Subgroups — #37581671

Order 24448.

354 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133197	C₂	1	1	yes	maximal, characteristic
2	#27132793	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₃₈₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132742	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132738	C₂	1	2	yes	maximal, complemented
6	#27132737	C₂	1	2	yes	maximal, complemented
7	C₈×D₇₆₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#27132723	C₂	1	2	yes	maximal, complemented
9	#27132688	C₂	1	2	yes	maximal, complemented
10	#27132687	C₂	1	1	yes	maximal, characteristic, complemented
11	#431×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15690271	C₂²	1	2	yes	—
13	C₂×#15690270	C₂²	1	2	yes	—
14	C₂×C₈×D₁₉₁	C₂²	1	2	yes	—
15	#20259316	C₄	1	2	yes	—
16	#20259314	C₄	1	1	yes	characteristic
17	#20259310	C₄	1	1	yes	characteristic
18	#20259305	C₄	1	2	yes	—
19	#20259301	C₄	1	1	yes	characteristic
20	#20259296	C₄	1	1	yes	characteristic
21	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
22	C₄×D₇₆₄	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₉₁	C₂²	1	2	yes	—
24	#20259279	C₂²	1	2	yes	—
25	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
26	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
28	Dic₁₉₁⋊C₈	C₂²	1	2	yes	—
29	Dic₁₉₁⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₁₉₁)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₁₉₁)⋊C₈	C₂²	1	2	yes	—
32	#20259232	C₂²	1	2	yes	—
33	C₈×Dic₁₉₁	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₈₂	C₂×C₄	1	1	yes	characteristic
42	C₂×D₇₆₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₉₁	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₉₁	C₂³	1	2	yes	—
45	#15690271	—	4	2	no	—
46	#15690270	—	4	2	no	—
47	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
48	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
49	C₈×D₁₉₁	—	4	2	no	—
50	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
52	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₉₁⋊C₄	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
57	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₉₁	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
69	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
72	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
73	C₂²×D₁₉₁	C₂²×C₄	1	2	yes	—
74	Dic₃₈₂	—	2	2	no	—
75	D₇₆₄	—	2	2	no	—
76	C₂²⋊D₁₉₁	—	4	2	no	—
77	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
80	C₄×D₁₉₁	—	2	2	no	—
81	C₄×D₁₉₁	—	2	2	no	—
82	C₁₉₁⋊C₈	—	2	2	no	—
83	C₁₉₁⋊C₈	—	2	2	no	—
84	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₉₁	—	2	2	no	—
86	C₂×C₄×C₁₉₁	—	2	1	no	—
87	C₂×C₄×C₁₉₁	—	2	1	no	—
88	C₂×C₄×C₁₉₁	—	4	1	no	—
89	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₉₁	—	2	1	no	—
96	C₈×C₁₉₁	—	2	1	no	—
97	C₈×C₁₉₁	—	2	1	no	—
98	C₈×C₁₉₁	—	2	1	no	—
99	C₂²×C₁₉₁	—	2	1	no	—
100	C₂²×C₁₉₁	—	2	1	no	—
101	C₂²×C₁₉₁	—	2	1	no	—
102	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₉₁	—	2	2	no	—
104	C₂×D₁₉₁	—	2	2	no	—
105	C₂×D₁₉₁	—	2	2	no	—
106	C₄×C₁₉₁	—	2	1	no	—
107	C₄×C₁₉₁	—	2	1	no	—
108	C₄×C₁₉₁	—	2	1	no	—
109	C₄×C₁₉₁	—	2	1	no	—
110	C₄×C₁₉₁	#131	1	2	yes	—
111	C₄×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₁₉₁	—	2	2	no	—
114	Dic₁₉₁	—	2	2	no	—
115	Dic₁₉₁	—	2	2	no	—
116	C₂×C₁₉₁	—	4	1	no	—
117	C₂×C₁₉₁	#520	1	1	yes	characteristic
118	C₂×C₁₉₁	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₉₁	#568	1	1	yes	characteristic
120	D₁₉₁	—	4	2	no	—
121	C₁₉₁	#2993	1	1	yes	characteristic, complemented, Sylow
122	#2993	—	191	1	no	maximal, Sylow
123	#529	—	191	1	no	—
