Subgroups — #37582669

Order 24448.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133210	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#15690288	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132990	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132986	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132946	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132942	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132861	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132857	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132811	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132807	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132776	C₂	1	1	yes	maximal, characteristic, complemented
12	#27132773	C₂	1	1	yes	maximal, characteristic, complemented
13	#27132761	C₂	1	1	yes	maximal, characteristic, complemented
14	#27132718	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15690288	C₂²	1	1	yes	characteristic
17	C₂×#15690287	—	2	1	no	—
18	C₂×#15690274	C₂²	1	1	yes	characteristic
19	C₂×D₁₅₂₈	C₂²	1	1	yes	characteristic
20	C₂×#15690272	C₂²	1	1	yes	characteristic
21	#20259321	C₂²	1	1	yes	characteristic
22	#20259318	C₂²	1	1	yes	characteristic
23	#20259308	—	2	1	no	—
24	#20259297	C₂²	1	1	yes	characteristic
25	#20259295	—	2	1	no	—
26	#20259293	C₂²	1	1	yes	characteristic
27	Q₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
28	C₄×D₇₆₄	—	2	1	no	—
29	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
30	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
31	#20259284	—	2	1	no	—
32	#20259284	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₉₁	—	2	1	no	—
34	#20259280	—	2	1	no	—
35	#20259280	C₂²	1	1	yes	characteristic
36	C₄²×D₁₉₁	—	2	1	no	—
37	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
38	#20259255	C₂²	1	1	yes	characteristic
39	#20259254	C₂²	1	1	yes	characteristic
40	#20259253	C₂²	1	1	yes	characteristic
41	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
43	#20259246	C₂²	1	1	yes	characteristic
44	#20259244	C₂²	1	1	yes	characteristic
45	#20259244	C₂²	1	1	yes	characteristic
46	#20259243	C₂²	1	1	yes	characteristic
47	#20259242	C₂²	1	1	yes	characteristic
48	#20259242	C₂²	1	1	yes	characteristic
49	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
51	#20259233	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₉₁	C₂²	1	1	yes	characteristic
55	#107×C₁₉₁	C₂²	1	1	yes	characteristic
56	#103×C₁₉₁	C₂²	1	1	yes	characteristic
57	#102×C₁₉₁	C₂²	1	1	yes	characteristic
58	#97×C₁₉₁	C₂²	1	1	yes	characteristic
59	#15690288	—	2	2	no	—
60	#15690288	—	2	2	no	—
61	#15690287	—	4	4	no	—
62	C₂×Dic₃₈₂	—	2	1	no	—
63	C₂×D₇₆₄	—	2	1	no	—
64	C₂×D₇₆₄	—	2	1	no	—
65	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
66	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
68	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
69	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
