Group #37532124 — subgroups

Subgroups — #37532124

Order 23168.

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

normal subgroups only

This group has 216 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	#26041332	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#13089678	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#13089678	C₂	1	4	yes	maximal, complemented
4	#533×C₁₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13089678	—	2	1	no	—
6	C₄⋊#13089678	—	2	1	no	—
7	C₄⋊#13089678	—	2	1	no	—
8	C₄⋊#13089678	C₂²	1	2	yes	—
9	C₂²⋊#13089678	—	2	1	no	—
10	C₄×#13089678	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₈₁	C₄	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₈₁	C₄	1	1	yes	characteristic
13	#125×C₁₈₁	—	2	1	no	—
14	C₂²⋊Q₈×C₁₈₁	—	2	1	no	—
15	C₂²⋊Q₈×C₁₈₁	C₄	1	2	yes	—
16	C₄×Q₈×C₁₈₁	—	2	1	no	—
17	C₄×Q₈×C₁₈₁	C₂²	1	2	yes	—
18	#117×C₁₈₁	—	2	1	no	—
19	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
20	C₂×#13089678	—	2	2	no	—
21	C₂×#13089678	—	2	2	no	—
22	C₂×#13089678	—	4	2	no	—
23	C₂×Q₈×C₁₈₁	—	2	1	no	—
24	C₂×Q₈×C₁₈₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₈₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₁₈₁	—	2	1	no	—
29	C₄⋊C₄×C₁₈₁	—	2	1	no	—
30	C₄⋊C₄×C₁₈₁	—	2	1	no	—
31	C₄⋊C₄×C₁₈₁	—	2	2	no	—
32	C₄⋊C₄×C₁₈₁	—	2	2	no	—
33	C₄⋊C₄×C₁₈₁	C₂×C₄	1	2	yes	—
34	C₂²⋊C₄×C₁₈₁	—	2	2	no	—
35	C₄²×C₁₈₁	—	2	2	no	—
36	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₈₁	D₄	1	1	yes	characteristic
38	C₄²×C₁₈₁	D₄	1	1	yes	characteristic
39	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
40	#13089678	—	4	2	no	—
41	#13089678	—	4	2	no	—
42	C₂³×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
43	Q₈×C₁₈₁	—	2	4	no	—
44	Q₈×C₁₈₁	—	4	2	no	—
45	C₂×C₄×C₁₈₁	—	2	2	no	—
46	C₂×C₄×C₁₈₁	—	2	2	no	—
47	C₂×C₄×C₁₈₁	—	2	2	no	—
48	C₂×C₄×C₁₈₁	—	4	1	no	—
49	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₈₁	SD₁₆	1	2	yes	—
53	C₂×C₄×C₁₈₁	Q₁₆	1	2	yes	—
54	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₁₈₁	—	2	1	no	—
58	C₂²×C₁₈₁	—	2	1	no	—
59	C₂²×C₁₈₁	#103	1	2	yes	—
60	C₂²×C₁₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₁₈₁	—	2	1	no	—
62	C₄×C₁₈₁	—	2	1	no	—
63	C₄×C₁₈₁	—	2	1	no	—
64	C₄×C₁₈₁	—	4	2	no	—
65	C₄×C₁₈₁	—	4	2	no	—
66	C₄×C₁₈₁	#103	1	2	yes	—
67	C₄×C₁₈₁	#131	1	2	yes	—
