Group #37571680 — subgroups

Subgroups — #37571680

Order 24448.

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

normal subgroups only

This group has 188 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	#27133391	C₂	1	1	yes	maximal, characteristic
2	#27133014	C₂	1	1	yes	maximal, characteristic
3	#27132959	C₂	1	1	yes	maximal, characteristic
4	#27132958	C₂	1	1	yes	maximal, characteristic
5	#27132408	C₂	1	1	yes	maximal, characteristic
6	#27132395	C₂	1	1	yes	maximal, characteristic
7	Q₈⋊C₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	#20259322	C₂²	1	1	yes	characteristic
9	#20259322	C₂²	1	1	yes	characteristic
10	#20259320	—	2	1	no	—
11	#20259314	—	2	1	no	—
12	#20259312	—	2	2	no	—
13	C₄×Dic₃₈₂	—	2	1	no	—
14	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
15	#20259287	—	2	1	no	—
16	#20259256	—	2	1	no	—
17	#20259255	—	2	1	no	—
18	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
19	#20259247	—	2	1	no	—
20	#20259247	—	2	2	no	—
21	#20259246	—	2	1	no	—
22	C₄×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₁₉₁	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₃₈₂	—	2	1	no	—
26	C₂×Dic₃₈₂	—	2	1	no	—
27	C₂×Dic₃₈₂	D₄	1	1	yes	characteristic
28	C₂×C₁₉₁⋊C₈	—	2	1	no	—
29	C₄⋊Dic₁₉₁	—	2	1	no	—
30	C₄⋊Dic₁₉₁	—	2	1	no	—
31	C₄⋊Dic₁₉₁	—	2	1	no	—
32	C₄⋊Dic₁₉₁	D₄	1	2	yes	—
33	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
34	Dic₁₉₁⋊C₄	—	2	2	no	—
35	Dic₁₉₁⋊C₄	—	2	2	no	—
36	Dic₁₉₁⋊C₄	—	2	2	no	—
37	Dic₁₉₁⋊C₄	—	2	1	no	—
38	C₄×Dic₁₉₁	—	2	2	no	—
39	C₄×Dic₁₉₁	—	2	1	no	—
40	C₂×Q₈×C₁₉₁	D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₉₁	—	2	1	no	—
42	C₂×C₈×C₁₉₁	—	2	1	no	—
43	C₄⋊C₄×C₁₉₁	—	2	1	no	—
44	C₄⋊C₄×C₁₉₁	D₄	1	1	yes	characteristic
45	C₄²×C₁₉₁	—	2	1	no	—
46	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
47	Dic₃₈₂	—	2	1	no	—
48	Dic₃₈₂	—	4	1	no	—
49	Dic₃₈₂	—	4	1	no	—
50	Dic₃₈₂	—	4	2	no	—
51	Dic₃₈₂	Q₁₆	1	2	yes	—
52	C₂×Dic₁₉₁	—	2	1	no	—
53	C₂×Dic₁₉₁	—	2	1	no	—
54	C₂×Dic₁₉₁	—	2	2	no	—
55	C₂×Dic₁₉₁	—	4	1	no	—
56	C₁₉₁⋊C₈	—	4	1	no	—
57	Q₈×C₁₉₁	—	2	1	no	—
58	Q₈×C₁₉₁	—	2	1	no	—
59	C₂×C₄×C₁₉₁	—	2	1	no	—
60	C₂×C₄×C₁₉₁	—	2	1	no	—
61	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
64	C₈×C₁₉₁	—	4	1	no	—
65	C₈×C₁₉₁	—	4	1	no	—
66	C₂²×C₁₉₁	#120	1	1	yes	characteristic
67	C₄×C₁₉₁	—	2	1	no	—
68	C₄×C₁₉₁	—	4	1	no	—
69	C₄×C₁₉₁	—	4	1	no	—
70	C₄×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₄×C₁₉₁	#135	1	1	yes	characteristic
72	C₄×C₁₉₁	#137	1	1	yes	characteristic
