Group #37546726 — subgroups

Subgroups — #37546726

Order 23168.

348 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 125 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	2	yes	maximal
2	#26041659	C₂	1	1	yes	maximal, characteristic
3	#26041657	C₂	1	2	yes	maximal
4	#26041489	C₂	1	2	yes	maximal
5	#26041488	C₂	1	4	yes	maximal
6	C₈×Dic₃₆₂	C₂	1	1	yes	maximal, characteristic
7	#26041373	C₂	1	2	yes	maximal
8	C₄×Q₁₆×C₁₈₁	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₇₂₄	C₂²	1	2	yes	—
10	C₂×Dic₇₂₄	C₂²	1	2	yes	—
11	C₂×#15668439	C₂²	1	4	yes	—
12	#19153816	C₂²	1	2	yes	—
13	#19153814	C₂²	1	4	yes	—
14	#19153808	—	2	2	no	—
15	#19153806	—	2	4	no	—
16	C₄×Dic₃₆₂	—	2	2	no	—
17	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
18	#19153781	C₂²	1	4	yes	—
19	C₄⋊#13089678	C₂²	1	1	yes	characteristic
20	#19153745	C₂²	1	2	yes	—
21	#19153739	C₂²	1	4	yes	—
22	C₈×Dic₁₈₁	C₂²	1	2	yes	—
23	C₂×Q₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₈₁	C₂²	1	2	yes	—
25	#107×C₁₈₁	C₂²	1	1	yes	characteristic
26	#103×C₁₈₁	C₂²	1	2	yes	—
27	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
28	C₂×Dic₃₆₂	D₄	1	1	yes	characteristic
29	C₂×Dic₃₆₂	C₂³	1	4	yes	—
30	Dic₇₂₄	—	2	4	no	—
31	Dic₇₂₄	—	4	2	no	—
32	#15668439	—	4	4	no	—
33	C₂×#13089678	C₂³	1	2	yes	—
34	C₄⋊Dic₁₈₁	—	2	2	no	—
35	C₄⋊Dic₁₈₁	—	2	2	no	—
36	C₄⋊Dic₁₈₁	D₄	1	1	yes	characteristic
37	#15668428	—	2	4	no	—
38	#15668428	—	2	4	no	—
39	#15668428	—	2	4	no	—
40	#15668428	D₄	1	2	yes	—
41	C₄×Dic₁₈₁	—	2	4	no	—
42	C₄×Dic₁₈₁	C₂³	1	2	yes	—
43	C₂×Q₈×C₁₈₁	C₂³	1	2	yes	—
44	Q₁₆×C₁₈₁	—	2	2	no	—
45	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
46	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₈₁	—	2	2	no	—
48	C₄⋊C₄×C₁₈₁	C₂³	1	2	yes	—
49	C₄²×C₁₈₁	—	2	2	no	—
50	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
51	Dic₃₆₂	—	2	4	no	—
52	Dic₃₆₂	—	2	4	no	—
53	Dic₃₆₂	Q₁₆	1	4	yes	complemented
54	C₂×Dic₁₈₁	—	2	4	no	—
55	C₂×Dic₁₈₁	C₂×D₄	1	2	yes	—
56	C₂×Dic₁₈₁	C₂×D₄	1	2	yes	—
57	#13089678	—	2	2	no	—
58	Q₈×C₁₈₁	—	2	2	no	—
59	Q₈×C₁₈₁	—	2	2	no	—
60	C₂×C₄×C₁₈₁	—	2	2	no	—
61	C₂×C₄×C₁₈₁	—	2	2	no	—
62	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
65	C₈×C₁₈₁	—	2	1	no	—
66	C₈×C₁₈₁	#41	1	2	yes	—
67	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₁₈₁	—	2	1	no	—
69	C₄×C₁₈₁	—	4	2	no	—
70	C₄×C₁₈₁	—	4	2	no	—
71	C₄×C₁₈₁	C₂×Q₁₆	1	2	yes	—
72	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
73	C₄×C₁₈₁	H(5,𝔽₂)	1	1	yes	characteristic
74	Dic₁₈₁	—	2	2	no	—
75	Dic₁₈₁	—	4	4	no	—
76	Dic₁₈₁	C₂×Q₁₆	1	4	yes	—
77	C₂×C₁₈₁	#550	1	1	yes	characteristic
78	C₂×C₁₈₁	C₂²×Q₁₆	1	1	yes	characteristic
79	C₂×C₁₈₁	#578	1	1	yes	characteristic
80	C₁₈₁	#3395	1	1	yes	characteristic, complemented, Sylow
81	#3395	—	181	1	no	maximal, Sylow
82	#557	—	181	2	no	—
83	#494	—	181	2	no	—
84	#494	—	181	1	no	—
85	#462	—	181	2	no	—
86	#462	—	181	4	no	—
87	C₈×Q₈	—	181	1	no	—
88	C₄×Q₁₆	—	181	2	no	—
89	C₄×Q₁₆	D₁₈₁	1	1	yes	characteristic
90	C₂×Q₁₆	—	181	4	no	—
91	C₂×Q₁₆	—	181	2	no	—
92	C₂×Q₁₆	—	181	2	no	—
93	C₂×Q₁₆	C₂×D₁₈₁	1	1	yes	characteristic
94	C₄⋊Q₈	—	181	4	no	—
95	C₄⋊Q₈	—	181	2	no	—
96	#125	—	362	2	no	—
97	#125	—	362	4	no	—
98	C₄×Q₈	—	362	2	no	—
99	C₄×Q₈	—	181	4	no	—
100	C₄×Q₈	—	181	1	no	—
101	C₄×Q₈	C₂×D₁₈₁	1	2	yes	—
102	#107	C₂×D₁₈₁	1	1	yes	characteristic
103	C₄⋊C₈	—	181	2	no	—
104	C₄⋊C₈	—	181	1	no	—
105	#103	—	181	4	no	—
106	#103	C₂×D₁₈₁	1	2	yes	—
107	C₄×C₈	—	181	2	no	—
108	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic
109	C₂×Q₈	—	181	4	no	—
110	C₂×Q₈	—	181	1	no	—
111	C₂×Q₈	C₂²×D₁₈₁	1	2	yes	—
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	362	4	no	—
114	Q₁₆	—	724	2	no	—
115	Q₁₆	—	724	4	no	—
116	C₂×C₈	—	181	2	no	—
117	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	362	4	no	—
121	C₄⋊C₄	—	362	4	no	—
122	C₄⋊C₄	—	362	4	no	—
123	C₄⋊C₄	—	362	2	no	—
124	C₄⋊C₄	—	362	2	no	—
125	C₄⋊C₄	—	181	2	no	—
126	C₄⋊C₄	—	181	1	no	—
127	C₄⋊C₄	C₂²×D₁₈₁	1	2	yes	—
128	C₄²	—	2	2	no	—
129	C₄²	—	362	4	no	—
130	C₄²	—	181	2	no	—
131	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
132	Q₈	—	2	2	no	—
133	Q₈	—	2	2	no	—
134	Q₈	—	362	4	no	—
135	Q₈	—	362	4	no	—
136	Q₈	—	181	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	362	4	no	—
140	C₂×C₄	—	181	2	no	—
141	C₂×C₄	—	181	2	no	—
142	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
143	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
144	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	362	2	no	—
147	C₈	#15668449	1	2	yes	—
148	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	362	2	no	—
153	C₄	—	181	4	no	—
154	C₄	—	724	4	no	—
155	C₄	Q₁₆×D₁₈₁	1	2	yes	—
156	C₄	C₂×#15668449	1	1	yes	characteristic
157	C₄	#19153862	1	1	yes	characteristic
158	C₂	#26041950	1	1	yes	characteristic
159	C₂	C₂×Q₁₆×D₁₈₁	1	1	yes	characteristic
160	C₂	#26042093	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26042007

