Group #37583030 — subgroups

Subgroups — #37583030

Order 24448.

356 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 157 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	#27133066	C₂	1	2	yes	maximal, complemented
3	#27133062	C₂	1	4	yes	maximal, complemented
4	#27133059	C₂	1	2	yes	maximal, complemented
5	#27132699	C₂	1	4	yes	maximal
6	C₂×C₄×C₁₉₁⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#533×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
9	#20259322	C₂²	1	2	yes	—
10	#20259314	C₂²	1	2	yes	—
11	#20259301	—	2	2	no	—
12	C₄×Dic₃₈₂	—	2	2	no	—
13	#20259281	C₂²	1	2	yes	—
14	#20259255	C₂²	1	8	yes	—
15	#20259253	C₂²	1	8	yes	—
16	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₁₉₁⋊C₈	C₂²	1	2	yes	—
18	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₉₁	C₂²	1	2	yes	—
20	#125×C₁₉₁	C₂²	1	2	yes	—
21	C₂²⋊Q₈×C₁₉₁	—	2	2	no	—
22	C₄×Q₈×C₁₉₁	—	2	2	no	—
23	#117×C₁₉₁	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
25	C₂×Dic₃₈₂	—	2	2	no	—
26	C₂×C₁₉₁⋊C₈	Q₈	1	8	yes	complemented
27	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
28	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₁₉₁	—	2	2	no	—
30	C₄⋊Dic₁₉₁	C₂³	1	4	yes	—
31	Dic₁₉₁⋊C₄	—	2	4	no	—
32	C₂²⋊Dic₁₉₁	—	2	2	no	—
33	C₄×Dic₁₉₁	—	2	2	no	—
34	C₂×Q₈×C₁₉₁	—	2	2	no	—
35	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
36	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₉₁	—	2	4	no	—
38	C₄⋊C₄×C₁₉₁	—	2	2	no	—
39	C₄⋊C₄×C₁₉₁	C₂³	1	4	yes	—
40	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
41	C₄²×C₁₉₁	—	2	2	no	—
42	C₄²×C₁₉₁	D₄	1	2	yes	—
43	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
44	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
45	Dic₃₈₂	—	4	4	no	—
46	C₂×Dic₁₉₁	—	2	4	no	—
47	C₁₉₁⋊C₈	C₂×Q₈	1	4	yes	—
48	C₁₉₁⋊C₈	C₂×Q₈	1	4	yes	—
49	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₁₉₁	—	4	4	no	—
51	C₂×C₄×C₁₉₁	—	2	4	no	—
52	C₂×C₄×C₁₉₁	—	2	4	no	—
53	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₁	C₂×Q₈	1	4	yes	—
57	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
58	C₂²×C₁₉₁	—	2	2	no	—
59	C₂²×C₁₉₁	C₄⋊Q₈	1	2	yes	—
60	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
61	C₄×C₁₉₁	—	2	2	no	—
62	C₄×C₁₉₁	—	4	4	no	—
63	C₄×C₁₉₁	C₄⋊Q₈	1	2	yes	—
64	C₄×C₁₉₁	#135	1	4	yes	—
65	C₄×C₁₉₁	C₂²×Q₈	1	2	yes	—
66	Dic₁₉₁	—	4	4	no	—
67	C₂×C₁₉₁	—	2	2	no	—
68	C₂×C₁₉₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₁₉₁	C₂×#135	1	2	yes	—
70	C₁₉₁	#3160	1	1	yes	characteristic, complemented, Sylow
71	#3160	—	191	1	no	maximal, Sylow
72	#533	—	191	1	no	—
73	#533	D₁₉₁	1	1	yes	characteristic
74	#500	—	191	2	no	—
75	#499	—	191	4	no	—
76	#498	—	191	2	no	—
77	#427	—	191	4	no	—
78	C₂×C₄×C₈	—	191	1	no	—
79	C₂²×C₈	—	191	2	no	—
80	C₄⋊Q₈	—	191	2	no	—
81	C₄⋊Q₈	C₂×D₁₉₁	1	2	yes	—
82	#125	—	191	2	no	—
83	#125	C₂×D₁₉₁	1	2	yes	—
84	C₂²⋊Q₈	—	2	2	no	—
85	C₂²⋊Q₈	—	382	2	no	—
86	C₄×Q₈	—	2	2	no	—
87	C₄×Q₈	—	382	2	no	—
88	#117	—	191	2	no	—
89	#117	C₂×D₁₉₁	1	2	yes	—
90	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
91	#107	—	191	8	no	—
92	#106	—	191	8	no	—
93	C₄×C₈	—	191	1	no	—
94	C₄×C₈	—	191	2	no	—
95	C₄×C₈	—	191	1	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	382	2	no	—
98	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
100	C₂×C₈	—	191	8	no	—
101	C₂×C₈	—	191	2	no	—
102	C₂×C₈	—	191	2	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	382	4	no	—
106	C₄⋊C₄	—	382	2	no	—
107	C₄⋊C₄	—	191	4	no	—
108	C₄⋊C₄	C₂²×D₁₉₁	1	4	yes	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	382	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	382	2	no	—
113	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
114	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
115	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
116	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	764	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	382	4	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₁₉₁	1	4	yes	—
126	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
127	C₈	—	191	4	no	—
128	C₈	—	191	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	#20259320	1	2	yes	—
131	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	4	4	no	—
134	C₄	—	764	4	no	—
135	C₄	#20259320	1	2	yes	—
136	C₄	#20259342	1	4	yes	—
137	C₄	C₂×Q₈×D₁₉₁	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	C₂×#20259320	1	1	yes	characteristic
140	C₂	C₂×#20259342	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133220

