Group #37571759 — subgroups

Subgroups — #37571759

Order 24448.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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	#27133388	C₂	1	1	yes	maximal, characteristic
2	#27132945	C₂	1	1	yes	maximal, characteristic
3	#27132870	C₂	1	1	yes	maximal, characteristic
4	#27132409	C₂	1	1	yes	maximal, characteristic
5	#27132395	C₂	1	1	yes	maximal, characteristic
6	#27132395	C₂	1	1	yes	maximal, characteristic
7	#500×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#15690277	—	2	1	no	—
9	#20259320	—	2	1	no	—
10	#20259313	—	2	1	no	—
11	#20259312	—	2	2	no	—
12	Q₈×Dic₁₉₁	—	2	1	no	—
13	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
14	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
15	#20259287	—	2	1	no	—
16	#20259253	—	2	1	no	—
17	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
19	#20259246	—	2	1	no	—
20	#20259245	—	2	1	no	—
21	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
22	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
23	#107×C₁₉₁	—	2	2	no	—
24	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₃₈₂	D₄	1	1	yes	characteristic
26	C₂×Dic₃₈₂	D₄	1	1	yes	characteristic
27	#15690277	—	4	2	no	—
28	C₂×C₁₉₁⋊C₈	—	2	1	no	—
29	C₂×C₁₉₁⋊C₈	—	2	1	no	—
30	C₄⋊Dic₁₉₁	—	2	2	no	—
31	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
32	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
33	Dic₁₉₁⋊C₄	—	2	2	no	—
34	Dic₁₉₁⋊C₄	—	2	2	no	—
35	Dic₁₉₁⋊C₄	—	2	1	no	—
36	Dic₁₉₁⋊C₄	—	2	1	no	—
37	C₄×Dic₁₉₁	—	2	2	no	—
38	C₄×Dic₁₉₁	—	2	1	no	—
39	C₄×Dic₁₉₁	—	2	1	no	—
40	C₂×Q₈×C₁₉₁	—	2	1	no	—
41	C₂×Q₈×C₁₉₁	—	2	1	no	—
42	C₂×C₈×C₁₉₁	—	2	1	no	—
43	C₄⋊C₄×C₁₉₁	—	2	1	no	—
44	C₄⋊C₄×C₁₉₁	—	2	1	no	—
45	C₄⋊C₄×C₁₉₁	—	2	1	no	—
46	C₄⋊C₄×C₁₉₁	D₄	1	2	yes	—
47	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
48	Dic₃₈₂	—	2	1	no	—
49	Dic₃₈₂	—	2	1	no	—
50	Dic₃₈₂	—	2	1	no	—
51	Dic₃₈₂	Q₁₆	1	2	yes	complemented
52	C₂×Dic₁₉₁	—	2	1	no	—
53	C₂×Dic₁₉₁	—	2	1	no	—
54	C₂×Dic₁₉₁	—	2	1	no	—
55	C₂×Dic₁₉₁	—	2	1	no	—
56	C₁₉₁⋊C₈	—	4	1	no	—
57	C₁₉₁⋊C₈	—	4	1	no	—
58	Q₈×C₁₉₁	—	4	1	no	—
59	Q₈×C₁₉₁	—	4	1	no	—
60	Q₈×C₁₉₁	—	4	1	no	—
61	Q₈×C₁₉₁	—	4	1	no	—
62	C₂×C₄×C₁₉₁	—	2	2	no	—
63	C₂×C₄×C₁₉₁	—	4	1	no	—
64	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
67	C₈×C₁₉₁	—	2	2	no	—
68	C₂²×C₁₉₁	#120	1	1	yes	characteristic
69	C₄×C₁₉₁	—	2	1	no	—
70	C₄×C₁₉₁	—	4	2	no	—
71	C₄×C₁₉₁	—	8	1	no	—
72	C₄×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₁₉₁	#135	1	1	yes	characteristic
