Group #37573341 — subgroups

Subgroups — #37573341

Order 24448.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20259246	C₂	1	4	yes	maximal
2	C₂×#20259233	C₂	1	1	yes	maximal, characteristic
3	#27132529	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
7	#20259246	—	2	8	no	—
8	#20259233	—	2	2	no	—
9	C₁₉₁⋊#95	—	2	2	no	—
10	C₂²×Q₈×C₁₉₁	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₉₁	—	2	2	no	—
12	C₄⋊Q₈×C₁₉₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₉₁	—	2	2	no	—
14	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₉₁	—	2	2	no	—
16	C₂×C₁₉₁⋊C₈	—	2	2	no	—
17	C₂×C₁₉₁⋊C₈	—	2	2	no	—
18	C₂×C₁₉₁⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₁₉₁	—	2	4	no	—
20	C₂×Q₈×C₁₉₁	—	2	2	no	—
21	C₂×Q₈×C₁₉₁	—	2	4	no	—
22	C₂×Q₈×C₁₉₁	—	2	4	no	—
23	C₂×Q₈×C₁₉₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₉₁	—	2	2	no	—
25	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
26	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₉₁	—	2	8	no	—
28	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
29	C₄²×C₁₉₁	D₄	1	2	yes	—
30	C₂×Dic₁₉₁	—	4	2	no	—
31	C₂×Dic₁₉₁	—	4	4	no	—
32	C₁₉₁⋊C₈	—	2	4	no	—
33	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₁₉₁	—	2	8	no	—
35	Q₈×C₁₉₁	—	4	4	no	—
36	C₂×C₄×C₁₉₁	—	2	4	no	—
37	C₂×C₄×C₁₉₁	—	2	4	no	—
38	C₂×C₄×C₁₉₁	—	4	2	no	—
39	C₂×C₄×C₁₉₁	—	4	2	no	—
40	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₁	#41	1	2	yes	—
45	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₉₁	#124	1	1	yes	characteristic
47	C₂²×C₁₉₁	#127	1	1	yes	characteristic
48	C₂²×C₁₉₁	#137	1	4	yes	—
49	C₄×C₁₉₁	—	2	4	no	—
50	C₄×C₁₉₁	—	4	4	no	—
51	C₄×C₁₉₁	C₄×D₄	1	2	yes	—
52	C₄×C₁₉₁	#121	1	2	yes	—
53	Dic₁₉₁	—	8	2	no	—
54	C₂×C₁₉₁	#390	1	1	yes	characteristic
55	C₂×C₁₉₁	#419	1	2	yes	—
56	C₂×C₁₉₁	#491	1	2	yes	—
57	C₂×C₁₉₁	#497	1	2	yes	—
58	C₁₉₁	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	191	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	D₁₉₁	1	1	yes	characteristic
61	C₂×#103	—	191	4	no	—
62	C₂×#97	—	191	1	no	—
63	#382	—	191	1	no	—
64	C₂²×Q₈	C₂×D₁₉₁	1	2	yes	—
65	C₂²×C₈	—	191	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₁₉₁	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	191	2	no	—
70	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
71	#103	—	382	8	no	—
72	#97	—	382	2	no	—
73	#95	—	382	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₁₉₁	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	382	2	no	—
80	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
82	C₂×C₈	—	382	2	no	—
83	C₂×C₈	—	382	2	no	—
84	C₂×C₈	—	191	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	382	4	no	—
87	C₄²	C₂×Dic₁₉₁	1	2	yes	—
88	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	764	4	no	—
97	C₂×C₄	—	764	2	no	—
98	C₂×C₄	C₂²⋊Dic₁₉₁	1	4	yes	—
99	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
102	C₂×C₄	#15690286	1	2	yes	—
103	C₈	—	382	4	no	—
104	C₂²	C₂×C₂²⋊Dic₁₉₁	1	1	yes	characteristic
105	C₂²	#20259309	1	1	yes	characteristic
106	C₂²	#20259318	1	1	yes	characteristic
107	C₂²	#20259353	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	1528	2	no	—
111	C₄	D₄×Dic₁₉₁	1	2	yes	—
112	C₄	#20259293	1	2	yes	—
113	C₂	#27132566	1	1	yes	characteristic
114	C₂	#27132672	1	2	yes	—
115	C₂	#27133031	1	2	yes	—
116	C₂	#27133055	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259246

C₂

yes

maximal

C₂×#20259233

C₂

yes

maximal, characteristic

#27132529

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

#20259246

—

#20259233

—

C₁₉₁⋊#95

—

C₂²×Q₈×C₁₉₁

C₂²

yes

—

C₄⋊Q₈×C₁₉₁

—

C₄⋊Q₈×C₁₉₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂²×Dic₁₉₁

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

D₄

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₄×D₄

yes

—

C₄×C₁₉₁

#121

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

#390

yes

characteristic

C₂×C₁₉₁

#419

yes

—

C₂×C₁₉₁

#491

yes

—

C₂×C₁₉₁

#497

yes

—

C₁₉₁

#1960

yes

characteristic, complemented, Sylow

#1960

—

191

maximal, Sylow

C₂×C₄⋊Q₈

D₁₉₁

yes

characteristic

C₂×#103

—

191

—

C₂×#97

—

191

—

#382

—

191

—

C₂²×Q₈

C₂×D₁₉₁

yes

—

C₂²×C₈

—

191

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₉₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#103

—

382

—

#97

—

382

—

#95

—

382

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₉₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄²

C₂×Dic₁₉₁

yes

—

C₄²

C₂²⋊D₁₉₁

yes

—

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

764

—

C₂×C₄

—

764

—

C₂×C₄

C₂²⋊Dic₁₉₁

yes

—

C₂×C₄

C₂²×Dic₁₉₁

yes

characteristic

100

C₂×C₄

D₄×D₁₉₁

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

102

C₂×C₄

#15690286

yes

—

103

C₈

—

382

—

104

C₂²

C₂×C₂²⋊Dic₁₉₁

yes

characteristic

105

C₂²

#20259309

yes

characteristic

106

C₂²

#20259318

yes

characteristic

107

C₂²

#20259353

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

1528

—

111

C₄

D₄×Dic₁₉₁

yes

—

112

C₄

#20259293

yes

—

113

C₂

#27132566

yes

characteristic

114

C₂

#27132672

yes

—

115

C₂

#27133031

yes

—

116

C₂

#27133055

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₂×C₄⋊Q₈×C₁₉₁	C₂
socle, prime radical	C₂³×C₁₉₁	C₂×D₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₉₁
hypocenter	C₁₉₁	#1960
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₂×C₄⋊Q₈×C₁₉₁

C₂

socle, prime radical

C₂³×C₁₉₁

C₂×D₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₉₁

hypocenter

C₁₉₁

#1960

LCS

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₉₁	C₄	semidirect
58. C₁₉₁	#1960	semidirect
67. C₄⋊Q₈	Dic₁₉₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₉₁

C₄

semidirect

58. C₁₉₁

#1960

semidirect

67. C₄⋊Q₈

Dic₁₉₁

semidirect

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

subgroup	index
C₂×#20259246	2
C₂×#20259233	2
#27132529	2
C₂×C₄⋊Q₈×C₁₉₁	2
#1960	191

subgroup

index

C₂×C₄⋊Q₈×C₁₉₁

191

Subgroups — #37573341

Conjugacy classes of subgroups

Complements

Maximal subgroups