Group #37581082 — subgroups

Subgroups — #37581082

Order 24448.

484 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 213 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×M₁₆×D₁₉₁	C₂	1	2	yes	maximal
2	C₂×#20259279	C₂	1	1	yes	maximal, characteristic
3	#27132688	C₂	1	8	yes	maximal, complemented
4	#27132687	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132686	C₂	1	2	yes	maximal, complemented
6	#423×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₁₉₁	C₂²	1	1	yes	characteristic
8	C₂×#15690271	C₂²	1	2	yes	—
9	C₂×#15690270	C₂²	1	4	yes	—
10	M₁₆×D₁₉₁	—	2	8	no	—
11	C₂×C₈×D₁₉₁	C₂²	1	4	yes	—
12	#20259279	C₄	1	4	yes	—
13	#20259279	C₂²	1	4	yes	—
14	C₂×Dic₁₉₁⋊C₄	C₄	1	2	yes	—
15	C₂×(C₂×D₁₉₁)⋊C₄	C₄	1	2	yes	—
16	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
17	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	4	yes	complemented
18	Dic₁₉₁⋊C₈	C₂²	1	8	yes	complemented
19	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₉₁	C₂²	1	4	yes	complemented
22	C₂³×D₁₉₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₉₁	C₂×C₄	1	2	yes	—
24	C₂²×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₁₉₁	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₁₉₁	D₄	1	4	yes	—
27	C₂×C₄×D₁₉₁	Q₈	1	4	yes	—
28	C₂×C₄×D₁₉₁	C₂³	1	2	yes	—
29	#15690271	—	2	4	no	—
30	#15690270	—	2	8	no	—
31	C₂×C₁₉₁⋊C₈	C₂³	1	4	yes	—
32	C₈×D₁₉₁	—	2	8	no	—
33	Dic₁₉₁⋊C₄	C₂×C₄	1	8	yes	—
34	(C₂×D₁₉₁)⋊C₄	C₂×C₄	1	8	yes	—
35	C₄×Dic₁₉₁	C₂×C₄	1	2	yes	—
36	C₄×Dic₁₉₁	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
39	M₁₆×C₁₉₁	—	2	4	no	—
40	C₂×C₈×C₁₉₁	C₂³	1	4	yes	—
41	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
42	C₄²×C₁₉₁	C₂³	1	2	yes	—
43	C₂²×D₁₉₁	—	4	2	no	—
44	C₂²×D₁₉₁	C₄⋊C₄	1	4	yes	—
45	C₂²×D₁₉₁	C₂²×C₄	1	2	yes	—
46	C₂×Dic₁₉₁	—	4	2	no	—
47	C₂×Dic₁₉₁	C₄⋊C₄	1	4	yes	—
48	C₂×Dic₁₉₁	C₂²×C₄	1	4	yes	—
49	C₂×Dic₁₉₁	C₂²×C₄	1	2	yes	—
50	C₄×D₁₉₁	C₄⋊C₄	1	8	yes	—
51	C₄×D₁₉₁	C₂×D₄	1	4	yes	—
52	C₄×D₁₉₁	C₂×Q₈	1	4	yes	—
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	4	yes	—
57	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	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	C₂×C₄⋊C₄	1	2	yes	—
64	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₁₉₁	—	2	4	no	—
66	C₂×D₁₉₁	—	4	4	no	—
67	C₂×D₁₉₁	C₂×C₄⋊C₄	1	4	yes	—
68	C₄×C₁₉₁	—	2	4	no	—
69	C₄×C₁₉₁	C₂×C₄⋊C₄	1	2	yes	—
70	C₄×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
