Group #37573766 — subgroups

Subgroups — #37573766

Order 24448.

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

normal subgroups only

This group has 298 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	C₂×#20259322	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259245	C₂	1	2	yes	maximal
3	C₂×#20259245	C₂	1	2	yes	maximal
4	#27132418	C₂	1	1	yes	maximal, characteristic
5	#389×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₈₂	C₂²	1	2	yes	—
7	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
8	#20259322	—	2	4	no	—
9	C₂×C₄⋊Dic₁₉₁	—	2	1	no	—
10	C₂×C₄⋊Dic₁₉₁	—	2	1	no	—
11	#20259245	—	2	4	no	—
12	#20259245	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₉₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
17	#95×C₁₉₁	—	2	1	no	—
18	C₂×Dic₃₈₂	—	2	2	no	—
19	C₂×Dic₃₈₂	—	2	2	no	—
20	C₂×Dic₃₈₂	—	2	2	no	—
21	C₂×Dic₃₈₂	—	2	2	no	—
22	C₂×Dic₃₈₂	—	2	2	no	—
23	C₂×Dic₃₈₂	D₄	1	4	yes	—
24	C₂²×Dic₁₉₁	—	2	2	no	—
25	C₂×C₁₉₁⋊C₈	—	2	2	no	—
26	C₂×C₁₉₁⋊C₈	—	2	2	no	—
27	C₂×C₁₉₁⋊C₈	—	2	2	no	—
28	C₄⋊Dic₁₉₁	—	2	4	no	—
29	C₄⋊Dic₁₉₁	—	2	4	no	—
30	C₂²×C₄×C₁₉₁	—	2	1	no	—
31	C₂²×C₄×C₁₉₁	—	2	1	no	—
32	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₉₁	—	2	1	no	—
36	C₄⋊C₄×C₁₉₁	—	2	2	no	—
37	C₄⋊C₄×C₁₉₁	—	4	2	no	—
38	C₄⋊C₄×C₁₉₁	D₄	1	2	yes	—
39	C₄²×C₁₉₁	—	2	2	no	—
40	Dic₃₈₂	—	2	4	no	—
41	Dic₃₈₂	—	2	4	no	—
42	Dic₃₈₂	—	4	2	no	—
43	Dic₃₈₂	—	4	2	no	—
44	C₂×Dic₁₉₁	—	2	4	no	—
45	C₂×Dic₁₉₁	—	4	2	no	—
46	C₂×Dic₁₉₁	—	4	2	no	—
47	C₁₉₁⋊C₈	—	4	2	no	—
48	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₁	—	2	1	no	—
50	C₂×C₄×C₁₉₁	—	2	1	no	—
51	C₂×C₄×C₁₉₁	—	2	1	no	—
52	C₂×C₄×C₁₉₁	—	2	1	no	—
53	C₂×C₄×C₁₉₁	—	2	2	no	—
54	C₂×C₄×C₁₉₁	—	4	1	no	—
55	C₂×C₄×C₁₉₁	—	4	1	no	—
56	C₂×C₄×C₁₉₁	—	4	1	no	—
57	C₂×C₄×C₁₉₁	—	4	1	no	—
58	C₂×C₄×C₁₉₁	—	4	1	no	—
59	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
60	C₂×C₄×C₁₉₁	Q₁₆	1	2	yes	—
61	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
64	C₂×C₄×C₁₉₁	#41	1	2	yes	—
65	C₂²×C₁₉₁	#120	1	1	yes	characteristic
66	C₂²×C₁₉₁	#121	1	1	yes	characteristic
67	C₂²×C₁₉₁	#124	1	1	yes	characteristic
68	C₂²×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₂²×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₁₉₁	#137	1	1	yes	characteristic
71	C₂²×C₁₉₁	#137	1	1	yes	characteristic
72	C₄×C₁₉₁	—	2	2	no	—
73	C₄×C₁₉₁	—	2	2	no	—
74	C₄×C₁₉₁	—	4	2	no	—
75	C₄×C₁₉₁	—	8	1	no	—
76	C₄×C₁₉₁	#121	1	2	yes	—
77	C₄×C₁₉₁	#123	1	2	yes	—
78	Dic₁₉₁	—	4	4	no	—
79	C₂×C₁₉₁	#394	1	1	yes	characteristic
