Group #44558357 — subgroups

Subgroups — #44558357

Order 32128.

432 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 149 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	#30631607	C₂	1	3	yes	maximal
2	#30631606	C₂	1	2	yes	maximal
3	#30631592	C₂	1	3	yes	maximal
4	#30631591	C₂	1	6	yes	maximal
5	#395×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
7	C₂²×Dic₅₀₂	C₂²	1	3	yes	—
8	C₂×Dic₂₅₁⋊C₄	C₂²	1	6	yes	—
9	C₂×C₄×Dic₂₅₁	C₂²	1	3	yes	—
10	C₂₅₁⋊#95	C₂²	1	6	yes	—
11	C₂₅₁⋊#95	C₂²	1	6	yes	—
12	C₂₅₁⋊#95	C₂²	1	3	yes	—
13	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	3	yes	—
15	#95×C₂₅₁	C₂²	1	3	yes	—
16	C₂×Dic₅₀₂	—	4	3	no	—
17	C₂×Dic₅₀₂	—	4	3	no	—
18	C₂×Dic₅₀₂	—	4	3	no	—
19	C₂×Dic₅₀₂	Q₈	1	12	yes	complemented
20	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
21	C₂²×Dic₂₅₁	C₂³	1	6	yes	—
22	Dic₂₅₁⋊C₄	—	4	6	no	—
23	C₄×Dic₂₅₁	—	4	3	no	—
24	C₂×Q₈×C₂₅₁	—	4	3	no	—
25	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
27	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
28	C₄⋊C₄×C₂₅₁	—	4	3	no	—
29	Dic₅₀₂	—	4	12	no	—
30	Dic₅₀₂	—	4	4	no	—
31	C₂×Dic₂₅₁	—	2	6	no	—
32	C₂×Dic₂₅₁	—	2	6	no	—
33	C₂×Dic₂₅₁	—	2	6	no	—
34	C₂×Dic₂₅₁	C₂×Q₈	1	12	yes	—
35	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
36	Q₈×C₂₅₁	—	4	4	no	—
37	C₂×C₄×C₂₅₁	—	2	3	no	—
38	C₂×C₄×C₂₅₁	—	2	3	no	—
39	C₂×C₄×C₂₅₁	—	2	3	no	—
40	C₂×C₄×C₂₅₁	—	2	3	no	—
41	C₂×C₄×C₂₅₁	—	2	6	no	—
42	C₂×C₄×C₂₅₁	C₂×Q₈	1	6	yes	—
43	C₂²×C₂₅₁	C₂²×Q₈	1	3	yes	—
44	C₂²×C₂₅₁	H(5,𝔽₂)	1	3	yes	—
45	C₂²×C₂₅₁	H(5,𝔽₂)	1	1	yes	characteristic
46	C₄×C₂₅₁	—	2	6	no	—
47	C₄×C₂₅₁	—	4	3	no	—
48	C₄×C₂₅₁	—	4	1	no	—
49	Dic₂₅₁	—	2	12	no	—
50	Dic₂₅₁	—	4	2	no	—
51	C₂×C₂₅₁	#543	1	3	yes	—
52	C₂×C₂₅₁	Q₈²	1	3	yes	—
53	C₂×C₂₅₁	H(3,𝔽₄)	1	1	yes	characteristic
54	C₂₅₁	#2811	1	1	yes	characteristic, complemented, Sylow
55	#2811	—	251	1	no	maximal, Sylow
56	#395	—	251	2	no	—
57	#395	—	251	3	no	—
58	#395	D₂₅₁	1	1	yes	characteristic
59	#391	—	251	3	no	—
60	#391	—	251	6	no	—
61	C₂²×Q₈	—	251	1	no	—
62	C₂²×Q₈	—	251	3	no	—
63	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
64	C₂×C₄⋊C₄	—	251	6	no	—
65	C₂×C₄⋊C₄	C₂×D₂₅₁	1	3	yes	—
66	C₂×C₄²	—	251	3	no	—
67	#95	—	251	6	no	—
68	#95	—	251	3	no	—
69	#95	—	251	6	no	—
70	#95	C₂×D₂₅₁	1	3	yes	—
71	C₂×Q₈	—	4	3	no	—
72	C₂×Q₈	—	1004	3	no	—
73	C₂×Q₈	—	1004	3	no	—
74	C₂×Q₈	—	1004	3	no	—
75	C₂×Q₈	—	251	12	no	—
76	C₂²×C₄	—	251	2	no	—
77	C₂²×C₄	—	251	6	no	—
78	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
80	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	1004	6	no	—
83	C₄²	—	1004	3	no	—
84	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
85	Q₈	—	4	4	no	—
86	Q₈	—	1004	12	no	—
87	Q₈	—	1004	4	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	6	no	—
93	C₂×C₄	—	502	6	no	—
94	C₂×C₄	—	502	6	no	—
95	C₂×C₄	—	502	6	no	—
96	C₂×C₄	—	251	12	no	—
97	C₂×C₄	Q₈×D₂₅₁	1	6	yes	—
98	C₂²	C₂×Q₈×D₂₅₁	1	3	yes	—
99	C₂²	#22548760	1	3	yes	—
100	C₂²	#22548761	1	1	yes	characteristic
101	C₄	—	2	6	no	—
102	C₄	—	4	3	no	—
103	C₄	—	4	1	no	—
104	C₄	—	1004	2	no	—
105	C₄	—	502	12	no	—
106	C₂	#30632312	1	3	yes	—
107	C₂	#30632417	1	3	yes	—
108	C₂	#30632430	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30631607

