Group #37571821 — subgroups

Subgroups — #37571821

Order 24448.

182 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₁₉₁⋊#564	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133027	C₂	1	2	yes	maximal
3	#27132405	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132404	C₂	1	2	yes	maximal, complemented
5	#490×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	#20259314	C₂²	1	2	yes	—
7	#20259313	—	2	2	no	—
8	#20259312	—	2	2	no	—
9	#20259312	—	2	2	no	—
10	#20259243	—	2	2	no	—
11	#20259243	—	2	2	no	—
12	#20259233	C₂²	1	2	yes	—
13	#127×C₁₉₁	C₂²	1	1	yes	characteristic
14	#125×C₁₉₁	C₂²	1	1	yes	characteristic
15	#102×C₁₉₁	—	2	2	no	—
16	#97×C₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₉₁⋊C₈	—	2	2	no	—
18	C₄⋊Dic₁₉₁	—	2	2	no	—
19	C₄⋊Dic₁₉₁	D₄	1	2	yes	—
20	C₄⋊Dic₁₉₁	D₄	1	2	yes	—
21	Dic₁₉₁⋊C₄	—	2	2	no	—
22	Dic₁₉₁⋊C₄	—	2	2	no	—
23	Dic₁₉₁⋊C₄	—	2	2	no	—
24	Dic₁₉₁⋊C₄	—	2	2	no	—
25	C₄×Dic₁₉₁	—	2	2	no	—
26	C₂×D₄×C₁₉₁	—	2	1	no	—
27	C₂×D₄×C₁₉₁	—	2	2	no	—
28	C₂×C₈×C₁₉₁	—	2	1	no	—
29	C₄⋊C₄×C₁₉₁	—	2	2	no	—
30	C₄⋊C₄×C₁₉₁	D₄	1	2	yes	—
31	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₉₁	—	2	2	no	—
33	C₂×Dic₁₉₁	—	2	2	no	—
34	C₁₉₁⋊C₈	—	4	2	no	—
35	C₂³×C₁₉₁	—	4	1	no	—
36	D₄×C₁₉₁	—	4	2	no	—
37	D₄×C₁₉₁	—	4	4	no	—
38	C₂×C₄×C₁₉₁	—	2	2	no	—
39	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
41	C₈×C₁₉₁	—	4	1	no	—
42	C₂²×C₁₉₁	—	8	2	no	—
43	C₂²×C₁₉₁	—	8	1	no	—
44	C₂²×C₁₉₁	#120	1	1	yes	characteristic
45	C₄×C₁₉₁	—	2	1	no	—
46	C₄×C₁₉₁	—	2	2	no	—
47	C₄×C₁₉₁	—	4	2	no	—
48	Dic₁₉₁	—	4	2	no	—
49	Dic₁₉₁	—	4	2	no	—
50	C₂×C₁₉₁	—	16	1	no	—
51	C₂×C₁₉₁	#454	1	2	yes	—
52	C₂×C₁₉₁	#454	1	1	yes	characteristic
53	C₁₉₁	#1658	1	1	yes	characteristic, complemented, Sylow
54	#1658	—	191	1	no	maximal, Sylow
55	#564	—	191	1	no	—
56	#490	—	191	2	no	—
57	#490	D₁₉₁	1	1	yes	characteristic
58	#330	—	191	1	no	—
59	#330	—	191	2	no	—
60	#127	C₂×D₁₉₁	1	1	yes	characteristic
61	#125	—	382	2	no	—
62	#125	—	382	2	no	—
63	#125	—	382	2	no	—
64	#125	—	191	2	no	—
65	#125	C₂×D₁₉₁	1	1	yes	characteristic
66	#102	—	2	2	no	—
67	#102	—	382	2	no	—
68	#102	—	382	2	no	—
69	#97	—	191	2	no	—
70	#97	C₂×D₁₉₁	1	1	yes	characteristic
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	382	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	382	2	no	—
77	C₄⋊C₄	—	382	2	no	—
78	C₄⋊C₄	—	382	2	no	—
79	C₄⋊C₄	—	382	2	no	—
80	C₄⋊C₄	—	382	2	no	—
81	C₄⋊C₄	—	191	2	no	—
82	C₄⋊C₄	—	191	2	no	—
83	C₄⋊C₄	C₂²⋊D₁₉₁	1	2	yes	—
84	C₄²	—	382	2	no	—
85	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
86	C₂³	—	4	1	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	382	2	no	—
91	C₂×C₄	—	382	2	no	—
92	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
94	C₈	—	4	1	no	—
95	C₈	—	764	2	no	—
96	C₂²	—	8	2	no	—
97	C₂²	—	8	1	no	—
98	C₂²	#20259290	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	—	4	2	no	—
102	C₄	—	764	2	no	—
103	C₄	—	764	2	no	—
104	C₂	—	16	1	no	—
105	C₂	#27132834	1	2	yes	—
106	C₂	#27132835	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₉₁⋊#564

