Subgroups — C₆₇×#11034 #30690285

Order 17152.

1366 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#421×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#413×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#100×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#344×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1738	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
11	C₂×AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
12	C₂×AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
13	C₂×AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
14	C₂×AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
15	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	4	yes	complemented
16	C₂²×SD₁₆×C₆₇	—	2	1	no	—
17	C₂²×D₈×C₆₇	—	2	1	no	—
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	#421×C₆₇	—	2	2	no	—
24	#421×C₆₇	—	2	2	no	—
25	C₂×#104×C₆₇	—	2	2	no	—
26	#418×C₆₇	—	2	2	no	—
27	#418×C₆₇	—	2	2	no	—
28	C₂×#102×C₆₇	—	2	2	no	—
29	#413×C₆₇	—	2	2	no	—
30	#413×C₆₇	—	2	2	no	—
31	C₂×#101×C₆₇	—	2	1	no	—
32	C₂×#100×C₆₇	—	2	2	no	—
33	C₂×#100×C₆₇	—	2	1	no	—
34	C₂×#100×C₆₇	—	2	2	no	—
35	C₂×#100×C₆₇	—	2	2	no	—
36	#344×C₆₇	—	2	2	no	—
37	#341×C₆₇	—	2	2	no	—
38	C₂⁵×C₆₇	—	2	1	no	—
39	C₂×#41×C₆₇	—	2	1	no	—
40	C₂×#41×C₆₇	—	2	1	no	—
41	C₂×#41×C₆₇	—	2	1	no	—
42	C₂×#41×C₆₇	—	2	1	no	—
43	C₂×#41×C₆₇	—	2	1	no	—
44	C₂×#41×C₆₇	—	2	1	no	—
45	C₂×#41×C₆₇	—	2	1	no	—
46	C₂×#41×C₆₇	—	2	1	no	—
47	C₂×#41×C₆₇	—	2	1	no	—
48	C₂×#41×C₆₇	—	2	1	no	—
49	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×#41×C₆₇	D₄	1	2	yes	—
51	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂²×D₄×C₆₇	—	2	1	no	—
53	C₂²×D₄×C₆₇	—	2	1	no	—
54	C₂²×D₄×C₆₇	—	2	1	no	—
55	C₂²×D₄×C₆₇	—	2	1	no	—
56	C₂²×D₄×C₆₇	—	2	1	no	—
57	C₂²×D₄×C₆₇	—	2	1	no	—
58	C₂²×D₄×C₆₇	—	2	1	no	—
59	C₂²×D₄×C₆₇	—	2	1	no	—
60	C₂²×D₄×C₆₇	—	2	1	no	—
61	C₂²×D₄×C₆₇	—	2	1	no	—
62	C₂²×D₄×C₆₇	—	2	1	no	—
63	C₂²×D₄×C₆₇	—	2	1	no	—
64	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
66	C₂²×D₄×C₆₇	D₄	1	2	yes	—
67	C₂²×D₄×C₆₇	D₄	1	2	yes	—
68	C₂³×C₄×C₆₇	—	2	1	no	—
69	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
70	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
71	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
72	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
73	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
74	C₂×SD₁₆×C₆₇	—	2	4	no	—
75	C₂×SD₁₆×C₆₇	—	2	4	no	—
76	C₂×SD₁₆×C₆₇	—	2	4	no	—
77	C₂×D₈×C₆₇	—	2	4	no	—
78	C₂×D₈×C₆₇	—	2	4	no	—
79	C₂×D₈×C₆₇	—	2	4	no	—
80	C₂×M₁₆×C₆₇	—	2	1	no	—
81	C₂×M₁₆×C₆₇	—	2	1	no	—
82	C₂×M₁₆×C₆₇	—	2	1	no	—
83	C₂×M₁₆×C₆₇	—	2	1	no	—
84	C₂×M₁₆×C₆₇	—	2	1	no	—
85	C₂×M₁₆×C₆₇	—	2	1	no	—
86	C₂×M₁₆×C₆₇	—	2	1	no	—
87	C₂×M₁₆×C₆₇	—	2	1	no	—
88	C₂×M₁₆×C₆₇	—	2	1	no	—
89	C₂×M₁₆×C₆₇	—	2	1	no	—
90	C₂×M₁₆×C₆₇	—	2	1	no	—
91	C₂×M₁₆×C₆₇	—	2	1	no	—
92	C₂×M₁₆×C₆₇	—	2	1	no	—
93	C₂×M₁₆×C₆₇	—	2	1	no	—
94	C₂×M₁₆×C₆₇	—	2	1	no	—
95	C₂×M₁₆×C₆₇	—	2	1	no	—
96	C₂×M₁₆×C₆₇	—	2	1	no	—
97	C₂×M₁₆×C₆₇	—	2	1	no	—
