Group #37573435 — subgroups

Subgroups — #37573435

Order 24448.

276 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×Dic₇₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259258	C₂	1	1	yes	maximal, characteristic
3	#27132671	C₂	1	2	yes	maximal
4	C₂×#20259240	C₂	1	2	yes	maximal
5	#405×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₈₂	C₂²	1	2	yes	—
7	C₂×Dic₇₆₄	—	2	1	no	—
8	C₂×Dic₇₆₄	—	2	1	no	—
9	C₂×Dic₇₆₄	—	2	2	no	—
10	C₂×Dic₇₆₄	C₄	1	4	yes	complemented
11	C₂×#15690271	C₂²	1	2	yes	—
12	#20259258	—	2	2	no	—
13	#20259245	—	2	4	no	—
14	#20259240	—	2	4	no	—
15	C₂×M₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
17	#117×C₁₉₁	C₂²	1	1	yes	characteristic
18	#97×C₁₉₁	—	2	1	no	—
19	C₄×C₈×C₁₉₁	—	2	1	no	—
20	C₂×Dic₃₈₂	—	2	2	no	—
21	C₂×Dic₃₈₂	—	2	2	no	—
22	C₂×Dic₃₈₂	—	2	2	no	—
23	C₂×Dic₃₈₂	—	2	2	no	—
24	C₂×Dic₃₈₂	—	2	2	no	—
25	C₂×Dic₃₈₂	C₂×C₄	1	4	yes	—
26	C₂²×Dic₁₉₁	—	2	2	no	—
27	Dic₇₆₄	—	2	4	no	—
28	Dic₇₆₄	—	2	4	no	—
29	#15690271	—	2	2	no	—
30	#15690271	—	2	2	no	—
31	C₂×C₁₉₁⋊C₈	—	2	2	no	—
32	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₉₁	—	2	1	no	—
34	M₁₆×C₁₉₁	D₄	1	2	yes	—
35	C₂×C₈×C₁₉₁	—	2	1	no	—
36	C₂×C₈×C₁₉₁	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
40	C₄⋊C₄×C₁₉₁	D₄	1	2	yes	—
41	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
42	C₄²×C₁₉₁	—	2	1	no	—
43	Dic₃₈₂	—	2	4	no	—
44	Dic₃₈₂	—	2	4	no	—
45	Dic₃₈₂	—	4	2	no	—
46	Dic₃₈₂	—	4	2	no	—
47	C₂×Dic₁₉₁	—	2	4	no	—
48	C₂×Dic₁₉₁	—	4	2	no	—
49	C₂×Dic₁₉₁	—	4	2	no	—
50	C₁₉₁⋊C₈	—	4	2	no	—
51	C₂³×C₁₉₁	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₉₁	—	2	2	no	—
53	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
59	C₈×C₁₉₁	—	2	2	no	—
60	C₈×C₁₉₁	C₂²⋊C₄	1	2	yes	—
61	C₈×C₁₉₁	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₁₉₁	—	2	2	no	—
63	C₂²×C₁₉₁	C₄×D₄	1	1	yes	characteristic
64	C₂²×C₁₉₁	#121	1	1	yes	characteristic
65	C₂²×C₁₉₁	#124	1	1	yes	characteristic
66	C₄×C₁₉₁	—	4	2	no	—
67	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₄×C₁₉₁	C₄×D₄	1	1	yes	characteristic
69	C₄×C₁₉₁	#121	1	1	yes	characteristic
70	C₄×C₁₉₁	#127	1	1	yes	characteristic
71	Dic₁₉₁	—	4	4	no	—
72	C₂×C₁₉₁	—	2	1	no	—
73	C₂×C₁₉₁	—	2	1	no	—
74	C₂×C₁₉₁	#390	1	1	yes	characteristic
75	C₂×C₁₉₁	#473	1	2	yes	—
76	C₁₉₁	#1979	1	1	yes	characteristic, complemented, Sylow
