Group #37571658 — subgroups

Subgroups — #37571658

Order 24448.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27133393	C₂	1	1	yes	maximal, characteristic
2	#27132955	C₂	1	2	yes	maximal
3	#27132412	C₂	1	1	yes	maximal, characteristic
4	#27132395	C₂	1	2	yes	maximal
5	#494×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	#20259321	—	2	2	no	—
7	#20259320	—	2	2	no	—
8	C₄×Dic₃₈₂	C₂²	1	2	yes	—
9	#20259287	—	2	2	no	—
10	#20259255	—	2	2	no	—
11	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	—
12	#20259245	—	2	2	no	—
13	C₂×Q₁₆×C₁₉₁	—	2	2	no	—
14	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₈₂	—	2	2	no	—
18	C₂×Dic₃₈₂	D₄	1	2	yes	—
19	C₂×C₁₉₁⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₉₁	D₄	1	2	yes	—
21	Dic₁₉₁⋊C₄	—	2	4	no	—
22	Dic₁₉₁⋊C₄	—	2	2	no	—
23	C₄×Dic₁₉₁	—	2	2	no	—
24	C₄×Dic₁₉₁	—	2	2	no	—
25	C₂×Q₈×C₁₉₁	—	2	1	no	—
26	C₂×Q₈×C₁₉₁	D₄	1	2	yes	—
27	Q₁₆×C₁₉₁	—	4	4	no	—
28	C₂×C₈×C₁₉₁	—	2	1	no	—
29	C₄⋊C₄×C₁₉₁	—	2	2	no	—
30	C₄⋊C₄×C₁₉₁	—	2	2	no	—
31	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
32	Dic₃₈₂	—	2	2	no	—
33	Dic₃₈₂	—	8	2	no	—
34	Dic₃₈₂	Q₁₆	1	4	yes	complemented
35	C₂×Dic₁₉₁	—	2	2	no	—
36	C₂×Dic₁₉₁	—	2	2	no	—
37	C₁₉₁⋊C₈	—	4	2	no	—
38	Q₈×C₁₉₁	—	4	2	no	—
39	Q₈×C₁₉₁	—	4	2	no	—
40	C₂×C₄×C₁₉₁	—	2	2	no	—
41	C₂×C₄×C₁₉₁	—	4	1	no	—
42	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₉₁	—	2	2	no	—
45	C₂²×C₁₉₁	#120	1	1	yes	characteristic
46	C₄×C₁₉₁	—	2	1	no	—
47	C₄×C₁₉₁	—	4	2	no	—
48	C₄×C₁₉₁	—	8	1	no	—
49	C₄×C₁₉₁	C₂×Q₁₆	1	2	yes	—
50	C₄×C₁₉₁	#137	1	2	yes	—
51	Dic₁₉₁	—	2	4	no	—
52	Dic₁₉₁	—	4	2	no	—
53	C₂×C₁₉₁	#451	1	2	yes	—
54	C₂×C₁₉₁	#453	1	1	yes	characteristic
55	C₁₉₁	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	191	1	no	maximal, Sylow
57	Q₈²	—	191	1	no	—
58	#494	D₁₉₁	1	1	yes	characteristic
59	#477	—	191	2	no	—
60	#333	—	191	1	no	—
61	Q₈⋊C₈	—	191	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	382	2	no	—
64	C₄⋊Q₈	—	382	2	no	—
65	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
67	C₄×Q₈	—	382	2	no	—
68	C₄×Q₈	—	191	2	no	—
69	#107	—	382	2	no	—
70	C₄⋊C₈	—	191	2	no	—
71	#103	—	382	2	no	—
72	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	382	2	no	—
75	C₂×Q₈	—	191	2	no	—
76	C₂×Q₈	C₂²⋊D₁₉₁	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	382	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	382	4	no	—
83	C₄⋊C₄	—	382	2	no	—
84	C₄⋊C₄	—	191	2	no	—
85	C₄²	—	382	2	no	—
86	C₄²	—	382	2	no	—
87	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	382	2	no	—
91	Q₈	—	191	4	no	—
92	Q₈	—	1528	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	382	2	no	—
96	C₂×C₄	—	382	2	no	—
97	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	764	2	no	—
101	C₂²	#20259290	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	382	4	no	—
106	C₄	—	764	2	no	—
107	C₄	Q₁₆×D₁₉₁	1	2	yes	—
108	C₄	#20259354	1	2	yes	—
109	C₂	#27132817	1	2	yes	—
110	C₂	#27132830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133393

C₂

yes

maximal, characteristic

#27132955

C₂

yes

maximal

#27132412

C₂

yes

maximal, characteristic

#27132395

C₂

yes

maximal

#494×C₁₉₁

C₂

yes

maximal, characteristic

#20259321

—

#20259320

—

C₄×Dic₃₈₂

C₂²

yes

—

#20259287

—

#20259255

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

—

#20259245

—

C₂×Q₁₆×C₁₉₁

—

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

D₄

yes

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

D₄

yes

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

D₄

yes

—

Q₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

Q₁₆

yes

complemented

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

#120

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×Q₁₆

yes

—

C₄×C₁₉₁

#137

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

#451

yes

—

C₂×C₁₉₁

#453

yes

characteristic

C₁₉₁

#1633

yes

characteristic, complemented, Sylow

#1633

—

191

maximal, Sylow

Q₈²

—

191

—

#494

D₁₉₁

yes

characteristic

#477

—

191

—

#333

—

191

—

Q₈⋊C₈

—

191

—

C₂×Q₁₆

—

C₄⋊Q₈

—

382

—

C₄⋊Q₈

—

382

—

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

C₄×Q₈

—

382

—

C₄×Q₈

—

191

—

#107

—

382

—

C₄⋊C₈

—

191

—

#103

—

382

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂×Q₈

C₂²⋊D₁₉₁

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

382

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

C₄²

—

382

—

C₄²

—

382

—

C₄²

C₂²×D₁₉₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

382

—

Q₈

—

191

—

Q₈

—

1528

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

D₄×D₁₉₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

C₈

—

100

C₈

—

764

—

101

C₂²

#20259290

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

382

—

106

C₄

—

764

—

107

C₄

Q₁₆×D₁₉₁

yes

—

108

C₄

#20259354

yes

—

109

C₂

#27132817

yes

—

110

C₂

#27132830

yes

characteristic

	subgroup	G/H
center	C₂²	#20259290
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#494×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	#120
hypercenter	#494	D₁₉₁
hypocenter	C₁₉₁	#1633
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#20259290

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#494×C₁₉₁

C₂

C₂²×C₁₉₁

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₃₈₂	Q₁₆	semidirect
55. C₁₉₁	#1633	semidirect

H ≅ G/N

type

34. Dic₃₈₂

Q₁₆

semidirect

55. C₁₉₁

#1633

semidirect

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

subgroup	index
#27133393	2
#27132955	2
#27132412	2
#27132395	2
#494×C₁₉₁	2
#1633	191

subgroup

index

#494×C₁₉₁

191

Subgroups — #37571658

Conjugacy classes of subgroups

Complements

Maximal subgroups