Group #45663399 — subgroups

Subgroups — #45663399

Order 32896.

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

normal subgroups only

This group has 228 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	#30657467	C₂	1	1	yes	maximal, characteristic, complemented
2	#30657111	C₂	1	1	yes	maximal, characteristic
3	#30656938	C₂	1	2	yes	maximal
4	#30656467	C₂	1	2	yes	maximal, complemented
5	#333×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16861607	—	2	2	no	—
7	C₂×#16861605	—	2	2	no	—
8	#22551591	—	2	2	no	—
9	#22551590	—	2	2	no	—
10	#22551589	C₂²	1	1	yes	characteristic
11	C₄×Dic₅₁₄	C₂²	1	2	yes	—
12	#22551557	—	2	2	no	—
13	#22551514	—	2	2	no	—
14	C₄×#14523339	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₅₇	C₂²	1	2	yes	—
17	C₂×Dic₅₁₄	—	2	2	no	—
18	C₂×Dic₅₁₄	D₄	1	2	yes	—
19	C₂×D₁₀₂₈	—	2	1	no	—
20	C₂×D₁₀₂₈	—	2	2	no	—
21	#16861607	—	4	4	no	—
22	#16861605	—	4	4	no	—
23	C₂×#14523339	—	2	1	no	—
24	C₄⋊Dic₂₅₇	D₄	1	2	yes	—
25	#16861597	—	2	4	no	—
26	#16861597	—	2	2	no	—
27	C₄×Dic₂₅₇	—	2	2	no	—
28	C₄×Dic₂₅₇	—	2	2	no	—
29	C₂×Q₈×C₂₅₇	D₄	1	2	yes	—
30	C₂×C₈×C₂₅₇	—	2	2	no	—
31	C₄⋊C₄×C₂₅₇	—	2	2	no	—
32	C₄²×C₂₅₇	C₂³	1	1	yes	characteristic
33	C₂²×D₂₅₇	—	4	1	no	—
34	Dic₅₁₄	—	2	2	no	—
35	Dic₅₁₄	—	8	2	no	—
36	Dic₅₁₄	SD₁₆	1	4	yes	complemented
37	D₁₀₂₈	—	4	2	no	—
38	D₁₀₂₈	—	4	2	no	—
39	D₁₀₂₈	—	4	2	no	—
40	C₂×Dic₂₅₇	—	2	2	no	—
41	C₂×Dic₂₅₇	—	2	2	no	—
42	#14523339	—	2	2	no	—
43	Q₈×C₂₅₇	—	4	2	no	—
44	C₂×C₄×C₂₅₇	—	2	2	no	—
45	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
47	C₈×C₂₅₇	—	4	2	no	—
48	C₂²×C₂₅₇	#120	1	1	yes	characteristic
49	C₂×D₂₅₇	—	8	2	no	—
50	C₂×D₂₅₇	—	8	1	no	—
51	C₄×C₂₅₇	—	2	1	no	—
52	C₄×C₂₅₇	—	4	2	no	—
53	C₄×C₂₅₇	C₂×SD₁₆	1	2	yes	—
54	C₄×C₂₅₇	#137	1	2	yes	—
55	Dic₂₅₇	—	2	4	no	—
56	Dic₂₅₇	—	4	2	no	—
57	C₂×C₂₅₇	#448	1	2	yes	—
58	C₂×C₂₅₇	#453	1	1	yes	characteristic
59	D₂₅₇	—	16	1	no	—
60	C₂₅₇	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	257	1	no	maximal, Sylow
62	Q₈²	—	257	1	no	—
63	#492	—	257	1	no	—
64	#460	—	257	2	no	—
65	#333	D₂₅₇	1	1	yes	characteristic, complemented
66	Q₈⋊C₈	—	257	2	no	—
67	C₂×SD₁₆	—	514	2	no	—
68	C₂×SD₁₆	—	514	2	no	—
69	C₄⋊Q₈	—	514	2	no	—
70	C₄⋊Q₈	—	514	2	no	—
71	C₄⋊Q₈	C₂×D₂₅₇	1	1	yes	characteristic
72	#127	—	257	1	no	—
73	C₄×Q₈	—	257	2	no	—
74	C₄×Q₈	—	514	2	no	—
75	C₄⋊C₈	C₂×D₂₅₇	1	2	yes	—
76	#102	—	514	2	no	—
77	C₄×C₈	—	257	1	no	—
78	C₂×Q₈	—	257	2	no	—
79	C₂×Q₈	—	514	2	no	—
80	C₂×Q₈	D₁₀₂₈	1	2	yes	—
81	C₂×D₄	—	514	2	no	—
82	C₂×D₄	—	514	1	no	—
83	SD₁₆	—	1028	4	no	—
84	SD₁₆	—	1028	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	514	1	no	—
87	C₄⋊C₄	—	257	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	514	4	no	—
90	C₄⋊C₄	—	514	2	no	—
91	C₄²	—	514	2	no	—
92	C₄²	—	514	2	no	—
93	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
94	C₂³	—	1028	1	no	—
95	Q₈	—	257	4	no	—
96	Q₈	—	514	2	no	—
97	Q₈	—	4	2	no	—
98	Q₈	—	2056	2	no	—
99	D₄	—	1028	2	no	—
100	D₄	—	1028	2	no	—
101	D₄	—	1028	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	514	2	no	—
104	C₂×C₄	—	514	2	no	—
105	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
106	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
107	C₈	—	514	2	no	—
108	C₈	—	4	2	no	—
109	C₂²	—	2056	2	no	—
110	C₂²	—	2056	1	no	—
111	C₂²	#22551562	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	514	4	no	—
114	C₄	—	4	2	no	—
115	C₄	—	1028	2	no	—
116	C₄	SD₁₆×D₂₅₇	1	2	yes	—
117	C₄	#22551624	1	2	yes	—
118	C₂	—	4112	1	no	—
119	C₂	#30656877	1	2	yes	—
120	C₂	#30656907	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30657467

