Group #45663460 — subgroups

Subgroups — #45663460

Order 32896.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#30657116	C₂	1	1	yes	maximal, characteristic
2	#30657032	C₂	1	2	yes	maximal
3	#30656486	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#14523339	C₂	1	2	yes	maximal
5	Q₈²×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#16861608	—	2	2	no	—
7	#22551592	C₂²	1	1	yes	characteristic
8	#22551525	—	2	2	no	—
9	C₄⋊#14523339	C₂²	1	2	yes	—
10	#22551516	—	2	2	no	—
11	C₄×#14523339	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₂₅₇	—	2	2	no	—
13	C₄⋊Q₈×C₂₅₇	—	2	2	no	—
14	C₄⋊Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₂₅₇	—	2	2	no	—
16	C₄×Q₈×C₂₅₇	C₂²	1	2	yes	—
17	C₂×Dic₅₁₄	—	2	1	no	—
18	#16861608	—	4	4	no	—
19	C₂×#14523339	—	2	1	no	—
20	C₂×#14523339	—	2	2	no	—
21	C₄⋊Dic₂₅₇	—	2	2	no	—
22	C₂×Q₈×C₂₅₇	—	2	2	no	—
23	C₂×Q₈×C₂₅₇	D₄	1	2	yes	—
24	C₂×Q₈×C₂₅₇	D₄	1	2	yes	—
25	C₄⋊C₄×C₂₅₇	—	2	4	no	—
26	C₄⋊C₄×C₂₅₇	—	2	2	no	—
27	C₄⋊C₄×C₂₅₇	—	2	2	no	—
28	C₄⋊C₄×C₂₅₇	D₄	1	2	yes	—
29	C₄²×C₂₅₇	—	2	2	no	—
30	C₄²×C₂₅₇	—	2	2	no	—
31	C₄²×C₂₅₇	C₂³	1	1	yes	characteristic
32	Dic₅₁₄	—	4	2	no	—
33	C₂×Dic₂₅₇	—	4	1	no	—
34	#14523339	—	2	2	no	—
35	#14523339	—	4	2	no	—
36	Q₈×C₂₅₇	—	2	2	no	—
37	Q₈×C₂₅₇	—	4	2	no	—
38	Q₈×C₂₅₇	—	8	2	no	—
39	Q₈×C₂₅₇	Q₁₆	1	4	yes	complemented
40	C₂×C₄×C₂₅₇	—	2	2	no	—
41	C₂×C₄×C₂₅₇	—	2	2	no	—
42	C₂×C₄×C₂₅₇	—	2	2	no	—
43	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
45	C₂²×C₂₅₇	#120	1	1	yes	characteristic
46	C₄×C₂₅₇	—	2	4	no	—
47	C₄×C₂₅₇	—	2	1	no	—
48	C₄×C₂₅₇	—	4	2	no	—
49	C₄×C₂₅₇	—	4	2	no	—
50	C₄×C₂₅₇	C₂×Q₁₆	1	2	yes	—
51	C₄×C₂₅₇	#137	1	2	yes	—
52	Dic₂₅₇	—	8	1	no	—
53	C₂×C₂₅₇	#451	1	2	yes	—
54	C₂×C₂₅₇	#453	1	1	yes	characteristic
55	C₂₅₇	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	257	1	no	maximal, Sylow
57	Q₈²	D₂₅₇	1	1	yes	characteristic
58	#494	—	257	1	no	—
59	#477	—	257	2	no	—
60	#333	—	257	1	no	—
61	Q₈⋊C₈	—	257	2	no	—
62	C₂×Q₁₆	—	514	2	no	—
63	C₄⋊Q₈	—	257	1	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	C₂×D₂₅₇	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₂₅₇	1	2	yes	—
69	#107	—	514	2	no	—
70	C₄⋊C₈	—	257	2	no	—
71	#103	—	514	2	no	—
72	C₄×C₈	—	257	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	514	1	no	—
75	C₂×Q₈	C₂²⋊D₂₅₇	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₂₅₇	1	2	yes	—
77	Q₁₆	—	1028	4	no	—
78	C₂×C₈	—	514	1	no	—
79	C₂×C₈	—	514	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	514	2	no	—
84	C₄⋊C₄	C₂²⋊D₂₅₇	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	1028	2	no	—
91	Q₈	—	8	2	no	—
92	Q₈	#16861608	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	1028	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	2	yes	—
99	C₈	—	514	2	no	—
100	C₈	—	1028	2	no	—
101	C₂²	#22551561	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	2056	1	no	—
107	C₄	C₂×#16861608	1	2	yes	—
108	C₄	#22551623	1	2	yes	—
109	C₂	#30656893	1	2	yes	—
110	C₂	#30656905	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30657116

