Group #37581002 — subgroups

Subgroups — #37581002

Order 24448.

704 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 249 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×#20259369	C₂	1	1	yes	maximal, characteristic
2	#27133430	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132645	C₂	1	6	yes	maximal, complemented
4	C₂×#20259240	C₂	1	6	yes	maximal, complemented
5	#568×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20259369	—	2	4	no	—
7	#20259369	—	2	4	no	—
8	C₂×#15690287	C₂²	1	3	yes	—
9	C₂×#15690286	C₄	1	6	yes	—
10	C₂²×Dic₃₈₂	C₂²	1	3	yes	—
11	C₂×Q₈×D₁₉₁	C₄	1	2	yes	—
12	#20259329	—	2	4	no	—
13	C₂×#15690271	C₂²	1	6	yes	—
14	#20259240	C₂²	1	12	yes	—
15	#20259238	C₂²	1	4	yes	—
16	C₂×#41×C₁₉₁	C₂²	1	1	yes	characteristic
17	#131×C₁₉₁	—	2	4	no	—
18	C₂×M₁₆×C₁₉₁	C₂²	1	6	yes	—
19	#15690287	—	2	6	no	—
20	#15690287	—	2	6	no	—
21	#15690286	—	2	24	no	—
22	C₂×Dic₃₈₂	—	2	6	no	—
23	C₂×Dic₃₈₂	—	2	6	no	—
24	C₂×Dic₃₈₂	C₂×C₄	1	6	yes	—
25	C₂×Dic₃₈₂	C₂³	1	3	yes	—
26	Q₈×D₁₉₁	—	2	8	no	—
27	C₂×D₇₆₄	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₁₉₁	C₂×C₄	1	6	yes	—
29	C₂²×Dic₁₉₁	C₂×C₄	1	6	yes	—
30	C₂×C₄×D₁₉₁	C₂×C₄	1	6	yes	—
31	#15690271	—	2	6	no	—
32	#15690271	C₂³	1	4	yes	—
33	C₂×C₁₉₁⋊C₈	—	2	6	no	—
34	#41×C₁₉₁	D₄	1	4	yes	—
35	#41×C₁₉₁	D₄	1	4	yes	—
36	C₂×Q₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₁₉₁	C₂×C₄	1	3	yes	—
38	C₂²×C₄×C₁₉₁	C₂³	1	3	yes	—
39	M₁₆×C₁₉₁	—	2	6	no	—
40	M₁₆×C₁₉₁	C₂³	1	4	yes	—
41	C₂×C₈×C₁₉₁	—	2	6	no	—
42	C₂²×D₁₉₁	C₂²×C₄	1	2	yes	—
43	Dic₃₈₂	—	2	12	no	—
44	Dic₃₈₂	—	2	3	no	—
45	Dic₃₈₂	—	2	3	no	—
46	D₇₆₄	—	2	1	no	—
47	D₇₆₄	—	2	1	no	—
48	C₂²⋊D₁₉₁	—	2	12	no	—
49	C₂×Dic₁₉₁	—	2	12	no	—
50	C₂×Dic₁₉₁	C₂²×C₄	1	6	yes	—
51	C₄×D₁₉₁	—	2	12	no	—
52	C₁₉₁⋊C₈	—	2	4	no	—
53	C₂³×C₁₉₁	C₂²×C₄	1	3	yes	—
54	Q₈×C₁₉₁	C₂²⋊C₄	1	4	yes	—
55	D₄×C₁₉₁	C₂²⋊C₄	1	12	yes	—
56	C₂×C₄×C₁₉₁	C₂²×C₄	1	3	yes	—
57	C₂×C₄×C₁₉₁	C₂×D₄	1	6	yes	—
58	C₂×C₄×C₁₉₁	C₂×D₄	1	6	yes	—
59	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
60	C₈×C₁₉₁	—	2	4	no	—
61	C₂²×C₁₉₁	—	4	3	no	—
62	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	6	yes	—
63	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
64	C₂×D₁₉₁	—	2	2	no	—
65	C₂×D₁₉₁	—	4	2	no	—
66	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	6	yes	—
67	C₄×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