98	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
99	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
100	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
101	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
102	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
103	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
104	C₂²×C₈×C₆₇	—	2	1	no	—
105	C₂²×C₈×C₆₇	—	2	1	no	—
106	C₂²×C₈×C₆₇	—	2	1	no	—
107	C₂²×C₈×C₆₇	—	2	1	no	—
108	#117×C₆₇	—	2	1	no	—
109	#117×C₆₇	—	2	1	no	—
110	#117×C₆₇	—	2	1	no	—
111	#117×C₆₇	D₄	1	2	yes	—
112	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
114	C₂×C₄²×C₆₇	—	2	1	no	—
115	#104×C₆₇	—	2	8	no	—
116	#102×C₆₇	—	2	8	no	—
117	#101×C₆₇	—	2	4	no	—
118	#100×C₆₇	—	2	8	no	—
119	#100×C₆₇	—	2	4	no	—
120	C₂⁴×C₆₇	—	2	2	no	—
121	C₂⁴×C₆₇	—	2	2	no	—
122	C₂⁴×C₆₇	—	2	1	no	—
123	C₂⁴×C₆₇	—	2	2	no	—
124	C₂⁴×C₆₇	—	4	1	no	—
125	C₂⁴×C₆₇	—	4	1	no	—
126	C₂⁴×C₆₇	—	4	1	no	—
127	C₂⁴×C₆₇	—	4	1	no	—
128	C₂⁴×C₆₇	—	8	4	no	—
129	C₂⁴×C₆₇	#41	1	1	yes	characteristic
130	#41×C₆₇	—	2	2	no	—
131	#41×C₆₇	—	2	2	no	—
132	#41×C₆₇	—	2	2	no	—
133	#41×C₆₇	—	2	2	no	—
134	#41×C₆₇	—	2	2	no	—
135	#41×C₆₇	—	2	2	no	—
136	#41×C₆₇	—	2	2	no	—
137	#41×C₆₇	—	2	2	no	—
138	#41×C₆₇	—	4	1	no	—
139	#41×C₆₇	—	4	1	no	—
140	#41×C₆₇	—	4	1	no	—
141	#41×C₆₇	—	4	1	no	—
142	#41×C₆₇	—	4	1	no	—
143	#41×C₆₇	—	4	1	no	—
144	#41×C₆₇	—	4	1	no	—
145	#41×C₆₇	—	4	1	no	—
146	C₂×Q₈×C₆₇	—	2	1	no	—
147	C₂×Q₈×C₆₇	—	2	1	no	—
148	C₂×Q₈×C₆₇	—	2	1	no	—
149	C₂×Q₈×C₆₇	—	2	1	no	—
150	C₂×Q₈×C₆₇	—	2	1	no	—
151	C₂×Q₈×C₆₇	C₂²⋊C₄	1	2	yes	—
152	C₂×D₄×C₆₇	—	2	2	no	—
153	C₂×D₄×C₆₇	—	2	2	no	—
154	C₂×D₄×C₆₇	—	2	2	no	—
155	C₂×D₄×C₆₇	—	2	2	no	—
156	C₂×D₄×C₆₇	—	2	2	no	—
157	C₂×D₄×C₆₇	—	2	2	no	—
158	C₂×D₄×C₆₇	—	2	2	no	—
159	C₂×D₄×C₆₇	—	2	2	no	—
160	C₂×D₄×C₆₇	—	2	1	no	—
161	C₂×D₄×C₆₇	—	2	1	no	—
162	C₂×D₄×C₆₇	—	2	2	no	—
163	C₂×D₄×C₆₇	—	2	2	no	—
164	C₂×D₄×C₆₇	—	2	2	no	—
165	C₂×D₄×C₆₇	—	2	2	no	—
166	C₂×D₄×C₆₇	—	2	2	no	—
167	C₂×D₄×C₆₇	—	2	1	no	—
168	C₂×D₄×C₆₇	—	2	2	no	—
169	C₂×D₄×C₆₇	—	2	2	no	—
170	C₂×D₄×C₆₇	—	2	1	no	—
171	C₂×D₄×C₆₇	—	2	1	no	—
172	C₂×D₄×C₆₇	—	2	2	no	—
173	C₂×D₄×C₆₇	—	2	2	no	—
174	C₂×D₄×C₆₇	—	4	1	no	—
175	C₂×D₄×C₆₇	—	4	1	no	—
176	C₂×D₄×C₆₇	—	4	1	no	—
177	C₂×D₄×C₆₇	—	4	1	no	—
178	C₂×D₄×C₆₇	—	4	1	no	—
179	C₂×D₄×C₆₇	—	4	1	no	—
180	C₂×D₄×C₆₇	—	4	1	no	—
181	C₂×D₄×C₆₇	—	4	1	no	—
182	C₂×D₄×C₆₇	—	4	1	no	—
183	C₂×D₄×C₆₇	—	4	1	no	—
184	C₂×D₄×C₆₇	—	4	1	no	—
185	C₂×D₄×C₆₇	—	4	1	no	—
186	C₂×D₄×C₆₇	—	4	1	no	—
187	C₂×D₄×C₆₇	—	4	1	no	—
188	C₂×D₄×C₆₇	—	4	1	no	—
189	C₂×D₄×C₆₇	—	4	1	no	—
190	C₂×D₄×C₆₇	C₂²⋊C₄	1	2	yes	—
191	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
192	C₂²×C₄×C₆₇	—	2	1	no	—
193	C₂²×C₄×C₆₇	—	2	2	no	—
194	C₂²×C₄×C₆₇	—	2	2	no	—
195	C₂²×C₄×C₆₇	—	4	1	no	—
196	C₂²×C₄×C₆₇	—	4	1	no	—
197	C₂²×C₄×C₆₇	—	4	1	no	—
198	C₂²×C₄×C₆₇	—	4	1	no	—
199	C₂²×C₄×C₆₇	—	4	1	no	—
200	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
204	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
205	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
