Group #45664606 — subgroups

Subgroups — #45664606

Order 32896.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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	#30656765	C₂	1	2	yes	maximal, complemented
2	#30656711	C₂	1	1	yes	maximal, characteristic
3	#30656704	C₂	1	1	yes	maximal, characteristic
4	#30656508	C₂	1	2	yes	maximal, complemented
5	#568×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#16861602	C₂²	1	1	yes	characteristic
7	C₂²×#14523339	C₂²	1	1	yes	characteristic
8	#22551551	C₂²	1	2	yes	—
9	#22551521	—	2	4	no	—
10	#22551507	C₄	1	4	yes	—
11	C₂²⋊#14523339	—	2	2	no	—
12	C₂×#41×C₂₅₇	C₂²	1	1	yes	characteristic
13	#131×C₂₅₇	—	2	4	no	—
14	C₂×M₁₆×C₂₅₇	—	2	2	no	—
15	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
16	#16861602	—	2	2	no	—
17	C₂×#14523339	—	2	1	no	—
18	C₂×#14523339	—	2	2	no	—
19	C₂×#14523339	—	2	1	no	—
20	C₄⋊Dic₂₅₇	—	2	2	no	—
21	C₂²⋊Dic₂₅₇	C₂×C₄	1	4	yes	—
22	C₄×Dic₂₅₇	—	2	2	no	—
23	#41×C₂₅₇	—	2	2	no	—
24	#41×C₂₅₇	—	2	2	no	—
25	C₂×Q₈×C₂₅₇	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₂₅₇	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₂₅₇	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
29	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
30	M₁₆×C₂₅₇	—	2	4	no	—
31	M₁₆×C₂₅₇	—	2	2	no	—
32	M₁₆×C₂₅₇	—	2	2	no	—
33	C₂×C₈×C₂₅₇	—	2	4	no	—
34	C₂×C₈×C₂₅₇	D₄	1	4	yes	—
35	C₂×Dic₂₅₇	—	2	4	no	—
36	#14523339	—	4	1	no	—
37	#14523339	—	4	1	no	—
38	C₂³×C₂₅₇	C₄⋊C₄	1	2	yes	—
39	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
40	Q₈×C₂₅₇	—	2	2	no	—
41	D₄×C₂₅₇	—	2	4	no	—
42	D₄×C₂₅₇	—	2	2	no	—
43	C₂×C₄×C₂₅₇	—	2	2	no	—
44	C₂×C₄×C₂₅₇	—	2	2	no	—
45	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₂₅₇	#41	1	2	yes	—
50	C₈×C₂₅₇	—	2	4	no	—
51	C₂²×C₂₅₇	—	2	2	no	—
52	C₂²×C₂₅₇	—	4	1	no	—
53	C₂²×C₂₅₇	—	4	2	no	—
54	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₅₇	C₄×D₄	1	2	yes	—
56	C₄×C₂₅₇	—	2	2	no	—
57	C₄×C₂₅₇	#120	1	1	yes	characteristic
58	C₄×C₂₅₇	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₂₅₇	#123	1	1	yes	characteristic
60	Dic₂₅₇	—	4	4	no	—
61	C₂×C₂₅₇	—	2	2	no	—
62	C₂×C₂₅₇	—	2	1	no	—
63	C₂×C₂₅₇	—	4	2	no	—
64	C₂×C₂₅₇	#385	1	1	yes	characteristic
65	C₂₅₇	#1868	1	1	yes	characteristic, complemented, Sylow
66	#1868	—	257	1	no	maximal, Sylow
67	#568	D₂₅₇	1	1	yes	characteristic
68	#424	—	257	2	no	—
69	#410	—	257	1	no	—
70	#408	—	257	1	no	—
71	#344	—	257	2	no	—
72	C₂×#41	C₂×D₂₅₇	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	257	1	no	—
75	C₂×M₁₆	—	2	2	no	—
76	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
77	C₂²×C₈	—	257	1	no	—
78	#117	—	257	2	no	—
79	C₄⋊C₈	—	514	4	no	—
80	#99	—	257	4	no	—
81	C₂²⋊C₈	—	514	2	no	—
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Dic₅₁₄	1	1	yes	characteristic
85	C₂×D₄	C₄×D₂₅₇	1	2	yes	—
86	C₂×D₄	Dic₅₁₄	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₂₅₇	1	2	yes	—
88	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
89	M₁₆	—	2	4	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	514	2	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	514	1	no	—
95	C₂×C₈	—	514	1	no	—
96	C₂×C₈	—	514	2	no	—
97	C₂×C₈	C₂²⋊D₂₅₇	1	4	yes	—
98	C₄⋊C₄	—	514	2	no	—
99	C₂²⋊C₄	—	257	4	no	—
100	C₄²	—	514	2	no	—
101	C₂³	#16861597	1	2	yes	—
102	C₂³	C₂×C₄×D₂₅₇	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	514	4	no	—
109	C₂×C₄	#16861597	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
112	C₂×C₄	C₂×Dic₅₁₄	1	1	yes	characteristic
113	C₂×C₄	#16861618	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	1028	1	no	—
116	C₈	—	1028	1	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×#16861597	1	1	yes	characteristic
121	C₂²	C₄×C₂²⋊D₂₅₇	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	1028	4	no	—
124	C₄	#22551561	1	1	yes	characteristic
125	C₄	#22551571	1	2	yes	—
126	C₄	#22551575	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	#30656610	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30656765

