Group #37535049 — subgroups

Subgroups — #37535049

Order 23168.

282 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 97 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×#19153816	C₂	1	1	yes	maximal, characteristic
2	C₂×#19153749	C₂	1	2	yes	maximal
3	#26041041	C₂	1	2	yes	maximal
4	#26041034	C₂	1	1	yes	maximal, characteristic
5	#389×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₆₂	—	2	1	no	—
7	C₂²×#13089678	C₂²	1	2	yes	—
8	#19153816	—	2	4	no	—
9	C₂×C₄⋊Dic₁₈₁	—	2	1	no	—
10	C₂×C₄⋊Dic₁₈₁	C₂²	1	2	yes	—
11	#19153749	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₈₁	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
16	#95×C₁₈₁	—	2	1	no	—
17	C₂×Dic₃₆₂	—	2	4	no	—
18	C₂×Dic₃₆₂	—	4	2	no	—
19	C₂×Dic₃₆₂	—	4	2	no	—
20	C₂²×Dic₁₈₁	—	2	2	no	—
21	C₂×#13089678	—	2	2	no	—
22	C₂×#13089678	—	2	2	no	—
23	C₂×#13089678	Q₈	1	4	yes	complemented
24	C₄⋊Dic₁₈₁	—	2	4	no	—
25	C₄⋊Dic₁₈₁	—	2	2	no	—
26	C₄⋊Dic₁₈₁	D₄	1	4	yes	—
27	C₂²×C₄×C₁₈₁	—	2	1	no	—
28	C₂²×C₄×C₁₈₁	—	2	1	no	—
29	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₁₈₁	—	2	1	no	—
33	C₄⋊C₄×C₁₈₁	—	2	2	no	—
34	C₄⋊C₄×C₁₈₁	—	4	2	no	—
35	C₄⋊C₄×C₁₈₁	Q₈	1	2	yes	complemented
36	C₄²×C₁₈₁	—	2	2	no	—
37	Dic₃₆₂	—	4	8	no	—
38	C₂×Dic₁₈₁	—	2	4	no	—
39	C₂×Dic₁₈₁	—	4	2	no	—
40	C₂×Dic₁₈₁	—	4	2	no	—
41	#13089678	—	2	4	no	—
42	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
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	1	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	1	no	—
51	C₂×C₄×C₁₈₁	—	4	1	no	—
52	C₂×C₄×C₁₈₁	—	4	1	no	—
53	C₂×C₄×C₁₈₁	D₈	1	2	yes	—
54	C₂×C₄×C₁₈₁	Q₁₆	1	2	yes	—
55	C₂×C₄×C₁₈₁	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×C₁₈₁	#41	1	1	yes	characteristic
59	C₂²×C₁₈₁	#120	1	1	yes	characteristic
60	C₂²×C₁₈₁	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₁₈₁	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₁₈₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₁₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₁₈₁	#137	1	1	yes	characteristic
66	C₄×C₁₈₁	—	2	2	no	—
67	C₄×C₁₈₁	—	2	2	no	—
68	C₄×C₁₈₁	—	4	2	no	—
69	C₄×C₁₈₁	—	8	1	no	—
70	C₄×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
71	C₄×C₁₈₁	C₄⋊Q₈	1	2	yes	—
72	Dic₁₈₁	—	4	4	no	—
73	C₂×C₁₈₁	#395	1	1	yes	characteristic
74	C₂×C₁₈₁	#447	1	1	yes	characteristic
75	C₂×C₁₈₁	#451	1	1	yes	characteristic
76	C₂×C₁₈₁	#474	1	1	yes	characteristic
77	C₂×C₁₈₁	#477	1	1	yes	characteristic
78	C₂×C₁₈₁	#500	1	1	yes	characteristic
79	C₂×C₁₈₁	#501	1	1	yes	characteristic
80	C₁₈₁	#2057	1	1	yes	characteristic, complemented, Sylow
81	#2057	—	181	1	no	maximal, Sylow
82	C₂×C₄⋊Q₈	—	181	1	no	—
83	C₂×#107	—	181	2	no	—
84	#389	D₁₈₁	1	1	yes	characteristic
85	#340	—	181	2	no	—
86	#336	—	181	1	no	—
87	C₂²×Q₈	—	362	1	no	—
88	C₂²×C₈	—	181	2	no	—
89	C₄⋊Q₈	—	362	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	362	1	no	—
92	C₂×C₄⋊C₄	—	181	2	no	—
93	C₂×C₄⋊C₄	C₂×D₁₈₁	1	1	yes	characteristic
94	C₂×C₄⋊C₄	C₂×D₁₈₁	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₁₈₁	1	1	yes	characteristic
96	#107	—	362	4	no	—
97	#95	—	2	1	no	—
98	C₂×Q₈	—	362	4	no	—
99	C₂×Q₈	—	724	2	no	—
100	C₂×Q₈	—	724	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	362	2	no	—
104	C₂²×C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
107	C₂×C₈	—	362	2	no	—
108	C₂×C₈	—	362	2	no	—
109	C₂×C₈	—	181	4	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	362	4	no	—
114	C₄⋊C₄	—	362	2	no	—
115	C₄⋊C₄	—	181	4	no	—
116	C₄⋊C₄	Dic₃₆₂	1	2	yes	complemented
117	C₄²	—	2	2	no	—
118	C₂³	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
119	Q₈	—	724	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	362	4	no	—
131	C₂×C₄	—	724	2	no	—
132	C₂×C₄	—	724	2	no	—
133	C₂×C₄	#15668434	1	2	yes	—
134	C₂×C₄	#15668439	1	2	yes	—
135	C₂×C₄	D₄×D₁₈₁	1	2	yes	—
136	C₂×C₄	Q₈×D₁₈₁	1	2	yes	—
137	C₂×C₄	C₂×Dic₃₆₂	1	1	yes	characteristic
138	C₂×C₄	#15668449	1	1	yes	characteristic
139	C₈	—	362	4	no	—
140	C₂²	#19153784	1	1	yes	characteristic
141	C₂²	#19153795	1	1	yes	characteristic
142	C₂²	#19153814	1	1	yes	characteristic
143	C₂²	C₂×#15668434	1	1	yes	characteristic
144	C₂²	C₂×#15668439	1	1	yes	characteristic
145	C₂²	#19153841	1	1	yes	characteristic
146	C₂²	#19153846	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	2	no	—
150	C₄	—	8	1	no	—
151	C₄	—	724	4	no	—
152	C₄	#19153794	1	2	yes	—
153	C₄	#19153815	1	2	yes	—
154	C₂	#26041199	1	1	yes	characteristic
155	C₂	#26041412	1	1	yes	characteristic
156	C₂	#26041434	1	1	yes	characteristic
157	C₂	#26041559	1	1	yes	characteristic
158	C₂	#26041574	1	1	yes	characteristic
159	C₂	#26041682	1	1	yes	characteristic
160	C₂	#26041689	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153816