C₂

yes

maximal

#26041659

C₂

yes

maximal, characteristic

#26041657

C₂

yes

maximal

#26041489

C₂

yes

maximal

#26041488

C₂

yes

maximal

C₈×Dic₃₆₂

C₂

yes

maximal, characteristic

#26041373

C₂

yes

maximal

C₄×Q₁₆×C₁₈₁

C₂

yes

maximal, characteristic

C₂×Dic₇₂₄

C₂²

yes

—

C₂×Dic₇₂₄

C₂²

yes

—

C₂×#15668439

C₂²

yes

—

#19153816

C₂²

yes

—

#19153814

C₂²

yes

—

#19153808

—

#19153806

—

C₄×Dic₃₆₂

—

C₄×Dic₃₆₂

C₂²

yes

characteristic

#19153781

C₂²

yes

—

C₄⋊#13089678

C₂²

yes

characteristic

#19153745

C₂²

yes

—

#19153739

C₂²

yes

—

C₈×Dic₁₈₁

C₂²

yes

—

C₂×Q₁₆×C₁₈₁

C₂²

yes

characteristic

C₄×Q₈×C₁₈₁

C₂²

yes

—

#107×C₁₈₁

C₂²

yes

characteristic

#103×C₁₈₁

C₂²

yes

—

C₄×C₈×C₁₈₁

C₂²

yes

characteristic

C₂×Dic₃₆₂

D₄

yes

characteristic

C₂×Dic₃₆₂

C₂³

yes

—

Dic₇₂₄

—

Dic₇₂₄

—

#15668439

—

C₂×#13089678

C₂³

yes

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

D₄

yes

—

—

—

D₄

yes

—

C₄×Dic₁₈₁

—

C₄×Dic₁₈₁

C₂³

yes

—

C₂×Q₈×C₁₈₁

C₂³

yes

—

Q₁₆×C₁₈₁

—

C₂×C₈×C₁₈₁

C₂³

yes

characteristic

C₂×C₈×C₁₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

C₂³

yes

—

C₄²×C₁₈₁

—

C₄²×C₁₈₁

C₂³

yes

characteristic

Dic₃₆₂

—

Dic₃₆₂

—

Dic₃₆₂

Q₁₆

yes

complemented

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

C₂×D₄

yes

—

C₂×Dic₁₈₁

C₂×D₄

yes

—

#13089678

—

Q₈×C₁₈₁

—

Q₈×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂⁴

yes

characteristic

C₈×C₁₈₁

—

C₈×C₁₈₁

#41

yes

—

C₂²×C₁₈₁

C₂²×D₄

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂×Q₁₆

yes

—

C₄×C₁₈₁

C₂×#41

yes

characteristic

C₄×C₁₈₁

H(5,𝔽₂)

