Group #37572514 — subgroups

Subgroups — #37572514

Order 24448.

272 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 95 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×Q₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
2	#27132686	C₂	1	2	yes	maximal
3	C₂×#20259241	C₂	1	1	yes	maximal, characteristic
4	#27132431	C₂	1	2	yes	maximal
5	C₂×#101×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15690271	C₂²	1	2	yes	—
7	Q₈×Dic₁₉₁	—	2	2	no	—
8	Q₈×Dic₁₉₁	—	2	2	no	—
9	C₂×C₄⋊Dic₁₉₁	C₄	1	2	yes	—
10	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₉₁	C₄	1	2	yes	—
12	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
13	Dic₁₉₁⋊C₈	—	2	2	no	—
14	Dic₁₉₁⋊C₈	—	2	2	no	—
15	#20259241	—	2	2	no	—
16	C₂²×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
18	#101×C₁₉₁	—	2	2	no	—
19	C₂²×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₉₁	C₂×C₄	1	2	yes	—
21	C₂²×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
22	#15690271	—	2	2	no	—
23	#15690271	—	2	2	no	—
24	C₂×C₁₉₁⋊C₈	—	2	2	no	—
25	C₄⋊Dic₁₉₁	—	2	4	no	—
26	C₄⋊Dic₁₉₁	—	2	2	no	—
27	C₄×Dic₁₉₁	—	2	2	no	—
28	C₄×Dic₁₉₁	—	2	2	no	—
29	C₄×Dic₁₉₁	D₄	1	2	yes	—
30	C₄×Dic₁₉₁	D₄	1	2	yes	—
31	C₂×Q₈×C₁₉₁	—	2	2	no	—
32	C₂×Q₈×C₁₉₁	—	2	2	no	—
33	C₂×Q₈×C₁₉₁	D₄	1	2	yes	—
34	C₂×Q₈×C₁₉₁	Q₈	1	2	yes	—
35	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₉₁	—	2	2	no	—
38	M₁₆×C₁₉₁	—	2	2	no	—
39	C₂×C₈×C₁₉₁	—	2	2	no	—
40	C₂×Dic₁₉₁	—	2	2	no	—
41	C₂×Dic₁₉₁	—	2	2	no	—
42	C₂×Dic₁₉₁	—	4	1	no	—
43	C₂×Dic₁₉₁	—	4	2	no	—
44	C₂×Dic₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×Dic₁₉₁	C₂²⋊C₄	1	2	yes	—
46	C₂×Dic₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₁₉₁⋊C₈	—	4	2	no	—
48	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₁₉₁	—	2	4	no	—
50	Q₈×C₁₉₁	—	2	4	no	—
51	C₂×C₄×C₁₉₁	—	2	2	no	—
52	C₂×C₄×C₁₉₁	—	2	2	no	—
53	C₂×C₄×C₁₉₁	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉₁	#41	1	2	yes	—
58	C₈×C₁₉₁	—	4	2	no	—
59	C₂²×C₁₉₁	—	2	1	no	—
60	C₂²×C₁₉₁	—	2	1	no	—
61	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₁₉₁	#117	1	1	yes	characteristic
64	C₄×C₁₉₁	—	2	4	no	—
65	C₄×C₁₉₁	#121	1	2	yes	—
66	C₄×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
67	Dic₁₉₁	—	2	1	no	—
68	Dic₁₉₁	—	2	2	no	—
69	Dic₁₉₁	—	4	2	no	—
70	Dic₁₉₁	#101	1	2	yes	complemented
71	C₂×C₁₉₁	—	2	1	no	—
72	C₂×C₁₉₁	—	2	1	no	—
73	C₂×C₁₉₁	#380	1	1	yes	characteristic
74	C₂×C₁₉₁	C₂×#101	1	1	yes	characteristic
75	C₂×C₁₉₁	#413	1	1	yes	characteristic
76	C₁₉₁	#1801	1	1	yes	characteristic, complemented, Sylow
77	#1801	—	191	1	no	maximal, Sylow
78	C₂×C₄×Q₈	—	191	1	no	—
79	#423	—	191	2	no	—
80	C₂×#101	—	191	1	no	—
81	C₂×#101	D₁₉₁	1	1	yes	characteristic
82	#343	—	191	2	no	—
83	C₂²×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
84	C₂×M₁₆	—	191	2	no	—
85	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
86	C₄×Q₈	—	382	2	no	—
87	C₄×Q₈	—	382	2	no	—
88	C₂×C₄⋊C₄	—	191	2	no	—
89	C₂×C₄⋊C₄	—	191	1	no	—
90	C₂×C₄²	—	191	2	no	—
91	C₂×C₄²	—	191	1	no	—
92	C₄⋊C₈	—	382	2	no	—
93	C₄⋊C₈	—	382	2	no	—
94	#101	—	2	2	no	—
95	#101	—	382	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂²⋊D₁₉₁	1	2	yes	—
99	C₂×Q₈	Dic₃₈₂	1	2	yes	—
100	C₂²×C₄	—	191	1	no	—
101	C₂²×C₄	—	191	2	no	—
102	C₂²×C₄	—	191	1	no	—
103	C₂²×C₄	C₄×D₁₉₁	1	2	yes	—
104	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	M₁₆	—	382	2	no	—
108	M₁₆	—	382	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	382	2	no	—
111	C₄⋊C₄	—	382	4	no	—
112	C₄⋊C₄	—	382	2	no	—
113	C₄²	—	382	2	no	—
114	C₄²	—	382	2	no	—
115	C₄²	—	191	2	no	—
116	C₄²	—	191	2	no	—
117	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	382	2	no	—
123	C₂×C₄	—	382	2	no	—
124	C₂×C₄	—	191	1	no	—
125	C₂×C₄	—	191	2	no	—
126	C₂×C₄	—	191	1	no	—
127	C₂×C₄	—	764	1	no	—
128	C₂×C₄	—	764	2	no	—
129	C₂×C₄	Dic₁₉₁⋊C₄	1	4	yes	—
130	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
132	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
133	C₂×C₄	#15690286	1	2	yes	—
134	C₈	—	4	2	no	—
135	C₈	—	764	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	C₂²⋊C₄×D₁₉₁	1	1	yes	characteristic
139	C₂²	C₂×Dic₁₉₁⋊C₄	1	1	yes	characteristic
140	C₂²	#20259278	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	382	1	no	—
143	C₄	—	382	2	no	—
144	C₄	—	191	2	no	—
145	C₄	—	764	2	no	—
146	C₄	#20259294	1	2	yes	—
147	C₄	#20259299	1	2	yes	—
148	C₂	—	2	1	no	—
149	C₂	—	2	1	no	—
150	C₂	#27132521	1	1	yes	characteristic
151	C₂	#101×D₁₉₁	1	1	yes	characteristic
152	C₂	#27132646	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₁₉₁

