Subgroups — #30863662

Order 17152.

750 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569144	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14524050	C₂	1	1	yes	maximal, characteristic
3	#22561154	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560985	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560810	C₂	1	2	yes	maximal, complemented
6	#22560486	C₂	1	2	yes	maximal
7	#22560229	C₂	1	1	yes	maximal, characteristic
8	#22560090	C₂	1	2	yes	maximal, complemented
9	#22560074	C₂	1	1	yes	maximal, characteristic, complemented
10	#22560049	C₂	1	2	yes	maximal
11	C₆₇×#3076	C₂	1	1	yes	maximal, characteristic
12	C₂×#14524050	C₂²	1	1	yes	characteristic
13	C₂×#14524050	C₂²	1	1	yes	characteristic
14	#16863527	—	2	1	no	—
15	#16863527	—	2	1	no	—
16	Q₈×C₆₇⋊C₈	—	2	1	no	—
17	Q₈×C₆₇⋊C₈	—	2	1	no	—
18	C₂×#14523979	C₂²	1	2	yes	—
19	#16863423	C₂²	1	2	yes	—
20	#16863420	C₂²	1	1	yes	characteristic
21	C₂×#14523970	C₂²	1	2	yes	—
22	#16863352	—	2	1	no	—
23	#16863352	—	2	1	no	—
24	#16863352	C₂²	1	1	yes	characteristic
25	#16863352	C₂²	1	2	yes	—
26	#16863352	C₂²	1	1	yes	characteristic
27	C₄×#11761612	—	2	1	no	—
28	C₄×#11761612	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
30	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
32	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
33	#16863161	C₂²	1	2	yes	—
34	#16863155	C₂²	1	2	yes	—
35	#16863155	C₂²	1	2	yes	—
36	#16863155	C₂²	1	2	yes	—
37	#16863150	C₂²	1	2	yes	—
38	#533×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
40	#479×C₆₇	C₄	1	2	yes	complemented
41	#478×C₆₇	C₄	1	2	yes	complemented
42	#477×C₆₇	C₄	1	1	yes	characteristic, complemented
43	#476×C₆₇	C₄	1	1	yes	characteristic, complemented
44	#475×C₆₇	C₄	1	1	yes	characteristic, complemented
45	#474×C₆₇	C₄	1	1	yes	characteristic, complemented
46	#427×C₆₇	C₂²	1	2	yes	—
47	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
48	#416×C₆₇	C₂²	1	2	yes	—
49	#14524050	—	2	2	no	—
50	#14524050	—	2	2	no	—
51	#14524050	—	2	2	no	—
52	#14524050	Q₈	1	4	yes	complemented
53	C₂×#11761612	—	2	1	no	—
54	C₂×#11761612	—	2	1	no	—
55	C₂×#11761612	C₂³	1	2	yes	—
56	C₂×#11761612	C₂³	1	1	yes	characteristic
57	C₂×#11761612	C₂³	1	1	yes	characteristic
58	C₂²×C₆₇⋊C₈	—	2	1	no	—
59	C₂²×C₆₇⋊C₈	—	2	1	no	—
60	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
61	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊Dic₆₇	—	2	1	no	—
63	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
64	#14523979	—	2	4	no	—
65	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
66	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
67	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
68	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
69	#14523970	—	2	4	no	—
70	#14523954	—	2	2	no	—
71	#14523954	—	2	1	no	—
72	#14523954	—	2	1	no	—
73	#14523954	—	2	1	no	—
74	C₄×C₆₇⋊C₈	—	2	2	no	—
75	C₄×C₆₇⋊C₈	—	2	1	no	—
76	C₄×C₆₇⋊C₈	—	2	1	no	—
77	C₄×C₆₇⋊C₈	—	2	1	no	—
78	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
79	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
80	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
