Group #37571416 — subgroups

Subgroups — #37571416

Order 24448.

152 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 65 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	#27132790	C₂	1	2	yes	maximal
2	C₄×Dic₇₆₄	C₂	1	1	yes	maximal, characteristic
3	#27132393	C₂	1	2	yes	maximal
4	C₈×C₁₉₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#335×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₆₄	C₄	1	1	yes	characteristic
7	C₄×Dic₃₈₂	C₂²	1	2	yes	—
8	#20259256	C₄	1	1	yes	characteristic
9	Dic₁₉₁⋊C₈	—	2	2	no	—
10	#20259247	C₄	1	2	yes	—
11	#20259232	—	2	2	no	—
12	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₉₁	C₂²	1	2	yes	—
15	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₈₂	C₂×C₄	1	2	yes	—
17	Dic₇₆₄	—	2	2	no	—
18	C₂×C₁₉₁⋊C₈	—	2	1	no	—
19	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
20	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
21	C₄⋊Dic₁₉₁	C₂×C₄	1	2	yes	—
22	Dic₁₉₁⋊C₄	—	2	2	no	—
23	C₄×Dic₁₉₁	—	2	2	no	—
24	C₂×C₈×C₁₉₁	—	2	2	no	—
25	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
28	Dic₃₈₂	—	2	2	no	—
29	Dic₃₈₂	—	2	2	no	—
30	C₂×Dic₁₉₁	—	2	2	no	—
31	C₂×Dic₁₉₁	—	2	2	no	—
32	C₁₉₁⋊C₈	—	2	2	no	—
33	C₁₉₁⋊C₈	—	2	1	no	—
34	C₁₉₁⋊C₈	Q₁₆	1	2	yes	complemented
35	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
38	C₈×C₁₉₁	—	2	1	no	—
39	C₈×C₁₉₁	—	4	2	no	—
40	C₈×C₁₉₁	M₁₆	1	2	yes	—
41	C₂²×C₁₉₁	C₄×D₄	1	1	yes	characteristic
42	C₄×C₁₉₁	—	2	1	no	—
43	C₄×C₁₉₁	C₂×M₁₆	1	1	yes	characteristic
44	C₄×C₁₉₁	#131	1	1	yes	characteristic
45	C₄×C₁₉₁	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₁₉₁	#135	1	1	yes	characteristic
47	Dic₁₉₁	—	4	2	no	—
48	Dic₁₉₁	—	4	2	no	—
49	C₂×C₁₉₁	#436	1	1	yes	characteristic
50	C₂×C₁₉₁	C₄×Q₁₆	1	1	yes	characteristic
51	C₂×C₁₉₁	#443	1	1	yes	characteristic
52	C₁₉₁	#1591	1	1	yes	characteristic, complemented, Sylow
53	#1591	—	191	1	no	maximal, Sylow
54	#446	—	191	2	no	—
55	C₄×Q₁₆	—	191	1	no	—
56	#335	D₁₉₁	1	1	yes	characteristic
57	Q₈⋊C₈	—	191	2	no	—
58	C₈²	—	191	1	no	—
59	C₂×Q₁₆	—	191	1	no	—
60	C₄×Q₈	—	191	2	no	—
61	#107	—	191	1	no	—
62	C₄⋊C₈	—	382	2	no	—
63	C₄⋊C₈	C₂×D₁₉₁	1	2	yes	—
64	#103	—	191	2	no	—
65	#97	—	382	2	no	—
66	C₄×C₈	—	191	1	no	—
67	C₄×C₈	—	191	1	no	—
68	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
69	C₂×Q₈	—	191	2	no	—
70	Q₁₆	—	382	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	382	1	no	—
73	C₂×C₈	—	191	1	no	—
74	C₂×C₈	—	191	1	no	—
75	C₂×C₈	C₄×D₁₉₁	1	1	yes	characteristic
76	C₂×C₈	C₄×D₁₉₁	1	1	yes	characteristic
77	C₄⋊C₄	—	382	2	no	—
78	C₄⋊C₄	—	191	2	no	—
79	C₄²	—	382	2	no	—
80	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
81	Q₈	—	382	2	no	—
82	Q₈	—	382	2	no	—
83	C₂×C₄	—	382	2	no	—
84	C₂×C₄	—	382	2	no	—
85	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
86	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
87	C₂×C₄	#15690288	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	4	2	no	—
90	C₈	—	382	2	no	—
91	C₈	—	382	1	no	—
92	C₈	—	191	2	no	—
93	C₈	#15690270	1	2	yes	—
94	C₂²	#20259284	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	764	2	no	—
97	C₄	—	764	2	no	—
98	C₄	C₂×#15690270	1	1	yes	characteristic
99	C₄	#20259330	1	1	yes	characteristic
100	C₄	Q₁₆×D₁₉₁	1	1	yes	characteristic
101	C₄	#20259341	1	1	yes	characteristic
102	C₂	#27132740	1	1	yes	characteristic
103	C₂	#27132757	1	1	yes	characteristic
104	C₂	#27132779	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132790