C₂

yes

maximal, characteristic

#27132686

C₂

yes

maximal

C₂×#20259241

C₂

yes

maximal, characteristic

#27132431

C₂

yes

maximal

C₂×#101×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₂²

yes

—

Q₈×Dic₁₉₁

—

Q₈×Dic₁₉₁

—

C₂×C₄⋊Dic₁₉₁

C₄

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₁

C₄

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

characteristic

Dic₁₉₁⋊C₈

—

Dic₁₉₁⋊C₈

—

#20259241

—

C₂²×Q₈×C₁₉₁

C₂²

yes

characteristic

C₂×M₁₆×C₁₉₁

C₂²

yes

—

#101×C₁₉₁

—

C₂²×Dic₁₉₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₁

C₂×C₄

yes

—

C₂²×Dic₁₉₁

C₂×C₄

yes

characteristic

#15690271

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

D₄

yes

—

C₄×Dic₁₉₁

D₄

yes

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

D₄

yes

—

C₂×Q₈×C₁₉₁

Q₈

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₉₁

C₂²⋊C₄

yes

—

C₂×Dic₁₉₁

C₂²⋊C₄

yes

characteristic

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₁

#117

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

#121

yes

—

C₄×C₁₉₁

C₂²⋊Q₈

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

#101

yes

complemented

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1801

—

191

maximal, Sylow

C₂×C₄×Q₈

—

191

—

#423

—

191

—

C₂×#101

—

191

—

C₂×#101

D₁₉₁

yes

characteristic

#343

—

191

—

C₂²×Q₈

C₂×D₁₉₁

yes

characteristic

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

C₄×Q₈

—

382

—

C₄×Q₈

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄²

—

191

—

C₂×C₄²

—

191

—

C₄⋊C₈

—

382

—

C₄⋊C₈

—

382

—

#101

—

#101

—

382

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₉₁

yes

—

C₂×Q₈

Dic₃₈₂

yes

—

100

C₂²×C₄

—

191

—

101

C₂²×C₄

—

191

—

102

C₂²×C₄

—

191

—

103

C₂²×C₄

C₄×D₁₉₁

yes

—

104

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

382

—

108

M₁₆

—

382

—

109

C₂×C₈

—

110

C₂×C₈

—

382

—

111

C₄⋊C₄

—

382

—

112

C₄⋊C₄

—

382

—

113

C₄²

—

382

—

114

C₄²

—

382

—

115

C₄²

—

191

—

116

C₄²

—

191

—

117

C₂³

C₂×C₄×D₁₉₁

yes

characteristic

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

382

—

123

C₂×C₄

—

382

—

124

C₂×C₄

—

191

—

125

C₂×C₄

—

191

—

126

C₂×C₄

—

191

—

127

C₂×C₄

—

764

—

128

C₂×C₄

—

764

—

129

C₂×C₄

Dic₁₉₁⋊C₄

yes

—

130

C₂×C₄

D₄×D₁₉₁

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

132

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

133

C₂×C₄

#15690286

yes

—

134

C₈

—

135

C₈

—

764

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂²⋊C₄×D₁₉₁

yes

characteristic

139

C₂²

C₂×Dic₁₉₁⋊C₄

yes

characteristic

140

C₂²

#20259278

yes

characteristic

141

C₄

—

142

C₄

—

382

—

143

C₄

—

382

—

144

C₄

—

191

—

145

C₄

—

764

—

146

C₄

#20259294

yes

—

147

C₄

#20259299

yes

—

148

C₂

—

149

C₂

—

150

C₂

#27132521

yes

characteristic

151

C₂

#101×D₁₉₁

yes

characteristic

152

C₂

#27132646

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₉₁
commutator, prime radical	C₂³×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₂×#101×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₂×#101	D₁₉₁
hypocenter	C₁₉₁	#1801
LCS	G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₉₁

commutator, prime radical

C₂³×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₂×#101×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
70. Dic₁₉₁	#101	semidirect
76. C₁₉₁	#1801	semidirect

H ≅ G/N

type

70. Dic₁₉₁

#101

semidirect

76. C₁₉₁

#1801

semidirect

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

subgroup	index
C₂×Q₈×Dic₁₉₁	2
#27132686	2
C₂×#20259241	2
#27132431	2
C₂×#101×C₁₉₁	2
#1801	191

subgroup

index

C₂×Q₈×Dic₁₉₁

C₂×#101×C₁₉₁

191

Subgroups — #37572514

Conjugacy classes of subgroups

Complements

Maximal subgroups