Group #37581172 — subgroups

Subgroups — #37581172

Order 24448.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₄×#15690287	C₂	1	1	yes	maximal, characteristic
2	#27132722	C₂	1	4	yes	maximal, complemented
3	#27132718	C₂	1	4	yes	maximal, complemented
4	#27132691	C₂	1	2	yes	maximal, complemented
5	#27132631	C₂	1	2	yes	maximal
6	C₂×#20259233	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15690287	C₂²	1	1	yes	characteristic
9	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₃₈₂	C₄	1	2	yes	—
11	C₄×D₇₆₄	C₄	1	2	yes	—
12	C₄×C₂²⋊D₁₉₁	C₄	1	4	yes	—
13	#20259281	C₂²	1	1	yes	characteristic
14	#20259279	C₂²	1	2	yes	—
15	C₄²×D₁₉₁	C₂²	1	2	yes	—
16	Dic₁₉₁⋊C₈	C₂²	1	8	yes	—
17	(C₂×D₁₉₁)⋊C₈	C₂²	1	8	yes	—
18	#20259233	C₂²	1	2	yes	—
19	#20259233	C₂²	1	2	yes	—
20	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
22	#97×C₁₉₁	C₂²	1	2	yes	—
23	C₄×C₈×C₁₉₁	C₂²	1	2	yes	—
24	#15690287	—	2	4	no	—
25	C₂×Dic₃₈₂	C₂×C₄	1	1	yes	characteristic
26	C₂×D₇₆₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₁₉₁	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₉₁	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₉₁	C₂³	1	2	yes	—
30	C₂×C₁₉₁⋊C₈	—	2	4	no	—
31	C₂×C₁₉₁⋊C₈	C₂³	1	4	yes	—
32	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
33	Dic₁₉₁⋊C₄	C₂×C₄	1	4	yes	—
34	C₂²⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
35	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	4	yes	—
36	C₄×Dic₁₉₁	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₉₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
40	M₁₆×C₁₉₁	—	2	4	no	—
41	C₂×C₈×C₁₉₁	C₂³	1	4	yes	—
42	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
43	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
44	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
45	C₂²×D₁₉₁	C₂²×C₄	1	2	yes	—
46	Dic₃₈₂	—	2	2	no	—
47	D₇₆₄	—	2	2	no	—
48	C₂²⋊D₁₉₁	—	2	4	no	—
49	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
50	C₂×Dic₁₉₁	C₂²×C₄	1	4	yes	—
51	C₄×D₁₉₁	—	2	4	no	—
52	C₄×D₁₉₁	—	4	4	no	—
53	C₁₉₁⋊C₈	—	2	4	no	—
54	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉₁	—	4	2	no	—
56	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₉₁	#41	1	4	yes	—
60	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₉₁	—	2	4	no	—
62	C₂²×C₁₉₁	—	2	2	no	—
63	C₂²×C₁₉₁	#117	1	2	yes	—
64	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₁₉₁	—	2	2	no	—
66	C₂×D₁₉₁	—	2	4	no	—
