Group #45664455 — subgroups

Subgroups — #45664455

Order 32896.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂×#22551503	C₂	1	1	yes	maximal, characteristic
2	#30656603	C₂	1	1	yes	maximal, characteristic
3	#30656499	C₂	1	4	yes	maximal
4	#384×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14523339	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
7	#22551503	C₄	1	4	yes	complemented
8	#22551499	—	2	2	no	—
9	C₂×C₄⋊C₄×C₂₅₇	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₅₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₅₇	—	2	2	no	—
13	C₂×#14523339	—	2	2	no	—
14	C₂×#14523339	—	2	2	no	—
15	C₂×#14523339	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₂₅₇	—	2	4	no	—
17	C₂²×C₄×C₂₅₇	—	2	2	no	—
18	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
19	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₂₅₇	—	2	4	no	—
21	C₄⋊C₄×C₂₅₇	—	4	4	no	—
22	C₄²×C₂₅₇	C₂×C₄	1	2	yes	—
23	C₄²×C₂₅₇	Q₈	1	2	yes	—
24	C₂×Dic₂₅₇	—	4	2	no	—
25	C₂×Dic₂₅₇	—	4	2	no	—
26	C₂×Dic₂₅₇	—	4	2	no	—
27	#14523339	—	2	4	no	—
28	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₅₇	—	2	2	no	—
30	C₂×C₄×C₂₅₇	—	2	2	no	—
31	C₂×C₄×C₂₅₇	—	4	2	no	—
32	C₂×C₄×C₂₅₇	—	4	4	no	—
33	C₂×C₄×C₂₅₇	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₅₇	#41	1	2	yes	—
37	C₂×C₄×C₂₅₇	#41	1	2	yes	—
38	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₂₅₇	#124	1	1	yes	characteristic
40	C₂²×C₂₅₇	#124	1	1	yes	characteristic
41	C₂²×C₂₅₇	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₂₅₇	#137	1	2	yes	—
43	C₄×C₂₅₇	—	2	4	no	—
44	C₄×C₂₅₇	—	8	2	no	—
45	C₄×C₂₅₇	#117	1	2	yes	—
46	C₄×C₂₅₇	#125	1	2	yes	—
47	Dic₂₅₇	—	8	2	no	—
48	C₂×C₂₅₇	#382	1	1	yes	characteristic
49	C₂×C₂₅₇	#428	1	2	yes	—
50	C₂×C₂₅₇	#489	1	2	yes	—
51	C₂×C₂₅₇	#490	1	1	yes	characteristic
52	C₂×C₂₅₇	#491	1	1	yes	characteristic
53	C₂₅₇	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	257	1	no	maximal, Sylow
55	C₂×#97	—	257	1	no	—
56	#384	D₂₅₇	1	1	yes	characteristic
57	#382	—	257	1	no	—
58	#340	—	257	4	no	—
59	C₂²×C₈	—	257	2	no	—
60	C₂×C₄⋊C₄	—	257	2	no	—
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	C₂×D₂₅₇	1	2	yes	—
63	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
64	#97	—	257	4	no	—
65	#95	—	514	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	514	2	no	—
68	C₂²×C₄	C₂²⋊D₂₅₇	1	2	yes	—
69	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
70	C₂×C₈	—	257	4	no	—
71	C₂×C₈	—	514	2	no	—
72	C₂×C₈	—	514	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	514	4	no	—
75	C₄⋊C₄	—	4	4	no	—
76	C₄²	C₄×D₂₅₇	1	2	yes	—
77	C₄²	Dic₅₁₄	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	1028	2	no	—
84	C₂×C₄	—	1028	2	no	—
85	C₂×C₄	—	1028	2	no	—
86	C₂×C₄	#16861597	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₅₁₄	1	1	yes	characteristic
89	C₂×C₄	#16861617	1	2	yes	—
90	C₂×C₄	#16861619	1	2	yes	—
91	C₈	—	514	4	no	—
92	C₂²	C₂×#16861597	1	1	yes	characteristic
93	C₂²	#22551579	1	1	yes	characteristic
94	C₂²	#22551581	1	1	yes	characteristic
95	C₂²	#22551618	1	2	yes	—
96	C₂²	#22551623	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	2056	2	no	—
100	C₄	#22551550	1	2	yes	—
101	C₄	#22551583	1	2	yes	—
102	C₂	#30656602	1	1	yes	characteristic
103	C₂	#30656777	1	2	yes	—
104	C₂	#30657099	1	2	yes	—
105	C₂	#30657101	1	1	yes	characteristic
106	C₂	#30657105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551503

