Group #37572027 — subgroups

Subgroups — #37572027

Order 24448.

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

normal subgroups only

This group has 146 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	#27133061	C₂	1	1	yes	maximal, characteristic
2	#27132959	C₂	1	1	yes	maximal, characteristic
3	#27132874	C₂	1	1	yes	maximal, characteristic
4	#27132793	C₂	1	1	yes	maximal, characteristic
5	#27132411	C₂	1	1	yes	maximal, characteristic
6	#27132410	C₂	1	1	yes	maximal, characteristic
7	#335×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₆₄	—	2	1	no	—
9	#20259322	C₂²	1	1	yes	characteristic
10	#20259322	C₂²	1	1	yes	characteristic
11	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
12	#20259256	—	2	1	no	—
13	#20259254	—	2	1	no	—
14	#20259254	—	2	1	no	—
15	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
16	Dic₁₉₁⋊C₈	—	2	1	no	—
17	#20259247	—	2	1	no	—
18	#20259247	—	2	1	no	—
19	#20259232	—	2	1	no	—
20	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₈₂	—	2	1	no	—
24	C₂×Dic₃₈₂	—	2	1	no	—
25	C₂×Dic₃₈₂	D₄	1	1	yes	characteristic
26	Dic₇₆₄	—	4	2	no	—
27	C₂×C₁₉₁⋊C₈	—	2	1	no	—
28	C₄⋊Dic₁₉₁	—	2	1	no	—
29	C₄⋊Dic₁₉₁	—	2	1	no	—
30	C₄⋊Dic₁₉₁	—	2	1	no	—
31	C₄⋊Dic₁₉₁	—	2	1	no	—
32	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
33	Dic₁₉₁⋊C₄	—	2	1	no	—
34	C₄×Dic₁₉₁	—	2	1	no	—
35	C₂×C₈×C₁₉₁	—	2	1	no	—
36	C₂×C₈×C₁₉₁	—	2	1	no	—
37	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
39	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
40	Dic₃₈₂	—	2	1	no	—
41	Dic₃₈₂	—	2	1	no	—
42	Dic₃₈₂	—	4	1	no	—
43	Dic₃₈₂	—	4	1	no	—
44	Dic₃₈₂	—	8	1	no	—
45	C₂×Dic₁₉₁	—	2	1	no	—
46	C₂×Dic₁₉₁	—	2	1	no	—
47	C₂×Dic₁₉₁	—	4	1	no	—
48	C₂×Dic₁₉₁	—	4	1	no	—
49	C₁₉₁⋊C₈	—	4	1	no	—
50	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
53	C₈×C₁₉₁	—	2	1	no	—
54	C₈×C₁₉₁	—	4	1	no	—
55	C₈×C₁₉₁	—	4	1	no	—
56	C₈×C₁₉₁	Q₁₆	1	2	yes	complemented
57	C₂²×C₁₉₁	#121	1	1	yes	characteristic
58	C₄×C₁₉₁	—	2	1	no	—
59	C₄×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₄×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₄×C₁₉₁	#137	1	1	yes	characteristic
62	C₄×C₁₉₁	#137	1	1	yes	characteristic
63	Dic₁₉₁	—	4	1	no	—
64	Dic₁₉₁	—	4	1	no	—
65	Dic₁₉₁	—	8	1	no	—
66	Dic₁₉₁	—	8	1	no	—
67	C₂×C₁₉₁	#462	1	1	yes	characteristic
68	C₂×C₁₉₁	#468	1	1	yes	characteristic
69	C₂×C₁₉₁	#471	1	1	yes	characteristic
70	C₁₉₁	#1692	1	1	yes	characteristic, complemented, Sylow
71	#1692	—	191	1	no	maximal, Sylow
72	#498	—	191	1	no	—
73	#477	—	191	1	no	—
74	#462	—	191	1	no	—
75	#446	—	191	1	no	—
76	#335	D₁₉₁	1	1	yes	characteristic
77	#333	—	191	1	no	—
78	#332	—	191	1	no	—
79	C₂×Q₁₆	—	382	1	no	—
80	C₄⋊Q₈	—	191	1	no	—
81	C₄⋊Q₈	—	191	1	no	—
82	C₄×Q₈	—	191	1	no	—
83	#107	—	382	1	no	—
84	#106	—	382	1	no	—
85	#106	—	382	1	no	—
86	C₄⋊C₈	—	382	1	no	—
87	C₄⋊C₈	—	191	1	no	—
88	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
90	#103	—	382	1	no	—
91	#103	—	382	1	no	—
92	#97	—	382	1	no	—
93	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
94	C₂×Q₈	—	382	1	no	—
95	C₂×Q₈	—	382	1	no	—
96	C₂×Q₈	—	191	1	no	—
97	Q₁₆	—	764	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	382	1	no	—
101	C₂×C₈	D₇₆₄	1	1	yes	characteristic
102	C₂×C₈	D₇₆₄	1	1	yes	characteristic
103	C₄⋊C₄	—	382	1	no	—
104	C₄⋊C₄	—	382	1	no	—
105	C₄⋊C₄	—	382	1	no	—
106	C₄⋊C₄	—	382	1	no	—
107	C₄⋊C₄	—	382	1	no	—
108	C₄⋊C₄	—	191	1	no	—
109	C₄²	—	382	1	no	—
110	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
111	Q₈	—	382	1	no	—
112	Q₈	—	382	1	no	—
113	Q₈	—	1528	1	no	—
114	Q₈	—	764	1	no	—
115	Q₈	—	764	1	no	—
116	C₂×C₄	—	382	1	no	—
117	C₂×C₄	—	382	1	no	—
118	C₂×C₄	—	764	1	no	—
119	C₂×C₄	—	764	1	no	—
120	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
121	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
122	C₂×C₄	#15690288	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	764	1	no	—
127	C₈	Dic₇₆₄	1	2	yes	complemented
128	C₂²	#20259297	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	1528	1	no	—
131	C₄	—	1528	1	no	—
132	C₄	—	764	1	no	—
133	C₄	—	764	1	no	—
134	C₄	C₂×Dic₇₆₄	1	1	yes	characteristic
135	C₄	#20259346	1	1	yes	characteristic
136	C₄	#20259354	1	1	yes	characteristic
137	C₄	#20259355	1	1	yes	characteristic
138	C₂	#27132874	1	1	yes	characteristic
139	C₂	#27132910	1	1	yes	characteristic
140	C₂	#27132927	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133061

