Subgroups — C₆₇×#9095 #30688346

Order 17152.

546 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1477	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1460	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1283	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	1	yes	characteristic
9	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₈×Q₈×C₆₇	—	2	1	no	—
12	#442×C₆₇	—	2	1	no	—
13	#440×C₆₇	—	2	1	no	—
14	C₄×SD₁₆×C₆₇	—	2	1	no	—
15	C₄×D₈×C₆₇	—	2	1	no	—
16	C₈×D₄×C₆₇	—	2	2	no	—
17	C₈×D₄×C₆₇	—	2	1	no	—
18	#431×C₆₇	—	2	1	no	—
19	#431×C₆₇	C₂²	1	1	yes	characteristic
20	#431×C₆₇	C₂²	1	1	yes	characteristic
21	#428×C₆₇	C₄	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#418×C₆₇	C₄	1	1	yes	characteristic
24	#417×C₆₇	C₄	1	1	yes	characteristic
25	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	—	2	1	no	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	Q₈⋊C₈×C₆₇	—	2	1	no	—
29	D₄⋊C₈×C₆₇	—	2	1	no	—
30	D₄⋊C₈×C₆₇	—	2	1	no	—
31	D₄⋊C₈×C₆₇	—	2	1	no	—
32	#323×C₆₇	—	2	2	no	—
33	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	—	2	1	no	—
36	AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
37	AGL(1,ℤ/8)×C₆₇	C₈	1	4	yes	complemented
38	C₂×SD₁₆×C₆₇	—	2	1	no	—
39	C₂×D₈×C₆₇	—	2	1	no	—
40	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	—	2	1	no	—
44	C₂²×C₈×C₆₇	—	2	1	no	—
45	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
47	C₄×D₄×C₆₇	—	2	1	no	—
48	C₄×D₄×C₆₇	—	2	1	no	—
49	C₄×D₄×C₆₇	—	2	1	no	—
50	C₄×D₄×C₆₇	—	2	1	no	—
51	C₄×D₄×C₆₇	—	2	1	no	—
52	C₄×D₄×C₆₇	—	2	1	no	—
53	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
54	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
55	#117×C₆₇	—	2	1	no	—
56	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄²×C₆₇	—	2	1	no	—
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	#107×C₆₇	—	2	1	no	—
62	#106×C₆₇	—	2	1	no	—
63	C₄⋊C₈×C₆₇	—	2	2	no	—
64	C₄⋊C₈×C₆₇	—	2	1	no	—
65	C₄⋊C₈×C₆₇	—	2	1	no	—
66	C₄⋊C₈×C₆₇	—	2	1	no	—
67	C₄⋊C₈×C₆₇	—	2	1	no	—
68	#103×C₆₇	—	2	1	no	—
69	#102×C₆₇	—	2	1	no	—
70	#102×C₆₇	—	2	1	no	—
71	#102×C₆₇	—	2	1	no	—
72	C₂²⋊C₈×C₆₇	—	2	2	no	—
73	C₂²⋊C₈×C₆₇	—	2	1	no	—
74	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
75	#97×C₆₇	—	2	1	no	—
76	C₄×C₈×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	—	2	1	no	—
78	C₄×C₈×C₆₇	—	2	1	no	—
79	C₄×C₈×C₆₇	—	2	1	no	—
80	C₄×C₈×C₆₇	—	2	1	no	—
81	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
82	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
83	C₂⁴×C₆₇	—	2	1	no	—
84	#41×C₆₇	—	2	1	no	—
85	#41×C₆₇	—	2	1	no	—
86	#41×C₆₇	C₂×C₈	1	2	yes	—
87	#41×C₆₇	M₁₆	1	2	yes	—
88	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×D₄×C₆₇	—	2	1	no	—
90	C₂×D₄×C₆₇	—	2	1	no	—
91	C₂×D₄×C₆₇	—	2	1	no	—
92	C₂×D₄×C₆₇	—	2	1	no	—
