Group #37571300 — subgroups

Subgroups — #37571300

Order 24448.

150 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 81 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	#27132720	C₂	1	1	yes	maximal, characteristic
2	#27132716	C₂	1	1	yes	maximal, characteristic
3	#27132416	C₂	1	2	yes	maximal, complemented
4	#27132414	C₂	1	2	yes	maximal, complemented
5	#431×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
7	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₁₉₁⋊C₈)	—	2	1	no	—
9	C₂²⋊(C₁₉₁⋊C₈)	—	2	1	no	—
10	#20259233	—	2	1	no	—
11	C₄×C₁₉₁⋊C₈	—	2	1	no	—
12	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₉₁	C₄	1	2	yes	—
14	C₂²⋊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	C₂×C₁₉₁⋊C₈	—	2	2	no	—
18	C₂×C₁₉₁⋊C₈	—	2	2	no	—
19	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₉₁	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₁₉₁	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₉₁	C₂×C₄	1	2	yes	—
24	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
26	C₄²×C₁₉₁	Q₈	1	1	yes	characteristic
27	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
28	C₁₉₁⋊C₈	—	4	2	no	—
29	C₁₉₁⋊C₈	—	4	2	no	—
30	C₂³×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₉₁	—	2	2	no	—
32	C₂×C₄×C₁₉₁	—	2	2	no	—
33	C₂×C₄×C₁₉₁	—	4	1	no	—
34	C₂×C₄×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₁₉₁	—	2	2	no	—
41	C₈×C₁₉₁	—	2	2	no	—
42	C₂²×C₁₉₁	—	2	1	no	—
43	C₂²×C₁₉₁	—	2	1	no	—
44	C₂²×C₁₉₁	—	2	1	no	—
45	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₁₉₁	—	2	1	no	—
47	C₄×C₁₉₁	—	2	1	no	—
48	C₄×C₁₉₁	—	2	1	no	—
49	C₄×C₁₉₁	—	2	1	no	—
50	C₄×C₁₉₁	#131	1	2	yes	—
51	C₄×C₁₉₁	#135	1	2	yes	—
52	C₂×C₁₉₁	—	4	1	no	—
53	C₂×C₁₉₁	#424	1	1	yes	characteristic
54	C₂×C₁₉₁	#427	1	1	yes	characteristic
55	C₂×C₁₉₁	#430	1	1	yes	characteristic
56	C₁₉₁	#1572	1	1	yes	characteristic, complemented, Sylow
57	#1572	—	191	1	no	maximal, Sylow
58	#432	—	191	1	no	—
59	#431	—	191	1	no	—
60	#431	D₁₉₁	1	1	yes	characteristic
61	#335	—	191	2	no	—
62	#334	—	191	2	no	—
63	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
64	C₄⋊C₈	—	191	2	no	—
65	C₄⋊C₈	—	191	2	no	—
66	C₄⋊C₈	Dic₁₉₁	1	2	yes	—
67	C₂²⋊C₈	—	382	1	no	—
68	C₂²⋊C₈	—	382	1	no	—
69	C₂²⋊C₈	Dic₁₉₁	1	2	yes	—
70	#97	—	382	1	no	—
71	C₄×C₈	—	382	1	no	—
72	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
73	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
75	C₂²×C₄	D₇₆₄	1	1	yes	characteristic
76	C₂²×C₄	Dic₃₈₂	1	1	yes	characteristic
77	C₂×C₈	—	382	2	no	—
78	C₂×C₈	—	382	2	no	—
79	C₂×C₈	C₂×Dic₁₉₁	1	2	yes	—
80	C₂×C₈	C₂×Dic₁₉₁	1	2	yes	—
81	C₄²	C₂×Dic₁₉₁	1	1	yes	characteristic
82	C₄²	D₇₆₄	1	1	yes	characteristic
83	C₄²	Dic₃₈₂	1	1	yes	characteristic
84	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
85	C₂³	C₄⋊Dic₁₉₁	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	C₄⋊Dic₁₉₁	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₁₉₁	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₁₉₁	1	1	yes	characteristic
92	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
93	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
95	C₈	—	2	2	no	—
96	C₈	—	2	2	no	—
97	C₈	—	764	2	no	—
98	C₈	—	764	2	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	C₂×C₄⋊Dic₁₉₁	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	#20259329	1	2	yes	—
108	C₄	#20259344	1	2	yes	—
109	C₂	—	4	1	no	—
110	C₂	#27132690	1	1	yes	characteristic
111	C₂	#27132701	1	1	yes	characteristic
112	C₂	#27132712	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132720

