Group #37571889 — subgroups

Subgroups — #37571889

Order 24448.

156 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 53 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	#27133042	C₂	1	1	yes	maximal, characteristic
2	#27132949	C₂	1	1	yes	maximal, characteristic
3	#27132945	C₂	1	1	yes	maximal, characteristic
4	#27132787	C₂	1	1	yes	maximal, characteristic
5	#27132415	C₂	1	1	yes	maximal, characteristic
6	#27132410	C₂	1	1	yes	maximal, characteristic
7	#332×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#15690277	—	2	1	no	—
9	C₂×#15690277	—	2	1	no	—
10	#20259322	C₂²	1	1	yes	characteristic
11	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
12	#20259254	—	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	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₁₉₁	—	2	1	no	—
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	#15690277	—	4	2	no	—
26	#15690277	—	4	2	no	—
27	C₂×C₁₉₁⋊C₈	—	2	1	no	—
28	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
29	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
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×Q₈×C₁₉₁	—	2	1	no	—
36	C₂×C₈×C₁₉₁	—	2	1	no	—
37	C₂×C₈×C₁₉₁	—	2	1	no	—
38	C₄⋊C₄×C₁₉₁	—	2	1	no	—
39	C₄⋊C₄×C₁₉₁	—	2	1	no	—
40	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
41	Dic₃₈₂	—	2	1	no	—
42	Dic₃₈₂	—	4	1	no	—
43	Dic₃₈₂	—	4	1	no	—
44	Dic₃₈₂	Q₁₆	1	2	yes	complemented
45	C₂×Dic₁₉₁	—	2	1	no	—
46	C₂×Dic₁₉₁	—	2	1	no	—
47	C₂×Dic₁₉₁	—	4	1	no	—
48	C₁₉₁⋊C₈	—	2	1	no	—
49	C₁₉₁⋊C₈	—	4	1	no	—
50	C₁₉₁⋊C₈	SD₁₆	1	2	yes	—
51	Q₈×C₁₉₁	—	4	1	no	—
52	Q₈×C₁₉₁	—	4	1	no	—
53	C₂×C₄×C₁₉₁	—	4	1	no	—
54	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
57	C₈×C₁₉₁	—	4	1	no	—
58	C₈×C₁₉₁	—	4	1	no	—
59	C₂²×C₁₉₁	#121	1	1	yes	characteristic
60	C₄×C₁₉₁	—	2	1	no	—
61	C₄×C₁₉₁	—	8	1	no	—
62	C₄×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₁₉₁	#135	1	1	yes	characteristic
65	C₄×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
66	Dic₁₉₁	—	2	2	no	—
67	Dic₁₉₁	—	4	1	no	—
68	Dic₁₉₁	—	8	1	no	—
69	C₂×C₁₉₁	#460	1	1	yes	characteristic
70	C₂×C₁₉₁	#467	1	1	yes	characteristic
71	C₂×C₁₉₁	#473	1	1	yes	characteristic
72	C₁₉₁	#1670	1	1	yes	characteristic, complemented, Sylow
73	#1670	—	191	1	no	maximal, Sylow
74	#494	—	191	1	no	—
75	#475	—	191	1	no	—
76	#475	—	191	1	no	—
77	C₈×Q₈	—	191	1	no	—
78	#335	—	191	1	no	—
79	#332	—	191	1	no	—
80	#332	D₁₉₁	1	1	yes	characteristic
81	C₂×Q₁₆	—	382	1	no	—
82	C₂×Q₁₆	—	382	1	no	—
83	C₄⋊Q₈	—	191	1	no	—
84	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
85	C₄×Q₈	—	191	1	no	—
86	#106	—	382	1	no	—
87	#106	—	382	1	no	—
88	C₄⋊C₈	—	382	1	no	—
89	C₄⋊C₈	—	191	1	no	—
90	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
91	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
92	#103	—	382	1	no	—
93	#103	—	382	1	no	—
94	C₄×C₈	—	382	1	no	—
95	C₄×C₈	—	191	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	382	1	no	—
98	C₂×Q₈	—	191	1	no	—
99	Q₁₆	—	764	2	no	—
100	Q₁₆	—	764	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	382	1	no	—
104	C₂×C₈	—	191	1	no	—
105	C₂×C₈	—	191	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	382	1	no	—
109	C₄⋊C₄	—	382	1	no	—
110	C₄⋊C₄	—	382	1	no	—
111	C₄⋊C₄	—	191	1	no	—
112	C₄²	—	382	1	no	—
113	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	382	1	no	—
117	Q₈	—	191	2	no	—
118	Q₈	—	764	1	no	—
119	Q₈	—	764	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	382	1	no	—
122	C₂×C₄	—	382	1	no	—
123	C₂×C₄	—	764	1	no	—
124	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
125	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
126	C₂×C₄	#15690287	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	382	1	no	—
130	C₈	—	191	2	no	—
131	C₈	—	764	1	no	—
132	C₂²	#20259295	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	382	2	no	—
136	C₄	—	1528	1	no	—
137	C₄	—	764	1	no	—
138	C₄	SD₁₆×D₁₉₁	1	1	yes	characteristic
139	C₄	Q₁₆×D₁₉₁	1	1	yes	characteristic
140	C₄	#20259343	1	1	yes	characteristic
141	C₄	#20259346	1	1	yes	characteristic
142	C₂	#27132862	1	1	yes	characteristic
143	C₂	#27132903	1	1	yes	characteristic
144	C₂	#27132937	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133042