93	C₂×D₄×C₆₇	—	2	1	no	—
94	C₂×D₄×C₆₇	—	2	1	no	—
95	C₂×D₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×D₄×C₆₇	C₂×C₈	1	2	yes	complemented
97	C₂²×C₄×C₆₇	—	2	1	no	—
98	C₂²×C₄×C₆₇	—	2	1	no	—
99	C₂²×C₄×C₆₇	—	2	1	no	—
100	C₂²×C₄×C₆₇	—	2	1	no	—
101	C₂²×C₄×C₆₇	—	4	1	no	—
102	C₂²×C₄×C₆₇	—	4	1	no	—
103	C₂²×C₄×C₆₇	—	8	1	no	—
104	C₂²×C₄×C₆₇	—	8	1	no	—
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
108	SD₁₆×C₆₇	—	2	4	no	—
109	D₈×C₆₇	—	2	4	no	—
110	M₁₆×C₆₇	—	2	1	no	—
111	M₁₆×C₆₇	—	2	1	no	—
112	M₁₆×C₆₇	—	2	1	no	—
113	M₁₆×C₆₇	C₂×C₈	1	2	yes	complemented
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	2	1	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	4	2	no	—
120	C₂×C₈×C₆₇	—	4	4	no	—
121	C₂×C₈×C₆₇	—	4	1	no	—
122	C₂×C₈×C₆₇	—	4	1	no	—
123	C₂×C₈×C₆₇	—	8	1	no	—
124	C₄⋊C₄×C₆₇	—	2	1	no	—
125	C₄⋊C₄×C₆₇	—	2	1	no	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	1	no	—
128	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
129	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
130	C₂²⋊C₄×C₆₇	—	2	1	no	—
131	C₂²⋊C₄×C₆₇	—	2	1	no	—
132	C₂²⋊C₄×C₆₇	—	2	1	no	—
133	C₂²⋊C₄×C₆₇	—	4	1	no	—
134	C₂²⋊C₄×C₆₇	—	4	1	no	—
135	C₄²×C₆₇	—	2	1	no	—
136	C₄²×C₆₇	—	2	1	no	—
137	C₄²×C₆₇	—	2	1	no	—
138	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₆₇	#41	1	1	yes	characteristic
142	C₂³×C₆₇	—	2	1	no	—
143	C₂³×C₆₇	—	2	2	no	—
144	C₂³×C₆₇	—	4	1	no	—
145	C₂³×C₆₇	—	4	1	no	—
146	C₂³×C₆₇	—	8	1	no	—
147	C₂³×C₆₇	—	8	1	no	—
148	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
149	Q₈×C₆₇	—	2	1	no	—
150	Q₈×C₆₇	C₂²⋊C₈	1	2	yes	—
151	D₄×C₆₇	—	2	2	no	—
152	D₄×C₆₇	—	2	2	no	—
153	D₄×C₆₇	—	2	2	no	—
154	D₄×C₆₇	—	2	1	no	—
155	D₄×C₆₇	—	2	2	no	—
156	D₄×C₆₇	—	4	1	no	—
157	D₄×C₆₇	—	4	1	no	—
158	D₄×C₆₇	C₂²⋊C₈	1	2	yes	—
159	C₂×C₄×C₆₇	—	2	1	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	1	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	8	2	no	—
172	C₂×C₄×C₆₇	—	8	1	no	—
173	C₂×C₄×C₆₇	—	8	1	no	—
174	C₂×C₄×C₆₇	—	8	1	no	—
175	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
178	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
179	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
180	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
182	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
183	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
184	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
185	C₈×C₆₇	—	2	2	no	—
186	C₈×C₆₇	—	2	2	no	—
187	C₈×C₆₇	—	4	2	no	—
188	C₈×C₆₇	—	4	1	no	—
189	C₈×C₆₇	—	4	1	no	—
190	C₈×C₆₇	—	8	1	no	—
191	C₈×C₆₇	—	8	1	no	—
192	C₂²×C₆₇	—	2	1	no	—
193	C₂²×C₆₇	—	2	1	no	—
194	C₂²×C₆₇	—	2	2	no	—
195	C₂²×C₆₇	—	4	1	no	—
196	C₂²×C₆₇	—	4	1	no	—
197	C₂²×C₆₇	—	4	2	no	—
198	C₂²×C₆₇	—	8	2	no	—
199	C₂²×C₆₇	—	8	2	no	—
200	C₂²×C₆₇	—	8	1	no	—
