Group #45666337 — subgroups

Subgroups — #45666337

Order 32896.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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	#30656909	C₂	1	3	yes	maximal, complemented
2	#30656534	C₂	1	3	yes	maximal, complemented
3	H(3,𝔽₄)×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#22551640	C₂²	1	1	yes	characteristic, complemented
5	#22551618	—	2	3	no	—
6	#22551520	—	2	3	no	—
7	#22551511	C₂²	1	3	yes	complemented
8	#124×C₂₅₇	—	2	3	no	—
9	#124×C₂₅₇	C₂²	1	3	yes	—
10	#120×C₂₅₇	—	2	3	no	—
11	#16861619	—	4	1	no	—
12	#16861618	—	2	3	no	—
13	Q₈×D₂₅₇	—	4	1	no	—
14	#16861606	—	4	3	no	—
15	#16861603	—	4	3	no	—
16	#16861602	—	2	3	no	—
17	C₂⁴×C₂₅₇	—	2	1	no	—
18	C₂×Q₈×C₂₅₇	—	2	1	no	—
19	C₂×Q₈×C₂₅₇	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₂₅₇	—	2	3	no	—
21	C₂×D₄×C₂₅₇	D₄	1	3	yes	—
22	C₂²⋊C₄×C₂₅₇	—	2	3	no	—
23	C₂²⋊C₄×C₂₅₇	—	2	6	no	—
24	C₄²×C₂₅₇	D₄	1	3	yes	—
25	Dic₅₁₄	—	2	3	no	—
26	D₁₀₂₈	—	2	3	no	—
27	C₂²⋊D₂₅₇	—	4	1	no	—
28	C₄×D₂₅₇	—	4	3	no	—
29	#14523339	—	4	3	no	—
30	C₂³×C₂₅₇	—	2	3	no	—
31	C₂³×C₂₅₇	—	8	2	no	—
32	C₂³×C₂₅₇	—	8	1	no	—
33	Q₈×C₂₅₇	—	4	1	no	—
34	Q₈×C₂₅₇	—	8	1	no	—
35	D₄×C₂₅₇	—	4	3	no	—
36	D₄×C₂₅₇	—	8	3	no	—
37	C₂×C₄×C₂₅₇	—	2	3	no	—
38	C₂×C₄×C₂₅₇	C₂×D₄	1	3	yes	—
39	C₂²×C₂₅₇	—	4	6	no	—
40	C₂²×C₂₅₇	—	4	3	no	—
41	C₂²×C₂₅₇	—	8	4	no	—
42	C₂²×C₂₅₇	#120	1	1	yes	characteristic
43	C₂×D₂₅₇	—	4	1	no	—
44	C₄×C₂₅₇	—	2	3	no	—
45	C₄×C₂₅₇	—	4	3	no	—
46	Dic₂₅₇	—	4	1	no	—
47	C₂×C₂₅₇	—	2	1	no	—
48	C₂×C₂₅₇	—	8	3	no	—
49	C₂×C₂₅₇	B(4,𝔽₂)	1	1	yes	characteristic
50	D₂₅₇	—	8	1	no	—
51	C₂₅₇	#2202	1	1	yes	characteristic, complemented, Sylow
52	#2202	—	257	1	no	maximal, Sylow
53	H(3,𝔽₄)	D₂₅₇	1	1	yes	characteristic, complemented
54	#454	—	257	3	no	—
55	#355	—	257	3	no	—
56	2₋¹⁺⁴	—	257	1	no	—
57	AGL(1,ℤ/8)	—	514	3	no	—
58	#124	—	2	3	no	—
59	#124	C₂×D₂₅₇	1	3	yes	—
60	#120	—	2	3	no	—
61	#104	—	514	3	no	—
62	#101	—	257	3	no	—
63	C₂⁴	—	2	1	no	—
64	#41	—	514	3	no	—
65	#41	—	1028	1	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	1028	1	no	—
68	C₂×Q₈	C₂²×D₂₅₇	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₂₅₇	1	3	yes	—
71	SD₁₆	—	1028	3	no	—
72	D₈	—	1028	3	no	—
73	M₁₆	—	514	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₂₅₇	1	3	yes	—
77	C₂³	—	2	3	no	—
78	C₂³	—	8	2	no	—
79	C₂³	—	8	1	no	—
80	Q₈	—	514	3	no	—
81	Q₈	—	4	1	no	—
82	Q₈	—	8	1	no	—
83	D₄	—	514	3	no	—
84	D₄	—	4	3	no	—
85	D₄	—	1028	1	no	—
86	D₄	—	8	3	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	1028	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	3	yes	—
90	C₈	—	1028	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	1028	1	no	—
94	C₂²	—	8	4	no	—
95	C₂²	#22551561	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	1028	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	2056	1	no	—
102	C₂	#30656924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30656909