69	Dic₁₉₁	—	2	6	no	—
70	C₂×C₁₉₁	—	2	1	no	—
71	C₂×C₁₉₁	—	2	6	no	—
72	C₂×C₁₉₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
73	D₁₉₁	—	4	2	no	—
74	C₁₉₁	#2889	1	1	yes	characteristic, complemented, Sylow
75	#2889	—	191	1	no	maximal, Sylow
76	C₂×2₋¹⁺⁴	—	191	1	no	—
77	#568	—	191	1	no	—
78	#568	D₁₉₁	1	1	yes	characteristic, complemented
79	#413	—	191	6	no	—
80	C₂×#101	—	191	6	no	—
81	2₋¹⁺⁴	—	382	4	no	—
82	2₋¹⁺⁴	—	382	4	no	—
83	C₂×#41	—	191	6	no	—
84	C₂×#41	—	191	3	no	—
85	C₂×#41	C₂×D₁₉₁	1	1	yes	characteristic
86	C₂²×Q₈	—	191	2	no	—
87	C₂²×Q₈	—	191	3	no	—
88	#131	—	2	4	no	—
89	#131	—	382	4	no	—
90	C₂×M₁₆	—	191	6	no	—
91	C₂×M₁₆	C₂×D₁₉₁	1	6	yes	—
92	#101	—	191	12	no	—
93	#100	—	191	4	no	—
94	#41	—	382	24	no	—
95	#41	—	382	6	no	—
96	#41	—	382	6	no	—
97	#41	C₂²⋊D₁₉₁	1	4	yes	—
98	#41	D₇₆₄	1	4	yes	—
99	C₂×Q₈	—	382	8	no	—
100	C₂×Q₈	—	382	6	no	—
101	C₂×Q₈	—	382	6	no	—
102	C₂×Q₈	—	191	6	no	—
103	C₂×Q₈	—	191	3	no	—
104	C₂×Q₈	C₄×D₁₉₁	1	1	yes	characteristic
105	C₂×D₄	—	191	6	no	—
106	C₂×D₄	—	191	1	no	—
107	C₂×D₄	C₄×D₁₉₁	1	3	yes	—
108	C₂²×C₄	—	191	6	no	—
109	C₂²×C₄	—	191	6	no	—
110	C₂²×C₄	C₂²×D₁₉₁	1	3	yes	—
111	M₁₆	—	2	6	no	—
112	M₁₆	—	382	6	no	—
113	M₁₆	—	191	4	no	—
114	M₁₆	C₂²×D₁₉₁	1	4	yes	—
115	C₂×C₈	—	2	6	no	—
116	C₂×C₈	—	382	6	no	—
117	C₂³	—	191	2	no	—
118	C₂³	C₂×C₄×D₁₉₁	1	3	yes	—
119	Q₈	—	382	3	no	—
120	Q₈	—	382	3	no	—
121	Q₈	—	382	12	no	—
122	Q₈	(C₂×D₁₉₁)⋊C₄	1	4	yes	—
123	D₄	—	382	12	no	—
124	D₄	—	382	1	no	—
125	D₄	—	382	1	no	—
126	D₄	(C₂×D₁₉₁)⋊C₄	1	12	yes	—
127	C₂×C₄	—	382	12	no	—
128	C₂×C₄	—	382	12	no	—
129	C₂×C₄	—	191	6	no	—
130	C₂×C₄	C₂×C₄×D₁₉₁	1	3	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	6	yes	—
132	C₂×C₄	C₂×D₇₆₄	1	6	yes	—
133	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
134	C₈	—	2	4	no	—
135	C₈	—	382	4	no	—
136	C₂²	—	4	3	no	—
137	C₂²	—	382	2	no	—
138	C₂²	—	764	2	no	—
139	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	6	yes	—
140	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
141	C₄	—	382	6	no	—
142	C₄	C₂×(C₂×D₁₉₁)⋊C₄	1	6	yes	—
143	C₄	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
144	C₄	C₂²×D₇₆₄	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	6	no	—
147	C₂	—	764	2	no	—
148	C₂	C₂²×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259369

