Group #37572586 — subgroups

Subgroups — #37572586

Order 24448.

380 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₁₉₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#27132417	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₉₁	C₄	1	1	yes	characteristic
6	C₂³×Dic₁₉₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₉₁	—	2	2	no	—
9	C₂×C₄×Dic₁₉₁	—	2	2	no	—
10	C₂×C₄×Dic₁₉₁	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
12	C₂²⋊(C₁₉₁⋊C₈)	—	2	2	no	—
13	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₉₁	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₉₁	—	2	2	no	—
16	C₂²×Dic₁₉₁	—	2	4	no	—
17	C₂²×Dic₁₉₁	—	2	2	no	—
18	C₂²×Dic₁₉₁	—	2	2	no	—
19	C₂²×Dic₁₉₁	—	2	2	no	—
20	C₂²×Dic₁₉₁	C₈	1	4	yes	complemented
21	C₂²×Dic₁₉₁	C₂×C₄	1	2	yes	—
22	C₂²×Dic₁₉₁	C₂×C₄	1	2	yes	—
23	C₂×C₁₉₁⋊C₈	—	2	2	no	—
24	C₂×C₁₉₁⋊C₈	—	2	4	no	—
25	C₄×Dic₁₉₁	—	2	4	no	—
26	C₄×Dic₁₉₁	—	2	2	no	—
27	C₄×Dic₁₉₁	—	2	2	no	—
28	C₂⁴×C₁₉₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₉₁	—	2	4	no	—
30	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
32	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
33	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₉₁	—	2	2	no	—
35	C₂×C₈×C₁₉₁	—	2	4	no	—
36	C₂×Dic₁₉₁	—	2	8	no	—
37	C₂×Dic₁₉₁	—	2	8	no	—
38	C₂×Dic₁₉₁	—	2	2	no	—
39	C₂×Dic₁₉₁	—	2	2	no	—
40	C₂×Dic₁₉₁	—	2	2	no	—
41	C₂×Dic₁₉₁	—	2	2	no	—
42	C₂×Dic₁₉₁	C₂×C₈	1	4	yes	—
43	C₂×Dic₁₉₁	M₁₆	1	4	yes	—
44	C₁₉₁⋊C₈	—	4	2	no	—
45	C₂³×C₁₉₁	—	2	2	no	—
46	C₂³×C₁₉₁	—	2	2	no	—
47	C₂³×C₁₉₁	C₂²⋊C₄	1	2	yes	—
48	C₂³×C₁₉₁	C₂×C₈	1	2	yes	—
49	C₂³×C₁₉₁	M₁₆	1	2	yes	—
50	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₁	—	2	2	no	—
52	C₂×C₄×C₁₉₁	—	2	4	no	—
53	C₂×C₄×C₁₉₁	—	2	4	no	—
54	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₉₁	#41	1	4	yes	—
58	C₈×C₁₉₁	—	4	2	no	—
59	C₂²×C₁₉₁	—	2	2	no	—
60	C₂²×C₁₉₁	—	2	2	no	—
61	C₂²×C₁₉₁	—	2	4	no	—
62	C₂²×C₁₉₁	—	2	4	no	—
63	C₂²×C₁₉₁	C₂²⋊C₈	1	4	yes	—
64	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₁₉₁	#117	1	2	yes	—
66	C₂²×C₁₉₁	C₂²×C₈	1	1	yes	characteristic
67	C₂²×C₁₉₁	C₂×M₁₆	1	2	yes	—
68	C₂²×C₁₉₁	C₂×M₁₆	1	1	yes	characteristic
69	C₄×C₁₉₁	—	2	2	no	—
70	C₄×C₁₉₁	#123	1	2	yes	—
71	C₄×C₁₉₁	#123	1	2	yes	—
72	Dic₁₉₁	—	2	4	no	—
73	Dic₁₉₁	—	2	4	no	—
74	C₂×C₁₉₁	—	2	4	no	—
75	C₂×C₁₉₁	#381	1	1	yes	characteristic
76	C₂×C₁₉₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₁₉₁	#407	1	1	yes	characteristic