C₂

yes

maximal, characteristic

#27132949

C₂

yes

maximal, characteristic

#27132945

C₂

yes

maximal, characteristic

#27132787

C₂

yes

maximal, characteristic

#27132415

C₂

yes

maximal, characteristic

#27132410

C₂

yes

maximal, characteristic

#332×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690277

—

C₂×#15690277

—

#20259322

C₂²

yes

characteristic

C₄×Dic₃₈₂

C₂²

yes

characteristic

#20259254

—

#20259253

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

characteristic

Dic₁₉₁⋊C₈

—

#20259247

—

#20259245

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₉₁

—

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

#15690277

—

#15690277

—

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

Q₁₆

yes

complemented

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

SD₁₆

yes

—

Q₈×C₁₉₁

—

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×SD₁₆

yes

characteristic

C₄×C₁₉₁

C₂×Q₁₆

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1670

—

191

maximal, Sylow

#494

—

191

—

#475

—

191

—

#475

—

191

—

C₈×Q₈

—

191

—

#335

—

191

—

#332

—

191

—

#332

D₁₉₁

yes

characteristic

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

382

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

C₄×Q₈

—

191

—

#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

—

C₄×C₈

—

382

—

C₄×C₈

—

191

—

C₂×Q₈

—

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

Q₁₆

—

764

—

100

Q₁₆

—

764

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

382

—

104

C₂×C₈

—

191

—

105

C₂×C₈

—

191

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

382

—

109

C₄⋊C₄

—

382

—

110

C₄⋊C₄

—

382

—

111

C₄⋊C₄

—

191

—

112

C₄²

—

382

—

113

C₄²

C₂²×D₁₉₁

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

382

—

117

Q₈

—

191

—

118

Q₈

—

764

—

119

Q₈

—

764

—

120

C₂×C₄

—

121

C₂×C₄

—

382

—

122

C₂×C₄

—

382

—

123

C₂×C₄

—

764

—

124

C₂×C₄

D₄×D₁₉₁

yes

characteristic

125

C₂×C₄

D₄×D₁₉₁

yes

characteristic

126

C₂×C₄

#15690287

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

382

—

130

C₈

—

191

—

131

C₈

—

764

—

132

C₂²

#20259295

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

382

—

136

C₄

—

1528

—

137

C₄

—

764

—

138

C₄

SD₁₆×D₁₉₁

yes

characteristic

139

C₄

Q₁₆×D₁₉₁

yes

characteristic

140

C₄

#20259343

yes

characteristic

141

C₄

#20259346

yes

characteristic

142

C₂

#27132862

yes

characteristic

143

C₂

#27132903

yes

characteristic

144

C₂

#27132937

yes

characteristic

	subgroup	G/H
center	C₂²	#20259295
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#332×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	#121
hypercenter	#332	D₁₉₁
hypocenter	C₁₉₁	#1670
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#20259295

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#332×C₁₉₁

C₂

C₂²×C₁₉₁

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
44. Dic₃₈₂	Q₁₆	semidirect
72. C₁₉₁	#1670	semidirect

H ≅ G/N

type

44. Dic₃₈₂

Q₁₆

semidirect

72. C₁₉₁

#1670

semidirect

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

subgroup	index
#27133042	2
#27132949	2
#27132945	2
#27132787	2
#27132415	2
#27132410	2
#332×C₁₉₁	2
#1670	191

subgroup

index

#332×C₁₉₁

191

Subgroups — #37571889

Conjugacy classes of subgroups

Complements

Maximal subgroups