Group #37583167 — subgroups

Subgroups — #37583167

Order 24448.

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

normal subgroups only

This group has 386 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	#27133267	C₂	1	2	yes	maximal, complemented
2	#27133065	C₂	1	1	yes	maximal, characteristic, complemented
3	#27133027	C₂	1	2	yes	maximal, complemented
4	#27132913	C₂	1	2	yes	maximal, complemented
5	#27132896	C₂	1	2	yes	maximal, complemented
6	#27132775	C₂	1	2	yes	maximal, complemented
7	D₈×Dic₁₉₁	C₂	1	2	yes	maximal, complemented
8	#27132617	C₂	1	1	yes	maximal, characteristic, complemented
9	#493×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15690270	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
12	#20259314	C₂²	1	2	yes	—
13	#20259303	—	2	4	no	—
14	#20259294	—	2	4	no	—
15	#20259293	C₂²	1	4	yes	—
16	D₄×Dic₁₉₁	C₂²	1	4	yes	—
17	#20259279	C₂²	1	1	yes	characteristic
18	#20259256	C₂²	1	2	yes	—
19	#20259254	C₂²	1	2	yes	—
20	#20259243	C₂²	1	4	yes	—
21	#20259243	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	C₂×D₈×C₁₉₁	C₂²	1	2	yes	—
26	C₂×D₈×C₁₉₁	C₂²	1	2	yes	—
27	#127×C₁₉₁	C₂²	1	2	yes	—
28	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊D₁₉₁	—	2	4	no	—
30	C₂²×Dic₁₉₁	—	2	4	no	—
31	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
32	#15690270	—	2	2	no	—
33	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
34	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
35	C₈×D₁₉₁	—	2	2	no	—
36	C₄⋊Dic₁₉₁	C₂³	1	4	yes	—
37	Dic₁₉₁⋊C₄	—	2	4	no	—
38	Dic₁₉₁⋊C₄	D₄	1	2	yes	—
39	C₂²⋊Dic₁₉₁	—	2	4	no	—
40	C₂²⋊Dic₁₉₁	—	2	4	no	—
41	(C₂×D₁₉₁)⋊C₄	D₄	1	2	yes	—
42	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
43	C₂×D₄×C₁₉₁	—	2	4	no	—
44	C₂×D₄×C₁₉₁	C₂³	1	4	yes	—
45	D₈×C₁₉₁	—	2	4	no	—
46	D₈×C₁₉₁	—	2	4	no	—
47	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
49	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
50	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
51	C₂²⋊D₁₉₁	—	8	4	no	—
52	C₂×Dic₁₉₁	—	2	4	no	—
53	C₂×Dic₁₉₁	—	8	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	C₂³×C₁₉₁	—	2	4	no	—
61	D₄×C₁₉₁	—	2	4	no	—
62	D₄×C₁₉₁	—	2	4	no	—
63	D₄×C₁₉₁	—	8	4	no	—
64	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
66	C₈×C₁₉₁	#41	1	2	yes	—
67	C₈×C₁₉₁	#41	1	2	yes	—
68	C₂²×C₁₉₁	—	4	4	no	—
69	C₂²×C₁₉₁	—	4	4	no	—
70	C₂²×C₁₉₁	—	4	4	no	—
71	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
72	C₂×D₁₉₁	—	2	1	no	—
73	C₂×D₁₉₁	—	2	2	no	—
74	C₄×C₁₉₁	—	2	2	no	—
75	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