67	C₄×C₁₉₁	—	2	2	no	—
68	C₄×C₁₉₁	—	2	2	no	—
69	C₄×C₁₉₁	#117	1	2	yes	—
70	C₄×C₁₉₁	C₂×#41	1	2	yes	—
71	Dic₁₉₁	—	2	2	no	—
72	Dic₁₉₁	—	2	4	no	—
73	C₂×C₁₉₁	—	2	2	no	—
74	C₂×C₁₉₁	C₂×#117	1	1	yes	characteristic
75	C₂×C₁₉₁	#568	1	2	yes	—
76	D₁₉₁	—	4	2	no	—
77	C₁₉₁	#2923	1	1	yes	characteristic, complemented, Sylow
78	#2923	—	191	1	no	maximal, Sylow
79	C₄×#41	—	191	1	no	—
80	#433	—	191	4	no	—
81	#432	—	191	4	no	—
82	#424	—	191	2	no	—
83	#408	—	191	2	no	—
84	C₄×M₁₆	D₁₉₁	1	1	yes	characteristic, complemented
85	C₂×#97	—	191	1	no	—
86	C₂×#41	—	191	1	no	—
87	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
88	C₂²×C₈	—	191	2	no	—
89	C₄×Q₈	—	191	2	no	—
90	C₄×D₄	—	191	4	no	—
91	C₄×D₄	—	191	2	no	—
92	#117	—	191	1	no	—
93	#117	—	191	2	no	—
94	C₂×C₄²	—	191	2	no	—
95	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
96	C₄⋊C₈	—	191	8	no	—
97	C₂²⋊C₈	—	191	8	no	—
98	#97	—	191	2	no	—
99	#97	—	191	2	no	—
100	#97	C₂×D₁₉₁	1	2	yes	—
101	C₄×C₈	C₂×D₁₉₁	1	2	yes	—
102	#41	—	382	4	no	—
103	C₂×Q₈	—	191	1	no	—
104	C₂×D₄	—	191	2	no	—
105	C₂×D₄	—	191	1	no	—
106	C₂²×C₄	—	191	2	no	—
107	C₂²×C₄	—	191	4	no	—
108	C₂²×C₄	C₄×D₁₉₁	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	382	4	no	—
112	C₂×C₈	—	191	4	no	—
113	C₂×C₈	C₂²×D₁₉₁	1	4	yes	—
114	C₄⋊C₄	—	191	4	no	—
115	C₄⋊C₄	—	191	2	no	—
116	C₂²⋊C₄	—	191	2	no	—
117	C₂²⋊C₄	—	191	4	no	—
118	C₄²	—	191	4	no	—
119	C₄²	—	191	2	no	—
120	C₄²	C₄×D₁₉₁	1	2	yes	—
121	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
122	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
123	C₂³	—	191	2	no	—
124	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
125	Q₈	—	382	2	no	—
126	D₄	—	382	4	no	—
127	D₄	—	382	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	382	4	no	—
130	C₂×C₄	—	191	4	no	—
131	C₂×C₄	—	191	2	no	—
132	C₂×C₄	—	764	4	no	—
133	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
136	C₂×C₄	#15690287	1	4	yes	—
137	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	382	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	382	2	no	—
142	C₂²	—	382	4	no	—
143	C₂²	#20259279	1	2	yes	—
144	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	382	2	no	—
148	C₄	—	382	4	no	—
149	C₄	#20259279	1	2	yes	—
150	C₄	C₂×#15690287	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	764	2	no	—
153	C₂	C₂×#20259279	1	1	yes	characteristic
154	C₂	#27133429	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#15690287

