Group #37572702 — subgroups

Subgroups — #37572702

Order 24448.

324 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 109 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×#15690271	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₉₁⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
4	#27132417	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15690271	—	2	4	no	—
7	C₂×#15690271	—	2	2	no	—
8	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₁₉₁	C₄	1	2	yes	—
10	C₂×C₄×Dic₁₉₁	C₂²	1	2	yes	—
11	Dic₁₉₁⋊C₈	—	2	4	no	—
12	C₁₉₁⋊#95	C₄	1	1	yes	characteristic
13	C₁₉₁⋊#95	C₄	1	1	yes	characteristic
14	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
17	C₂²⋊C₈×C₁₉₁	—	2	2	no	—
18	C₂²×Dic₁₉₁	C₂×C₄	1	2	yes	—
19	C₂²×Dic₁₉₁	C₂×C₄	1	2	yes	—
20	#15690271	—	2	8	no	—
21	C₂×C₁₉₁⋊C₈	—	2	2	no	—
22	C₂×C₁₉₁⋊C₈	—	2	2	no	—
23	C₂×C₁₉₁⋊C₈	D₄	1	4	yes	—
24	C₂²⋊Dic₁₉₁	—	2	4	no	—
25	C₄×Dic₁₉₁	—	2	2	no	—
26	C₄×Dic₁₉₁	—	2	2	no	—
27	C₂⁴×C₁₉₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉₁	—	2	2	no	—
29	C₂²×C₄×C₁₉₁	—	2	2	no	—
30	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
32	C₂²×C₄×C₁₉₁	Q₈	1	2	yes	—
33	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₉₁	—	2	1	no	—
35	C₂×C₈×C₁₉₁	—	2	1	no	—
36	C₂×C₈×C₁₉₁	—	2	2	no	—
37	C₂×C₈×C₁₉₁	—	2	2	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	—	2	2	no	—
43	C₂×Dic₁₉₁	M₁₆	1	4	yes	complemented
44	C₁₉₁⋊C₈	—	2	4	no	—
45	C₂³×C₁₉₁	—	2	2	no	—
46	C₂³×C₁₉₁	—	2	2	no	—
47	C₂³×C₁₉₁	—	2	1	no	—
48	C₂³×C₁₉₁	—	2	1	no	—
49	C₂³×C₁₉₁	C₄⋊C₄	1	2	yes	—
50	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₁	—	2	2	no	—
52	C₂×C₄×C₁₉₁	—	2	2	no	—
53	C₂×C₄×C₁₉₁	—	2	4	no	—
54	C₂×C₄×C₁₉₁	—	2	2	no	—
55	C₂×C₄×C₁₉₁	C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₉₁	#41	1	2	yes	—
60	C₈×C₁₉₁	—	4	1	no	—
61	C₈×C₁₉₁	—	4	1	no	—
62	C₂²×C₁₉₁	—	2	2	no	—
63	C₂²×C₁₉₁	—	2	2	no	—
64	C₂²×C₁₉₁	—	2	2	no	—
65	C₂²×C₁₉₁	—	4	2	no	—
66	C₂²×C₁₉₁	—	4	2	no	—
67	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₁₉₁	C₄×D₄	1	2	yes	—
69	C₂²×C₁₉₁	C₂×M₁₆	1	2	yes	—
70	C₂²×C₁₉₁	#131	1	1	yes	characteristic
71	C₂²×C₁₉₁	#131	1	1	yes	characteristic
72	C₄×C₁₉₁	—	2	2	no	—
73	C₄×C₁₉₁	#120	1	1	yes	characteristic
74	C₄×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
75	C₄×C₁₉₁	#123	1	1	yes	characteristic
76	Dic₁₉₁	—	2	4	no	—
77	Dic₁₉₁	—	4	2	no	—
78	C₂×C₁₉₁	—	4	2	no	—