80	C₂×C₁₉₁	#448	1	1	yes	characteristic
81	C₂×C₁₉₁	#451	1	1	yes	characteristic
82	C₂×C₁₉₁	#461	1	1	yes	characteristic
83	C₂×C₁₉₁	#462	1	1	yes	characteristic
84	C₂×C₁₉₁	#487	1	1	yes	characteristic
85	C₂×C₁₉₁	#491	1	1	yes	characteristic
86	C₁₉₁	#2043	1	1	yes	characteristic, complemented, Sylow
87	#2043	—	191	1	no	maximal, Sylow
88	C₂×C₄⋊Q₈	—	191	1	no	—
89	C₂×#103	—	191	2	no	—
90	C₂×#103	—	191	2	no	—
91	#389	D₁₉₁	1	1	yes	characteristic
92	#336	—	191	1	no	—
93	C₂²×Q₈	—	191	2	no	—
94	C₂²×C₈	—	191	2	no	—
95	C₄⋊Q₈	—	382	4	no	—
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄⋊C₄	—	382	1	no	—
98	C₂×C₄⋊C₄	—	382	1	no	—
99	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
102	#103	—	382	4	no	—
103	#103	—	382	4	no	—
104	#95	—	2	1	no	—
105	C₂×Q₈	—	382	2	no	—
106	C₂×Q₈	—	382	2	no	—
107	C₂×Q₈	—	382	2	no	—
108	C₂×Q₈	—	382	2	no	—
109	C₂×Q₈	—	382	2	no	—
110	C₂×Q₈	—	191	4	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	382	2	no	—
114	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
117	C₂×C₈	—	382	2	no	—
118	C₂×C₈	—	382	2	no	—
119	C₂×C₈	—	382	2	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	382	4	no	—
124	C₄⋊C₄	—	382	4	no	—
125	C₄⋊C₄	D₇₆₄	1	2	yes	—
126	C₄²	—	2	2	no	—
127	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
128	Q₈	—	382	4	no	—
129	Q₈	—	382	4	no	—
130	Q₈	—	764	2	no	—
131	Q₈	—	764	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	382	4	no	—
143	C₂×C₄	—	764	2	no	—
144	C₂×C₄	—	764	2	no	—
145	C₂×C₄	#15690275	1	2	yes	—
146	C₂×C₄	#15690277	1	2	yes	—
147	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
148	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
149	C₂×C₄	#15690287	1	1	yes	characteristic
150	C₂×C₄	#15690288	1	2	yes	—
151	C₈	—	764	2	no	—
152	C₂²	#20259290	1	1	yes	characteristic
153	C₂²	#20259296	1	1	yes	characteristic
154	C₂²	#20259311	1	1	yes	characteristic
155	C₂²	C₂×#15690275	1	1	yes	characteristic
156	C₂²	C₂×#15690277	1	1	yes	characteristic
157	C₂²	#20259352	1	1	yes	characteristic
158	C₂²	#20259352	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	2	no	—
162	C₄	—	8	1	no	—
163	C₄	—	764	4	no	—
164	C₄	#20259297	1	2	yes	—
165	C₄	#20259307	1	2	yes	—
166	C₂	#27132580	1	1	yes	characteristic
167	C₂	#27132801	1	1	yes	characteristic
168	C₂	#27132818	1	1	yes	characteristic
169	C₂	#27132868	1	1	yes	characteristic
170	C₂	#27132873	1	1	yes	characteristic
171	C₂	#27133015	1	1	yes	characteristic
172	C₂	#27133033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259322

