Subgroups — #37576122

Order 24448.

564 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259298	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259298	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132596	C₂	1	2	yes	maximal, complemented
5	#27132585	C₂	1	2	yes	maximal
6	#27132570	C₂	1	1	yes	maximal, characteristic, complemented
7	#27132561	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132558	C₂	1	2	yes	maximal, complemented
9	#27132553	C₂	1	2	yes	maximal
10	C₄×(C₂×D₁₉₁)⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄⋊Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₉₁	C₂²	1	1	yes	characteristic
13	#20259298	—	2	4	no	—
14	#20259298	—	2	4	no	—
15	Q₈×Dic₁₉₁	—	2	4	no	—
16	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
20	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₁₉₁⋊C₄	C₂²	1	2	yes	—
22	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	2	yes	—
24	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	2	yes	—
25	C₂×(C₂×D₁₉₁)⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
28	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
29	C₁₉₁⋊#95	C₂²	1	1	yes	characteristic
30	C₁₉₁⋊#95	C₂²	1	2	yes	—
31	C₁₉₁⋊#95	C₂²	1	2	yes	—
32	C₁₉₁⋊#95	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
34	C₂²×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
35	C₄⋊Q₈×C₁₉₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
37	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
38	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
39	C₂³×D₁₉₁	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
43	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
44	C₂²×Dic₁₉₁	C₂³	1	1	yes	characteristic
45	C₂×C₄×D₁₉₁	—	2	2	no	—
46	C₂×C₄×D₁₉₁	—	2	2	no	—
47	C₂×C₄×D₁₉₁	—	2	2	no	—
48	C₄⋊Dic₁₉₁	—	2	4	no	—
49	C₄⋊Dic₁₉₁	—	2	2	no	—
50	C₄⋊Dic₁₉₁	D₄	1	4	yes	—
51	Dic₁₉₁⋊C₄	—	2	4	no	—
52	Dic₁₉₁⋊C₄	—	2	4	no	—
53	Dic₁₉₁⋊C₄	—	2	4	no	—
54	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
55	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
56	(C₂×D₁₉₁)⋊C₄	—	2	2	no	—
57	(C₂×D₁₉₁)⋊C₄	Q₈	1	4	yes	complemented
58	C₄×Dic₁₉₁	—	2	4	no	—
59	C₄×Dic₁₉₁	—	2	2	no	—
60	C₂×Q₈×C₁₉₁	—	2	2	no	—
61	C₂×Q₈×C₁₉₁	—	2	2	no	—
62	C₂×Q₈×C₁₉₁	—	2	2	no	—
63	C₂×Q₈×C₁₉₁	—	2	2	no	—
64	C₂×Q₈×C₁₉₁	—	2	2	no	—
65	C₂×Q₈×C₁₉₁	D₄	1	4	yes	—
66	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
67	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
68	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₉₁	—	2	4	no	—
72	C₄⋊C₄×C₁₉₁	—	2	4	no	—
73	C₄²×C₁₉₁	—	2	2	no	—
74	C₂²×D₁₉₁	—	2	1	no	—
75	C₂²×D₁₉₁	—	2	1	no	—
76	C₂²×D₁₉₁	—	2	1	no	—
77	C₂²×D₁₉₁	—	2	1	no	—
78	C₂²×D₁₉₁	—	2	1	no	—
79	C₂²×D₁₉₁	—	2	1	no	—
80	C₂²×D₁₉₁	C₂×Q₈	1	2	yes	—
81	C₂×Dic₁₉₁	—	2	1	no	—
82	C₂×Dic₁₉₁	—	2	2	no	—
83	C₂×Dic₁₉₁	—	2	2	no	—
84	C₂×Dic₁₉₁	—	2	1	no	—
85	C₂×Dic₁₉₁	—	2	1	no	—
86	C₂×Dic₁₉₁	—	2	2	no	—
87	C₂×Dic₁₉₁	—	2	1	no	—
88	C₂×Dic₁₉₁	—	2	1	no	—
89	C₂×Dic₁₉₁	—	2	1	no	—
90	C₂×Dic₁₉₁	—	2	2	no	—
91	C₂×Dic₁₉₁	—	2	1	no	—
92	C₂×Dic₁₉₁	—	2	1	no	—
93	C₂×Dic₁₉₁	—	2	2	no	—
94	C₂×Dic₁₉₁	C₂×D₄	1	4	yes	—
95	C₂×Dic₁₉₁	C₂×Q₈	1	2	yes	—
96	C₄×D₁₉₁	—	4	4	no	—
97	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
