Group #45664439 — subgroups

Subgroups — #45664439

Order 32896.

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

normal subgroups only

This group has 216 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×C₄⋊#14523339	C₂	1	1	yes	maximal, characteristic
2	#30656583	C₂	1	1	yes	maximal, characteristic
3	#30656491	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14523339	C₂²	1	2	yes	—
6	C₂×C₄×Dic₂₅₇	C₂²	1	2	yes	—
7	C₄⋊#14523339	—	2	2	no	—
8	#22551499	C₄	1	2	yes	—
9	#22551499	C₄	1	2	yes	—
10	C₂²×C₈×C₂₅₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₂₅₇	—	2	2	no	—
13	C₂²×Dic₂₅₇	C₂×C₄	1	4	yes	—
14	C₂×#14523339	—	2	2	no	—
15	C₂×#14523339	—	2	4	no	—
16	C₄×Dic₂₅₇	—	2	2	no	—
17	C₄×Dic₂₅₇	—	2	2	no	—
18	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₂₅₇	—	2	2	no	—
22	C₂×C₈×C₂₅₇	—	2	4	no	—
23	C₄²×C₂₅₇	D₄	1	2	yes	—
24	C₄²×C₂₅₇	Q₈	1	2	yes	—
25	C₂×Dic₂₅₇	—	2	4	no	—
26	C₂×Dic₂₅₇	—	2	4	no	—
27	C₂×Dic₂₅₇	—	2	4	no	—
28	#14523339	—	4	2	no	—
29	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₅₇	—	2	2	no	—
31	C₂×C₄×C₂₅₇	—	2	2	no	—
32	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₂₅₇	C₂×D₄	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	C₈×C₂₅₇	—	4	2	no	—
39	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₂₅₇	#117	1	2	yes	—
41	C₂²×C₂₅₇	#131	1	2	yes	—
42	C₂²×C₂₅₇	#131	1	2	yes	—
43	C₄×C₂₅₇	—	2	2	no	—
44	C₄×C₂₅₇	—	2	2	no	—
45	C₄×C₂₅₇	#124	1	1	yes	characteristic
46	C₄×C₂₅₇	#124	1	1	yes	characteristic
47	C₄×C₂₅₇	#125	1	2	yes	—
48	Dic₂₅₇	—	4	4	no	—
49	C₂×C₂₅₇	#382	1	1	yes	characteristic
50	C₂×C₂₅₇	#424	1	1	yes	characteristic
51	C₂×C₂₅₇	#424	1	1	yes	characteristic
52	C₂×C₂₅₇	#433	1	4	yes	—
53	C₂₅₇	#1832	1	1	yes	characteristic, complemented, Sylow
54	#1832	—	257	1	no	maximal, Sylow
55	C₂×C₄⋊C₈	—	257	1	no	—
56	C₂×C₄⋊C₈	D₂₅₇	1	1	yes	characteristic
57	#376	—	257	1	no	—
58	#336	—	257	4	no	—
59	C₂²×C₈	—	257	2	no	—
60	C₂²×C₈	C₂×D₂₅₇	1	2	yes	—
61	C₂×C₄²	—	257	2	no	—
62	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	514	2	no	—
65	#95	—	257	2	no	—
66	#95	—	257	2	no	—
67	C₂²×C₄	—	257	4	no	—
68	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
69	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	514	2	no	—
74	C₂×C₈	—	514	4	no	—
75	C₄²	—	514	2	no	—
76	C₄²	—	514	2	no	—
77	C₄²	C₂²⋊D₂₅₇	1	2	yes	—
78	C₄²	Dic₅₁₄	1	2	yes	—
79	C₂³	C₂×C₄×D₂₅₇	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	514	4	no	—
83	C₂×C₄	—	514	4	no	—
84	C₂×C₄	—	514	4	no	—
85	C₂×C₄	#16861597	1	2	yes	—
86	C₂×C₄	#16861597	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊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	—	4	2	no	—
92	C₈	—	1028	2	no	—
93	C₂²	C₂×#16861597	1	1	yes	characteristic
94	C₂²	#22551550	1	2	yes	—
95	C₂²	#22551598	1	2	yes	—
96	C₂²	#22551600	1	2	yes	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	1028	4	no	—
100	C₄	#22551579	1	1	yes	characteristic
101	C₄	#22551581	1	1	yes	characteristic
102	C₄	#22551583	1	2	yes	—
103	C₂	#30656602	1	1	yes	characteristic
104	C₂	#30656763	1	1	yes	characteristic
105	C₂	#30656765	1	1	yes	characteristic
106	C₂	#30656797	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#14523339