209	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
210	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
211	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
212	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
213	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
214	SD₁₆×C₆₇	—	2	16	no	—
215	D₈×C₆₇	—	2	16	no	—
216	M₁₆×C₆₇	—	2	2	no	—
217	M₁₆×C₆₇	—	2	2	no	—
218	M₁₆×C₆₇	—	2	2	no	—
219	M₁₆×C₆₇	—	2	2	no	—
220	M₁₆×C₆₇	—	2	2	no	—
221	M₁₆×C₆₇	—	2	2	no	—
222	M₁₆×C₆₇	—	2	2	no	—
223	M₁₆×C₆₇	—	2	2	no	—
224	M₁₆×C₆₇	—	2	2	no	—
225	M₁₆×C₆₇	—	4	1	no	—
226	M₁₆×C₆₇	—	4	1	no	—
227	M₁₆×C₆₇	—	4	1	no	—
228	M₁₆×C₆₇	—	4	1	no	—
229	M₁₆×C₆₇	—	4	1	no	—
230	M₁₆×C₆₇	—	4	1	no	—
231	M₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
232	C₂×C₈×C₆₇	—	2	2	no	—
233	C₂×C₈×C₆₇	—	2	2	no	—
234	C₂×C₈×C₆₇	—	2	2	no	—
235	C₂×C₈×C₆₇	—	2	2	no	—
236	C₂×C₈×C₆₇	—	2	2	no	—
237	C₂×C₈×C₆₇	—	2	2	no	—
238	C₂×C₈×C₆₇	—	4	1	no	—
239	C₂×C₈×C₆₇	—	4	1	no	—
240	C₂×C₈×C₆₇	—	4	1	no	—
241	C₂×C₈×C₆₇	—	4	1	no	—
242	C₂×C₈×C₆₇	—	4	1	no	—
243	C₂×C₈×C₆₇	—	4	1	no	—
244	C₄⋊C₄×C₆₇	—	2	2	no	—
245	C₄⋊C₄×C₆₇	—	4	1	no	—
246	C₂²⋊C₄×C₆₇	—	2	2	no	—
247	C₂²⋊C₄×C₆₇	—	4	1	no	—
248	C₄²×C₆₇	—	2	2	no	—
249	C₄²×C₆₇	—	4	1	no	—
250	C₂³×C₆₇	—	2	2	no	—
251	C₂³×C₆₇	—	2	4	no	—
252	C₂³×C₆₇	—	4	2	no	—
253	C₂³×C₆₇	—	4	2	no	—
254	C₂³×C₆₇	—	4	2	no	—
255	C₂³×C₆₇	—	4	2	no	—
256	C₂³×C₆₇	—	4	1	no	—
257	C₂³×C₆₇	—	4	2	no	—
258	C₂³×C₆₇	—	4	2	no	—
259	C₂³×C₆₇	—	4	1	no	—
260	C₂³×C₆₇	—	4	2	no	—
261	C₂³×C₆₇	—	8	2	no	—
262	C₂³×C₆₇	—	8	4	no	—
263	C₂³×C₆₇	—	8	4	no	—
264	C₂³×C₆₇	—	8	4	no	—
265	C₂³×C₆₇	—	8	4	no	—
266	C₂³×C₆₇	—	8	4	no	—
267	C₂³×C₆₇	—	8	4	no	—
268	C₂³×C₆₇	—	8	4	no	—
269	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
270	C₂³×C₆₇	#121	1	1	yes	characteristic
271	C₂³×C₆₇	#121	1	1	yes	characteristic
272	C₂³×C₆₇	#121	1	1	yes	characteristic
273	C₂³×C₆₇	#123	1	1	yes	characteristic
274	C₂³×C₆₇	#123	1	1	yes	characteristic
275	C₂³×C₆₇	#124	1	1	yes	characteristic
276	Q₈×C₆₇	—	2	2	no	—
277	Q₈×C₆₇	—	2	2	no	—
278	Q₈×C₆₇	—	4	1	no	—
279	Q₈×C₆₇	—	4	1	no	—
280	D₄×C₆₇	—	2	2	no	—
281	D₄×C₆₇	—	2	4	no	—
282	D₄×C₆₇	—	2	4	no	—
283	D₄×C₆₇	—	2	2	no	—
284	D₄×C₆₇	—	4	2	no	—
285	D₄×C₆₇	—	4	2	no	—
286	D₄×C₆₇	—	4	2	no	—
287	D₄×C₆₇	—	4	2	no	—
288	D₄×C₆₇	—	4	2	no	—
289	D₄×C₆₇	—	4	2	no	—
290	D₄×C₆₇	—	4	2	no	—
291	D₄×C₆₇	—	4	2	no	—
292	D₄×C₆₇	—	4	2	no	—
293	D₄×C₆₇	—	4	1	no	—
294	D₄×C₆₇	—	4	1	no	—
295	D₄×C₆₇	—	4	2	no	—
296	C₂×C₄×C₆₇	—	2	2	no	—
297	C₂×C₄×C₆₇	—	4	1	no	—
298	C₂×C₄×C₆₇	—	4	1	no	—
299	C₂×C₄×C₆₇	—	4	2	no	—
300	C₂×C₄×C₆₇	—	4	2	no	—
301	C₂×C₄×C₆₇	—	4	2	no	—
302	C₂×C₄×C₆₇	—	4	2	no	—
303	C₂×C₄×C₆₇	—	4	2	no	—
304	C₂×C₄×C₆₇	—	8	2	no	—
305	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
306	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
307	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
308	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
309	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
310	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
