Group #37583176 — subgroups

Subgroups — #37583176

Order 24448.

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

normal subgroups only

This group has 338 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	#27133274	C₂	1	2	yes	maximal
2	#27133065	C₂	1	1	yes	maximal, characteristic, complemented
3	#27133031	C₂	1	2	yes	maximal, complemented
4	#27132918	C₂	1	2	yes	maximal
5	#27132901	C₂	1	2	yes	maximal
6	#27132770	C₂	1	2	yes	maximal, complemented
7	Q₁₆×Dic₁₉₁	C₂	1	2	yes	maximal, complemented
8	#27132617	C₂	1	1	yes	maximal, characteristic
9	#494×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15690270	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
12	#20259317	—	2	4	no	—
13	#20259314	C₂²	1	2	yes	—
14	#20259312	—	2	4	no	—
15	#20259298	C₂²	1	4	yes	—
16	Q₈×Dic₁₉₁	C₂²	1	4	yes	—
17	#20259279	C₂²	1	1	yes	characteristic
18	#20259256	C₂²	1	2	yes	—
19	#20259254	C₂²	1	2	yes	—
20	#20259246	C₂²	1	4	yes	—
21	#20259246	C₂²	1	4	yes	—
22	#20259233	C₂²	1	1	yes	characteristic
23	#20259232	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
25	C₂×Q₁₆×C₁₉₁	C₂²	1	2	yes	—
26	C₂×Q₁₆×C₁₉₁	C₂²	1	2	yes	—
27	C₄⋊Q₈×C₁₉₁	C₂²	1	2	yes	—
28	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₁₉₁	C₂³	1	1	yes	characteristic
30	#15690270	—	2	2	no	—
31	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
32	C₂×C₁₉₁⋊C₈	C₂³	1	1	yes	characteristic
33	C₈×D₁₉₁	—	2	2	no	—
34	C₄⋊Dic₁₉₁	—	2	4	no	—
35	C₄⋊Dic₁₉₁	C₂³	1	4	yes	—
36	Dic₁₉₁⋊C₄	—	2	4	no	—
37	Dic₁₉₁⋊C₄	—	2	4	no	—
38	Dic₁₉₁⋊C₄	D₄	1	2	yes	—
39	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
40	(C₂×D₁₉₁)⋊C₄	D₄	1	2	yes	—
41	C₄×Dic₁₉₁	—	2	4	no	—
42	C₄×Dic₁₉₁	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₁₉₁	C₂³	1	4	yes	—
44	Q₁₆×C₁₉₁	—	2	4	no	—
45	Q₁₆×C₁₉₁	—	2	4	no	—
46	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₁₉₁	—	2	4	no	—
49	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
50	C₂²×D₁₉₁	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₁₉₁	—	2	4	no	—
52	C₂×Dic₁₉₁	C₂×D₄	1	2	yes	—
53	C₂×Dic₁₉₁	C₂×D₄	1	1	yes	characteristic
54	C₄×D₁₉₁	—	2	1	no	—
55	C₄×D₁₉₁	—	2	1	no	—
56	C₁₉₁⋊C₈	—	2	1	no	—
57	C₁₉₁⋊C₈	—	2	1	no	—
58	Q₈×C₁₉₁	—	2	4	no	—
59	Q₈×C₁₉₁	—	2	4	no	—
60	C₂×C₄×C₁₉₁	—	2	4	no	—
61	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
63	C₈×C₁₉₁	#41	1	2	yes	—
64	C₈×C₁₉₁	#41	1	2	yes	—
65	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₁₉₁	—	2	1	no	—
67	C₂×D₁₉₁	—	2	2	no	—
68	C₄×C₁₉₁	—	2	2	no	—
69	C₄×C₁₉₁	—	4	4	no	—