yes

characteristic

Dic₁₈₁

—

Dic₁₈₁

—

Dic₁₈₁

C₂×Q₁₆

yes

—

C₂×C₁₈₁

#550

yes

characteristic

C₂×C₁₈₁

C₂²×Q₁₆

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, complemented, Sylow

#3395

—

181

maximal, Sylow

#557

—

181

—

#494

—

181

—

#494

—

181

—

#462

—

181

—

#462

—

181

—

C₈×Q₈

—

181

—

C₄×Q₁₆

—

181

—

C₄×Q₁₆

D₁₈₁

yes

characteristic

C₂×Q₁₆

—

181

—

C₂×Q₁₆

—

181

—

C₂×Q₁₆

—

181

—

C₂×Q₁₆

C₂×D₁₈₁

yes

characteristic

C₄⋊Q₈

—

181

—

C₄⋊Q₈

—

181

—

#125

—

362

—

#125

—

362

—

C₄×Q₈

—

362

—

C₄×Q₈

—

181

—

100

C₄×Q₈

—

181

—

101

C₄×Q₈

C₂×D₁₈₁

yes

—

102

#107

C₂×D₁₈₁

yes

characteristic

103

C₄⋊C₈

—

181

—

104

C₄⋊C₈

—

181

—

105

#103

—

181

—

106

#103

C₂×D₁₈₁

yes

—

107

C₄×C₈

—

181

—

108

C₄×C₈

C₂×D₁₈₁

yes

characteristic

109

C₂×Q₈

—

181

—

110

C₂×Q₈

—

181

—

111

C₂×Q₈

C₂²×D₁₈₁

yes

—

112

Q₁₆

—

113

Q₁₆

—

362

—

114

Q₁₆

—

724

—

115

Q₁₆

—

724

—

116

C₂×C₈

—

181

—

117

C₂×C₈

C₂²×D₁₈₁

yes

characteristic

118

C₂×C₈

C₂²×D₁₈₁

yes

characteristic

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

—

128

C₄²

—

129

C₄²

—

362

—

130

C₄²

—

181

—

131

C₄²

C₂²×D₁₈₁

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

362

—

135

Q₈

—

362

—

136

Q₈

—

181

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

362

—

140

C₂×C₄

—

181

—

141

C₂×C₄

—

181

—

142

C₂×C₄

D₄×D₁₈₁

yes

characteristic

143

C₂×C₄

D₄×D₁₈₁

yes

characteristic

144

C₂×C₄

C₂³×D₁₈₁

yes

characteristic

145

C₈

—

146

C₈

—

362

—

147

C₈

#15668449

yes

—

148

C₂²

C₂×D₄×D₁₈₁

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

362

—

153

C₄

—

181

—

154

C₄

—

724

—

155

C₄

Q₁₆×D₁₈₁

yes

—

156

C₄

C₂×#15668449

yes

characteristic

157

C₄

#19153862

yes

characteristic

158

C₂

#26041950

yes

characteristic

159

C₂

C₂×Q₁₆×D₁₈₁

yes

characteristic

160

C₂

#26042093

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	C₄×Q₁₆×C₁₈₁	C₂
socle, prime radical	C₂²×C₁₈₁	C₂²×D₄
hypercenter	C₄×Q₁₆	D₁₈₁
hypocenter	C₁₈₁	#3395
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₁₈₁

commutator, cosocle

C₂×C₄×C₁₈₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₈₁

Fitting

C₄×Q₁₆×C₁₈₁

C₂

socle, prime radical

C₂²×C₁₈₁

C₂²×D₄

hypercenter

C₄×Q₁₆

D₁₈₁

hypocenter

C₁₈₁

#3395

LCS

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
53. Dic₃₆₂	Q₁₆	semidirect
80. C₁₈₁	#3395	semidirect

H ≅ G/N

type

53. Dic₃₆₂

Q₁₆

semidirect

80. C₁₈₁

#3395

semidirect

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

subgroup	index
#26042007	2
#26041659	2
#26041657	2
#26041489	2
#26041488	2
C₈×Dic₃₆₂	2
#26041373	2
C₄×Q₁₆×C₁₈₁	2
#3395	181

subgroup

index

C₈×Dic₃₆₂

C₄×Q₁₆×C₁₈₁

181

Subgroups — #37546726

Conjugacy classes of subgroups

Complements

Maximal subgroups