201	C₂²×C₆₇	—	8	1	no	—
202	C₂²×C₆₇	#386	1	1	yes	characteristic
203	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
204	C₂²×C₆₇	#435	1	1	yes	characteristic
205	C₄×C₆₇	—	2	2	no	—
206	C₄×C₆₇	—	2	1	no	—
207	C₄×C₆₇	—	2	1	no	—
208	C₄×C₆₇	—	2	1	no	—
209	C₄×C₆₇	—	4	1	no	—
210	C₄×C₆₇	—	4	1	no	—
211	C₄×C₆₇	—	8	1	no	—
212	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
213	C₄×C₆₇	#408	1	1	yes	characteristic
214	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
215	C₄×C₆₇	#435	1	1	yes	characteristic
216	C₄×C₆₇	#471	1	1	yes	characteristic
217	C₄×C₆₇	#472	1	1	yes	characteristic
218	#9095	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂×C₆₇	—	2	1	no	—
220	C₂×C₆₇	—	2	1	no	—
221	C₂×C₆₇	—	4	2	no	—
222	C₂×C₆₇	—	8	2	no	—
223	C₂×C₆₇	#1869	1	1	yes	characteristic
224	C₂×C₆₇	#1906	1	1	yes	characteristic
225	C₂×C₆₇	#1907	1	1	yes	characteristic
226	C₈×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
227	C₄×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic
228	#1477	C₂×C₆₇	1	1	yes	characteristic, complemented
229	C₂×D₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#1460	C₂×C₆₇	1	1	yes	characteristic, complemented
231	#1283	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#1278	C₂×C₆₇	1	1	yes	characteristic, complemented
233	C₆₇	#9095	1	1	yes	characteristic, direct product factor, Sylow
234	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	1	yes	characteristic
235	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
237	C₈×Q₈	—	2	1	no	—
238	#442	—	2	1	no	—
239	#440	—	2	1	no	—
240	C₄×SD₁₆	—	2	1	no	—
241	C₄×D₈	—	2	1	no	—
242	C₈×D₄	—	2	1	no	—
243	C₈×D₄	—	2	2	no	—
244	#431	—	2	1	no	—
245	#431	C₂²×C₆₇	1	1	yes	characteristic
246	#431	C₂²×C₆₇	1	1	yes	characteristic
247	#428	C₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
249	#418	C₄×C₆₇	1	1	yes	characteristic
250	#417	C₄×C₆₇	1	1	yes	characteristic
251	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×C₄×C₈	—	2	1	no	—
253	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
254	Q₈⋊C₈	—	2	1	no	—
255	D₄⋊C₈	—	2	1	no	—
256	D₄⋊C₈	—	2	1	no	—
257	D₄⋊C₈	—	2	1	no	—
258	#323	—	2	2	no	—
259	C₂×#41	C₂×C₄×C₆₇	1	1	yes	characteristic
260	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
261	C₂³×C₄	—	2	1	no	—
262	AGL(1,ℤ/8)	—	2	2	no	—
263	AGL(1,ℤ/8)	C₈×C₆₇	1	4	yes	complemented
264	C₂×SD₁₆	—	2	1	no	—
265	C₂×D₈	—	2	1	no	—
266	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
267	C₂×M₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
268	C₂²×C₈	—	2	2	no	—
269	C₂²×C₈	—	2	1	no	—
270	C₂²×C₈	—	2	1	no	—
271	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
272	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
273	C₄×D₄	—	2	1	no	—
274	C₄×D₄	—	2	1	no	—
