Group #37574036 — subgroups

Subgroups — #37574036

Order 24448.

222 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 65 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	#27132712	C₂	1	1	yes	maximal, characteristic, complemented
2	#27132711	C₂	1	2	yes	maximal, complemented
3	#27132627	C₂	1	2	yes	maximal
4	C₁₉₁⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	#407×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15690271	C₂²	1	2	yes	—
7	C₂×#15690271	C₂²	1	2	yes	—
8	#20259258	—	2	2	no	—
9	#20259257	—	2	4	no	—
10	C₂²⋊(C₁₉₁⋊C₈)	—	2	4	no	—
11	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₂²⋊C₈×C₁₉₁	—	2	2	no	—
15	#15690271	—	2	2	no	—
16	#15690271	—	2	2	no	—
17	#15690271	—	2	2	no	—
18	#15690271	—	2	2	no	—
19	C₂×C₁₉₁⋊C₈	—	2	2	no	—
20	C₂×C₁₉₁⋊C₈	D₄	1	4	yes	—
21	C₂⁴×C₁₉₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₉₁	—	2	1	no	—
23	C₂²×C₄×C₁₉₁	—	2	2	no	—
24	C₂²×C₄×C₁₉₁	—	2	2	no	—
25	C₂²×C₄×C₁₉₁	—	2	1	no	—
26	C₂²×C₄×C₁₉₁	Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₉₁	—	2	2	no	—
29	M₁₆×C₁₉₁	—	2	2	no	—
30	C₂×C₈×C₁₉₁	—	2	1	no	—
31	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
32	C₁₉₁⋊C₈	—	2	4	no	—
33	C₁₉₁⋊C₈	—	4	2	no	—
34	C₂³×C₁₉₁	—	2	2	no	—
35	C₂³×C₁₉₁	—	2	1	no	—
36	C₂³×C₁₉₁	—	4	2	no	—
37	C₂³×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₉₁	—	2	2	no	—
39	C₂×C₄×C₁₉₁	—	2	1	no	—
40	C₂×C₄×C₁₉₁	—	4	1	no	—
41	C₂×C₄×C₁₉₁	—	4	2	no	—
42	C₂×C₄×C₁₉₁	—	4	1	no	—
43	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₁	#41	1	2	yes	—
46	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
47	C₈×C₁₉₁	—	2	2	no	—
48	C₈×C₁₉₁	—	4	1	no	—
49	C₂²×C₁₉₁	—	2	4	no	—
50	C₂²×C₁₉₁	—	2	2	no	—
51	C₂²×C₁₉₁	—	4	1	no	—
52	C₂²×C₁₉₁	—	4	2	no	—
53	C₂²×C₁₉₁	—	4	1	no	—
54	C₂²×C₁₉₁	—	4	1	no	—
55	C₂²×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
56	C₂²×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₁₉₁	—	4	1	no	—
58	C₄×C₁₉₁	#123	1	1	yes	characteristic
59	C₄×C₁₉₁	#123	1	2	yes	—
60	C₄×C₁₉₁	#127	1	1	yes	characteristic
61	C₂×C₁₉₁	—	2	2	no	—
62	C₂×C₁₉₁	—	2	1	no	—
63	C₂×C₁₉₁	—	4	2	no	—
64	C₂×C₁₉₁	#399	1	1	yes	characteristic
65	C₁₉₁	#2091	1	1	yes	characteristic, complemented, Sylow
66	#2091	—	191	1	no	maximal, Sylow
67	#430	—	191	2	no	—
68	#430	—	191	1	no	—
69	#407	—	191	2	no	—
70	#407	D₁₉₁	1	1	yes	characteristic
71	#338	—	191	1	no	—
72	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
73	C₂×M₁₆	—	191	2	no	—
74	C₂×M₁₆	—	191	2	no	—
75	C₂×M₁₆	C₂×D₁₉₁	1	1	yes	characteristic
76	C₂×M₁₆	C₂×D₁₉₁	1	1	yes	characteristic
77	#108	—	382	4	no	—
78	#108	—	382	2	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	—	382	4	no	—
81	C₂⁴	Dic₃₈₂	1	1	yes	characteristic
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	Dic₃₈₂	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	382	2	no	—
91	M₁₆	—	382	2	no	—
92	M₁₆	—	382	2	no	—
93	M₁₆	—	382	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	382	2	no	—
96	C₂×C₈	—	191	4	no	—
97	C₂×C₈	C₂²⋊D₁₉₁	1	2	yes	—
98	C₂³	—	2	2	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	4	2	no	—
101	C₂³	C₂×Dic₃₈₂	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
109	C₂×C₄	#15690286	1	2	yes	—
110	C₂×C₄	#15690287	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	382	4	no	—
114	C₈	—	764	2	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	4	1	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	1	no	—
121	C₂²	#20259299	1	2	yes	—
122	C₂²	#20259301	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	#20259303	1	1	yes	characteristic
125	C₄	#20259304	1	2	yes	—
126	C₄	#20259318	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#27132599	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132712

