Group #45664446 — subgroups

Subgroups — #45664446

Order 32896.

248 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₈×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
2	#30656606	C₂	1	2	yes	maximal
3	#30656500	C₂	1	1	yes	maximal, characteristic
4	#30656499	C₂	1	2	yes	maximal
5	#340×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14523339	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
8	C₂×#16861597	—	2	2	no	—
9	C₂×#16861597	—	2	2	no	—
10	C₂×C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
11	C₈×Dic₂₅₇	C₄	1	4	yes	complemented
12	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₅₇	C₂²	1	2	yes	—
14	C₂²×Dic₂₅₇	—	2	2	no	—
15	C₂²×Dic₂₅₇	D₄	1	1	yes	characteristic
16	C₂²×Dic₂₅₇	D₄	1	1	yes	characteristic
17	C₂×#14523339	—	2	2	no	—
18	C₂×#14523339	C₂×C₄	1	2	yes	—
19	C₄⋊Dic₂₅₇	—	2	2	no	—
20	C₄⋊Dic₂₅₇	—	2	2	no	—
21	#16861597	—	4	4	no	—
22	#16861597	—	4	4	no	—
23	C₄×Dic₂₅₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₂₅₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₅₇	—	2	2	no	—
26	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₅₇	—	2	2	no	—
28	C₂×C₈×C₂₅₇	C₂×C₄	1	2	yes	—
29	C₄⋊C₄×C₂₅₇	—	2	2	no	—
30	C₄⋊C₄×C₂₅₇	—	2	2	no	—
31	C₂×Dic₂₅₇	—	4	2	no	—
32	C₂×Dic₂₅₇	—	4	2	no	—
33	C₂×Dic₂₅₇	—	4	2	no	—
34	C₂×Dic₂₅₇	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₂₅₇	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₂₅₇	D₈	1	2	yes	—
37	C₂×Dic₂₅₇	SD₁₆	1	4	yes	—
38	C₂×Dic₂₅₇	Q₁₆	1	2	yes	—
39	#14523339	—	2	2	no	—
40	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₅₇	—	4	2	no	—
42	C₂×C₄×C₂₅₇	—	4	2	no	—
43	C₂×C₄×C₂₅₇	—	4	2	no	—
44	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₅₇	#41	1	2	yes	—
47	C₂×C₄×C₂₅₇	#41	1	2	yes	—
48	C₈×C₂₅₇	—	2	2	no	—
49	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₅₇	#124	1	2	yes	—
51	C₂²×C₂₅₇	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₂₅₇	C₂×SD₁₆	1	2	yes	—
53	C₂²×C₂₅₇	C₂×Q₁₆	1	1	yes	characteristic
54	C₄×C₂₅₇	—	8	2	no	—
55	C₄×C₂₅₇	#117	1	1	yes	characteristic
56	C₄×C₂₅₇	#117	1	1	yes	characteristic
57	C₄×C₂₅₇	#125	1	2	yes	—
58	Dic₂₅₇	—	8	2	no	—
59	Dic₂₅₇	#106	1	4	yes	—
