Group #37532997 — subgroups

Subgroups — #37532997

Order 23168.

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	#26042007	C₂	1	1	yes	maximal, characteristic
2	#26041675	C₂	1	1	yes	maximal, characteristic
3	#26041486	C₂	1	1	yes	maximal, characteristic
4	#26041026	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊#13089678	C₂	1	1	yes	maximal, characteristic
6	#26041009	C₂	1	1	yes	maximal, characteristic
7	#475×C₁₈₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#15668439	—	2	1	no	—
9	#19153816	C₂²	1	1	yes	characteristic
10	#19153814	—	2	1	no	—
11	#19153808	—	2	1	no	—
12	#19153806	—	2	2	no	—
13	C₄×Dic₃₆₂	—	2	1	no	—
14	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
15	#19153781	—	2	1	no	—
16	#19153747	—	2	2	no	—
17	C₄⋊#13089678	C₂²	1	1	yes	characteristic
18	#19153740	—	2	1	no	—
19	C₄×#13089678	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
22	#106×C₁₈₁	—	2	1	no	—
23	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
24	#103×C₁₈₁	—	2	1	no	—
25	C₂×Dic₃₆₂	—	2	1	no	—
26	C₂×Dic₃₆₂	D₄	1	1	yes	characteristic
27	#15668439	—	4	2	no	—
28	C₂×#13089678	—	2	1	no	—
29	C₂×#13089678	—	2	1	no	—
30	C₄⋊Dic₁₈₁	—	2	1	no	—
31	C₄⋊Dic₁₈₁	D₄	1	1	yes	characteristic
32	C₄⋊Dic₁₈₁	D₄	1	2	yes	—
33	#15668428	—	2	2	no	—
34	#15668428	—	2	2	no	—
35	#15668428	—	2	2	no	—
36	#15668428	—	2	1	no	—
37	C₄×Dic₁₈₁	—	2	2	no	—
38	C₄×Dic₁₈₁	—	2	1	no	—
39	C₂×Q₈×C₁₈₁	—	2	1	no	—
40	C₂×Q₈×C₁₈₁	D₄	1	1	yes	characteristic
41	C₂×C₈×C₁₈₁	—	2	1	no	—
42	C₄⋊C₄×C₁₈₁	—	2	1	no	—
43	C₄⋊C₄×C₁₈₁	—	2	1	no	—
44	C₄⋊C₄×C₁₈₁	—	2	1	no	—
45	C₄⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
46	C₄²×C₁₈₁	—	2	1	no	—
47	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
48	Dic₃₆₂	—	2	1	no	—
49	Dic₃₆₂	—	4	1	no	—
50	Dic₃₆₂	—	4	1	no	—
51	Dic₃₆₂	SD₁₆	1	2	yes	—
52	C₂×Dic₁₈₁	—	2	1	no	—
53	C₂×Dic₁₈₁	—	2	1	no	—
54	C₂×Dic₁₈₁	—	2	2	no	—
55	#13089678	—	2	2	no	—
56	#13089678	—	4	1	no	—
57	Q₈×C₁₈₁	—	2	1	no	—
58	Q₈×C₁₈₁	—	2	1	no	—
59	Q₈×C₁₈₁	—	4	1	no	—
60	Q₈×C₁₈₁	—	4	1	no	—
61	C₂×C₄×C₁₈₁	—	2	1	no	—
62	C₂×C₄×C₁₈₁	—	2	1	no	—
63	C₂×C₄×C₁₈₁	—	4	1	no	—
64	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
67	C₈×C₁₈₁	—	4	1	no	—
68	C₂²×C₁₈₁	#120	1	1	yes	characteristic
69	C₄×C₁₈₁	—	2	1	no	—
70	C₄×C₁₈₁	—	4	1	no	—
71	C₄×C₁₈₁	—	4	1	no	—
72	C₄×C₁₈₁	—	8	1	no	—
73	C₄×C₁₈₁	C₂×SD₁₆	1	1	yes	characteristic
74	C₄×C₁₈₁	#135	1	1	yes	characteristic
75	C₄×C₁₈₁	#137	1	1	yes	characteristic
