Group #37581614 — subgroups

Subgroups — #37581614

Order 24448.

322 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 161 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	#27133220	C₂	1	1	yes	maximal, characteristic
2	#27132793	C₂	1	2	yes	maximal, complemented
3	#27132792	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₃₈₂	C₂	1	2	yes	maximal, complemented
5	#27132721	C₂	1	4	yes	maximal
6	#27132720	C₂	1	1	yes	maximal, characteristic
7	#431×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	#20259322	C₄	1	1	yes	characteristic
9	#20259322	C₄	1	1	yes	characteristic
10	#20259314	C₄	1	2	yes	—
11	#20259301	C₄	1	2	yes	—
12	#20259301	C₄	1	2	yes	—
13	C₄×Dic₃₈₂	C₂²	1	2	yes	—
14	C₄×Dic₃₈₂	C₂²	1	2	yes	—
15	#20259281	C₂²	1	2	yes	—
16	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
17	Dic₁₉₁⋊C₈	C₂²	1	4	yes	—
18	Dic₁₉₁⋊C₈	C₂²	1	4	yes	—
19	C₂²⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
20	#20259233	C₂²	1	2	yes	—
21	#20259232	C₂²	1	4	yes	—
22	C₈×Dic₁₉₁	C₂²	1	4	yes	—
23	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₁₉₁	C₂²	1	2	yes	—
25	C₂²⋊C₈×C₁₉₁	C₂²	1	2	yes	—
26	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
27	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
28	C₂×Dic₃₈₂	C₂×C₄	1	2	yes	—
29	C₂×Dic₃₈₂	C₂×C₄	1	2	yes	—
30	C₂×C₁₉₁⋊C₈	C₂³	1	4	yes	—
31	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
34	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
35	Dic₁₉₁⋊C₄	C₂×C₄	1	4	yes	—
36	Dic₁₉₁⋊C₄	C₂×C₄	1	4	yes	—
37	C₂²⋊Dic₁₉₁	C₂×C₄	1	4	yes	—
38	C₄×Dic₁₉₁	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
43	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
44	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
47	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
48	Dic₃₈₂	—	2	4	no	—
49	Dic₃₈₂	—	4	2	no	—
50	C₂×Dic₁₉₁	C₂²×C₄	1	4	yes	—
51	C₂×Dic₁₉₁	C₂²×C₄	1	4	yes	—
52	C₁₉₁⋊C₈	—	2	4	no	—
53	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₁	—	2	2	no	—
55	C₂×C₄×C₁₉₁	—	2	2	no	—
56	C₂×C₄×C₁₉₁	—	4	1	no	—
57	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
63	C₈×C₁₉₁	—	2	2	no	—
64	C₈×C₁₉₁	—	2	2	no	—
65	C₂²×C₁₉₁	—	2	1	no	—
66	C₂²×C₁₉₁	—	2	1	no	—
67	C₂²×C₁₉₁	—	2	1	no	—
68	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₁₉₁	—	2	1	no	—
70	C₄×C₁₉₁	—	2	1	no	—
71	C₄×C₁₉₁	—	2	1	no	—
72	C₄×C₁₉₁	—	2	1	no	—
73	C₄×C₁₉₁	#131	1	2	yes	—
74	C₄×C₁₉₁	2₋¹⁺⁴	1	2	yes	—
75	Dic₁₉₁	—	2	4	no	—
76	Dic₁₉₁	—	2	4	no	—
77	C₂×C₁₉₁	—	4	1	no	—
78	C₂×C₁₉₁	#519	1	1	yes	characteristic
79	C₂×C₁₉₁	C₂×#131	1	1	yes	characteristic
80	C₂×C₁₉₁	#568	1	1	yes	characteristic
81	C₁₉₁	#2985	1	1	yes	characteristic, complemented, Sylow
82	#2985	—	191	1	no	maximal, Sylow
83	#533	—	191	1	no	—
84	#446	—	191	4	no	—
85	#446	—	191	2	no	—
86	C₈×Q₈	—	191	2	no	—
87	#433	—	191	4	no	—
88	#432	—	191	1	no	—
89	#431	D₁₉₁	1	1	yes	characteristic
90	C₄⋊Q₈	—	191	1	no	—
91	C₄⋊Q₈	—	191	1	no	—
92	#125	—	191	2	no	—
93	C₂²⋊Q₈	—	191	2	no	—
94	C₂²⋊Q₈	—	191	2	no	—
95	C₄×Q₈	—	191	2	no	—
96	C₄×Q₈	—	191	2	no	—
97	#117	—	191	2	no	—
98	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
99	C₄⋊C₈	—	191	4	no	—
100	C₄⋊C₈	—	191	4	no	—
101	C₄⋊C₈	—	191	2	no	—
102	C₄⋊C₈	C₂×D₁₉₁	1	2	yes	—
103	C₂²⋊C₈	—	191	2	no	—
104	C₂²⋊C₈	C₂×D₁₉₁	1	2	yes	—
105	#97	—	191	4	no	—
106	#97	—	191	2	no	—
107	C₄×C₈	—	191	4	no	—
108	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
109	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
110	C₂×Q₈	—	191	2	no	—
111	C₂×Q₈	—	191	2	no	—
112	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
115	C₂×C₈	—	191	4	no	—
116	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
117	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
118	C₄⋊C₄	—	191	4	no	—
119	C₄⋊C₄	—	191	4	no	—
120	C₄⋊C₄	—	191	2	no	—
121	C₄⋊C₄	—	191	2	no	—
122	C₄⋊C₄	—	191	2	no	—
123	C₄⋊C₄	—	191	2	no	—
124	C₂²⋊C₄	—	191	4	no	—
125	C₄²	—	191	4	no	—
126	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
127	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
128	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
129	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
130	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
131	Q₈	—	382	4	no	—
132	Q₈	—	764	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	191	4	no	—
137	C₂×C₄	—	191	4	no	—
138	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	2	2	no	—
146	C₈	—	382	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	382	4	no	—
156	C₄	—	382	4	no	—
157	C₄	#20259328	1	2	yes	—
158	C₄	#20259369	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#27133140	1	1	yes	characteristic
161	C₂	C₂×#20259328	1	1	yes	characteristic
162	C₂	#27133429	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133220