C₂

yes

maximal, characteristic, complemented

#27132711

C₂

yes

maximal, complemented

#27132627

C₂

yes

maximal

C₁₉₁⋊#338

C₂

yes

maximal, characteristic, complemented

#407×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₂²

yes

—

C₂×#15690271

C₂²

yes

—

#20259258

—

#20259257

—

C₂²⋊(C₁₉₁⋊C₈)

—

C₂³×C₄×C₁₉₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₉₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₉₁

C₂²

yes

characteristic

C₂²⋊C₈×C₁₉₁

—

—

—

—

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₂⁴×C₁₉₁

Q₈

yes

characteristic

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

Q₈

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂²⋊Q₈

yes

—

C₂²×C₁₉₁

C₂²⋊Q₈

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

#123

yes

characteristic

C₄×C₁₉₁

#123

yes

—

C₄×C₁₉₁

#127

yes

characteristic

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#399

yes

characteristic

C₁₉₁

#2091

yes

characteristic, complemented, Sylow

#2091

—

191

maximal, Sylow

#430

—

191

—

#430

—

191

—

#407

—

191

—

#407

D₁₉₁

yes

characteristic

#338

—

191

—

C₂³×C₄

C₂×D₁₉₁

yes

characteristic

C₂×M₁₆

—

191

—

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

characteristic

C₂×M₁₆

C₂×D₁₉₁

yes

characteristic

#108

—

382

—

#108

—

382

—

C₂²⋊C₈

—

C₂²⋊C₈

—

382

—

C₂⁴

Dic₃₈₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₃₈₂

yes

characteristic

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

382

—

M₁₆

—

382

—

M₁₆

—

382

—

M₁₆

—

382

—

C₂×C₈

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₂×C₈

C₂²⋊D₁₉₁

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×Dic₃₈₂

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

D₄×D₁₉₁

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

109

C₂×C₄

#15690286

yes

—

110

C₂×C₄

#15690287

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

382

—

114

C₈

—

764

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#20259299

yes

—

122

C₂²

#20259301

yes

characteristic

123

C₄

—

124

C₄

#20259303

yes

characteristic

125

C₄

#20259304

yes

—

126

C₄

#20259318

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#27132599

yes

characteristic

	subgroup	G/H
center	C₄	#20259318
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#407×C₁₉₁	C₂
socle	C₂×C₁₉₁	#399
prime radical	C₂⁴×C₁₉₁	Q₈
hypercenter	#407	D₁₉₁
hypocenter	C₁₉₁	#2091
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #407

subgroup

G/H

center

C₄

#20259318

commutator, cosocle

C₂²×C₄×C₁₉₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#407×C₁₉₁

C₂

socle

C₂×C₁₉₁

#399

prime radical

C₂⁴×C₁₉₁

Q₈

D₁₉₁

C₁₉₁

G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #407

N	H ≅ G/N	type
1. #27132712	C₂	semidirect
2. #27132711	C₂	semidirect
4. C₁₉₁⋊#338	C₂	semidirect
65. C₁₉₁	#2091	semidirect

H ≅ G/N

type

1. #27132712

C₂

semidirect

2. #27132711

C₂

semidirect

4. C₁₉₁⋊#338

C₂

semidirect

65. C₁₉₁

#2091

semidirect

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

subgroup	index
#27132712	2
#27132711	2
#27132627	2
C₁₉₁⋊#338	2
#407×C₁₉₁	2
#2091	191

subgroup

index

#27132712

#27132711

#27132627

C₁₉₁⋊#338

#407×C₁₉₁

#2091

191

Subgroups — #37574036

Conjugacy classes of subgroups

Complements

Maximal subgroups