70	C₂×C₄×D₁₉₁	—	2	1	no	—
71	C₂×C₄×D₁₉₁	—	2	1	no	—
72	C₂×C₄×D₁₉₁	—	2	1	no	—
73	C₂×C₄×D₁₉₁	—	2	1	no	—
74	C₂×C₄×D₁₉₁	D₄	1	1	yes	characteristic
75	C₂×C₄×D₁₉₁	D₄	1	1	yes	characteristic
76	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
77	#15690274	—	2	2	no	—
78	D₁₅₂₈	—	2	2	no	—
79	#15690272	—	2	2	no	—
80	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₁₉₁	—	2	1	no	—
83	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₁₉₁	C₂³	1	1	yes	characteristic
85	Dic₁₉₁⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₁₉₁	—	2	1	no	—
87	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
90	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
91	(C₂×D₁₉₁)⋊C₄	—	2	1	no	—
92	C₄×Dic₁₉₁	—	2	1	no	—
93	C₄×Dic₁₉₁	—	2	1	no	—
94	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
95	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
96	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₁₉₁	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₉₁	—	2	1	no	—
100	SD₁₆×C₁₉₁	—	2	2	no	—
101	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₉₁	—	2	1	no	—
104	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
107	C₄²×C₁₉₁	—	2	1	no	—
108	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
109	C₂²×D₁₉₁	—	2	1	no	—
110	C₂²×D₁₉₁	—	2	1	no	—
111	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
112	Dic₃₈₂	—	4	2	no	—
113	D₇₆₄	—	2	4	no	—
114	D₇₆₄	—	2	1	no	—
115	D₇₆₄	—	2	1	no	—
116	D₇₆₄	—	2	1	no	—
117	D₇₆₄	—	4	2	no	—
118	D₇₆₄	#41	1	2	yes	complemented
119	C₂²⋊D₁₉₁	—	4	2	no	—
120	C₂²⋊D₁₉₁	—	4	2	no	—
121	C₂²⋊D₁₉₁	—	4	2	no	—
122	C₂×Dic₁₉₁	—	2	1	no	—
123	C₂×Dic₁₉₁	—	2	1	no	—
124	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
127	C₄×D₁₉₁	—	2	4	no	—
128	C₄×D₁₉₁	—	2	1	no	—
129	C₄×D₁₉₁	—	2	1	no	—
130	C₄×D₁₉₁	—	2	2	no	—
131	C₄×D₁₉₁	—	4	1	no	—
132	C₄×D₁₉₁	—	4	2	no	—
133	C₄×D₁₉₁	—	4	2	no	—
134	C₄×D₁₉₁	—	4	2	no	—
135	C₁₉₁⋊C₈	—	2	1	no	—
136	C₁₉₁⋊C₈	—	2	1	no	—
137	C₂³×C₁₉₁	—	2	1	no	—
138	Q₈×C₁₉₁	—	2	1	no	—
139	Q₈×C₁₉₁	#41	1	2	yes	complemented
140	D₄×C₁₉₁	—	2	1	no	—
141	D₄×C₁₉₁	—	2	1	no	—
142	C₂×C₄×C₁₉₁	—	2	1	no	—
143	C₂×C₄×C₁₉₁	—	2	1	no	—
144	C₂×C₄×C₁₉₁	—	2	1	no	—
145	C₂×C₄×C₁₉₁	—	4	2	no	—
146	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
149	C₈×C₁₉₁	—	2	1	no	—
150	C₈×C₁₉₁	—	2	1	no	—
151	C₂²×C₁₉₁	—	4	1	no	—
152	C₂²×C₁₉₁	—	4	1	no	—
153	C₂²×C₁₉₁	—	4	1	no	—
154	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₉₁	—	2	1	no	—
156	C₂×D₁₉₁	—	2	2	no	—
157	C₂×D₁₉₁	—	2	1	no	—
158	C₂×D₁₉₁	—	2	1	no	—
159	C₂×D₁₉₁	—	4	1	no	—
160	C₂×D₁₉₁	—	4	1	no	—
161	C₂×D₁₉₁	—	4	1	no	—
162	C₂×D₁₉₁	—	4	1	no	—
163	C₂×D₁₉₁	—	4	1	no	—
164	C₄×C₁₉₁	—	2	2	no	—
165	C₄×C₁₉₁	—	2	1	no	—
166	C₄×C₁₉₁	—	2	1	no	—
167	C₄×C₁₉₁	—	4	1	no	—
168	C₄×C₁₉₁	—	4	1	no	—
169	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
170	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
171	Dic₁₉₁	—	2	1	no	—
172	Dic₁₉₁	—	2	1	no	—
173	Dic₁₉₁	—	4	1	no	—
