Group #37529967 — subgroups

Subgroups — #2024×C₁₈₁ #37529967

Order 23168.

290 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 108 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	C₂×C₄⋊Q₈×C₁₈₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#103×C₁₈₁	C₂	1	2	yes	maximal
3	#391×C₁₈₁	C₂	1	1	yes	maximal, characteristic
4	#340×C₁₈₁	C₂	1	2	yes	maximal
5	#336×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₈₁	—	2	1	no	—
7	C₂²×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₈₁	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₈₁	—	2	4	no	—
10	C₂×C₄⋊C₄×C₁₈₁	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
14	#103×C₁₈₁	—	2	4	no	—
15	#95×C₁₈₁	—	2	1	no	—
16	#95×C₁₈₁	—	2	1	no	—
17	C₂×Q₈×C₁₈₁	—	2	4	no	—
18	C₂×Q₈×C₁₈₁	—	2	1	no	—
19	C₂×Q₈×C₁₈₁	—	2	2	no	—
20	C₂×Q₈×C₁₈₁	—	2	2	no	—
21	C₂×Q₈×C₁₈₁	—	4	2	no	—
22	C₂×Q₈×C₁₈₁	—	4	2	no	—
23	C₂×Q₈×C₁₈₁	Q₈	1	2	yes	complemented
24	C₂²×C₄×C₁₈₁	—	2	1	no	—
25	C₂²×C₄×C₁₈₁	—	2	2	no	—
26	C₂²×C₄×C₁₈₁	—	2	1	no	—
27	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₈₁	—	2	2	no	—
31	C₂×C₈×C₁₈₁	—	2	2	no	—
32	C₂×C₈×C₁₈₁	—	2	2	no	—
33	C₄⋊C₄×C₁₈₁	—	2	4	no	—
34	C₄⋊C₄×C₁₈₁	—	2	2	no	—
35	C₄⋊C₄×C₁₈₁	—	2	2	no	—
36	C₄⋊C₄×C₁₈₁	D₄	1	4	yes	—
37	C₄²×C₁₈₁	—	2	2	no	—
38	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₁₈₁	—	2	4	no	—
40	Q₈×C₁₈₁	—	4	8	no	—
41	Q₈×C₁₈₁	—	4	2	no	—
42	C₂×C₄×C₁₈₁	—	2	1	no	—
43	C₂×C₄×C₁₈₁	—	2	1	no	—
44	C₂×C₄×C₁₈₁	—	2	1	no	—
45	C₂×C₄×C₁₈₁	—	2	1	no	—
46	C₂×C₄×C₁₈₁	—	2	4	no	—
47	C₂×C₄×C₁₈₁	—	2	2	no	—
48	C₂×C₄×C₁₈₁	—	4	1	no	—
49	C₂×C₄×C₁₈₁	—	4	1	no	—
50	C₂×C₄×C₁₈₁	—	4	2	no	—
51	C₂×C₄×C₁₈₁	—	4	2	no	—
52	C₂×C₄×C₁₈₁	—	4	1	no	—
53	C₂×C₄×C₁₈₁	—	4	1	no	—
54	C₂×C₄×C₁₈₁	—	4	1	no	—
55	C₂×C₄×C₁₈₁	SD₁₆	1	2	yes	—
56	C₂×C₄×C₁₈₁	Q₁₆	1	2	yes	—
57	C₂×C₄×C₁₈₁	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₈₁	#41	1	2	yes	—
60	C₂×C₄×C₁₈₁	#41	1	1	yes	characteristic
61	C₈×C₁₈₁	—	4	2	no	—
62	C₂²×C₁₈₁	#120	1	1	yes	characteristic
63	C₂²×C₁₈₁	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₁₈₁	#124	1	1	yes	characteristic
65	C₂²×C₁₈₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₁₈₁	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₁₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₁₈₁	#137	1	1	yes	characteristic
69	C₄×C₁₈₁	—	2	2	no	—
70	C₄×C₁₈₁	—	2	2	no	—
71	C₄×C₁₈₁	—	4	2	no	—
72	C₄×C₁₈₁	—	4	4	no	—
73	C₄×C₁₈₁	—	8	1	no	—
74	C₄×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
75	C₄×C₁₈₁	#124	1	2	yes	—
