Group #37572140 — subgroups

Subgroups — #37572140

Order 24448.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#27133066	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133059	C₂	1	1	yes	maximal, characteristic, complemented
3	#27133035	C₂	1	1	yes	maximal, characteristic
4	#27133014	C₂	1	1	yes	maximal, characteristic, complemented
5	#27133009	C₂	1	1	yes	maximal, characteristic
6	#27132385	C₂	1	1	yes	maximal, characteristic, complemented
7	#492×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#15690275	—	2	2	no	—
9	#20259322	C₂²	1	1	yes	characteristic
10	#20259322	C₂²	1	1	yes	characteristic
11	#20259255	—	2	2	no	—
12	#20259253	—	2	2	no	—
13	#20259247	—	2	2	no	—
14	#20259243	—	2	2	no	—
15	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₁₉₁	—	2	2	no	—
18	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
19	#127×C₁₉₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₈₂	—	2	1	no	—
22	C₂×Dic₃₈₂	—	2	1	no	—
23	#15690275	—	4	4	no	—
24	C₂×C₁₉₁⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₉₁⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₁₉₁	—	2	1	no	—
27	C₄⋊Dic₁₉₁	—	2	1	no	—
28	C₄⋊Dic₁₉₁	—	2	1	no	—
29	C₄⋊Dic₁₉₁	—	2	1	no	—
30	C₂×Q₈×C₁₉₁	—	2	1	no	—
31	C₂×D₄×C₁₉₁	—	2	1	no	—
32	C₂×D₄×C₁₉₁	—	2	1	no	—
33	C₂×D₄×C₁₉₁	—	2	1	no	—
34	SD₁₆×C₁₉₁	—	4	4	no	—
35	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
36	C₄⋊C₄×C₁₉₁	—	2	1	no	—
37	C₄⋊C₄×C₁₉₁	—	2	1	no	—
38	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
39	Dic₃₈₂	—	4	2	no	—
40	Dic₃₈₂	—	4	2	no	—
41	C₂×Dic₁₉₁	—	4	1	no	—
42	C₂×Dic₁₉₁	—	4	1	no	—
43	C₁₉₁⋊C₈	—	2	2	no	—
44	C₁₉₁⋊C₈	SD₁₆	1	4	yes	complemented
45	C₂³×C₁₉₁	—	4	1	no	—
46	Q₈×C₁₉₁	—	4	2	no	—
47	D₄×C₁₉₁	—	4	2	no	—
48	D₄×C₁₉₁	—	4	2	no	—
49	D₄×C₁₉₁	—	4	2	no	—
50	C₂×C₄×C₁₉₁	—	4	1	no	—
51	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
54	C₈×C₁₉₁	—	2	2	no	—
55	C₂²×C₁₉₁	—	8	1	no	—
56	C₂²×C₁₉₁	—	8	1	no	—
57	C₂²×C₁₉₁	—	8	1	no	—
58	C₂²×C₁₉₁	#127	1	1	yes	characteristic
59	C₄×C₁₉₁	—	8	1	no	—
60	C₄×C₁₉₁	C₂×SD₁₆	1	2	yes	—
61	C₄×C₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₁₉₁	#137	1	2	yes	—
63	Dic₁₉₁	—	8	1	no	—
64	Dic₁₉₁	—	8	1	no	—
65	C₂×C₁₉₁	—	16	1	no	—
66	C₂×C₁₉₁	#492	1	1	yes	characteristic
67	C₂×C₁₉₁	#496	1	1	yes	characteristic
68	C₂×C₁₉₁	#497	1	1	yes	characteristic
69	C₁₉₁	#1714	1	1	yes	characteristic, complemented, Sylow
70	#1714	—	191	1	no	maximal, Sylow
71	#500	—	191	1	no	—
72	#498	—	191	1	no	—
73	#492	—	191	1	no	—
74	#492	D₁₉₁	1	1	yes	characteristic
75	#487	—	191	1	no	—
76	#486	—	191	1	no	—
77	#322	—	191	1	no	—
78	C₂×SD₁₆	—	2	2	no	—
79	C₂×SD₁₆	—	382	2	no	—
80	C₄⋊Q₈	—	191	1	no	—
81	C₄⋊Q₈	—	191	1	no	—
82	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
83	#127	C₂×D₁₉₁	1	1	yes	characteristic
84	#107	—	382	2	no	—
85	#106	—	382	2	no	—
86	#103	—	382	2	no	—
87	#102	—	382	2	no	—
88	C₄×C₈	—	191	1	no	—
89	C₄×C₈	—	191	1	no	—
90	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	382	1	no	—
93	C₂×Q₈	—	382	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	4	4	no	—
98	SD₁₆	—	764	4	no	—
99	C₂×C₈	—	191	2	no	—
100	C₂×C₈	—	191	2	no	—
101	C₂×C₈	C₂²⋊D₁₉₁	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	382	1	no	—
105	C₄⋊C₄	—	382	1	no	—
106	C₄⋊C₄	—	382	1	no	—
107	C₄⋊C₄	—	382	1	no	—
108	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	764	2	no	—
112	Q₈	—	764	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	764	1	no	—
118	C₂×C₄	—	764	1	no	—
119	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
120	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	382	2	no	—
124	C₈	—	191	4	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#20259318	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	1528	1	no	—
131	C₄	—	1528	1	no	—
132	C₄	SD₁₆×D₁₉₁	1	2	yes	—
133	C₄	#20259348	1	2	yes	—
134	C₄	#20259351	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#27133034	1	1	yes	characteristic
137	C₂	#27133050	1	1	yes	characteristic
138	C₂	#27133054	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133066

