Group #37572023 — subgroups

Subgroups — #37572023

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	#27132955	C₂	1	1	yes	maximal, characteristic
2	#27132870	C₂	1	1	yes	maximal, characteristic
3	#27132793	C₂	1	1	yes	maximal, characteristic
4	#27132416	C₂	1	1	yes	maximal, characteristic
5	#27132412	C₂	1	1	yes	maximal, characteristic
6	#27132409	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#15690277	—	2	1	no	—
9	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
10	#20259255	—	2	1	no	—
11	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
13	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
14	Dic₁₉₁⋊C₈	—	2	1	no	—
15	#20259246	—	2	1	no	—
16	#20259245	—	2	1	no	—
17	#20259232	—	2	1	no	—
18	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
20	#106×C₁₉₁	—	2	1	no	—
21	#106×C₁₉₁	—	2	1	no	—
22	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₈₂	D₄	1	1	yes	characteristic
24	#15690277	—	4	2	no	—
25	C₂×C₁₉₁⋊C₈	—	2	1	no	—
26	C₂×C₁₉₁⋊C₈	—	2	1	no	—
27	C₂×C₁₉₁⋊C₈	—	2	1	no	—
28	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
29	Dic₁₉₁⋊C₄	—	2	1	no	—
30	C₄×Dic₁₉₁	—	2	1	no	—
31	C₂×Q₈×C₁₉₁	—	2	1	no	—
32	C₂×Q₈×C₁₉₁	—	2	1	no	—
33	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₁₉₁	—	2	1	no	—
36	C₄⋊C₄×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	C₂×Dic₁₉₁	—	2	1	no	—
43	C₂×Dic₁₉₁	—	2	1	no	—
44	C₁₉₁⋊C₈	—	4	1	no	—
45	C₁₉₁⋊C₈	—	4	1	no	—
46	C₁₉₁⋊C₈	—	4	1	no	—
47	Q₈×C₁₉₁	—	4	1	no	—
48	Q₈×C₁₉₁	—	4	1	no	—
49	Q₈×C₁₉₁	—	8	1	no	—
50	C₂×C₄×C₁₉₁	—	4	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	—	2	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	—	8	1	no	—
60	C₄×C₁₉₁	—	8	1	no	—
61	C₄×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₄×C₁₉₁	#137	1	1	yes	characteristic
64	C₄×C₁₉₁	#137	1	1	yes	characteristic
65	Dic₁₉₁	—	4	1	no	—
66	Dic₁₉₁	—	4	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	D₁₉₁	1	1	yes	characteristic
73	#477	—	191	1	no	—
74	#462	—	191	1	no	—
75	#446	—	191	1	no	—
76	#335	—	191	1	no	—
77	#333	—	191	1	no	—
78	#332	—	191	1	no	—
79	C₂×Q₁₆	—	382	1	no	—
80	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
81	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
82	C₄×Q₈	—	191	1	no	—
83	#107	—	382	1	no	—
84	#106	—	2	1	no	—
85	#106	—	2	1	no	—
86	C₄⋊C₈	—	382	1	no	—
87	C₄⋊C₈	—	191	1	no	—
88	C₄⋊C₈	—	191	1	no	—
89	C₄⋊C₈	—	191	1	no	—
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	—	2	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	191	1	no	—
97	Q₁₆	—	764	2	no	—
98	C₂×C₈	—	382	1	no	—
99	C₂×C₈	—	382	1	no	—
100	C₂×C₈	—	382	1	no	—
101	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	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	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	8	1	no	—
114	Q₈	—	382	1	no	—
115	Q₈	—	382	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	382	1	no	—
119	C₂×C₄	—	382	1	no	—
120	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
122	C₂×C₄	#15690286	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	764	1	no	—
125	C₈	—	764	1	no	—
126	C₈	—	764	1	no	—
127	C₈	#15690277	1	2	yes	complemented
128	C₂²	#20259293	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	764	1	no	—
133	C₄	—	764	1	no	—
134	C₄	C₂×#15690277	1	1	yes	characteristic
135	C₄	#20259349	1	1	yes	characteristic
136	C₄	#20259351	1	1	yes	characteristic
137	C₄	#20259353	1	1	yes	characteristic
138	C₂	#27132870	1	1	yes	characteristic
139	C₂	#27132906	1	1	yes	characteristic
140	C₂	#27132923	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132955