76	C₂×C₁₈₁	#393	1	1	yes	characteristic
77	C₂×C₁₈₁	#450	1	1	yes	characteristic
78	C₂×C₁₈₁	#451	1	1	yes	characteristic
79	C₂×C₁₈₁	#475	1	1	yes	characteristic
80	C₂×C₁₈₁	#477	1	1	yes	characteristic
81	C₂×C₁₈₁	#487	1	1	yes	characteristic
82	C₂×C₁₈₁	#489	1	1	yes	characteristic
83	C₁₈₁	#2024	1	1	yes	characteristic, direct product factor, Sylow
84	#2024	C₁₈₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂×C₄⋊Q₈	C₂×C₁₈₁	1	1	yes	characteristic
86	C₂×#103	C₂×C₁₈₁	1	2	yes	—
87	#391	C₂×C₁₈₁	1	1	yes	characteristic
88	#340	C₂×C₁₈₁	1	2	yes	—
89	#336	C₂×C₁₈₁	1	1	yes	characteristic
90	C₂²×Q₈	—	2	1	no	—
91	C₂²×Q₈	C₂²×C₁₈₁	1	1	yes	characteristic
92	C₂²×C₈	C₂²×C₁₈₁	1	2	yes	—
93	C₄⋊Q₈	—	2	4	no	—
94	C₂×C₄⋊C₄	—	2	1	no	—
95	C₂×C₄⋊C₄	C₂²×C₁₈₁	1	1	yes	characteristic
96	C₂×C₄⋊C₄	C₂²×C₁₈₁	1	2	yes	—
97	C₂×C₄²	C₂²×C₁₈₁	1	1	yes	characteristic
98	#103	—	2	4	no	—
99	#95	—	2	1	no	—
100	#95	—	2	1	no	—
101	C₂×Q₈	—	2	4	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	4	2	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	Q₈×C₁₈₁	1	2	yes	complemented
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	D₄×C₁₈₁	1	1	yes	characteristic
112	C₂²×C₄	D₄×C₁₈₁	1	1	yes	characteristic
113	C₂²×C₄	C₂³×C₁₈₁	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	D₄×C₁₈₁	1	4	yes	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×D₄×C₁₈₁	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	8	no	—
125	Q₈	—	4	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	SD₁₆×C₁₈₁	1	2	yes	—
140	C₂×C₄	Q₁₆×C₁₈₁	1	2	yes	—
141	C₂×C₄	C₂×D₄×C₁₈₁	1	2	yes	—
142	C₂×C₄	C₂×Q₈×C₁₈₁	1	1	yes	characteristic
143	C₂×C₄	#41×C₁₈₁	1	2	yes	—
144	C₂×C₄	#41×C₁₈₁	1	1	yes	characteristic
145	C₈	—	4	2	no	—
146	C₂²	#120×C₁₈₁	1	1	yes	characteristic
147	C₂²	C₂²⋊Q₈×C₁₈₁	1	1	yes	characteristic
148	C₂²	#124×C₁₈₁	1	1	yes	characteristic
149	C₂²	C₂×SD₁₆×C₁₈₁	1	1	yes	characteristic
150	C₂²	C₂×Q₁₆×C₁₈₁	1	1	yes	characteristic
151	C₂²	AGL(1,ℤ/8)×C₁₈₁	1	1	yes	characteristic
152	C₂²	#137×C₁₈₁	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	4	no	—
156	C₄	—	4	2	no	—
157	C₄	—	8	1	no	—
158	C₄	C₂²⋊Q₈×C₁₈₁	1	2	yes	—
159	C₄	#124×C₁₈₁	1	2	yes	—
160	C₂	#393×C₁₈₁	1	1	yes	characteristic
161	C₂	#450×C₁₈₁	1	1	yes	characteristic
162	C₂	#451×C₁₈₁	1	1	yes	characteristic
163	C₂	#475×C₁₈₁	1	1	yes	characteristic
164	C₂	#477×C₁₈₁	1	1	yes	characteristic
165	C₂	#487×C₁₈₁	1	1	yes	characteristic
166	C₂	#489×C₁₈₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#103×C₁₈₁