C₂

yes

maximal, characteristic

C₂×#19153749

C₂

yes

maximal

#26041041

C₂

yes

maximal

#26041034

C₂

yes

maximal, characteristic

#389×C₁₈₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₆₂

—

C₂²×#13089678

C₂²

yes

—

#19153816

—

C₂×C₄⋊Dic₁₈₁

—

C₂×C₄⋊Dic₁₈₁

C₂²

yes

—

#19153749

—

C₂×C₄⋊C₄×C₁₈₁

—

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

characteristic

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

#95×C₁₈₁

—

C₂×Dic₃₆₂

—

C₂×Dic₃₆₂

—

C₂×Dic₃₆₂

—

C₂²×Dic₁₈₁

—

C₂×#13089678

—

C₂×#13089678

—

C₂×#13089678

Q₈

yes

complemented

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

D₄

yes

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

D₄

yes

characteristic

C₂²×C₄×C₁₈₁

D₄

yes

characteristic

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

Q₈

yes

complemented

C₄²×C₁₈₁

—

Dic₃₆₂

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

—

C₂³×C₁₈₁

C₂×D₄

yes

characteristic

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

D₈

yes

—

C₂×C₄×C₁₈₁

Q₁₆

yes

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

—

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

C₂×Q₈

yes

C₂×C₄×C₁₈₁

yes

C₂²×C₁₈₁

yes

C₂²×C₁₈₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₈₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₈₁