C₂

yes

maximal, characteristic

#27132793

C₂

yes

maximal, complemented

#27132792

C₂

yes

maximal, complemented

C₈×Dic₃₈₂

C₂

yes

maximal, complemented

#27132721

C₂

yes

maximal

#27132720

C₂

yes

maximal, characteristic

#431×C₁₉₁

C₂

yes

maximal, characteristic

#20259322

C₄

yes

characteristic

#20259322

C₄

yes

characteristic

#20259314

C₄

yes

—

#20259301

C₄

yes

—

#20259301

C₄

yes

—

C₄×Dic₃₈₂

C₂²

yes

—

C₄×Dic₃₈₂

C₂²

yes

—

#20259281

C₂²

yes

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

—

Dic₁₉₁⋊C₈

C₂²

yes

—

Dic₁₉₁⋊C₈

C₂²

yes

—

C₂²⋊(C₁₉₁⋊C₈)

C₂²

yes

—

#20259233

C₂²

yes

—

#20259232

C₂²

yes

—

C₈×Dic₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

—

C₂²⋊C₈×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

C₂×C₄

yes

—

C₂×Dic₃₈₂

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

Dic₁₉₁⋊C₄

C₂×C₄

yes

—

Dic₁₉₁⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₁₉₁

C₂×C₄

yes

—

C₄×Dic₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

—

C₂×C₈×C₁₉₁

C₂³

yes

—

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

C₂²×C₄

yes

—

C₂×Dic₁₉₁

C₂²×C₄

yes

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂³×C₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#131

