Group #37572015 — subgroups

Subgroups — #37572015

Order 24448.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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	#27133043	C₂	1	1	yes	maximal, characteristic
2	#27132949	C₂	1	2	yes	maximal
3	#27132793	C₂	1	1	yes	maximal, characteristic
4	#27132408	C₂	1	2	yes	maximal
5	#334×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₆₄	—	2	1	no	—
7	C₂×Dic₇₆₄	—	2	1	no	—
8	#20259322	C₂²	1	2	yes	—
9	C₄×Dic₃₈₂	C₂²	1	1	yes	characteristic
10	#20259254	—	2	2	no	—
11	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₁₉₁⋊C₈	—	2	1	no	—
13	#20259247	—	2	2	no	—
14	#20259232	—	2	1	no	—
15	C₄⋊C₈×C₁₉₁	C₂²	1	2	yes	—
16	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₈₂	—	2	2	no	—
18	C₂×Dic₃₈₂	D₄	1	1	yes	characteristic
19	Dic₇₆₄	—	4	2	no	—
20	Dic₇₆₄	—	4	2	no	—
21	C₂×C₁₉₁⋊C₈	—	2	1	no	—
22	C₄⋊Dic₁₉₁	—	2	2	no	—
23	C₄⋊Dic₁₉₁	—	2	2	no	—
24	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
25	Dic₁₉₁⋊C₄	—	2	1	no	—
26	C₄×Dic₁₉₁	—	2	1	no	—
27	C₂×C₈×C₁₉₁	—	2	2	no	—
28	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
30	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
31	Dic₃₈₂	—	2	2	no	—
32	Dic₃₈₂	—	4	2	no	—
33	Dic₃₈₂	—	4	2	no	—
34	C₂×Dic₁₉₁	—	2	1	no	—
35	C₂×Dic₁₉₁	—	2	1	no	—
36	C₂×Dic₁₉₁	—	4	2	no	—
37	C₁₉₁⋊C₈	—	4	1	no	—
38	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
41	C₈×C₁₉₁	—	2	1	no	—
42	C₈×C₁₉₁	—	4	2	no	—
43	C₈×C₁₉₁	SD₁₆	1	2	yes	—
44	C₂²×C₁₉₁	#121	1	1	yes	characteristic
45	C₄×C₁₉₁	—	2	1	no	—
46	C₄×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
47	C₄×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
48	C₄×C₁₉₁	#137	1	2	yes	—
49	Dic₁₉₁	—	4	1	no	—
50	Dic₁₉₁	—	4	1	no	—
51	Dic₁₉₁	—	8	2	no	—
52	C₂×C₁₉₁	#460	1	1	yes	characteristic
53	C₂×C₁₉₁	#470	1	1	yes	characteristic
54	C₂×C₁₉₁	#473	1	1	yes	characteristic
55	C₁₉₁	#1690	1	1	yes	characteristic, complemented, Sylow
56	#1690	—	191	1	no	maximal, Sylow
57	#494	—	191	1	no	—
58	#475	—	191	2	no	—
59	#446	—	191	1	no	—
60	#334	D₁₉₁	1	1	yes	characteristic
61	#332	—	191	2	no	—
62	C₂×Q₁₆	—	382	1	no	—
63	C₂×Q₁₆	—	382	1	no	—
64	C₄⋊Q₈	—	191	2	no	—
65	C₄×Q₈	—	191	1	no	—
66	#106	—	382	2	no	—
67	C₄⋊C₈	—	382	1	no	—
68	C₄⋊C₈	—	191	1	no	—
69	C₄⋊C₈	C₂×D₁₉₁	1	2	yes	—
70	#103	—	382	2	no	—
71	#97	—	382	1	no	—
72	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
73	C₂×Q₈	—	382	2	no	—
74	C₂×Q₈	—	191	1	no	—
75	Q₁₆	—	764	2	no	—
76	Q₁₆	—	764	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	382	1	no	—
79	C₂×C₈	D₇₆₄	1	1	yes	characteristic
80	C₂×C₈	D₇₆₄	1	1	yes	characteristic
81	C₄⋊C₄	—	382	1	no	—
82	C₄⋊C₄	—	382	2	no	—
83	C₄⋊C₄	—	382	2	no	—
84	C₄⋊C₄	—	191	1	no	—
85	C₄²	—	382	1	no	—
86	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
87	Q₈	—	382	2	no	—
88	Q₈	—	764	2	no	—
89	Q₈	—	764	2	no	—
90	C₂×C₄	—	382	1	no	—
91	C₂×C₄	—	382	1	no	—
92	C₂×C₄	—	764	2	no	—
93	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
94	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
95	C₂×C₄	#15690288	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	4	2	no	—
98	C₈	—	764	1	no	—
99	C₈	#15690276	1	2	yes	—
100	C₂²	#20259297	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	1528	2	no	—
103	C₄	—	764	1	no	—
104	C₄	—	764	1	no	—
105	C₄	C₂×#15690276	1	1	yes	characteristic
106	C₄	#20259350	1	1	yes	characteristic
107	C₄	#20259351	1	2	yes	—
108	C₂	#27132864	1	1	yes	characteristic
109	C₂	#27132922	1	1	yes	characteristic
110	C₂	#27132939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133043