76	C₄×C₁₈₁	#137	1	1	yes	characteristic
77	Dic₁₈₁	—	2	2	no	—
78	Dic₁₈₁	—	4	2	no	—
79	Dic₁₈₁	—	4	1	no	—
80	C₂×C₁₈₁	#448	1	1	yes	characteristic
81	C₂×C₁₈₁	#452	1	1	yes	characteristic
82	C₂×C₁₈₁	#456	1	1	yes	characteristic
83	C₁₈₁	#1653	1	1	yes	characteristic, complemented, Sylow
84	#1653	—	181	1	no	maximal, Sylow
85	#557	—	181	1	no	—
86	#498	—	181	1	no	—
87	#475	D₁₈₁	1	1	yes	characteristic
88	#462	—	181	1	no	—
89	#332	—	181	1	no	—
90	Q₈⋊C₈	—	181	1	no	—
91	Q₈⋊C₈	—	181	1	no	—
92	C₂×Q₁₆	—	362	1	no	—
93	C₄⋊Q₈	—	362	1	no	—
94	C₄⋊Q₈	—	181	1	no	—
95	C₄⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
96	#125	—	362	1	no	—
97	#125	—	362	2	no	—
98	C₄×Q₈	—	362	1	no	—
99	C₄×Q₈	—	362	1	no	—
100	C₄×Q₈	—	181	1	no	—
101	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
102	#106	—	2	1	no	—
103	#106	—	362	2	no	—
104	C₄⋊C₈	—	181	1	no	—
105	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
106	#103	—	2	1	no	—
107	#103	—	362	1	no	—
108	C₄×C₈	—	181	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	362	1	no	—
111	C₂×Q₈	—	181	1	no	—
112	C₂×Q₈	C₂²⋊D₁₈₁	1	1	yes	characteristic
113	Q₁₆	—	724	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	362	1	no	—
116	C₂×C₈	—	362	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	362	2	no	—
121	C₄⋊C₄	—	362	2	no	—
122	C₄⋊C₄	—	362	2	no	—
123	C₄⋊C₄	—	362	1	no	—
124	C₄⋊C₄	—	362	1	no	—
125	C₄⋊C₄	—	181	1	no	—
126	C₄⋊C₄	—	181	2	no	—
127	C₄⋊C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	362	2	no	—
130	C₄²	—	362	1	no	—
131	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	2	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	4	1	no	—
136	Q₈	—	362	1	no	—
137	Q₈	—	181	2	no	—
138	Q₈	—	724	1	no	—
139	Q₈	—	724	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	362	1	no	—
144	C₂×C₄	—	362	1	no	—
145	C₂×C₄	—	362	2	no	—
146	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
147	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	362	2	no	—
151	C₈	—	724	1	no	—
152	C₂²	#19153784	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	362	2	no	—
158	C₄	—	724	2	no	—
159	C₄	—	724	1	no	—
160	C₄	SD₁₆×D₁₈₁	1	1	yes	characteristic
161	C₄	#19153836	1	1	yes	characteristic
162	C₄	#19153845	1	1	yes	characteristic
163	C₄	#19153847	1	1	yes	characteristic
164	C₂	#26041416	1	1	yes	characteristic
165	C₂	#26041439	1	1	yes	characteristic
166	C₂	#26041460	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26042007

