Group #37582176 — subgroups

Subgroups — #37582176

Order 24448.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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, complemented
2	C₂×#20259328	C₂	1	1	yes	maximal, characteristic
3	#27133239	C₂	1	2	yes	maximal
4	#27132922	C₂	1	4	yes	maximal, complemented
5	#27132905	C₂	1	4	yes	maximal, complemented
6	#27132671	C₂	1	2	yes	maximal, complemented
7	#427×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15690287	C₂²	1	1	yes	characteristic
9	C₂²×Dic₃₈₂	C₂²	1	2	yes	—
10	C₂×Dic₇₆₄	—	2	4	no	—
11	C₂×Dic₇₆₄	C₂²	1	2	yes	—
12	C₂×Dic₇₆₄	C₂²	1	2	yes	—
13	#20259328	—	2	4	no	—
14	C₂×#15690271	C₂²	1	1	yes	characteristic
15	C₂×#15690270	C₂²	1	2	yes	—
16	C₂×C₈×D₁₉₁	C₂²	1	2	yes	—
17	#20259322	—	2	2	no	—
18	#20259310	—	2	2	no	—
19	#20259306	—	2	4	no	—
20	#20259302	C₂²	1	8	yes	—
21	#20259245	C₂²	1	8	yes	—
22	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
23	#117×C₁₉₁	C₂²	1	2	yes	—
24	#107×C₁₉₁	C₂²	1	2	yes	—
25	#106×C₁₉₁	C₂²	1	2	yes	—
26	#15690287	—	2	2	no	—
27	#15690287	—	2	2	no	—
28	C₂×Dic₃₈₂	—	2	2	no	—
29	C₂×Dic₃₈₂	—	2	4	no	—
30	C₂×Dic₃₈₂	—	2	4	no	—
31	C₂×Dic₃₈₂	D₄	1	1	yes	characteristic
32	C₂×Dic₃₈₂	C₂³	1	4	yes	—
33	C₂×D₇₆₄	D₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₉₁	D₄	1	2	yes	—
35	C₂²×Dic₁₉₁	—	2	2	no	—
36	C₂×C₄×D₁₉₁	C₂³	1	2	yes	—
37	Dic₇₆₄	—	2	4	no	—
38	Dic₇₆₄	—	2	4	no	—
39	#15690271	—	2	2	no	—
40	#15690270	—	2	4	no	—
41	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
42	C₈×D₁₉₁	—	2	4	no	—
43	C₄⋊Dic₁₉₁	—	2	8	no	—
44	(C₂×D₁₉₁)⋊C₄	—	2	8	no	—
45	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₁₉₁	D₄	1	4	yes	—
47	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₁₉₁	C₂³	1	4	yes	—
50	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
51	C₄²×C₁₉₁	—	2	2	no	—
52	C₂²×D₁₉₁	C₂×D₄	1	2	yes	—
53	Dic₃₈₂	—	2	2	no	—
54	Dic₃₈₂	—	2	4	no	—
55	Dic₃₈₂	—	2	4	no	—
56	Dic₃₈₂	—	4	4	no	—
57	D₇₆₄	—	2	2	no	—
58	C₂²⋊D₁₉₁	—	2	4	no	—
59	C₂×Dic₁₉₁	—	2	4	no	—
60	C₂×Dic₁₉₁	—	4	4	no	—
61	C₂×Dic₁₉₁	C₂×D₄	1	2	yes	—
62	C₄×D₁₉₁	—	2	2	no	—
63	C₄×D₁₉₁	—	2	2	no	—
64	C₁₉₁⋊C₈	—	2	2	no	—
65	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₉₁	—	2	4	no	—
67	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₁₉₁	#41	1	2	yes	—
70	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
71	C₈×C₁₉₁	C₂×D₄	1	2	yes	—
72	C₈×C₁₉₁	C₂×D₄	1	2	yes	—
73	C₂²×C₁₉₁	—	2	2	no	—
74	C₂²×C₁₉₁	#121	1	2	yes	—
75	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
76	C₂×D₁₉₁	—	2	2	no	—
77	C₂×D₁₉₁	—	2	4	no	—
78	C₄×C₁₉₁	—	4	4	no	—
79	C₄×C₁₉₁	#121	1	2	yes	—
80	C₄×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
81	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