76	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
77	Dic₁₉₁	—	2	1	no	—
78	Dic₁₉₁	—	2	2	no	—
79	Dic₁₉₁	—	4	4	no	—
80	C₂×C₁₉₁	—	8	4	no	—
81	C₂×C₁₉₁	#532	1	1	yes	characteristic
82	C₂×C₁₉₁	#576	1	2	yes	—
83	D₁₉₁	—	4	1	no	—
84	C₁₉₁	#3176	1	1	yes	characteristic, complemented, Sylow
85	#3176	—	191	1	no	maximal, Sylow
86	#540	—	191	2	no	—
87	#499	—	191	1	no	—
88	#493	D₁₉₁	1	1	yes	characteristic, complemented
89	#490	—	191	2	no	—
90	#469	—	191	2	no	—
91	#466	—	191	2	no	—
92	#442	—	191	2	no	—
93	C₄×D₈	—	191	2	no	—
94	#405	—	191	1	no	—
95	C₂×D₈	C₂×D₁₉₁	1	2	yes	—
96	C₂×D₈	C₂×D₁₉₁	1	2	yes	—
97	C₂×M₁₆	—	191	1	no	—
98	C₂²×C₈	—	191	1	no	—
99	#127	C₂×D₁₉₁	1	2	yes	—
100	#125	—	191	2	no	—
101	#123	—	382	4	no	—
102	#121	—	382	4	no	—
103	#121	—	191	4	no	—
104	C₄×D₄	—	191	4	no	—
105	#117	—	191	1	no	—
106	#107	—	191	2	no	—
107	#106	—	191	2	no	—
108	#102	—	191	4	no	—
109	#102	—	191	4	no	—
110	#97	—	191	1	no	—
111	#97	—	191	1	no	—
112	C₄×C₈	—	191	1	no	—
113	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	—	382	4	no	—
116	C₂×D₄	C₂²×D₁₉₁	1	4	yes	—
117	C₂²×C₄	—	382	4	no	—
118	C₂²×C₄	—	191	1	no	—
119	D₈	—	2	4	no	—
120	D₈	—	2	4	no	—
121	M₁₆	—	382	2	no	—
122	C₂×C₈	—	382	2	no	—
123	C₂×C₈	—	191	1	no	—
124	C₂×C₈	—	191	1	no	—
125	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
126	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
127	C₄⋊C₄	—	382	4	no	—
128	C₄⋊C₄	—	191	4	no	—
129	C₄⋊C₄	—	191	2	no	—
130	C₂²⋊C₄	—	382	4	no	—
131	C₂²⋊C₄	—	382	4	no	—
132	C₂²⋊C₄	—	191	2	no	—
133	C₄²	—	191	1	no	—
134	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
135	C₂³	—	2	4	no	—
136	C₂³	—	191	1	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	8	4	no	—
140	D₄	—	1528	4	no	—
141	C₂×C₄	—	382	1	no	—
142	C₂×C₄	—	382	1	no	—
143	C₂×C₄	—	382	4	no	—
144	C₂×C₄	—	191	2	no	—
145	C₂×C₄	—	191	1	no	—
146	C₂×C₄	—	1528	4	no	—
147	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
148	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
149	C₈	—	382	1	no	—
150	C₈	—	382	1	no	—
151	C₈	#15690286	1	2	yes	—
152	C₈	#15690286	1	2	yes	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	382	2	no	—
157	C₂²	—	382	1	no	—
158	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	382	2	no	—
161	C₄	—	382	1	no	—
162	C₄	—	764	4	no	—
163	C₄	C₂×#15690286	1	1	yes	characteristic
164	C₄	C₂×#15690286	1	1	yes	characteristic
165	C₂	—	8	4	no	—
166	C₂	—	764	1	no	—
167	C₂	#27133211	1	1	yes	characteristic
168	C₂	#27133464	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133267

