Group #45665294 — subgroups

Subgroups — #45665294

Order 32896.

276 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×C₄⋊Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14523339	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
7	#22551503	—	2	2	no	—
8	#22551499	—	2	1	no	—
9	#22551499	—	2	1	no	—
10	C₂²×Q₈×C₂₅₇	—	2	1	no	—
11	C₄⋊Q₈×C₂₅₇	—	2	2	no	—
12	C₄⋊Q₈×C₂₅₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₂₅₇	—	2	1	no	—
14	C₂×C₄⋊C₄×C₂₅₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₂₅₇	—	2	2	no	—
17	C₂×#14523339	—	2	2	no	—
18	C₂×#14523339	—	2	2	no	—
19	C₂×#14523339	Q₈	1	4	yes	complemented
20	C₄⋊Dic₂₅₇	—	2	4	no	—
21	C₂×Q₈×C₂₅₇	—	2	4	no	—
22	C₂×Q₈×C₂₅₇	—	4	4	no	—
23	C₂²×C₄×C₂₅₇	—	2	2	no	—
24	C₂²×C₄×C₂₅₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₅₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₅₇	—	2	4	no	—
28	C₄⋊C₄×C₂₅₇	—	2	2	no	—
29	C₄⋊C₄×C₂₅₇	C₂×C₄	1	4	yes	—
30	C₄²×C₂₅₇	C₂×C₄	1	2	yes	—
31	C₄²×C₂₅₇	D₄	1	2	yes	—
32	C₂×Dic₂₅₇	—	4	2	no	—
33	C₂×Dic₂₅₇	—	4	2	no	—
34	C₂×Dic₂₅₇	—	4	2	no	—
35	#14523339	—	2	4	no	—
36	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₂₅₇	—	4	8	no	—
38	C₂×C₄×C₂₅₇	—	2	2	no	—
39	C₂×C₄×C₂₅₇	—	2	2	no	—
40	C₂×C₄×C₂₅₇	—	2	4	no	—
41	C₂×C₄×C₂₅₇	—	4	4	no	—
42	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₅₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₂₅₇	#41	1	2	yes	—
48	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₅₇	#124	1	1	yes	characteristic
50	C₂²×C₂₅₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₂₅₇	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₂₅₇	#137	1	2	yes	—
53	C₄×C₂₅₇	—	2	2	no	—
54	C₄×C₂₅₇	—	2	2	no	—
55	C₄×C₂₅₇	—	4	4	no	—
56	C₄×C₂₅₇	C₄×Q₈	1	2	yes	—
57	C₄×C₂₅₇	C₂²⋊Q₈	1	2	yes	—
58	Dic₂₅₇	—	8	2	no	—
59	C₂×C₂₅₇	#391	1	1	yes	characteristic
60	C₂×C₂₅₇	#418	1	1	yes	characteristic
61	C₂×C₂₅₇	#419	1	1	yes	characteristic
62	C₂×C₂₅₇	#489	1	2	yes	—
63	C₂×C₂₅₇	#501	1	2	yes	—
64	C₂₅₇	#1993	1	1	yes	characteristic, complemented, Sylow
65	#1993	—	257	1	no	maximal, Sylow
66	C₂×C₄⋊Q₈	D₂₅₇	1	1	yes	characteristic
67	C₂×#97	—	257	1	no	—
68	#382	—	257	1	no	—
69	#340	—	257	4	no	—
70	C₂²×Q₈	—	2	1	no	—
71	C₂²×C₈	—	257	2	no	—
72	C₄⋊Q₈	—	2	2	no	—
73	C₄⋊Q₈	Dic₂₅₇	1	4	yes	complemented
74	C₂×C₄⋊C₄	—	257	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	C₂×D₂₅₇	1	2	yes	—
77	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
78	#97	—	514	2	no	—
79	#95	—	514	1	no	—
80	#95	—	514	1	no	—
81	C₂×Q₈	—	2	4	no	—
82	C₂×Q₈	—	4	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	514	2	no	—
85	C₂²×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₂₅₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
88	C₂×C₈	—	257	4	no	—
89	C₂×C₈	—	514	2	no	—
90	C₂×C₈	—	514	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	514	4	no	—
94	C₄⋊C₄	C₂×Dic₂₅₇	1	4	yes	—
95	C₄²	C₂×Dic₂₅₇	1	2	yes	—
96	C₄²	C₂²⋊D₂₅₇	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
98	Q₈	—	4	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	1028	2	no	—
104	C₂×C₄	—	1028	2	no	—
105	C₂×C₄	—	1028	2	no	—
106	C₂×C₄	C₂²⋊Dic₂₅₇	1	2	yes	—
107	C₂×C₄	C₂²⋊Dic₂₅₇	1	2	yes	—
108	C₂×C₄	C₂²×Dic₂₅₇	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₂₅₇	1	2	yes	—
111	C₂×C₄	#16861619	1	2	yes	—
112	C₈	—	514	4	no	—
113	C₂²	C₂×C₂²⋊Dic₂₅₇	1	1	yes	characteristic
114	C₂²	#22551581	1	1	yes	characteristic
115	C₂²	#22551590	1	1	yes	characteristic
116	C₂²	#22551618	1	2	yes	—
117	C₂²	#22551623	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	4	4	no	—
121	C₄	—	2056	2	no	—
122	C₄	Q₈×Dic₂₅₇	1	2	yes	—
123	C₄	#22551568	1	2	yes	—
124	C₂	#30656644	1	1	yes	characteristic
125	C₂	#30656742	1	1	yes	characteristic
126	C₂	#30656746	1	1	yes	characteristic
127	C₂	#30657099	1	2	yes	—
128	C₂	#30657143	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551503