C₂

yes

maximal, complemented

#30656534

C₂

yes

maximal, complemented

H(3,𝔽₄)×C₂₅₇

C₂

yes

maximal, characteristic, complemented

#22551640

C₂²

yes

characteristic, complemented

#22551618

—

#22551520

—

#22551511

C₂²

yes

complemented

#124×C₂₅₇

—

#124×C₂₅₇

C₂²

yes

—

#120×C₂₅₇

—

#16861619

—

#16861618

—

Q₈×D₂₅₇

—

#16861606

—

#16861603

—

#16861602

—

C₂⁴×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

C₂³

yes

characteristic

C₂×D₄×C₂₅₇

—

C₂×D₄×C₂₅₇

D₄

yes

—

C₂²⋊C₄×C₂₅₇

—

C₂²⋊C₄×C₂₅₇

—

C₄²×C₂₅₇

D₄

yes

—

Dic₅₁₄

—

D₁₀₂₈

—

C₂²⋊D₂₅₇

—

C₄×D₂₅₇

—

#14523339

—

C₂³×C₂₅₇

—

C₂³×C₂₅₇

—

C₂³×C₂₅₇

—

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

D₄×C₂₅₇

—

D₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

#120

yes

characteristic

C₂×D₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

B(4,𝔽₂)

yes

characteristic

D₂₅₇

—

C₂₅₇

#2202

yes

characteristic, complemented, Sylow

#2202

—

257

maximal, Sylow

H(3,𝔽₄)

D₂₅₇

yes

characteristic, complemented

#454

—

257

—

#355

—

257

—

2₋¹⁺⁴

—

257

—

AGL(1,ℤ/8)

—

514

—

#124

—

#124

C₂×D₂₅₇

yes

—

#120

—

#104

—

514

—

#101

—

257

—

C₂⁴

—

#41

—

514

—

#41

—

1028

—

C₂×Q₈

—

C₂×Q₈

—

1028

—

C₂×Q₈

C₂²×D₂₅₇

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₂₅₇

yes

—

SD₁₆

—

1028

—

D₈

—

1028

—

M₁₆

—

514

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₂₅₇

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

514

—

Q₈

—

Q₈

—

D₄

—

514

—

D₄

—

D₄

—

1028

—

D₄

—

C₂×C₄

—

C₂×C₄

—

1028

—

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

—

C₈

—

1028

—

C₂²

—

C₂²

—

C₂²

—

1028

—

C₂²

—

C₂²

#22551561

yes

characteristic

C₄

—

C₄

—

C₄

—

1028

—

C₂

—

100

C₂

—

101

C₂

—

2056

—

102

C₂

#30656924

yes

characteristic

	subgroup	G/H
center	C₂	#30656924
commutator, cosocle	C₂×Q₈×C₂₅₇	C₂³
Frattini	C₂×Q₈	C₂²×D₂₅₇
Fitting	H(3,𝔽₄)×C₂₅₇	C₂
socle	C₂×C₂₅₇	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₂₅₇
hypocenter	C₂₅₇	#2202
LCS	G ▷ C₂×Q₈×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#30656924

commutator, cosocle

C₂×Q₈×C₂₅₇

C₂³

Frattini

C₂×Q₈

C₂²×D₂₅₇

Fitting

H(3,𝔽₄)×C₂₅₇

C₂

C₂×C₂₅₇

D₂₅₇

C₂₅₇

G ▷ C₂×Q₈×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #30656909	C₂	semidirect
2. #30656534	C₂	semidirect
3. H(3,𝔽₄)×C₂₅₇	C₂	semidirect
4. #22551640	C₂²	semidirect
7. #22551511	C₂²	semidirect
51. C₂₅₇	#2202	semidirect
53. H(3,𝔽₄)	D₂₅₇	semidirect

H ≅ G/N

type

1. #30656909

C₂

semidirect

2. #30656534

C₂

semidirect

3. H(3,𝔽₄)×C₂₅₇

C₂

semidirect

4. #22551640

C₂²

semidirect

7. #22551511

C₂²

semidirect

51. C₂₅₇

#2202

semidirect

53. H(3,𝔽₄)

D₂₅₇

semidirect

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

subgroup	index
#30656909	2
#30656534	2
H(3,𝔽₄)×C₂₅₇	2
#2202	257

subgroup

index

#30656909

#30656534

H(3,𝔽₄)×C₂₅₇

#2202

257

Subgroups — #45666337

Conjugacy classes of subgroups

Complements

Maximal subgroups