C₂

yes

maximal, characteristic

#27132949

C₂

yes

maximal

#27132793

C₂

yes

maximal, characteristic

#27132408

C₂

yes

maximal

#334×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

—

C₂×Dic₇₆₄

—

#20259322

C₂²

yes

—

C₄×Dic₃₈₂

C₂²

yes

characteristic

#20259254

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

characteristic

Dic₁₉₁⋊C₈

—

#20259247

—

#20259232

—

C₄⋊C₈×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

D₄

yes

characteristic

Dic₇₆₄

—

Dic₇₆₄

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

D₄

yes

characteristic

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₈×C₁₉₁

SD₁₆

yes

—

C₂²×C₁₉₁

#121

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×SD₁₆

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1690

—

191

maximal, Sylow

#494

—

191

—

#475

—

191

—

#446

—

191

—

#334

D₁₉₁

yes

characteristic

#332

—

191

—

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

382

—

C₄⋊Q₈

—

191

—

C₄×Q₈

—

191

—

#106

—

382

—

C₄⋊C₈

—

382

—

C₄⋊C₈

—

191

—

C₄⋊C₈

C₂×D₁₉₁

yes

—

#103

—

382

—

#97

—

382

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

Q₁₆

—

764

—

Q₁₆

—

764

—

C₂×C₈

—

C₂×C₈

—

382

—

C₂×C₈

D₇₆₄

yes

characteristic

C₂×C₈

D₇₆₄

yes

characteristic

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

C₄²

—

382

—

C₄²

C₂²×D₁₉₁

yes

characteristic

Q₈

—

382

—

Q₈

—

764

—

Q₈

—

764

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

—

764

—

C₂×C₄

D₄×D₁₉₁

yes

characteristic

C₂×C₄

C₂×D₇₆₄

yes

characteristic

C₂×C₄

#15690288

yes

characteristic

C₈

—

C₈

—

C₈

—

764

—

C₈

#15690276

yes

—

100

C₂²

#20259297

yes

characteristic

101

C₄

—

102

C₄

—

1528

—

103

C₄

—

764

—

104

C₄

—

764

—

105

C₄

C₂×#15690276

yes

characteristic

106

C₄

#20259350

yes

characteristic

107

C₄

#20259351

yes

—

108

C₂

#27132864

yes

characteristic

109

C₂

#27132922

yes

characteristic

110

C₂

#27132939

yes

characteristic

	subgroup	G/H
center	C₂²	#20259297
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#334×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	#121
hypercenter	#334	D₁₉₁
hypocenter	C₁₉₁	#1690
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#20259297

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#334×C₁₉₁

C₂

C₂²×C₁₉₁

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
55. C₁₉₁	#1690	semidirect

H ≅ G/N

type

55. C₁₉₁

#1690

semidirect

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

subgroup	index
#27133043	2
#27132949	2
#27132793	2
#27132408	2
#334×C₁₉₁	2
#1690	191

subgroup

index

#334×C₁₉₁

191

Subgroups — #37572015

Conjugacy classes of subgroups

Complements

Maximal subgroups