Group #37583180 — subgroups

Subgroups — #37583180

Order 24448.

338 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 117 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	#27133276	C₂	1	2	yes	maximal
2	#27133043	C₂	1	1	yes	maximal, characteristic, complemented
3	#27133033	C₂	1	2	yes	maximal, complemented
4	#27132922	C₂	1	2	yes	maximal
5	#27132905	C₂	1	2	yes	maximal
6	#27132769	C₂	1	2	yes	maximal, complemented
7	#27132757	C₂	1	2	yes	maximal, complemented
8	#27132617	C₂	1	1	yes	maximal, characteristic
9	#499×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₇₆₄	C₂²	1	2	yes	—
11	C₂×Dic₇₆₄	C₂²	1	2	yes	—
12	C₂×#15690270	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
14	#20259322	C₂²	1	2	yes	—
15	#20259317	—	2	4	no	—
16	#20259313	—	2	4	no	—
17	#20259302	C₂²	1	4	yes	—
18	#20259287	C₂²	1	4	yes	—
19	#20259279	C₂²	1	1	yes	characteristic
20	#20259245	C₂²	1	4	yes	—
21	#20259245	C₂²	1	4	yes	—
22	#20259233	C₂²	1	1	yes	characteristic
23	#20259232	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
25	#125×C₁₉₁	C₂²	1	2	yes	—
26	#107×C₁₉₁	C₂²	1	2	yes	—
27	#106×C₁₉₁	C₂²	1	2	yes	—
28	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
29	C₂×Dic₃₈₂	C₂³	1	4	yes	—
30	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
31	Dic₇₆₄	—	2	4	no	—
32	Dic₇₆₄	—	2	4	no	—
33	#15690270	—	2	2	no	—
34	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
35	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
36	C₈×D₁₉₁	—	2	2	no	—
37	C₄⋊Dic₁₉₁	—	2	4	no	—
38	C₄⋊Dic₁₉₁	—	2	4	no	—
39	Dic₁₉₁⋊C₄	—	2	4	no	—
40	Dic₁₉₁⋊C₄	D₄	1	2	yes	—
41	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
42	(C₂×D₁₉₁)⋊C₄	D₄	1	2	yes	—
43	C₄×Dic₁₉₁	—	2	4	no	—
44	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₉₁	—	2	4	no	—
48	C₄⋊C₄×C₁₉₁	C₂³	1	4	yes	—
49	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
50	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
51	Dic₃₈₂	—	2	4	no	—
52	Dic₃₈₂	—	2	4	no	—
53	C₂×Dic₁₉₁	—	2	4	no	—
54	C₂×Dic₁₉₁	C₂×D₄	1	2	yes	—
55	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
56	C₄×D₁₉₁	—	2	1	no	—
57	C₄×D₁₉₁	—	2	1	no	—
58	C₁₉₁⋊C₈	—	2	1	no	—
59	C₁₉₁⋊C₈	—	2	1	no	—
60	C₂×C₄×C₁₉₁	—	2	4	no	—
61	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
63	C₈×C₁₉₁	#41	1	2	yes	—
64	C₈×C₁₉₁	#41	1	2	yes	—
65	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₁₉₁	—	2	1	no	—
67	C₂×D₁₉₁	—	2	2	no	—
68	C₄×C₁₉₁	—	2	2	no	—