70	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
71	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
72	Dic₁₉₁	—	2	1	no	—
73	Dic₁₉₁	—	2	2	no	—
74	Dic₁₉₁	—	4	4	no	—
75	C₂×C₁₉₁	#532	1	1	yes	characteristic
76	C₂×C₁₉₁	#578	1	2	yes	—
77	D₁₉₁	—	4	1	no	—
78	C₁₉₁	#3177	1	1	yes	characteristic, complemented, Sylow
79	#3177	—	191	1	no	maximal, Sylow
80	#541	—	191	2	no	—
81	#499	—	191	1	no	—
82	#494	D₁₉₁	1	1	yes	characteristic, complemented
83	#491	—	191	2	no	—
84	#470	—	191	2	no	—
85	#467	—	191	2	no	—
86	#441	—	191	2	no	—
87	C₄×Q₁₆	—	191	2	no	—
88	#405	—	191	1	no	—
89	C₂×Q₁₆	C₂×D₁₉₁	1	2	yes	—
90	C₂×Q₁₆	C₂×D₁₉₁	1	2	yes	—
91	C₂×M₁₆	—	191	1	no	—
92	C₂²×C₈	—	191	1	no	—
93	C₄⋊Q₈	C₂×D₁₉₁	1	2	yes	—
94	#126	—	382	4	no	—
95	#125	—	382	4	no	—
96	#125	—	191	2	no	—
97	C₂²⋊Q₈	—	191	4	no	—
98	C₄×Q₈	—	191	4	no	—
99	#117	—	191	1	no	—
100	#107	—	191	2	no	—
101	#106	—	191	2	no	—
102	#103	—	191	4	no	—
103	#103	—	191	4	no	—
104	#97	—	191	1	no	—
105	#97	—	191	1	no	—
106	C₄×C₈	—	191	1	no	—
107	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
108	C₂×Q₈	C₂²×D₁₉₁	1	4	yes	—
109	C₂²×C₄	—	191	1	no	—
110	Q₁₆	—	2	4	no	—
111	Q₁₆	—	2	4	no	—
112	M₁₆	—	382	2	no	—
113	C₂×C₈	—	382	2	no	—
114	C₂×C₈	—	191	1	no	—
115	C₂×C₈	—	191	1	no	—
116	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
117	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
118	C₄⋊C₄	—	2	4	no	—
119	C₄⋊C₄	—	382	4	no	—
120	C₄⋊C₄	—	382	4	no	—
121	C₄⋊C₄	—	382	4	no	—
122	C₄⋊C₄	—	191	4	no	—
123	C₄⋊C₄	—	191	2	no	—
124	C₂²⋊C₄	—	382	4	no	—
125	C₂²⋊C₄	—	191	2	no	—
126	C₄²	—	382	4	no	—
127	C₄²	—	191	1	no	—
128	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
129	C₂³	—	191	1	no	—
130	Q₈	—	2	4	no	—
131	Q₈	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	382	1	no	—
134	C₂×C₄	—	382	1	no	—
135	C₂×C₄	—	382	4	no	—
136	C₂×C₄	—	191	2	no	—
137	C₂×C₄	—	191	1	no	—
138	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
139	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
140	C₈	—	382	1	no	—
141	C₈	—	382	1	no	—
142	C₈	#15690286	1	2	yes	—
143	C₈	#15690286	1	2	yes	—
144	C₂²	—	382	1	no	—
145	C₂²	—	382	2	no	—
146	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	4	4	no	—
149	C₄	—	382	1	no	—
150	C₄	—	382	2	no	—
151	C₄	—	764	4	no	—
152	C₄	C₂×#15690286	1	1	yes	characteristic
153	C₄	C₂×#15690286	1	1	yes	characteristic
154	C₂	—	764	1	no	—
155	C₂	#27133211	1	1	yes	characteristic
156	C₂	#27133476	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133274

