Group #37571234 — subgroups

Subgroups — #37571234

Order 24448.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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	#27132720	C₂	1	2	yes	maximal, complemented
2	#27132407	C₂	1	4	yes	maximal, complemented
3	C₂×#127×C₁₉₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	4	yes	complemented
5	C₂²⋊(C₁₉₁⋊C₈)	—	2	2	no	—
6	#20259233	—	2	2	no	—
7	C₂²×D₄×C₁₉₁	—	2	2	no	—
8	C₂²×D₄×C₁₉₁	C₄	1	2	yes	—
9	#127×C₁₉₁	—	2	2	no	—
10	#127×C₁₉₁	C₄	1	2	yes	—
11	#127×C₁₉₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
13	C₂×C₁₉₁⋊C₈	—	2	4	no	—
14	C₂⁴×C₁₉₁	—	2	2	no	—
15	C₂×D₄×C₁₉₁	—	2	2	no	—
16	C₂×D₄×C₁₉₁	—	2	4	no	—
17	C₂×D₄×C₁₉₁	—	2	4	no	—
18	C₂×D₄×C₁₉₁	—	4	4	no	—
19	C₂×D₄×C₁₉₁	—	4	8	no	—
20	C₂×D₄×C₁₉₁	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
23	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₉₁	D₄	1	2	yes	—
25	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
26	C₁₉₁⋊C₈	—	4	4	no	—
27	C₂³×C₁₉₁	—	2	4	no	—
28	C₂³×C₁₉₁	—	4	8	no	—
29	C₂³×C₁₉₁	—	4	4	no	—
30	C₂³×C₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
31	D₄×C₁₉₁	—	4	4	no	—
32	D₄×C₁₉₁	—	4	4	no	—
33	D₄×C₁₉₁	—	4	8	no	—
34	D₄×C₁₉₁	—	4	8	no	—
35	C₂×C₄×C₁₉₁	—	2	2	no	—
36	C₂×C₄×C₁₉₁	—	2	4	no	—
37	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
41	C₂²×C₁₉₁	—	2	2	no	—
42	C₂²×C₁₉₁	—	4	4	no	—
43	C₂²×C₁₉₁	—	4	8	no	—
44	C₂²×C₁₉₁	—	8	8	no	—
45	C₂²×C₁₉₁	#100	1	2	yes	—
46	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₁₉₁	—	2	1	no	—
48	C₄×C₁₉₁	—	2	1	no	—
49	C₄×C₁₉₁	—	2	2	no	—
50	C₄×C₁₉₁	AGL(1,ℤ/8)	1	4	yes	—
51	C₂×C₁₉₁	—	2	2	no	—
52	C₂×C₁₉₁	—	8	4	no	—
53	C₂×C₁₉₁	C₂×#100	1	1	yes	characteristic
54	C₂×C₁₉₁	#418	1	2	yes	—
55	C₁₉₁	#1555	1	1	yes	characteristic, complemented, Sylow
56	#1555	—	191	1	no	maximal, Sylow
57	C₂×#127	D₁₉₁	1	1	yes	characteristic
58	#432	—	191	2	no	—
59	#331	—	191	4	no	—
60	C₂²×D₄	—	2	2	no	—
61	C₂²×D₄	Dic₁₉₁	1	2	yes	—
62	#127	—	2	2	no	—
63	#127	Dic₁₉₁	1	2	yes	—
64	#127	C₂×D₁₉₁	1	2	yes	—
65	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
66	C₄⋊C₈	—	191	4	no	—
67	C₂²⋊C₈	—	382	2	no	—
68	#97	—	382	2	no	—
69	C₂⁴	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	—	2	4	no	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	—	4	4	no	—
74	C₂×D₄	—	4	8	no	—
75	C₂×D₄	C₂×Dic₁₉₁	1	4	yes	—
76	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
78	C₂×C₈	—	382	4	no	—
79	C₄²	C₂×Dic₁₉₁	1	1	yes	characteristic
80	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
81	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
82	C₂³	—	2	4	no	—
83	C₂³	—	4	8	no	—
84	C₂³	—	4	4	no	—
85	C₂³	C₂²⋊Dic₁₉₁	1	1	yes	characteristic
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	8	no	—
89	D₄	—	4	8	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	C₂²⋊Dic₁₉₁	1	2	yes	—
93	C₂×C₄	C₂²⋊Dic₁₉₁	1	1	yes	characteristic
94	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
95	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
96	C₈	—	764	4	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	4	4	no	—
99	C₂²	—	4	8	no	—
100	C₂²	—	8	8	no	—
101	C₂²	#20259239	1	2	yes	—
102	C₂²	C₂×C₂²⋊Dic₁₉₁	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	2	no	—
106	C₄	#20259348	1	4	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	8	4	no	—
109	C₂	C₂×#20259239	1	1	yes	characteristic
110	C₂	#27132668	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27132720

