Group #37572001 — subgroups

Subgroups — #37572001

Order 24448.

184 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 55 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	#27132860	C₂	1	2	yes	maximal, complemented
2	#27132793	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132414	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132407	C₂	1	2	yes	maximal, complemented
5	#493×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15690275	—	2	2	no	—
7	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
8	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
10	Dic₁₉₁⋊C₈	—	2	1	no	—
11	#20259243	—	2	2	no	—
12	#20259232	—	2	1	no	—
13	C₂×D₈×C₁₉₁	—	2	1	no	—
14	C₂×D₈×C₁₉₁	—	2	1	no	—
15	#127×C₁₉₁	C₂²	1	2	yes	—
16	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₈₂	D₄	1	1	yes	characteristic
18	#15690275	—	4	4	no	—
19	C₂×C₁₉₁⋊C₈	—	2	2	no	—
20	C₂×C₁₉₁⋊C₈	—	2	1	no	—
21	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
22	Dic₁₉₁⋊C₄	—	2	1	no	—
23	C₄×Dic₁₉₁	—	2	1	no	—
24	C₂×D₄×C₁₉₁	—	2	2	no	—
25	C₂×D₄×C₁₉₁	—	2	2	no	—
26	C₂×D₄×C₁₉₁	—	2	2	no	—
27	D₈×C₁₉₁	—	4	2	no	—
28	D₈×C₁₉₁	—	4	2	no	—
29	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
31	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
32	Dic₃₈₂	—	2	2	no	—
33	C₂×Dic₁₉₁	—	2	1	no	—
34	C₂×Dic₁₉₁	—	2	1	no	—
35	C₁₉₁⋊C₈	—	4	1	no	—
36	C₁₉₁⋊C₈	—	4	2	no	—
37	C₂³×C₁₉₁	—	4	2	no	—
38	D₄×C₁₉₁	—	4	2	no	—
39	D₄×C₁₉₁	—	4	2	no	—
40	D₄×C₁₉₁	—	4	2	no	—
41	D₄×C₁₉₁	—	4	2	no	—
42	D₄×C₁₉₁	—	8	2	no	—
43	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
46	C₈×C₁₉₁	—	2	1	no	—
47	C₈×C₁₉₁	SD₁₆	1	2	yes	complemented
48	C₂²×C₁₉₁	—	8	2	no	—
49	C₂²×C₁₉₁	—	8	2	no	—
50	C₂²×C₁₉₁	—	8	2	no	—
51	C₂²×C₁₉₁	#121	1	1	yes	characteristic
52	C₄×C₁₉₁	—	2	1	no	—
53	C₄×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
56	Dic₁₉₁	—	4	1	no	—
57	Dic₁₉₁	—	4	1	no	—
58	C₂×C₁₉₁	—	16	2	no	—
59	C₂×C₁₉₁	#460	1	1	yes	characteristic
60	C₂×C₁₉₁	#469	1	1	yes	characteristic
61	C₂×C₁₉₁	#472	1	1	yes	characteristic
62	C₁₉₁	#1688	1	1	yes	characteristic, complemented, Sylow
63	#1688	—	191	1	no	maximal, Sylow
64	#493	D₁₉₁	1	1	yes	characteristic
65	#460	—	191	2	no	—
66	#446	—	191	1	no	—
67	#334	—	191	1	no	—
68	#331	—	191	2	no	—
69	C₂×SD₁₆	—	382	2	no	—
70	C₂×D₈	—	2	1	no	—
71	C₂×D₈	—	2	1	no	—
72	#127	C₂×D₁₉₁	1	2	yes	—
73	C₄×Q₈	—	191	1	no	—
74	C₄⋊C₈	—	382	1	no	—
75	C₄⋊C₈	—	191	1	no	—
76	C₄⋊C₈	—	191	2	no	—
77	#102	—	382	2	no	—
78	#97	—	382	1	no	—
79	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
80	C₂×Q₈	—	191	1	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	2	no	—
83	C₂×D₄	—	2	2	no	—
84	SD₁₆	—	764	4	no	—
85	D₈	—	4	2	no	—
86	D₈	—	4	2	no	—
87	C₂×C₈	—	382	2	no	—
88	C₂×C₈	—	382	1	no	—
89	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
91	C₄⋊C₄	—	382	1	no	—
92	C₄⋊C₄	—	191	1	no	—
93	C₄²	—	382	1	no	—
94	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
95	C₂³	—	4	2	no	—
96	Q₈	—	382	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	8	2	no	—
102	C₂×C₄	—	382	1	no	—
103	C₂×C₄	—	382	1	no	—
104	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
106	C₂×C₄	#15690286	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	764	2	no	—
109	C₈	—	764	1	no	—
110	C₈	#15690275	1	2	yes	complemented
111	C₂²	—	8	2	no	—
112	C₂²	—	8	2	no	—
113	C₂²	—	8	2	no	—
114	C₂²	#20259293	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	764	1	no	—
117	C₄	—	764	1	no	—
118	C₄	C₂×#15690275	1	1	yes	characteristic
119	C₄	#20259346	1	2	yes	—
120	C₄	#20259348	1	1	yes	characteristic
121	C₂	—	16	2	no	—
122	C₂	#27132860	1	1	yes	characteristic
123	C₂	#27132913	1	1	yes	characteristic
124	C₂	#27132930	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132860