79	C₂×C₁₉₁	#385	1	1	yes	characteristic
80	C₂×C₁₉₁	#423	1	1	yes	characteristic
81	C₂×C₁₉₁	#424	1	1	yes	characteristic
82	C₂×C₁₉₁	#435	1	2	yes	—
83	C₂×C₁₉₁	#436	1	2	yes	—
84	C₁₉₁	#1851	1	1	yes	characteristic, complemented, Sylow
85	#1851	—	191	1	no	maximal, Sylow
86	C₂²×M₁₆	—	191	1	no	—
87	C₂×C₄⋊C₈	—	191	2	no	—
88	C₂×C₂²⋊C₈	D₁₉₁	1	1	yes	characteristic
89	C₄×C₂²⋊C₄	—	191	1	no	—
90	#336	—	191	2	no	—
91	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
92	C₂×M₁₆	—	382	2	no	—
93	C₂×M₁₆	—	382	4	no	—
94	C₂²×C₈	—	191	2	no	—
95	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
97	C₂×C₂²⋊C₄	—	191	2	no	—
98	C₂×C₄²	—	191	2	no	—
99	C₄⋊C₈	—	382	4	no	—
100	C₂²⋊C₈	—	2	2	no	—
101	#95	—	191	1	no	—
102	#95	—	191	1	no	—
103	C₂⁴	C₄×D₁₉₁	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	191	2	no	—
107	C₂²×C₄	—	191	2	no	—
108	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
110	C₂²×C₄	Dic₃₈₂	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
112	M₁₆	—	382	8	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	382	2	no	—
118	C₂×C₈	—	382	2	no	—
119	C₂×C₈	—	191	4	no	—
120	C₂²⋊C₄	—	382	4	no	—
121	C₄²	—	382	2	no	—
122	C₄²	—	382	2	no	—
123	C₂³	—	2	1	no	—
124	C₂³	—	2	1	no	—
125	C₂³	—	2	2	no	—
126	C₂³	—	2	2	no	—
127	C₂³	Dic₁₉₁⋊C₄	1	2	yes	—
128	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	382	2	no	—
134	C₂×C₄	—	382	2	no	—
135	C₂×C₄	—	382	2	no	—
136	C₂×C₄	—	382	2	no	—
137	C₂×C₄	—	382	2	no	—
138	C₂×C₄	—	191	4	no	—
139	C₂×C₄	Dic₁₉₁⋊C₄	1	2	yes	—
140	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
142	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
143	C₂×C₄	#15690286	1	2	yes	—
144	C₈	—	4	1	no	—
145	C₈	—	4	1	no	—
146	C₈	—	382	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂×Dic₁₉₁⋊C₄	1	1	yes	characteristic
153	C₂²	(C₂²⋊D₁₉₁)⋊C₄	1	2	yes	—
154	C₂²	M₁₆×D₁₉₁	1	2	yes	—
155	C₂²	#20259328	1	1	yes	characteristic
156	C₂²	#20259328	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	382	4	no	—
159	C₄	—	764	2	no	—
160	C₄	#20259290	1	1	yes	characteristic
161	C₄	#20259299	1	2	yes	—
162	C₄	#20259303	1	1	yes	characteristic
163	C₂	—	4	2	no	—
164	C₂	#27132535	1	1	yes	characteristic
165	C₂	#27132686	1	1	yes	characteristic
166	C₂	#27132689	1	1	yes	characteristic
167	C₂	#27132737	1	2	yes	—
168	C₂	#27132738	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#15690271

