Group #37572607 — subgroups

Subgroups — #37572607

Order 24448.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₄⋊Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
2	#27132523	C₂	1	1	yes	maximal, characteristic
3	#27132426	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₉₁	—	2	1	no	—
6	C₂×C₄⋊Dic₁₉₁	—	2	2	no	—
7	C₂×C₄⋊Dic₁₉₁	—	2	1	no	—
8	C₂×C₄⋊Dic₁₉₁	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₁₉₁	—	2	1	no	—
12	C₁₉₁⋊#95	—	2	1	no	—
13	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₉₁	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₉₁	—	2	2	no	—
16	C₂²×Dic₁₉₁	—	2	4	no	—
17	C₂²×Dic₁₉₁	—	2	2	no	—
18	C₂²×Dic₁₉₁	—	2	2	no	—
19	C₂²×Dic₁₉₁	—	4	2	no	—
20	C₂²×Dic₁₉₁	—	4	2	no	—
21	C₄⋊Dic₁₉₁	—	2	4	no	—
22	C₄⋊Dic₁₉₁	—	2	2	no	—
23	C₄⋊Dic₁₉₁	—	2	4	no	—
24	C₄⋊Dic₁₉₁	C₂×C₄	1	4	yes	—
25	C₂²⋊Dic₁₉₁	—	4	2	no	—
26	C₂⁴×C₁₉₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₁	—	2	4	no	—
28	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
30	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₉₁	—	2	2	no	—
33	C₂×C₈×C₁₉₁	—	2	4	no	—
34	C₂×Dic₁₉₁	—	2	4	no	—
35	C₂×Dic₁₉₁	—	4	8	no	—
36	C₂×Dic₁₉₁	—	4	2	no	—
37	C₂×Dic₁₉₁	—	4	2	no	—
38	C₂×Dic₁₉₁	—	4	2	no	—
39	C₂×Dic₁₉₁	—	4	2	no	—
40	C₂×Dic₁₉₁	—	4	2	no	—
41	C₂³×C₁₉₁	—	2	2	no	—
42	C₂³×C₁₉₁	—	2	2	no	—
43	C₂³×C₁₉₁	C₂²⋊C₄	1	2	yes	—
44	C₂³×C₁₉₁	D₈	1	2	yes	—
45	C₂³×C₁₉₁	SD₁₆	1	2	yes	—
46	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉₁	—	2	2	no	—
48	C₂×C₄×C₁₉₁	—	2	4	no	—
49	C₂×C₄×C₁₉₁	—	2	4	no	—
50	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉₁	#41	1	4	yes	—
54	C₈×C₁₉₁	—	4	2	no	—
55	C₂²×C₁₉₁	—	2	2	no	—
56	C₂²×C₁₉₁	—	2	2	no	—
57	C₂²×C₁₉₁	—	2	4	no	—
58	C₂²×C₁₉₁	—	2	4	no	—
59	C₂²×C₁₉₁	#102	1	4	yes	—
60	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₁₉₁	#123	1	2	yes	—
62	C₂²×C₁₉₁	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₁₉₁	#137	1	2	yes	—
65	C₄×C₁₉₁	—	2	2	no	—
66	C₄×C₁₉₁	#117	1	2	yes	—
67	C₄×C₁₉₁	#123	1	2	yes	—
68	Dic₁₉₁	—	4	4	no	—
69	Dic₁₉₁	—	8	2	no	—
70	C₂×C₁₉₁	—	2	4	no	—
71	C₂×C₁₉₁	#381	1	1	yes	characteristic
72	C₂×C₁₉₁	C₂×#102	1	1	yes	characteristic
73	C₂×C₁₉₁	#419	1	1	yes	characteristic
74	C₂×C₁₉₁	#480	1	2	yes	—
75	C₂×C₁₉₁	#483	1	2	yes	—