275	C₄×D₄	—	2	1	no	—
276	C₄×D₄	—	2	1	no	—
277	C₄×D₄	—	2	1	no	—
278	C₄×D₄	—	2	1	no	—
279	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
280	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
281	#117	—	2	1	no	—
282	#117	C₂×C₄×C₆₇	1	1	yes	characteristic
283	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
284	C₂×C₂²⋊C₄	—	2	1	no	—
285	C₂×C₄²	—	2	1	no	—
286	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
287	#107	—	2	1	no	—
288	#106	—	2	1	no	—
289	C₄⋊C₈	—	2	2	no	—
290	C₄⋊C₈	—	2	1	no	—
291	C₄⋊C₈	—	2	1	no	—
292	C₄⋊C₈	—	2	1	no	—
293	C₄⋊C₈	—	2	1	no	—
294	#103	—	2	1	no	—
295	#102	—	2	1	no	—
296	#102	—	2	1	no	—
297	#102	—	2	1	no	—
298	C₂²⋊C₈	—	2	2	no	—
299	C₂²⋊C₈	—	2	1	no	—
300	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
301	#97	—	2	1	no	—
302	C₄×C₈	—	2	2	no	—
303	C₄×C₈	—	2	1	no	—
304	C₄×C₈	—	2	1	no	—
305	C₄×C₈	—	2	1	no	—
306	C₄×C₈	—	2	1	no	—
307	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
308	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
309	C₂⁴	—	2	1	no	—
310	#41	—	2	1	no	—
311	#41	—	2	1	no	—
312	#41	C₂×C₈×C₆₇	1	2	yes	—
313	#41	M₁₆×C₆₇	1	2	yes	—
314	C₂×Q₈	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
315	C₂×D₄	—	2	1	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
322	C₂×D₄	C₂×C₈×C₆₇	1	2	yes	complemented
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	4	1	no	—
328	C₂²×C₄	—	4	1	no	—
329	C₂²×C₄	—	8	1	no	—
330	C₂²×C₄	—	8	1	no	—
331	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
333	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
334	SD₁₆	—	2	4	no	—
335	D₈	—	2	4	no	—
336	M₁₆	—	2	1	no	—
337	M₁₆	—	2	1	no	—
338	M₁₆	—	2	1	no	—
339	M₁₆	C₂×C₈×C₆₇	1	2	yes	complemented
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	4	2	no	—
346	C₂×C₈	—	4	4	no	—
347	C₂×C₈	—	4	1	no	—
348	C₂×C₈	—	4	1	no	—
349	C₂×C₈	—	8	1	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
355	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
356	C₂²⋊C₄	—	2	1	no	—
357	C₂²⋊C₄	—	2	1	no	—
358	C₂²⋊C₄	—	2	1	no	—
359	C₂²⋊C₄	—	4	1	no	—
360	C₂²⋊C₄	—	4	1	no	—
361	C₄²	—	2	1	no	—
362	C₄²	—	2	1	no	—
363	C₄²	—	2	1	no	—
364	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
365	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
366	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
367	C₄²	#41×C₆₇	1	1	yes	characteristic
368	C₂³	—	2	1	no	—
369	C₂³	—	2	2	no	—
370	C₂³	—	4	1	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	8	1	no	—
373	C₂³	—	8	1	no	—
374	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
375	Q₈	—	2	1	no	—
376	Q₈	C₂²⋊C₈×C₆₇	1	2	yes	—
377	D₄	—	2	2	no	—
378	D₄	—	2	2	no	—
379	D₄	—	2	2	no	—
380	D₄	—	2	1	no	—
381	D₄	—	2	2	no	—
382	D₄	—	4	1	no	—
383	D₄	—	4	1	no	—
