Group #37535193 — subgroups

Subgroups — #37535193

Order 23168.

230 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 91 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×#19153749	C₂	1	2	yes	maximal
2	#26041177	C₂	1	1	yes	maximal, characteristic
3	#26041041	C₂	1	2	yes	maximal
4	#26041034	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13089678	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₈₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₈₁	C₂²	1	1	yes	characteristic
9	C₂×#15668428	—	2	1	no	—
10	#19153749	—	2	4	no	—
11	#19153723	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₈₁	—	2	1	no	—
13	C₂×C₄⋊C₄×C₁₈₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₈₁	—	2	1	no	—
17	C₂²×Dic₁₈₁	—	2	1	no	—
18	C₂×#13089678	—	2	2	no	—
19	C₂×#13089678	—	2	2	no	—
20	C₂×#13089678	Q₈	1	4	yes	—
21	C₄⋊Dic₁₈₁	—	2	1	no	—
22	C₄⋊Dic₁₈₁	—	2	2	no	—
23	C₄⋊Dic₁₈₁	D₄	1	2	yes	—
24	#15668428	—	4	2	no	—
25	C₂²×C₄×C₁₈₁	—	2	2	no	—
26	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₈₁	—	2	2	no	—
30	C₄⋊C₄×C₁₈₁	—	2	2	no	—
31	C₄⋊C₄×C₁₈₁	—	4	2	no	—
32	C₄⋊C₄×C₁₈₁	—	4	2	no	—
33	C₄²×C₁₈₁	—	2	2	no	—
34	C₂×Dic₁₈₁	—	2	2	no	—
35	C₂×Dic₁₈₁	—	4	1	no	—
36	C₂×Dic₁₈₁	—	4	1	no	—
37	C₂×Dic₁₈₁	—	4	1	no	—
38	C₂×Dic₁₈₁	—	4	1	no	—
39	C₂×Dic₁₈₁	—	4	1	no	—
40	#13089678	—	2	4	no	—
41	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
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	—	4	2	no	—
47	C₂×C₄×C₁₈₁	—	4	2	no	—
48	C₂×C₄×C₁₈₁	—	4	2	no	—
49	C₂×C₄×C₁₈₁	D₈	1	2	yes	—
50	C₂×C₄×C₁₈₁	Q₁₆	1	2	yes	—
51	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₁₈₁	#41	1	1	yes	characteristic
54	C₂×C₄×C₁₈₁	#41	1	2	yes	—
55	C₂²×C₁₈₁	#121	1	1	yes	characteristic
56	C₂²×C₁₈₁	#123	1	1	yes	characteristic
57	C₂²×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
58	C₂²×C₁₈₁	C₂×D₈	1	1	yes	characteristic
59	C₂²×C₁₈₁	C₂×Q₁₆	1	1	yes	characteristic
60	C₂²×C₁₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂²×C₁₈₁	#137	1	1	yes	characteristic
62	C₄×C₁₈₁	—	2	2	no	—
63	C₄×C₁₈₁	—	2	2	no	—
64	C₄×C₁₈₁	—	8	2	no	—
65	C₄×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₁₈₁	#125	1	2	yes	—
67	Dic₁₈₁	—	4	2	no	—
68	Dic₁₈₁	—	8	1	no	—
69	C₂×C₁₈₁	#398	1	1	yes	characteristic
70	C₂×C₁₈₁	#459	1	1	yes	characteristic
71	C₂×C₁₈₁	#462	1	1	yes	characteristic
72	C₂×C₁₈₁	#480	1	1	yes	characteristic
73	C₂×C₁₈₁	#484	1	1	yes	characteristic
74	C₂×C₁₈₁	#500	1	1	yes	characteristic
75	C₂×C₁₈₁	#501	1	1	yes	characteristic
76	C₁₈₁	#2083	1	1	yes	characteristic, complemented, Sylow
77	#2083	—	181	1	no	maximal, Sylow
78	C₂×#107	—	181	2	no	—
79	#389	—	181	1	no	—
80	#384	D₁₈₁	1	1	yes	characteristic
81	#340	—	181	2	no	—
82	#336	—	181	1	no	—
83	C₂²×C₈	—	181	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	362	1	no	—
87	C₂×C₄⋊C₄	—	181	1	no	—
88	C₂×C₄⋊C₄	—	181	1	no	—
89	C₂×C₄⋊C₄	C₂×D₁₈₁	1	2	yes	—
90	C₂×C₄²	C₂×D₁₈₁	1	1	yes	characteristic
91	#107	—	362	4	no	—
92	#95	—	362	1	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	362	1	no	—
95	C₂²×C₄	—	362	1	no	—
96	C₂²×C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
99	C₂×C₈	—	362	2	no	—
100	C₂×C₈	—	362	2	no	—
101	C₂×C₈	—	181	4	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	362	1	no	—
107	C₄⋊C₄	—	362	2	no	—
108	C₄⋊C₄	—	181	2	no	—
109	C₄⋊C₄	—	724	2	no	—
110	C₄²	—	2	2	no	—
111	C₂³	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	362	2	no	—
120	C₂×C₄	—	724	1	no	—
121	C₂×C₄	—	724	1	no	—
122	C₂×C₄	—	724	1	no	—
123	C₂×C₄	—	724	1	no	—
124	C₂×C₄	—	724	1	no	—
125	C₂×C₄	#15668434	1	2	yes	—
126	C₂×C₄	#15668439	1	2	yes	—
127	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₈₁	1	2	yes	—
129	C₂×C₄	#15668448	1	1	yes	characteristic
130	C₂×C₄	#15668449	1	2	yes	—
131	C₈	—	362	4	no	—
132	C₂²	#19153787	1	1	yes	characteristic
133	C₂²	#19153799	1	1	yes	characteristic
134	C₂²	#19153814	1	1	yes	characteristic
135	C₂²	C₂×#15668434	1	1	yes	characteristic
136	C₂²	C₂×#15668439	1	1	yes	characteristic
137	C₂²	#19153842	1	1	yes	characteristic
138	C₂²	#19153847	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	8	2	no	—
142	C₄	—	1448	1	no	—
143	C₄	—	724	2	no	—
144	C₄	#19153794	1	2	yes	—
145	C₄	#19153806	1	2	yes	—
146	C₂	#26041210	1	1	yes	characteristic
147	C₂	#26041471	1	1	yes	characteristic
148	C₂	#26041486	1	1	yes	characteristic
149	C₂	#26041592	1	1	yes	characteristic
150	C₂	#26041614	1	1	yes	characteristic
151	C₂	#26041682	1	1	yes	characteristic
152	C₂	#26041685	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153749