311	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
312	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
313	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
314	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
315	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
316	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
317	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
318	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
319	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
320	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
321	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
322	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
323	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
324	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
325	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
326	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
327	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
328	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
329	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
330	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
331	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
332	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
333	C₈×C₆₇	—	2	4	no	—
334	C₈×C₆₇	—	4	2	no	—
335	C₈×C₆₇	—	4	2	no	—
336	C₈×C₆₇	—	4	2	no	—
337	C₂²×C₆₇	—	2	2	no	—
338	C₂²×C₆₇	—	2	2	no	—
339	C₂²×C₆₇	—	2	2	no	—
340	C₂²×C₆₇	—	4	1	no	—
341	C₂²×C₆₇	—	4	1	no	—
342	C₂²×C₆₇	—	4	1	no	—
343	C₂²×C₆₇	—	4	1	no	—
344	C₂²×C₆₇	—	4	2	no	—
345	C₂²×C₆₇	—	4	4	no	—
346	C₂²×C₆₇	—	8	2	no	—
347	C₂²×C₆₇	—	8	4	no	—
348	C₂²×C₆₇	—	8	2	no	—
349	C₂²×C₆₇	—	8	4	no	—
350	C₂²×C₆₇	—	8	4	no	—
351	C₂²×C₆₇	—	8	4	no	—
352	C₂²×C₆₇	—	8	4	no	—
353	C₂²×C₆₇	—	8	4	no	—
354	C₂²×C₆₇	—	8	2	no	—
355	C₂²×C₆₇	—	8	4	no	—
356	C₂²×C₆₇	#386	1	1	yes	characteristic
357	C₂²×C₆₇	#386	1	1	yes	characteristic
358	C₂²×C₆₇	#390	1	1	yes	characteristic
359	C₂²×C₆₇	#392	1	1	yes	characteristic
360	C₂²×C₆₇	#394	1	1	yes	characteristic
361	C₂²×C₆₇	#394	1	1	yes	characteristic
362	C₂²×C₆₇	#394	1	1	yes	characteristic
363	C₄×C₆₇	—	4	2	no	—
364	C₄×C₆₇	—	8	2	no	—
365	C₄×C₆₇	#379	1	1	yes	characteristic
366	C₄×C₆₇	#386	1	1	yes	characteristic
367	C₄×C₆₇	#386	1	1	yes	characteristic
368	C₄×C₆₇	#390	1	1	yes	characteristic
369	C₄×C₆₇	#392	1	1	yes	characteristic
370	C₄×C₆₇	#392	1	1	yes	characteristic
371	C₄×C₆₇	#392	1	1	yes	characteristic
372	C₄×C₆₇	#394	1	1	yes	characteristic
373	#11034	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
374	C₂×C₆₇	—	2	1	no	—
375	C₂×C₆₇	—	2	1	no	—
376	C₂×C₆₇	—	2	1	no	—
377	C₂×C₆₇	—	2	1	no	—
378	C₂×C₆₇	—	2	1	no	—
379	C₂×C₆₇	—	2	1	no	—
380	C₂×C₆₇	—	8	2	no	—
381	C₂×C₆₇	—	8	4	no	—
382	C₂×C₆₇	#1438	1	1	yes	characteristic
383	C₂×C₆₇	#2048	1	2	yes	—
384	C₂²×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic
385	C₂×#421	C₂×C₆₇	1	1	yes	characteristic, complemented
386	C₂×#418	C₂×C₆₇	1	1	yes	characteristic, complemented
