Group #37571156 — subgroups

Subgroups — #37571156

Order 24448.

194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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	#27133219	C₂	1	1	yes	maximal, characteristic
2	#27132718	C₂	1	2	yes	maximal
3	#27132405	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132404	C₂	1	2	yes	maximal, complemented
5	#330×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20259321	C₄	1	2	yes	—
7	#20259314	C₂²	1	1	yes	characteristic
8	#20259300	—	2	2	no	—
9	#20259287	—	2	2	no	—
10	#20259280	C₄	1	2	yes	—
11	C₄²×D₁₉₁	C₂²	1	1	yes	characteristic
12	Dic₁₉₁⋊C₈	—	2	2	no	—
13	(C₂×D₁₉₁)⋊C₈	—	2	2	no	—
14	#20259233	C₂²	1	2	yes	—
15	#125×C₁₉₁	C₂²	1	1	yes	characteristic
16	#97×C₁₉₁	C₂²	1	2	yes	—
17	C₂×Dic₃₈₂	—	2	2	no	—
18	C₂×C₄×D₁₉₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₁₉₁	D₄	1	2	yes	—
20	C₂×C₁₉₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
22	Dic₁₉₁⋊C₄	—	2	2	no	—
23	Dic₁₉₁⋊C₄	—	2	2	no	—
24	(C₂×D₁₉₁)⋊C₄	—	2	2	no	—
25	C₄×Dic₁₉₁	—	2	2	no	—
26	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₉₁	D₄	1	2	yes	—
28	C₂×C₈×C₁₉₁	—	2	2	no	—
29	C₄⋊C₄×C₁₉₁	—	2	2	no	—
30	C₄⋊C₄×C₁₉₁	C₂×C₄	1	2	yes	—
31	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
32	C₂²×D₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₃₈₂	—	4	4	no	—
34	C₂×Dic₁₉₁	—	2	2	no	—
35	C₂×Dic₁₉₁	C₂²⋊C₄	1	2	yes	—
36	C₂×Dic₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×D₁₉₁	—	2	2	no	—
38	C₄×D₁₉₁	—	4	2	no	—
39	C₁₉₁⋊C₈	—	4	2	no	—
40	C₂×C₄×C₁₉₁	—	2	2	no	—
41	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
43	C₈×C₁₉₁	—	4	2	no	—
44	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×D₁₉₁	—	2	1	no	—
46	C₂×D₁₉₁	—	2	2	no	—
47	C₄×C₁₉₁	—	2	1	no	—
48	C₄×C₁₉₁	—	2	2	no	—
49	C₄×C₁₉₁	—	4	2	no	—
50	Dic₁₉₁	—	2	2	no	—
51	Dic₁₉₁	—	4	2	no	—
52	Dic₁₉₁	#101	1	2	yes	—
53	C₂×C₁₉₁	C₂×#101	1	1	yes	characteristic
54	C₂×C₁₉₁	#421	1	2	yes	—
55	D₁₉₁	—	4	1	no	—
56	C₁₉₁	#1537	1	1	yes	characteristic, complemented, Sylow
57	#1537	—	191	1	no	maximal, Sylow
58	#533	—	191	1	no	—
59	#432	—	191	2	no	—
60	#330	—	191	1	no	—
61	#330	—	191	2	no	—
62	#330	D₁₉₁	1	1	yes	characteristic, complemented
63	C₄⋊Q₈	—	191	2	no	—
64	#125	—	191	1	no	—
65	#125	C₂×D₁₉₁	1	1	yes	characteristic
66	C₂²⋊Q₈	—	382	2	no	—
67	C₄×Q₈	—	382	2	no	—
68	#117	—	191	2	no	—
69	C₂×C₄²	—	191	1	no	—
70	C₄⋊C₈	—	382	2	no	—
71	C₂²⋊C₈	—	382	2	no	—
72	#97	—	191	2	no	—
73	#97	C₂×D₁₉₁	1	2	yes	—
74	C₂×Q₈	—	382	2	no	—
75	C₂²×C₄	—	191	1	no	—
76	C₂²×C₄	—	191	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	382	2	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	382	2	no	—
81	C₄⋊C₄	—	382	2	no	—
82	C₄⋊C₄	—	191	2	no	—
83	C₄⋊C₄	C₄×D₁₉₁	1	2	yes	—
84	C₂²⋊C₄	—	382	2	no	—
85	C₄²	—	382	2	no	—
86	C₄²	—	191	1	no	—
87	C₄²	—	191	2	no	—
88	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
89	C₂³	—	191	1	no	—
90	Q₈	—	764	4	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	382	2	no	—
93	C₂×C₄	—	382	2	no	—
94	C₂×C₄	—	191	2	no	—
95	C₂×C₄	—	191	1	no	—
96	C₂×C₄	—	764	2	no	—
97	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
98	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
99	C₈	—	4	2	no	—
100	C₈	—	764	2	no	—
101	C₂²	—	382	2	no	—
102	C₂²	—	382	1	no	—
103	C₂²	C₂²⋊C₄×D₁₉₁	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	2	no	—
107	C₄	—	382	2	no	—
108	C₄	—	191	2	no	—
109	C₄	—	764	2	no	—
110	C₂	—	764	1	no	—
111	C₂	#101×D₁₉₁	1	1	yes	characteristic
112	C₂	#27132678	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133219

