Group #45664074 — subgroups

Subgroups — #45664074

Order 32896.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#30656695	C₂	1	1	yes	maximal, characteristic
2	#30656693	C₂	1	2	yes	maximal
3	#30656495	C₂	1	1	yes	maximal, characteristic
4	#30656493	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#16861602	C₂²	1	2	yes	—
7	C₂²×#14523339	C₂²	1	1	yes	characteristic
8	#22551551	C₂²	1	1	yes	characteristic
9	#22551503	C₄	1	4	yes	complemented
10	#22551502	C₄	1	2	yes	complemented
11	C₄×#14523339	C₄	1	4	yes	complemented
12	C₈×Dic₂₅₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
14	#117×C₂₅₇	C₂²	1	2	yes	—
15	#16861602	—	2	2	no	—
16	#16861602	—	2	2	no	—
17	C₂×#14523339	C₂×C₄	1	4	yes	—
18	C₂×#14523339	C₂×C₄	1	1	yes	characteristic
19	C₂×#14523339	C₂×C₄	1	2	yes	—
20	C₂×#14523339	C₂×C₄	1	1	yes	characteristic
21	C₂×#14523339	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₂₅₇	—	2	1	no	—
23	C₂²⋊Dic₂₅₇	—	2	1	no	—
24	C₄×Dic₂₅₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₅₇	—	2	2	no	—
27	C₂×C₈×C₂₅₇	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₂₅₇	—	2	2	no	—
29	C₂²⋊C₄×C₂₅₇	—	2	2	no	—
30	C₄²×C₂₅₇	C₂×C₄	1	4	yes	—
31	C₂×Dic₂₅₇	—	2	2	no	—
32	#14523339	—	2	4	no	—
33	#14523339	C₄²	1	1	yes	characteristic
34	#14523339	C₄²	1	1	yes	characteristic
35	#14523339	C₄²	1	2	yes	—
36	C₂³×C₂₅₇	#41	1	1	yes	characteristic
37	C₂×C₄×C₂₅₇	—	2	4	no	—
38	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₅₇	#41	1	2	yes	—
41	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
42	C₈×C₂₅₇	—	2	2	no	—
43	C₂²×C₂₅₇	—	4	1	no	—
44	C₂²×C₂₅₇	#117	1	2	yes	—
45	C₂²×C₂₅₇	#117	1	1	yes	characteristic
46	C₄×C₂₅₇	—	4	4	no	—
47	C₄×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₂₅₇	#117	1	1	yes	characteristic
49	C₄×C₂₅₇	#117	1	2	yes	—
50	Dic₂₅₇	—	4	2	no	—
51	C₂×C₂₅₇	—	2	1	no	—
52	C₂×C₂₅₇	—	2	2	no	—
53	C₂×C₂₅₇	#376	1	1	yes	characteristic
54	C₂₅₇	#1747	1	1	yes	characteristic, complemented, Sylow
55	#1747	—	257	1	no	maximal, Sylow
56	#405	—	257	1	no	—
57	#405	—	257	2	no	—
58	#338	—	257	1	no	—
59	H(3,ℤ/4)	—	257	2	no	—
60	H(3,ℤ/4)	D₂₅₇	1	1	yes	characteristic
61	C₂×M₁₆	—	257	2	no	—
62	C₂×M₁₆	C₂×D₂₅₇	1	1	yes	characteristic
63	C₂²×C₈	—	257	1	no	—
64	#117	—	257	1	no	—
65	#117	C₂×D₂₅₇	1	2	yes	—
66	#97	—	257	2	no	—
67	#97	—	257	4	no	—
68	C₄×C₈	—	257	2	no	—
69	C₄×C₈	—	257	4	no	—
70	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	514	2	no	—
73	M₁₆	—	514	2	no	—
74	C₂×C₈	—	257	1	no	—
75	C₂×C₈	—	257	2	no	—
76	C₂×C₈	—	257	1	no	—
77	C₂×C₈	—	257	2	no	—
78	C₂×C₈	—	257	4	no	—
79	C₂×C₈	C₄×D₂₅₇	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	514	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	514	1	no	—
84	C₄²	—	257	2	no	—
85	C₄²	C₄×D₂₅₇	1	4	yes	—
86	C₂³	#16861617	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	514	2	no	—
89	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₂₅₇	1	2	yes	—
91	C₂×C₄	#16861617	1	2	yes	—
92	C₂×C₄	#16861619	1	1	yes	characteristic
93	C₈	—	257	1	no	—
94	C₈	—	257	1	no	—
95	C₈	—	257	2	no	—
96	C₈	—	2	2	no	—
97	C₈	—	514	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#22551548	1	2	yes	—
100	C₂²	#22551550	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	1028	2	no	—
103	C₄	C₄²×D₂₅₇	1	1	yes	characteristic
104	C₄	#22551548	1	1	yes	characteristic
105	C₄	#22551550	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#30656582	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30656695