C₂

yes

maximal, complemented

#27132793

C₂

yes

maximal, characteristic, complemented

#27132414

C₂

yes

maximal, characteristic, complemented

#27132407

C₂

yes

maximal, complemented

#493×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690275

—

C₄×Dic₃₈₂

C₂²

yes

characteristic

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

—

Dic₁₉₁⋊C₈

—

#20259243

—

#20259232

—

C₂×D₈×C₁₉₁

—

C₂×D₈×C₁₉₁

—

#127×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

D₄

yes

characteristic

#15690275

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

D₄

yes

characteristic

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

D₈×C₁₉₁

—

D₈×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

SD₁₆

yes

complemented

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#121

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₄×C₁₉₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₁

AGL(1,ℤ/8)

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1688

—

191

maximal, Sylow

#493

D₁₉₁

yes

characteristic

#460

—

191

—

#446

—

191

—

#334

—

191

—

#331

—

191

—

C₂×SD₁₆

—

382

—

C₂×D₈

—

C₂×D₈

—

#127

C₂×D₁₉₁

yes

—

C₄×Q₈

—

191

—

C₄⋊C₈

—

382

—

C₄⋊C₈

—

191

—

C₄⋊C₈

—

191

—

#102

—

382

—

#97

—

382

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

191

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

764

—

D₈

—

D₈

—

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

C₂²⋊D₁₉₁

yes

characteristic

C₂×C₈

C₂²⋊D₁₉₁

yes

characteristic

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

C₄²

—

382

—

C₄²

C₂²×D₁₉₁

yes

characteristic

C₂³

—

Q₈

—

382

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

382

—

103

C₂×C₄

—

382

—

104

C₂×C₄

D₄×D₁₉₁

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

106

C₂×C₄

#15690286

yes

characteristic

107

C₈

—

108

C₈

—

764

—

109

C₈

—

764

—

110

C₈

#15690275

yes

complemented

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#20259293

yes

characteristic

115

C₄

—

116

C₄

—

764

—

117

C₄

—

764

—

118

C₄

C₂×#15690275

yes

characteristic

119

C₄

#20259346

yes

—

120

C₄

#20259348

yes

characteristic

121

C₂

—

122

C₂

#27132860

yes

characteristic

123

C₂

#27132913

yes

characteristic

124

C₂

#27132930

yes

characteristic

	subgroup	G/H
center	C₂²	#20259293
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting, prime radical	#493×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#121
hypercenter	#493	D₁₉₁
hypocenter	C₁₉₁	#1688
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#20259293

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting, prime radical

#493×C₁₉₁

C₂

C₂²×C₁₉₁

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #27132860	C₂	semidirect
2. #27132793	C₂	semidirect
3. #27132414	C₂	semidirect
4. #27132407	C₂	semidirect
47. C₈×C₁₉₁	SD₁₆	semidirect
62. C₁₉₁	#1688	semidirect
110. C₈	#15690275	semidirect

H ≅ G/N

type

1. #27132860

C₂

semidirect

2. #27132793

C₂

semidirect

3. #27132414

C₂

semidirect

4. #27132407

C₂

semidirect

47. C₈×C₁₉₁

SD₁₆

semidirect

62. C₁₉₁

#1688

semidirect

110. C₈

#15690275

semidirect

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

subgroup	index
#27132860	2
#27132793	2
#27132414	2
#27132407	2
#493×C₁₉₁	2
#1688	191

subgroup

index

#493×C₁₉₁

191

Subgroups — #37572001

Conjugacy classes of subgroups

Complements

Maximal subgroups