Subgroups — #37581513

Order 24448.

372 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133200	C₂	1	1	yes	maximal, characteristic
2	#27132794	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132736	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132736	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132733	C₂	1	1	yes	maximal, characteristic, complemented
6	#27132733	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₇₆₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132728	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132723	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132723	C₂	1	1	yes	maximal, characteristic, complemented
11	#27132715	C₂	1	1	yes	maximal, characteristic, complemented
12	#27132620	C₂	1	1	yes	maximal, characteristic, complemented
13	#27132617	C₂	1	1	yes	maximal, characteristic, complemented
14	#97×D₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#446×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15690270	C₂²	1	1	yes	characteristic
17	C₂×#15690270	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
22	#20259317	C₄	1	1	yes	characteristic
23	#20259317	C₄	1	1	yes	characteristic
24	#20259313	C₄	1	1	yes	characteristic
25	#20259311	C₄	1	1	yes	characteristic
26	#20259307	C₄	1	1	yes	characteristic
27	#20259307	C₄	1	1	yes	characteristic
28	#20259298	C₄	1	1	yes	characteristic
29	#20259297	C₄	1	1	yes	characteristic
30	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
31	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
32	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
33	#20259279	C₂²	1	1	yes	characteristic
34	#20259279	C₂²	1	1	yes	characteristic
35	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
36	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
37	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
39	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
40	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
41	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
42	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
43	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
44	(C₂×D₁₉₁)⋊C₈	C₂²	1	1	yes	characteristic
45	#20259233	C₂²	1	1	yes	characteristic
46	#20259233	C₂²	1	1	yes	characteristic
47	#20259232	C₂²	1	1	yes	characteristic
48	#20259232	C₂²	1	1	yes	characteristic
49	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
56	#97×C₁₉₁	C₂²	1	1	yes	characteristic
57	#97×C₁₉₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
59	C₂×D₇₆₄	C₂×C₄	1	1	yes	characteristic
60	C₂×D₇₆₄	C₂×C₄	1	1	yes	characteristic
61	C₂×D₇₆₄	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₉₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₉₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
67	#15690270	—	2	2	no	—
68	#15690270	—	2	2	no	—
69	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₈×D₁₉₁	—	2	2	no	—
74	C₈×D₁₉₁	—	2	2	no	—
75	C₈×D₁₉₁	—	2	2	no	—
76	C₈×D₁₉₁	—	2	2	no	—
77	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
78	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
79	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
80	Dic₁₉₁⋊C₄	C₂×C₄	1	1	yes	characteristic
81	Dic₁₉₁⋊C₄	C₂×C₄	1	1	yes	characteristic
82	Dic₁₉₁⋊C₄	C₂×C₄	1	1	yes	characteristic
83	Dic₁₉₁⋊C₄	C₂×C₄	1	1	yes	characteristic
84	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
85	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
88	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
89	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
90	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
