Group #37581158 — subgroups

Subgroups — #37581158

Order 24448.

392 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 213 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	#27132724	C₂	1	8	yes	maximal, complemented
2	#27132687	C₂	1	2	yes	maximal, complemented
3	#27132627	C₂	1	2	yes	maximal
4	C₄×#15690271	C₂	1	2	yes	maximal, complemented
5	C₂×#117×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15690271	C₂²	1	4	yes	—
7	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	8	yes	complemented
8	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	8	yes	—
9	#20259233	C₂²	1	4	yes	complemented
10	C₄×C₁₉₁⋊C₈	C₂²	1	4	yes	complemented
11	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
12	#117×C₁₉₁	C₄	1	4	yes	—
13	#117×C₁₉₁	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₁₉₁	C₄	1	2	yes	—
15	C₂×C₂²⋊C₄×C₁₉₁	C₄	1	2	yes	—
16	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
18	#15690271	—	2	8	no	—
19	C₂×C₁₉₁⋊C₈	C₂³	1	8	yes	—
20	C₂⁴×C₁₉₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₉₁	—	2	4	no	—
22	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₉₁	C₂×C₄	1	8	yes	—
29	C₂²⋊C₄×C₁₉₁	C₂×C₄	1	8	yes	—
30	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
31	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
32	C₄²×C₁₉₁	C₂³	1	2	yes	—
33	C₄²×C₁₉₁	C₂³	1	2	yes	—
34	C₁₉₁⋊C₈	—	2	8	no	—
35	C₂³×C₁₉₁	—	2	2	no	—
36	C₂³×C₁₉₁	—	4	2	no	—
37	C₂³×C₁₉₁	C₂²×C₄	1	2	yes	—
38	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₁	—	2	2	no	—
40	C₂×C₄×C₁₉₁	—	2	4	no	—
41	C₂×C₄×C₁₉₁	—	2	4	no	—
42	C₂×C₄×C₁₉₁	—	4	2	no	—
43	C₂×C₄×C₁₉₁	—	4	2	no	—
44	C₂×C₄×C₁₉₁	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₁₉₁	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₁	#41	1	4	yes	—
49	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
50	C₂²×C₁₉₁	—	2	2	no	—
51	C₂²×C₁₉₁	—	2	2	no	—
52	C₂²×C₁₉₁	—	2	4	no	—
53	C₂²×C₁₉₁	—	4	4	no	—
54	C₂²×C₁₉₁	#117	1	2	yes	—
55	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₁₉₁	—	2	2	no	—
57	C₄×C₁₉₁	—	2	4	no	—
58	C₄×C₁₉₁	—	2	4	no	—
59	C₄×C₁₉₁	#117	1	2	yes	—
60	C₄×C₁₉₁	C₂×#41	1	2	yes	—
61	C₂×C₁₉₁	—	2	2	no	—
62	C₂×C₁₉₁	—	4	2	no	—
63	C₂×C₁₉₁	C₂×#117	1	1	yes	characteristic
64	C₂×C₁₉₁	#568	1	2	yes	—
65	C₁₉₁	#2921	1	1	yes	characteristic, complemented, Sylow
66	#2921	—	191	1	no	maximal, Sylow
67	C₂×#117	D₁₉₁	1	1	yes	characteristic
68	#433	—	191	8	no	—
69	#423	—	191	2	no	—
70	#407	—	191	2	no	—
71	C₄×M₁₆	—	191	2	no	—
72	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
73	C₂×M₁₆	—	191	4	no	—
74	#117	Dic₁₉₁	1	4	yes	—
75	#117	C₂×D₁₉₁	1	4	yes	—
76	C₂×C₄⋊C₄	Dic₁₉₁	1	2	yes	—
77	C₂×C₂²⋊C₄	Dic₁₉₁	1	2	yes	—
78	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
80	C₄⋊C₈	—	191	8	no	—
81	C₂²⋊C₈	—	191	8	no	—
82	#97	—	191	4	no	—
83	C₄×C₈	—	191	4	no	—
84	C₂⁴	C₂×Dic₁₉₁	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₁₉₁	1	2	yes	—
88	C₂²×C₄	C₂×Dic₁₉₁	1	2	yes	—
89	C₂²×C₄	C₂×Dic₁₉₁	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
92	M₁₆	—	382	8	no	—
93	C₂×C₈	—	191	8	no	—
94	C₄⋊C₄	C₂×Dic₁₉₁	1	8	yes	—
95	C₂²⋊C₄	C₂×Dic₁₉₁	1	8	yes	—
96	C₄²	C₂×Dic₁₉₁	1	2	yes	—
97	C₄²	C₂×Dic₁₉₁	1	2	yes	—
98	C₄²	C₂²×D₁₉₁	1	2	yes	—
99	C₄²	C₂²×D₁₉₁	1	2	yes	—
100	C₂³	—	2	2	no	—
101	C₂³	—	4	2	no	—
102	C₂³	C₂²×Dic₁₉₁	1	2	yes	—
103	C₂³	C₂²×Dic₁₉₁	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	C₂²×Dic₁₉₁	1	4	yes	—
110	C₂×C₄	C₂²×Dic₁₉₁	1	4	yes	—
111	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
112	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
113	C₂×C₄	#15690287	1	4	yes	—
114	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
115	C₈	—	382	8	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	4	4	no	—
120	C₂²	#20259281	1	2	yes	—
121	C₂²	C₂³×Dic₁₉₁	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	4	no	—
125	C₄	#20259281	1	2	yes	—
126	C₄	C₂×#15690287	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	4	2	no	—
129	C₂	C₂×#20259281	1	1	yes	characteristic
130	C₂	#27133430	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27132724