C₂

yes

maximal, characteristic

#27132718

C₂

yes

maximal

#27132405

C₂

yes

maximal, characteristic, complemented

#27132404

C₂

yes

maximal, complemented

#330×C₁₉₁

C₂

yes

maximal, characteristic, complemented

#20259321

C₄

yes

—

#20259314

C₂²

yes

—

—

C₄

yes

—

C₄²×D₁₉₁

C₂²

yes

characteristic

Dic₁₉₁⋊C₈

—

(C₂×D₁₉₁)⋊C₈

—

#20259233

C₂²

yes

—

#125×C₁₉₁

C₂²

yes

characteristic

#97×C₁₉₁

C₂²

yes

—

C₂×Dic₃₈₂

—

C₂×C₄×D₁₉₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₉₁

D₄

yes

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

(C₂×D₁₉₁)⋊C₄

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

C₂×C₄

yes

characteristic

C₄×Dic₁₉₁

D₄

yes

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

C₂²⋊C₄

yes

characteristic

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂²⋊C₄

yes

—

C₂×Dic₁₉₁

C₂²⋊C₄

yes

characteristic

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

#101

yes

—

C₂×C₁₉₁

C₂×#101

yes

characteristic

C₂×C₁₉₁

#421

yes

—

D₁₉₁

—

C₁₉₁

#1537

yes

characteristic, complemented, Sylow

#1537

—

191

maximal, Sylow

#533

—

191

—

#432

—

191

—

#330

—

191

—

#330

—

191

—

#330

D₁₉₁

yes

characteristic, complemented

C₄⋊Q₈

—

191

—

#125

—

191

—

#125

C₂×D₁₉₁

yes

characteristic

C₂²⋊Q₈

—

382

—

C₄×Q₈

—

382

—

#117

—

191

—

C₂×C₄²

—

191

—

C₄⋊C₈

—

382

—

C₂²⋊C₈

—

382

—

#97

—

191

—

#97

C₂×D₁₉₁

yes

—

C₂×Q₈

—

382

—

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

C₂×C₈

—

C₂×C₈

—

382

—

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

C₄⋊C₄

C₄×D₁₉₁

yes

—

C₂²⋊C₄

—

382

—

C₄²

—

382

—

C₄²

—

191

—

C₄²

—

191

—

C₄²

C₂²×D₁₉₁

yes

characteristic

C₂³

—

191

—

Q₈

—

764

—

C₂×C₄

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

—

191

—

C₂×C₄

—

191

—

C₂×C₄

—

764

—

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

C₂×C₄

D₄×D₁₉₁

yes

—

C₈

—

100

C₈

—

764

—

101

C₂²

—

382

—

102

C₂²

—

382

—

103

C₂²

C₂²⋊C₄×D₁₉₁

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

382

—

108

C₄

—

191

—

109

C₄

—

764

—

110

C₂

—

764

—

111

C₂

#101×D₁₉₁

yes

characteristic

112

C₂

#27132678

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#330×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₉₁	C₂²⋊C₄
cosocle	C₄²×C₁₉₁	C₂³
hypercenter	#330	D₁₉₁
hypocenter	C₁₉₁	#1537
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#330×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₉₁

C₂²⋊C₄

cosocle

C₄²×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
3. #27132405	C₂	semidirect
4. #27132404	C₂	semidirect
5. #330×C₁₉₁	C₂	semidirect
56. C₁₉₁	#1537	semidirect
62. #330	D₁₉₁	semidirect

H ≅ G/N

type

3. #27132405

C₂

semidirect

4. #27132404

C₂

semidirect

5. #330×C₁₉₁

C₂

semidirect

56. C₁₉₁

#1537

semidirect

62. #330

D₁₉₁

semidirect

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

subgroup	index
#27133219	2
#27132718	2
#27132405	2
#27132404	2
#330×C₁₉₁	2
#1537	191

subgroup

index

#330×C₁₉₁

191

Subgroups — #37571156

Conjugacy classes of subgroups

Complements

Maximal subgroups