C₂

yes

maximal, characteristic

#27132716

C₂

yes

maximal, characteristic

#27132416

C₂

yes

maximal, complemented

#27132414

C₂

yes

maximal, complemented

#431×C₁₉₁

C₂

yes

maximal, characteristic

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₉₁⋊C₈)

—

C₂²⋊(C₁₉₁⋊C₈)

—

#20259233

—

C₄×C₁₉₁⋊C₈

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₄

yes

—

C₂²⋊C₈×C₁₉₁

C₄

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

Q₈

yes

characteristic

C₂×C₈×C₁₉₁

C₂×C₄

yes

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

D₄

yes

characteristic

C₄²×C₁₉₁

Q₈

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#131

yes

—

C₄×C₁₉₁

#135

yes

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1572

—

191

maximal, Sylow

#432

—

191

—

#431

—

191

—

#431

D₁₉₁

yes

characteristic

#335

—

191

—

#334

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

191

—

C₄⋊C₈

—

191

—

C₄⋊C₈

Dic₁₉₁

yes

—

C₂²⋊C₈

—

382

—

C₂²⋊C₈

—

382

—

C₂²⋊C₈

Dic₁₉₁

yes

—

#97

—

382

—

C₄×C₈

—

382

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₉₁

yes

characteristic

C₂²×C₄

D₇₆₄

yes

characteristic

C₂²×C₄

Dic₃₈₂

yes

characteristic

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

C₂×Dic₁₉₁

yes

—

C₂×C₈

C₂×Dic₁₉₁

yes

—

C₄²

C₂×Dic₁₉₁

yes

characteristic

C₄²

D₇₆₄

yes

characteristic

C₄²

Dic₃₈₂

yes

characteristic

C₄²

C₂²×D₁₉₁

yes

characteristic

C₂³

C₄⋊Dic₁₉₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₉₁

yes

characteristic

C₂×C₄

C₄⋊Dic₁₉₁

yes

characteristic

C₂×C₄

C₄⋊Dic₁₉₁

yes

characteristic

C₂×C₄

C₂²×Dic₁₉₁

yes

characteristic

C₂×C₄

C₂×D₇₆₄

yes

characteristic

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

C₈

—

C₈

—

C₈

—

764

—

C₈

—

764

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×C₄⋊Dic₁₉₁

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#20259329

yes

—

108

C₄

#20259344

yes

—

109

C₂

—

110

C₂

#27132690

yes

characteristic

111

C₂

#27132701

yes

characteristic

112

C₂

#27132712

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₆₄
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#431×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₄⋊C₄
prime radical	C₂³×C₁₉₁	C₄⋊C₄
cosocle	C₄²×C₁₉₁	C₂³
hypercenter	#431	D₁₉₁
hypocenter	C₁₉₁	#1572
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₇₆₄

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#431×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₄⋊C₄

prime radical

C₂³×C₁₉₁

C₄⋊C₄

cosocle

C₄²×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #27132416	C₂	semidirect
4. #27132414	C₂	semidirect
56. C₁₉₁	#1572	semidirect

H ≅ G/N

type

3. #27132416

C₂

semidirect

4. #27132414

C₂

semidirect

56. C₁₉₁

#1572

semidirect

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

subgroup	index
#27132720	2
#27132716	2
#27132416	2
#27132414	2
#431×C₁₉₁	2
#1572	191

subgroup

index

#431×C₁₉₁

191

Subgroups — #37571300

Conjugacy classes of subgroups

Complements

Maximal subgroups