C₂

yes

maximal, characteristic

#26041675

C₂

yes

maximal, characteristic

#26041486

C₂

yes

maximal, characteristic

#26041026

C₂

yes

maximal, characteristic

Q₈⋊#13089678

C₂

yes

maximal, characteristic

#26041009

C₂

yes

maximal, characteristic

#475×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668439

—

#19153816

C₂²

yes

—

—

—

C₄×Dic₃₆₂

—

C₄×Dic₃₆₂

C₂²

yes

—

—

C₂²

yes

characteristic

#19153740

—

C₄×#13089678

C₂²

yes

characteristic

C₄⋊Q₈×C₁₈₁

C₂²

yes

characteristic

C₄×Q₈×C₁₈₁

C₂²

yes

characteristic

#106×C₁₈₁

—

C₄⋊C₈×C₁₈₁

C₂²

yes

characteristic

#103×C₁₈₁

—

C₂×Dic₃₆₂

—

C₂×Dic₃₆₂

D₄

yes

—

—

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

D₄

yes

characteristic

C₄⋊Dic₁₈₁

D₄

yes

—

—

—

—

—

C₄×Dic₁₈₁

—

C₄×Dic₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂×Q₈×C₁₈₁

D₄

yes

characteristic

C₂×C₈×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

D₄

yes

characteristic

C₄²×C₁₈₁

—

C₄²×C₁₈₁

C₂³

yes

characteristic

Dic₃₆₂

—

Dic₃₆₂

—

Dic₃₆₂

—

Dic₃₆₂

SD₁₆

yes

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

—

#13089678

—

Q₈×C₁₈₁

—

Q₈×C₁₈₁

—

Q₈×C₁₈₁

—

Q₈×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₈×C₁₈₁

—

C₂²×C₁₈₁

#120

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂×SD₁₆

yes

C₄×C₁₈₁

yes

C₄×C₁₈₁

yes

C₄×C₁₈₁

yes

Dic₁₈₁

—

Dic₁₈₁

—

Dic₁₈₁

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, complemented, Sylow

#1653

—

181

maximal, Sylow

#557

—

181

—

#498

—

181

—

#475

D₁₈₁

yes

characteristic

#462

—

181

—

#332

—

181

—

Q₈⋊C₈

—

181

—

Q₈⋊C₈

—

181

—

C₂×Q₁₆

—

362

—

C₄⋊Q₈

—

362

—

C₄⋊Q₈

—

181

—

C₄⋊Q₈

C₂×D₁₈₁

yes

characteristic

#125

—

362

—

#125

—

362

—

C₄×Q₈

—

362

—

C₄×Q₈

—

362

—

100

C₄×Q₈

—

181

—

101

C₄×Q₈

C₂×D₁₈₁

yes

characteristic

102

#106

—

103

#106

—

362

—

104

C₄⋊C₈

—

181

—

105

C₄⋊C₈

C₂×D₁₈₁

yes

characteristic

106

#103

—

107

#103

—

362

—

108

C₄×C₈

—

181

—

109

C₂×Q₈

—

110

C₂×Q₈

—

362

—

111

C₂×Q₈

—

181

—

112

C₂×Q₈

C₂²⋊D₁₈₁

yes

characteristic

113

Q₁₆

—

724

—

114

C₂×C₈

—

115

C₂×C₈

—

362

—

116

C₂×C₈

—

362

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

362

—

121

C₄⋊C₄

—

362

—

122

C₄⋊C₄

—

362

—

123

C₄⋊C₄

—

362

—

124

C₄⋊C₄

—

362

—

125

C₄⋊C₄

—

181

—

126

C₄⋊C₄

—

181

—

127

C₄⋊C₄

C₂²⋊D₁₈₁

yes

characteristic

128

C₄²

—

129

C₄²

—

362

—

130

C₄²

—

362

—

131

C₄²

C₂²×D₁₈₁

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

362

—

137

Q₈

—

181

—

138

Q₈

—

724

—

139

Q₈

—

724

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

362

—

144

C₂×C₄

—

362

—

145

C₂×C₄

—

362

—

146

C₂×C₄

D₄×D₁₈₁

yes

characteristic

147

C₂×C₄

D₄×D₁₈₁

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

characteristic

149

C₈

—

150

C₈

—

362

—

151

C₈

—

724

—

152

C₂²

#19153784

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

362

—

158

C₄

—

724

—

159

C₄

—

724

—

160

C₄

SD₁₆×D₁₈₁

yes

characteristic

161

C₄

#19153836

yes

characteristic

162

C₄

#19153845

yes

characteristic

163

C₄

#19153847

yes

characteristic

164

C₂

#26041416

yes

characteristic

165

C₂

#26041439

yes

characteristic

166

C₂

#26041460

yes

characteristic

	subgroup	G/H
center	C₂²	#19153784
commutator, cosocle	C₄²×C₁₈₁	C₂³
Frattini	C₄²	C₂²×D₁₈₁
Fitting	#475×C₁₈₁	C₂
socle, prime radical	C₂²×C₁₈₁	#120
hypercenter	#475	D₁₈₁
hypocenter	C₁₈₁	#1653
LCS	G ▷ C₄²×C₁₈₁ ▷ C₂²×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#19153784

commutator, cosocle

C₄²×C₁₈₁

C₂³

Frattini

C₄²

C₂²×D₁₈₁

Fitting

#475×C₁₈₁

C₂

C₂²×C₁₈₁

D₁₈₁

C₁₈₁

G ▷ C₄²×C₁₈₁ ▷ C₂²×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
83. C₁₈₁	#1653	semidirect

H ≅ G/N

type

83. C₁₈₁

#1653

semidirect

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

subgroup	index
#26042007	2
#26041675	2
#26041486	2
#26041026	2
Q₈⋊#13089678	2
#26041009	2
#475×C₁₈₁	2
#1653	181

subgroup

index

#475×C₁₈₁

181

Subgroups — #37532997

Conjugacy classes of subgroups

Complements

Maximal subgroups