C₂

yes

maximal, characteristic

#30656603

C₂

yes

maximal, characteristic

#30656499

C₂

yes

maximal

C₂×C₄⋊Q₈×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×#14523339

C₂²

yes

—

C₂×C₄⋊Dic₂₅₇

C₂²

yes

—

#22551503

—

#22551499

—

#22551499

—

C₂²×Q₈×C₂₅₇

—

C₄⋊Q₈×C₂₅₇

—

C₄⋊Q₈×C₂₅₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₅₇

—

C₂×C₄⋊C₄×C₂₅₇

C₂²

yes

—

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

C₂²×Dic₂₅₇

—

C₂×#14523339

—

C₂×#14523339

—

C₂×#14523339

Q₈

yes

complemented

C₄⋊Dic₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂²×C₄×C₂₅₇

—

C₂²×C₄×C₂₅₇

D₄

yes

characteristic

C₂²×C₄×C₂₅₇

D₄

yes

characteristic

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

C₂×C₄

yes

—

C₄²×C₂₅₇

C₂×C₄

yes

—

C₄²×C₂₅₇

D₄

yes

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

C₂×D₄

yes

characteristic

Q₈×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×Q₈

yes

—

C₂×C₄×C₂₅₇

#41

yes

—

C₂²×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₇

#124

yes

characteristic

C₂²×C₂₅₇

C₄⋊Q₈

yes

characteristic

C₂²×C₂₅₇

AGL(1,ℤ/8)

yes

—

C₂²×C₂₅₇

#137

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

C₄×Q₈

yes

—

C₄×C₂₅₇

C₂²⋊Q₈

yes

—

Dic₂₅₇

—

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

—

C₂×C₂₅₇

#501

yes

—

C₂₅₇

#1993

yes

characteristic, complemented, Sylow

#1993

—

257

maximal, Sylow

C₂×C₄⋊Q₈

D₂₅₇

yes

characteristic

C₂×#97

—

257

—

#382

—

257

—

#340

—

257

—

C₂²×Q₈

—

C₂²×C₈

—

257

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₂₅₇

yes

complemented

C₂×C₄⋊C₄

—

257

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₅₇

yes

—

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

#97

—

514

—

#95

—

514

—

#95

—

514

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

514

—

C₂²×C₄

C₂²⋊D₂₅₇

yes

characteristic

C₂²×C₄

C₂²⋊D₂₅₇

yes

characteristic

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

C₂×C₈

—

257

—

C₂×C₈

—

514

—

C₂×C₈

—

514

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

514

—

C₄⋊C₄

C₂×Dic₂₅₇

yes

—

C₄²

C₂×Dic₂₅₇

yes

—

C₄²

C₂²⋊D₂₅₇

yes

—

C₂³

C₂×C₂²⋊D₂₅₇

yes

characteristic

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

1028

—

104

C₂×C₄

—

1028

—

105

C₂×C₄

—

1028

—

106

C₂×C₄

C₂²⋊Dic₂₅₇

yes

—

107

C₂×C₄

C₂²⋊Dic₂₅₇

yes

—

108

C₂×C₄

C₂²×Dic₂₅₇

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

110

C₂×C₄

Q₈×D₂₅₇

yes

—

111

C₂×C₄

#16861619

yes

—

112

C₈

—

514

—

113

C₂²

C₂×C₂²⋊Dic₂₅₇

yes

characteristic

114

C₂²

#22551581

yes

characteristic

115

C₂²

#22551590

yes

characteristic

116

C₂²

#22551618

yes

—

117

C₂²

#22551623

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

2056

—

122

C₄

Q₈×Dic₂₅₇

yes

—

123

C₄

#22551568

yes

—

124

C₂

#30656644

yes

characteristic

125

C₂

#30656742

yes

characteristic

126

C₂

#30656746

yes

characteristic

127

C₂

#30657099

yes

—

128

C₂

#30657143

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₅₇
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	C₂×C₄⋊Q₈×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂×D₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₅₇
hypocenter	C₂₅₇	#1993
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₅₇

commutator

C₂×C₄×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

C₂×C₄⋊Q₈×C₂₅₇

C₂

socle, prime radical

C₂³×C₂₅₇

C₂×D₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₂₅₇

hypocenter

C₂₅₇

#1993

LCS

G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₂₅₇	C₄	semidirect
19. C₂×#14523339	Q₈	semidirect
64. C₂₅₇	#1993	semidirect
73. C₄⋊Q₈	Dic₂₅₇	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₂₅₇

C₄

semidirect

19. C₂×#14523339

Q₈

semidirect

64. C₂₅₇

#1993

semidirect

73. C₄⋊Q₈

Dic₂₅₇

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
C₂×C₄⋊Q₈×C₂₅₇	2
#1993	257

subgroup

index

C₂×C₄⋊Q₈×C₂₅₇

257

Subgroups — #45665294

Conjugacy classes of subgroups

Complements

Maximal subgroups