77	#1979	—	191	1	no	maximal, Sylow
78	C₂²×Q₁₆	—	191	1	no	—
79	C₂×#108	—	191	1	no	—
80	#419	—	191	2	no	—
81	C₂×#101	—	191	2	no	—
82	#405	D₁₉₁	1	1	yes	characteristic
83	C₂²×Q₈	—	191	2	no	—
84	C₂×Q₁₆	—	382	1	no	—
85	C₂×Q₁₆	—	382	1	no	—
86	C₂×Q₁₆	—	382	2	no	—
87	C₂×Q₁₆	—	191	4	no	—
88	C₂×M₁₆	—	191	2	no	—
89	C₂×M₁₆	C₂×D₁₉₁	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
91	#117	C₂×D₁₉₁	1	1	yes	characteristic
92	#108	—	382	2	no	—
93	#103	—	382	4	no	—
94	#101	—	382	4	no	—
95	#97	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₂×Q₈	—	382	2	no	—
98	C₂×Q₈	—	382	2	no	—
99	C₂×Q₈	—	382	2	no	—
100	C₂×Q₈	—	382	2	no	—
101	C₂×Q₈	—	382	2	no	—
102	C₂×Q₈	—	191	4	no	—
103	C₂²×C₄	—	382	2	no	—
104	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
105	Q₁₆	—	382	4	no	—
106	Q₁₆	—	382	4	no	—
107	M₁₆	—	2	1	no	—
108	M₁₆	—	382	2	no	—
109	M₁₆	—	382	2	no	—
110	M₁₆	D₇₆₄	1	2	yes	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	382	2	no	—
113	C₂×C₈	C₄×D₁₉₁	1	2	yes	—
114	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
116	C₂×C₈	D₇₆₄	1	2	yes	—
117	C₄⋊C₄	D₇₆₄	1	2	yes	—
118	C₂²⋊C₄	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₂³	#15690287	1	1	yes	characteristic
121	Q₈	—	382	4	no	—
122	Q₈	—	382	4	no	—
123	Q₈	—	764	2	no	—
124	Q₈	—	764	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	382	4	no	—
127	C₂×C₄	—	764	2	no	—
128	C₂×C₄	—	764	2	no	—
129	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
131	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
132	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
133	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
134	C₂×C₄	#15690287	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	764	2	no	—
137	C₈	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
138	C₈	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	C₄×D₇₆₄	1	1	yes	characteristic
141	C₂²	#20259296	1	1	yes	characteristic
142	C₂²	#20259310	1	1	yes	characteristic
143	C₄	—	4	2	no	—
144	C₄	—	764	4	no	—
145	C₄	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
146	C₄	C₄×D₇₆₄	1	1	yes	characteristic
147	C₄	#20259296	1	1	yes	characteristic
148	C₄	#20259319	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#27132565	1	1	yes	characteristic
152	C₂	#27132938	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₆₄

