Group #37574112 — subgroups

Subgroups — M₃₂×Dic₁₉₁ #37574112

Order 24448.

196 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂×#20259262	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
3	#27132438	C₂	1	2	yes	maximal, complemented
4	C₁₆×Dic₁₉₁	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₉₁⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₉₁	C₄	1	1	yes	characteristic
8	#20259262	C₄	1	4	yes	complemented
9	C₂×C₁₉₁⋊C₁₆	C₂²	1	2	yes	—
10	C₈×Dic₁₉₁	C₄	1	1	yes	characteristic
11	C₈×Dic₁₉₁	C₄	1	1	yes	characteristic
12	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
15	M₃₂×C₁₉₁	C₄	1	4	yes	complemented
16	C₂×C₁₆×C₁₉₁	C₂²	1	2	yes	—
17	C₂²×Dic₁₉₁	C₈	1	2	yes	—
18	C₂×C₁₉₁⋊C₈	C₈	1	4	yes	—
19	C₂×C₁₉₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₉₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₉₁	C₈	1	2	yes	—
22	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
24	C₁₉₁⋊C₁₆	C₂×C₄	1	4	yes	complemented
25	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₉₁	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₁₉₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
30	C₁₆×C₁₉₁	C₂×C₄	1	4	yes	complemented
31	C₂×Dic₁₉₁	—	2	4	no	—
32	C₂×Dic₁₉₁	C₂×C₈	1	2	yes	—
33	C₂×Dic₁₉₁	C₂×C₈	1	2	yes	—
34	C₁₉₁⋊C₈	C₂×C₈	1	2	yes	—
35	C₁₉₁⋊C₈	C₂×C₈	1	2	yes	—
36	C₂³×C₁₉₁	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₁₉₁	C₄²	1	2	yes	—
38	C₂×C₄×C₁₉₁	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×C₁₉₁	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
41	C₈×C₁₉₁	C₄²	1	2	yes	—
42	C₈×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₁₉₁	—	2	2	no	—
45	C₂²×C₁₉₁	C₄×C₈	1	2	yes	—
46	C₂²×C₁₉₁	C₂²×C₈	1	1	yes	characteristic
47	C₄×C₁₉₁	C₄×C₈	1	2	yes	—
48	C₄×C₁₉₁	C₂×C₄²	1	1	yes	characteristic
49	C₄×C₁₉₁	C₂²×C₈	1	1	yes	characteristic
50	Dic₁₉₁	—	2	2	no	—
51	Dic₁₉₁	M₃₂	1	4	yes	direct product factor
52	C₂×C₁₉₁	—	2	2	no	—
53	C₂×C₁₉₁	C₂×C₄×C₈	1	1	yes	characteristic
54	C₂×C₁₉₁	C₂×M₃₂	1	2	yes	—
55	C₁₉₁	C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
56	C₄×M₃₂	—	191	1	no	maximal, Sylow
57	C₂×M₃₂	—	191	1	no	—
58	C₂×M₃₂	D₁₉₁	1	1	yes	characteristic
59	C₂×C₄×C₈	—	191	1	no	—
60	#346	—	191	2	no	—
61	C₄×C₁₆	—	191	2	no	—
62	C₂²×C₈	—	191	1	no	—
63	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
64	C₂×C₄²	—	191	1	no	—
65	M₃₂	—	191	4	no	—
66	M₃₂	Dic₁₉₁	1	4	yes	direct product factor
67	C₂×C₁₆	—	191	2	no	—
68	C₂×C₁₆	C₂×D₁₉₁	1	2	yes	—
69	C₄×C₈	—	191	1	no	—
70	C₄×C₈	—	191	1	no	—
71	C₄×C₈	—	191	1	no	—
72	C₄×C₈	—	191	1	no	—
73	C₂²×C₄	—	191	2	no	—
74	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
75	C₂×C₈	—	191	4	no	—
76	C₂×C₈	—	191	1	no	—
77	C₂×C₈	—	191	1	no	—
78	C₂×C₈	C₄×D₁₉₁	1	1	yes	characteristic
79	C₂×C₈	C₄×D₁₉₁	1	2	yes	—
80	C₂×C₈	C₂×Dic₁₉₁	1	2	yes	—
81	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
82	C₄²	—	191	2	no	—
83	C₄²	—	191	1	no	—
84	C₄²	—	191	1	no	—
85	C₁₆	—	191	4	no	—
86	C₁₆	C₂×Dic₁₉₁	1	4	yes	complemented
87	C₂³	C₈×D₁₉₁	1	1	yes	characteristic
88	C₂×C₄	—	382	4	no	—
89	C₂×C₄	—	191	2	no	—
90	C₂×C₄	—	191	2	no	—
91	C₂×C₄	C₄×Dic₁₉₁	1	2	yes	—
92	C₂×C₄	C₈×D₁₉₁	1	1	yes	characteristic
93	C₂×C₄	C₈×D₁₉₁	1	2	yes	—
94	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
95	C₈	—	191	2	no	—
96	C₈	—	191	2	no	—
97	C₈	C₄×Dic₁₉₁	1	2	yes	—
98	C₈	C₂×C₄×D₁₉₁	1	1	yes	characteristic
99	C₈	C₂²×Dic₁₉₁	1	1	yes	characteristic
100	C₂²	—	2	2	no	—
101	C₂²	C₈×Dic₁₉₁	1	2	yes	—
102	C₂²	C₂×C₈×D₁₉₁	1	1	yes	characteristic
103	C₄	—	382	2	no	—
104	C₄	—	191	4	no	—
105	C₄	C₈×Dic₁₉₁	1	2	yes	—
106	C₄	C₂×C₄×Dic₁₉₁	1	1	yes	characteristic
107	C₄	C₂×C₈×D₁₉₁	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	C₂×C₈×Dic₁₉₁	1	1	yes	characteristic
110	C₂	M₃₂×D₁₉₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259262