91	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
92	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
93	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
104	C₄⋊C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
105	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
106	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
107	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₉₁	C₂²×C₄	1	1	yes	characteristic
109	C₂²×D₁₉₁	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₁₉₁	C₂²×C₄	1	1	yes	characteristic
111	D₇₆₄	—	2	2	no	—
112	D₇₆₄	—	2	2	no	—
113	D₇₆₄	—	4	1	no	—
114	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₁₉₁	C₂²×C₄	1	1	yes	characteristic
119	C₄×D₁₉₁	—	2	2	no	—
120	C₄×D₁₉₁	—	2	1	no	—
121	C₄×D₁₉₁	—	2	1	no	—
122	C₄×D₁₉₁	—	2	1	no	—
123	C₄×D₁₉₁	—	2	1	no	—
124	C₄×D₁₉₁	—	2	1	no	—
125	C₄×D₁₉₁	—	2	1	no	—
126	C₄×D₁₉₁	—	4	1	no	—
127	C₁₉₁⋊C₈	—	2	1	no	—
128	C₁₉₁⋊C₈	—	2	1	no	—
129	C₁₉₁⋊C₈	—	2	1	no	—
130	C₁₉₁⋊C₈	#41	1	2	yes	—
131	Q₈×C₁₉₁	—	4	1	no	—
132	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₉₁	—	2	1	no	—
140	C₈×C₁₉₁	—	2	1	no	—
141	C₈×C₁₉₁	—	2	1	no	—
142	C₈×C₁₉₁	#41	1	2	yes	—
143	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
144	C₂×D₁₉₁	—	2	1	no	—
145	C₂×D₁₉₁	—	2	1	no	—
146	C₂×D₁₉₁	—	2	1	no	—
147	C₂×D₁₉₁	—	2	1	no	—
148	C₂×D₁₉₁	—	2	1	no	—
149	C₂×D₁₉₁	—	2	1	no	—
150	C₂×D₁₉₁	—	2	1	no	—
151	C₂×D₁₉₁	—	2	1	no	—
152	C₂×D₁₉₁	#131	1	2	yes	—
153	C₄×C₁₉₁	—	2	1	no	—
154	C₄×C₁₉₁	—	2	1	no	—
155	C₄×C₁₉₁	—	2	1	no	—
156	C₄×C₁₉₁	—	2	1	no	—
157	C₄×C₁₉₁	—	2	1	no	—
158	C₄×C₁₉₁	—	2	1	no	—
159	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
161	Dic₁₉₁	—	2	1	no	—
162	Dic₁₉₁	—	2	1	no	—
163	Dic₁₉₁	—	2	1	no	—
164	Dic₁₉₁	—	2	1	no	—
165	Dic₁₉₁	—	2	1	no	—
166	C₂×C₁₉₁	C₄×#41	1	1	yes	characteristic
167	C₂×C₁₉₁	C₂×#131	1	1	yes	characteristic
168	C₂×C₁₉₁	#568	1	1	yes	characteristic
169	D₁₉₁	—	2	2	no	—
170	D₁₉₁	—	4	1	no	—
171	D₁₉₁	—	4	1	no	—
172	C₁₉₁	#2968	1	1	yes	characteristic, complemented, Sylow
173	#2968	—	191	1	no	maximal, Sylow
174	#529	—	191	1	no	—
175	#446	—	191	1	no	—
176	#446	D₁₉₁	1	1	yes	characteristic, complemented
177	#435	—	191	1	no	—
178	#435	—	191	1	no	—
179	#435	—	191	1	no	—
180	#435	—	191	1	no	—
181	C₈×D₄	—	191	1	no	—
182	C₈×D₄	—	191	1	no	—
183	#433	—	191	1	no	—
184	#433	—	191	1	no	—
185	#431	—	191	1	no	—
186	#405	—	191	1	no	—
187	#405	—	191	1	no	—
188	C₂×#97	—	191	1	no	—
189	C₂×M₁₆	—	191	1	no	—
190	C₂×M₁₆	—	191	1	no	—
191	C₂²×C₈	—	191	1	no	—
192	C₂²×C₈	—	191	1	no	—
193	C₂²×C₈	—	191	1	no	—
194	C₂²×C₈	—	191	1	no	—
195	#126	—	191	1	no	—
196	#126	—	191	1	no	—
197	#125	—	191	1	no	—
198	#124	—	191	1	no	—
199	#123	—	191	1	no	—
200	#123	—	191	1	no	—
201	C₂²⋊Q₈	—	191	1	no	—
202	#121	—	191	1	no	—
203	C₄×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
204	C₄×D₄	—	191	1	no	—
205	C₄×D₄	—	191	1	no	—
206	C₄×D₄	—	191	1	no	—
207	#117	—	191	1	no	—
208	#117	—	191	1	no	—
209	C₂×C₄²	—	191	1	no	—
210	C₄⋊C₈	—	191	1	no	—
211	C₄⋊C₈	—	191	1	no	—
212	C₄⋊C₈	—	191	1	no	—
213	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
214	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
216	C₂²⋊C₈	—	191	1	no	—
217	C₂²⋊C₈	—	191	1	no	—
218	C₂²⋊C₈	—	191	1	no	—
219	C₂²⋊C₈	—	191	1	no	—
220	C₂²⋊C₈	—	191	1	no	—
221	C₂²⋊C₈	—	191	1	no	—
222	#97	—	191	1	no	—
223	#97	—	191	1	no	—
224	#97	—	191	1	no	—
225	#97	—	191	1	no	—
226	#97	C₂×D₁₉₁	1	1	yes	characteristic
227	#97	C₂×D₁₉₁	1	1	yes	characteristic
228	C₄×C₈	—	191	1	no	—
229	C₄×C₈	—	191	1	no	—
230	C₄×C₈	—	191	1	no	—
231	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