82	Dic₁₉₁	—	2	2	no	—
83	Dic₁₉₁	—	4	4	no	—
84	C₂×C₁₉₁	—	2	2	no	—
85	C₂×C₁₉₁	C₂×#121	1	1	yes	characteristic
86	C₂×C₁₉₁	#578	1	2	yes	—
87	D₁₉₁	—	4	2	no	—
88	C₁₉₁	#3059	1	1	yes	characteristic, complemented, Sylow
89	#3059	—	191	1	no	maximal, Sylow
90	C₂²×Q₁₆	—	191	1	no	—
91	C₂×#131	—	191	1	no	—
92	#536	—	191	2	no	—
93	#470	—	191	4	no	—
94	#467	—	191	4	no	—
95	#427	D₁₉₁	1	1	yes	characteristic, complemented
96	#419	—	191	2	no	—
97	C₂×#41	—	191	1	no	—
98	C₂²×Q₈	—	191	2	no	—
99	C₂×Q₁₆	—	382	4	no	—
100	C₂×Q₁₆	—	191	2	no	—
101	C₂×Q₁₆	—	191	2	no	—
102	#131	—	382	4	no	—
103	C₂×M₁₆	—	191	1	no	—
104	C₂×M₁₆	—	191	2	no	—
105	C₂²×C₈	—	191	2	no	—
106	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
107	C₄⋊Q₈	—	382	2	no	—
108	#124	—	382	2	no	—
109	#123	—	382	4	no	—
110	C₂²⋊Q₈	—	191	8	no	—
111	#117	C₂×D₁₉₁	1	2	yes	—
112	#107	C₂×D₁₉₁	1	2	yes	—
113	#106	C₂×D₁₉₁	1	2	yes	—
114	#103	—	191	8	no	—
115	#41	—	382	2	no	—
116	#41	—	382	2	no	—
117	C₂×Q₈	—	382	2	no	—
118	C₂×Q₈	—	382	4	no	—
119	C₂×Q₈	—	382	4	no	—
120	C₂×Q₈	—	191	4	no	—
121	C₂×Q₈	—	191	1	no	—
122	C₂×D₄	—	191	2	no	—
123	C₂×D₄	—	191	1	no	—
124	C₂²×C₄	—	382	2	no	—
125	C₂²×C₄	—	191	2	no	—
126	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
127	Q₁₆	—	382	4	no	—
128	Q₁₆	—	382	4	no	—
129	M₁₆	—	382	2	no	—
130	M₁₆	—	382	4	no	—
131	C₂×C₈	—	382	4	no	—
132	C₂×C₈	—	191	2	no	—
133	C₂×C₈	D₇₆₄	1	4	yes	—
134	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
136	C₄⋊C₄	—	382	8	no	—
137	C₄⋊C₄	C₂²×D₁₉₁	1	4	yes	—
138	C₂²⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	382	8	no	—
140	C₄²	—	2	2	no	—
141	C₂³	—	191	2	no	—
142	C₂³	D₄×D₁₉₁	1	1	yes	characteristic
143	Q₈	—	382	2	no	—
144	Q₈	—	382	4	no	—
145	Q₈	—	382	4	no	—
146	Q₈	—	764	4	no	—
147	D₄	—	382	4	no	—
148	D₄	—	382	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	382	2	no	—
151	C₂×C₄	—	382	2	no	—
152	C₂×C₄	—	382	4	no	—
153	C₂×C₄	—	191	2	no	—
154	C₂×C₄	—	764	4	no	—
155	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
156	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
157	C₂×C₄	#15690288	1	2	yes	—
158	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
159	C₈	—	382	2	no	—
160	C₈	C₂×D₇₆₄	1	2	yes	—
161	C₈	C₂×D₇₆₄	1	2	yes	—
162	C₂²	—	2	2	no	—
163	C₂²	—	382	4	no	—
164	C₂²	—	382	2	no	—
165	C₂²	#20259297	1	2	yes	—
166	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
167	C₄	—	4	4	no	—
168	C₄	—	382	2	no	—
169	C₄	—	764	4	no	—
170	C₄	#20259297	1	2	yes	—
171	C₄	C₂²×D₇₆₄	1	1	yes	characteristic
172	C₄	C₂×#15690288	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	—	764	2	no	—
175	C₂	C₂×#20259297	1	1	yes	characteristic
176	C₂	#27133477	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₇₆₄