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₉₁

C₂

yes

maximal, characteristic

#27132438

C₂

yes

maximal, complemented

C₁₆×Dic₁₉₁

C₂

yes

maximal, complemented

C₂×M₃₂×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₁₉₁

C₄

yes

characteristic

#20259262

C₄

yes

complemented

C₂×C₁₉₁⋊C₁₆

C₂²

yes

—

C₈×Dic₁₉₁

C₄

yes

characteristic

C₈×Dic₁₉₁

C₄

yes

characteristic

C₈×Dic₁₉₁

C₂²

yes

characteristic

C₈×Dic₁₉₁

C₂²

yes

characteristic

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

M₃₂×C₁₉₁

C₄

yes

complemented

C₂×C₁₆×C₁₉₁

C₂²

yes

—

C₂²×Dic₁₉₁

C₈

yes

—

C₂×C₁₉₁⋊C₈

C₈

yes

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

characteristic

C₄×Dic₁₉₁

C₈

yes

—

C₄×Dic₁₉₁

C₂×C₄

yes

characteristic

C₄×Dic₁₉₁

C₂×C₄

yes

characteristic

C₁₉₁⋊C₁₆

C₂×C₄

yes

complemented

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂×C₄

yes

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

—

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₁₆×C₁₉₁

C₂×C₄

yes

complemented

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂×C₈

yes

—

C₂×Dic₁₉₁

C₂×C₈

yes

—

C₁₉₁⋊C₈

C₂×C₈

yes

—

C₁₉₁⋊C₈

C₂×C₈

yes

—

C₂³×C₁₉₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₁

C₄²

yes

—

C₂×C₄×C₁₉₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₁

C₂×C₈

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₈×C₁₉₁

C₄²

yes

—

C₈×C₁₉₁

C₂²×C₄

yes

characteristic

C₈×C₁₉₁

C₂²×C₄

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₄×C₈

yes

—

C₂²×C₁₉₁

C₂²×C₈

yes

characteristic

C₄×C₁₉₁

C₄×C₈

yes

—

C₄×C₁₉₁

C₂×C₄²

yes

characteristic

C₄×C₁₉₁

C₂²×C₈

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

M₃₂

yes

direct product factor

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×C₄×C₈

yes

characteristic

C₂×C₁₉₁

C₂×M₃₂

yes

—

C₁₉₁

C₄×M₃₂

yes

characteristic, complemented, Sylow

C₄×M₃₂

—

191

maximal, Sylow

C₂×M₃₂

—

191

—

C₂×M₃₂

D₁₉₁

yes

characteristic

C₂×C₄×C₈

—

191

—

#346

—

191

—

C₄×C₁₆

—

191

—

C₂²×C₈

—

191

—

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

C₂×C₄²

—

191

—

M₃₂

—

191

—

M₃₂

Dic₁₉₁

yes

direct product factor

C₂×C₁₆

—

191

—

C₂×C₁₆

C₂×D₁₉₁

yes

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₂×C₈

C₄×D₁₉₁

yes

characteristic

C₂×C₈

C₄×D₁₉₁

yes

—

C₂×C₈

C₂×Dic₁₉₁

yes

—

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

