Group #37581591 — subgroups

Subgroups — #37581591

Order 24448.

394 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 197 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	#27132735	C₂	1	8	yes	maximal, complemented
2	#27132720	C₂	1	2	yes	maximal, complemented
3	#27132630	C₂	1	4	yes	maximal, complemented
4	#532×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#15690271	C₂²	1	4	yes	—
6	C₂²×C₁₉₁⋊C₈	C₂²	1	4	yes	—
7	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	4	yes	—
8	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	4	yes	—
9	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	8	yes	—
10	#20259233	C₂²	1	4	yes	—
11	C₂×#41×C₁₉₁	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₉₁	C₄	1	1	yes	characteristic
13	#127×C₁₉₁	C₄	1	1	yes	characteristic
14	#124×C₁₉₁	C₄	1	2	yes	—
15	#121×C₁₉₁	C₄	1	4	yes	—
16	C₄×D₄×C₁₉₁	C₂²	1	4	yes	—
17	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
18	#15690271	—	4	4	no	—
19	C₂×C₁₉₁⋊C₈	—	4	4	no	—
20	C₂×C₁₉₁⋊C₈	C₂³	1	8	yes	—
21	#41×C₁₉₁	—	4	2	no	—
22	#41×C₁₉₁	—	4	2	no	—
23	C₂×Q₈×C₁₉₁	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₁₉₁	C₂×C₄	1	4	yes	—
25	C₂×D₄×C₁₉₁	C₂×C₄	1	4	yes	—
26	C₂×D₄×C₁₉₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₉₁	C₂³	1	4	yes	—
30	C₄⋊C₄×C₁₉₁	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₉₁	C₂×C₄	1	8	yes	—
32	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₉₁	C₂³	1	2	yes	—
35	C₁₉₁⋊C₈	—	2	8	no	—
36	C₂³×C₁₉₁	C₂²×C₄	1	4	yes	—
37	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
38	Q₈×C₁₉₁	—	4	2	no	—
39	D₄×C₁₉₁	—	4	4	no	—
40	D₄×C₁₉₁	—	4	2	no	—
41	D₄×C₁₉₁	—	4	4	no	—
42	C₂×C₄×C₁₉₁	—	2	2	no	—
43	C₂×C₄×C₁₉₁	—	2	4	no	—
44	C₂×C₄×C₁₉₁	—	2	4	no	—
45	C₂×C₄×C₁₉₁	—	4	2	no	—
46	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₉₁	C₂²×C₄	1	4	yes	—
50	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
51	C₂²×C₁₉₁	—	2	1	no	—
52	C₂²×C₁₉₁	—	2	4	no	—
53	C₂²×C₁₉₁	—	2	2	no	—
54	C₂²×C₁₉₁	—	2	8	no	—
55	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₁₉₁	—	2	1	no	—
57	C₄×C₁₉₁	—	2	4	no	—
58	C₄×C₁₉₁	—	2	2	no	—
59	C₄×C₁₉₁	—	2	2	no	—
60	C₄×C₁₉₁	H(5,𝔽₂)	1	2	yes	—
61	C₂×C₁₉₁	—	4	1	no	—
62	C₂×C₁₉₁	—	4	4	no	—
63	C₂×C₁₉₁	#518	1	1	yes	characteristic
64	C₂×C₁₉₁	#568	1	2	yes	—
65	C₁₉₁	#2980	1	1	yes	characteristic, complemented, Sylow
66	#2980	—	191	1	no	maximal, Sylow
67	#532	D₁₉₁	1	1	yes	characteristic
68	#435	—	191	8	no	—
69	#432	—	191	2	no	—
70	#408	—	191	4	no	—
71	C₂×#41	C₂×D₁₉₁	1	2	yes	—
72	C₂×M₁₆	—	191	4	no	—
73	C₂²×C₈	—	191	4	no	—
74	C₄⋊Q₈	Dic₁₉₁	1	1	yes	characteristic
75	#127	Dic₁₉₁	1	1	yes	characteristic
76	#124	Dic₁₉₁	1	2	yes	—
77	#121	Dic₁₉₁	1	4	yes	—
78	C₄×D₄	C₂×D₁₉₁	1	4	yes	—
79	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
80	C₄⋊C₈	—	191	4	no	—
81	C₂²⋊C₈	—	191	4	no	—
82	C₂²⋊C₈	—	191	8	no	—
83	#97	—	191	4	no	—
84	#41	—	4	2	no	—
85	#41	—	4	2	no	—
86	C₂×Q₈	C₂×Dic₁₉₁	1	2	yes	—
87	C₂×D₄	C₂×Dic₁₉₁	1	4	yes	—
88	C₂×D₄	C₂×Dic₁₉₁	1	4	yes	—
89	C₂×D₄	C₂×Dic₁₉₁	1	2	yes	—
90	C₂²×C₄	C₂×Dic₁₉₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₁₉₁	1	4	yes	—
93	M₁₆	—	764	4	no	—
94	C₂×C₈	—	191	8	no	—
95	C₂×C₈	—	764	4	no	—
96	C₄⋊C₄	C₂×Dic₁₉₁	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₁₉₁	1	8	yes	—
98	C₄²	C₂×Dic₁₉₁	1	1	yes	characteristic
99	C₄²	C₂×Dic₁₉₁	1	1	yes	characteristic
100	C₄²	C₂²×D₁₉₁	1	2	yes	—
101	C₂³	C₂²×Dic₁₉₁	1	4	yes	—
102	C₂³	C₂²×Dic₁₉₁	1	1	yes	characteristic
103	Q₈	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	4	no	—
106	D₄	—	4	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
113	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
114	C₂×C₄	C₂²×Dic₁₉₁	1	4	yes	—
115	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
116	C₈	—	382	8	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	8	no	—
121	C₂²	C₂³×Dic₁₉₁	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	#20259367	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	—	4	4	no	—
129	C₂	#27133138	1	1	yes	characteristic
130	C₂	#27133430	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27132735