81	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
82	C₈×Dic₆₇	—	2	4	no	—
83	C₆₇⋊#95	—	2	1	no	—
84	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
86	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
88	#125×C₆₇	C₂×C₄	1	2	yes	—
89	C₂²⋊Q₈×C₆₇	—	2	1	no	—
90	C₂²⋊Q₈×C₆₇	—	2	1	no	—
91	C₄×Q₈×C₆₇	—	2	1	no	—
92	C₄×Q₈×C₆₇	—	2	1	no	—
93	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
95	C₄×D₄×C₆₇	—	2	1	no	—
96	C₄×D₄×C₆₇	—	2	1	no	—
97	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
99	#117×C₆₇	—	2	1	no	—
100	#117×C₆₇	C₂³	1	1	yes	characteristic
101	#117×C₆₇	C₂³	1	2	yes	—
102	C₂×C₄²×C₆₇	—	2	1	no	—
103	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
104	#107×C₆₇	C₂×C₄	1	2	yes	—
105	#107×C₆₇	C₂×C₄	1	2	yes	—
106	#106×C₆₇	C₂×C₄	1	2	yes	—
107	#106×C₆₇	C₂×C₄	1	2	yes	—
108	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
109	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
110	C₄⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
111	#103×C₆₇	C₂×C₄	1	2	yes	—
112	#103×C₆₇	C₂×C₄	1	2	yes	—
113	#102×C₆₇	C₂×C₄	1	2	yes	—
114	#102×C₆₇	C₂×C₄	1	2	yes	—
115	C₂²×Dic₆₇	—	2	2	no	—
116	#11761612	—	2	4	no	—
117	#11761612	—	2	1	no	—
118	#11761612	—	2	1	no	—
119	#11761612	—	2	4	no	—
120	#11761612	—	2	2	no	—
121	#11761612	—	2	2	no	—
122	#11761612	C₂×Q₈	1	2	yes	—
123	#11761612	#41	1	2	yes	—
124	C₂×C₆₇⋊C₈	—	2	4	no	—
125	C₂×C₆₇⋊C₈	—	2	1	no	—
126	C₂×C₆₇⋊C₈	—	2	1	no	—
127	C₂×C₆₇⋊C₈	—	2	1	no	—
128	C₂×C₆₇⋊C₈	—	2	1	no	—
129	C₂×C₆₇⋊C₈	—	2	4	no	—
130	C₂×C₆₇⋊C₈	—	2	2	no	—
131	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
136	C₂×C₆₇⋊C₈	#41	1	2	yes	—
137	C₄⋊Dic₆₇	—	2	4	no	—
138	C₄×Dic₆₇	—	2	2	no	—
139	C₄×Dic₆₇	—	2	2	no	—
140	#41×C₆₇	—	2	1	no	—
141	#41×C₆₇	—	2	1	no	—
142	#41×C₆₇	C₂×Q₈	1	2	yes	—
143	#41×C₆₇	#41	1	2	yes	—
144	C₂×Q₈×C₆₇	—	2	1	no	—
145	C₂×Q₈×C₆₇	—	2	1	no	—
146	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
147	C₂×D₄×C₆₇	—	2	1	no	—
148	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	—	2	1	no	—
150	C₂²×C₄×C₆₇	—	2	1	no	—
151	C₂²×C₄×C₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
158	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
159	C₄⋊C₄×C₆₇	—	2	1	no	—
160	C₄⋊C₄×C₆₇	—	2	1	no	—
161	C₄⋊C₄×C₆₇	—	2	2	no	—
162	C₄⋊C₄×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	—	2	1	no	—
164	C₄⋊C₄×C₆₇	—	2	1	no	—
165	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
166	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
167	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
169	C₂²⋊C₄×C₆₇	—	2	1	no	—
170	C₂²⋊C₄×C₆₇	—	2	2	no	—
171	C₂²⋊C₄×C₆₇	—	2	1	no	—
172	C₂²⋊C₄×C₆₇	—	2	1	no	—
173	C₄²×C₆₇	—	2	1	no	—
174	C₄²×C₆₇	—	2	1	no	—
175	C₄²×C₆₇	—	2	1	no	—
176	C₄²×C₆₇	—	2	2	no	—
177	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
181	C₂³×C₆₇	—	2	1	no	—
182	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
183	Q₈×C₆₇	—	2	1	no	—
184	Q₈×C₆₇	—	4	2	no	—
185	Q₈×C₆₇	—	4	2	no	—
186	Q₈×C₆₇	C₄×Q₈	1	2	yes	—
187	D₄×C₆₇	—	2	2	no	—
188	D₄×C₆₇	—	2	2	no	—
189	D₄×C₆₇	—	2	1	no	—
190	D₄×C₆₇	C₄×Q₈	1	2	yes	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	2	no	—
199	C₂×C₄×C₆₇	—	4	1	no	—