69	C₄×C₁₉₁	—	4	4	no	—
70	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
71	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
72	Dic₁₉₁	—	2	1	no	—
73	Dic₁₉₁	—	2	2	no	—
74	Dic₁₉₁	—	4	4	no	—
75	C₂×C₁₉₁	#532	1	1	yes	characteristic
76	C₂×C₁₉₁	#578	1	2	yes	—
77	D₁₉₁	—	4	1	no	—
78	C₁₉₁	#3177	1	1	yes	characteristic, complemented, Sylow
79	#3177	—	191	1	no	maximal, Sylow
80	#541	—	191	2	no	—
81	#499	D₁₉₁	1	1	yes	characteristic, complemented
82	#494	—	191	1	no	—
83	#491	—	191	2	no	—
84	#470	—	191	2	no	—
85	#467	—	191	2	no	—
86	#441	—	191	2	no	—
87	C₄×Q₁₆	—	191	2	no	—
88	#405	—	191	1	no	—
89	C₂×Q₁₆	—	191	2	no	—
90	C₂×Q₁₆	—	191	2	no	—
91	C₂×M₁₆	—	191	1	no	—
92	C₂²×C₈	—	191	1	no	—
93	C₄⋊Q₈	—	191	2	no	—
94	#126	—	382	4	no	—
95	#125	—	382	4	no	—
96	#125	C₂×D₁₉₁	1	2	yes	—
97	C₂²⋊Q₈	—	191	4	no	—
98	C₄×Q₈	—	191	4	no	—
99	#117	—	191	1	no	—
100	#107	C₂×D₁₉₁	1	2	yes	—
101	#106	C₂×D₁₉₁	1	2	yes	—
102	#103	—	191	4	no	—
103	#103	—	191	4	no	—
104	#97	—	191	1	no	—
105	#97	—	191	1	no	—
106	C₄×C₈	—	191	1	no	—
107	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
108	C₂×Q₈	—	191	4	no	—
109	C₂²×C₄	—	191	1	no	—
110	Q₁₆	—	382	4	no	—
111	Q₁₆	—	382	4	no	—
112	M₁₆	—	382	2	no	—
113	C₂×C₈	—	382	2	no	—
114	C₂×C₈	—	191	1	no	—
115	C₂×C₈	—	191	1	no	—
116	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	382	4	no	—
120	C₄⋊C₄	—	382	4	no	—
121	C₄⋊C₄	—	382	4	no	—
122	C₄⋊C₄	—	191	2	no	—
123	C₄⋊C₄	C₂²×D₁₉₁	1	4	yes	—
124	C₂²⋊C₄	—	382	4	no	—
125	C₂²⋊C₄	—	191	2	no	—
126	C₄²	—	382	4	no	—
127	C₄²	—	191	1	no	—
128	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
129	C₂³	—	191	1	no	—
130	Q₈	—	382	4	no	—
131	Q₈	—	382	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	382	1	no	—
134	C₂×C₄	—	382	1	no	—
135	C₂×C₄	—	382	4	no	—
136	C₂×C₄	—	191	2	no	—
137	C₂×C₄	—	191	1	no	—
138	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
139	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
140	C₈	—	382	1	no	—
141	C₈	—	382	1	no	—
142	C₈	#15690288	1	2	yes	—
143	C₈	#15690288	1	2	yes	—
144	C₂²	—	382	1	no	—
145	C₂²	—	382	2	no	—
146	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	4	no	—
149	C₄	—	382	1	no	—
150	C₄	—	382	2	no	—
151	C₄	—	764	4	no	—
152	C₄	C₂×#15690288	1	1	yes	characteristic
153	C₄	C₂×#15690288	1	1	yes	characteristic
154	C₂	—	764	1	no	—
155	C₂	#27133214	1	1	yes	characteristic
156	C₂	#27133477	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133276