C₂

yes

maximal

C₄×Dic₇₆₄

C₂

yes

maximal, characteristic

#27132393

C₂

yes

maximal

C₈×C₁₉₁⋊C₈

C₂

yes

maximal, characteristic

#335×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

C₄

yes

characteristic

C₄×Dic₃₈₂

C₂²

yes

—

#20259256

C₄

yes

characteristic

Dic₁₉₁⋊C₈

—

#20259247

C₄

yes

—

#20259232

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

C₂×C₄

yes

—

Dic₇₆₄

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

D₄

yes

characteristic

C₂×C₁₉₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

Q₁₆

yes

complemented

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₈×C₁₉₁

M₁₆

yes

—

C₂²×C₁₉₁

C₄×D₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×M₁₆

yes

characteristic

C₄×C₁₉₁

#131

yes

characteristic

C₄×C₁₉₁

C₂×Q₁₆

yes

characteristic

C₄×C₁₉₁

#135

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

#436

yes

characteristic

C₂×C₁₉₁

C₄×Q₁₆

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1591

—

191

maximal, Sylow

#446

—

191

—

C₄×Q₁₆

—

191

—

#335

D₁₉₁

yes

characteristic

Q₈⋊C₈

—

191

—

C₈²

—

191

—

C₂×Q₁₆

—

191

—

C₄×Q₈

—

191

—

#107

—

191

—

C₄⋊C₈

—

382

—

C₄⋊C₈

C₂×D₁₉₁

yes

—

#103

—

191

—

#97

—

382

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

191

—

Q₁₆

—

382

—

C₂×C₈

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₂×C₈

C₄×D₁₉₁

yes

characteristic

C₂×C₈

C₄×D₁₉₁

yes

characteristic

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

C₄²

—

382

—

C₄²

C₂²×D₁₉₁

yes

characteristic

Q₈

—

382

—

Q₈

—

382

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

C₂×C₄

D₄×D₁₉₁

yes

characteristic

C₂×C₄

#15690288

yes

characteristic

C₈

—

C₈

—

C₈

—

382

—

C₈

—

382

—

C₈

—

191

—

C₈

#15690270

yes

—

C₂²

#20259284

yes

characteristic

C₄

—

C₄

—

764

—

C₄

—

764

—

C₄

yes

C₄

yes

100

C₄

Q₁₆×D₁₉₁

yes

characteristic

101

C₄

#20259341

yes

characteristic

102

C₂

#27132740

yes

characteristic

103

C₂

#27132757

yes

characteristic

104

C₂

#27132779

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#335×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	C₄×D₄
cosocle	C₄²×C₁₉₁	C₂³
hypercenter	#335	D₁₉₁
hypocenter	C₁₉₁	#1591
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#335×C₁₉₁

C₂

socle, prime radical

C₂²×C₁₉₁

C₄×D₄

cosocle

C₄²×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
34. C₁₉₁⋊C₈	Q₁₆	semidirect
52. C₁₉₁	#1591	semidirect

H ≅ G/N

type

34. C₁₉₁⋊C₈

Q₁₆

semidirect

52. C₁₉₁

#1591

semidirect

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

subgroup	index
#27132790	2
C₄×Dic₇₆₄	2
#27132393	2
C₈×C₁₉₁⋊C₈	2
#335×C₁₉₁	2
#1591	191

subgroup

index

#27132790

C₄×Dic₇₆₄

#27132393

C₈×C₁₉₁⋊C₈

#335×C₁₉₁

#1591

191

Subgroups — #37571416

Conjugacy classes of subgroups

Complements

Maximal subgroups