60	Dic₂₅₇	#107	1	4	yes	—
61	C₂×C₂₅₇	#382	1	1	yes	characteristic
62	C₂×C₂₅₇	C₂×#106	1	1	yes	characteristic
63	C₂×C₂₅₇	C₂×#107	1	1	yes	characteristic
64	C₂×C₂₅₇	#486	1	2	yes	—
65	C₂×C₂₅₇	#487	1	2	yes	—
66	C₂₅₇	#1834	1	1	yes	characteristic, complemented, Sylow
67	#1834	—	257	1	no	maximal, Sylow
68	C₂×C₄×C₈	—	257	1	no	—
69	#384	—	257	2	no	—
70	#340	—	257	1	no	—
71	#340	—	257	2	no	—
72	#340	D₂₅₇	1	1	yes	characteristic
73	C₂²×C₈	—	257	1	no	—
74	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	257	2	no	—
76	C₂×C₄⋊C₄	—	514	2	no	—
77	C₂×C₄⋊C₄	—	514	2	no	—
78	C₂×C₄⋊C₄	C₂×D₂₅₇	1	2	yes	—
79	C₂×C₄²	—	257	1	no	—
80	C₄×C₈	—	257	4	no	—
81	C₂²×C₄	—	257	1	no	—
82	C₂²×C₄	—	257	1	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	514	2	no	—
85	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
86	C₂×C₈	—	257	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	514	2	no	—
89	C₂×C₈	C₄×D₂₅₇	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	514	2	no	—
93	C₄⋊C₄	—	514	2	no	—
94	C₄⋊C₄	—	1028	4	no	—
95	C₄⋊C₄	—	1028	4	no	—
96	C₄²	—	257	2	no	—
97	C₄²	—	257	2	no	—
98	C₂³	D₄×D₂₅₇	1	1	yes	characteristic
99	C₂×C₄	—	257	2	no	—
100	C₂×C₄	—	257	4	no	—
101	C₂×C₄	—	257	2	no	—
102	C₂×C₄	—	257	2	no	—
103	C₂×C₄	—	257	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	1028	2	no	—
108	C₂×C₄	—	1028	2	no	—
109	C₂×C₄	—	1028	2	no	—
110	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₂₅₇	1	1	yes	characteristic
112	C₂×C₄	#16861617	1	2	yes	—
113	C₂×C₄	#16861618	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	514	2	no	—
116	C₂²	C₄⋊C₄×D₂₅₇	1	1	yes	characteristic
117	C₂²	#22551578	1	2	yes	—
118	C₂²	D₈×D₂₅₇	1	1	yes	characteristic
119	C₂²	SD₁₆×D₂₅₇	1	2	yes	—
120	C₂²	Q₁₆×D₂₅₇	1	1	yes	characteristic
121	C₄	—	257	4	no	—
122	C₄	—	257	4	no	—
123	C₄	—	8	2	no	—
124	C₄	—	2056	2	no	—
125	C₄	#22551548	1	1	yes	characteristic
126	C₄	#22551549	1	1	yes	characteristic
127	C₄	#22551582	1	2	yes	—
128	C₂	#30656601	1	1	yes	characteristic
129	C₂	#106×D₂₅₇	1	1	yes	characteristic
130	C₂	#107×D₂₅₇	1	1	yes	characteristic
131	C₂	#30657082	1	2	yes	—
132	C₂	#30657086	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₂₅₇

