Subgroups — #37581180

Order 24448.

402 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259281	C₂	1	1	yes	maximal, characteristic
2	#27132721	C₂	1	4	yes	maximal, complemented
3	#27132717	C₂	1	2	yes	maximal, complemented
4	#27132713	C₂	1	2	yes	maximal, complemented
5	#27132691	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊(C₁₉₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#27132621	C₂	1	2	yes	maximal, complemented
8	#407×C₁₉₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
10	#20259281	C₄	1	2	yes	—
11	#20259281	C₄	1	1	yes	characteristic
12	#20259281	C₄	1	1	yes	characteristic
13	#20259281	C₂²	1	1	yes	characteristic
14	#20259281	C₂²	1	2	yes	—
15	#20259281	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₉₁	C₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₉₁	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₁₉₁	C₄	1	2	yes	—
19	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
20	Dic₁₉₁⋊C₈	C₂²	1	4	yes	complemented
21	Dic₁₉₁⋊C₈	C₂²	1	4	yes	complemented
22	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
25	#20259232	C₂²	1	4	yes	complemented
26	C₈×Dic₁₉₁	C₂²	1	4	yes	complemented
27	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
29	C₂²⋊C₈×C₁₉₁	C₂²	1	2	yes	—
30	C₂²⋊C₈×C₁₉₁	C₂²	1	2	yes	—
31	C₂²×Dic₁₉₁	C₂×C₄	1	2	yes	—
32	C₂²×Dic₁₉₁	C₂×C₄	1	2	yes	—
33	C₂×C₁₉₁⋊C₈	—	2	4	no	—
34	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
35	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
36	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
39	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
40	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
42	C₂²⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
43	C₂²⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
44	C₂²⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
46	C₄×Dic₁₉₁	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₉₁	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₉₁	C₂³	1	2	yes	—
49	C₄×Dic₁₉₁	C₂³	1	2	yes	—
50	C₂⁴×C₁₉₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉₁	—	2	4	no	—
52	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
58	M₁₆×C₁₉₁	—	2	4	no	—
59	C₂×C₈×C₁₉₁	C₂³	1	4	yes	—
60	C₂×Dic₁₉₁	—	2	4	no	—
61	C₂×Dic₁₉₁	—	4	2	no	—
62	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
63	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
64	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
65	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
66	C₁₉₁⋊C₈	—	2	2	no	—
67	C₁₉₁⋊C₈	—	2	2	no	—
68	C₂³×C₁₉₁	—	2	2	no	—
69	C₂³×C₁₉₁	—	2	2	no	—
70	C₂³×C₁₉₁	—	4	1	no	—
71	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
72	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
73	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₉₁	—	2	2	no	—
75	C₂×C₄×C₁₉₁	—	2	2	no	—
76	C₂×C₄×C₁₉₁	—	2	2	no	—
77	C₂×C₄×C₁₉₁	—	2	2	no	—
78	C₂×C₄×C₁₉₁	—	2	2	no	—
79	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₉₁	#41	1	4	yes	—
83	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
84	C₈×C₁₉₁	—	2	4	no	—
85	C₂²×C₁₉₁	—	2	1	no	—
86	C₂²×C₁₉₁	—	2	1	no	—
87	C₂²×C₁₉₁	—	2	1	no	—
88	C₂²×C₁₉₁	—	2	1	no	—
89	C₂²×C₁₉₁	—	2	1	no	—
90	C₂²×C₁₉₁	—	2	1	no	—
91	C₂²×C₁₉₁	—	2	1	no	—
92	C₂²×C₁₉₁	—	4	4	no	—
93	C₂²×C₁₉₁	#117	1	2	yes	—
94	C₂²×C₁₉₁	#131	1	2	yes	—
95	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
96	C₄×C₁₉₁	—	2	1	no	—
97	C₄×C₁₉₁	—	2	1	no	—
98	C₄×C₁₉₁	#117	1	2	yes	—
99	C₄×C₁₉₁	C₂×#41	1	2	yes	—
100	Dic₁₉₁	—	2	2	no	—
101	Dic₁₉₁	—	2	2	no	—
102	Dic₁₉₁	—	2	2	no	—
103	Dic₁₉₁	—	2	2	no	—
104	C₂×C₁₉₁	—	2	2	no	—
105	C₂×C₁₉₁	—	2	2	no	—
106	C₂×C₁₉₁	—	4	1	no	—