C₂

yes

maximal, characteristic

#27132722

C₂

yes

maximal, complemented

#27132718

C₂

yes

maximal, complemented

#27132691

C₂

yes

maximal, complemented

#27132631

C₂

yes

maximal

C₂×#20259233

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690287

C₂²

yes

characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₄×Dic₃₈₂

C₄

yes

—

C₄×D₇₆₄

C₄

yes

—

C₄×C₂²⋊D₁₉₁

C₄

yes

—

#20259281

C₂²

yes

characteristic

#20259279

C₂²

yes

—

C₄²×D₁₉₁

C₂²

yes

—

Dic₁₉₁⋊C₈

C₂²

yes

—

(C₂×D₁₉₁)⋊C₈

C₂²

yes

—

#20259233

C₂²

yes

—

#20259233

C₂²

yes

—

C₂×M₁₆×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#97×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

—

#15690287

—

C₂×Dic₃₈₂

C₂×C₄

yes

characteristic

C₂×D₇₆₄

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₁₉₁

C₂×C₄

yes

—

C₂×C₄×D₁₉₁

C₂×C₄

yes

—

C₂×C₄×D₁₉₁

C₂³

yes

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

Dic₁₉₁⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₁₉₁

C₂×C₄

yes

—

(C₂×D₁₉₁)⋊C₄

C₂×C₄

yes

—

C₄×Dic₁₉₁

C₂×C₄

yes

—

C₄×Dic₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

C₂²×C₄

yes

—

Dic₃₈₂

—

D₇₆₄

—

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

C₂²×C₄

yes

—

C₂×Dic₁₉₁

C₂²×C₄

yes

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#117

yes

—

C₂²×C₁₉₁

C₂³×C₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#117

yes

—

C₄×C₁₉₁

C₂×#41

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×#117

yes

characteristic

C₂×C₁₉₁

#568

yes

—

D₁₉₁

—

C₁₉₁

#2923

yes

characteristic, complemented, Sylow

#2923

—

191

maximal, Sylow

C₄×#41

—

191

—

#433

—

191

—

#432

—

191

—

#424

—

191

—

#408

—

191

—

C₄×M₁₆

D₁₉₁

yes

characteristic, complemented

C₂×#97

—

191

—

C₂×#41

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

C₂²×C₈

—

191

—

C₄×Q₈

—

191

—

C₄×D₄

—

191

—

C₄×D₄

—

191

—

#117

—

191

—

#117

—

191

—

C₂×C₄²

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

191

—

C₂²⋊C₈

—

191

—

#97

—

191

—

#97

—

191

—

100

#97

C₂×D₁₉₁

yes

—

101

C₄×C₈

C₂×D₁₉₁

yes

—

102

#41

—

382

—

103

C₂×Q₈

—

191

—

104

C₂×D₄

—

191

—

105

C₂×D₄

—

191

—

106

C₂²×C₄

—

191

—

107

C₂²×C₄

—

191

—

108

C₂²×C₄

C₄×D₁₉₁

yes

—

109

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

382

—

112

C₂×C₈

—

191

—

113

C₂×C₈

C₂²×D₁₉₁

yes

—

114

C₄⋊C₄

—

191

—

115

C₄⋊C₄

—

191

—

116

C₂²⋊C₄

—

191

—

117

C₂²⋊C₄

—

191

—

118

C₄²

—

191

—

119

C₄²

—

191

—

120

C₄²

C₄×D₁₉₁

yes

—

121

C₄²

C₂²×D₁₉₁

yes

characteristic

122

C₄²

C₂²×D₁₉₁

yes

characteristic

123

C₂³

—

191

—

124

C₂³

C₂×C₄×D₁₉₁

yes

characteristic

125

Q₈

—

382

—

126

D₄

—

382

—

127

D₄

—

382

—

128

C₂×C₄

—

129

C₂×C₄

—

382

—

130

C₂×C₄

—

191

—

131

C₂×C₄

—

191

—

132

C₂×C₄

—

764

—

133

C₂×C₄

C₂×C₄×D₁₉₁

yes

—

134

C₂×C₄

C₂×C₄×D₁₉₁

yes

—

135

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

136

C₂×C₄

#15690287

yes

—

137

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

138

C₈

—

139

C₈

—

382

—

140

C₂²

—

141

C₂²

—

382

—

142

C₂²

—

382

—

143

C₂²

#20259279

yes

—

144

C₂²

C₂²×C₄×D₁₉₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

382

—

148

C₄

—

382

—

149

C₄

#20259279

yes

—

150

C₄

C₂×#15690287

yes

—

151

C₂

—

152

C₂

—

764

—

153

C₂

C₂×#20259279

yes

characteristic

154

C₂

#27133429

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Fitting	C₄×M₁₆×C₁₉₁	C₂
prime radical	C₂×#15690287	C₂²
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	C₄×M₁₆	D₁₉₁
hypocenter	C₁₉₁	#2923
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₉₁

commutator, socle

C₂²×C₁₉₁

C₂³×C₄

Fitting

C₄×M₁₆×C₁₉₁

C₂

prime radical

C₂×#15690287

C₂²

cosocle

C₂×C₄×C₁₉₁

C₂⁴

hypercenter

C₄×M₁₆

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #27132722	C₂	semidirect
3. #27132718	C₂	semidirect
4. #27132691	C₂	semidirect
6. C₂×#20259233	C₂	semidirect
7. C₄×M₁₆×C₁₉₁	C₂	semidirect
77. C₁₉₁	#2923	semidirect
84. C₄×M₁₆	D₁₉₁	semidirect

H ≅ G/N

type

2. #27132722

C₂

semidirect

3. #27132718

C₂

semidirect

4. #27132691

C₂

semidirect

6. C₂×#20259233

C₂

semidirect

7. C₄×M₁₆×C₁₉₁

C₂

semidirect

77. C₁₉₁

#2923

semidirect

84. C₄×M₁₆

D₁₉₁

semidirect

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

subgroup	index
C₄×#15690287	2
#27132722	2
#27132718	2
#27132691	2
#27132631	2
C₂×#20259233	2
C₄×M₁₆×C₁₉₁	2
#2923	191

subgroup

index

C₄×M₁₆×C₁₉₁

191

Subgroups — #37581172

Conjugacy classes of subgroups

Complements

Maximal subgroups