C₂

yes

maximal, characteristic

#30657032

C₂

yes

maximal

#30656486

C₂

yes

maximal, characteristic

Q₈⋊#14523339

C₂

yes

maximal

Q₈²×C₂₅₇

C₂

yes

maximal, characteristic

C₂×#16861608

—

#22551592

C₂²

yes

characteristic

#22551525

—

C₄⋊#14523339

C₂²

yes

—

#22551516

—

C₄×#14523339

C₂²

yes

characteristic

C₄⋊Q₈×C₂₅₇

—

C₄⋊Q₈×C₂₅₇

—

C₄⋊Q₈×C₂₅₇

C₂²

yes

characteristic

C₄×Q₈×C₂₅₇

—

C₄×Q₈×C₂₅₇

C₂²

yes

—

C₂×Dic₅₁₄

—

#16861608

—

C₂×#14523339

—

C₂×#14523339

—

C₄⋊Dic₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

D₄

yes

—

C₂×Q₈×C₂₅₇

D₄

yes

—

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

D₄

yes

—

C₄²×C₂₅₇

—

C₄²×C₂₅₇

—

C₄²×C₂₅₇

C₂³

yes

characteristic

Dic₅₁₄

—

C₂×Dic₂₅₇

—

#14523339

—

#14523339

—

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

Q₈×C₂₅₇

Q₁₆

yes

complemented

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂²×C₂₅₇

#120

yes

characteristic

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

C₂×Q₁₆

yes

—

C₄×C₂₅₇

#137

yes

—

Dic₂₅₇

—

C₂×C₂₅₇

#451

yes

—

C₂×C₂₅₇

#453

yes

characteristic

C₂₅₇

#1633

yes

characteristic, complemented, Sylow

#1633

—

257

maximal, Sylow

Q₈²

D₂₅₇

yes

characteristic

#494

—

257

—

#477

—

257

—

#333

—

257

—

Q₈⋊C₈

—

257

—

C₂×Q₁₆

—

514

—

C₄⋊Q₈

—

257

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₅₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₂₅₇

yes

—

#107

—

514

—

C₄⋊C₈

—

257

—

#103

—

514

—

C₄×C₈

—

257

—

C₂×Q₈

—

C₂×Q₈

—

514

—

C₂×Q₈

C₂²⋊D₂₅₇

yes

—

C₂×Q₈

C₂²⋊D₂₅₇

yes

—

Q₁₆

—

1028

—

C₂×C₈

—

514

—

C₂×C₈

—

514

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

514

—

C₄⋊C₄

C₂²⋊D₂₅₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₂₅₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

1028

—

Q₈

—

Q₈

#16861608

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1028

—

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

—

C₈

—

514

—

100

C₈

—

1028

—

101

C₂²

#22551561

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

2056

—

107

C₄

C₂×#16861608

yes

—

108

C₄

#22551623

yes

—

109

C₂

#30656893

yes

—

110

C₂

#30656905

yes

characteristic

	subgroup	G/H
center	C₂²	#22551561
commutator, cosocle	C₄²×C₂₅₇	C₂³
Frattini	C₄²	C₂²×D₂₅₇
Fitting	Q₈²×C₂₅₇	C₂
socle, prime radical	C₂²×C₂₅₇	#120
hypercenter	Q₈²	D₂₅₇
hypocenter	C₂₅₇	#1633
LCS	G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#22551561

commutator, cosocle

C₄²×C₂₅₇

C₂³

Frattini

C₄²

C₂²×D₂₅₇

Fitting

Q₈²×C₂₅₇

C₂

socle, prime radical

C₂²×C₂₅₇

#120

hypercenter

Q₈²

D₂₅₇

hypocenter

C₂₅₇

#1633

LCS

G ▷ C₄²×C₂₅₇ ▷ C₂²×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
39. Q₈×C₂₅₇	Q₁₆	semidirect
55. C₂₅₇	#1633	semidirect
92. Q₈	#16861608	semidirect

H ≅ G/N

type

39. Q₈×C₂₅₇

Q₁₆

semidirect

55. C₂₅₇

#1633

semidirect

92. Q₈

#16861608

semidirect

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

subgroup	index
#30657116	2
#30657032	2
#30656486	2
Q₈⋊#14523339	2
Q₈²×C₂₅₇	2
#1633	257

subgroup

index

Q₈²×C₂₅₇

257

Subgroups — #45663460

Conjugacy classes of subgroups

Complements

Maximal subgroups