C₂

yes

maximal, characteristic

C₂×Dic₁₉₁⋊C₈

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₁₉₁

C₂

yes

maximal, characteristic

#27132417

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

—

C₂×#15690271

—

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₂²⋊Dic₁₉₁

C₄

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

—

Dic₁₉₁⋊C₈

—

C₁₉₁⋊#95

C₄

yes

characteristic

C₁₉₁⋊#95

C₄

yes

characteristic

C₂³×C₄×C₁₉₁

C₂²

yes

characteristic

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

C₂²⋊C₈×C₁₉₁

—

C₂²×Dic₁₉₁

C₂×C₄

yes

—

C₂²×Dic₁₉₁

C₂×C₄

yes

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₂²⋊Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂⁴×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

Q₈

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

M₁₆

yes

complemented

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₄⋊C₄

yes

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₄×D₄

yes

—

C₂²×C₁₉₁

C₂×M₁₆

yes

—

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₄×C₁₉₁

—

C₄×C₁₉₁

#120

yes

characteristic

C₄×C₁₉₁

C₂²⋊Q₈

yes

—

C₄×C₁₉₁

#123

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

—

C₂×C₁₉₁

#436

yes

—

C₁₉₁

#1851

yes

characteristic, complemented, Sylow

#1851

—

191

maximal, Sylow

C₂²×M₁₆

—

191

—

C₂×C₄⋊C₈

—

191

—

C₂×C₂²⋊C₈

D₁₉₁

yes

characteristic

C₄×C₂²⋊C₄

—

191

—

#336

—

191

—

C₂³×C₄

C₂×D₁₉₁

yes

characteristic

C₂×M₁₆

—

382

—

C₂×M₁₆

—

382

—

C₂²×C₈

—

191

—

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

C₂×C₂²⋊C₄

—

191

—

C₂×C₄²

—

191

—

C₄⋊C₈

—

382

—

100

C₂²⋊C₈

—

101

#95

—

191

—

102

#95

—

191

—

103

C₂⁴

C₄×D₁₉₁

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

191

—

107

C₂²×C₄

—

191

—

108

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

109

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

110

C₂²×C₄

Dic₃₈₂

yes

—

111

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

112

M₁₆

—

382

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

382

—

118

C₂×C₈

—

382

—

119

C₂×C₈

—

191

—

120

C₂²⋊C₄

—

382

—

121

C₄²

—

382

—

122

C₄²

—

382

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

Dic₁₉₁⋊C₄

yes

—

128

C₂³

C₂×C₄×D₁₉₁

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

382

—

134

C₂×C₄

—

382

—

135

C₂×C₄

—

382

—

136

C₂×C₄

—

382

—

137

C₂×C₄

—

382

—

138

C₂×C₄

—

191

—

139

C₂×C₄

Dic₁₉₁⋊C₄

yes

—

140

C₂×C₄

D₄×D₁₉₁

yes

—

141

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

142

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

143

C₂×C₄

#15690286

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

382

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×Dic₁₉₁⋊C₄

yes

characteristic

153

C₂²

(C₂²⋊D₁₉₁)⋊C₄

yes

—

154

C₂²

M₁₆×D₁₉₁

yes

—

155

C₂²

#20259328

yes

characteristic

156

C₂²

#20259328

yes

characteristic

157

C₄

—

158

C₄

—

382

—

159

C₄

—

764

—

160

C₄

#20259290

yes

characteristic

161

C₄

#20259299

yes

—

162

C₄

#20259303

yes

characteristic

163

C₂

—

164

C₂

#27132535

yes

characteristic

165

C₂

#27132686

yes

characteristic

166

C₂

#27132689

yes

characteristic

167

C₂

#27132737

yes

—

168

C₂

#27132738

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₁
commutator, socle	C₂³×C₁₉₁	C₂²×C₄
Fitting	C₂×C₂²⋊C₈×C₁₉₁	C₂
prime radical	C₂⁴×C₁₉₁	C₂×C₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₂×C₂²⋊C₈	D₁₉₁
hypocenter	C₁₉₁	#1851
LCS	G ▷ C₂³×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₁

commutator, socle

C₂³×C₁₉₁

C₂²×C₄

Fitting

C₂×C₂²⋊C₈×C₁₉₁

C₂

prime radical

C₂⁴×C₁₉₁

C₂×C₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

hypercenter

C₂×C₂²⋊C₈

D₁₉₁

C₁₉₁

G ▷ C₂³×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×Dic₁₉₁⋊C₈	C₂	semidirect
4. #27132417	C₂	semidirect
43. C₂×Dic₁₉₁	M₁₆	semidirect
84. C₁₉₁	#1851	semidirect

H ≅ G/N

type

2. C₂×Dic₁₉₁⋊C₈

C₂

semidirect

4. #27132417

C₂

semidirect

43. C₂×Dic₁₉₁

M₁₆

semidirect

84. C₁₉₁

#1851

semidirect

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

subgroup	index
C₂²×#15690271	2
C₂×Dic₁₉₁⋊C₈	2
C₄×C₂²⋊Dic₁₉₁	2
#27132417	2
C₂×C₂²⋊C₈×C₁₉₁	2
#1851	191

subgroup

index

C₂²×#15690271

C₂×Dic₁₉₁⋊C₈

C₄×C₂²⋊Dic₁₉₁

#27132417

C₂×C₂²⋊C₈×C₁₉₁

#1851

191

Subgroups — #37572702

Conjugacy classes of subgroups

Complements

Maximal subgroups