C₂

yes

maximal, characteristic

#27132870

C₂

yes

maximal, characteristic

#27132793

C₂

yes

maximal, characteristic

#27132416

C₂

yes

maximal, characteristic

#27132412

C₂

yes

maximal, characteristic

#27132409

C₂

yes

maximal, characteristic

#498×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690277

—

C₄×Dic₃₈₂

C₂²

yes

characteristic

#20259255

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

characteristic

Dic₁₉₁⋊C₈

—

#20259246

—

#20259245

—

#20259232

—

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

#106×C₁₉₁

—

#106×C₁₉₁

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

D₄

yes

characteristic

#15690277

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

D₄

yes

characteristic

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×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

Q₁₆

yes

complemented

C₂²×C₁₉₁

#121

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×Q₁₆

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1692

—

191

maximal, Sylow

#498

D₁₉₁

yes

characteristic

#477

—

191

—

#462

—

191

—

#446

—

191

—

#335

—

191

—

#333

—

191

—

#332

—

191

—

C₂×Q₁₆

—

382

—

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

C₄×Q₈

—

191

—

#107

—

382

—

#106

—

#106

—

C₄⋊C₈

—

382

—

C₄⋊C₈

—

191

—

C₄⋊C₈

—

191

—

C₄⋊C₈

—

191

—

#103

—

382

—

#103

—

382

—

#97

—

382

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

191

—

Q₁₆

—

764

—

C₂×C₈

—

382

—

C₂×C₈

—

382

—

100

C₂×C₈

—

382

—

101

C₂×C₈

C₂²⋊D₁₉₁

yes

characteristic

102

C₂×C₈

C₂²⋊D₁₉₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

382

—

108

C₄⋊C₄

—

191

—

109

C₄²

—

382

—

110

C₄²

C₂²×D₁₉₁

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

382

—

115

Q₈

—

382

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

382

—

119

C₂×C₄

—

382

—

120

C₂×C₄

D₄×D₁₉₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

122

C₂×C₄

#15690286

yes

characteristic

123

C₈

—

124

C₈

—

764

—

125

C₈

—

764

—

126

C₈

—

764

—

127

C₈

#15690277

yes

complemented

128

C₂²

#20259293

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

764

—

133

C₄

—

764

—

134

C₄

C₂×#15690277

yes

characteristic

135

C₄

#20259349

yes

characteristic

136

C₄

#20259351

yes

characteristic

137

C₄

#20259353

yes

characteristic

138

C₂

#27132870

yes

characteristic

139

C₂

#27132906

yes

characteristic

140

C₂

#27132923

yes

characteristic

	subgroup	G/H
center	C₂²	#20259293
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#498×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	#121
hypercenter	#498	D₁₉₁
hypocenter	C₁₉₁	#1692
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#20259293

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#498×C₁₉₁

C₂

C₂²×C₁₉₁

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
56. C₈×C₁₉₁	Q₁₆	semidirect
70. C₁₉₁	#1692	semidirect
127. C₈	#15690277	semidirect

H ≅ G/N

type

56. C₈×C₁₉₁

Q₁₆

semidirect

70. C₁₉₁

#1692

semidirect

127. C₈

#15690277

semidirect

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

subgroup	index
#27132955	2
#27132870	2
#27132793	2
#27132416	2
#27132412	2
#27132409	2
#498×C₁₉₁	2
#1692	191

subgroup

index

#498×C₁₉₁

191

Subgroups — #37572023

Conjugacy classes of subgroups

Complements

Maximal subgroups