C₂

yes

maximal

#26041177

C₂

yes

maximal, characteristic

#26041041

C₂

yes

maximal

#26041034

C₂

yes

maximal, characteristic

#384×C₁₈₁

C₂

yes

maximal, characteristic

C₂²×#13089678

C₂²

yes

—

C₂×C₄⋊Dic₁₈₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₈₁

C₂²

yes

—

—

—

C₂×C₄⋊C₄×C₁₈₁

—

C₂×C₄⋊C₄×C₁₈₁

—

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

—

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

C₂²×Dic₁₈₁

—

C₂²×Dic₁₈₁

—

C₂×#13089678

—

C₂×#13089678

—

C₂×#13089678

Q₈

yes

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

D₄

yes

—

#15668428

—

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

—

C₄²×C₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

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

D₈

yes

—

C₂×C₄×C₁₈₁

Q₁₆

yes

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

#41

yes

characteristic

C₂×C₄×C₁₈₁

#41

yes

—

C₂²×C₁₈₁

yes

C₂²×C₁₈₁

yes

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

C₂²⋊Q₈

yes

—

C₄×C₁₈₁

#125

yes

—

Dic₁₈₁

—

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

#2083

—

181

maximal, Sylow

C₂×#107

—

181

—

#389

—

181

—

#384

D₁₈₁

yes

characteristic

#340

—

181

—

#336

—

181

—

C₂²×C₈

—

181

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

362

—

C₂×C₄⋊C₄

—

181

—

C₂×C₄⋊C₄

—

181

—

C₂×C₄⋊C₄

C₂×D₁₈₁

yes

—

C₂×C₄²

C₂×D₁₈₁

yes

characteristic

#107

—

362

—

#95

—

362

—

C₂²×C₄

—

C₂²×C₄

—

362

—

C₂²×C₄

—

362

—

C₂²×C₄

C₂²⋊D₁₈₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₈₁

yes

characteristic

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

C₂×C₈

—

362

—

100

C₂×C₈

—

362

—

101

C₂×C₈

—

181

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

362

—

107

C₄⋊C₄

—

362

—

108

C₄⋊C₄

—

181

—

109

C₄⋊C₄

—

724

—

110

C₄²

—

111

C₂³

C₂×C₂²⋊D₁₈₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

362

—

120

C₂×C₄

—

724

—

121

C₂×C₄

—

724

—

122

C₂×C₄

—

724

—

123

C₂×C₄

—

724

—

124

C₂×C₄

—

724

—

125

C₂×C₄

#15668434

yes

—

126

C₂×C₄

#15668439

yes

—

127

C₂×C₄

D₄×D₁₈₁

yes

characteristic

128

C₂×C₄

Q₈×D₁₈₁

yes

—

129

C₂×C₄

#15668448

yes

characteristic

130

C₂×C₄

#15668449

yes

—

131

C₈

—

362

—

132

C₂²

#19153787

yes

characteristic

133

C₂²

#19153799

yes

characteristic

134

C₂²

#19153814

yes

characteristic

135

C₂²

C₂×#15668434

yes

characteristic

136

C₂²

C₂×#15668439

yes

characteristic

137

C₂²

#19153842

yes

characteristic

138

C₂²

#19153847

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

1448

—

143

C₄

—

724

—

144

C₄

#19153794

yes

—

145

C₄

#19153806

yes

—

146

C₂

#26041210

yes

characteristic

147

C₂

#26041471

yes

characteristic

148

C₂

#26041486

yes

characteristic

149

C₂

#26041592

yes

characteristic

150

C₂

#26041614

yes

characteristic

151

C₂

#26041682

yes

characteristic

152

C₂

#26041685

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₈₁
commutator, cosocle	C₂²×C₄×C₁₈₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	#384×C₁₈₁	C₂
socle, prime radical	C₂³×C₁₈₁	C₂×D₄
hypercenter	#384	D₁₈₁
hypocenter	C₁₈₁	#2083
LCS	G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₈₁

commutator, cosocle

C₂²×C₄×C₁₈₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

#384×C₁₈₁

C₂

socle, prime radical

C₂³×C₁₈₁

C₂×D₄

D₁₈₁

C₁₈₁

G ▷ C₂²×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
76. C₁₈₁	#2083	semidirect

H ≅ G/N

type

76. C₁₈₁

#2083

semidirect

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

subgroup	index
C₂×#19153749	2
#26041177	2
#26041041	2
#26041034	2
#384×C₁₈₁	2
#2083	181

subgroup

index

#384×C₁₈₁

181

Subgroups — #37535193

Conjugacy classes of subgroups

Complements

Maximal subgroups