73	C₄×C₁₉₁	#137	1	1	yes	characteristic
74	Dic₁₉₁	—	2	2	no	—
75	Dic₁₉₁	—	4	2	no	—
76	Dic₁₉₁	—	4	1	no	—
77	Dic₁₉₁	—	8	1	no	—
78	C₂×C₁₉₁	#451	1	1	yes	characteristic
79	C₂×C₁₉₁	#452	1	1	yes	characteristic
80	C₂×C₁₉₁	#454	1	1	yes	characteristic
81	C₁₉₁	#1636	1	1	yes	characteristic, complemented, Sylow
82	#1636	—	191	1	no	maximal, Sylow
83	#557	—	191	1	no	—
84	#487	—	191	1	no	—
85	#477	—	191	1	no	—
86	#477	—	191	1	no	—
87	#332	—	191	1	no	—
88	Q₈⋊C₈	—	191	1	no	—
89	Q₈⋊C₈	D₁₉₁	1	1	yes	characteristic
90	C₄⋊Q₈	—	382	1	no	—
91	C₄⋊Q₈	—	191	1	no	—
92	C₄⋊Q₈	—	191	1	no	—
93	#125	—	382	1	no	—
94	#125	—	382	2	no	—
95	C₄×Q₈	—	382	1	no	—
96	C₄×Q₈	—	382	1	no	—
97	C₄×Q₈	—	191	1	no	—
98	C₄×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
99	#107	—	382	1	no	—
100	#107	—	382	1	no	—
101	C₄⋊C₈	—	191	1	no	—
102	C₄⋊C₈	C₂×D₁₉₁	1	1	yes	characteristic
103	#103	—	382	1	no	—
104	#103	—	382	1	no	—
105	#103	—	382	2	no	—
106	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
107	C₂×Q₈	—	382	1	no	—
108	C₂×Q₈	—	382	1	no	—
109	C₂×Q₈	—	191	1	no	—
110	C₂×Q₈	D₇₆₄	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	382	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	382	2	no	—
116	C₄⋊C₄	—	382	2	no	—
117	C₄⋊C₄	—	382	2	no	—
118	C₄⋊C₄	—	382	1	no	—
119	C₄⋊C₄	—	382	1	no	—
120	C₄⋊C₄	—	382	1	no	—
121	C₄⋊C₄	—	382	1	no	—
122	C₄⋊C₄	—	191	2	no	—
123	C₄⋊C₄	—	191	1	no	—
124	C₄⋊C₄	D₇₆₄	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	—	382	2	no	—
127	C₄²	—	382	1	no	—
128	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	2	1	no	—
131	Q₈	—	382	1	no	—
132	Q₈	—	191	2	no	—
133	Q₈	—	764	2	no	—
134	Q₈	—	764	1	no	—
135	Q₈	—	764	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	382	1	no	—
139	C₂×C₄	—	382	1	no	—
140	C₂×C₄	—	382	2	no	—
141	C₂×C₄	—	764	1	no	—
142	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
143	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
144	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	4	1	no	—
147	C₈	—	764	1	no	—
148	C₂²	#20259292	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	382	2	no	—
153	C₄	—	1528	1	no	—
154	C₄	—	764	2	no	—
155	C₄	—	764	1	no	—
156	C₄	Q₁₆×D₁₉₁	1	1	yes	characteristic
157	C₄	#20259344	1	1	yes	characteristic
158	C₄	#20259351	1	1	yes	characteristic
159	C₄	#20259355	1	1	yes	characteristic
160	C₂	#27132820	1	1	yes	characteristic
161	C₂	#27132827	1	1	yes	characteristic
162	C₂	#27132837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133391