72	Dic₁₉₁	—	2	4	no	—
73	Dic₁₉₁	C₂×C₄⋊C₄	1	4	yes	—
74	C₂×C₁₉₁	—	2	2	no	—
75	C₂×C₁₉₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₁₉₁	#568	1	2	yes	—
77	D₁₉₁	—	2	4	no	—
78	C₁₉₁	#2905	1	1	yes	characteristic, complemented, Sylow
79	#2905	—	191	1	no	maximal, Sylow
80	C₂²×M₁₆	—	191	2	no	—
81	C₂×#117	—	191	1	no	—
82	#424	—	191	8	no	—
83	#423	—	191	2	no	—
84	#423	—	191	1	no	—
85	#423	D₁₉₁	1	1	yes	characteristic, complemented
86	C₂³×C₄	—	191	1	no	—
87	C₂×M₁₆	—	382	8	no	—
88	C₂×M₁₆	—	191	2	no	—
89	C₂×M₁₆	—	191	4	no	—
90	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
91	C₂²×C₈	—	191	4	no	—
92	#117	—	191	4	no	—
93	#117	—	191	4	no	—
94	C₂×C₄⋊C₄	—	191	2	no	—
95	C₂×C₂²⋊C₄	—	191	2	no	—
96	C₂×C₄²	—	191	1	no	—
97	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
98	C₄⋊C₈	—	191	8	no	—
99	C₄⋊C₈	—	191	4	no	—
100	C₄⋊C₈	C₂×D₁₉₁	1	4	yes	complemented
101	C₂⁴	—	191	1	no	—
102	C₂²×C₄	—	191	1	no	—
103	C₂²×C₄	—	191	2	no	—
104	C₂²×C₄	—	191	4	no	—
105	C₂²×C₄	—	191	4	no	—
106	C₂²×C₄	—	191	2	no	—
107	C₂²×C₄	—	191	2	no	—
108	C₂²×C₄	C₄×D₁₉₁	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	M₁₆	—	382	8	no	—
112	M₁₆	—	382	4	no	—
113	C₂×C₈	—	382	8	no	—
114	C₂×C₈	—	191	4	no	—
115	C₂×C₈	C₂²×D₁₉₁	1	4	yes	—
116	C₄⋊C₄	—	191	8	no	—
117	C₂²⋊C₄	—	191	8	no	—
118	C₄²	—	191	2	no	—
119	C₄²	—	191	2	no	—
120	C₄²	C₄×D₁₉₁	1	2	yes	—
121	C₄²	C₂²×D₁₉₁	1	2	yes	—
122	C₂³	—	191	2	no	—
123	C₂³	—	191	4	no	—
124	C₂³	—	764	2	no	—
125	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	191	4	no	—
128	C₂×C₄	—	191	2	no	—
129	C₂×C₄	—	191	4	no	—
130	C₂×C₄	—	191	4	no	—
131	C₂×C₄	—	191	4	no	—
132	C₂×C₄	—	191	8	no	—
133	C₂×C₄	—	764	2	no	—
134	C₂×C₄	C₂×C₄×D₁₉₁	1	4	yes	—
135	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
136	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
137	C₂×C₄	Q₈×D₁₉₁	1	2	yes	—
138	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	382	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	382	4	no	—
143	C₂²	—	191	4	no	—
144	C₂²	—	764	4	no	—
145	C₂²	C₄⋊C₄×D₁₉₁	1	2	yes	—
146	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	382	4	no	—
149	C₄	—	191	4	no	—
150	C₄	C₄⋊C₄×D₁₉₁	1	2	yes	—
151	C₄	C₂×D₄×D₁₉₁	1	1	yes	characteristic
152	C₄	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	382	4	no	—
155	C₂	C₂×C₄⋊C₄×D₁₉₁	1	1	yes	characteristic
156	C₂	#27133429	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×D₁₉₁