74	C₄×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₁₉₁	#137	1	1	yes	characteristic
76	Dic₁₉₁	—	2	2	no	—
77	Dic₁₉₁	—	4	1	no	—
78	Dic₁₉₁	—	4	1	no	—
79	Dic₁₉₁	—	4	1	no	—
80	C₂×C₁₉₁	#449	1	1	yes	characteristic
81	C₂×C₁₉₁	#451	1	1	yes	characteristic
82	C₂×C₁₉₁	#456	1	1	yes	characteristic
83	C₁₉₁	#1648	1	1	yes	characteristic, complemented, Sylow
84	#1648	—	191	1	no	maximal, Sylow
85	#557	—	191	1	no	—
86	#500	D₁₉₁	1	1	yes	characteristic
87	#475	—	191	1	no	—
88	#462	—	191	1	no	—
89	#332	—	191	1	no	—
90	Q₈⋊C₈	—	191	1	no	—
91	Q₈⋊C₈	—	191	1	no	—
92	C₂×Q₁₆	—	382	1	no	—
93	C₄⋊Q₈	—	382	1	no	—
94	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
95	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
96	#125	—	382	1	no	—
97	#125	—	382	2	no	—
98	C₄×Q₈	—	382	1	no	—
99	C₄×Q₈	—	382	1	no	—
100	C₄×Q₈	—	191	1	no	—
101	C₄×Q₈	—	191	1	no	—
102	#107	—	2	2	no	—
103	#106	—	382	1	no	—
104	C₄⋊C₈	—	191	1	no	—
105	C₄⋊C₈	—	191	1	no	—
106	#103	—	382	1	no	—
107	#103	—	382	1	no	—
108	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	2	1	no	—
111	C₂×Q₈	—	191	1	no	—
112	C₂×Q₈	—	191	1	no	—
113	Q₁₆	—	764	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	382	1	no	—
116	C₂×C₈	—	382	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	382	2	no	—
121	C₄⋊C₄	—	382	2	no	—
122	C₄⋊C₄	—	382	2	no	—
123	C₄⋊C₄	—	382	1	no	—
124	C₄⋊C₄	—	382	1	no	—
125	C₄⋊C₄	—	191	1	no	—
126	C₄⋊C₄	—	191	1	no	—
127	C₄⋊C₄	C₂²⋊D₁₉₁	1	2	yes	—
128	C₄²	—	382	2	no	—
129	C₄²	—	382	1	no	—
130	C₄²	—	382	1	no	—
131	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
132	Q₈	—	4	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	4	1	no	—
136	Q₈	—	382	1	no	—
137	Q₈	—	382	1	no	—
138	Q₈	—	382	1	no	—
139	Q₈	—	191	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	382	1	no	—
143	C₂×C₄	—	382	1	no	—
144	C₂×C₄	—	382	1	no	—
145	C₂×C₄	—	382	1	no	—
146	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
147	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	764	1	no	—
151	C₈	—	764	1	no	—
152	C₂²	#20259290	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	2	no	—
155	C₄	—	8	1	no	—
156	C₄	—	382	2	no	—
157	C₄	—	764	1	no	—
158	C₄	—	764	1	no	—
159	C₄	—	764	1	no	—
160	C₄	Q₁₆×D₁₉₁	1	1	yes	characteristic
161	C₄	#20259341	1	1	yes	characteristic
162	C₄	#20259346	1	1	yes	characteristic
163	C₄	#20259354	1	1	yes	characteristic
164	C₂	#27132805	1	1	yes	characteristic
165	C₂	#27132817	1	1	yes	characteristic
166	C₂	#27132845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133388