76	C₁₉₁	#1823	1	1	yes	characteristic, complemented, Sylow
77	#1823	—	191	1	no	maximal, Sylow
78	C₂²×C₄⋊C₄	—	191	1	no	—
79	C₂×C₂²⋊C₈	D₁₉₁	1	1	yes	characteristic
80	#380	—	191	1	no	—
81	#340	—	191	4	no	—
82	C₂³×C₄	—	382	1	no	—
83	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
84	C₂²×C₈	C₂×D₁₉₁	1	2	yes	—
85	C₂×C₄⋊C₄	—	382	2	no	—
86	C₂×C₄⋊C₄	—	382	1	no	—
87	C₂×C₄⋊C₄	—	191	2	no	—
88	C₂×C₄⋊C₄	—	191	4	no	—
89	C₂×C₄⋊C₄	—	191	2	no	—
90	C₂×C₂²⋊C₄	—	382	1	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	#95	—	382	1	no	—
93	C₂⁴	D₇₆₄	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	382	4	no	—
96	C₂²×C₄	—	382	2	no	—
97	C₂²×C₄	—	382	2	no	—
98	C₂²×C₄	—	764	2	no	—
99	C₂²×C₄	—	764	2	no	—
100	C₂²×C₄	C₄×D₁₉₁	1	2	yes	—
101	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
102	C₂²×C₄	D₇₆₄	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	4	no	—
106	C₄⋊C₄	—	382	4	no	—
107	C₄⋊C₄	—	382	2	no	—
108	C₄⋊C₄	—	382	4	no	—
109	C₄⋊C₄	—	191	4	no	—
110	C₂²⋊C₄	—	764	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
114	C₂³	D₁₅₂₈	1	2	yes	—
115	C₂³	#15690276	1	2	yes	—
116	C₂³	C₂×D₇₆₄	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	382	4	no	—
121	C₂×C₄	—	764	2	no	—
122	C₂×C₄	—	764	2	no	—
123	C₂×C₄	—	764	2	no	—
124	C₂×C₄	—	764	2	no	—
125	C₂×C₄	—	764	2	no	—
126	C₂×C₄	—	764	8	no	—
127	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
128	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
130	C₂×C₄	#15690286	1	4	yes	—
131	C₈	—	4	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	4	no	—
134	C₂²	—	2	4	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#20259244	1	4	yes	—
137	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
138	C₂²	#20259306	1	2	yes	—
139	C₂²	C₂×D₁₅₂₈	1	1	yes	characteristic
140	C₂²	C₂×#15690276	1	1	yes	characteristic
141	C₂²	#20259355	1	2	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	1528	2	no	—
144	C₄	—	764	4	no	—
145	C₄	#20259278	1	2	yes	—
146	C₄	#20259303	1	2	yes	—
147	C₂	—	2	4	no	—
148	C₂	#27132524	1	1	yes	characteristic
149	C₂	C₂×#20259244	1	1	yes	characteristic
150	C₂	#27132673	1	1	yes	characteristic
151	C₂	#27132977	1	2	yes	—
152	C₂	#27132994	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₁₉₁