387	C₂×#413	C₂×C₆₇	1	1	yes	characteristic, complemented
388	C₂²×#100	C₂×C₆₇	1	1	yes	characteristic, complemented
389	C₂×#344	C₂×C₆₇	1	1	yes	characteristic, complemented
390	#1738	C₂×C₆₇	1	1	yes	characteristic, complemented
391	C₆₇	#11034	1	1	yes	characteristic, direct product factor, Sylow
392	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
393	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
394	C₂×AGL(1,ℤ/8)	—	2	2	no	—
395	C₂×AGL(1,ℤ/8)	—	2	2	no	—
396	C₂×AGL(1,ℤ/8)	—	2	2	no	—
397	C₂×AGL(1,ℤ/8)	—	2	2	no	—
398	C₂×AGL(1,ℤ/8)	—	2	2	no	—
399	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	4	yes	complemented
400	C₂²×SD₁₆	—	2	1	no	—
401	C₂²×D₈	—	2	1	no	—
402	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
403	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
404	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
405	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
406	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
407	#421	—	2	2	no	—
408	#421	—	2	2	no	—
409	C₂×#104	—	2	2	no	—
410	#418	—	2	2	no	—
411	#418	—	2	2	no	—
412	C₂×#102	—	2	2	no	—
413	#413	—	2	2	no	—
414	#413	—	2	2	no	—
415	C₂×#101	—	2	1	no	—
416	C₂×#100	—	2	2	no	—
417	C₂×#100	—	2	1	no	—
418	C₂×#100	—	2	2	no	—
419	C₂×#100	—	2	2	no	—
420	#344	—	2	2	no	—
421	#341	—	2	2	no	—
422	C₂⁵	—	2	1	no	—
423	C₂×#41	—	2	1	no	—
424	C₂×#41	—	2	1	no	—
425	C₂×#41	—	2	1	no	—
426	C₂×#41	—	2	1	no	—
427	C₂×#41	—	2	1	no	—
428	C₂×#41	—	2	1	no	—
429	C₂×#41	—	2	1	no	—
430	C₂×#41	—	2	1	no	—
431	C₂×#41	—	2	1	no	—
432	C₂×#41	—	2	1	no	—
433	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
434	C₂×#41	D₄×C₆₇	1	2	yes	—
435	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
436	C₂²×D₄	—	2	1	no	—
437	C₂²×D₄	—	2	1	no	—
438	C₂²×D₄	—	2	1	no	—
439	C₂²×D₄	—	2	1	no	—
440	C₂²×D₄	—	2	1	no	—
441	C₂²×D₄	—	2	1	no	—
442	C₂²×D₄	—	2	1	no	—
443	C₂²×D₄	—	2	1	no	—
444	C₂²×D₄	—	2	1	no	—
445	C₂²×D₄	—	2	1	no	—
446	C₂²×D₄	—	2	1	no	—
447	C₂²×D₄	—	2	1	no	—
448	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
449	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
450	C₂²×D₄	D₄×C₆₇	1	2	yes	—
451	C₂²×D₄	D₄×C₆₇	1	2	yes	—
452	C₂³×C₄	—	2	1	no	—
453	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
454	AGL(1,ℤ/8)	—	2	8	no	—
455	AGL(1,ℤ/8)	—	2	8	no	—
456	AGL(1,ℤ/8)	—	2	8	no	—
457	AGL(1,ℤ/8)	—	2	8	no	—
458	C₂×SD₁₆	—	2	4	no	—
459	C₂×SD₁₆	—	2	4	no	—
460	C₂×SD₁₆	—	2	4	no	—
461	C₂×D₈	—	2	4	no	—
462	C₂×D₈	—	2	4	no	—
463	C₂×D₈	—	2	4	no	—
464	C₂×M₁₆	—	2	1	no	—
465	C₂×M₁₆	—	2	1	no	—
466	C₂×M₁₆	—	2	1	no	—
467	C₂×M₁₆	—	2	1	no	—
468	C₂×M₁₆	—	2	1	no	—
469	C₂×M₁₆	—	2	1	no	—
470	C₂×M₁₆	—	2	1	no	—
471	C₂×M₁₆	—	2	1	no	—
472	C₂×M₁₆	—	2	1	no	—
473	C₂×M₁₆	—	2	1	no	—
474	C₂×M₁₆	—	2	1	no	—
475	C₂×M₁₆	—	2	1	no	—
476	C₂×M₁₆	—	2	1	no	—
477	C₂×M₁₆	—	2	1	no	—
478	C₂×M₁₆	—	2	1	no	—
479	C₂×M₁₆	—	2	1	no	—
480	C₂×M₁₆	—	2	1	no	—
481	C₂×M₁₆	—	2	1	no	—
482	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
483	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