C₂

yes

maximal, characteristic

#30656606

C₂

yes

maximal

#30656500

C₂

yes

maximal, characteristic

#30656499

C₂

yes

maximal

#340×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×#14523339

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₅₇

C₂²

yes

—

C₂×#16861597

—

C₂×#16861597

—

C₂×C₄×Dic₂₅₇

C₂²

yes

characteristic

C₈×Dic₂₅₇

C₄

yes

complemented

C₂²×C₈×C₂₅₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₅₇

C₂²

yes

—

C₂²×Dic₂₅₇

—

C₂²×Dic₂₅₇

D₄

yes

characteristic

C₂²×Dic₂₅₇

D₄

yes

characteristic

C₂×#14523339

—

C₂×#14523339

C₂×C₄

yes

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

#16861597

—

#16861597

—

C₄×Dic₂₅₇

C₂×C₄

yes

—

C₄×Dic₂₅₇

Q₈

yes

—

C₂²×C₄×C₂₅₇

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

C₂×C₄

yes

—

C₄⋊C₄×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

C₄⋊C₄

yes

—

C₂×Dic₂₅₇

C₄⋊C₄

yes

—

C₂×Dic₂₅₇

D₈

yes

—

C₂×Dic₂₅₇

SD₁₆

yes

—

C₂×Dic₂₅₇

Q₁₆

yes

—

#14523339

—

C₂³×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²×C₄

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

#124

yes

—

C₂²×C₂₅₇

C₂×D₈

yes

characteristic

C₂²×C₂₅₇

C₂×SD₁₆

yes

—

C₂²×C₂₅₇

C₂×Q₁₆

yes

characteristic

C₄×C₂₅₇

—

C₄×C₂₅₇

yes

C₄×C₂₅₇

yes

C₄×C₂₅₇

yes

—

Dic₂₅₇

—

Dic₂₅₇

#106

yes

—

Dic₂₅₇

#107

yes

—

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

—

C₂×C₂₅₇

#487

yes

—

C₂₅₇

#1834

yes

characteristic, complemented, Sylow

#1834

—

257

maximal, Sylow

C₂×C₄×C₈

—

257

—

#384

—

257

—

#340

—

257

—

#340

—

257

—

#340

D₂₅₇

yes

characteristic

C₂²×C₈

—

257

—

C₂²×C₈

C₂×D₂₅₇

yes

characteristic

C₂×C₄⋊C₄

—

257

—

C₂×C₄⋊C₄

—

514

—

C₂×C₄⋊C₄

—

514

—

C₂×C₄⋊C₄

C₂×D₂₅₇

yes

—

C₂×C₄²

—

257

—

C₄×C₈

—

257

—

C₂²×C₄

—

257

—

C₂²×C₄

—

257

—

C₂²×C₄

—

C₂²×C₄

—

514

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

C₂×C₈

—

257

—

C₂×C₈

—

C₂×C₈

—

514

—

C₂×C₈

C₄×D₂₅₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

1028

—

C₄⋊C₄

—

1028

—

C₄²

—

257

—

C₄²

—

257

—

C₂³

D₄×D₂₅₇

yes

characteristic

C₂×C₄

—

257

—

100

C₂×C₄

—

257

—

101

C₂×C₄

—

257

—

102

C₂×C₄

—

257

—

103

C₂×C₄

—

257

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

1028

—

108

C₂×C₄

—

1028

—

109

C₂×C₄

—

1028

—

110

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

111

C₂×C₄

Q₈×D₂₅₇

yes

characteristic

112

C₂×C₄

#16861617

yes

—

113

C₂×C₄

#16861618

yes

—

114

C₈

—

115

C₈

—

514

—

116

C₂²

C₄⋊C₄×D₂₅₇

yes

characteristic

117

C₂²

#22551578

yes

—

118

C₂²

D₈×D₂₅₇

yes

characteristic

119

C₂²

SD₁₆×D₂₅₇

yes

—

120

C₂²

Q₁₆×D₂₅₇

yes

characteristic

121

C₄

—

257

—

122

C₄

—

257

—

123

C₄

—

124

C₄

—

2056

—

125

C₄

#22551548

yes

characteristic

126

C₄

#22551549

yes

characteristic

127

C₄

#22551582

yes

—

128

C₂

#30656601

yes

characteristic

129

C₂

#106×D₂₅₇

yes

characteristic

130

C₂

#107×D₂₅₇

yes

characteristic

131

C₂

#30657082

yes

—

132

C₂

#30657086

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₅₇
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#340×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂×D₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#340	D₂₅₇
hypocenter	C₂₅₇	#1834
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₂₅₇

commutator

C₂×C₄×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#340×C₂₅₇

C₂

socle, prime radical

C₂³×C₂₅₇

C₂×D₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
11. C₈×Dic₂₅₇	C₄	semidirect
66. C₂₅₇	#1834	semidirect

H ≅ G/N

type

11. C₈×Dic₂₅₇

C₄

semidirect

66. C₂₅₇

#1834

semidirect

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

subgroup	index
C₂×C₈×Dic₂₅₇	2
#30656606	2
#30656500	2
#30656499	2
#340×C₂₅₇	2
#1834	257

subgroup

index

C₂×C₈×Dic₂₅₇

#340×C₂₅₇

257

Subgroups — #45664446

Conjugacy classes of subgroups

Complements

Maximal subgroups