C₂

yes

maximal, characteristic

C₂×#20259245

C₂

yes

maximal

C₂×#20259245

C₂

yes

maximal

#27132418

C₂

yes

maximal, characteristic

#389×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₈₂

C₂²

yes

—

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

#20259322

—

C₂×C₄⋊Dic₁₉₁

—

C₂×C₄⋊Dic₁₉₁

—

#20259245

—

#20259245

—

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

characteristic

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#95×C₁₉₁

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

D₄

yes

—

C₂²×Dic₁₉₁

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂×C₄×C₁₉₁

Q₁₆

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

C₂×C₄×C₁₉₁

yes

C₂×C₄×C₁₉₁

yes

—

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₁

C₂×Q₁₆

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#121

yes

—

C₄×C₁₉₁

#123

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#2043

—

191

maximal, Sylow

C₂×C₄⋊Q₈

—

191

—

C₂×#103

—

191

—

C₂×#103

—

191

—

#389

D₁₉₁

yes

characteristic

#336

—

191

—

C₂²×Q₈

—

191

—

C₂²×C₈

—

191

—

C₄⋊Q₈

—

382

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

characteristic

100

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

characteristic

101

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

102

#103

—

382

—

103

#103

—

382

—

104

#95

—

105

C₂×Q₈

—

382

—

106

C₂×Q₈

—

382

—

107

C₂×Q₈

—

382

—

108

C₂×Q₈

—

382

—

109

C₂×Q₈

—

382

—

110

C₂×Q₈

—

191

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

382

—

114

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

115

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

116

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

117

C₂×C₈

—

382

—

118

C₂×C₈

—

382

—

119

C₂×C₈

—

382

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

382

—

124

C₄⋊C₄

—

382

—

125

C₄⋊C₄

D₇₆₄

yes

—

126

C₄²

—

127

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

128

Q₈

—

382

—

129

Q₈

—

382

—

130

Q₈

—

764

—

131

Q₈

—

764

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

382

—

143

C₂×C₄

—

764

—

144

C₂×C₄

—

764

—

145

C₂×C₄

#15690275

yes

—

146

C₂×C₄

#15690277

yes

—

147

C₂×C₄

D₄×D₁₉₁

yes

—

148

C₂×C₄

C₂×D₇₆₄

yes

characteristic

149

C₂×C₄

#15690287

yes

characteristic

150

C₂×C₄

#15690288

yes

—

151

C₈

—

764

—

152

C₂²

#20259290

yes

characteristic

153

C₂²

#20259296

yes

characteristic

154

C₂²

#20259311

yes

characteristic

155

C₂²

C₂×#15690275

yes

characteristic

156

C₂²

C₂×#15690277

yes

characteristic

157

C₂²

#20259352

yes

characteristic

158

C₂²

#20259352

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

764

—

164

C₄

#20259297

yes

—

165

C₄

#20259307

yes

—

166

C₂

#27132580

yes

characteristic

167

C₂

#27132801

yes

characteristic

168

C₂

#27132818

yes

characteristic

169

C₂

#27132868

yes

characteristic

170

C₂

#27132873

yes

characteristic

171

C₂

#27133015

yes

characteristic

172

C₂

#27133033

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₁
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#389×C₁₉₁	C₂
socle, prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	#389	D₁₉₁
hypocenter	C₁₉₁	#2043
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₁

commutator, cosocle

C₂²×C₄×C₁₉₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#389×C₁₉₁

C₂

socle, prime radical

C₂³×C₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
86. C₁₉₁	#2043	semidirect

H ≅ G/N

type

86. C₁₉₁

#2043

semidirect

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

subgroup	index
C₂×#20259322	2
C₂×#20259245	2
#27132418	2
#389×C₁₉₁	2
#2043	191

subgroup

index

#389×C₁₉₁

191

Subgroups — #37573766

Conjugacy classes of subgroups

Complements

Maximal subgroups