C₂×D₈

yes

characteristic

C₂²×C₁₈₁

C₂×Q₁₆

yes

C₂²×C₁₈₁

yes

C₂²×C₁₈₁

yes

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂²⋊Q₈

yes

—

C₄×C₁₈₁

C₄⋊Q₈

yes

—

Dic₁₈₁

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, complemented, Sylow

#2057

—

181

maximal, Sylow

C₂×C₄⋊Q₈

—

181

—

C₂×#107

—

181

—

#389

D₁₈₁

yes

characteristic

#340

—

181

—

#336

—

181

—

C₂²×Q₈

—

362

—

C₂²×C₈

—

181

—

C₄⋊Q₈

—

362

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

362

—

C₂×C₄⋊C₄

—

181

—

C₂×C₄⋊C₄

C₂×D₁₈₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₈₁

yes

characteristic

C₂×C₄²

C₂×D₁₈₁

yes

characteristic

#107

—

362

—

#95

—

C₂×Q₈

—

362

—

C₂×Q₈

—

724

—

100

C₂×Q₈

—

724

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

362

—

104

C₂²×C₄

C₂²⋊D₁₈₁

yes

characteristic

105

C₂²×C₄

C₂²⋊D₁₈₁

yes

characteristic

106

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

107

C₂×C₈

—

362

—

108

C₂×C₈

—

362

—

109

C₂×C₈

—

181

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

362

—

114

C₄⋊C₄

—

362

—

115

C₄⋊C₄

—

181

—

116

C₄⋊C₄

Dic₃₆₂

yes

complemented

117

C₄²

—

118

C₂³

C₂×C₂²⋊D₁₈₁

yes

characteristic

119

Q₈

—

724

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

362

—

131

C₂×C₄

—

724

—

132

C₂×C₄

—

724

—

133

C₂×C₄

#15668434

yes

—

134

C₂×C₄

#15668439

yes

—

135

C₂×C₄

D₄×D₁₈₁

yes

—

136

C₂×C₄

Q₈×D₁₈₁

yes

—

137

C₂×C₄

C₂×Dic₃₆₂

yes

characteristic

138

C₂×C₄

#15668449

yes

characteristic

139

C₈

—

362

—

140

C₂²

#19153784

yes

characteristic

141

C₂²

#19153795

yes

characteristic

142

C₂²

#19153814

yes

characteristic

143

C₂²

C₂×#15668434

yes

characteristic

144

C₂²

C₂×#15668439

yes

characteristic

145

C₂²

#19153841

yes

characteristic

146

C₂²

#19153846

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

724

—

152

C₄

#19153794

yes

—

153

C₄

#19153815

yes

—

154

C₂

#26041199

yes

characteristic

155

C₂

#26041412

yes

characteristic

156

C₂

#26041434

yes

characteristic

157

C₂

#26041559

yes

characteristic

158

C₂

#26041574

yes

characteristic

159

C₂

#26041682

yes

characteristic

160

C₂

#26041689

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₈₁
commutator, cosocle	C₂²×C₄×C₁₈₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	#389×C₁₈₁	C₂
socle, prime radical	C₂³×C₁₈₁	C₂×D₄
hypercenter	#389	D₁₈₁
hypocenter	C₁₈₁	#2057
LCS	G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₈₁

commutator, cosocle

C₂²×C₄×C₁₈₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

#389×C₁₈₁

C₂

socle, prime radical

C₂³×C₁₈₁

C₂×D₄

D₁₈₁

C₁₈₁

G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
23. C₂×#13089678	Q₈	semidirect
35. C₄⋊C₄×C₁₈₁	Q₈	semidirect
80. C₁₈₁	#2057	semidirect
116. C₄⋊C₄	Dic₃₆₂	semidirect

H ≅ G/N

type

23. C₂×#13089678

Q₈

semidirect

35. C₄⋊C₄×C₁₈₁

Q₈

semidirect

80. C₁₈₁

#2057

semidirect

116. C₄⋊C₄

Dic₃₆₂

semidirect

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

subgroup	index
C₂×#19153816	2
C₂×#19153749	2
#26041041	2
#26041034	2
#389×C₁₈₁	2
#2057	181

subgroup

index

#389×C₁₈₁

181

Subgroups — #37535049

Conjugacy classes of subgroups

Complements

Maximal subgroups