C₂

yes

maximal, characteristic

#27132523

C₂

yes

maximal, characteristic

#27132426

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂³×Dic₁₉₁

—

C₂×C₄⋊Dic₁₉₁

—

C₂×C₄⋊Dic₁₉₁

—

C₂×C₄⋊Dic₁₉₁

C₄

yes

complemented

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×C₂²⋊Dic₁₉₁

—

C₁₉₁⋊#95

—

C₂³×C₄×C₁₉₁

C₂²

yes

characteristic

C₂²×C₈×C₁₉₁

C₂²

yes

—

C₂²⋊C₈×C₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

C₂²⋊Dic₁₉₁

—

C₂⁴×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂²⋊C₄

yes

—

C₂³×C₁₉₁

D₈

yes

—

C₂³×C₁₉₁

SD₁₆

yes

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#102

yes

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₁

#123

yes

—

C₂²×C₁₉₁

C₂×D₈

yes

characteristic

C₂²×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₁

#137

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#117

yes

—

C₄×C₁₉₁

#123

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

—

C₂×C₁₉₁

#483

yes

—

C₁₉₁

#1823

yes

characteristic, complemented, Sylow

#1823

—

191

maximal, Sylow

C₂²×C₄⋊C₄

—

191

—

C₂×C₂²⋊C₈

D₁₉₁

yes

characteristic

#380

—

191

—

#340

—

191

—

C₂³×C₄

—

382

—

C₂³×C₄

C₂×D₁₉₁

yes

characteristic

C₂²×C₈

C₂×D₁₉₁

yes

—

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₂²⋊C₄

—

382

—

C₂²⋊C₈

—

#95

—

382

—

C₂⁴

D₇₆₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

—

764

—

C₂²×C₄

—

764

—

100

C₂²×C₄

C₄×D₁₉₁

yes

—

101

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

102

C₂²×C₄

D₇₆₄

yes

characteristic

103

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

382

—

107

C₄⋊C₄

—

382

—

108

C₄⋊C₄

—

382

—

109

C₄⋊C₄

—

191

—

110

C₂²⋊C₄

—

764

—

111

C₂³

—

112

C₂³

—

113

C₂³

(C₂×D₁₉₁)⋊C₄

yes

—

114

C₂³

D₁₅₂₈

yes

—

115

C₂³

#15690276

yes

—

116

C₂³

C₂×D₇₆₄

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

382

—

121

C₂×C₄

—

764

—

122

C₂×C₄

—

764

—

123

C₂×C₄

—

764

—

124

C₂×C₄

—

764

—

125

C₂×C₄

—

764

—

126

C₂×C₄

—

764

—

127

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

128

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

130

C₂×C₄

#15690286

yes

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#20259244

yes

—

137

C₂²

C₂×(C₂×D₁₉₁)⋊C₄

yes

characteristic

138

C₂²

#20259306

yes

—

139

C₂²

C₂×D₁₅₂₈

yes

characteristic

140

C₂²

C₂×#15690276

yes

characteristic

141

C₂²

#20259355

yes

—

142

C₄

—

143

C₄

—

1528

—

144

C₄

—

764

—

145

C₄

#20259278

yes

—

146

C₄

#20259303

yes

—

147

C₂

—

148

C₂

#27132524

yes

characteristic

149

C₂

C₂×#20259244

yes

characteristic

150

C₂

#27132673

yes

characteristic

151

C₂

#27132977

yes

—

152

C₂

#27132994

yes

—

	subgroup	G/H
center	C₂³	C₂×D₇₆₄
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₂×C₂²⋊C₈×C₁₉₁	C₂
socle	C₂³×C₁₉₁	C₂×D₄
prime radical	C₂⁴×C₁₉₁	D₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₂×C₂²⋊C₈	D₁₉₁
hypocenter	C₁₉₁	#1823
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₇₆₄

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₂×C₂²⋊C₈×C₁₉₁

C₂

socle

C₂³×C₁₉₁

C₂×D₄

prime radical

C₂⁴×C₁₉₁

D₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

hypercenter

C₂×C₂²⋊C₈

D₁₉₁

hypocenter

C₁₉₁

#1823

LCS

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #27132426	C₂	semidirect
8. C₂×C₄⋊Dic₁₉₁	C₄	semidirect
76. C₁₉₁	#1823	semidirect

H ≅ G/N

type

3. #27132426

C₂

semidirect

8. C₂×C₄⋊Dic₁₉₁

C₄

semidirect

76. C₁₉₁

#1823

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₁₉₁	2
#27132523	2
#27132426	2
C₂×C₂²⋊C₈×C₁₉₁	2
#1823	191

subgroup

index

C₂²×C₄⋊Dic₁₉₁

#27132523

#27132426

C₂×C₂²⋊C₈×C₁₉₁

#1823

191

Subgroups — #37572607

Conjugacy classes of subgroups

Complements

Maximal subgroups