C₂

yes

maximal, characteristic, complemented

C₂×#20259328

C₂

yes

maximal, characteristic

#27133239

C₂

yes

maximal

#27132922

C₂

yes

maximal, complemented

#27132905

C₂

yes

maximal, complemented

#27132671

C₂

yes

maximal, complemented

#427×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690287

C₂²

yes

characteristic

C₂²×Dic₃₈₂

C₂²

yes

—

C₂×Dic₇₆₄

—

C₂×Dic₇₆₄

C₂²

yes

—

C₂×Dic₇₆₄

C₂²

yes

—

#20259328

—

C₂×#15690271

C₂²

yes

characteristic

C₂×#15690270

C₂²

yes

—

C₂×C₈×D₁₉₁

C₂²

yes

—

—

—

—

C₂²

yes

—

#20259245

C₂²

yes

—

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

#117×C₁₉₁

C₂²

yes

—

#107×C₁₉₁

C₂²

yes

—

#106×C₁₉₁

C₂²

yes

—

#15690287

—

#15690287

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

D₄

yes

characteristic

C₂×Dic₃₈₂

C₂³

yes

—

C₂×D₇₆₄

D₄

yes

characteristic

C₂×C₂²⋊D₁₉₁

D₄

yes

—

C₂²×Dic₁₉₁

—

C₂×C₄×D₁₉₁

C₂³

yes

—

Dic₇₆₄

—

Dic₇₆₄

—

#15690271

—

#15690270

—

C₂×C₁₉₁⋊C₈

C₂³

yes

—

C₈×D₁₉₁

—

C₄⋊Dic₁₉₁

—

(C₂×D₁₉₁)⋊C₄

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

D₄

yes

—

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

C₂³

yes

—

C₂²⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

C₂²×D₁₉₁

C₂×D₄

yes

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

D₇₆₄

—

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂×D₄

yes

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

C₂×D₄

yes

—

C₈×C₁₉₁

C₂×D₄

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#121

yes

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#121

yes

—

C₄×C₁₉₁

C₂²×D₄

yes

characteristic

C₄×C₁₉₁

C₂×#41

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×#121

yes

characteristic

C₂×C₁₉₁

#578

yes

—

D₁₉₁

—

C₁₉₁

#3059

yes

characteristic, complemented, Sylow

#3059

—

191

maximal, Sylow

C₂²×Q₁₆

—

191

—

C₂×#131

—

191

—

#536

—

191

—

#470

—

191

—

#467

—

191

—

#427

D₁₉₁

yes

characteristic, complemented

#419

—

191

—

C₂×#41

—

191

—

C₂²×Q₈

—

191

—

C₂×Q₁₆

—

382

—

100

C₂×Q₁₆

—

191

—

101

C₂×Q₁₆

—

191

—

102

#131

—

382

—

103

C₂×M₁₆

—

191

—

104

C₂×M₁₆

—

191

—

105

C₂²×C₈

—

191

—

106

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

107

C₄⋊Q₈

—

382

—

108

#124

—

382

—

109

#123

—

382

—

110

C₂²⋊Q₈

—

191

—

111

#117

C₂×D₁₉₁

yes

—

112

#107

C₂×D₁₉₁

yes

—

113

#106

C₂×D₁₉₁

yes

—

114

#103

—

191

—

115

#41

—

382

—

116

#41

—

382

—

117

C₂×Q₈

—

382

—

118

C₂×Q₈

—

382

—

119

C₂×Q₈

—

382

—

120

C₂×Q₈

—