484	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
485	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
486	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
487	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
488	C₂²×C₈	—	2	1	no	—
489	C₂²×C₈	—	2	1	no	—
490	C₂²×C₈	—	2	1	no	—
491	C₂²×C₈	—	2	1	no	—
492	#117	—	2	1	no	—
493	#117	—	2	1	no	—
494	#117	—	2	1	no	—
495	#117	D₄×C₆₇	1	2	yes	—
496	C₂×C₄⋊C₄	—	2	1	no	—
497	C₂×C₂²⋊C₄	—	2	1	no	—
498	C₂×C₄²	—	2	1	no	—
499	#104	—	2	8	no	—
500	#102	—	2	8	no	—
501	#101	—	2	4	no	—
502	#100	—	2	8	no	—
503	#100	—	2	4	no	—
504	C₂⁴	—	2	2	no	—
505	C₂⁴	—	2	2	no	—
506	C₂⁴	—	2	1	no	—
507	C₂⁴	—	2	2	no	—
508	C₂⁴	—	4	1	no	—
509	C₂⁴	—	4	1	no	—
510	C₂⁴	—	4	1	no	—
511	C₂⁴	—	4	1	no	—
512	C₂⁴	—	8	4	no	—
513	C₂⁴	#41×C₆₇	1	1	yes	characteristic
514	#41	—	2	2	no	—
515	#41	—	2	2	no	—
516	#41	—	2	2	no	—
517	#41	—	2	2	no	—
518	#41	—	2	2	no	—
519	#41	—	2	2	no	—
520	#41	—	2	2	no	—
521	#41	—	2	2	no	—
522	#41	—	4	1	no	—
523	#41	—	4	1	no	—
524	#41	—	4	1	no	—
525	#41	—	4	1	no	—
526	#41	—	4	1	no	—
527	#41	—	4	1	no	—
528	#41	—	4	1	no	—
529	#41	—	4	1	no	—
530	C₂×Q₈	—	2	1	no	—
531	C₂×Q₈	—	2	1	no	—
532	C₂×Q₈	—	2	1	no	—
533	C₂×Q₈	—	2	1	no	—
534	C₂×Q₈	—	2	1	no	—
535	C₂×Q₈	C₂²⋊C₄×C₆₇	1	2	yes	—
536	C₂×D₄	—	2	2	no	—
537	C₂×D₄	—	2	2	no	—
538	C₂×D₄	—	2	2	no	—
539	C₂×D₄	—	2	2	no	—
540	C₂×D₄	—	2	2	no	—
541	C₂×D₄	—	2	2	no	—
542	C₂×D₄	—	2	2	no	—
543	C₂×D₄	—	2	2	no	—
544	C₂×D₄	—	2	1	no	—
545	C₂×D₄	—	2	1	no	—
546	C₂×D₄	—	2	2	no	—
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	1	no	—
552	C₂×D₄	—	2	2	no	—
553	C₂×D₄	—	2	2	no	—
554	C₂×D₄	—	2	1	no	—
555	C₂×D₄	—	2	1	no	—
556	C₂×D₄	—	2	2	no	—
557	C₂×D₄	—	2	2	no	—
558	C₂×D₄	—	4	1	no	—
559	C₂×D₄	—	4	1	no	—
560	C₂×D₄	—	4	1	no	—
561	C₂×D₄	—	4	1	no	—
562	C₂×D₄	—	4	1	no	—
563	C₂×D₄	—	4	1	no	—
564	C₂×D₄	—	4	1	no	—
565	C₂×D₄	—	4	1	no	—
566	C₂×D₄	—	4	1	no	—
567	C₂×D₄	—	4	1	no	—
568	C₂×D₄	—	4	1	no	—
569	C₂×D₄	—	4	1	no	—
570	C₂×D₄	—	4	1	no	—
571	C₂×D₄	—	4	1	no	—
572	C₂×D₄	—	4	1	no	—
573	C₂×D₄	—	4	1	no	—
574	C₂×D₄	C₂²⋊C₄×C₆₇	1	2	yes	—
575	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
576	C₂²×C₄	—	2	1	no	—
577	C₂²×C₄	—	2	2	no	—
578	C₂²×C₄	—	2	2	no	—
579	C₂²×C₄	—	4	1	no	—
580	C₂²×C₄	—	4	1	no	—
581	C₂²×C₄	—	4	1	no	—
582	C₂²×C₄	—	4	1	no	—
583	C₂²×C₄	—	4	1	no	—
584	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
585	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
586	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
587	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
588	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
589	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
590	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
591	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
592	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
593	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
594	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
595	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