C₂

yes

maximal, characteristic

C₂×#20259258

C₂

yes

maximal, characteristic

#27132671

C₂

yes

maximal

C₂×#20259240

C₂

yes

maximal

#405×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₈₂

C₂²

yes

—

C₂×Dic₇₆₄

—

C₂×Dic₇₆₄

—

C₂×Dic₇₆₄

—

C₂×Dic₇₆₄

C₄

yes

complemented

C₂×#15690271

C₂²

yes

—

#20259258

—

#20259245

—

#20259240

—

C₂×M₁₆×C₁₉₁

C₂²

yes

characteristic

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

#117×C₁₉₁

C₂²

yes

characteristic

#97×C₁₉₁

—

C₄×C₈×C₁₉₁

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

C₂×C₄

yes

—

C₂²×Dic₁₉₁

—

Dic₇₆₄

—

Dic₇₆₄

—

#15690271

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

D₄

yes

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

—

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

D₄

yes

—

C₄⋊C₄×C₁₉₁

D₄

yes

—

C₂²⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

#41

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

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⋊C₄

yes

—

C₈×C₁₉₁

C₂²⋊C₄

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₄×D₄

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₉₁

C₄×D₄

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#390

yes

characteristic

C₂×C₁₉₁

#473

yes

—

C₁₉₁

#1979

yes

characteristic, complemented, Sylow

#1979

—

191

maximal, Sylow

C₂²×Q₁₆

—

191

—

C₂×#108

—

191

—

#419

—

191

—

C₂×#101

—

191

—

#405

D₁₉₁

yes

characteristic

C₂²×Q₈

—

191

—

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

191

—

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

characteristic

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

#117

C₂×D₁₉₁

yes

characteristic

#108

—

382

—

#103

—

382

—

#101

—

382

—

#97

—

C₄×C₈

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

100

C₂×Q₈

—

382

—

101

C₂×Q₈

—

382

—

102

C₂×Q₈

—

191

—

103

C₂²×C₄

—

382

—

104

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

105

Q₁₆

—

382

—

106

Q₁₆

—

382

—

107

M₁₆

—

108

M₁₆

—

382

—

109

M₁₆

—

382

—

110

M₁₆

D₇₆₄

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

382

—

113

C₂×C₈

C₄×D₁₉₁

yes

—

114

C₂×C₈

C₂²⋊D₁₉₁

yes

characteristic

115

C₂×C₈

C₂²⋊D₁₉₁

yes

characteristic

116

C₂×C₈

D₇₆₄

yes

—

117

C₄⋊C₄

D₇₆₄

yes

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂³

#15690287

yes

characteristic

121

Q₈

—

382

—

122

Q₈

—

382

—

123

Q₈

—

764

—

124

Q₈

—

764

—

125

C₂×C₄

—

126

C₂×C₄

—

382

—

127

C₂×C₄

—

764

—

128

C₂×C₄

—

764

—

129

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

131

C₂×C₄

C₂×D₇₆₄

yes

characteristic

132

C₂×C₄

C₂×D₇₆₄

yes

characteristic

133

C₂×C₄

C₂×D₇₆₄

yes

characteristic

134

C₂×C₄

#15690287

yes

characteristic

135

C₈

—

136

C₈

—

764

—

137

C₈

(C₂×D₁₉₁)⋊C₄

yes

—

138

C₈

(C₂×D₁₉₁)⋊C₄

yes

—

139

C₂²

—

140

C₂²

C₄×D₇₆₄

yes

characteristic

141

C₂²

#20259296

yes

characteristic

142

C₂²

#20259310

yes

characteristic

143

C₄

—

144

C₄

—

764

—

145

C₄

C₂×(C₂×D₁₉₁)⋊C₄

yes

characteristic

146

C₄

C₄×D₇₆₄

yes

characteristic

147

C₄

#20259296

yes

characteristic

148

C₄

#20259319

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#27132565

yes

characteristic

152

C₂

#27132938

yes

—

	subgroup	G/H
center	C₂²	#20259296
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#405×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#121
prime radical	C₂³×C₁₉₁	#41
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	#405	D₁₉₁
hypocenter	C₁₉₁	#1979
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#20259296

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#405×C₁₉₁

C₂

C₂²×C₁₉₁

C₂³×C₁₉₁

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
10. C₂×Dic₇₆₄	C₄	semidirect
76. C₁₉₁	#1979	semidirect

H ≅ G/N

type

10. C₂×Dic₇₆₄

C₄

semidirect

76. C₁₉₁

#1979

semidirect

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

subgroup	index
C₂²×Dic₇₆₄	2
C₂×#20259258	2
#27132671	2
C₂×#20259240	2
#405×C₁₉₁	2
#1979	191

subgroup

index

C₂²×Dic₇₆₄

#405×C₁₉₁

191

Subgroups — #37573435

Conjugacy classes of subgroups

Complements

Maximal subgroups