191

—

121

C₂×Q₈

—

191

—

122

C₂×D₄

—

191

—

123

C₂×D₄

—

191

—

124

C₂²×C₄

—

382

—

125

C₂²×C₄

—

191

—

126

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

127

Q₁₆

—

382

—

128

Q₁₆

—

382

—

129

M₁₆

—

382

—

130

M₁₆

—

382

—

131

C₂×C₈

—

382

—

132

C₂×C₈

—

191

—

133

C₂×C₈

D₇₆₄

yes

—

134

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

135

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

136

C₄⋊C₄

—

382

—

137

C₄⋊C₄

C₂²×D₁₉₁

yes

—

138

C₂²⋊C₄

—

139

C₂²⋊C₄

—

382

—

140

C₄²

—

141

C₂³

—

191

—

142

C₂³

D₄×D₁₉₁

yes

characteristic

143

Q₈

—

382

—

144

Q₈

—

382

—

145

Q₈

—

382

—

146

Q₈

—

764

—

147

D₄

—

382

—

148

D₄

—

382

—

149

C₂×C₄

—

150

C₂×C₄

—

382

—

151

C₂×C₄

—

382

—

152

C₂×C₄

—

382

—

153

C₂×C₄

—

191

—

154

C₂×C₄

—

764

—

155

C₂×C₄

D₄×D₁₉₁

yes

characteristic

156

C₂×C₄

C₂×D₇₆₄

yes

—

157

C₂×C₄

#15690288

yes

—

158

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

159

C₈

—

382

—

160

C₈

C₂×D₇₆₄

yes

—

161

C₈

C₂×D₇₆₄

yes

—

162

C₂²

—

163

C₂²

—

382

—

164

C₂²

—

382

—

165

C₂²

#20259297

yes

—

166

C₂²

C₂×D₄×D₁₉₁

yes

characteristic

167

C₄

—

168

C₄

—

382

—

169

C₄

—

764

—

170

C₄

#20259297

yes

—

171

C₄

C₂²×D₇₆₄

yes

characteristic

172

C₄

C₂×#15690288

yes

characteristic

173

C₂

—

174

C₂

—

764

—

175

C₂

C₂×#20259297

yes

characteristic

176

C₂

#27133477

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#427×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂×#15690287	C₂²
hypercenter	#427	D₁₉₁
hypocenter	C₁₉₁	#3059
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₁₉₁

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#427×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

C₂²

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
1. C₂²×Dic₇₆₄	C₂	semidirect
4. #27132922	C₂	semidirect
5. #27132905	C₂	semidirect
6. #27132671	C₂	semidirect
7. #427×C₁₉₁	C₂	semidirect
88. C₁₉₁	#3059	semidirect
95. #427	D₁₉₁	semidirect

H ≅ G/N

type

1. C₂²×Dic₇₆₄

C₂

semidirect

4. #27132922

C₂

semidirect

5. #27132905

C₂

semidirect

6. #27132671

C₂

semidirect

7. #427×C₁₉₁

C₂

semidirect

88. C₁₉₁

#3059

semidirect

95. #427

D₁₉₁

semidirect

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

subgroup	index
C₂²×Dic₇₆₄	2
C₂×#20259328	2
#27133239	2
#27132922	2
#27132905	2
#27132671	2
#427×C₁₉₁	2
#3059	191

subgroup

index

C₂²×Dic₇₆₄

#427×C₁₉₁

191

Subgroups — #37582176

Conjugacy classes of subgroups

Complements

Maximal subgroups