C₂

yes

maximal

#30631606

C₂

yes

maximal

#30631592

C₂

yes

maximal

#30631591

C₂

yes

maximal

#395×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₀₂

C₂²

yes

characteristic

C₂²×Dic₅₀₂

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

C₂²

yes

—

C₂×C₄×Dic₂₅₁

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂²×Q₈×C₂₅₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

#95×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

Q₈

yes

complemented

C₂²×Dic₂₅₁

C₂³

yes

—

C₂²×Dic₂₅₁

C₂³

yes

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₄⋊C₄×C₂₅₁

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

C₂×Q₈

yes

—

C₂³×C₂₅₁

C₂⁴

yes

characteristic

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×Q₈

yes

—

C₂²×C₂₅₁

C₂²×Q₈

yes

—

C₂²×C₂₅₁

H(5,𝔽₂)

yes

—

C₂²×C₂₅₁

H(5,𝔽₂)

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

#543

yes

—

C₂×C₂₅₁

Q₈²

yes

—

C₂×C₂₅₁

H(3,𝔽₄)

yes

characteristic

C₂₅₁

#2811

yes

characteristic, complemented, Sylow

#2811

—

251

maximal, Sylow

#395

—

251

—

#395

—

251

—

#395

D₂₅₁

yes

characteristic

#391

—

251

—

#391

—

251

—

C₂²×Q₈

—

251

—

C₂²×Q₈

—

251

—

C₂²×Q₈

C₂×D₂₅₁

yes

characteristic

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₄²

—

251

—

#95

—

251

—

#95

—

251

—

#95

—

251

—

#95

C₂×D₂₅₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

1004

—

C₂×Q₈

—

1004

—

C₂×Q₈

—

1004

—

C₂×Q₈

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₂²×C₄

C₂²×D₂₅₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

1004

—

C₄²

—

1004

—

C₂³

C₂³×D₂₅₁

yes

characteristic

Q₈

—

Q₈

—

1004

—

Q₈

—

1004

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

—

502

—

C₂×C₄

—

251

—

C₂×C₄

Q₈×D₂₅₁

yes

—

C₂²

C₂×Q₈×D₂₅₁

yes

—

C₂²

#22548760

yes

—

100

C₂²

#22548761

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

1004

—

105

C₄

—

502

—

106

C₂

#30632312

yes

—

107

C₂

#30632417

yes

—

108

C₂

#30632430

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, prime radical, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	#395×C₂₅₁	C₂
hypercenter	#395	D₂₅₁
hypocenter	C₂₅₁	#2811
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₅₁

commutator, socle, prime radical, cosocle

C₂³×C₂₅₁

C₂⁴

Fitting

#395×C₂₅₁

C₂

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
19. C₂×Dic₅₀₂	Q₈	semidirect
54. C₂₅₁	#2811	semidirect

H ≅ G/N

type

19. C₂×Dic₅₀₂

Q₈

semidirect

54. C₂₅₁

#2811

semidirect

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

subgroup	index
#30631607	2
#30631606	2
#30631592	2
#30631591	2
#395×C₂₅₁	2
#2811	251

subgroup

index

#395×C₂₅₁

251

Subgroups — #44558357

Conjugacy classes of subgroups

Complements

Maximal subgroups