C₂

yes

maximal, characteristic

#27132959

C₂

yes

maximal, characteristic

#27132874

C₂

yes

maximal, characteristic

#27132793

C₂

yes

maximal, characteristic

#27132411

C₂

yes

maximal, characteristic

#27132410

C₂

yes

maximal, characteristic

#335×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

—

#20259322

C₂²

yes

characteristic

#20259322

C₂²

yes

characteristic

C₄×Dic₃₈₂

C₂²

yes

—

—

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

characteristic

Dic₁₉₁⋊C₈

—

#20259247

—

#20259247

—

#20259232

—

C₄⋊C₈×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

D₄

yes

characteristic

Dic₇₆₄

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

D₄

yes

characteristic

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₈×C₁₉₁

Q₁₆

yes

complemented

C₂²×C₁₉₁

#121

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×Q₁₆

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1692

—

191

maximal, Sylow

#498

—

191

—

#477

—

191

—

#462

—

191

—

#446

—

191

—

#335

D₁₉₁

yes

characteristic

#333

—

191

—

#332

—

191

—

C₂×Q₁₆

—

382

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

—

191

—

C₄×Q₈

—

191

—

#107

—

382

—

#106

—

382

—

#106

—

382

—

C₄⋊C₈

—

382

—

C₄⋊C₈

—

191

—

C₄⋊C₈

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

C₂×D₁₉₁

yes

characteristic

#103

—

382

—

#103

—

382

—

#97

—

382

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

Q₁₆

—

764

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

382

—

101

C₂×C₈

D₇₆₄

yes

characteristic

102

C₂×C₈

D₇₆₄

yes

characteristic

103

C₄⋊C₄

—

382

—

104

C₄⋊C₄

—

382

—

105

C₄⋊C₄

—

382

—

106

C₄⋊C₄

—

382

—

107

C₄⋊C₄

—

382

—

108

C₄⋊C₄

—

191

—

109

C₄²

—

382

—

110

C₄²

C₂²×D₁₉₁

yes

characteristic

111

Q₈

—

382

—

112

Q₈

—

382

—

113

Q₈

—

1528

—

114

Q₈

—

764

—

115

Q₈

—

764

—

116

C₂×C₄

—

382

—

117

C₂×C₄

—

382

—

118

C₂×C₄

—

764

—

119

C₂×C₄

—

764

—

120

C₂×C₄

D₄×D₁₉₁

yes

characteristic

121

C₂×C₄

C₂×D₇₆₄

yes

characteristic

122

C₂×C₄

#15690288

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

764

—

127

C₈

Dic₇₆₄

yes

complemented

128

C₂²

#20259297

yes

characteristic

129

C₄

—

130

C₄

—

1528

—

131

C₄

—

1528

—

132

C₄

—

764

—

133

C₄

—

764

—

134

C₄

C₂×Dic₇₆₄

yes

characteristic

135

C₄

#20259346

yes

characteristic

136

C₄

#20259354

yes

characteristic

137

C₄

#20259355

yes

characteristic

138

C₂

#27132874

yes

characteristic

139

C₂

#27132910

yes

characteristic

140

C₂

#27132927

yes

characteristic

	subgroup	G/H
center	C₂²	#20259297
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#335×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	#121
hypercenter	#335	D₁₉₁
hypocenter	C₁₉₁	#1692
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#20259297

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#335×C₁₉₁

C₂

C₂²×C₁₉₁

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
56. C₈×C₁₉₁	Q₁₆	semidirect
70. C₁₉₁	#1692	semidirect
127. C₈	Dic₇₆₄	semidirect

H ≅ G/N

type

56. C₈×C₁₉₁

Q₁₆

semidirect

70. C₁₉₁

#1692

semidirect

127. C₈

Dic₇₆₄

semidirect

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

subgroup	index
#27133061	2
#27132959	2
#27132874	2
#27132793	2
#27132411	2
#27132410	2
#335×C₁₉₁	2
#1692	191

subgroup

index

#335×C₁₉₁

191

Subgroups — #37572027

Conjugacy classes of subgroups

Complements

Maximal subgroups