C₂

yes

maximal, characteristic

#27132945

C₂

yes

maximal, characteristic

#27132870

C₂

yes

maximal, characteristic

#27132409

C₂

yes

maximal, characteristic

#27132395

C₂

yes

maximal, characteristic

#27132395

C₂

yes

maximal, characteristic

#500×C₁₉₁

C₂

yes

maximal, characteristic

—

—

—

—

Q₈×Dic₁₉₁

—

C₄×Dic₃₈₂

C₂²

yes

characteristic

C₄×Dic₃₈₂

C₂²

yes

characteristic

#20259287

—

#20259253

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

characteristic

#20259246

—

#20259245

—

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

#107×C₁₉₁

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

D₄

yes

characteristic

C₂×Dic₃₈₂

D₄

yes

characteristic

#15690277

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

D₄

yes

characteristic

C₄⋊Dic₁₉₁

D₄

yes

characteristic

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

Q₁₆

yes

complemented

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

#120

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×Q₁₆

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1648

—

191

maximal, Sylow

#557

—

191

—

#500

D₁₉₁

yes

characteristic

#475

—

191

—

#462

—

191

—

#332

—

191

—

Q₈⋊C₈

—

191

—

Q₈⋊C₈

—

191

—

C₂×Q₁₆

—

382

—

C₄⋊Q₈

—

382

—

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

#125

—

382

—

#125

—

382

—

C₄×Q₈

—

382

—

C₄×Q₈

—

382

—

100

C₄×Q₈

—

191

—

101

C₄×Q₈

—

191

—

102

#107

—

103

#106

—

382

—

104

C₄⋊C₈

—

191

—

105

C₄⋊C₈

—

191

—

106

#103

—

382

—

107

#103

—

382

—

108

C₄×C₈

C₂×D₁₉₁

yes

characteristic

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

191

—

112

C₂×Q₈

—

191

—

113

Q₁₆

—

764

—

114

C₂×C₈

—

115

C₂×C₈

—

382

—

116

C₂×C₈

—

382

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

382

—

121

C₄⋊C₄

—

382

—

122

C₄⋊C₄

—

382

—

123

C₄⋊C₄

—

382

—

124

C₄⋊C₄

—

382

—

125

C₄⋊C₄

—

191

—

126

C₄⋊C₄

—

191

—

127

C₄⋊C₄

C₂²⋊D₁₉₁

yes

—

128

C₄²

—

382

—

129

C₄²

—

382

—

130

C₄²

—

382

—

131

C₄²

C₂²×D₁₉₁

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

382

—

137

Q₈

—

382

—

138

Q₈

—

382

—

139

Q₈

—

191

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

382

—

143

C₂×C₄

—

382

—

144

C₂×C₄

—

382

—

145

C₂×C₄

—

382

—

146

C₂×C₄

D₄×D₁₉₁

yes

characteristic

147

C₂×C₄

D₄×D₁₉₁

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

149

C₈

—

150

C₈

—

764

—

151

C₈

—

764

—

152

C₂²

#20259290

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

382

—

157

C₄

—

764

—

158

C₄

—

764

—

159

C₄

—

764

—

160

C₄

Q₁₆×D₁₉₁

yes

characteristic

161

C₄

#20259341

yes

characteristic

162

C₄

#20259346

yes

characteristic

163

C₄

#20259354

yes

characteristic

164

C₂

#27132805

yes

characteristic

165

C₂

#27132817

yes

characteristic

166

C₂

#27132845

yes

characteristic

	subgroup	G/H
center	C₂²	#20259290
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#500×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	#120
hypercenter	#500	D₁₉₁
hypocenter	C₁₉₁	#1648
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#20259290

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#500×C₁₉₁

C₂

C₂²×C₁₉₁

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
51. Dic₃₈₂	Q₁₆	semidirect
83. C₁₉₁	#1648	semidirect

H ≅ G/N

type

51. Dic₃₈₂

Q₁₆

semidirect

83. C₁₉₁

#1648

semidirect

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

subgroup	index
#27133388	2
#27132945	2
#27132870	2
#27132409	2
#27132395	2
#500×C₁₉₁	2
#1648	191

subgroup

index

#500×C₁₉₁

191

Subgroups — #37571759

Conjugacy classes of subgroups

Complements

Maximal subgroups