596	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
597	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
598	SD₁₆	—	2	16	no	—
599	D₈	—	2	16	no	—
600	M₁₆	—	2	2	no	—
601	M₁₆	—	2	2	no	—
602	M₁₆	—	2	2	no	—
603	M₁₆	—	2	2	no	—
604	M₁₆	—	2	2	no	—
605	M₁₆	—	2	2	no	—
606	M₁₆	—	2	2	no	—
607	M₁₆	—	2	2	no	—
608	M₁₆	—	2	2	no	—
609	M₁₆	—	4	1	no	—
610	M₁₆	—	4	1	no	—
611	M₁₆	—	4	1	no	—
612	M₁₆	—	4	1	no	—
613	M₁₆	—	4	1	no	—
614	M₁₆	—	4	1	no	—
615	M₁₆	C₂²⋊C₄×C₆₇	1	4	yes	—
616	C₂×C₈	—	2	2	no	—
617	C₂×C₈	—	2	2	no	—
618	C₂×C₈	—	2	2	no	—
619	C₂×C₈	—	2	2	no	—
620	C₂×C₈	—	2	2	no	—
621	C₂×C₈	—	2	2	no	—
622	C₂×C₈	—	4	1	no	—
623	C₂×C₈	—	4	1	no	—
624	C₂×C₈	—	4	1	no	—
625	C₂×C₈	—	4	1	no	—
626	C₂×C₈	—	4	1	no	—
627	C₂×C₈	—	4	1	no	—
628	C₄⋊C₄	—	2	2	no	—
629	C₄⋊C₄	—	4	1	no	—
630	C₂²⋊C₄	—	2	2	no	—
631	C₂²⋊C₄	—	4	1	no	—
632	C₄²	—	2	2	no	—
633	C₄²	—	4	1	no	—
634	C₂³	—	2	2	no	—
635	C₂³	—	2	4	no	—
636	C₂³	—	4	2	no	—
637	C₂³	—	4	2	no	—
638	C₂³	—	4	2	no	—
639	C₂³	—	4	2	no	—
640	C₂³	—	4	1	no	—
641	C₂³	—	4	1	no	—
642	C₂³	—	4	2	no	—
643	C₂³	—	4	2	no	—
644	C₂³	—	4	2	no	—
645	C₂³	—	8	4	no	—
646	C₂³	—	8	2	no	—
647	C₂³	—	8	4	no	—
648	C₂³	—	8	4	no	—
649	C₂³	—	8	4	no	—
650	C₂³	—	8	4	no	—
651	C₂³	—	8	4	no	—
652	C₂³	—	8	4	no	—
653	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
654	C₂³	#121×C₆₇	1	1	yes	characteristic
655	C₂³	#121×C₆₇	1	1	yes	characteristic
656	C₂³	#121×C₆₇	1	1	yes	characteristic
657	C₂³	#123×C₆₇	1	1	yes	characteristic
658	C₂³	#123×C₆₇	1	1	yes	characteristic
659	C₂³	#124×C₆₇	1	1	yes	characteristic
660	Q₈	—	2	2	no	—
661	Q₈	—	2	2	no	—
662	Q₈	—	4	1	no	—
663	Q₈	—	4	1	no	—
664	D₄	—	2	2	no	—
665	D₄	—	2	4	no	—
666	D₄	—	2	4	no	—
667	D₄	—	2	2	no	—
668	D₄	—	4	2	no	—
669	D₄	—	4	2	no	—
670	D₄	—	4	2	no	—
671	D₄	—	4	2	no	—
672	D₄	—	4	2	no	—
673	D₄	—	4	2	no	—
674	D₄	—	4	2	no	—
675	D₄	—	4	2	no	—
676	D₄	—	4	2	no	—
677	D₄	—	4	1	no	—
678	D₄	—	4	1	no	—
679	D₄	—	4	2	no	—
680	C₂×C₄	—	2	2	no	—
681	C₂×C₄	—	4	1	no	—
682	C₂×C₄	—	4	1	no	—
683	C₂×C₄	—	4	2	no	—
684	C₂×C₄	—	4	2	no	—
685	C₂×C₄	—	4	2	no	—
686	C₂×C₄	—	4	2	no	—
687	C₂×C₄	—	4	2	no	—
688	C₂×C₄	—	8	2	no	—
689	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
690	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
691	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
692	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
693	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
694	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
695	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
696	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
697	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
698	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
699	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
700	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
701	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
702	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
703	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