200	C₂×C₄×C₆₇	—	4	2	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
203	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
204	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
207	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
212	C₂×Dic₆₇	—	2	4	no	—
213	C₂×Dic₆₇	—	4	2	no	—
214	C₂×Dic₆₇	—	4	2	no	—
215	C₈×C₆₇	—	2	2	no	—
216	C₈×C₆₇	—	2	2	no	—
217	C₆₇⋊C₈	—	2	2	no	—
218	C₆₇⋊C₈	—	2	1	no	—
219	C₆₇⋊C₈	—	2	2	no	—
220	C₆₇⋊C₈	—	2	1	no	—
221	C₆₇⋊C₈	—	4	2	no	—
222	C₆₇⋊C₈	C₂²⋊Q₈	1	2	yes	—
223	C₆₇⋊C₈	C₂²⋊Q₈	1	2	yes	—
224	C₂²×C₆₇	—	2	1	no	—
225	C₂²×C₆₇	—	2	1	no	—
226	C₂²×C₆₇	—	2	2	no	—
227	C₂²×C₆₇	—	4	1	no	—
228	C₂²×C₆₇	—	4	1	no	—
229	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
230	C₂²×C₆₇	#529	1	1	yes	characteristic
231	C₂²×C₆₇	#533	1	1	yes	characteristic
232	C₄×C₆₇	—	2	1	no	—
233	C₄×C₆₇	—	2	1	no	—
234	C₄×C₆₇	—	2	1	no	—
235	C₄×C₆₇	—	2	2	no	—
236	C₄×C₆₇	—	4	1	no	—
237	C₄×C₆₇	—	4	2	no	—
238	C₄×C₆₇	—	4	1	no	—
239	C₄×C₆₇	—	4	2	no	—
240	C₄×C₆₇	#443	1	2	yes	—
241	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
242	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
243	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
244	C₄×C₆₇	#525	1	1	yes	characteristic
245	Dic₆₇	—	4	4	no	—
246	#21627	—	67	1	no	maximal, Sylow
247	C₂×C₆₇	—	2	1	no	—
248	C₂×C₆₇	—	2	1	no	—
249	C₂×C₆₇	—	4	2	no	—
250	C₂×C₆₇	C₄×C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
252	C₂×C₆₇	#2955	1	1	yes	characteristic
253	#3076	D₆₇	1	1	yes	characteristic
254	#2960	—	67	1	no	—
255	C₄×#131	—	67	1	no	—
256	#1986	—	67	1	no	—
257	#1950	—	67	1	no	—
258	#1922	—	67	2	no	—
259	#1866	—	67	2	no	—
260	#1806	—	67	1	no	—
261	#1772	—	67	2	no	—
262	C₈×C₄⋊C₄	—	67	1	no	—
263	#1764	—	67	2	no	—
264	C₆₇	#21627	1	1	yes	characteristic, complemented, Sylow
265	C₂×#131	—	67	1	no	—
266	C₂×#131	—	67	1	no	—
267	#533	C₂×D₆₇	1	1	yes	characteristic
268	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
269	#479	Dic₆₇	1	2	yes	complemented
270	#478	Dic₆₇	1	2	yes	complemented
271	#477	Dic₆₇	1	1	yes	characteristic, complemented
272	#476	Dic₆₇	1	1	yes	characteristic, complemented
273	#475	Dic₆₇	1	1	yes	characteristic, complemented
274	#474	Dic₆₇	1	1	yes	characteristic, complemented
275	#446	—	134	1	no	—
276	#446	—	134	1	no	—
277	C₈×Q₈	—	134	1	no	—
278	C₈×Q₈	—	134	1	no	—
279	C₂×#108	—	67	2	no	—
280	#427	C₂×D₆₇	1	2	yes	—
281	#424	—	67	2	no	—
282	#423	—	67	1	no	—
283	C₂×C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
284	C₂×#104	—	67	2	no	—
285	#416	C₂×D₆₇	1	2	yes	—
286	#405	—	67	1	no	—
287	#405	—	67	1	no	—
288	#405	—	67	2	no	—
289	#405	—	134	1	no	—
290	#405	—	134	1	no	—
291	C₄×M₁₆	—	67	1	no	—
292	C₄×M₁₆	—	134	1	no	—
293	C₂×C₄×C₈	—	67	1	no	—
294	C₂×C₄×C₈	—	67	2	no	—
295	C₂×C₄×C₈	—	134	1	no	—
296	C₄×C₄⋊C₄	—	67	1	no	—
297	#341	—	67	2	no	—
298	#339	—	67	2	no	—
299	#339	—	67	2	no	—
300	#339	—	67	2	no	—
301	#336	—	67	2	no	—
302	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
303	#131	—	67	4	no	—
304	#131	—	134	2	no	—
305	#131	—	134	2	no	—
306	#131	—	134	2	no	—
307	C₂×M₁₆	—	67	1	no	—
308	C₂×M₁₆	—	67	1	no	—
309	C₂×M₁₆	—	67	2	no	—
310	C₂×M₁₆	—	134	1	no	—