174	Dic₁₉₁	—	4	1	no	—
175	Dic₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
176	C₂×C₁₉₁	—	8	1	no	—
177	C₂×C₁₉₁	#525	1	1	yes	characteristic
178	C₂×C₁₉₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₂×C₁₉₁	#575	1	1	yes	characteristic
180	D₁₉₁	—	4	1	no	—
181	D₁₉₁	—	4	2	no	—
182	D₁₉₁	—	8	1	no	—
183	C₁₉₁	#3115	1	1	yes	characteristic, complemented, Sylow
184	#3115	—	191	1	no	maximal, Sylow
185	#532	—	191	1	no	—
186	C₄×#41	—	191	1	no	—
187	#482	—	191	1	no	—
188	#482	—	191	1	no	—
189	#475	—	191	1	no	—
190	#474	—	191	1	no	—
191	#460	—	191	1	no	—
192	#459	—	191	1	no	—
193	#449	—	191	1	no	—
194	#449	—	191	1	no	—
195	#442	—	191	1	no	—
196	#442	—	191	1	no	—
197	#440	—	191	1	no	—
198	#440	D₁₉₁	1	1	yes	characteristic, complemented
199	#432	—	191	1	no	—
200	C₂×#41	—	382	1	no	—
201	C₂×#41	—	191	1	no	—
202	C₂×SD₁₆	—	191	1	no	—
203	C₂×SD₁₆	C₂×D₁₉₁	1	1	yes	characteristic
204	C₂×D₈	—	191	1	no	—
205	C₂×D₈	—	191	1	no	—
206	C₄⋊Q₈	—	191	1	no	—
207	#127	—	191	1	no	—
208	#124	—	382	1	no	—
209	#121	—	382	1	no	—
210	#121	—	191	1	no	—
211	#121	—	191	1	no	—
212	C₄×Q₈	—	191	1	no	—
213	C₄×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
214	C₄×D₄	—	382	1	no	—
215	C₄×D₄	—	382	1	no	—
216	C₄×D₄	—	382	1	no	—
217	C₄×D₄	—	191	1	no	—
218	C₄×D₄	—	191	1	no	—
219	C₄×D₄	—	191	1	no	—
220	C₄×D₄	C₂×D₁₉₁	1	1	yes	characteristic
221	#117	—	382	1	no	—
222	#117	—	191	1	no	—
223	C₂×C₄²	—	382	1	no	—
224	C₂×C₄²	—	191	1	no	—
225	#107	—	191	1	no	—
226	#107	C₂×D₁₉₁	1	1	yes	characteristic
227	#106	—	191	1	no	—
228	#106	—	191	1	no	—
229	C₄⋊C₈	—	191	1	no	—
230	C₄⋊C₈	—	191	1	no	—
231	#103	—	191	1	no	—
232	#103	C₂×D₁₉₁	1	1	yes	characteristic
233	#102	—	191	1	no	—
234	#102	—	191	1	no	—
235	#102	—	191	1	no	—
236	#102	—	191	1	no	—
237	#102	—	191	1	no	—
238	#102	C₂×D₁₉₁	1	1	yes	characteristic
239	C₂²⋊C₈	—	191	1	no	—
240	C₂²⋊C₈	—	191	1	no	—
241	#97	—	191	1	no	—
242	#97	C₂×D₁₉₁	1	1	yes	characteristic
243	#41	—	382	2	no	—
244	#41	—	382	2	no	—
245	#41	—	764	4	no	—
246	C₂×Q₈	—	382	1	no	—
247	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
248	C₂×D₄	—	382	1	no	—
249	C₂×D₄	—	382	1	no	—
250	C₂×D₄	—	382	1	no	—
251	C₂×D₄	—	382	1	no	—
252	C₂×D₄	—	382	1	no	—
253	C₂×D₄	—	191	1	no	—
254	C₂×D₄	—	191	1	no	—
255	C₂×D₄	C₂²×D₁₉₁	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	382	1	no	—
258	C₂²×C₄	—	382	1	no	—
259	C₂²×C₄	—	382	1	no	—
260	C₂²×C₄	—	382	1	no	—
261	C₂²×C₄	—	191	1	no	—
262	C₂²×C₄	—	191	1	no	—
263	C₂²×C₄	—	191	1	no	—
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	382	2	no	—
266	D₈	—	382	2	no	—
267	D₈	—	382	2	no	—
268	C₂×C₈	—	191	1	no	—
269	C₂×C₈	—	191	1	no	—
270	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	382	1	no	—
274	C₄⋊C₄	—	382	1	no	—
275	C₄⋊C₄	—	191	1	no	—
276	C₄⋊C₄	—	191	1	no	—
277	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	382	1	no	—
281	C₂²⋊C₄	—	382	1	no	—
282	C₂²⋊C₄	—	382	1	no	—
283	C₂²⋊C₄	—	382	1	no	—