98	Q₈×C₁₉₁	—	2	8	no	—
99	Q₈×C₁₉₁	—	4	4	no	—
100	C₂×C₄×C₁₉₁	—	2	1	no	—
101	C₂×C₄×C₁₉₁	—	2	1	no	—
102	C₂×C₄×C₁₉₁	—	2	2	no	—
103	C₂×C₄×C₁₉₁	—	2	2	no	—
104	C₂×C₄×C₁₉₁	—	2	1	no	—
105	C₂×C₄×C₁₉₁	—	2	1	no	—
106	C₂×C₄×C₁₉₁	—	2	2	no	—
107	C₂×C₄×C₁₉₁	—	2	2	no	—
108	C₂×C₄×C₁₉₁	—	2	2	no	—
109	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₉₁	C₂×D₄	1	4	yes	—
111	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
113	C₂×C₄×C₁₉₁	#41	1	2	yes	—
114	C₂×C₄×C₁₉₁	#41	1	2	yes	—
115	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
116	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
117	C₂²×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
118	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
119	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
120	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
121	C₂²×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×D₁₉₁	—	2	2	no	—
123	C₂×D₁₉₁	—	2	2	no	—
124	C₂×D₁₉₁	—	2	2	no	—
125	C₂×D₁₉₁	—	4	1	no	—
126	C₂×D₁₉₁	—	4	1	no	—
127	C₂×D₁₉₁	—	4	1	no	—
128	C₂×D₁₉₁	—	4	1	no	—
129	C₄×C₁₉₁	—	2	4	no	—
130	C₄×C₁₉₁	—	2	2	no	—
131	C₄×C₁₉₁	—	2	2	no	—
132	C₄×C₁₉₁	—	4	2	no	—
133	C₄×C₁₉₁	#121	1	4	yes	—
134	Dic₁₉₁	—	2	4	no	—
135	Dic₁₉₁	—	2	2	no	—
136	Dic₁₉₁	—	4	1	no	—
137	Dic₁₉₁	—	4	1	no	—
138	Dic₁₉₁	—	4	2	no	—
139	C₂×C₁₉₁	C₂×#121	1	1	yes	characteristic
140	C₂×C₁₉₁	#533	1	1	yes	characteristic
141	C₂×C₁₉₁	#541	1	1	yes	characteristic
142	C₂×C₁₉₁	#547	1	1	yes	characteristic
143	C₂×C₁₉₁	#548	1	1	yes	characteristic
144	C₂×C₁₉₁	D₄×Q₈	1	1	yes	characteristic
145	C₂×C₁₉₁	D₄×Q₈	1	1	yes	characteristic
146	D₁₉₁	—	4	2	no	—
147	C₁₉₁	#2429	1	1	yes	characteristic, complemented, Sylow
148	#2429	—	191	1	no	maximal, Sylow
149	C₂×C₄⋊Q₈	D₁₉₁	1	1	yes	characteristic, complemented
150	C₂×C₂²⋊Q₈	—	191	1	no	—
151	C₂×C₂²⋊Q₈	—	191	1	no	—
152	C₂×C₄×Q₈	—	191	1	no	—
153	#398	—	191	2	no	—
154	#396	—	191	2	no	—
155	#391	—	191	1	no	—
156	#389	—	191	2	no	—
157	#389	—	191	1	no	—
158	#388	—	191	2	no	—
159	C₄×C₂²⋊C₄	—	191	1	no	—
160	C₂²×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
161	C₂²×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
162	C₂³×C₄	—	191	1	no	—
163	C₄⋊Q₈	—	2	4	no	—
164	C₂²⋊Q₈	—	382	4	no	—
165	C₂²⋊Q₈	—	382	4	no	—
166	C₄×Q₈	—	382	4	no	—
167	C₂×C₄⋊C₄	—	191	2	no	—
168	C₂×C₄⋊C₄	—	191	1	no	—
169	C₂×C₄⋊C₄	—	191	2	no	—
170	C₂×C₄⋊C₄	—	191	2	no	—
171	C₂×C₄⋊C₄	—	191	2	no	—
172	C₂×C₄⋊C₄	—	191	1	no	—
173	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
174	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
175	C₂×C₂²⋊C₄	—	191	2	no	—
176	C₂×C₂²⋊C₄	—	191	2	no	—
177	C₂×C₂²⋊C₄	—	191	1	no	—
178	C₂×C₂²⋊C₄	—	191	1	no	—
179	C₂×C₄²	—	191	1	no	—
180	C₂×C₄²	—	191	2	no	—
181	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
182	#95	—	191	1	no	—
183	#95	—	191	1	no	—
184	#95	—	191	2	no	—
185	#95	—	191	2	no	—
186	#95	—	191	2	no	—
187	C₂⁴	—	191	1	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	C₂²⋊D₁₉₁	1	4	yes	—
194	C₂²×C₄	—	382	2	no	—
195	C₂²×C₄	—	382	2	no	—
196	C₂²×C₄	—	382	2	no	—
197	C₂²×C₄	—	191	1	no	—
198	C₂²×C₄	—	191	2	no	—
199	C₂²×C₄	—	191	1	no	—
200	C₂²×C₄	—	191	1	no	—
201	C₂²×C₄	—	191	2	no	—