C₂

yes

maximal, characteristic, complemented

#27133027

C₂

yes

maximal

#27132405

C₂

yes

maximal, characteristic, complemented

#27132404

C₂

yes

maximal, complemented

#490×C₁₉₁

C₂

yes

maximal, characteristic

#20259314

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

#127×C₁₉₁

C₂²

yes

characteristic

#125×C₁₉₁

C₂²

yes

characteristic

#102×C₁₉₁

—

#97×C₁₉₁

C₂²

yes

characteristic

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

D₄

yes

—

C₄⋊Dic₁₉₁

D₄

yes

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#120

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#454

yes

—

C₂×C₁₉₁

#454

yes

characteristic

C₁₉₁

#1658

yes

characteristic, complemented, Sylow

#1658

—

191

maximal, Sylow

#564

—

191

—

#490

—

191

—

#490

D₁₉₁

yes

characteristic

#330

—

191

—

#330

—

191

—

#127

C₂×D₁₉₁

yes

characteristic

#125

—

382

—

#125

—

382

—

#125

—

382

—

#125

—

191

—

#125

C₂×D₁₉₁

yes

characteristic

#102

—

#102

—

382

—

#102

—

382

—

#97

—

191

—

#97

C₂×D₁₉₁

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×C₈

—

C₂×C₈

—

382

—

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

C₂²⋊D₁₉₁

yes

—

C₄²

—

382

—

C₄²

C₂²×D₁₉₁

yes

characteristic

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

D₄×D₁₉₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

C₈

—

C₈

—

764

—

C₂²

—

C₂²

—

C₂²

#20259290

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

764

—

103

C₄

—

764

—

104

C₂

—

105

C₂

#27132834

yes

—

106

C₂

#27132835

yes

characteristic

	subgroup	G/H
center	C₂²	#20259290
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#490×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#120
prime radical	#127×C₁₉₁	C₂²
hypercenter	#490	D₁₉₁
hypocenter	C₁₉₁	#1658
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #490

subgroup

G/H

center

C₂²

#20259290

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#490×C₁₉₁

C₂

socle

C₂²×C₁₉₁

#120

prime radical

#127×C₁₉₁

C₂²

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #490

N	H ≅ G/N	type
1. C₁₉₁⋊#564	C₂	semidirect
3. #27132405	C₂	semidirect
4. #27132404	C₂	semidirect
53. C₁₉₁	#1658	semidirect

H ≅ G/N

type

1. C₁₉₁⋊#564

C₂

semidirect

3. #27132405

C₂

semidirect

4. #27132404

C₂

semidirect

53. C₁₉₁

#1658

semidirect

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

subgroup	index
C₁₉₁⋊#564	2
#27133027	2
#27132405	2
#27132404	2
#490×C₁₉₁	2
#1658	191

subgroup

index

C₁₉₁⋊#564

#490×C₁₉₁

191

Subgroups — #37571821

Conjugacy classes of subgroups

Complements

Maximal subgroups