384	D₄	C₂²⋊C₈×C₆₇	1	2	yes	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	8	2	no	—
398	C₂×C₄	—	8	1	no	—
399	C₂×C₄	—	8	1	no	—
400	C₂×C₄	—	8	1	no	—
401	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
402	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
403	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
404	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
405	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
406	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
407	C₂×C₄	C₂²×C₈×C₆₇	1	1	yes	characteristic
408	C₂×C₄	C₂×M₁₆×C₆₇	1	1	yes	characteristic
409	C₂×C₄	#131×C₆₇	1	1	yes	characteristic
410	C₂×C₄	#131×C₆₇	1	1	yes	characteristic
411	C₈	—	2	2	no	—
412	C₈	—	2	2	no	—
413	C₈	—	4	2	no	—
414	C₈	—	4	1	no	—
415	C₈	—	4	1	no	—
416	C₈	—	8	1	no	—
417	C₈	—	8	1	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	2	no	—
421	C₂²	—	4	1	no	—
422	C₂²	—	4	1	no	—
423	C₂²	—	4	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	2	no	—
426	C₂²	—	8	1	no	—
427	C₂²	—	8	1	no	—
428	C₂²	#386×C₆₇	1	1	yes	characteristic
429	C₂²	C₈×D₄×C₆₇	1	1	yes	characteristic
430	C₂²	#435×C₆₇	1	1	yes	characteristic
431	C₄	—	2	2	no	—
432	C₄	—	2	1	no	—
433	C₄	—	2	1	no	—
434	C₄	—	2	1	no	—
435	C₄	—	4	1	no	—
436	C₄	—	4	1	no	—
437	C₄	—	8	1	no	—
438	C₄	C₂×C₂²⋊C₈×C₆₇	1	1	yes	characteristic
439	C₄	#408×C₆₇	1	1	yes	characteristic
440	C₄	C₈×D₄×C₆₇	1	1	yes	characteristic
441	C₄	#435×C₆₇	1	1	yes	characteristic
442	C₄	#471×C₆₇	1	1	yes	characteristic
443	C₄	#472×C₆₇	1	1	yes	characteristic
444	C₂	—	2	1	no	—
445	C₂	—	2	1	no	—
446	C₂	—	4	2	no	—
447	C₂	—	8	2	no	—
448	C₂	C₆₇×#1869	1	1	yes	characteristic
449	C₂	C₆₇×#1906	1	1	yes	characteristic
450	C₂	C₆₇×#1907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	#121
commutator	C₂×C₄	C₂²×C₈×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
socle	C₂²×C₆₇	#386
prime radical	C₂×AGL(1,ℤ/8)×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈×#41×C₆₇	C₂	semidirect
3. C₆₇×#1477	C₂	semidirect
4. C₂×D₄⋊C₈×C₆₇	C₂	semidirect
5. C₆₇×#1460	C₂	semidirect
6. C₆₇×#1283	C₂	semidirect
7. C₆₇×#1278	C₂	semidirect
37. AGL(1,ℤ/8)×C₆₇	C₈	semidirect
96. C₂×D₄×C₆₇	C₂×C₈	semidirect
113. M₁₆×C₆₇	C₂×C₈	semidirect
218. #9095	C₆₇	direct (N × H)
226. C₈×#41	C₂×C₆₇	semidirect
228. #1477	C₂×C₆₇	semidirect
229. C₂×D₄⋊C₈	C₂×C₆₇	semidirect
230. #1460	C₂×C₆₇	semidirect
231. #1283	C₂×C₆₇	semidirect
232. #1278	C₂×C₆₇	semidirect
233. C₆₇	#9095	direct (N × H)
263. AGL(1,ℤ/8)	C₈×C₆₇	semidirect
322. C₂×D₄	C₂×C₈×C₆₇	semidirect
339. M₁₆	C₂×C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41×C₆₇	2
C₄×AGL(1,ℤ/8)×C₆₇	2
C₆₇×#1477	2
C₂×D₄⋊C₈×C₆₇	2
C₆₇×#1460	2
C₆₇×#1283	2
C₆₇×#1278	2
#9095	67

Subgroups — C67×#9095 #30688346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#9095 #30688346