311	C₂×M₁₆	—	134	1	no	—
312	C₂²×C₈	—	67	1	no	—
313	C₂²×C₈	—	67	1	no	—
314	C₂²×C₈	—	134	1	no	—
315	C₂²×C₈	—	134	1	no	—
316	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
317	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
318	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
319	#125	C₂×Dic₆₇	1	2	yes	—
320	C₂²⋊Q₈	—	2	1	no	—
321	C₂²⋊Q₈	—	2	1	no	—
322	C₄×Q₈	—	2	1	no	—
323	C₄×Q₈	—	2	1	no	—
324	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
325	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
326	C₄×D₄	—	2	1	no	—
327	C₄×D₄	—	2	1	no	—
328	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
329	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
330	#117	—	2	1	no	—
331	#117	C₂²×D₆₇	1	1	yes	characteristic
332	#117	C₂²×D₆₇	1	2	yes	—
333	C₂×C₄⋊C₄	—	134	1	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	—	67	2	no	—
336	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
337	#108	—	134	4	no	—
338	#107	C₂×Dic₆₇	1	2	yes	—
339	#107	C₂×Dic₆₇	1	2	yes	—
340	#106	C₂×Dic₆₇	1	2	yes	—
341	#106	C₂×Dic₆₇	1	2	yes	—
342	C₄⋊C₈	—	134	2	no	—
343	C₄⋊C₈	—	134	2	no	—
344	C₄⋊C₈	—	134	1	no	—
345	C₄⋊C₈	—	134	1	no	—
346	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
347	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
348	C₄⋊C₈	C₂×Dic₆₇	1	2	yes	—
349	#104	—	134	4	no	—
350	#103	C₂×Dic₆₇	1	2	yes	—
351	#103	C₂×Dic₆₇	1	2	yes	—
352	#102	C₂×Dic₆₇	1	2	yes	—
353	#102	C₂×Dic₆₇	1	2	yes	—
354	#97	—	134	1	no	—
355	#97	—	134	1	no	—
356	#97	—	134	2	no	—
357	#97	—	134	1	no	—
358	C₄×C₈	—	67	1	no	—
359	C₄×C₈	—	67	1	no	—
360	C₄×C₈	—	67	1	no	—
361	C₄×C₈	—	67	1	no	—
362	C₄×C₈	—	134	1	no	—
363	C₄×C₈	—	134	1	no	—
364	C₄×C₈	—	134	2	no	—
365	C₄×C₈	—	134	4	no	—
366	C₄×C₈	—	134	1	no	—
367	#95	—	134	1	no	—
368	#41	—	2	1	no	—
369	#41	—	2	1	no	—
370	#41	Q₈×D₆₇	1	2	yes	—
371	#41	#11761629	1	2	yes	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	134	2	no	—
381	C₂²×C₄	#11761627	1	1	yes	characteristic
382	C₂²×C₄	#11761627	1	1	yes	characteristic
383	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
384	M₁₆	—	67	2	no	—
385	M₁₆	—	67	2	no	—
386	M₁₆	—	134	2	no	—
387	M₁₆	—	134	2	no	—
388	M₁₆	—	134	4	no	—
389	M₁₆	—	134	4	no	—
390	M₁₆	—	134	1	no	—
391	M₁₆	—	134	1	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	67	1	no	—
395	C₂×C₈	—	67	1	no	—
396	C₂×C₈	—	67	2	no	—
397	C₂×C₈	—	67	2	no	—
398	C₂×C₈	—	67	1	no	—
399	C₂×C₈	—	67	1	no	—
400	C₂×C₈	—	134	2	no	—
401	C₂×C₈	—	134	4	no	—
402	C₂×C₈	—	134	4	no	—
403	C₂×C₈	—	134	1	no	—
404	C₂×C₈	—	134	1	no	—
405	C₂×C₈	—	134	1	no	—
406	C₂×C₈	—	134	1	no	—
407	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
408	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	134	4	no	—
416	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
417	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
418	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
419	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
420	C₂²⋊C₄	—	2	1	no	—
421	C₂²⋊C₄	—	2	2	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	134	2	no	—
429	C₄²	—	134	2	no	—
430	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