C₂

yes

maximal, characteristic

#27133014

C₂

yes

maximal, characteristic

#27132959

C₂

yes

maximal, characteristic

#27132958

C₂

yes

maximal, characteristic

#27132408

C₂

yes

maximal, characteristic

#27132395

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₁₉₁

C₂

yes

maximal, characteristic

#20259322

C₂²

yes

characteristic

#20259322

C₂²

yes

—

—

—

C₄×Dic₃₈₂

—

C₄×Dic₃₈₂

C₂²

yes

—

—

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

—

—

—

C₄×Q₈×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

D₄

yes

characteristic

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

D₄

yes

—

C₄⋊Dic₁₉₁

D₄

yes

characteristic

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂×Q₈×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

D₄

yes

characteristic

C₄²×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

Q₁₆

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

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

—

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

#1636

—

191

maximal, Sylow

#557

—

191

—

#487

—

191

—

#477

—

191

—

#477

—

191

—

#332

—

191

—

Q₈⋊C₈

—

191

—

Q₈⋊C₈

D₁₉₁

yes

characteristic

C₄⋊Q₈

—

382

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

—

191

—

#125

—

382

—

#125

—

382

—

C₄×Q₈

—

382

—

C₄×Q₈

—

382

—

C₄×Q₈

—

191

—

C₄×Q₈

C₂×D₁₉₁

yes

characteristic

#107

—

382

—

100

#107

—

382

—

101

C₄⋊C₈

—

191

—

102

C₄⋊C₈

C₂×D₁₉₁

yes

characteristic

103

#103

—

382

—

104

#103

—

382

—

105

#103

—

382

—

106

C₄×C₈

C₂×D₁₉₁

yes

characteristic

107

C₂×Q₈

—

382

—

108

C₂×Q₈

—

382

—

109

C₂×Q₈

—

191

—

110

C₂×Q₈

D₇₆₄

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

382

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

382

—

116

C₄⋊C₄

—

382

—

117

C₄⋊C₄

—

382

—

118

C₄⋊C₄

—

382

—

119

C₄⋊C₄

—

382

—

120

C₄⋊C₄

—

382

—

121

C₄⋊C₄

—

382

—

122

C₄⋊C₄

—

191

—

123

C₄⋊C₄

—

191

—

124

C₄⋊C₄

D₇₆₄

yes

characteristic

125

C₄²

—

126

C₄²

—

382

—

127

C₄²

—

382

—

128

C₄²

C₂²×D₁₉₁

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

382

—

132

Q₈

—

191

—

133

Q₈

—

764

—

134

Q₈

—

764

—

135

Q₈

—

764

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

382

—

139

C₂×C₄

—

382

—

140

C₂×C₄

—

382

—

141

C₂×C₄

—

764

—

142

C₂×C₄

D₄×D₁₉₁

yes

characteristic

143

C₂×C₄

D₄×D₁₉₁

yes

characteristic

144

C₂×C₄

C₂×D₇₆₄

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

764

—

148

C₂²

#20259292

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

382

—

153

C₄

—

1528

—

154

C₄

—

764

—

155

C₄

—

764

—

156

C₄

Q₁₆×D₁₉₁

yes

characteristic

157

C₄

#20259344

yes

characteristic

158

C₄

#20259351

yes

characteristic

159

C₄

#20259355

yes

characteristic

160

C₂

#27132820

yes

characteristic

161

C₂

#27132827

yes

characteristic

162

C₂

#27132837

yes

characteristic

	subgroup	G/H
center	C₂²	#20259292
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	Q₈⋊C₈×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	#120
hypercenter	Q₈⋊C₈	D₁₉₁
hypocenter	C₁₉₁	#1636
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#20259292

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

Q₈⋊C₈×C₁₉₁

C₂

socle, prime radical

C₂²×C₁₉₁

#120

hypercenter

Q₈⋊C₈

D₁₉₁

hypocenter

C₁₉₁

#1636

LCS

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
81. C₁₉₁	#1636	semidirect

H ≅ G/N

type

81. C₁₉₁

#1636

semidirect

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

subgroup	index
#27133391	2
#27133014	2
#27132959	2
#27132958	2
#27132408	2
#27132395	2
Q₈⋊C₈×C₁₉₁	2
#1636	191

subgroup

index

Q₈⋊C₈×C₁₉₁

191

Subgroups — #37571680

Conjugacy classes of subgroups

Complements

Maximal subgroups