Group #37573683 — subgroups

Subgroups — #37573683

Order 24448.

466 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 71 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	#27132518	C₂	1	3	yes	maximal
2	#27132430	C₂	1	3	yes	maximal, complemented
3	#543×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₁₉₁	C₂²	1	1	yes	characteristic
5	C₂×C₂²⋊Dic₁₉₁	C₂²	1	3	yes	—
6	C₂×(C₂×D₁₉₁)⋊C₄	—	2	6	no	—
7	C₂²⋊Q₈×C₁₉₁	—	2	6	no	—
8	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	3	yes	—
9	C₂³×D₁₉₁	—	2	3	no	—
10	C₂³×D₁₉₁	—	2	3	no	—
11	C₂³×D₁₉₁	—	4	4	no	—
12	C₂³×D₁₉₁	Q₈	1	2	yes	complemented
13	C₂²×Dic₁₉₁	—	2	3	no	—
14	C₂²⋊Dic₁₉₁	—	2	6	no	—
15	(C₂×D₁₉₁)⋊C₄	—	4	12	no	—
16	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
17	C₂×Q₈×C₁₉₁	—	2	2	no	—
18	C₂²×C₄×C₁₉₁	D₄	1	6	yes	—
19	C₄⋊C₄×C₁₉₁	—	2	6	no	—
20	C₂²⋊C₄×C₁₉₁	—	2	6	no	—
21	C₂²×D₁₉₁	—	2	12	no	—
22	C₂²×D₁₉₁	—	2	6	no	—
23	C₂²×D₁₉₁	—	4	4	no	—
24	C₂²×D₁₉₁	—	4	12	no	—
25	C₂²×D₁₉₁	—	4	6	no	—
26	C₂²×D₁₉₁	—	4	1	no	—
27	C₂²×D₁₉₁	—	4	3	no	—
28	C₂×Dic₁₉₁	—	4	6	no	—
29	C₂×Dic₁₉₁	—	4	3	no	—
30	C₂³×C₁₉₁	—	4	2	no	—
31	C₂³×C₁₉₁	C₂×D₄	1	3	yes	—
32	C₂³×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
33	C₂³×C₁₉₁	#41	1	3	yes	—
34	Q₈×C₁₉₁	—	8	2	no	—
35	C₂×C₄×C₁₉₁	—	2	6	no	—
36	C₂×C₄×C₁₉₁	—	4	6	no	—
37	C₂²×C₁₉₁	—	2	6	no	—
38	C₂²×C₁₉₁	—	4	1	no	—
39	C₂²×C₁₉₁	—	4	3	no	—
40	C₂²×C₁₉₁	#120	1	1	yes	characteristic
41	C₂²×C₁₉₁	C₂²⋊Q₈	1	3	yes	—
42	C₂²×C₁₉₁	#124	1	3	yes	—
43	C₂×D₁₉₁	—	2	12	no	—
44	C₂×D₁₉₁	—	4	12	no	—
45	C₂×D₁₉₁	—	4	6	no	—
46	C₂×D₁₉₁	—	4	2	no	—
47	C₂×D₁₉₁	—	4	4	no	—
48	C₄×C₁₉₁	—	4	6	no	—
49	Dic₁₉₁	—	8	3	no	—
50	C₂×C₁₉₁	—	2	3	no	—
51	C₂×C₁₉₁	—	2	3	no	—
52	C₂×C₁₉₁	#393	1	1	yes	characteristic
53	C₂×C₁₉₁	B(4,𝔽₂)	1	2	yes	—
54	D₁₉₁	—	4	4	no	—
55	C₁₉₁	#2029	1	1	yes	characteristic, complemented, Sylow
56	#2029	—	191	1	no	maximal, Sylow
57	#543	D₁₉₁	1	1	yes	characteristic, complemented
58	#379	—	191	3	no	—
59	#342	—	191	3	no	—
60	C₂⁵	—	191	1	no	—
61	C₂²⋊Q₈	—	2	6	no	—
62	C₂×C₂²⋊C₄	—	382	6	no	—
63	C₂×C₂²⋊C₄	—	191	3	no	—
64	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	3	yes	—
65	C₂⁴	—	382	3	no	—
66	C₂⁴	—	382	3	no	—
67	C₂⁴	—	191	2	no	—
68	C₂⁴	—	764	4	no	—
69	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
70	C₂×Q₈	—	2	2	no	—
71	C₂²×C₄	—	382	3	no	—
72	C₂²×C₄	C₂²⋊D₁₉₁	1	6	yes	—
73	C₄⋊C₄	—	2	6	no	—
74	C₂²⋊C₄	—	2	6	no	—
75	C₂²⋊C₄	—	382	6	no	—
76	C₂²⋊C₄	—	764	12	no	—
77	C₂³	—	4	2	no	—
78	C₂³	—	382	12	no	—
79	C₂³	—	382	6	no	—
80	C₂³	—	764	6	no	—
81	C₂³	—	764	4	no	—
82	C₂³	—	764	12	no	—
83	C₂³	—	764	1	no	—
84	C₂³	—	764	3	no	—
85	C₂³	C₂×C₂²⋊D₁₉₁	1	3	yes	—
86	C₂³	Q₈×D₁₉₁	1	1	yes	characteristic
87	C₂³	#15690288	1	3	yes	—
88	Q₈	—	8	2	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	4	6	no	—
91	C₂×C₄	—	764	6	no	—
92	C₂×C₄	—	764	3	no	—
93	C₂²	—	2	6	no	—
94	C₂²	—	4	1	no	—
95	C₂²	—	4	3	no	—
96	C₂²	—	382	12	no	—
97	C₂²	—	764	12	no	—
98	C₂²	—	764	6	no	—
99	C₂²	—	764	2	no	—
100	C₂²	—	764	4	no	—
101	C₂²	#20259291	1	1	yes	characteristic
102	C₂²	#20259298	1	3	yes	—
103	C₂²	#20259311	1	3	yes	—
104	C₄	—	4	6	no	—
105	C₄	—	1528	3	no	—
106	C₂	—	2	3	no	—
107	C₂	—	2	3	no	—
108	C₂	—	764	4	no	—
109	C₂	#27132574	1	1	yes	characteristic
110	C₂	#27132850	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27132518

