Group #37571290 — subgroups

Subgroups — #37571290

Order 24448.

168 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 89 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	#27132687	C₂	1	1	yes	maximal, characteristic
3	#27132413	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊C₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#15690271	—	2	1	no	—
6	C₄⋊(C₁₉₁⋊C₈)	—	2	1	no	—
7	C₄⋊(C₁₉₁⋊C₈)	C₄	1	2	yes	—
8	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
9	C₂²⋊(C₁₉₁⋊C₈)	C₄	1	2	yes	—
10	C₄×C₁₉₁⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₈×C₁₉₁	—	2	1	no	—
12	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₉₁	—	2	1	no	—
14	C₄⋊C₈×C₁₉₁	C₂²	1	2	yes	—
15	#15690271	—	4	2	no	—
16	C₂×C₁₉₁⋊C₈	—	2	2	no	—
17	C₂×C₁₉₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉₁	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₁₉₁	—	2	2	no	—
22	C₂×C₈×C₁₉₁	—	4	2	no	—
23	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
25	C₄²×C₁₉₁	Q₈	1	1	yes	characteristic
26	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
27	C₁₉₁⋊C₈	—	2	4	no	—
28	C₁₉₁⋊C₈	—	4	2	no	—
29	C₂³×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₉₁	—	2	2	no	—
31	C₂×C₄×C₁₉₁	—	4	1	no	—
32	C₂×C₄×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
36	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
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	—	4	2	no	—
41	C₂²×C₁₉₁	—	2	1	no	—
42	C₂²×C₁₉₁	—	2	1	no	—
43	C₂²×C₁₉₁	#106	1	2	yes	—
44	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₁₉₁	—	2	1	no	—
46	C₄×C₁₉₁	—	2	1	no	—
47	C₄×C₁₉₁	—	2	1	no	—
48	C₄×C₁₉₁	#106	1	2	yes	—
49	C₄×C₁₉₁	#131	1	2	yes	—
50	C₄×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×C₁₉₁	—	2	2	no	—
53	C₂×C₁₉₁	#424	1	1	yes	characteristic
54	C₂×C₁₉₁	C₂×#106	1	1	yes	characteristic
55	C₂×C₁₉₁	#430	1	1	yes	characteristic
56	C₁₉₁	#1570	1	1	yes	characteristic, complemented, Sylow
57	#1570	—	191	1	no	maximal, Sylow
58	#431	—	191	1	no	—
59	#423	—	191	1	no	—
60	C₂×C₄⋊C₈	D₁₉₁	1	1	yes	characteristic
61	#334	—	191	4	no	—
62	C₂×M₁₆	—	382	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	382	1	no	—
67	C₄⋊C₈	—	191	2	no	—
68	C₄⋊C₈	—	191	2	no	—
69	C₄⋊C₈	C₂×D₁₉₁	1	2	yes	—
70	C₂²⋊C₈	—	191	2	no	—
71	C₄×C₈	—	191	2	no	—
72	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
74	C₂²×C₄	Dic₃₈₂	1	1	yes	characteristic
75	M₁₆	—	764	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	382	2	no	—
79	C₂×C₈	—	191	4	no	—
80	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
81	C₄²	C₂²⋊D₁₉₁	1	1	yes	characteristic
82	C₄²	Dic₃₈₂	1	1	yes	characteristic
83	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
84	C₂³	Dic₁₉₁⋊C₄	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	Dic₁₉₁⋊C₄	1	1	yes	characteristic
88	C₂×C₄	Dic₁₉₁⋊C₄	1	1	yes	characteristic
89	C₂×C₄	Dic₁₉₁⋊C₄	1	1	yes	characteristic
90	C₂×C₄	#15690274	1	2	yes	—
91	C₂×C₄	#15690275	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	382	4	no	—
97	C₈	—	764	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#20259253	1	2	yes	—
101	C₂²	C₂×Dic₁₉₁⋊C₄	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#20259253	1	2	yes	—
106	C₄	#20259328	1	2	yes	—
107	C₄	C₂×#15690274	1	1	yes	characteristic
108	C₄	C₂×#15690275	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#27132689	1	1	yes	characteristic
111	C₂	C₂×#20259253	1	1	yes	characteristic
112	C₂	#27132711	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132716

C₂

yes

maximal, characteristic

#27132687

C₂

yes

maximal, characteristic

#27132413

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

—

C₄⋊(C₁₉₁⋊C₈)

—

C₄⋊(C₁₉₁⋊C₈)

C₄

yes

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₉₁⋊C₈)

C₄

yes

—

C₄×C₁₉₁⋊C₈

C₂²

yes

—

C₂²×C₈×C₁₉₁

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

—

C₄⋊C₈×C₁₉₁

C₂²

yes

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

Q₈

yes

characteristic

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

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

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#106

yes

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#106

yes

—

C₄×C₁₉₁

#131

yes

—

C₄×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₄×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1570

—

191

maximal, Sylow

#431

—

191

—

#423

—

191

—

C₂×C₄⋊C₈

D₁₉₁

yes

characteristic

#334

—

191

—

C₂×M₁₆

—

382

—

C₂²×C₈

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

382

—

C₄⋊C₈

—

191

—

C₄⋊C₈

—

191

—

C₄⋊C₈

C₂×D₁₉₁

yes

—

C₂²⋊C₈

—

191

—

C₄×C₈

—

191

—

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

C₂²×C₄

Dic₃₈₂

yes

characteristic

M₁₆

—

764

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₄²

C₄×D₁₉₁

yes

characteristic

C₄²

C₂²⋊D₁₉₁

yes

characteristic

C₄²

Dic₃₈₂

yes

characteristic

C₄²

C₂²×D₁₉₁

yes

characteristic

C₂³

Dic₁₉₁⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₁₉₁⋊C₄

yes

characteristic

C₂×C₄

Dic₁₉₁⋊C₄

yes

characteristic

C₂×C₄

Dic₁₉₁⋊C₄

yes

characteristic

C₂×C₄

#15690274

yes

—

C₂×C₄

#15690275

yes

—

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

C₈

—

C₈

—

382

—

C₈

—

764

—

C₂²

—

C₂²

—

100

C₂²

#20259253

yes

—

101

C₂²

C₂×Dic₁₉₁⋊C₄

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#20259253

yes

—

106

C₄

#20259328

yes

—

107

C₄

C₂×#15690274

yes

characteristic

108

C₄

C₂×#15690275

yes

characteristic

109

C₂

—

110

C₂

#27132689

yes

characteristic

111

C₂

C₂×#20259253

yes

characteristic

112

C₂

#27132711

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₁
Fitting	C₂×C₄⋊C₈×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₄⋊C₄
prime radical	C₂³×C₁₉₁	C₄⋊C₄
cosocle	C₄²×C₁₉₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₉₁
hypocenter	C₁₉₁	#1570
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₁

Fitting

C₂×C₄⋊C₈×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₄⋊C₄

prime radical

C₂³×C₁₉₁

C₄⋊C₄

cosocle

C₄²×C₁₉₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₉₁

hypocenter

C₁₉₁

#1570

LCS

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #27132413	C₂	semidirect
56. C₁₉₁	#1570	semidirect

H ≅ G/N

type

3. #27132413

C₂

semidirect

56. C₁₉₁

#1570

semidirect

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

subgroup	index
#27132716	2
#27132687	2
#27132413	2
C₂×C₄⋊C₈×C₁₉₁	2
#1570	191

subgroup

index

C₂×C₄⋊C₈×C₁₉₁

191

Subgroups — #37571290

Conjugacy classes of subgroups

Complements

Maximal subgroups