68	C₄×C₁₈₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₁₈₁	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₁₈₁	—	2	2	no	—
71	C₂×C₁₈₁	#408	1	1	yes	characteristic
72	C₂×C₁₈₁	C₂×#103	1	1	yes	characteristic
73	C₂×C₁₈₁	#421	1	1	yes	characteristic
74	C₁₈₁	#1481	1	1	yes	characteristic, complemented, Sylow
75	#1481	—	181	1	no	maximal, Sylow
76	#533	D₁₈₁	1	1	yes	characteristic
77	#431	—	181	1	no	—
78	C₂×C₄⋊C₈	—	181	1	no	—
79	Q₈⋊C₈	—	181	4	no	—
80	C₂²×C₈	—	362	1	no	—
81	C₄⋊Q₈	Dic₁₈₁	1	1	yes	characteristic
82	C₄⋊Q₈	Dic₁₈₁	1	1	yes	characteristic
83	#125	—	2	1	no	—
84	C₂²⋊Q₈	—	2	1	no	—
85	C₂²⋊Q₈	Dic₁₈₁	1	2	yes	—
86	C₄×Q₈	—	2	1	no	—
87	C₄×Q₈	C₂×D₁₈₁	1	2	yes	—
88	#117	—	2	1	no	—
89	C₂×C₄²	C₂×D₁₈₁	1	1	yes	characteristic
90	C₄⋊C₈	—	362	1	no	—
91	C₄⋊C₈	—	362	1	no	—
92	C₄⋊C₈	—	181	2	no	—
93	C₂²⋊C₈	—	362	1	no	—
94	C₄×C₈	—	181	2	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	C₂×Dic₁₈₁	1	2	yes	—
97	C₂²×C₄	C₂×Dic₁₈₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
100	C₂×C₈	—	362	2	no	—
101	C₂×C₈	—	362	2	no	—
102	C₂×C₈	—	724	2	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	C₂×Dic₁₈₁	1	2	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	C₂×Dic₁₈₁	1	1	yes	characteristic
112	C₄²	C₂²⋊D₁₈₁	1	1	yes	characteristic
113	C₄²	C₂²⋊D₁₈₁	1	1	yes	characteristic
114	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
115	C₂³	C₂²⋊Dic₁₈₁	1	1	yes	characteristic
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	C₂²⋊Dic₁₈₁	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊Dic₁₈₁	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊Dic₁₈₁	1	1	yes	characteristic
125	C₂×C₄	#15668436	1	2	yes	—
126	C₂×C₄	#15668439	1	2	yes	—
127	C₂×C₄	C₂²×Dic₁₈₁	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
130	C₈	—	724	2	no	—
131	C₈	—	724	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	#19153740	1	2	yes	—
135	C₂²	C₂×C₂²⋊Dic₁₈₁	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	4	2	no	—
140	C₄	—	4	2	no	—
141	C₄	#19153740	1	2	yes	—
142	C₄	#19153823	1	2	yes	—
143	C₄	C₂×#15668436	1	1	yes	characteristic
144	C₄	C₂×#15668439	1	1	yes	characteristic
145	C₂	—	2	2	no	—
146	C₂	#26041246	1	1	yes	characteristic
147	C₂	C₂×#19153740	1	1	yes	characteristic
148	C₂	#26041296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26041332