107	C₂×C₁₉₁	C₂×#117	1	1	yes	characteristic
108	C₂×C₁₉₁	C₂×#131	1	1	yes	characteristic
109	C₂×C₁₉₁	#568	1	1	yes	characteristic
110	C₁₉₁	#2924	1	1	yes	characteristic, complemented, Sylow
111	#2924	—	191	1	no	maximal, Sylow
112	C₂×#117	—	191	1	no	—
113	#433	—	191	4	no	—
114	#432	—	191	2	no	—
115	#431	—	191	2	no	—
116	#424	—	191	2	no	—
117	#407	D₁₉₁	1	1	yes	characteristic
118	C₂×C₂²⋊C₈	—	191	1	no	—
119	#405	—	191	2	no	—
120	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
121	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
122	C₂²×C₈	—	191	2	no	—
123	#117	—	191	1	no	—
124	#117	—	191	1	no	—
125	#117	—	191	2	no	—
126	#117	—	191	1	no	—
127	#117	—	191	1	no	—
128	#117	—	191	2	no	—
129	C₂×C₄⋊C₄	—	191	1	no	—
130	C₂×C₄⋊C₄	—	191	1	no	—
131	C₂×C₂²⋊C₄	—	191	2	no	—
132	C₂×C₄²	—	191	2	no	—
133	C₄⋊C₈	—	191	4	no	—
134	C₄⋊C₈	—	191	4	no	—
135	C₂²⋊C₈	—	191	1	no	—
136	C₂²⋊C₈	—	191	1	no	—
137	C₂²⋊C₈	—	191	2	no	—
138	C₂²⋊C₈	C₂×D₁₉₁	1	2	yes	—
139	C₂²⋊C₈	C₂×D₁₉₁	1	2	yes	—
140	#97	—	191	4	no	—
141	C₄×C₈	—	191	4	no	—
142	C₂⁴	C₄×D₁₉₁	1	1	yes	characteristic
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	191	2	no	—
145	C₂²×C₄	—	191	2	no	—
146	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
148	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
152	M₁₆	—	2	4	no	—
153	C₂×C₈	—	382	4	no	—
154	C₂×C₈	—	191	2	no	—
155	C₂×C₈	—	191	2	no	—
156	C₂×C₈	C₂²×D₁₉₁	1	4	yes	—
157	C₄⋊C₄	—	191	2	no	—
158	C₄⋊C₄	—	191	2	no	—
159	C₄⋊C₄	—	191	1	no	—
160	C₄⋊C₄	—	191	1	no	—
161	C₄⋊C₄	—	191	1	no	—
162	C₄⋊C₄	—	191	1	no	—
163	C₂²⋊C₄	—	191	2	no	—
164	C₂²⋊C₄	—	191	2	no	—
165	C₂²⋊C₄	—	191	2	no	—
166	C₂²⋊C₄	—	191	2	no	—
167	C₄²	—	191	2	no	—
168	C₄²	—	191	2	no	—
169	C₄²	—	191	2	no	—
170	C₄²	—	191	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	4	1	no	—
174	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
175	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
176	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	382	4	no	—
183	C₂×C₄	—	191	2	no	—
184	C₂×C₄	—	191	2	no	—
185	C₂×C₄	—	191	2	no	—
186	C₂×C₄	—	191	2	no	—
187	C₂×C₄	—	764	2	no	—
188	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
189	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
190	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
191	C₂×C₄	#15690286	1	4	yes	—
192	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
193	C₈	—	2	4	no	—
194	C₈	—	382	2	no	—
195	C₈	—	382	2	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	4	no	—
204	C₂²	#20259278	1	2	yes	—
205	C₂²	#20259330	1	2	yes	—
206	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	382	2	no	—
210	C₄	—	382	2	no	—
211	C₄	—	382	2	no	—
212	C₄	—	382	2	no	—
213	C₄	#20259278	1	2	yes	—
214	C₄	C₂×#15690286	1	2	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	2	2	no	—
217	C₂	—	4	1	no	—
218	C₂	C₂×#20259278	1	1	yes	characteristic
219	C₂	C₂×#20259330	1	1	yes	characteristic
220	C₂	#27133429	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Fitting	#407×C₁₉₁	C₂
prime radical	C₂⁴×C₁₉₁	C₂×C₄
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	#407	D₁₉₁
hypocenter	C₁₉₁	#2924
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. #27132721	C₂	semidirect
3. #27132717	C₂	semidirect
4. #27132713	C₂	semidirect
5. #27132691	C₂	semidirect
7. #27132621	C₂	semidirect
20. Dic₁₉₁⋊C₈	C₂²	semidirect
21. Dic₁₉₁⋊C₈	C₂²	semidirect
25. #20259232	C₂²	semidirect
26. C₈×Dic₁₉₁	C₂²	semidirect
110. C₁₉₁	#2924	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20259281	2
#27132721	2
#27132717	2
#27132713	2
#27132691	2
C₂×C₂²⋊(C₁₉₁⋊C₈)	2
#27132621	2
#407×C₁₉₁	2
#2924	191