Group #37570341 — subgroups

Subgroups — #37570341

Order 24448.

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

normal subgroups only

This group has 202 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	#27132418	C₂	1	2	yes	maximal, complemented
2	#390×C₁₉₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₉₁⋊C₈	—	2	2	no	—
4	C₂²×D₄×C₁₉₁	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₁₉₁	C₄	1	1	yes	characteristic
6	C₂×C₂²⋊C₄×C₁₉₁	—	2	2	no	—
7	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
8	C₂×C₁₉₁⋊C₈	—	2	4	no	—
9	C₂×C₁₉₁⋊C₈	—	4	2	no	—
10	C₂×C₁₉₁⋊C₈	—	4	2	no	—
11	C₂⁴×C₁₉₁	—	2	1	no	—
12	C₂×D₄×C₁₉₁	—	2	1	no	—
13	C₂×D₄×C₁₉₁	—	4	1	no	—
14	C₂×D₄×C₁₉₁	—	4	1	no	—
15	C₂×D₄×C₁₉₁	C₈	1	2	yes	complemented
16	C₂²×C₄×C₁₉₁	—	2	1	no	—
17	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₉₁	—	4	1	no	—
20	C₂²⋊C₄×C₁₉₁	—	4	2	no	—
21	C₂²⋊C₄×C₁₉₁	—	4	2	no	—
22	C₄²×C₁₉₁	—	4	1	no	—
23	C₁₉₁⋊C₈	—	4	4	no	—
24	C₂³×C₁₉₁	—	2	2	no	—
25	C₂³×C₁₉₁	—	4	1	no	—
26	C₂³×C₁₉₁	—	4	1	no	—
27	C₂³×C₁₉₁	—	4	2	no	—
28	C₂³×C₁₉₁	—	4	2	no	—
29	C₂³×C₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
30	D₄×C₁₉₁	—	4	2	no	—
31	D₄×C₁₉₁	—	4	2	no	—
32	C₂×C₄×C₁₉₁	—	2	1	no	—
33	C₂×C₄×C₁₉₁	—	2	2	no	—
34	C₂×C₄×C₁₉₁	—	2	1	no	—
35	C₂×C₄×C₁₉₁	—	2	2	no	—
36	C₂×C₄×C₁₉₁	—	4	1	no	—
37	C₂×C₄×C₁₉₁	—	4	2	no	—
38	C₂×C₄×C₁₉₁	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×C₁₉₁	M₁₆	1	1	yes	characteristic
40	C₂×C₄×C₁₉₁	D₈	1	2	yes	—
41	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
42	C₂²×C₁₉₁	—	4	2	no	—
43	C₂²×C₁₉₁	—	4	4	no	—
44	C₂²×C₁₉₁	—	8	2	no	—
45	C₂²×C₁₉₁	—	8	1	no	—
46	C₂²×C₁₉₁	—	8	1	no	—
47	C₂²×C₁₉₁	C₂²⋊C₈	1	1	yes	characteristic
48	C₂²×C₁₉₁	#99	1	1	yes	characteristic
49	C₂²×C₁₉₁	#100	1	1	yes	characteristic
50	C₂²×C₁₉₁	#102	1	2	yes	—
51	C₂²×C₁₉₁	#104	1	2	yes	—
52	C₄×C₁₉₁	—	2	1	no	—
53	C₄×C₁₉₁	—	2	1	no	—
54	C₄×C₁₉₁	—	2	2	no	—
55	C₄×C₁₉₁	—	2	2	no	—
56	C₄×C₁₉₁	—	8	1	no	—
57	C₂×C₁₉₁	—	8	2	no	—
58	C₂×C₁₉₁	#323	1	1	yes	characteristic
59	C₂×C₁₉₁	D₄⋊C₈	1	2	yes	—
60	C₂×C₁₉₁	#327	1	2	yes	—
61	C₂×C₁₉₁	#329	1	1	yes	characteristic
62	C₂×C₁₉₁	#331	1	1	yes	characteristic
63	C₁₉₁	#1270	1	1	yes	characteristic, complemented, Sylow
64	#1270	—	191	1	no	maximal, Sylow
65	#390	D₁₉₁	1	1	yes	characteristic
66	#336	—	191	2	no	—
67	C₂²×D₄	Dic₁₉₁	1	1	yes	characteristic
68	C₂²×C₈	—	382	2	no	—
69	C₂×C₄⋊C₄	Dic₁₉₁	1	1	yes	characteristic
70	C₂×C₂²⋊C₄	—	2	2	no	—
71	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
72	C₂⁴	—	2	1	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	—	4	1	no	—
75	C₂×D₄	—	4	1	no	—
76	C₂×D₄	C₁₉₁⋊C₈	1	2	yes	complemented
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
80	C₂×C₈	—	382	4	no	—
81	C₂×C₈	—	764	2	no	—
82	C₂×C₈	—	764	2	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₂²⋊C₄	—	4	2	no	—
85	C₂²⋊C₄	—	4	2	no	—
86	C₄²	—	4	1	no	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	1	no	—
89	C₂³	—	4	1	no	—
90	C₂³	—	4	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	C₂²⋊Dic₁₉₁	1	1	yes	characteristic
93	D₄	—	4	2	no	—
94	D₄	—	4	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	C₂×C₁₉₁⋊C₈	1	1	yes	characteristic
102	C₂×C₄	#15690271	1	1	yes	characteristic
103	C₂×C₄	#15690272	1	2	yes	—
104	C₂×C₄	#15690275	1	2	yes	—
105	C₈	—	764	4	no	—
106	C₂²	—	4	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	8	2	no	—
109	C₂²	—	8	1	no	—
110	C₂²	—	8	1	no	—
111	C₂²	C₂²⋊(C₁₉₁⋊C₈)	1	1	yes	characteristic
112	C₂²	#20259237	1	1	yes	characteristic
113	C₂²	#20259239	1	1	yes	characteristic
114	C₂²	#20259243	1	2	yes	—
115	C₂²	#20259249	1	2	yes	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	1	no	—
121	C₂	—	8	2	no	—
122	C₂	#27132387	1	1	yes	characteristic
123	C₂	D₄⋊(C₁₉₁⋊C₈)	1	2	yes	—
124	C₂	#27132397	1	2	yes	—
125	C₂	#27132403	1	1	yes	characteristic
126	C₂	#27132407	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132418

