Group #37570809 — subgroups

Subgroups — #37570809

Order 24448.

208 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 125 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	#27132716	C₂	1	1	yes	maximal, characteristic
2	C₄×#15690271	C₂	1	1	yes	maximal, characteristic
3	#27132385	C₂	1	2	yes	maximal, complemented
4	#27132384	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#15690271	C₄	1	2	yes	—
7	C₄⋊(C₁₉₁⋊C₈)	C₄	1	2	yes	—
8	C₂²⋊(C₁₉₁⋊C₈)	C₄	1	2	yes	—
9	#20259233	C₄	1	2	yes	—
10	C₄×C₁₉₁⋊C₈	C₂²	1	2	yes	—
11	C₄×C₁₉₁⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₉₁	C₄	1	2	yes	—
13	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₉₁	C₄	1	2	yes	—
15	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
17	#15690271	—	2	4	no	—
18	C₂×C₁₉₁⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₁₉₁⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₉₁	—	2	4	no	—
24	C₂×C₈×C₁₉₁	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₉₁	C₂×C₄	1	2	yes	—
26	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
30	C₁₉₁⋊C₈	—	2	4	no	—
31	C₁₉₁⋊C₈	—	2	4	no	—
32	C₂³×C₁₉₁	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₁₉₁	—	2	2	no	—
34	C₂×C₄×C₁₉₁	—	2	2	no	—
35	C₂×C₄×C₁₉₁	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₁₉₁	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₁₉₁	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₁₉₁	M₁₆	1	4	yes	—
39	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
42	C₈×C₁₉₁	—	2	2	no	—
43	C₈×C₁₉₁	M₁₆	1	4	yes	complemented
44	C₂²×C₁₉₁	—	2	1	no	—
45	C₂²×C₁₉₁	—	2	1	no	—
46	C₂²×C₁₉₁	#97	1	2	yes	—
47	C₂²×C₁₉₁	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₉₁	—	2	1	no	—
49	C₄×C₁₉₁	—	2	1	no	—
50	C₄×C₁₉₁	#97	1	2	yes	—
51	C₄×C₁₉₁	C₂×M₁₆	1	2	yes	—
52	C₄×C₁₉₁	C₂×M₁₆	1	2	yes	—
53	C₄×C₁₉₁	#131	1	2	yes	—
54	C₂×C₁₉₁	—	2	2	no	—
55	C₂×C₁₉₁	C₂×#97	1	1	yes	characteristic
56	C₂×C₁₉₁	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₁₉₁	#405	1	1	yes	characteristic
58	C₁₉₁	#1451	1	1	yes	characteristic, complemented, Sylow
59	#1451	—	191	1	no	maximal, Sylow
60	#431	—	191	1	no	—
61	C₄×M₁₆	—	191	1	no	—
62	C₂×C₄×C₈	D₁₉₁	1	1	yes	characteristic
63	#322	—	191	2	no	—
64	#322	—	191	2	no	—
65	C₂×M₁₆	—	191	2	no	—
66	C₂²×C₈	Dic₁₉₁	1	2	yes	—
67	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
68	C₄⋊C₈	—	191	2	no	—
69	C₂²⋊C₈	—	191	2	no	—
70	#97	—	191	2	no	—
71	C₄×C₈	—	191	2	no	—
72	C₄×C₈	—	191	2	no	—
73	C₄×C₈	Dic₁₉₁	1	2	yes	—
74	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
75	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
76	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
77	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
78	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
79	M₁₆	—	382	4	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	191	4	no	—
82	C₂×C₈	—	191	4	no	—
83	C₂×C₈	C₂×Dic₁₉₁	1	2	yes	—
84	C₂×C₈	C₂×Dic₁₉₁	1	2	yes	—
85	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
86	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
87	C₄²	C₂×Dic₁₉₁	1	1	yes	characteristic
88	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
89	C₂³	C₄×Dic₁₉₁	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	C₄×Dic₁₉₁	1	1	yes	characteristic
93	C₂×C₄	C₄×Dic₁₉₁	1	1	yes	characteristic
94	C₂×C₄	C₄×Dic₁₉₁	1	1	yes	characteristic
95	C₂×C₄	#15690270	1	4	yes	—
96	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	382	4	no	—
101	C₈	—	382	4	no	—
102	C₈	#15690271	1	4	yes	complemented
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#20259232	1	2	yes	—
106	C₂²	C₂×C₄×Dic₁₉₁	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	#20259232	1	2	yes	—
110	C₄	C₂×#15690270	1	2	yes	—
111	C₄	C₂×#15690271	1	2	yes	—
112	C₄	#20259328	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	C₂×#20259232	1	1	yes	characteristic
115	C₂	C₄×#15690271	1	1	yes	characteristic
116	C₂	#27132618	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132716

