Group #37572029 — subgroups

Subgroups — #37572029

Order 24448.

146 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 51 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	#27133059	C₂	1	1	yes	maximal, characteristic
2	#27132955	C₂	1	1	yes	maximal, characteristic
3	#27132874	C₂	1	1	yes	maximal, characteristic
4	#27132790	C₂	1	1	yes	maximal, characteristic
5	#27132415	C₂	1	1	yes	maximal, characteristic
6	#27132408	C₂	1	1	yes	maximal, characteristic
7	#333×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₆₄	—	2	1	no	—
9	#20259322	C₂²	1	1	yes	characteristic
10	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
11	#20259255	—	2	1	no	—
12	#20259253	—	2	1	no	—
13	#20259253	—	2	1	no	—
14	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₁₉₁⋊C₈	—	2	1	no	—
16	#20259247	—	2	1	no	—
17	#20259245	—	2	1	no	—
18	#20259232	—	2	1	no	—
19	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×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	D₄	1	1	yes	characteristic
25	Dic₇₆₄	—	4	2	no	—
26	C₂×C₁₉₁⋊C₈	—	2	1	no	—
27	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
29	C₄⋊Dic₁₉₁	—	2	1	no	—
30	C₄⋊Dic₁₉₁	—	2	1	no	—
31	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
32	Dic₁₉₁⋊C₄	—	2	1	no	—
33	C₄×Dic₁₉₁	—	2	1	no	—
34	C₂×Q₈×C₁₉₁	—	2	1	no	—
35	C₂×C₈×C₁₉₁	—	2	1	no	—
36	C₂×C₈×C₁₉₁	—	2	1	no	—
37	C₄⋊C₄×C₁₉₁	—	2	1	no	—
38	C₄⋊C₄×C₁₉₁	—	2	1	no	—
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	C₂×Dic₁₉₁	—	2	1	no	—
45	C₂×Dic₁₉₁	—	2	1	no	—
46	C₂×Dic₁₉₁	—	4	1	no	—
47	C₁₉₁⋊C₈	—	2	1	no	—
48	C₁₉₁⋊C₈	—	4	1	no	—
49	C₁₉₁⋊C₈	Q₁₆	1	2	yes	complemented
50	Q₈×C₁₉₁	—	8	1	no	—
51	C₂×C₄×C₁₉₁	—	4	1	no	—
52	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
55	C₈×C₁₉₁	—	4	1	no	—
56	C₈×C₁₉₁	—	4	1	no	—
57	C₂²×C₁₉₁	#121	1	1	yes	characteristic
58	C₄×C₁₉₁	—	2	1	no	—
59	C₄×C₁₉₁	—	8	1	no	—
60	C₄×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₄×C₁₉₁	#137	1	1	yes	characteristic
63	C₄×C₁₉₁	#137	1	1	yes	characteristic
64	Dic₁₉₁	—	4	1	no	—
65	Dic₁₉₁	—	4	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	—	191	1	no	—
77	#333	D₁₉₁	1	1	yes	characteristic
78	#332	—	191	1	no	—
79	C₂×Q₁₆	—	382	1	no	—
80	C₄⋊Q₈	—	191	1	no	—
81	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
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	—	191	1	no	—
94	C₂×Q₈	—	2	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	—	191	1	no	—
102	C₂×C₈	—	191	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	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	—	8	1	no	—
112	Q₈	—	382	1	no	—
113	Q₈	—	382	1	no	—
114	Q₈	—	764	1	no	—
115	Q₈	—	764	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	382	1	no	—
118	C₂×C₄	—	382	1	no	—
119	C₂×C₄	—	764	1	no	—
120	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
121	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
122	C₂×C₄	#15690287	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	382	1	no	—
126	C₈	—	191	2	no	—
127	C₈	—	764	1	no	—
128	C₂²	#20259295	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	1528	1	no	—
132	C₄	—	764	1	no	—
133	C₄	—	764	1	no	—
134	C₄	Q₁₆×D₁₉₁	1	1	yes	characteristic
135	C₄	#20259349	1	1	yes	characteristic
136	C₄	#20259351	1	1	yes	characteristic
137	C₄	#20259354	1	1	yes	characteristic
138	C₂	#27132872	1	1	yes	characteristic
139	C₂	#27132912	1	1	yes	characteristic
140	C₂	#27132925	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133059

C₂

yes

maximal, characteristic

#27132955

C₂

yes

maximal, characteristic

#27132874

C₂

yes

maximal, characteristic

#27132790

C₂

yes

maximal, characteristic

#27132415

C₂

yes

maximal, characteristic

#27132408

C₂

yes

maximal, characteristic

#333×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

—

#20259322

C₂²

yes

characteristic

C₄×Dic₃₈₂

C₂²

yes

—

—

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

characteristic

Dic₁₉₁⋊C₈

—

#20259247

—

#20259245

—

#20259232

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

D₄

yes

characteristic

Dic₇₆₄

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

D₄

yes

characteristic

C₂×C₁₉₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

D₄

yes

characteristic

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

Q₁₆

yes

complemented

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

#121

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×Q₁₆

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

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

—

191

—

#333

D₁₉₁

yes

characteristic

#332

—

191

—

C₂×Q₁₆

—

382

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

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

—

191

—

C₂×Q₈

—

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

Q₁₆

—

764

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

382

—

101

C₂×C₈

—

191

—

102

C₂×C₈

—

191

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

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

—

112

Q₈

—

382

—

113

Q₈

—

382

—

114

Q₈

—

764

—

115

Q₈

—

764

—

116

C₂×C₄

—

117

C₂×C₄

—

382

—

118

C₂×C₄

—

382

—

119

C₂×C₄

—

764

—

120

C₂×C₄

D₄×D₁₉₁

yes

characteristic

121

C₂×C₄

D₄×D₁₉₁

yes

characteristic

122

C₂×C₄

#15690287

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

382

—

126

C₈

—

191

—

127

C₈

—

764

—

128

C₂²

#20259295

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

1528

—

132

C₄

—

764

—

133

C₄

—

764

—

134

C₄

Q₁₆×D₁₉₁

yes

characteristic

135

C₄

#20259349

yes

characteristic

136

C₄

#20259351

yes

characteristic

137

C₄

#20259354

yes

characteristic

138

C₂

#27132872

yes

characteristic

139

C₂

#27132912

yes

characteristic

140

C₂

#27132925

yes

characteristic

	subgroup	G/H
center	C₂²	#20259295
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#333×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	#121
hypercenter	#333	D₁₉₁
hypocenter	C₁₉₁	#1692
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#20259295

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#333×C₁₉₁

C₂

C₂²×C₁₉₁

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
49. C₁₉₁⋊C₈	Q₁₆	semidirect
70. C₁₉₁	#1692	semidirect

H ≅ G/N

type

49. C₁₉₁⋊C₈

Q₁₆

semidirect

70. C₁₉₁

#1692

semidirect

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

subgroup	index
#27133059	2
#27132955	2
#27132874	2
#27132790	2
#27132415	2
#27132408	2
#333×C₁₉₁	2
#1692	191

subgroup

index

#333×C₁₉₁

191

Subgroups — #37572029

Conjugacy classes of subgroups

Complements

Maximal subgroups