C₄²

—

191

—

C₄²

—

191

—

C₄²

—

191

—

C₁₆

—

191

—

C₁₆

C₂×Dic₁₉₁

yes

complemented

C₂³

C₈×D₁₉₁

yes

characteristic

C₂×C₄

—

382

—

C₂×C₄

—

191

—

C₂×C₄

—

191

—

C₂×C₄

C₄×Dic₁₉₁

yes

—

C₂×C₄

C₈×D₁₉₁

yes

characteristic

C₂×C₄

C₈×D₁₉₁

yes

—

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

C₈

—

191

—

C₈

—

191

—

C₈

C₄×Dic₁₉₁

yes

—

C₈

C₂×C₄×D₁₉₁

yes

characteristic

C₈

C₂²×Dic₁₉₁

yes

characteristic

100

C₂²

—

101

C₂²

C₈×Dic₁₉₁

yes

—

102

C₂²

C₂×C₈×D₁₉₁

yes

characteristic

103

C₄

—

382

—

104

C₄

—

191

—

105

C₄

C₈×Dic₁₉₁

yes

—

106

C₄

C₂×C₄×Dic₁₉₁

yes

characteristic

107

C₄

C₂×C₈×D₁₉₁

yes

characteristic

108

C₂

—

109

C₂

C₂×C₈×Dic₁₉₁

yes

characteristic

110

C₂

M₃₂×D₁₉₁

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₉₁
commutator	C₂×C₁₉₁	C₂×C₄×C₈
Fitting	C₂×M₃₂×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×C₈
prime radical	C₂³×C₁₉₁	C₂×C₈
cosocle	C₂×C₈×C₁₉₁	C₂³
hypercenter	C₂×M₃₂	D₁₉₁
hypocenter	C₁₉₁	C₄×M₃₂
LCS	G ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₉₁

commutator

C₂×C₁₉₁

C₂×C₄×C₈

Fitting

C₂×M₃₂×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×C₈

prime radical

C₂³×C₁₉₁

C₂×C₈

cosocle

C₂×C₈×C₁₉₁

C₂³

hypercenter

C₂×M₃₂

D₁₉₁

hypocenter

C₁₉₁

C₄×M₃₂

LCS

G ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #27132438	C₂	semidirect
4. C₁₆×Dic₁₉₁	C₂	semidirect
8. #20259262	C₄	semidirect
15. M₃₂×C₁₉₁	C₄	semidirect
24. C₁₉₁⋊C₁₆	C₂×C₄	semidirect
30. C₁₆×C₁₉₁	C₂×C₄	semidirect
51. Dic₁₉₁	M₃₂	direct (N × H)
55. C₁₉₁	C₄×M₃₂	semidirect
66. M₃₂	Dic₁₉₁	direct (N × H)
86. C₁₆	C₂×Dic₁₉₁	semidirect

H ≅ G/N

type

3. #27132438

C₂

semidirect

4. C₁₆×Dic₁₉₁

C₂

semidirect

8. #20259262

C₄

semidirect

15. M₃₂×C₁₉₁

C₄

semidirect

24. C₁₉₁⋊C₁₆

C₂×C₄

semidirect

30. C₁₆×C₁₉₁

C₂×C₄

semidirect

51. Dic₁₉₁

M₃₂

direct (N × H)

55. C₁₉₁

C₄×M₃₂

semidirect

66. M₃₂

Dic₁₉₁

direct (N × H)

86. C₁₆

C₂×Dic₁₉₁

semidirect

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

subgroup	index
C₂×#20259262	2
C₂×C₈×Dic₁₉₁	2
#27132438	2
C₁₆×Dic₁₉₁	2
C₂×M₃₂×C₁₉₁	2
C₄×M₃₂	191

subgroup

index

C₂×#20259262

C₂×C₈×Dic₁₉₁

#27132438

C₁₆×Dic₁₉₁

C₂×M₃₂×C₁₉₁

C₄×M₃₂

191

Subgroups — M₃₂×Dic₁₉₁ #37574112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M32×Dic191 #37574112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂×Dic₁₉₁ #37574112