C₂

yes

maximal, complemented

#27133065

C₂

yes

maximal, characteristic, complemented

#27133027

C₂

yes

maximal, complemented

#27132913

C₂

yes

maximal, complemented

#27132896

C₂

yes

maximal, complemented

#27132775

C₂

yes

maximal, complemented

D₈×Dic₁₉₁

C₂

yes

maximal, complemented

#27132617

C₂

yes

maximal, characteristic, complemented

#493×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690270

C₂²

yes

characteristic

C₂×C₈×D₁₉₁

C₂²

yes

characteristic

#20259314

C₂²

yes

—

#20259303

—

#20259294

—

#20259293

C₂²

yes

—

D₄×Dic₁₉₁

C₂²

yes

—

#20259279

C₂²

yes

characteristic

#20259256

C₂²

yes

—

#20259254

C₂²

yes

—

#20259243

C₂²

yes

—

#20259243

C₂²

yes

—

#20259233

C₂²

yes

characteristic

#20259232

C₂²

yes

characteristic

C₈×Dic₁₉₁

C₂²

yes

characteristic

C₂×D₈×C₁₉₁

C₂²

yes

—

C₂×D₈×C₁₉₁

C₂²

yes

—

#127×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×C₂²⋊D₁₉₁

—

C₂²×Dic₁₉₁

—

C₂×C₄×D₁₉₁

C₂³

yes

characteristic

#15690270

—

C₂×C₁₉₁⋊C₈

C₂³

yes

characteristic

C₂×C₁₉₁⋊C₈

C₂³

yes

characteristic

C₈×D₁₉₁

—

C₄⋊Dic₁₉₁

C₂³

yes

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

D₄

yes

—

C₂²⋊Dic₁₉₁

—

C₂²⋊Dic₁₉₁

—

(C₂×D₁₉₁)⋊C₄

D₄

yes

—

C₄×Dic₁₉₁

C₂³

yes

characteristic

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

C₂³

yes

—

D₈×C₁₉₁

—

D₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

C₂×D₄

yes

characteristic

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂×D₄

yes

—

C₂×Dic₁₉₁

C₂×D₄

yes

characteristic

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

#41

yes

—

C₈×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×#41

yes

characteristic

C₄×C₁₉₁

C₂×#41

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#532

yes

characteristic

C₂×C₁₉₁

#576

yes

—

D₁₉₁

—

C₁₉₁

#3176

yes

characteristic, complemented, Sylow

#3176

—

191

maximal, Sylow

#540

—

191

—

#499

—

191

—

#493

D₁₉₁

yes

characteristic, complemented

#490

—

191

—

#469

—

191

—

#466

—

191

—

#442

—

191

—

C₄×D₈

—

191

—

#405

—

191

—

C₂×D₈

C₂×D₁₉₁

yes

—

C₂×D₈

C₂×D₁₉₁

yes

—

C₂×M₁₆

—

191

—

C₂²×C₈

—

191

—

#127

C₂×D₁₉₁

yes

—

100

#125

—

191

—

101

#123

—

382

—

102

#121

—

382

—

103

#121

—

191

—

104

C₄×D₄

—

191

—

105

#117

—

191

—

106

#107

—

191

—

107

#106

—

191

—

108

#102

—

191

—

109

#102

—

191

—

110

#97

—

191

—

111

#97

—

191

—

112

C₄×C₈

—

191

—

113

C₄×C₈

C₂×D₁₉₁

yes

characteristic

114

C₂×D₄

—

115

C₂×D₄

—

382

—

116

C₂×D₄

C₂²×D₁₉₁

yes

—

117

C₂²×C₄

—

382

—

118

C₂²×C₄

—

191

—

119

D₈

—

120

D₈

—

121

M₁₆

—

382

—

122

C₂×C₈

—

382

—

123

C₂×C₈

—

191

—

124

C₂×C₈

—

191

—

125

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

126

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

127

C₄⋊C₄

—

382

—

128

C₄⋊C₄

—

191

—

129

C₄⋊C₄

—

191

—

130

C₂²⋊C₄

—

382

—

131

C₂²⋊C₄

—

382

—

132

C₂²⋊C₄

—

191

—

133

C₄²

—

191

—

134

C₄²

C₂²×D₁₉₁

yes

characteristic

135

C₂³

—

136

C₂³

—

191

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

1528

—

141

C₂×C₄

—

382

—

142

C₂×C₄

—

382

—

143

C₂×C₄

—

382

—

144

C₂×C₄

—

191

—

145

C₂×C₄

—

191

—

146

C₂×C₄

—

1528

—

147

C₂×C₄

D₄×D₁₉₁

yes

—

148

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

149

C₈

—

382

—

150

C₈

—

382

—

151

C₈

#15690286

yes

—

152

C₈

#15690286

yes

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

382

—

157

C₂²

—

382

—

158

C₂²

C₂×D₄×D₁₉₁

yes

characteristic

159

C₄

—

160

C₄

—

382

—

161

C₄

—

382

—

162

C₄

—

764

—

163

C₄

C₂×#15690286

yes

characteristic

164

C₄

C₂×#15690286

yes

characteristic

165

C₂

—

166

C₂

—

764

—

167

C₂

#27133211

yes

characteristic

168

C₂

#27133464

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#493×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
hypercenter	#493	D₁₉₁
hypocenter	C₁₉₁	#3176
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₁₉₁

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#493×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #27133267	C₂	semidirect
2. #27133065	C₂	semidirect
3. #27133027	C₂	semidirect
4. #27132913	C₂	semidirect
5. #27132896	C₂	semidirect
6. #27132775	C₂	semidirect
7. D₈×Dic₁₉₁	C₂	semidirect
8. #27132617	C₂	semidirect
9. #493×C₁₉₁	C₂	semidirect
84. C₁₉₁	#3176	semidirect
88. #493	D₁₉₁	semidirect

H ≅ G/N

type

1. #27133267

C₂

semidirect

2. #27133065

C₂

semidirect

3. #27133027

C₂

semidirect

4. #27132913

C₂

semidirect

5. #27132896

C₂

semidirect

6. #27132775

C₂

semidirect

7. D₈×Dic₁₉₁

C₂

semidirect

8. #27132617

C₂

semidirect

9. #493×C₁₉₁

C₂

semidirect

84. C₁₉₁

#3176

semidirect

88. #493

D₁₉₁

semidirect

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

subgroup	index
#27133267	2
#27133065	2
#27133027	2
#27132913	2
#27132896	2
#27132775	2
D₈×Dic₁₉₁	2
#27132617	2
#493×C₁₉₁	2
#3176	191

subgroup

index

D₈×Dic₁₉₁

#493×C₁₉₁

191

Subgroups — #37583167

Conjugacy classes of subgroups

Complements

Maximal subgroups