284	C₂²⋊C₄	—	382	1	no	—
285	C₂²⋊C₄	—	382	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	382	1	no	—
288	C₄²	—	382	1	no	—
289	C₄²	—	191	1	no	—
290	C₄²	—	191	1	no	—
291	C₄²	—	191	1	no	—
292	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
293	C₂³	—	2	1	no	—
294	C₂³	—	382	1	no	—
295	C₂³	—	382	1	no	—
296	C₂³	—	191	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	764	2	no	—
299	Q₈	#15690287	1	2	yes	complemented
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	382	4	no	—
303	D₄	—	382	1	no	—
304	D₄	—	382	1	no	—
305	D₄	—	382	1	no	—
306	D₄	—	191	2	no	—
307	D₄	—	764	2	no	—
308	D₄	—	764	2	no	—
309	D₄	—	764	2	no	—
310	D₄	—	764	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	382	4	no	—
316	C₂×C₄	—	382	2	no	—
317	C₂×C₄	—	382	1	no	—
318	C₂×C₄	—	382	1	no	—
319	C₂×C₄	—	382	1	no	—
320	C₂×C₄	—	382	1	no	—
321	C₂×C₄	—	191	1	no	—
322	C₂×C₄	—	191	1	no	—
323	C₂×C₄	—	191	1	no	—
324	C₂×C₄	—	764	1	no	—
325	C₂×C₄	—	764	2	no	—
326	C₂×C₄	—	764	2	no	—
327	C₂×C₄	—	764	2	no	—
328	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
329	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	382	1	no	—
334	C₈	—	382	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	382	1	no	—
339	C₂²	—	382	1	no	—
340	C₂²	—	382	1	no	—
341	C₂²	—	382	2	no	—
342	C₂²	—	764	1	no	—
343	C₂²	—	764	1	no	—
344	C₂²	—	764	1	no	—
345	C₂²	—	764	1	no	—
346	C₂²	—	764	1	no	—
347	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	2	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	382	1	no	—
354	C₄	—	382	1	no	—
355	C₄	—	191	2	no	—
356	C₄	—	764	1	no	—
357	C₄	—	764	1	no	—
358	C₄	#41×D₁₉₁	1	1	yes	characteristic
359	C₄	C₂×#15690287	1	1	yes	characteristic
360	C₂	—	8	1	no	—
361	C₂	—	1528	1	no	—
362	C₂	—	764	1	no	—
363	C₂	—	764	2	no	—
364	C₂	#27133172	1	1	yes	characteristic
365	C₂	AGL(1,ℤ/8)×D₁₉₁	1	1	yes	characteristic
366	C₂	#27133462	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#440×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
hypercenter	#440	D₁₉₁
hypocenter	C₁₉₁	#3115
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #27133210	C₂	semidirect
2. C₄×#15690288	C₂	semidirect
3. #27132990	C₂	semidirect
4. #27132986	C₂	semidirect
5. #27132946	C₂	semidirect
6. #27132942	C₂	semidirect
7. #27132861	C₂	semidirect
8. #27132857	C₂	semidirect
9. #27132811	C₂	semidirect
10. #27132807	C₂	semidirect
11. #27132776	C₂	semidirect
12. #27132773	C₂	semidirect
13. #27132761	C₂	semidirect
14. #27132718	C₂	semidirect
15. #440×C₁₉₁	C₂	semidirect
118. D₇₆₄	#41	semidirect
139. Q₈×C₁₉₁	#41	semidirect
183. C₁₉₁	#3115	semidirect
198. #440	D₁₉₁	semidirect
299. Q₈	#15690287	semidirect

Maximal subgroups

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

subgroup	index
#27133210	2
C₄×#15690288	2
#27132990	2
#27132986	2
#27132946	2
#27132942	2
#27132861	2
#27132857	2
#27132811	2
#27132807	2
#27132776	2
#27132773	2
#27132761	2
#27132718	2
#440×C₁₉₁	2
#3115	191