C₂

yes

maximal, characteristic

#27133066

C₂

yes

maximal, complemented

#27133062

C₂

yes

maximal, complemented

#27133059

C₂

yes

maximal, complemented

#27132699

C₂

yes

maximal

C₂×C₄×C₁₉₁⋊C₈

C₂

yes

maximal, characteristic

#533×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

#20259322

C₂²

yes

—

#20259314

C₂²

yes

—

#20259301

—

C₄×Dic₃₈₂

—

#20259281

C₂²

yes

—

#20259255

C₂²

yes

—

#20259253

C₂²

yes

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₉₁⋊C₈

C₂²

yes

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₁

C₂²

yes

—

#125×C₁₉₁

C₂²

yes

—

C₂²⋊Q₈×C₁₉₁

—

C₄×Q₈×C₁₉₁

—

#117×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×C₁₉₁⋊C₈

Q₈

yes

complemented

C₂×C₁₉₁⋊C₈

C₂³

yes

—

C₂×C₁₉₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

C₂³

yes

—

Dic₁₉₁⋊C₄

—

C₂²⋊Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

C₂³

yes

—

C₂²⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

C₂×Q₈

yes

—

C₁₉₁⋊C₈

C₂×Q₈

yes

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₄⋊Q₈

yes

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₄⋊Q₈

yes

—

C₄×C₁₉₁

#135

yes

—

C₄×C₁₉₁

C₂²×Q₈

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₉₁

C₂×#135

yes

—

C₁₉₁

#3160

yes

characteristic, complemented, Sylow

#3160

—

191

maximal, Sylow

#533

—

191

—

#533

D₁₉₁

yes

characteristic

#500

—

191

—

#499

—

191

—

#498

—

191

—

#427

—

191

—

C₂×C₄×C₈

—

191

—

C₂²×C₈

—

191

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

C₂×D₁₉₁

yes

—

#125

—

191

—

#125

C₂×D₁₉₁

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

382

—

C₄×Q₈

—

C₄×Q₈

—

382

—

#117

—

191

—

#117

C₂×D₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#107

—

191

—

#106

—

191

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₂×Q₈

—

C₂×Q₈

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

100

C₂×C₈

—

191

—

101

C₂×C₈

—

191

—

102

C₂×C₈

—

191

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

382

—

106

C₄⋊C₄

—

382

—

107

C₄⋊C₄

—

191

—

108

C₄⋊C₄

C₂²×D₁₉₁

yes

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

382

—

111

C₄²

—

112

C₄²

—

382

—

113

C₄²

C₂²⋊D₁₉₁

yes

—

114

C₄²

C₂²×D₁₉₁

yes

characteristic

115

C₄²

C₂²×D₁₉₁

yes

characteristic

116

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

117

Q₈

—

118

Q₈

—

764

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

382

—

122

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

124

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

125

C₂×C₄

Q₈×D₁₉₁

yes

—

126

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

127

C₈

—

191

—

128

C₈

—

191

—

129

C₂²

—

130

C₂²

#20259320

yes

—

131

C₂²

C₂²×C₂²⋊D₁₉₁

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

764

—

135

C₄

#20259320

yes

—

136

C₄

#20259342

yes

—

137

C₄

C₂×Q₈×D₁₉₁

yes

—

138

C₂

—

139

C₂

C₂×#20259320

yes

characteristic

140

C₂

C₂×#20259342

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#533×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	#533	D₁₉₁
hypocenter	C₁₉₁	#3160
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₉₁

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#533×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

prime radical

C₂³×C₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #27133066	C₂	semidirect
3. #27133062	C₂	semidirect
4. #27133059	C₂	semidirect
26. C₂×C₁₉₁⋊C₈	Q₈	semidirect
70. C₁₉₁	#3160	semidirect

H ≅ G/N

type

2. #27133066

C₂

semidirect

3. #27133062

C₂

semidirect

4. #27133059

C₂

semidirect

26. C₂×C₁₉₁⋊C₈

Q₈

semidirect

70. C₁₉₁

#3160

semidirect

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

subgroup	index
#27133220	2
#27133066	2
#27133062	2
#27133059	2
#27132699	2
C₂×C₄×C₁₉₁⋊C₈	2
#533×C₁₉₁	2
#3160	191

subgroup

index

C₂×C₄×C₁₉₁⋊C₈

#533×C₁₉₁

#3160

191

Subgroups — #37583030

Conjugacy classes of subgroups

Complements

Maximal subgroups