202	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
203	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
204	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
205	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	382	4	no	—
210	C₄⋊C₄	—	382	4	no	—
211	C₄⋊C₄	—	382	4	no	—
212	C₄⋊C₄	—	382	2	no	—
213	C₄⋊C₄	—	382	4	no	—
214	C₄⋊C₄	—	191	4	no	—
215	C₂²⋊C₄	—	382	4	no	—
216	C₂²⋊C₄	—	382	4	no	—
217	C₂²⋊C₄	—	382	2	no	—
218	C₂²⋊C₄	—	191	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	382	4	no	—
221	C₄²	—	382	2	no	—
222	C₂³	—	382	1	no	—
223	C₂³	—	382	1	no	—
224	C₂³	—	382	1	no	—
225	C₂³	—	382	1	no	—
226	C₂³	—	382	1	no	—
227	C₂³	—	382	1	no	—
228	C₂³	—	191	2	no	—
229	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
230	Q₈	—	2	8	no	—
231	Q₈	—	4	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	382	1	no	—
242	C₂×C₄	—	382	2	no	—
243	C₂×C₄	—	382	2	no	—
244	C₂×C₄	—	382	1	no	—
245	C₂×C₄	—	382	1	no	—
246	C₂×C₄	—	382	1	no	—
247	C₂×C₄	—	382	2	no	—
248	C₂×C₄	—	382	1	no	—
249	C₂×C₄	—	382	1	no	—
250	C₂×C₄	—	382	1	no	—
251	C₂×C₄	—	382	2	no	—
252	C₂×C₄	—	382	1	no	—
253	C₂×C₄	—	382	2	no	—
254	C₂×C₄	—	191	4	no	—
255	C₂×C₄	—	191	2	no	—
256	C₂×C₄	—	764	4	no	—
257	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
258	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	4	yes	—
259	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
260	C₂×C₄	Q₈×D₁₉₁	1	2	yes	—
261	C₂×C₄	#15690286	1	2	yes	—
262	C₂×C₄	#15690288	1	2	yes	—
263	C₂²	—	382	2	no	—
264	C₂²	—	382	2	no	—
265	C₂²	—	382	2	no	—
266	C₂²	—	764	1	no	—
267	C₂²	—	764	1	no	—
268	C₂²	—	764	1	no	—
269	C₂²	—	764	1	no	—
270	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
271	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
272	C₂²	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
273	C₂²	C₂×#15690286	1	1	yes	characteristic
274	C₂²	C₂×#15690288	1	1	yes	characteristic
275	C₂²	#20259370	1	1	yes	characteristic
276	C₂²	#20259370	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	4	2	no	—
281	C₄	—	382	4	no	—
282	C₄	—	382	2	no	—
283	C₄	—	764	1	no	—
284	C₄	—	764	2	no	—
285	C₄	—	764	1	no	—
286	C₄	#20259294	1	4	yes	—
287	C₂	—	764	2	no	—
288	C₂	C₂×#20259294	1	1	yes	characteristic
289	C₂	#27133218	1	1	yes	characteristic
290	C₂	#27133274	1	1	yes	characteristic
291	C₂	#27133316	1	1	yes	characteristic
292	C₂	#27133324	1	1	yes	characteristic
293	C₂	Q₈×C₂²⋊D₁₉₁	1	1	yes	characteristic
294	C₂	#27133328	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator, socle, cosocle	C₂³×C₁₉₁	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₁₉₁	C₂
prime radical	C₂³×D₁₉₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₉₁
hypocenter	C₁₉₁	#2429
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₂×Q₈×Dic₁₉₁	C₂	semidirect
4. #27132596	C₂	semidirect
6. #27132570	C₂	semidirect
7. #27132561	C₂	semidirect
8. #27132558	C₂	semidirect
11. C₂×C₄⋊Q₈×C₁₉₁	C₂	semidirect
57. (C₂×D₁₉₁)⋊C₄	Q₈	semidirect
147. C₁₉₁	#2429	semidirect
149. C₂×C₄⋊Q₈	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259298	2
C₂×Q₈×Dic₁₉₁	2
#27132596	2
#27132585	2
#27132570	2
#27132561	2
#27132558	2
#27132553	2
C₄×(C₂×D₁₉₁)⋊C₄	2
C₂×C₄⋊Q₈×C₁₉₁	2
#2429	191