C₂

yes

maximal, characteristic

#27133430

C₂

yes

maximal, characteristic, complemented

#27132645

C₂

yes

maximal, complemented

C₂×#20259240

C₂

yes

maximal, complemented

#568×C₁₉₁

C₂

yes

maximal, characteristic, complemented

#20259369

—

#20259369

—

C₂×#15690287

C₂²

yes

—

C₂×#15690286

C₄

yes

—

C₂²×Dic₃₈₂

C₂²

yes

—

C₂×Q₈×D₁₉₁

C₄

yes

—

#20259329

—

C₂×#15690271

C₂²

yes

—

#20259240

C₂²

yes

—

#20259238

C₂²

yes

—

C₂×#41×C₁₉₁

C₂²

yes

characteristic

#131×C₁₉₁

—

C₂×M₁₆×C₁₉₁

C₂²

yes

—

#15690287

—

#15690287

—

#15690286

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

C₂×C₄

yes

—

C₂×Dic₃₈₂

C₂³

yes

—

Q₈×D₁₉₁

—

C₂×D₇₆₄

C₂³

yes

characteristic

C₂×C₂²⋊D₁₉₁

C₂×C₄

yes

—

C₂²×Dic₁₉₁

C₂×C₄

yes

—

C₂×C₄×D₁₉₁

C₂×C₄

yes

—

#15690271

—

#15690271

C₂³

yes

—

C₂×C₁₉₁⋊C₈

—

#41×C₁₉₁

D₄

yes

—

#41×C₁₉₁

D₄

yes

—

C₂×Q₈×C₁₉₁

C₂×C₄

yes

characteristic

C₂×D₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

—

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

C₂³

yes

—

C₂×C₈×C₁₉₁

—

C₂²×D₁₉₁

C₂²×C₄

yes

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

D₇₆₄

—

D₇₆₄

—

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂²×C₄

yes

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

—

Q₈×C₁₉₁

C₂²⋊C₄

yes

—

D₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₉₁

C₂³×C₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

C₂×C₂²⋊C₄

yes

—

C₄×C₁₉₁

C₂²×D₄

yes

characteristic

C₄×C₁₉₁

C₂²×D₄

yes

characteristic

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂²×C₂²⋊C₄

yes

characteristic

D₁₉₁

—

C₁₉₁

#2889

yes

characteristic, complemented, Sylow

#2889

—

191

maximal, Sylow

C₂×2₋¹⁺⁴

—

191

—

#568

—

191

—

#568

D₁₉₁

yes

characteristic, complemented

#413

—

191

—

C₂×#101

—

191

—

2₋¹⁺⁴

—

382

—

2₋¹⁺⁴

—

382

—

C₂×#41

—

191

—

C₂×#41

—

191

—

C₂×#41

C₂×D₁₉₁

yes

characteristic

C₂²×Q₈

—

191

—

C₂²×Q₈

—

191

—

#131

—

#131

—

382

—

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

#101

—

191

—

#100

—

191

—

#41

—

382

—

#41

—

382

—

#41

—

382

—

#41

C₂²⋊D₁₉₁

yes

—

#41

D₇₆₄

yes

—

C₂×Q₈

—

382

—

100

C₂×Q₈

—

382

—

101

C₂×Q₈

—

382

—

102

C₂×Q₈

—

191

—

103

C₂×Q₈

—

191

—

104

C₂×Q₈

C₄×D₁₉₁

yes

characteristic

105

C₂×D₄

—

191

—

106

C₂×D₄

—

191

—

107

C₂×D₄

C₄×D₁₉₁

yes

—

108

C₂²×C₄

—

191

—

109

C₂²×C₄

—

191

—

110

C₂²×C₄

C₂²×D₁₉₁

yes

—

111

M₁₆

—

112

M₁₆

—

382

—

113

M₁₆

—

191

—

114

M₁₆

C₂²×D₁₉₁

yes

—

115

C₂×C₈

—

116

C₂×C₈

—

382

—

117

C₂³

—

191

—

118

C₂³

C₂×C₄×D₁₉₁

yes

—

119

Q₈

—

382

—

120

Q₈

—

382

—

121

Q₈

—

382

—

122

Q₈

(C₂×D₁₉₁)⋊C₄

yes

—

123

D₄

—

382

—

124

D₄

—

382

—

125

D₄

—

382

—

126

D₄

(C₂×D₁₉₁)⋊C₄

yes

—

127

C₂×C₄

—

382

—

128

C₂×C₄

—

382

—

129

C₂×C₄

—

191

—

130

C₂×C₄

C₂×C₄×D₁₉₁

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

132

C₂×C₄

C₂×D₇₆₄

yes

—

133

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

134

C₈

—

135

C₈

—

382

—

136

C₂²

—

137

C₂²

—

382

—

138

C₂²

—

764

—

139

C₂²

C₂×(C₂×D₁₉₁)⋊C₄

yes

—

140

C₂²

C₂²×C₄×D₁₉₁

yes

characteristic

141

C₄

—

382

—

142

C₄

C₂×(C₂×D₁₉₁)⋊C₄

yes

—

143

C₄

C₂²×C₂²⋊D₁₉₁

yes

characteristic

144

C₄

C₂²×D₇₆₄

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

764

—

148

C₂

C₂²×(C₂×D₁₉₁)⋊C₄

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×(C₂×D₁₉₁)⋊C₄
commutator	C₂²×C₁₉₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#568×C₁₉₁	C₂
socle	C₂×C₁₉₁	C₂²×C₂²⋊C₄
prime radical	C₂×#20259369	C₂
cosocle	C₂×C₄×C₁₉₁	C₂⁴
hypercenter	#568	D₁₉₁
hypocenter	C₁₉₁	#2889
LCS	G ▷ C₂²×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₂

C₂²×(C₂×D₁₉₁)⋊C₄

commutator

C₂²×C₁₉₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#568×C₁₉₁

C₂

socle

C₂×C₁₉₁

C₂²×C₂²⋊C₄

prime radical

C₂×#20259369

C₂

cosocle

C₂×C₄×C₁₉₁

C₂⁴

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
2. #27133430	C₂	semidirect
3. #27132645	C₂	semidirect
4. C₂×#20259240	C₂	semidirect
5. #568×C₁₉₁	C₂	semidirect
74. C₁₉₁	#2889	semidirect
78. #568	D₁₉₁	semidirect

H ≅ G/N

type

2. #27133430

C₂

semidirect

3. #27132645

C₂

semidirect

4. C₂×#20259240

C₂

semidirect

5. #568×C₁₉₁

C₂

semidirect

74. C₁₉₁

#2889

semidirect

78. #568

D₁₉₁

semidirect

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

subgroup	index
C₂×#20259369	2
#27133430	2
#27132645	2
C₂×#20259240	2
#568×C₁₉₁	2
#2889	191

subgroup

index

#568×C₁₉₁

191

Subgroups — #37581002

Conjugacy classes of subgroups

Complements

Maximal subgroups