C₂

yes

maximal

#27132430

C₂

yes

maximal, complemented

#543×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₁₉₁

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₉₁

C₂²

yes

—

C₂×(C₂×D₁₉₁)⋊C₄

—

C₂²⋊Q₈×C₁₉₁

—

C₂×C₂²⋊C₄×C₁₉₁

C₂²

yes

—

C₂³×D₁₉₁

—

C₂³×D₁₉₁

—

C₂³×D₁₉₁

—

C₂³×D₁₉₁

Q₈

yes

complemented

C₂²×Dic₁₉₁

—

C₂²⋊Dic₁₉₁

—

(C₂×D₁₉₁)⋊C₄

—

C₂⁴×C₁₉₁

C₂³

yes

characteristic

C₂×Q₈×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₄⋊C₄×C₁₉₁

—

C₂²⋊C₄×C₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

—

C₂³×C₁₉₁

C₂×Q₈

yes

characteristic

C₂³×C₁₉₁

#41

yes

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#120

yes

characteristic

C₂²×C₁₉₁

C₂²⋊Q₈

yes

—

C₂²×C₁₉₁

#124

yes

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#393

yes

characteristic

C₂×C₁₉₁

B(4,𝔽₂)

yes

—

D₁₉₁

—

C₁₉₁

#2029

yes

characteristic, complemented, Sylow

#2029

—

191

maximal, Sylow

#543

D₁₉₁

yes

characteristic, complemented

#379

—

191

—

#342

—

191

—

C₂⁵

—

191

—

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

—

382

—

C₂×C₂²⋊C₄

—

191

—

C₂×C₂²⋊C₄

C₂×D₁₉₁

yes

—

C₂⁴

—

382

—

C₂⁴

—

382

—

C₂⁴

—

191

—

C₂⁴

—

764

—

C₂⁴

C₂²×D₁₉₁

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

382

—

C₂²⋊C₄

—

764

—

C₂³

—

C₂³

—

382

—

C₂³

—

382

—

C₂³

—

764

—

C₂³

—

764

—

C₂³

—

764

—

C₂³

—

764

—

C₂³

—

764

—

C₂³

C₂×C₂²⋊D₁₉₁

yes

—

C₂³

Q₈×D₁₉₁

yes

characteristic

C₂³

#15690288

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

764

—

C₂×C₄

—

764

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

382

—

C₂²

—

764

—

C₂²

—

764

—

C₂²

—

764

—

100

C₂²

—

764

—

101

C₂²

#20259291

yes

characteristic

102

C₂²

#20259298

yes

—

103

C₂²

#20259311

yes

—

104

C₄

—

105

C₄

—

1528

—

106

C₂

—

107

C₂

—

108

C₂

—

764

—

109

C₂

#27132574

yes

characteristic

110

C₂

#27132850

yes

—

	subgroup	G/H
center	C₂²	#20259291
commutator, cosocle	C₂⁴×C₁₉₁	C₂³
Frattini	C₂⁴	C₂²×D₁₉₁
Fitting	#543×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#120
prime radical	C₂⁴×D₁₉₁	C₂²
hypercenter	#543	D₁₉₁
hypocenter	C₁₉₁	#2029
LCS	G ▷ C₂⁴×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂²

#20259291

commutator, cosocle

C₂⁴×C₁₉₁

C₂³

Frattini

C₂⁴

C₂²×D₁₉₁

Fitting

#543×C₁₉₁

C₂

socle

C₂²×C₁₉₁

#120

prime radical

C₂⁴×D₁₉₁

C₂²

D₁₉₁

C₁₉₁

G ▷ C₂⁴×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
2. #27132430	C₂	semidirect
3. #543×C₁₉₁	C₂	semidirect
12. C₂³×D₁₉₁	Q₈	semidirect
55. C₁₉₁	#2029	semidirect
57. #543	D₁₉₁	semidirect

H ≅ G/N

type

2. #27132430

C₂

semidirect

3. #543×C₁₉₁

C₂

semidirect

12. C₂³×D₁₉₁

Q₈

semidirect

55. C₁₉₁

#2029

semidirect

57. #543

D₁₉₁

semidirect

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

subgroup	index
#27132518	2
#27132430	2
#543×C₁₉₁	2
#2029	191

subgroup

index

#27132518

#27132430

#543×C₁₉₁

#2029

191

Subgroups — #37573683

Conjugacy classes of subgroups

Complements

Maximal subgroups