C₂

yes

maximal, characteristic

#30656693

C₂

yes

maximal

#30656495

C₂

yes

maximal, characteristic

#30656493

C₂

yes

maximal

H(3,ℤ/4)×C₂₅₇

C₂

yes

maximal, characteristic

C₂×#16861602

C₂²

yes

—

C₂²×#14523339

C₂²

yes

characteristic

#22551551

C₂²

yes

characteristic

#22551503

C₄

yes

complemented

#22551502

C₄

yes

complemented

C₄×#14523339

C₄

yes

complemented

C₈×Dic₂₅₇

C₄

yes

complemented

C₂×M₁₆×C₂₅₇

C₂²

yes

characteristic

#117×C₂₅₇

C₂²

yes

—

#16861602

—

#16861602

—

C₂×#14523339

C₂×C₄

yes

—

C₂×#14523339

C₂×C₄

yes

characteristic

C₂×#14523339

C₂×C₄

yes

—

C₂×#14523339

C₂×C₄

yes

characteristic

C₂×#14523339

C₂×C₄

yes

—

C₄⋊Dic₂₅₇

—

C₂²⋊Dic₂₅₇

—

C₄×Dic₂₅₇

C₂×C₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

C₂×C₄

yes

—

C₄⋊C₄×C₂₅₇

—

C₂²⋊C₄×C₂₅₇

—

C₄²×C₂₅₇

C₂×C₄

yes

—

C₂×Dic₂₅₇

—

#14523339

—

#14523339

C₄²

yes

characteristic

#14523339

C₄²

yes

characteristic

#14523339

C₄²

yes

—

C₂³×C₂₅₇

#41

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₂×C₄×C₂₅₇

#41

yes

characteristic

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

#117

yes

—

C₂²×C₂₅₇

#117

yes

characteristic

C₄×C₂₅₇

—

C₄×C₂₅₇

C₂×C₄²

yes

C₄×C₂₅₇

yes

C₄×C₂₅₇

yes

—

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

#376

yes

characteristic

C₂₅₇

#1747

yes

characteristic, complemented, Sylow

#1747

—

257

maximal, Sylow

#405

—

257

—

#405

—

257

—

#338

—

257

—

H(3,ℤ/4)

—

257

—

H(3,ℤ/4)

D₂₅₇

yes

characteristic

C₂×M₁₆

—

257

—

C₂×M₁₆

C₂×D₂₅₇

yes

characteristic

C₂²×C₈

—

257

—

#117

—

257

—

#117

C₂×D₂₅₇

yes

—

#97

—

257

—

#97

—

257

—

C₄×C₈

—

257

—

C₄×C₈

—

257

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

M₁₆

—

M₁₆

—

514

—

M₁₆

—

514

—

C₂×C₈

—

257

—

C₂×C₈

—

257

—

C₂×C₈

—

257

—

C₂×C₈

—

257

—

C₂×C₈

—

257

—

C₂×C₈

C₄×D₂₅₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

514

—

C₂²⋊C₄

—

C₂²⋊C₄

—

514

—

C₄²

—

257

—

C₄²

C₄×D₂₅₇

yes

—

C₂³

#16861617

yes

characteristic

C₂×C₄

—

C₂×C₄

—

514

—

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

C₂×C₄

C₂×C₄×D₂₅₇

yes

—

C₂×C₄

#16861617

yes

—

C₂×C₄

#16861619

yes

characteristic

C₈

—

257

—

C₈

—

257

—

C₈

—

257

—

C₈

—

C₈

—

514

—

C₂²

—

C₂²

#22551548

yes

—

100

C₂²

#22551550

yes

characteristic

101

C₄

—

102

C₄

—

1028

—

103

C₄

C₄²×D₂₅₇

yes

characteristic

104

C₄

#22551548

yes

characteristic

105

C₄

#22551550

yes

—

106

C₂

—

107

C₂

—

108

C₂

#30656582

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₂₅₇
commutator	C₄×C₂₅₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	H(3,ℤ/4)×C₂₅₇	C₂
socle	C₂×C₂₅₇	#376
prime radical	C₂³×C₂₅₇	#41
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	H(3,ℤ/4)	D₂₅₇
hypocenter	C₂₅₇	#1747
LCS	G ▷ C₄×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₂₅₇

commutator

C₄×C₂₅₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

H(3,ℤ/4)×C₂₅₇

C₂

C₂×C₂₅₇

C₂³×C₂₅₇

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₄×C₂₅₇ ▷ C₂₅₇

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #22551503	C₄	semidirect
10. #22551502	C₄	semidirect
11. C₄×#14523339	C₄	semidirect
12. C₈×Dic₂₅₇	C₄	semidirect
54. C₂₅₇	#1747	semidirect

H ≅ G/N

type

9. #22551503

C₄

semidirect

10. #22551502

C₄

semidirect

11. C₄×#14523339

C₄

semidirect

12. C₈×Dic₂₅₇

C₄

semidirect

54. C₂₅₇

#1747

semidirect

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

subgroup	index
#30656695	2
#30656693	2
#30656495	2
#30656493	2
H(3,ℤ/4)×C₂₅₇	2
#1747	257

subgroup

index

H(3,ℤ/4)×C₂₅₇

257

Subgroups — #45664074

Conjugacy classes of subgroups

Complements

Maximal subgroups