C₂

yes

maximal

#391×C₁₈₁

C₂

yes

maximal, characteristic

#340×C₁₈₁

C₂

yes

maximal

#336×C₁₈₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₈₁

—

C₂²×Q₈×C₁₈₁

C₂²

yes

characteristic

C₂²×C₈×C₁₈₁

C₂²

yes

—

C₄⋊Q₈×C₁₈₁

—

C₂×C₄⋊C₄×C₁₈₁

—

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

—

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

#103×C₁₈₁

—

#95×C₁₈₁

—

#95×C₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂×Q₈×C₁₈₁

Q₈

yes

complemented

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

D₄

yes

characteristic

C₂²×C₄×C₁₈₁

D₄

yes

characteristic

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

D₄

yes

—

C₄²×C₁₈₁

—

C₂³×C₁₈₁

C₂×D₄

yes

characteristic

Q₈×C₁₈₁

—

Q₈×C₁₈₁

—

Q₈×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

SD₁₆

yes

—

C₂×C₄×C₁₈₁

Q₁₆

yes

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

—

C₂×C₄×C₁₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₈₁

#41

yes

—

C₂×C₄×C₁₈₁

#41

yes

characteristic

C₈×C₁₈₁

—

C₂²×C₁₈₁

#120

yes

characteristic

C₂²×C₁₈₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₈₁

#124

yes

characteristic

C₂²×C₁₈₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₈₁

C₂×Q₁₆

yes

C₂²×C₁₈₁

yes

C₂²×C₁₈₁

yes

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂²⋊Q₈

yes

—

C₄×C₁₈₁

#124

yes

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, direct product factor, Sylow

#2024

C₁₈₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₁₈₁

yes

characteristic

C₂×#103

C₂×C₁₈₁

yes

—

#391

C₂×C₁₈₁

yes

characteristic

#340

C₂×C₁₈₁

yes

—

#336

C₂×C₁₈₁

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

C₂²×C₁₈₁

yes

characteristic

C₂²×C₈

C₂²×C₁₈₁

yes

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₈₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₈₁

yes

—

C₂×C₄²

C₂²×C₁₈₁

yes

characteristic

#103

—

#95

—

100

#95

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

Q₈×C₁₈₁

yes

complemented

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

D₄×C₁₈₁

yes

characteristic

112

C₂²×C₄

D₄×C₁₈₁

yes

characteristic

113

C₂²×C₄

C₂³×C₁₈₁

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

D₄×C₁₈₁

yes

—

121

C₄²

—

122

C₂³

C₂×D₄×C₁₈₁

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

SD₁₆×C₁₈₁

yes

—

140

C₂×C₄

Q₁₆×C₁₈₁

yes

—

141

C₂×C₄

C₂×D₄×C₁₈₁

yes

—

142

C₂×C₄

C₂×Q₈×C₁₈₁

yes

characteristic

143

C₂×C₄

#41×C₁₈₁

yes

—

144

C₂×C₄

#41×C₁₈₁

yes

characteristic

145

C₈

—

146

C₂²

#120×C₁₈₁

yes

characteristic

147

C₂²

C₂²⋊Q₈×C₁₈₁

yes

characteristic

148

C₂²

#124×C₁₈₁

yes

characteristic

149

C₂²

C₂×SD₁₆×C₁₈₁

yes

characteristic

150

C₂²

C₂×Q₁₆×C₁₈₁

yes

characteristic

151

C₂²

AGL(1,ℤ/8)×C₁₈₁

yes

characteristic

152

C₂²

#137×C₁₈₁

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

C₂²⋊Q₈×C₁₈₁

yes

—

159

C₄

#124×C₁₈₁

yes

—

160

C₂

#393×C₁₈₁

yes

characteristic

161

C₂

#450×C₁₈₁

yes

characteristic

162

C₂

#451×C₁₈₁

yes

characteristic

163

C₂

#475×C₁₈₁

yes

characteristic

164

C₂

#477×C₁₈₁

yes

characteristic

165

C₂

#487×C₁₈₁

yes

characteristic

166

C₂

#489×C₁₈₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₈₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₈₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₈₁ ◁ C₂×C₄²×C₁₈₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₈₁

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₈₁

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₈₁ ◁ C₂×C₄²×C₁₈₁ ◁ G

N	H ≅ G/N	type
23. C₂×Q₈×C₁₈₁	Q₈	semidirect
83. C₁₈₁	#2024	direct (N × H)
84. #2024	C₁₈₁	direct (N × H)
107. C₂×Q₈	Q₈×C₁₈₁	semidirect

H ≅ G/N

type

23. C₂×Q₈×C₁₈₁

Q₈

semidirect

83. C₁₈₁

#2024

direct (N × H)

84. #2024

C₁₈₁

direct (N × H)

107. C₂×Q₈

Q₈×C₁₈₁

semidirect

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

subgroup	index
C₂×C₄⋊Q₈×C₁₈₁	2
C₂×#103×C₁₈₁	2
#391×C₁₈₁	2
#340×C₁₈₁	2
#336×C₁₈₁	2
#2024	181

subgroup

index

C₂×C₄⋊Q₈×C₁₈₁

C₂×#103×C₁₈₁

#391×C₁₈₁

#340×C₁₈₁

#336×C₁₈₁

#2024

181

Subgroups — #2024×C₁₈₁ #37529967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2024×C181 #37529967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2024×C₁₈₁ #37529967