C₂

yes

maximal, complemented

#390×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

—

C₂²×D₄×C₁₉₁

C₄

yes

characteristic

C₂×C₄⋊C₄×C₁₉₁

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₁₉₁

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂⁴×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

C₈

yes

complemented

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

—

C₄⋊C₄×C₁₉₁

—

C₂²⋊C₄×C₁₉₁

—

C₂²⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂²⋊C₄

yes

characteristic

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₁

M₁₆

yes

characteristic

C₂×C₄×C₁₉₁

D₈

yes

—

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂²⋊C₈

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

—

C₂²×C₁₉₁

#104

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#323

yes

characteristic

C₂×C₁₉₁

D₄⋊C₈

yes

—

C₂×C₁₉₁

#327

yes

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1270

—

191

maximal, Sylow

#390

D₁₉₁

yes

characteristic

#336

—

191

—

C₂²×D₄

Dic₁₉₁

yes

characteristic

C₂²×C₈

—

382

—

C₂×C₄⋊C₄

Dic₁₉₁

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₁₉₁⋊C₈

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₉₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂×C₈

—

382

—

C₂×C₈

—

764

—

C₂×C₈

—

764

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₁₉₁

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×C₁₉₁⋊C₈

yes

characteristic

102

C₂×C₄

#15690271

yes

characteristic

103

C₂×C₄

#15690272

yes

—

104

C₂×C₄

#15690275

yes

—

105

C₈

—

764

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂²⋊(C₁₉₁⋊C₈)

yes

characteristic

112

C₂²

#20259237

yes

characteristic

113

C₂²

#20259239

yes

characteristic

114

C₂²

#20259243

yes

—

115

C₂²

#20259249

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

#27132387

yes

characteristic

123

C₂

D₄⋊(C₁₉₁⋊C₈)

yes

—

124

C₂

#27132397

yes

—

125

C₂

#27132403

yes

characteristic

126

C₂

#27132407

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂×C₈
Frattini	C₂×C₄²	C₂×D₁₉₁
Fitting	#390×C₁₉₁	C₂
socle	C₂³×C₁₉₁	C₂²⋊C₄
prime radical	C₂²×D₄×C₁₉₁	C₄
cosocle	C₂×C₄²×C₁₉₁	C₂²
hypercenter	#390	D₁₉₁
hypocenter	C₁₉₁	#1270
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

subgroup

G/H

center

C₂³

C₂²⋊Dic₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂×C₈

Frattini

C₂×C₄²

C₂×D₁₉₁

Fitting

#390×C₁₉₁

C₂

socle

C₂³×C₁₉₁

C₂²⋊C₄

prime radical

C₂²×D₄×C₁₉₁

C₄

cosocle

C₂×C₄²×C₁₉₁

C₂²

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #390

N	H ≅ G/N	type
1. #27132418	C₂	semidirect
15. C₂×D₄×C₁₉₁	C₈	semidirect
63. C₁₉₁	#1270	semidirect
76. C₂×D₄	C₁₉₁⋊C₈	semidirect

H ≅ G/N

type

1. #27132418

C₂

semidirect

15. C₂×D₄×C₁₉₁

C₈

semidirect

63. C₁₉₁

#1270

semidirect

76. C₂×D₄

C₁₉₁⋊C₈

semidirect

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

subgroup	index
#27132418	2
#390×C₁₉₁	2
#1270	191

subgroup

index

#27132418

#390×C₁₉₁

#1270

191

Subgroups — #37570341

Conjugacy classes of subgroups

Complements

Maximal subgroups