704	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
705	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
706	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
707	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
708	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
709	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
710	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
711	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
712	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
713	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
714	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
715	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
716	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
717	C₈	—	2	4	no	—
718	C₈	—	4	2	no	—
719	C₈	—	4	2	no	—
720	C₈	—	4	2	no	—
721	C₂²	—	2	2	no	—
722	C₂²	—	2	2	no	—
723	C₂²	—	2	2	no	—
724	C₂²	—	4	1	no	—
725	C₂²	—	4	1	no	—
726	C₂²	—	4	1	no	—
727	C₂²	—	4	1	no	—
728	C₂²	—	4	2	no	—
729	C₂²	—	4	4	no	—
730	C₂²	—	8	4	no	—
731	C₂²	—	8	2	no	—
732	C₂²	—	8	4	no	—
733	C₂²	—	8	4	no	—
734	C₂²	—	8	4	no	—
735	C₂²	—	8	4	no	—
736	C₂²	—	8	4	no	—
737	C₂²	—	8	2	no	—
738	C₂²	—	8	4	no	—
739	C₂²	—	8	2	no	—
740	C₂²	#386×C₆₇	1	1	yes	characteristic
741	C₂²	#386×C₆₇	1	1	yes	characteristic
742	C₂²	#390×C₆₇	1	1	yes	characteristic
743	C₂²	#392×C₆₇	1	1	yes	characteristic
744	C₂²	#394×C₆₇	1	1	yes	characteristic
745	C₂²	#394×C₆₇	1	1	yes	characteristic
746	C₂²	#394×C₆₇	1	1	yes	characteristic
747	C₄	—	4	2	no	—
748	C₄	—	8	2	no	—
749	C₄	#379×C₆₇	1	1	yes	characteristic
750	C₄	#386×C₆₇	1	1	yes	characteristic
751	C₄	#386×C₆₇	1	1	yes	characteristic
752	C₄	#390×C₆₇	1	1	yes	characteristic
753	C₄	#392×C₆₇	1	1	yes	characteristic
754	C₄	#392×C₆₇	1	1	yes	characteristic
755	C₄	#392×C₆₇	1	1	yes	characteristic
756	C₄	#394×C₆₇	1	1	yes	characteristic
757	C₂	—	2	1	no	—
758	C₂	—	2	1	no	—
759	C₂	—	2	1	no	—
760	C₂	—	2	1	no	—
761	C₂	—	2	1	no	—
762	C₂	—	2	1	no	—
763	C₂	—	8	4	no	—
764	C₂	—	8	2	no	—
765	C₂	C₆₇×#1438	1	1	yes	characteristic
766	C₂	C₆₇×#2048	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	#394
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂²×AGL(1,ℤ/8)×C₆₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#421×C₆₇	C₂	semidirect
3. C₂×#418×C₆₇	C₂	semidirect
4. C₂×#413×C₆₇	C₂	semidirect
5. C₂²×#100×C₆₇	C₂	semidirect
6. C₂×#344×C₆₇	C₂	semidirect
7. C₆₇×#1738	C₂	semidirect
15. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
373. #11034	C₆₇	direct (N × H)
385. C₂×#421	C₂×C₆₇	semidirect
386. C₂×#418	C₂×C₆₇	semidirect
387. C₂×#413	C₂×C₆₇	semidirect
388. C₂²×#100	C₂×C₆₇	semidirect
389. C₂×#344	C₂×C₆₇	semidirect
390. #1738	C₂×C₆₇	semidirect
391. C₆₇	#11034	direct (N × H)
399. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×AGL(1,ℤ/8)×C₆₇	2
C₂×#421×C₆₇	2
C₂×#418×C₆₇	2
C₂×#413×C₆₇	2
C₂²×#100×C₆₇	2
C₂×#344×C₆₇	2
C₆₇×#1738	2
#11034	67

Subgroups — C67×#11034 #30690285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11034 #30690285