yes

—

C₄×C₁₉₁

2₋¹⁺⁴

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#2985

—

191

maximal, Sylow

#533

—

191

—

#446

—

191

—

#446

—

191

—

C₈×Q₈

—

191

—

#433

—

191

—

#432

—

191

—

#431

D₁₉₁

yes

characteristic

C₄⋊Q₈

—

191

—

C₄⋊Q₈

—

191

—

#125

—

191

—

C₂²⋊Q₈

—

191

—

C₂²⋊Q₈

—

191

—

C₄×Q₈

—

191

—

C₄×Q₈

—

191

—

#117

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

191

—

100

C₄⋊C₈

—

191

—

101

C₄⋊C₈

—

191

—

102

C₄⋊C₈

C₂×D₁₉₁

yes

—

103

C₂²⋊C₈

—

191

—

104

C₂²⋊C₈

C₂×D₁₉₁

yes

—

105

#97

—

191

—

106

#97

—

191

—

107

C₄×C₈

—

191

—

108

C₄×C₈

C₂×D₁₉₁

yes

characteristic

109

C₄×C₈

C₂×D₁₉₁

yes

characteristic

110

C₂×Q₈

—

191

—

111

C₂×Q₈

—

191

—

112

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

113

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

114

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

115

C₂×C₈

—

191

—

116

C₂×C₈

C₂²×D₁₉₁

yes

—

117

C₂×C₈

C₂²×D₁₉₁

yes

—

118

C₄⋊C₄

—

191

—

119

C₄⋊C₄

—

191

—

120

C₄⋊C₄

—

191

—

121

C₄⋊C₄

—

191

—

122

C₄⋊C₄

—

191

—

123

C₄⋊C₄

—

191

—

124

C₂²⋊C₄

—

191

—

125

C₄²

—

191

—

126

C₄²

C₄×D₁₉₁

yes

characteristic

127

C₄²

C₄×D₁₉₁

yes

characteristic

128

C₄²

C₂²×D₁₉₁

yes

characteristic

129

C₄²

C₂²×D₁₉₁

yes

characteristic

130

C₂³

C₂×C₄×D₁₉₁

yes

characteristic

131

Q₈

—

382

—

132

Q₈

—

764

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

191

—

137

C₂×C₄

—

191

—

138

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

139

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

140

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

141

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

142

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

143

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

382

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂²×C₄×D₁₉₁

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

382

—

156

C₄

—

382

—

157

C₄

#20259328

yes

—

158

C₄

#20259369

yes

—

159

C₂

—

160

C₂

#27133140

yes

characteristic

161

C₂

C₂×#20259328

yes

characteristic

162

C₂

#27133429

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Fitting	#431×C₁₉₁	C₂
prime radical	C₂³×C₁₉₁	C₂²×C₄
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	#431	D₁₉₁
hypocenter	C₁₉₁	#2985
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₉₁

commutator, socle

C₂²×C₁₉₁

C₂³×C₄

Fitting

#431×C₁₉₁

C₂

prime radical

C₂³×C₁₉₁

C₂²×C₄

cosocle

C₂×C₄×C₁₉₁

C₂⁴

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #27132793	C₂	semidirect
3. #27132792	C₂	semidirect
4. C₈×Dic₃₈₂	C₂	semidirect
81. C₁₉₁	#2985	semidirect

H ≅ G/N

type

2. #27132793

C₂

semidirect

3. #27132792

C₂

semidirect

4. C₈×Dic₃₈₂

C₂

semidirect

81. C₁₉₁

#2985

semidirect

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

subgroup	index
#27133220	2
#27132793	2
#27132792	2
C₈×Dic₃₈₂	2
#27132721	2
#27132720	2
#431×C₁₉₁	2
#2985	191

subgroup

index

C₈×Dic₃₈₂

#431×C₁₉₁

191

Subgroups — #37581614

Conjugacy classes of subgroups

Complements

Maximal subgroups