C₂

yes

maximal, characteristic

C₄×#15690271

C₂

yes

maximal, characteristic

#27132385

C₂

yes

maximal, complemented

#27132384

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₄

yes

—

C₄⋊(C₁₉₁⋊C₈)

C₄

yes

—

C₂²⋊(C₁₉₁⋊C₈)

C₄

yes

—

#20259233

C₄

yes

—

C₄×C₁₉₁⋊C₈

C₂²

yes

—

C₄×C₁₉₁⋊C₈

C₂²

yes

—

C₂²×C₈×C₁₉₁

C₄

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₄

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

#15690271

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₄²

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₄²

yes

characteristic

C₂×C₄×C₁₉₁

C₄²

yes

characteristic

C₂×C₄×C₁₉₁

C₄²

yes

characteristic

C₂×C₄×C₁₉₁

M₁₆

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

M₁₆

yes

complemented

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#97

yes

—

C₂²×C₁₉₁

C₂×C₄²

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#97

yes

—

C₄×C₁₉₁

C₂×M₁₆

yes

—

C₄×C₁₉₁

C₂×M₁₆

yes

—

C₄×C₁₉₁

#131

yes

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×#97

yes

characteristic

C₂×C₁₉₁

C₄×M₁₆

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1451

—

191

maximal, Sylow

#431

—

191

—

C₄×M₁₆

—

191

—

C₂×C₄×C₈

D₁₉₁

yes

characteristic

#322

—

191

—

#322

—

191

—

C₂×M₁₆

—

191

—

C₂²×C₈

Dic₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

191

—

C₂²⋊C₈

—

191

—

#97

—

191

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₄×C₈

Dic₁₉₁

yes

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₉₁

yes

characteristic

M₁₆

—

382

—

C₂×C₈

—

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₂×C₈

C₂×Dic₁₉₁

yes

—

C₂×C₈

C₂×Dic₁₉₁

yes

—

C₄²

C₄×D₁₉₁

yes

characteristic

C₄²

C₄×D₁₉₁

yes

characteristic

C₄²

C₂×Dic₁₉₁

yes

characteristic

C₄²

C₂²×D₁₉₁

yes

characteristic

C₂³

C₄×Dic₁₉₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₉₁

yes

characteristic

C₂×C₄

C₄×Dic₁₉₁

yes

characteristic

C₂×C₄

C₄×Dic₁₉₁

yes

characteristic

C₂×C₄

#15690270

yes

—

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

C₂×C₄

C₂²×Dic₁₉₁

yes

characteristic

C₈

—

100

C₈

—

382

—

101

C₈

—

382

—

102

C₈

#15690271

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

#20259232

yes

—

106

C₂²

C₂×C₄×Dic₁₉₁

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

#20259232

yes

—

110

C₄

C₂×#15690270

yes

—

111

C₄

C₂×#15690271

yes

—

112

C₄

#20259328

yes

—

113

C₂

—

114

C₂

C₂×#20259232

yes

characteristic

115

C₂

C₄×#15690271

yes

characteristic

116

C₂

#27132618

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂×C₄²
Fitting	C₂×C₄×C₈×C₁₉₁	C₂
prime radical	C₂³×C₁₉₁	C₄²
cosocle	C₄²×C₁₉₁	C₂³
hypercenter	C₂×C₄×C₈	D₁₉₁
hypocenter	C₁₉₁	#1451
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₄²

C₂²×D₁₉₁

commutator, socle

C₂²×C₁₉₁

C₂×C₄²

Fitting

C₂×C₄×C₈×C₁₉₁

C₂

prime radical

C₂³×C₁₉₁

C₄²

cosocle

C₄²×C₁₉₁

C₂³

hypercenter

C₂×C₄×C₈

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #27132385	C₂	semidirect
4. #27132384	C₂	semidirect
43. C₈×C₁₉₁	M₁₆	semidirect
58. C₁₉₁	#1451	semidirect
102. C₈	#15690271	semidirect

H ≅ G/N

type

3. #27132385

C₂

semidirect

4. #27132384

C₂

semidirect

43. C₈×C₁₉₁

M₁₆

semidirect

58. C₁₉₁

#1451

semidirect

102. C₈

#15690271

semidirect

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

subgroup	index
#27132716	2
C₄×#15690271	2
#27132385	2
#27132384	2
C₂×C₄×C₈×C₁₉₁	2
#1451	191

subgroup

index

C₂×C₄×C₈×C₁₉₁

191

Subgroups — #37570809

Conjugacy classes of subgroups

Complements

Maximal subgroups