C₂

yes

maximal

#27133065

C₂

yes

maximal, characteristic, complemented

#27133031

C₂

yes

maximal, complemented

#27132918

C₂

yes

maximal

#27132901

C₂

yes

maximal

#27132770

C₂

yes

maximal, complemented

Q₁₆×Dic₁₉₁

C₂

yes

maximal, complemented

#27132617

C₂

yes

maximal, characteristic

#494×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690270

C₂²

yes

characteristic

C₂×C₈×D₁₉₁

C₂²

yes

characteristic

#20259317

—

#20259314

C₂²

yes

—

#20259312

—

#20259298

C₂²

yes

—

Q₈×Dic₁₉₁

C₂²

yes

—

#20259279

C₂²

yes

characteristic

#20259256

C₂²

yes

—

#20259254

C₂²

yes

—

#20259246

C₂²

yes

—

#20259246

C₂²

yes

—

#20259233

C₂²

yes

characteristic

#20259232

C₂²

yes

characteristic

C₈×Dic₁₉₁

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉₁

C₂²

yes

—

C₂×Q₁₆×C₁₉₁

C₂²

yes

—

C₄⋊Q₈×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×C₄×D₁₉₁

C₂³

yes

characteristic

#15690270

—

C₂×C₁₉₁⋊C₈

C₂³

yes

characteristic

C₂×C₁₉₁⋊C₈

C₂³

yes

characteristic

C₈×D₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

C₂³

yes

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

D₄

yes

—

(C₂×D₁₉₁)⋊C₄

—

(C₂×D₁₉₁)⋊C₄

D₄

yes

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

C₂³

yes

characteristic

C₂×Q₈×C₁₉₁

C₂³

yes

—

Q₁₆×C₁₉₁

—

Q₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

C₂×D₄

yes

characteristic

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂×D₄

yes

—

C₂×Dic₁₉₁

C₂×D₄

yes

characteristic

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

#41

yes

—

C₈×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×#41

yes

characteristic

C₄×C₁₉₁

C₂×#41

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

#532

yes

characteristic

C₂×C₁₉₁

#578

yes

—

D₁₉₁

—

C₁₉₁

#3177

yes

characteristic, complemented, Sylow

#3177

—

191

maximal, Sylow

#541

—

191

—

#499

—

191

—

#494

D₁₉₁

yes

characteristic, complemented

#491

—

191

—

#470

—

191

—

#467

—

191

—

#441

—

191

—

C₄×Q₁₆

—

191

—

#405

—

191

—

C₂×Q₁₆

C₂×D₁₉₁

yes

—

C₂×Q₁₆

C₂×D₁₉₁

yes

—

C₂×M₁₆

—

191

—

C₂²×C₈

—

191

—

C₄⋊Q₈

C₂×D₁₉₁

yes

—

#126

—

382

—

#125

—

382

—

#125

—

191

—

C₂²⋊Q₈

—

191

—

C₄×Q₈

—

191

—

#117

—

191

—

100

#107

—

191

—

101

#106

—

191

—

102

#103

—

191

—

103

#103

—

191

—

104

#97

—

191

—

105

#97

—

191

—

106

C₄×C₈

—

191

—

107

C₄×C₈

C₂×D₁₉₁

yes

characteristic

108

C₂×Q₈

C₂²×D₁₉₁

yes

—

109

C₂²×C₄

—

191

—

110

Q₁₆

—

111

Q₁₆

—

112

M₁₆

—

382

—

113

C₂×C₈

—

382

—

114

C₂×C₈

—

191

—

115

C₂×C₈

—

191

—

116

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

117

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

382

—

120

C₄⋊C₄

—

382

—

121

C₄⋊C₄

—

382

—

122

C₄⋊C₄

—

191

—

123

C₄⋊C₄

—

191

—

124

C₂²⋊C₄

—

382

—

125

C₂²⋊C₄

—

191

—

126

C₄²

—

382

—

127

C₄²

—

191

—

128

C₄²

C₂²×D₁₉₁

yes

characteristic

129

C₂³

—

191

—

130

Q₈

—

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

382

—

134

C₂×C₄

—

382

—

135

C₂×C₄

—

382

—

136

C₂×C₄

—

191

—

137

C₂×C₄

—

191

—

138

C₂×C₄

D₄×D₁₉₁

yes

—

139

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

140

C₈

—

382

—

141

C₈

—

382

—

142

C₈

#15690286

yes

—

143

C₈

#15690286

yes

—

144

C₂²

—

382

—

145

C₂²

—

382

—

146

C₂²

C₂×D₄×D₁₉₁

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

382

—

150

C₄

—

382

—

151

C₄

—

764

—

152

C₄

C₂×#15690286

yes

characteristic

153

C₄

C₂×#15690286

yes

characteristic

154

C₂

—

764

—

155

C₂

#27133211

yes

characteristic

156

C₂

#27133476

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#494×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂²×D₁₉₁	C₂×D₄
hypercenter	#494	D₁₉₁
hypocenter	C₁₉₁	#3177
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₁₉₁

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#494×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

prime radical

C₂²×D₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #27133065	C₂	semidirect
3. #27133031	C₂	semidirect
6. #27132770	C₂	semidirect
7. Q₁₆×Dic₁₉₁	C₂	semidirect
9. #494×C₁₉₁	C₂	semidirect
78. C₁₉₁	#3177	semidirect
82. #494	D₁₉₁	semidirect

H ≅ G/N

type

2. #27133065

C₂

semidirect

3. #27133031

C₂

semidirect

6. #27132770

C₂

semidirect

7. Q₁₆×Dic₁₉₁

C₂

semidirect

9. #494×C₁₉₁

C₂

semidirect

78. C₁₉₁

#3177

semidirect

82. #494

D₁₉₁

semidirect

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

subgroup	index
#27133274	2
#27133065	2
#27133031	2
#27132918	2
#27132901	2
#27132770	2
Q₁₆×Dic₁₉₁	2
#27132617	2
#494×C₁₉₁	2
#3177	191

subgroup

index

Q₁₆×Dic₁₉₁

#494×C₁₉₁

191

Subgroups — #37583176

Conjugacy classes of subgroups

Complements

Maximal subgroups