431	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
432	C₄²	D₄×D₆₇	1	1	yes	characteristic
433	C₄²	D₄×D₆₇	1	1	yes	characteristic
434	C₂³	—	2	1	no	—
435	C₂³	C₂×#11761627	1	1	yes	characteristic
436	Q₈	—	2	1	no	—
437	Q₈	—	4	2	no	—
438	Q₈	—	4	2	no	—
439	Q₈	Q₈×Dic₆₇	1	2	yes	—
440	D₄	—	2	2	no	—
441	D₄	—	2	2	no	—
442	D₄	—	2	1	no	—
443	D₄	Q₈×Dic₆₇	1	2	yes	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	268	2	no	—
456	C₂×C₄	—	268	2	no	—
457	C₂×C₄	—	134	4	no	—
458	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
459	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
460	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
461	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
462	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
463	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
464	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
465	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
466	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
467	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
468	C₈	—	2	2	no	—
469	C₈	—	2	2	no	—
470	C₈	—	268	2	no	—
471	C₈	—	67	2	no	—
472	C₈	—	67	2	no	—
473	C₈	—	134	2	no	—
474	C₈	—	134	1	no	—
475	C₈	—	134	1	no	—
476	C₈	—	134	2	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	4	1	no	—
481	C₂²	—	4	1	no	—
482	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
483	C₂²	#16863932	1	1	yes	characteristic
484	C₂²	#16863948	1	1	yes	characteristic
485	C₄	—	2	1	no	—
486	C₄	—	2	1	no	—
487	C₄	—	2	1	no	—
488	C₄	—	2	2	no	—
489	C₄	—	4	1	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	2	no	—
493	C₄	—	268	4	no	—
494	C₄	#16863513	1	2	yes	—
495	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
496	C₄	#41×Dic₆₇	1	1	yes	characteristic
497	C₄	C₂²⋊Q₈×D₆₇	1	1	yes	characteristic
498	C₄	#16863909	1	1	yes	characteristic
499	C₂	—	2	1	no	—
500	C₂	—	2	1	no	—
501	C₂	—	4	2	no	—
502	C₂	C₂²⋊Q₈×Dic₆₇	1	1	yes	characteristic
503	C₂	C₂×#16863513	1	1	yes	characteristic
504	C₂	#22569112	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#3076	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#41×C₆₇	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3076	D₆₇
hypocenter	C₆₇	#21627
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3076

Complements

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

N	H ≅ G/N	type
1. #22569144	C₂	semidirect
3. #22561154	C₂	semidirect
4. #22560985	C₂	semidirect
5. #22560810	C₂	semidirect
8. #22560090	C₂	semidirect
9. #22560074	C₂	semidirect
40. #479×C₆₇	C₄	semidirect
41. #478×C₆₇	C₄	semidirect
42. #477×C₆₇	C₄	semidirect
43. #476×C₆₇	C₄	semidirect
44. #475×C₆₇	C₄	semidirect
45. #474×C₆₇	C₄	semidirect
52. #14524050	Q₈	semidirect
264. C₆₇	#21627	semidirect
269. #479	Dic₆₇	semidirect
270. #478	Dic₆₇	semidirect
271. #477	Dic₆₇	semidirect
272. #476	Dic₆₇	semidirect
273. #475	Dic₆₇	semidirect
274. #474	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22569144	2
C₄×#14524050	2
#22561154	2
#22560985	2
#22560810	2
#22560486	2
#22560229	2
#22560090	2
#22560074	2
#22560049	2
C₆₇×#3076	2
#21627	67