C₂

yes

maximal, complemented

#27132720

C₂

yes

maximal, complemented

#27132630

C₂

yes

maximal, complemented

#532×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₂²

yes

—

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₉₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₉₁⋊C₈)

C₂²

yes

—

#20259233

C₂²

yes

—

C₂×#41×C₁₉₁

C₂²

yes

—

C₄⋊Q₈×C₁₉₁

C₄

yes

characteristic

#127×C₁₉₁

C₄

yes

characteristic

#124×C₁₉₁

C₄

yes

—

#121×C₁₉₁

C₄

yes

—

C₄×D₄×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂³

yes

—

#41×C₁₉₁

—

#41×C₁₉₁

—

C₂×Q₈×C₁₉₁

C₂×C₄

yes

—

C₂×D₄×C₁₉₁

C₂×C₄

yes

—

C₂×D₄×C₁₉₁

C₂×C₄

yes

—

C₂×D₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

—

C₄⋊C₄×C₁₉₁

C₂×C₄

yes

—

C₂²⋊C₄×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

Q₈×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂³×C₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

H(5,𝔽₂)

yes

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#518

yes

characteristic

C₂×C₁₉₁

#568

yes

—

C₁₉₁

#2980

yes

characteristic, complemented, Sylow

#2980

—

191

maximal, Sylow

#532

D₁₉₁

yes

characteristic

#435

—

191

—

#432

—

191

—

#408

—

191

—

C₂×#41

C₂×D₁₉₁

yes

—

C₂×M₁₆

—

191

—

C₂²×C₈

—

191

—

C₄⋊Q₈

Dic₁₉₁

yes

characteristic

#127

Dic₁₉₁

yes

characteristic

#124

Dic₁₉₁

yes

—

#121

Dic₁₉₁

yes

—

C₄×D₄

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

191

—

C₂²⋊C₈

—

191

—

C₂²⋊C₈

—

191

—

#97

—

191

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₁₉₁

yes

—

C₂×D₄

C₂×Dic₁₉₁

yes

—

C₂×D₄

C₂×Dic₁₉₁

yes

—

C₂×D₄

C₂×Dic₁₉₁

yes

—

C₂²×C₄

C₂×Dic₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂²×C₄

C₂²×D₁₉₁

yes

—

M₁₆

—

764

—

C₂×C₈

—

191

—

C₂×C₈

—

764

—

C₄⋊C₄

C₂×Dic₁₉₁

yes

—

C₂²⋊C₄

C₂×Dic₁₉₁

yes

—

C₄²

C₂×Dic₁₉₁

yes

characteristic

C₄²

C₂×Dic₁₉₁

yes

characteristic

100

C₄²

C₂²×D₁₉₁

yes

—

101

C₂³

C₂²×Dic₁₉₁

yes

—

102

C₂³

C₂²×Dic₁₉₁

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×Dic₁₉₁

yes

characteristic

112

C₂×C₄

C₂²×Dic₁₉₁

yes

—

113

C₂×C₄

C₂²×Dic₁₉₁

yes

—

114

C₂×C₄

C₂²×Dic₁₉₁

yes

—

115

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

116

C₈

—

382

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×Dic₁₉₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#20259367

yes

—

127

C₂

—

128

C₂

—

129

C₂

#27133138

yes

characteristic

130

C₂

#27133430

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Fitting, prime radical	#532×C₁₉₁	C₂
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	#532	D₁₉₁
hypocenter	C₁₉₁	#2980
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₉₁

commutator, socle

C₂²×C₁₉₁

C₂³×C₄

Fitting, prime radical

#532×C₁₉₁

C₂

cosocle

C₂×C₄×C₁₉₁

C₂⁴

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #27132735	C₂	semidirect
2. #27132720	C₂	semidirect
3. #27132630	C₂	semidirect
65. C₁₉₁	#2980	semidirect

H ≅ G/N

type

1. #27132735

C₂

semidirect

2. #27132720

C₂

semidirect

3. #27132630

C₂

semidirect

65. C₁₉₁

#2980

semidirect

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

subgroup	index
#27132735	2
#27132720	2
#27132630	2
#532×C₁₉₁	2
#2980	191

subgroup

index

#532×C₁₉₁

191

Subgroups — #37581591

Conjugacy classes of subgroups

Complements

Maximal subgroups