C₂

yes

maximal, complemented

#27132407

C₂

yes

maximal, complemented

C₂×#127×C₁₉₁

C₂

yes

maximal, characteristic

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

complemented

C₂²⋊(C₁₉₁⋊C₈)

—

#20259233

—

C₂²×D₄×C₁₉₁

—

C₂²×D₄×C₁₉₁

C₄

yes

—

#127×C₁₉₁

—

#127×C₁₉₁

C₄

yes

—

#127×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×C₁₉₁⋊C₈

—

C₂⁴×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂²⋊C₄

yes

characteristic

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#100

yes

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

AGL(1,ℤ/8)

yes

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×#100

yes

characteristic

C₂×C₁₉₁

#418

yes

—

C₁₉₁

#1555

yes

characteristic, complemented, Sylow

#1555

—

191

maximal, Sylow

C₂×#127

D₁₉₁

yes

characteristic

#432

—

191

—

#331

—

191

—

C₂²×D₄

—

C₂²×D₄

Dic₁₉₁

yes

—

#127

—

#127

Dic₁₉₁

yes

—

#127

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

191

—

C₂²⋊C₈

—

382

—

#97

—

382

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₉₁

yes

—

C₂²×C₄

C₂×Dic₁₉₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂×C₈

—

382

—

C₄²

C₂×Dic₁₉₁

yes

characteristic

C₄²

C₂²⋊D₁₉₁

yes

—

C₄²

C₂²×D₁₉₁

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₁₉₁

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₉₁

yes

—

C₂×C₄

C₂²⋊Dic₁₉₁

yes

characteristic

C₂×C₄

C₂²×Dic₁₉₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

C₈

—

764

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#20259239

yes

—

102

C₂²

C₂×C₂²⋊Dic₁₉₁

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#20259348

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₂×#20259239

yes

characteristic

110

C₂

#27132668

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₁
Fitting, prime radical	C₂×#127×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
cosocle	C₄²×C₁₉₁	C₂³
hypercenter	C₂×#127	D₁₉₁
hypocenter	C₁₉₁	#1555
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₁

Fitting, prime radical

C₂×#127×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

cosocle

C₄²×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #27132720	C₂	semidirect
2. #27132407	C₂	semidirect
4. C₄⋊(C₁₉₁⋊C₈)	C₂²	semidirect
55. C₁₉₁	#1555	semidirect

H ≅ G/N

type

1. #27132720

C₂

semidirect

2. #27132407

C₂

semidirect

4. C₄⋊(C₁₉₁⋊C₈)

C₂²

semidirect

55. C₁₉₁

#1555

semidirect

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

subgroup	index
#27132720	2
#27132407	2
C₂×#127×C₁₉₁	2
#1555	191

subgroup

index

#27132720

#27132407

C₂×#127×C₁₉₁

#1555

191

Subgroups — #37571234

Conjugacy classes of subgroups

Complements

Maximal subgroups