C₂

yes

maximal, complemented

#30656711

C₂

yes

maximal, characteristic

#30656704

C₂

yes

maximal, characteristic

#30656508

C₂

yes

maximal, complemented

#568×C₂₅₇

C₂

yes

maximal, characteristic

C₂×#16861602

C₂²

yes

characteristic

C₂²×#14523339

C₂²

yes

characteristic

#22551551

C₂²

yes

—

#22551521

—

#22551507

C₄

yes

—

C₂²⋊#14523339

—

C₂×#41×C₂₅₇

C₂²

yes

characteristic

#131×C₂₅₇

—

C₂×M₁₆×C₂₅₇

—

C₂×M₁₆×C₂₅₇

C₂²

yes

—

—

—

—

—

C₄⋊Dic₂₅₇

—

C₂²⋊Dic₂₅₇

C₂×C₄

yes

—

C₄×Dic₂₅₇

—

#41×C₂₅₇

—

#41×C₂₅₇

—

C₂×Q₈×C₂₅₇

Q₈

yes

characteristic

C₂×D₄×C₂₅₇

C₂×C₄

yes

—

C₂×D₄×C₂₅₇

Q₈

yes

characteristic

C₂²×C₄×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

M₁₆×C₂₅₇

—

M₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

D₄

yes

—

C₂×Dic₂₅₇

—

#14523339

—

#14523339

—

C₂³×C₂₅₇

C₄⋊C₄

yes

—

C₂³×C₂₅₇

C₂²×C₄

yes

characteristic

Q₈×C₂₅₇

—

D₄×C₂₅₇

—

D₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₇

C₄×D₄

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#120

yes

characteristic

C₄×C₂₅₇

C₂²⋊Q₈

yes

—

C₄×C₂₅₇

#123

yes

characteristic

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

#385

yes

characteristic

C₂₅₇

#1868

yes

characteristic, complemented, Sylow

#1868

—

257

maximal, Sylow

#568

D₂₅₇

yes

characteristic

#424

—

257

—

#410

—

257

—

#408

—

257

—

#344

—

257

—

C₂×#41

C₂×D₂₅₇

yes

characteristic

#131

—

C₂×M₁₆

—

257

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₅₇

yes

—

C₂²×C₈

—

257

—

#117

—

257

—

C₄⋊C₈

—

514

—

#99

—

257

—

C₂²⋊C₈

—

514

—

#41

—

#41

—

C₂×Q₈

Dic₅₁₄

yes

characteristic

C₂×D₄

C₄×D₂₅₇

yes

—

C₂×D₄

Dic₅₁₄

yes

characteristic

C₂²×C₄

C₂²⋊D₂₅₇

yes

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

514

—

C₂×C₈

—

C₂×C₈

—

514

—

C₂×C₈

—

514

—

C₂×C₈

—

514

—

C₂×C₈

C₂²⋊D₂₅₇

yes

—

C₄⋊C₄

—

514

—

C₂²⋊C₄

—

257

—

100

C₄²

—

514

—

101

C₂³

#16861597

yes

—

102

C₂³

C₂×C₄×D₂₅₇

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

514

—

109

C₂×C₄

#16861597

yes

—

110

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

—

111

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

112

C₂×C₄

C₂×Dic₅₁₄

yes

characteristic

113

C₂×C₄

#16861618

yes

—

114

C₈

—

115

C₈

—

1028

—

116

C₈

—

1028

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×#16861597

yes

characteristic

121

C₂²

C₄×C₂²⋊D₂₅₇

yes

—

122

C₄

—

123

C₄

—

1028

—

124

C₄

#22551561

yes

characteristic

125

C₄

#22551571

yes

—

126

C₄

#22551575

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#30656610

yes

characteristic

	subgroup	G/H
center	C₄	#22551561
commutator	C₂³×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#568×C₂₅₇	C₂
socle	C₂×C₂₅₇	#385
prime radical	C₂×#41×C₂₅₇	C₂²
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#568	D₂₅₇
hypocenter	C₂₅₇	#1868
LCS	G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₄

#22551561

commutator

C₂³×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#568×C₂₅₇

C₂

socle

C₂×C₂₅₇

#385

prime radical

C₂×#41×C₂₅₇

C₂²

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
1. #30656765	C₂	semidirect
4. #30656508	C₂	semidirect
65. C₂₅₇	#1868	semidirect

H ≅ G/N

type

1. #30656765

C₂

semidirect

4. #30656508

C₂

semidirect

65. C₂₅₇

#1868

semidirect

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

subgroup	index
#30656765	2
#30656711	2
#30656704	2
#30656508	2
#568×C₂₅₇	2
#1868	257

subgroup

index

#568×C₂₅₇

257

Subgroups — #45664606

Conjugacy classes of subgroups

Complements

Maximal subgroups