	—
549	C₂×D₄	—	2	2	no	—
550	C₂×D₄	—	2	2	no	—
551	C₂×D₄	—	2	2	no	—
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	—	4	2	no	—
554	C₂×D₄	—	4	4	no	—
555	C₂×D₄	—	4	2	no	—
556	C₂×D₄	—	4	2	no	—
557	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
558	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
559	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
560	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
561	C₂²×C₄	—	2	1	no	—
562	C₂²×C₄	—	2	4	no	—
563	C₂²×C₄	—	2	2	no	—
564	C₂²×C₄	—	2	1	no	—
565	C₂²×C₄	—	2	2	no	—
566	C₂²×C₄	—	2	2	no	—
567	C₂²×C₄	—	2	1	no	—
568	C₂²×C₄	—	2	1	no	—
569	C₂²×C₄	—	2	1	no	—
570	C₂²×C₄	—	2	2	no	—
571	C₂²×C₄	—	2	2	no	—
572	C₂²×C₄	—	4	4	no	—
573	C₂²×C₄	—	4	2	no	—
574	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
575	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
576	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
577	Q₁₆	—	2	4	no	—
578	Q₁₆	—	2	8	no	—
579	Q₁₆	—	4	4	no	—
580	SD₁₆	—	2	4	no	—
581	SD₁₆	—	2	4	no	—
582	SD₁₆	—	2	8	no	—
583	SD₁₆	—	4	4	no	—
584	D₈	—	2	4	no	—
585	M₁₆	—	2	4	no	—
586	M₁₆	—	2	2	no	—
587	M₁₆	—	2	2	no	—
588	M₁₆	—	2	2	no	—
589	M₁₆	—	2	2	no	—
590	C₂×C₈	—	2	4	no	—
591	C₂×C₈	—	2	2	no	—
592	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
593	C₄⋊C₄	—	2	4	no	—
594	C₄⋊C₄	—	2	4	no	—
595	C₄⋊C₄	—	2	2	no	—
596	C₄⋊C₄	—	2	4	no	—
597	C₄⋊C₄	—	2	2	no	—
598	C₄⋊C₄	—	2	1	no	—
599	C₄⋊C₄	—	2	1	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	4	2	no	—
602	C₂²⋊C₄	—	2	2	no	—
603	C₂²⋊C₄	—	2	4	no	—
604	C₂²⋊C₄	—	2	2	no	—
605	C₂²⋊C₄	—	2	4	no	—
606	C₂²⋊C₄	—	2	2	no	—
607	C₂²⋊C₄	—	2	1	no	—
608	C₂²⋊C₄	—	2	2	no	—
609	C₂²⋊C₄	—	2	2	no	—
610	C₂²⋊C₄	—	4	2	no	—
611	C₂²⋊C₄	—	4	2	no	—
612	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
613	C₄²	—	2	4	no	—
614	C₄²	—	2	1	no	—
615	C₄²	—	2	2	no	—
616	C₄²	C₂×D₄×C₆₇	1	2	yes	—
617	C₂³	—	2	1	no	—
618	C₂³	—	2	2	no	—
619	C₂³	—	2	1	no	—
620	C₂³	—	2	2	no	—
621	C₂³	—	4	2	no	—
622	C₂³	—	4	2	no	—
623	C₂³	#120×C₆₇	1	2	yes	—
624	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
625	Q₈	—	2	2	no	—
626	Q₈	—	2	4	no	—
627	Q₈	—	2	2	no	—
628	Q₈	—	2	1	no	—
629	Q₈	—	2	2	no	—
630	Q₈	—	2	1	no	—
631	Q₈	—	2	4	no	—
632	Q₈	—	2	2	no	—
633	Q₈	—	4	4	no	—
634	Q₈	—	4	8	no	—
635	Q₈	—	4	2	no	—
636	Q₈	—	4	1	no	—
637	Q₈	—	4	1	no	—
638	Q₈	#120×C₆₇	1	4	yes	—
639	D₄	—	2	4	no	—
640	D₄	—	2	1	no	—
641	D₄	—	2	2	no	—
642	D₄	—	2	1	no	—
643	D₄	—	2	2	no	—
644	D₄	—	2	2	no	—
645	D₄	—	4	4	no	—
646	D₄	—	4	2	no	—
647	D₄	—	4	4	no	—
648	D₄	—	4	4	no	—
649	D₄	—	4	4	no	—
650	D₄	—	4	2	no	—
651	D₄	—	4	2	no	—
652	D₄	—	4	2	no	—
653	D₄	—	8	4	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	4	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	2	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	1	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	2	2	no	—
665	C₂×C₄	—	4	4	no	—
666	C₂×C₄	—	4	4	no	—
667	C₂×C₄	—	4	8	no	—
668	C₂×C₄	—	4	4	no	—
669	C₂×C₄	—	4	2	no	—
670	C₂×C₄	—	4	2	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	2	no	—
674	C₂×C₄	—	4	2	no	—
675	C₂×C₄	—	4	2	no	—
676	C₂×C₄	—	4	2	no	—
677	C₂×C₄	—	4	2	no	—