78	C₂×C₁₉₁	#431	1	2	yes	—
79	C₂×C₁₉₁	#432	1	2	yes	—
80	C₁₉₁	#1817	1	1	yes	characteristic, complemented, Sylow
81	#1817	—	191	1	no	maximal, Sylow
82	C₂²×C₄²	—	191	1	no	—
83	C₂×C₂²⋊C₈	—	191	1	no	—
84	C₂×C₂²⋊C₈	D₁₉₁	1	1	yes	characteristic
85	#336	—	191	4	no	—
86	C₂³×C₄	—	191	1	no	—
87	C₂³×C₄	—	191	1	no	—
88	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
89	C₂²×C₈	—	191	2	no	—
90	C₂²×C₈	C₂×D₁₉₁	1	2	yes	—
91	C₂×C₄²	—	382	2	no	—
92	C₂×C₄²	—	382	2	no	—
93	C₂×C₄²	—	191	2	no	—
94	C₂×C₄²	—	191	2	no	—
95	C₂²⋊C₈	—	2	2	no	—
96	C₂²⋊C₈	—	382	2	no	—
97	C₂⁴	C₄×D₁₉₁	1	1	yes	characteristic
98	C₂²×C₄	—	2	4	no	—
99	C₂²×C₄	—	382	2	no	—
100	C₂²×C₄	—	382	2	no	—
101	C₂²×C₄	—	382	2	no	—
102	C₂²×C₄	—	382	4	no	—
103	C₂²×C₄	—	191	4	no	—
104	C₂²×C₄	—	191	2	no	—
105	C₂²×C₄	—	191	2	no	—
106	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
107	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
108	C₂²×C₄	D₇₆₄	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	382	2	no	—
113	C₂×C₈	—	382	4	no	—
114	C₄²	—	382	4	no	—
115	C₄²	—	382	2	no	—
116	C₄²	—	382	2	no	—
117	C₂³	—	2	2	no	—
118	C₂³	—	2	2	no	—
119	C₂³	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
120	C₂³	C₈×D₁₉₁	1	2	yes	—
121	C₂³	#15690270	1	2	yes	—
122	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	382	8	no	—
127	C₂×C₄	—	382	8	no	—
128	C₂×C₄	—	382	2	no	—
129	C₂×C₄	—	382	2	no	—
130	C₂×C₄	—	382	2	no	—
131	C₂×C₄	—	382	2	no	—
132	C₂×C₄	—	191	4	no	—
133	C₂×C₄	—	191	4	no	—
134	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
136	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
137	C₂×C₄	#15690286	1	4	yes	—
138	C₈	—	4	2	no	—
139	C₈	—	764	2	no	—
140	C₂²	—	2	4	no	—
141	C₂²	—	2	4	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	(C₂×D₁₉₁)⋊C₈	1	4	yes	—
145	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
146	C₂²	#20259278	1	2	yes	—
147	C₂²	C₂×C₈×D₁₉₁	1	1	yes	characteristic
148	C₂²	M₁₆×D₁₉₁	1	2	yes	—
149	C₂²	C₂×#15690270	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	382	4	no	—
152	C₄	—	382	4	no	—
153	C₄	#20259303	1	2	yes	—
154	C₄	#20259306	1	2	yes	—
155	C₂	—	2	4	no	—
156	C₂	#27132524	1	1	yes	characteristic
157	C₂	C₂×(C₂×D₁₉₁)⋊C₈	1	1	yes	characteristic
158	C₂	#27132626	1	1	yes	characteristic
159	C₂	#27132713	1	2	yes	—
160	C₂	#27132717	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×Dic₁₉₁