232	C₂×Q₈	C₄×D₁₉₁	1	1	yes	characteristic
233	C₂×D₄	—	191	1	no	—
234	C₂×D₄	—	191	1	no	—
235	C₂×D₄	—	191	1	no	—
236	C₂²×C₄	—	191	1	no	—
237	C₂²×C₄	—	191	1	no	—
238	C₂²×C₄	—	191	1	no	—
239	C₂²×C₄	—	191	1	no	—
240	C₂²×C₄	—	191	1	no	—
241	M₁₆	—	382	2	no	—
242	M₁₆	—	382	2	no	—
243	C₂×C₈	—	382	2	no	—
244	C₂×C₈	—	382	2	no	—
245	C₂×C₈	—	382	2	no	—
246	C₂×C₈	—	382	2	no	—
247	C₂×C₈	—	191	1	no	—
248	C₂×C₈	—	191	1	no	—
249	C₂×C₈	—	191	1	no	—
250	C₂×C₈	—	191	1	no	—
251	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
254	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
255	C₄⋊C₄	—	191	1	no	—
256	C₄⋊C₄	—	191	1	no	—
257	C₄⋊C₄	—	191	1	no	—
258	C₄⋊C₄	—	191	1	no	—
259	C₄⋊C₄	—	191	1	no	—
260	C₄⋊C₄	—	191	1	no	—
261	C₄⋊C₄	—	191	1	no	—
262	C₄⋊C₄	C₄×D₁₉₁	1	1	yes	characteristic
263	C₄⋊C₄	C₄×D₁₉₁	1	1	yes	characteristic
264	C₄⋊C₄	C₄×D₁₉₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	191	1	no	—
266	C₂²⋊C₄	—	191	1	no	—
267	C₂²⋊C₄	—	191	1	no	—
268	C₂²⋊C₄	—	191	1	no	—
269	C₂²⋊C₄	—	191	1	no	—
270	C₂²⋊C₄	—	191	1	no	—
271	C₂²⋊C₄	—	191	1	no	—
272	C₂²⋊C₄	—	191	1	no	—
273	C₂²⋊C₄	—	191	1	no	—
274	C₂²⋊C₄	—	191	1	no	—
275	C₄²	—	191	1	no	—
276	C₄²	—	191	1	no	—
277	C₄²	—	191	1	no	—
278	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
279	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
280	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
281	C₂³	—	191	1	no	—
282	C₂³	—	191	1	no	—
283	C₂³	—	191	1	no	—
284	Q₈	—	4	1	no	—
285	D₄	—	382	2	no	—
286	D₄	—	382	2	no	—
287	D₄	—	764	1	no	—
288	C₂×C₄	—	382	2	no	—
289	C₂×C₄	—	382	1	no	—
290	C₂×C₄	—	382	1	no	—
291	C₂×C₄	—	382	1	no	—
292	C₂×C₄	—	382	1	no	—
293	C₂×C₄	—	382	1	no	—
294	C₂×C₄	—	382	1	no	—
295	C₂×C₄	—	191	1	no	—
296	C₂×C₄	—	191	1	no	—
297	C₂×C₄	—	191	1	no	—
298	C₂×C₄	—	191	1	no	—
299	C₂×C₄	—	191	1	no	—
300	C₂×C₄	—	764	1	no	—
301	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
304	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
306	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	382	1	no	—
312	C₈	—	382	1	no	—
313	C₈	—	382	1	no	—
314	C₈	—	191	2	no	—
315	C₈	#15690288	1	2	yes	—
316	C₂²	—	382	1	no	—
317	C₂²	—	382	1	no	—
318	C₂²	—	382	1	no	—
319	C₂²	—	382	1	no	—
320	C₂²	—	382	1	no	—
321	C₂²	—	382	1	no	—
322	C₂²	—	382	1	no	—
323	C₂²	—	382	1	no	—
324	C₂²	—	191	2	no	—
325	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	382	1	no	—
333	C₄	—	382	1	no	—
334	C₄	—	382	1	no	—
335	C₄	—	382	1	no	—
336	C₄	—	382	1	no	—
337	C₄	#41×D₁₉₁	1	1	yes	characteristic
338	C₄	C₂×#15690288	1	1	yes	characteristic
339	C₂	—	382	2	no	—
340	C₂	—	764	1	no	—
341	C₂	—	764	1	no	—
342	C₂	C₄×#15690288	1	1	yes	characteristic
343	C₂	#131×D₁₉₁	1	1	yes	characteristic
344	C₂	#27133429	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Fitting	#446×C₁₉₁	C₂
prime radical	#20259297	C₄
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	#446	D₁₉₁
hypocenter	C₁₉₁	#2968
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #27132794	C₂	semidirect
3. #27132736	C₂	semidirect
4. #27132736	C₂	semidirect
5. #27132733	C₂	semidirect
6. #27132733	C₂	semidirect
7. C₈×D₇₆₄	C₂	semidirect
8. #27132728	C₂	semidirect
9. #27132723	C₂	semidirect
10. #27132723	C₂	semidirect
11. #27132715	C₂	semidirect
12. #27132620	C₂	semidirect
13. #27132617	C₂	semidirect
14. #97×D₁₉₁	C₂	semidirect
15. #446×C₁₉₁	C₂	semidirect
172. C₁₉₁	#2968	semidirect
176. #446	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133200	2
#27132794	2
#27132736	2
#27132733	2
C₈×D₇₆₄	2
#27132728	2
#27132723	2
#27132715	2
#27132620	2
#27132617	2
#97×D₁₉₁	2
#446×C₁₉₁	2
#2968	191