678	C₂×C₄	#120×C₆₇	1	2	yes	—
679	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
680	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
681	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
682	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
683	C₈	—	2	4	no	—
684	C₈	—	4	2	no	—
685	C₂²	—	2	2	no	—
686	C₂²	—	2	4	no	—
687	C₂²	—	2	2	no	—
688	C₂²	—	4	1	no	—
689	C₂²	—	4	2	no	—
690	C₂²	—	4	1	no	—
691	C₂²	—	4	2	no	—
692	C₂²	—	4	2	no	—
693	C₂²	—	4	1	no	—
694	C₂²	—	4	1	no	—
695	C₂²	—	4	2	no	—
696	C₂²	—	8	2	no	—
697	C₂²	—	16	2	no	—
698	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
699	C₂²	D₄²×C₆₇	1	2	yes	—
700	C₄	—	2	4	no	—
701	C₄	—	2	2	no	—
702	C₄	—	4	2	no	—
703	C₄	—	4	2	no	—
704	C₄	—	4	1	no	—
705	C₄	—	4	2	no	—
706	C₄	—	4	1	no	—
707	C₄	—	4	2	no	—
708	C₄	—	8	2	no	—
709	C₄	C₂×#120×C₆₇	1	1	yes	characteristic
710	C₄	#534×C₆₇	1	1	yes	characteristic
711	C₄	D₄²×C₆₇	1	2	yes	—
712	C₂	—	2	1	no	—
713	C₂	—	2	2	no	—
714	C₂	—	4	2	no	—
715	C₂	—	4	2	no	—
716	C₂	—	8	1	no	—
717	C₂	—	8	2	no	—
718	C₂	C₆₇×#2403	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#2403
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×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×#3486	C₂	semidirect
3. C₆₇×#3025	C₂	semidirect
4. C₆₇×#3022	C₂	semidirect
5. C₆₇×#3004	C₂	semidirect
6. C₆₇×#2010	C₂	semidirect
7. C₆₇×#2009	C₂	semidirect
8. C₆₇×#1909	C₂	semidirect
9. C₆₇×#1905	C₂	semidirect
10. C₆₇×#1891	C₂	semidirect
11. C₆₇×#1796	C₂	semidirect
14. C₂³×Q₈×C₆₇	C₂²	semidirect
29. #536×C₆₇	C₂²	semidirect
33. #525×C₆₇	C₂²	semidirect
34. #519×C₆₇	C₂²	semidirect
42. #421×C₆₇	C₂²	semidirect
43. #419×C₆₇	C₂²	semidirect
44. #419×C₆₇	C₂²	semidirect
48. #407×C₆₇	C₂²	semidirect
49. #344×C₆₇	C₂²	semidirect
50. H(3,ℤ/4)×C₆₇	C₂²	semidirect
62. C₂×#41×C₆₇	D₄	semidirect
72. C₂²×Q₈×C₆₇	D₄	semidirect
92. C₂×Q₁₆×C₆₇	D₄	semidirect
95. C₂×SD₁₆×C₆₇	D₄	semidirect
102. C₄⋊Q₈×C₆₇	D₄	semidirect
105. #124×C₆₇	D₄	semidirect
352. #23885	C₆₇	direct (N × H)
360. #3586	C₂×C₆₇	semidirect
361. #3486	C₂×C₆₇	semidirect
362. #3025	C₂×C₆₇	semidirect
363. #3022	C₂×C₆₇	semidirect
364. #3004	C₂×C₆₇	semidirect
365. #2010	C₂×C₆₇	semidirect
366. #2009	C₂×C₆₇	semidirect
367. #1909	C₂×C₆₇	semidirect
368. #1905	C₂×C₆₇	semidirect
369. #1891	C₂×C₆₇	semidirect
370. #1796	C₂×C₆₇	semidirect
371. C₆₇	#23885	direct (N × H)
374. C₂³×Q₈	C₂²×C₆₇	semidirect
389. #536	C₂²×C₆₇	semidirect
393. #525	C₂²×C₆₇	semidirect
394. #519	C₂²×C₆₇	semidirect
402. #421	C₂²×C₆₇	semidirect
403. #419	C₂²×C₆₇	semidirect
404. #419	C₂²×C₆₇	semidirect
408. #407	C₂²×C₆₇	semidirect
409. #344	C₂²×C₆₇	semidirect
410. H(3,ℤ/4)	C₂²×C₆₇	semidirect
422. C₂×#41	D₄×C₆₇	semidirect
432. C₂²×Q₈	D₄×C₆₇	semidirect
452. C₂×Q₁₆	D₄×C₆₇	semidirect
455. C₂×SD₁₆	D₄×C₆₇	semidirect
462. C₄⋊Q₈	D₄×C₆₇	semidirect
465. #124	D₄×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×#3486	2
C₆₇×#3025	2
C₆₇×#3022	2
C₆₇×#3004	2
C₆₇×#2010	2
C₆₇×#2009	2
C₆₇×#1909	2
C₆₇×#1905	2
C₆₇×#1891	2
C₆₇×#1796	2
#23885	67

Subgroups — C67×#23885 #30703136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#23885 #30703136