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₁₉₁⋊C₈)

C₂

yes

maximal, characteristic

#27132417

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂³×Dic₁₉₁

C₄

yes

characteristic

C₂³×Dic₁₉₁

C₄

yes

characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₉₁

—

C₂×C₄×Dic₁₉₁

—

C₂×C₄×Dic₁₉₁

C₄

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

—

C₂²⋊(C₁₉₁⋊C₈)

—

C₂³×C₄×C₁₉₁

C₂²

yes

characteristic

C₂²×C₈×C₁₉₁

C₂²

yes

—

C₂²⋊C₈×C₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

C₈

yes

complemented

C₂²×Dic₁₉₁

C₂×C₄

yes

—

C₂²×Dic₁₉₁

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂⁴×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂×C₈

yes

—

C₂×Dic₁₉₁

M₁₆

yes

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂²⋊C₄

yes

—

C₂³×C₁₉₁

C₂×C₈

yes

—

C₂³×C₁₉₁

M₁₆

yes

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂²⋊C₈

yes

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₁

#117

yes

—

C₂²×C₁₉₁

C₂²×C₈

yes

characteristic

C₂²×C₁₉₁

C₂×M₁₆

yes

—

C₂²×C₁₉₁

C₂×M₁₆

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

#123

yes

—

C₄×C₁₉₁

#123

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#381

yes

characteristic

C₂×C₁₉₁

C₂×C₂²⋊C₈

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

—

C₂×C₁₉₁

#432

yes

—

C₁₉₁

#1817

yes

characteristic, complemented, Sylow

#1817

—

191

maximal, Sylow

C₂²×C₄²

—

191

—

C₂×C₂²⋊C₈

—

191

—

C₂×C₂²⋊C₈

D₁₉₁

yes

characteristic

#336

—

191

—

C₂³×C₄

—

191

—

C₂³×C₄

—

191

—

C₂³×C₄

C₂×D₁₉₁

yes

characteristic

C₂²×C₈

—

191

—

C₂²×C₈

C₂×D₁₉₁

yes

—

C₂×C₄²

—

382

—

C₂×C₄²

—

382

—

C₂×C₄²

—

191

—

C₂×C₄²

—

191

—

C₂²⋊C₈

—

C₂²⋊C₈

—

382

—

C₂⁴

C₄×D₁₉₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

382

—

100

C₂²×C₄

—

382

—

101

C₂²×C₄

—

382

—

102

C₂²×C₄

—

382

—

103

C₂²×C₄

—

191

—

104

C₂²×C₄

—

191

—

105

C₂²×C₄

—

191

—

106

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

107

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

108

C₂²×C₄

D₇₆₄

yes

—

109

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

382

—

113

C₂×C₈

—

382

—

114

C₄²

—

382

—

115

C₄²

—

382

—

116

C₄²

—

382

—

117

C₂³

—

118

C₂³

—

119

C₂³

(C₂×D₁₉₁)⋊C₄

yes

—

120

C₂³

C₈×D₁₉₁

yes

—

121

C₂³

#15690270

yes

—

122

C₂³

C₂×C₄×D₁₉₁

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

382

—

127

C₂×C₄

—

382

—

128

C₂×C₄

—

382

—

129

C₂×C₄

—

382

—

130

C₂×C₄

—

382

—

131

C₂×C₄

—

382

—

132

C₂×C₄

—

191

—

133

C₂×C₄

—

191

—

134

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

135

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

136

C₂×C₄

C₂×D₇₆₄

yes

characteristic

137

C₂×C₄

#15690286

yes

—

138

C₈

—

139

C₈

—

764

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

(C₂×D₁₉₁)⋊C₈

yes

—

145

C₂²

C₂×(C₂×D₁₉₁)⋊C₄

yes

characteristic

146

C₂²

#20259278

yes

—

147

C₂²

C₂×C₈×D₁₉₁

yes

characteristic

148

C₂²

M₁₆×D₁₉₁

yes

—

149

C₂²

C₂×#15690270

yes

characteristic

150

C₄

—

151

C₄

—

382

—

152

C₄

—

382

—

153

C₄

#20259303

yes

—

154

C₄

#20259306

yes

—

155

C₂

—

156

C₂

#27132524

yes

characteristic

157

C₂

C₂×(C₂×D₁₉₁)⋊C₈

yes

characteristic

158

C₂

#27132626

yes

characteristic

159

C₂

#27132713

yes

—

160

C₂

#27132717

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₁
commutator	C₂²×C₁₉₁	C₂²×C₈
Fitting	C₂×C₂²⋊C₈×C₁₉₁	C₂
socle	C₂³×C₁₉₁	C₂²×C₄
prime radical	C₂⁴×C₁₉₁	C₂×C₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₂×C₂²⋊C₈	D₁₉₁
hypocenter	C₁₉₁	#1817
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₁

commutator

C₂²×C₁₉₁

C₂²×C₈

Fitting

C₂×C₂²⋊C₈×C₁₉₁

C₂

socle

C₂³×C₁₉₁

C₂²×C₄

prime radical

C₂⁴×C₁₉₁

C₂×C₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

hypercenter

C₂×C₂²⋊C₈

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #27132417	C₂	semidirect
20. C₂²×Dic₁₉₁	C₈	semidirect
80. C₁₉₁	#1817	semidirect

H ≅ G/N

type

3. #27132417

C₂

semidirect

20. C₂²×Dic₁₉₁

C₈

semidirect

80. C₁₉₁

#1817

semidirect

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

subgroup	index
C₂²×C₄×Dic₁₉₁	2
C₂×C₂²⋊(C₁₉₁⋊C₈)	2
#27132417	2
C₂×C₂²⋊C₈×C₁₉₁	2
#1817	191

subgroup

index

C₂²×C₄×Dic₁₉₁

C₂×C₂²⋊(C₁₉₁⋊C₈)

#27132417

C₂×C₂²⋊C₈×C₁₉₁

#1817

191

Subgroups — #37572586

Conjugacy classes of subgroups

Complements

Maximal subgroups