124	#446	—	191	1	no	—
125	C₈×Q₈	—	191	1	no	—
126	#436	—	191	2	no	—
127	#436	—	191	1	no	—
128	#435	—	191	2	no	—
129	C₈×D₄	—	191	1	no	—
130	#433	—	191	2	no	—
131	#431	D₁₉₁	1	1	yes	characteristic, complemented
132	#424	—	191	2	no	—
133	#423	—	191	1	no	—
134	C₂×M₁₆	—	191	2	no	—
135	C₂×M₁₆	—	191	2	no	—
136	C₂²×C₈	—	191	2	no	—
137	#126	—	191	2	no	—
138	#125	—	191	1	no	—
139	#124	—	191	1	no	—
140	#123	—	191	2	no	—
141	C₂²⋊Q₈	—	191	1	no	—
142	#121	—	191	1	no	—
143	C₄×Q₈	—	191	1	no	—
144	C₄×D₄	—	191	2	no	—
145	C₄×D₄	—	191	1	no	—
146	#117	—	191	2	no	—
147	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
148	C₄⋊C₈	—	191	2	no	—
149	C₄⋊C₈	—	191	2	no	—
150	C₄⋊C₈	—	191	2	no	—
151	C₄⋊C₈	—	191	1	no	—
152	C₄⋊C₈	—	191	1	no	—
153	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
155	C₂²⋊C₈	—	191	2	no	—
156	C₂²⋊C₈	—	191	2	no	—
157	C₂²⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
159	#97	—	191	2	no	—
160	C₄×C₈	—	191	2	no	—
161	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
162	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
163	C₂×Q₈	—	191	1	no	—
164	C₂×D₄	—	191	2	no	—
165	C₂×D₄	—	191	1	no	—
166	C₂²×C₄	—	191	2	no	—
167	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
170	M₁₆	—	764	2	no	—
171	M₁₆	—	764	2	no	—
172	C₂×C₈	—	191	2	no	—
173	C₂×C₈	—	191	2	no	—
174	C₂×C₈	—	764	2	no	—
175	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
179	C₄⋊C₄	—	191	2	no	—
180	C₄⋊C₄	—	191	2	no	—
181	C₄⋊C₄	—	191	2	no	—
182	C₄⋊C₄	—	191	2	no	—
183	C₄⋊C₄	—	191	1	no	—
184	C₄⋊C₄	—	191	1	no	—
185	C₂²⋊C₄	—	191	2	no	—
186	C₂²⋊C₄	—	191	2	no	—
187	C₂²⋊C₄	—	191	2	no	—
188	C₂²⋊C₄	—	191	2	no	—
189	C₂²⋊C₄	—	191	2	no	—
190	C₄²	—	191	2	no	—
191	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
192	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
193	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
194	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
195	C₂³	—	191	2	no	—
196	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
197	Q₈	—	382	2	no	—
198	D₄	—	382	2	no	—
199	D₄	—	764	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	382	2	no	—
205	C₂×C₄	—	382	2	no	—
206	C₂×C₄	—	191	2	no	—
207	C₂×C₄	—	191	2	no	—
208	C₂×C₄	—	191	2	no	—
209	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	382	2	no	—
220	C₈	—	382	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	382	2	no	—
225	C₂²	—	382	2	no	—
226	C₂²	—	382	2	no	—
227	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	382	2	no	—
233	C₄	—	382	2	no	—
234	C₄	—	382	2	no	—
235	C₄	#20259328	1	2	yes	—
236	C₄	#20259368	1	1	yes	characteristic
237	C₄	#20259369	1	1	yes	characteristic
238	C₂	—	4	1	no	—
239	C₂	—	764	2	no	—
240	C₂	#27133143	1	1	yes	characteristic
241	C₂	C₂×#20259328	1	1	yes	characteristic
242	C₂	#27133429	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Fitting	#431×C₁₉₁	C₂
prime radical	#20259296	C₄
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	#431	D₁₉₁
hypocenter	C₁₉₁	#2993
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #27132793	C₂	semidirect
3. C₈×Dic₃₈₂	C₂	semidirect
4. #27132742	C₂	semidirect
5. #27132738	C₂	semidirect
6. #27132737	C₂	semidirect
7. C₈×D₇₆₄	C₂	semidirect
8. #27132723	C₂	semidirect
9. #27132688	C₂	semidirect
10. #27132687	C₂	semidirect
11. #431×C₁₉₁	C₂	semidirect
121. C₁₉₁	#2993	semidirect
131. #431	D₁₉₁	semidirect

Maximal subgroups

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

subgroup	index
#27133197	2
#27132793	2
C₈×Dic₃₈₂	2
#27132742	2
#27132738	2
#27132737	2
C₈×D₇₆₄	2
#27132723	2
#27132688	2
#27132687	2
#431×C₁₉₁	2
#2993	191