C₂

yes

maximal, complemented

#27132687

C₂

yes

maximal, complemented

#27132627

C₂

yes

maximal

C₄×#15690271

C₂

yes

maximal, complemented

C₂×#117×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₂²

yes

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

complemented

C₂²⋊(C₁₉₁⋊C₈)

C₂²

yes

—

#20259233

C₂²

yes

complemented

C₄×C₁₉₁⋊C₈

C₂²

yes

complemented

C₂³×C₄×C₁₉₁

C₂²

yes

characteristic

#117×C₁₉₁

C₄

yes

—

#117×C₁₉₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₁

C₄

yes

—

C₂×C₂²⋊C₄×C₁₉₁

C₄

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#15690271

—

C₂×C₁₉₁⋊C₈

C₂³

yes

—

C₂⁴×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

C₂×C₄

yes

—

C₂²⋊C₄×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

C₂³

yes

—

C₄²×C₁₉₁

C₂³

yes

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂²×C₄

yes

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×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

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#117

yes

—

C₂²×C₁₉₁

C₂³×C₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#117

yes

—

C₄×C₁₉₁

C₂×#41

yes

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×#117

yes

characteristic

C₂×C₁₉₁

#568

yes

—

C₁₉₁

#2921

yes

characteristic, complemented, Sylow

#2921

—

191

maximal, Sylow

C₂×#117

D₁₉₁

yes

characteristic

#433

—

191

—

#423

—

191

—

#407

—

191

—

C₄×M₁₆

—

191

—

C₂³×C₄

C₂×D₁₉₁

yes

characteristic

C₂×M₁₆

—

191

—

#117

Dic₁₉₁

yes

—

#117

C₂×D₁₉₁

yes

—

C₂×C₄⋊C₄

Dic₁₉₁

yes

—

C₂×C₂²⋊C₄

Dic₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

191

—

C₂²⋊C₈

—

191

—

#97

—

191

—

C₄×C₈

—

191

—

C₂⁴

C₂×Dic₁₉₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₉₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₉₁

yes

—

C₂²×C₄

C₂×Dic₁₉₁

yes

—

C₂²×C₄

C₂×Dic₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

M₁₆

—

382

—

C₂×C₈

—

191

—

C₄⋊C₄

C₂×Dic₁₉₁

yes

—

C₂²⋊C₄

C₂×Dic₁₉₁

yes

—

C₄²

C₂×Dic₁₉₁

yes

—

C₄²

C₂×Dic₁₉₁

yes

—

C₄²

C₂²×D₁₉₁

yes

—

C₄²

C₂²×D₁₉₁

yes

—

100

C₂³

—

101

C₂³

—

102

C₂³

C₂²×Dic₁₉₁

yes

—

103

C₂³

C₂²×Dic₁₉₁

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²×Dic₁₉₁

yes

—

110

C₂×C₄

C₂²×Dic₁₉₁

yes

—

111

C₂×C₄

C₂²×Dic₁₉₁

yes

—

112

C₂×C₄

C₂²×Dic₁₉₁

yes

characteristic

113

C₂×C₄

#15690287

yes

—

114

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

115

C₈

—

382

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

#20259281

yes

—

121

C₂²

C₂³×Dic₁₉₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#20259281

yes

—

126

C₄

C₂×#15690287

yes

—

127

C₂

—

128

C₂

—

129

C₂

C₂×#20259281

yes

characteristic

130

C₂

#27133430

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Fitting	C₂×#117×C₁₉₁	C₂
prime radical	C₂⁴×C₁₉₁	C₂×C₄
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	C₂×#117	D₁₉₁
hypocenter	C₁₉₁	#2921
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₉₁

commutator, socle

C₂²×C₁₉₁

C₂³×C₄

Fitting

C₂×#117×C₁₉₁

C₂

prime radical

C₂⁴×C₁₉₁

C₂×C₄

cosocle

C₂×C₄×C₁₉₁

C₂⁴

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂×C₄ ◁ C₂×#117

N	H ≅ G/N	type
1. #27132724	C₂	semidirect
2. #27132687	C₂	semidirect
4. C₄×#15690271	C₂	semidirect
7. C₄⋊(C₁₉₁⋊C₈)	C₂²	semidirect
9. #20259233	C₂²	semidirect
10. C₄×C₁₉₁⋊C₈	C₂²	semidirect
65. C₁₉₁	#2921	semidirect

H ≅ G/N

type

1. #27132724

C₂

semidirect

2. #27132687

C₂

semidirect

4. C₄×#15690271

C₂

semidirect

7. C₄⋊(C₁₉₁⋊C₈)

C₂²

semidirect

9. #20259233

C₂²

semidirect

10. C₄×C₁₉₁⋊C₈

C₂²

semidirect

65. C₁₉₁

#2921

semidirect

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

subgroup	index
#27132724	2
#27132687	2
#27132627	2
C₄×#15690271	2
C₂×#117×C₁₉₁	2
#2921	191

subgroup

index

C₂×#117×C₁₉₁

191

Subgroups — #37581158

Conjugacy classes of subgroups

Complements

Maximal subgroups