C₂

yes

maximal, characteristic, complemented

#30657111

C₂

yes

maximal, characteristic

#30656938

C₂

yes

maximal

#30656467

C₂

yes

maximal, complemented

#333×C₂₅₇

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₅₁₄

C₂²

yes

—

#22551557

—

#22551514

—

C₄×#14523339

C₂²

yes

characteristic

C₄⋊Q₈×C₂₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₇

C₂²

yes

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

D₄

yes

—

C₂×D₁₀₂₈

—

C₂×D₁₀₂₈

—

#16861607

—

#16861605

—

C₂×#14523339

—

C₄⋊Dic₂₅₇

D₄

yes

—

#16861597

—

#16861597

—

C₄×Dic₂₅₇

—

C₄×Dic₂₅₇

—

C₂×Q₈×C₂₅₇

D₄

yes

—

C₂×C₈×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

C₄²×C₂₅₇

C₂³

yes

characteristic

C₂²×D₂₅₇

—

Dic₅₁₄

—

Dic₅₁₄

—

Dic₅₁₄

SD₁₆

yes

complemented

D₁₀₂₈

—

D₁₀₂₈

—

D₁₀₂₈

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

Q₈×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₈×C₂₅₇

—

C₂²×C₂₅₇

#120

yes

characteristic

C₂×D₂₅₇

—

C₂×D₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

C₂×SD₁₆

yes

—

C₄×C₂₅₇

#137

yes

—

Dic₂₅₇

—

Dic₂₅₇

—

C₂×C₂₅₇

#448

yes

—

C₂×C₂₅₇

#453

yes

characteristic

D₂₅₇

—

C₂₅₇

#1623

yes

characteristic, complemented, Sylow

#1623

—

257

maximal, Sylow

Q₈²

—

257

—

#492

—

257

—

#460

—

257

—

#333

D₂₅₇

yes

characteristic, complemented

Q₈⋊C₈

—

257

—

C₂×SD₁₆

—

514

—

C₂×SD₁₆

—

514

—

C₄⋊Q₈

—

514

—

C₄⋊Q₈

—

514

—

C₄⋊Q₈

C₂×D₂₅₇

yes

characteristic

#127

—

257

—

C₄×Q₈

—

257

—

C₄×Q₈

—

514

—

C₄⋊C₈

C₂×D₂₅₇

yes

—

#102

—

514

—

C₄×C₈

—

257

—

C₂×Q₈

—

257

—

C₂×Q₈

—

514

—

C₂×Q₈

D₁₀₂₈

yes

—

C₂×D₄

—

514

—

C₂×D₄

—

514

—

SD₁₆

—

1028

—

SD₁₆

—

1028

—

C₂×C₈

—

C₂×C₈

—

514

—

C₄⋊C₄

—

257

—

C₄⋊C₄

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

514

—

C₄²

—

514

—

C₄²

—

514

—

C₄²

C₂²×D₂₅₇

yes

characteristic

C₂³

—

1028

—

Q₈

—

257

—

Q₈

—

514

—

Q₈

—

Q₈

—

2056

—

D₄

—

1028

—

100

D₄

—

1028

—

101

D₄

—

1028

—

102

C₂×C₄

—

103

C₂×C₄

—

514

—

104

C₂×C₄

—

514

—

105

C₂×C₄

D₄×D₂₅₇

yes

—

106

C₂×C₄

C₂×D₁₀₂₈

yes

characteristic

107

C₈

—

514

—

108

C₈

—

109

C₂²

—

2056

—

110

C₂²

—

2056

—

111

C₂²

#22551562

yes

characteristic

112

C₄

—

113

C₄

—

514

—

114

C₄

—

115

C₄

—

1028

—

116

C₄

SD₁₆×D₂₅₇

yes

—

117

C₄

#22551624

yes

—

118

C₂

—

4112

—

119

C₂

#30656877

yes

—

120

C₂

#30656907

yes

characteristic

	subgroup	G/H
center	C₂²	#22551562
commutator, cosocle	C₄²×C₂₅₇	C₂³
Frattini	C₄²	C₂²×D₂₅₇
Fitting	#333×C₂₅₇	C₂
socle	C₂²×C₂₅₇	#120
prime radical	#22551589	C₂²
hypercenter	#333	D₂₅₇
hypocenter	C₂₅₇	#1623
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#22551562

commutator, cosocle

C₄²×C₂₅₇

C₂³

Frattini

C₄²

C₂²×D₂₅₇

Fitting

#333×C₂₅₇

C₂

C₂²×C₂₅₇

C₂²

D₂₅₇

C₂₅₇

G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #30657467	C₂	semidirect
4. #30656467	C₂	semidirect
5. #333×C₂₅₇	C₂	semidirect
36. Dic₅₁₄	SD₁₆	semidirect
60. C₂₅₇	#1623	semidirect
65. #333	D₂₅₇	semidirect

H ≅ G/N

type

1. #30657467

C₂

semidirect

4. #30656467

C₂

semidirect

5. #333×C₂₅₇

C₂

semidirect

36. Dic₅₁₄

SD₁₆

semidirect

60. C₂₅₇

#1623

semidirect

65. #333

D₂₅₇

semidirect

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

subgroup	index
#30657467	2
#30657111	2
#30656938	2
#30656467	2
#333×C₂₅₇	2
#1623	257

subgroup

index

#333×C₂₅₇

257

Subgroups — #45663399

Conjugacy classes of subgroups

Complements

Maximal subgroups