C₂

yes

maximal, characteristic

#30656583

C₂

yes

maximal, characteristic

#30656491

C₂

yes

maximal

C₂×C₄⋊C₈×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×#14523339

C₂²

yes

—

C₂×C₄×Dic₂₅₇

C₂²

yes

—

C₄⋊#14523339

—

#22551499

C₄

yes

—

#22551499

C₄

yes

—

C₂²×C₈×C₂₅₇

C₂²

yes

—

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₇

—

C₂²×Dic₂₅₇

C₂×C₄

yes

—

C₂×#14523339

—

C₂×#14523339

—

C₄×Dic₂₅₇

—

C₄×Dic₂₅₇

—

C₂²×C₄×C₂₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₄²×C₂₅₇

D₄

yes

—

C₄²×C₂₅₇

Q₈

yes

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₂×C₄×C₂₅₇

#41

yes

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₇

#117

yes

—

C₂²×C₂₅₇

#131

yes

—

C₂²×C₂₅₇

#131

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

yes

C₄×C₂₅₇

yes

C₄×C₂₅₇

yes

—

Dic₂₅₇

—

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

—

C₂₅₇

#1832

yes

characteristic, complemented, Sylow

#1832

—

257

maximal, Sylow

C₂×C₄⋊C₈

—

257

—

C₂×C₄⋊C₈

D₂₅₇

yes

characteristic

#376

—

257

—

#336

—

257

—

C₂²×C₈

—

257

—

C₂²×C₈

C₂×D₂₅₇

yes

—

C₂×C₄²

—

257

—

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

514

—

#95

—

257

—

#95

—

257

—

C₂²×C₄

—

257

—

C₂²×C₄

C₄×D₂₅₇

yes

characteristic

C₂²×C₄

C₄×D₂₅₇

yes

characteristic

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

514

—

C₂×C₈

—

514

—

C₄²

—

514

—

C₄²

—

514

—

C₄²

C₂²⋊D₂₅₇

yes

—

C₄²

Dic₅₁₄

yes

—

C₂³

C₂×C₄×D₂₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

514

—

C₂×C₄

—

514

—

C₂×C₄

—

514

—

C₂×C₄

#16861597

yes

—

C₂×C₄

#16861597

yes

—

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

C₂×C₄

C₂×Dic₅₁₄

yes

characteristic

C₂×C₄

#16861617

yes

—

C₂×C₄

#16861619

yes

—

C₈

—

C₈

—

1028

—

C₂²

C₂×#16861597

yes

characteristic

C₂²

#22551550

yes

—

C₂²

#22551598

yes

—

C₂²

#22551600

yes

—

C₄

—

C₄

—

C₄

—

1028

—

100

C₄

#22551579

yes

characteristic

101

C₄

#22551581

yes

characteristic

102

C₄

#22551583

yes

—

103

C₂

#30656602

yes

characteristic

104

C₂

#30656763

yes

characteristic

105

C₂

#30656765

yes

characteristic

106

C₂

#30656797

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₅₇
commutator, socle, prime radical	C₂³×C₂₅₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₂₅₇	C₂
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₅₇
hypocenter	C₂₅₇	#1832
LCS	G ▷ C₂³×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₅₇

commutator, socle, prime radical

C₂³×C₂₅₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₂₅₇

C₂

cosocle

C₂²×C₄×C₂₅₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₅₇

C₂₅₇

G ▷ C₂³×C₂₅₇ ▷ C₂₅₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
53. C₂₅₇	#1832	semidirect

H ≅ G/N

type

53. C₂₅₇

#1832

semidirect

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

subgroup	index
C₂×C₄⋊#14523339	2
#30656583	2
#30656491	2
C₂×C₄⋊C₈×C₂₅₇	2
#1832	257

subgroup

index

C₂×C₄⋊C₈×C₂₅₇

257

Subgroups — #45664439

Conjugacy classes of subgroups

Complements

Maximal subgroups