C₂

yes

maximal, characteristic

C₂×C₄⋊#13089678

C₂

yes

maximal, characteristic

Q₈⋊#13089678

C₂

yes

maximal, complemented

#533×C₁₈₁

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂²⋊#13089678

—

C₄×#13089678

C₂²

yes

—

C₄⋊Q₈×C₁₈₁

C₄

yes

characteristic

C₄⋊Q₈×C₁₈₁

C₄

yes

characteristic

#125×C₁₈₁

—

C₂²⋊Q₈×C₁₈₁

—

C₂²⋊Q₈×C₁₈₁

C₄

yes

—

C₄×Q₈×C₁₈₁

—

C₄×Q₈×C₁₈₁

C₂²

yes

—

#117×C₁₈₁

—

C₂×C₄²×C₁₈₁

C₂²

yes

—

—

—

C₂×Q₈×C₁₈₁

—

C₂×Q₈×C₁₈₁

C₂×C₄

yes

—

C₂²×C₄×C₁₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₈₁

D₄

yes

characteristic

C₂²×C₄×C₁₈₁

D₄

yes

characteristic

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

C₂×C₄

yes

—

C₂²⋊C₄×C₁₈₁

—

C₄²×C₁₈₁

—

C₄²×C₁₈₁

C₂×C₄

yes

characteristic

C₄²×C₁₈₁

D₄

yes

characteristic

C₄²×C₁₈₁

D₄

yes

characteristic

C₄²×C₁₈₁

C₂³

yes

characteristic

#13089678

—

#13089678

—

C₂³×C₁₈₁

C₂²⋊C₄

yes

characteristic

Q₈×C₁₈₁

—

Q₈×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

SD₁₆

yes

—

C₂×C₄×C₁₈₁

Q₁₆

yes

—

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

#103

yes

—

C₂²×C₁₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

#103

yes

—

C₄×C₁₈₁

#131

yes

—

C₄×C₁₈₁

C₂×SD₁₆

yes

characteristic

C₄×C₁₈₁

C₂×Q₁₆

yes

characteristic

C₂×C₁₈₁

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, complemented, Sylow

#1481

—

181

maximal, Sylow

#533

D₁₈₁

yes

characteristic

#431

—

181

—

C₂×C₄⋊C₈

—

181

—

Q₈⋊C₈

—

181

—

C₂²×C₈

—

362

—

C₄⋊Q₈

Dic₁₈₁

yes

characteristic

C₄⋊Q₈

Dic₁₈₁

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₁₈₁

yes

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₈₁

yes

—

#117

—

C₂×C₄²

C₂×D₁₈₁

yes

characteristic

C₄⋊C₈

—

362

—

C₄⋊C₈

—

362

—

C₄⋊C₈

—

181

—

C₂²⋊C₈

—

362

—

C₄×C₈

—

181

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₈₁

yes

—

C₂²×C₄

C₂×Dic₁₈₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₈₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₈₁

yes

characteristic

100

C₂×C₈

—

362

—

101

C₂×C₈

—

362

—

102

C₂×C₈

—

724

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂×Dic₁₈₁

yes

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₂×Dic₁₈₁

yes

characteristic

112

C₄²

C₂²⋊D₁₈₁

yes

characteristic

113

C₄²

C₂²⋊D₁₈₁

yes

characteristic

114

C₄²

C₂²×D₁₈₁

yes

characteristic

115

C₂³

C₂²⋊Dic₁₈₁

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂²⋊Dic₁₈₁

yes

characteristic

123

C₂×C₄

C₂²⋊Dic₁₈₁

yes

characteristic

124

C₂×C₄

C₂²⋊Dic₁₈₁

yes

characteristic

125

C₂×C₄

#15668436

yes

—

126

C₂×C₄

#15668439

yes

—

127

C₂×C₄

C₂²×Dic₁₈₁

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

characteristic

130

C₈

—

724

—

131

C₈

—

724

—

132

C₂²

—

133

C₂²

—

134

C₂²

#19153740

yes

—

135

C₂²

C₂×C₂²⋊Dic₁₈₁

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#19153740

yes

—

142

C₄

#19153823

yes

—

143

C₄

C₂×#15668436

yes

characteristic

144

C₄

C₂×#15668439

yes

characteristic

145

C₂

—

146

C₂

#26041246

yes

characteristic

147

C₂

C₂×#19153740

yes

characteristic

148

C₂

#26041296

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₈₁
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₈₁
Fitting	#533×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₈₁	C₂²⋊C₄
cosocle	C₄²×C₁₈₁	C₂³
hypercenter	#533	D₁₈₁
hypocenter	C₁₈₁	#1481
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₈₁

commutator

C₂×C₄×C₁₈₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₈₁

Fitting

#533×C₁₈₁

C₂

socle

C₂²×C₁₈₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₈₁

C₂²⋊C₄

cosocle

C₄²×C₁₈₁

C₂³

D₁₈₁

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. Q₈⋊#13089678	C₂	semidirect
74. C₁₈₁	#1481	semidirect

H ≅ G/N

type

3. Q₈⋊#13089678

C₂

semidirect

74. C₁₈₁

#1481

semidirect

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

subgroup	index
#26041332	2
C₂×C₄⋊#13089678	2
Q₈⋊#13089678	2
#533×C₁₈₁	2
#1481	181

subgroup

index

#533×C₁₈₁

181

Subgroups — #37532124

Conjugacy classes of subgroups

Complements

Maximal subgroups