C₂

yes

maximal

C₂×#20259279

C₂

yes

maximal, characteristic

#27132688

C₂

yes

maximal, complemented

#27132687

C₂

yes

maximal, characteristic, complemented

#27132686

C₂

yes

maximal, complemented

#423×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₉₁

C₂²

yes

characteristic

C₂×#15690271

C₂²

yes

—

C₂×#15690270

C₂²

yes

—

M₁₆×D₁₉₁

—

C₂×C₈×D₁₉₁

C₂²

yes

—

#20259279

C₄

yes

—

#20259279

C₂²

yes

—

C₂×Dic₁₉₁⋊C₄

C₄

yes

—

C₂×(C₂×D₁₉₁)⋊C₄

C₄

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

characteristic

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

complemented

Dic₁₉₁⋊C₈

C₂²

yes

complemented

C₂×M₁₆×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

complemented

C₂³×D₁₉₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₁

C₂×C₄

yes

—

C₂²×Dic₁₉₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₉₁

C₂×C₄

yes

—

C₂×C₄×D₁₉₁

D₄

yes

—

C₂×C₄×D₁₉₁

Q₈

yes

—

C₂×C₄×D₁₉₁

C₂³

yes

—

#15690271

—

#15690270

—

C₂×C₁₉₁⋊C₈

C₂³

yes

—

C₈×D₁₉₁

—

Dic₁₉₁⋊C₄

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

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

C₄⋊C₄

yes

—

C₂²×D₁₉₁

C₂²×C₄

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₄⋊C₄

yes

—

C₂×Dic₁₉₁

C₂²×C₄

yes

—

C₂×Dic₁₉₁

C₂²×C₄

yes

—

C₄×D₁₉₁

C₄⋊C₄

yes

—

C₄×D₁₉₁

C₂×D₄

yes

—

C₄×D₁₉₁

C₂×Q₈

yes

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

—

C₂²×C₁₉₁

C₂³×C₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

C₂×C₄⋊C₄

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×C₄⋊C₄

yes

—

C₄×C₁₉₁

C₂²×D₄

yes

characteristic

C₄×C₁₉₁

C₂²×Q₈

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

C₂×C₄⋊C₄

yes

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₁₉₁

#568

yes

—

D₁₉₁

—

C₁₉₁

#2905

yes

characteristic, complemented, Sylow

#2905

—

191

maximal, Sylow

C₂²×M₁₆

—

191

—

C₂×#117

—

191

—

#424

—

191

—

#423

—

191

—

#423

—

191

—

#423

D₁₉₁

yes

characteristic, complemented

C₂³×C₄

—

191

—

C₂×M₁₆

—

382

—

C₂×M₁₆

—

191

—

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

C₂²×C₈

—

191

—

#117

—

191

—

#117

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₂²⋊C₄

—

191

—

C₂×C₄²

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

191

—

C₄⋊C₈

—

191

—

100

C₄⋊C₈

C₂×D₁₉₁

yes

complemented

101

C₂⁴

—

191

—

102

C₂²×C₄

—

191

—

103

C₂²×C₄

—

191

—

104

C₂²×C₄

—

191

—

105

C₂²×C₄

—

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

M₁₆

—

382

—

112

M₁₆

—

382

—

113

C₂×C₈

—

382

—

114

C₂×C₈

—

191

—

115

C₂×C₈

C₂²×D₁₉₁

yes

—

116

C₄⋊C₄

—

191

—

117

C₂²⋊C₄

—

191

—

118

C₄²

—

191

—

119

C₄²

—

191

—

120

C₄²

C₄×D₁₉₁

yes

—

121

C₄²

C₂²×D₁₉₁

yes

—

122

C₂³

—

191

—

123

C₂³

—

191

—

124

C₂³

—

764

—

125

C₂³

C₂×C₄×D₁₉₁

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

191

—

128

C₂×C₄

—

191

—

129

C₂×C₄

—

191

—

130

C₂×C₄

—

191

—

131

C₂×C₄

—

191

—

132

C₂×C₄

—

191

—

133

C₂×C₄

—

764

—

134

C₂×C₄

C₂×C₄×D₁₉₁

yes

—

135

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

136

C₂×C₄

D₄×D₁₉₁

yes

—

137

C₂×C₄

Q₈×D₁₉₁

yes

—

138

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

139

C₈

—

140

C₈

—

382

—

141

C₂²

—

142

C₂²

—

382

—

143

C₂²

—

191

—

144

C₂²

—

764

—

145

C₂²

C₄⋊C₄×D₁₉₁

yes

—

146

C₂²

C₂²×C₄×D₁₉₁

yes

characteristic

147

C₄

—

148

C₄

—

382

—

149

C₄

—

191

—

150

C₄

C₄⋊C₄×D₁₉₁

yes

—

151

C₄

C₂×D₄×D₁₉₁

yes

characteristic

152

C₄

C₂×Q₈×D₁₉₁

yes

characteristic

153

C₂

—

154

C₂

—

382

—

155

C₂

C₂×C₄⋊C₄×D₁₉₁

yes

characteristic

156

C₂

#27133429

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂³×C₄
Fitting	#423×C₁₉₁	C₂
prime radical	C₂³×D₁₉₁	C₂×C₄
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	#423	D₁₉₁
hypocenter	C₁₉₁	#2905
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₉₁

commutator, socle

C₂²×C₁₉₁

C₂³×C₄

Fitting

#423×C₁₉₁

C₂

prime radical

C₂³×D₁₉₁

C₂×C₄

cosocle

C₂×C₄×C₁₉₁

C₂⁴

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
3. #27132688	C₂	semidirect
4. #27132687	C₂	semidirect
5. #27132686	C₂	semidirect
6. #423×C₁₉₁	C₂	semidirect
17. C₄⋊(C₁₉₁⋊C₈)	C₂²	semidirect
18. Dic₁₉₁⋊C₈	C₂²	semidirect
21. C₄⋊C₈×C₁₉₁	C₂²	semidirect
78. C₁₉₁	#2905	semidirect
85. #423	D₁₉₁	semidirect
100. C₄⋊C₈	C₂×D₁₉₁	semidirect

H ≅ G/N

type

3. #27132688

C₂

semidirect

4. #27132687

C₂

semidirect

5. #27132686

C₂

semidirect

6. #423×C₁₉₁

C₂

semidirect

17. C₄⋊(C₁₉₁⋊C₈)

C₂²

semidirect

18. Dic₁₉₁⋊C₈

C₂²

semidirect

21. C₄⋊C₈×C₁₉₁

C₂²

semidirect

78. C₁₉₁

#2905

semidirect

85. #423

D₁₉₁

semidirect

100. C₄⋊C₈

C₂×D₁₉₁

semidirect

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

subgroup	index
C₂×M₁₆×D₁₉₁	2
C₂×#20259279	2
#27132688	2
#27132687	2
#27132686	2
#423×C₁₉₁	2
#2905	191

subgroup

index

C₂×M₁₆×D₁₉₁

#423×C₁₉₁

191

Subgroups — #37581082

Conjugacy classes of subgroups

Complements

Maximal subgroups