C₂

yes

maximal, characteristic, complemented

#27133059

C₂

yes

maximal, characteristic, complemented

#27133035

C₂

yes

maximal, characteristic

#27133014

C₂

yes

maximal, characteristic, complemented

#27133009

C₂

yes

maximal, characteristic

#27132385

C₂

yes

maximal, characteristic, complemented

#492×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690275

—

#20259322

C₂²

yes

characteristic

#20259322

C₂²

yes

—

—

—

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₁₉₁

—

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

#127×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

#15690275

—

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

SD₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

SD₁₆

yes

complemented

C₂³×C₁₉₁

—

Q₈×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#127

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×SD₁₆

yes

—

C₄×C₁₉₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₁

#137

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1714

—

191

maximal, Sylow

#500

—

191

—

#498

—

191

—

#492

—

191

—

#492

D₁₉₁

yes

characteristic

#487

—

191

—

#486

—

191

—

#322

—

191

—

C₂×SD₁₆

—

C₂×SD₁₆

—

382

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

#127

C₂×D₁₉₁

yes

characteristic

#107

—

382

—

#106

—

382

—

#103

—

382

—

#102

—

382

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

764

—

C₂×C₈

—

191

—

100

C₂×C₈

—

191

—

101

C₂×C₈

C₂²⋊D₁₉₁

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

382

—

105

C₄⋊C₄

—

382

—

106

C₄⋊C₄

—

382

—

107

C₄⋊C₄

—

382

—

108

C₄²

C₂²×D₁₉₁

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

764

—

112

Q₈

—

764

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

764

—

118

C₂×C₄

—

764

—

119

C₂×C₄

D₄×D₁₉₁

yes

characteristic

120

C₂×C₄

D₄×D₁₉₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

122

C₈

—

123

C₈

—

382

—

124

C₈

—

191

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#20259318

yes

characteristic

129

C₄

—

130

C₄

—

1528

—

131

C₄

—

1528

—

132

C₄

SD₁₆×D₁₉₁

yes

—

133

C₄

#20259348

yes

—

134

C₄

#20259351

yes

—

135

C₂

—

136

C₂

#27133034

yes

characteristic

137

C₂

#27133050

yes

characteristic

138

C₂

#27133054

yes

characteristic

	subgroup	G/H
center	C₂²	#20259318
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#492×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#127
prime radical	#127×C₁₉₁	C₂²
hypercenter	#492	D₁₉₁
hypocenter	C₁₉₁	#1714
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#20259318

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#492×C₁₉₁

C₂

socle

C₂²×C₁₉₁

#127

prime radical

#127×C₁₉₁

C₂²

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #27133066	C₂	semidirect
2. #27133059	C₂	semidirect
4. #27133014	C₂	semidirect
6. #27132385	C₂	semidirect
44. C₁₉₁⋊C₈	SD₁₆	semidirect
69. C₁₉₁	#1714	semidirect

H ≅ G/N

type

1. #27133066

C₂

semidirect

2. #27133059

C₂

semidirect

4. #27133014

C₂

semidirect

6. #27132385

C₂

semidirect

44. C₁₉₁⋊C₈

SD₁₆

semidirect

69. C₁₉₁

#1714

semidirect

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

subgroup	index
#27133066	2
#27133059	2
#27133035	2
#27133014	2
#27133009	2
#27132385	2
#492×C₁₉₁	2
#1714	191

subgroup

index

#492×C₁₉₁

191

Subgroups — #37572140

Conjugacy classes of subgroups

Complements

Maximal subgroups