C₂

yes

maximal, characteristic

#30656603

C₂

yes

maximal, characteristic

#30656499

C₂

yes

maximal

#384×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×#14523339

C₂²

yes

—

C₂×C₄⋊Dic₂₅₇

C₂²

yes

—

#22551503

C₄

yes

complemented

#22551499

—

C₂×C₄⋊C₄×C₂₅₇

—

C₂×C₄⋊C₄×C₂₅₇

C₂²

yes

—

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

C₂²×Dic₂₅₇

—

C₂×#14523339

—

C₂×#14523339

—

C₂×#14523339

C₂×C₄

yes

—

C₄⋊Dic₂₅₇

—

C₂²×C₄×C₂₅₇

—

C₂²×C₄×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

C₄²×C₂₅₇

C₂×C₄

yes

—

C₄²×C₂₅₇

Q₈

yes

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₂×C₄×C₂₅₇

#41

yes

—

C₂²×C₂₅₇

C₂×C₄⋊C₄

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

yes

—

C₂²×C₂₅₇

#137

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#117

yes

—

C₄×C₂₅₇

#125

yes

—

Dic₂₅₇

—

C₂×C₂₅₇

#382

yes

characteristic

C₂×C₂₅₇

#428

yes

—

C₂×C₂₅₇

#489

yes

—

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂₅₇

yes

characteristic, complemented, Sylow

#1836

—

257

maximal, Sylow

C₂×#97

—

257

—

#384

D₂₅₇

yes

characteristic

#382

—

257

—

#340

—

257

—

C₂²×C₈

—

257

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₅₇

yes

—

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

#97

—

257

—

#95

—

514

—

C₂²×C₄

—

C₂²×C₄

—

514

—

C₂²×C₄

C₂²⋊D₂₅₇

yes

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

C₂×C₈

—

257

—

C₂×C₈

—

514

—

C₂×C₈

—

514

—

C₄⋊C₄

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

C₄²

C₄×D₂₅₇

yes

—

C₄²

Dic₅₁₄

yes

—

C₂³

C₂×C₂²⋊D₂₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1028

—

C₂×C₄

—

1028

—

C₂×C₄

—

1028

—

C₂×C₄

#16861597

yes

—

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

C₂×C₄

C₂×Dic₅₁₄

yes

characteristic

C₂×C₄

#16861617

yes

—

C₂×C₄

#16861619

yes

—

C₈

—

514

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#22551623

yes

—

C₄

—

C₄

—

C₄

—

2056

—

100

C₄

#22551550

yes

—

101

C₄

#22551583

yes

—

102

C₂

#30656602

yes

characteristic

103

C₂

#30656777

yes

—

104

C₂

#30657099

yes

—

105

C₂

#30657101

yes

characteristic

106

C₂

#30657105

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₅₇
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#384×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂×D₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#384	D₂₅₇
hypocenter	C₂₅₇	#1836
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₅₇

commutator

C₂×C₄×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#384×C₂₅₇

C₂

socle, prime radical

C₂³×C₂₅₇

C₂×D₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #22551503	C₄	semidirect
53. C₂₅₇	#1836	semidirect

H ≅ G/N

type

7. #22551503

C₄

semidirect

53. C₂₅₇

#1836

semidirect

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

subgroup	index
C₂×#22551503	2
#30656603	2
#30656499	2
#384×C₂₅₇	2
#1836	257

subgroup

index

#384×C₂₅₇

257

Subgroups — #45664455

Conjugacy classes of subgroups

Complements

Maximal subgroups