C₂

yes

maximal

#27133043

C₂

yes

maximal, characteristic, complemented

#27133033

C₂

yes

maximal, complemented

#27132922

C₂

yes

maximal

#27132905

C₂

yes

maximal

#27132769

C₂

yes

maximal, complemented

#27132757

C₂

yes

maximal, complemented

#27132617

C₂

yes

maximal, characteristic

#499×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₆₄

C₂²

yes

—

C₂×Dic₇₆₄

C₂²

yes

—

C₂×#15690270

C₂²

yes

characteristic

C₂×C₈×D₁₉₁

C₂²

yes

characteristic

#20259322

C₂²

yes

—

#20259317

—

#20259313

—

#20259302

C₂²

yes

—

#20259287

C₂²

yes

—

#20259279

C₂²

yes

characteristic

#20259245

C₂²

yes

—

#20259245

C₂²

yes

—

#20259233

C₂²

yes

characteristic

#20259232

C₂²

yes

characteristic

C₈×Dic₁₉₁

C₂²

yes

characteristic

#125×C₁₉₁

C₂²

yes

—

#107×C₁₉₁

C₂²

yes

—

#106×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

C₂³

yes

—

C₂×C₄×D₁₉₁

C₂³

yes

characteristic

Dic₇₆₄

—

Dic₇₆₄

—

#15690270

—

C₂×C₁₉₁⋊C₈

C₂³

yes

characteristic

C₂×C₁₉₁⋊C₈

C₂³

yes

characteristic

C₈×D₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

D₄

yes

—

(C₂×D₁₉₁)⋊C₄

—

(C₂×D₁₉₁)⋊C₄

D₄

yes

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

C₂³

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

C₂×D₄

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂×D₄

yes

—

C₂×Dic₁₉₁

C₂×D₄

yes

characteristic

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

#41

yes

—

C₈×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×#41

yes

characteristic

C₄×C₁₉₁

C₂×#41

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

#532

yes

characteristic

C₂×C₁₉₁

#578

yes

—

D₁₉₁

—

C₁₉₁

#3177

yes

characteristic, complemented, Sylow

#3177

—

191

maximal, Sylow

#541

—

191

—

#499

D₁₉₁

yes

characteristic, complemented

#494

—

191

—

#491

—

191

—

#470

—

191

—

#467

—

191

—

#441

—

191

—

C₄×Q₁₆

—

191

—

#405

—

191

—

C₂×Q₁₆

—

191

—

C₂×Q₁₆

—

191

—

C₂×M₁₆

—

191

—

C₂²×C₈

—

191

—

C₄⋊Q₈

—

191

—

#126

—

382

—

#125

—

382

—

#125

C₂×D₁₉₁

yes

—

C₂²⋊Q₈

—

191

—

C₄×Q₈

—

191

—

#117

—

191

—

100

#107

C₂×D₁₉₁

yes

—

101

#106

C₂×D₁₉₁

yes

—

102

#103

—

191

—

103

#103

—

191

—

104

#97

—

191

—

105

#97

—

191

—

106

C₄×C₈

—

191

—

107

C₄×C₈

C₂×D₁₉₁

yes

characteristic

108

C₂×Q₈

—

191

—

109

C₂²×C₄

—

191

—

110

Q₁₆

—

382

—

111

Q₁₆

—

382

—

112

M₁₆

—

382

—

113

C₂×C₈

—

382

—

114

C₂×C₈

—

191

—

115

C₂×C₈

—

191

—

116

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

117

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

382

—

120

C₄⋊C₄

—

382

—

121

C₄⋊C₄

—

382

—

122

C₄⋊C₄

—

191

—

123

C₄⋊C₄

C₂²×D₁₉₁

yes

—

124

C₂²⋊C₄

—

382

—

125

C₂²⋊C₄

—

191

—

126

C₄²

—

382

—

127

C₄²

—

191

—

128

C₄²

C₂²×D₁₉₁

yes

characteristic

129

C₂³

—

191

—

130

Q₈

—

382

—

131

Q₈

—

382

—

132

C₂×C₄

—

133

C₂×C₄

—

382

—

134

C₂×C₄

—

382

—

135

C₂×C₄

—

382

—

136

C₂×C₄

—

191

—

137

C₂×C₄

—

191

—

138

C₂×C₄

D₄×D₁₉₁

yes

—

139

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

140

C₈

—

382

—

141

C₈

—

382

—

142

C₈

#15690288

yes

—

143

C₈

#15690288

yes

—

144

C₂²

—

382

—

145

C₂²

—

382

—

146

C₂²

C₂×D₄×D₁₉₁

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

382

—

150

C₄

—

382

—

151

C₄

—

764

—

152

C₄

C₂×#15690288

yes

characteristic

153

C₄

C₂×#15690288

yes

characteristic

154

C₂

—

764

—

155

C₂

#27133214

yes

characteristic

156

C₂

#27133477

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#499×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂²×D₁₉₁	C₂×D₄
hypercenter	#499	D₁₉₁
hypocenter	C₁₉₁	#3177
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₁₉₁

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#499×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

prime radical

C₂²×D₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
2. #27133043	C₂	semidirect
3. #27133033	C₂	semidirect
6. #27132769	C₂	semidirect
7. #27132757	C₂	semidirect
9. #499×C₁₉₁	C₂	semidirect
78. C₁₉₁	#3177	semidirect
81. #499	D₁₉₁	semidirect

H ≅ G/N

type

2. #27133043

C₂

semidirect

3. #27133033

C₂

semidirect

6. #27132769

C₂

semidirect

7. #27132757

C₂

semidirect

9. #499×C₁₉₁

C₂

semidirect

78. C₁₉₁

#3177

semidirect

81. #499

D₁₉₁

semidirect

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

subgroup	index
#27133276	2
#27133043	2
#27133033	2
#27132922	2
#27132905	2
#27132769	2
#27132757	2
#27132617	2
#499×C₁₉₁	2
#3177	191

subgroup

index

#499×C₁₉₁

191

Subgroups — #37583180

Conjugacy classes of subgroups

Complements

Maximal subgroups