Group #45665472 — subgroups

Subgroups — #45665472

Order 32896.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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×#22551584	C₂	1	1	yes	maximal, characteristic
2	C₂×#22551517	C₂	1	2	yes	maximal
3	#30656643	C₂	1	1	yes	maximal, characteristic
4	#30656500	C₂	1	2	yes	maximal
5	#336×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₁₄	C₂²	1	1	yes	characteristic
7	#22551584	—	2	4	no	—
8	C₂×C₄⋊Dic₂₅₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
10	C₂×#16861597	—	2	2	no	—
11	#22551517	—	2	4	no	—
12	#22551499	—	2	2	no	—
13	C₂²×C₈×C₂₅₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₁₄	—	2	2	no	—
16	C₂×Dic₅₁₄	—	2	1	no	—
17	C₂×Dic₅₁₄	—	2	2	no	—
18	C₂×Dic₅₁₄	Q₈	1	2	yes	—
19	C₂²×Dic₂₅₇	—	2	1	no	—
20	C₂²×Dic₂₅₇	—	2	2	no	—
21	C₂²×Dic₂₅₇	—	2	1	no	—
22	C₄⋊Dic₂₅₇	—	2	2	no	—
23	C₄⋊Dic₂₅₇	—	2	2	no	—
24	C₄⋊Dic₂₅₇	D₄	1	4	yes	—
25	#16861597	—	2	8	no	—
26	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
27	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₂₅₇	—	2	2	no	—
29	C₂×C₈×C₂₅₇	—	2	2	no	—
30	C₂×C₈×C₂₅₇	—	2	2	no	—
31	C₄²×C₂₅₇	—	2	2	no	—
32	Dic₅₁₄	—	2	4	no	—
33	Dic₅₁₄	—	4	2	no	—
34	C₂×Dic₂₅₇	—	2	4	no	—
35	C₂×Dic₂₅₇	—	2	2	no	—
36	C₂×Dic₂₅₇	—	4	1	no	—
37	C₂×Dic₂₅₇	—	4	1	no	—
38	C₂×Dic₂₅₇	—	4	2	no	—
39	C₂×Dic₂₅₇	—	4	2	no	—
40	C₂×Dic₂₅₇	—	4	1	no	—
41	C₂×Dic₂₅₇	—	4	2	no	—
42	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₅₇	—	2	2	no	—
44	C₂×C₄×C₂₅₇	—	2	2	no	—
45	C₂×C₄×C₂₅₇	—	2	2	no	—
46	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
49	C₂×C₄×C₂₅₇	#41	1	2	yes	—
50	C₈×C₂₅₇	—	4	2	no	—
51	C₂²×C₂₅₇	#120	1	1	yes	characteristic
52	C₂²×C₂₅₇	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₂₅₇	#124	1	1	yes	characteristic
54	C₂²×C₂₅₇	#135	1	2	yes	—
55	C₂²×C₂₅₇	#137	1	2	yes	—
56	C₄×C₂₅₇	—	4	2	no	—
57	C₄×C₂₅₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₂₅₇	#124	1	2	yes	—
59	Dic₂₅₇	—	4	2	no	—
60	Dic₂₅₇	—	4	4	no	—
61	Dic₂₅₇	—	8	1	no	—
62	C₂×C₂₅₇	#393	1	1	yes	characteristic
63	C₂×C₂₅₇	#452	1	2	yes	—
64	C₂×C₂₅₇	#479	1	2	yes	—
65	C₂×C₂₅₇	#488	1	1	yes	characteristic
66	C₂×C₂₅₇	#491	1	1	yes	characteristic
67	C₂₅₇	#2026	1	1	yes	characteristic, complemented, Sylow
68	#2026	—	257	1	no	maximal, Sylow
69	C₂×#125	—	257	1	no	—
70	C₂×#103	—	257	2	no	—
71	#391	—	257	1	no	—
72	#340	—	257	2	no	—
73	#336	D₂₅₇	1	1	yes	characteristic
74	C₂²×Q₈	—	257	1	no	—
75	C₂²×C₈	C₂×D₂₅₇	1	2	yes	—
76	#125	—	514	4	no	—
77	C₂×C₄⋊C₄	—	257	2	no	—
78	C₂×C₄⋊C₄	—	257	1	no	—
79	C₂×C₄⋊C₄	—	514	2	no	—
80	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
81	#103	—	514	4	no	—
82	#95	—	514	2	no	—
83	C₂×Q₈	—	257	2	no	—
84	C₂×Q₈	—	514	1	no	—
85	C₂×Q₈	—	514	2	no	—
86	C₂×Q₈	—	514	2	no	—
87	C₂²×C₄	—	514	1	no	—
88	C₂²×C₄	—	514	2	no	—
89	C₂²×C₄	—	514	1	no	—
90	C₂²×C₄	D₁₀₂₈	1	2	yes	—
91	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₄⋊C₄	—	257	4	no	—
96	C₄⋊C₄	—	514	8	no	—
97	C₄⋊C₄	—	514	2	no	—
98	C₄⋊C₄	—	514	2	no	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×D₁₀₂₈	1	1	yes	characteristic
101	Q₈	—	514	4	no	—
102	Q₈	—	1028	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	514	4	no	—
107	C₂×C₄	—	514	2	no	—
108	C₂×C₄	—	1028	1	no	—
109	C₂×C₄	—	1028	1	no	—
110	C₂×C₄	—	1028	2	no	—
111	C₂×C₄	—	1028	2	no	—
112	C₂×C₄	—	1028	1	no	—
113	C₂×C₄	—	1028	2	no	—
114	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
115	C₂×C₄	Q₈×D₂₅₇	1	1	yes	characteristic
116	C₂×C₄	#16861617	1	1	yes	characteristic
117	C₂×C₄	#16861618	1	2	yes	—
118	C₈	—	4	2	no	—
119	C₂²	#22551562	1	1	yes	characteristic
120	C₂²	#22551572	1	1	yes	characteristic
121	C₂²	#22551580	1	1	yes	characteristic
122	C₂²	#22551614	1	2	yes	—
123	C₂²	#22551625	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	1028	4	no	—
126	C₄	—	1028	2	no	—
127	C₄	—	2056	1	no	—
128	C₄	#22551570	1	2	yes	—
129	C₄	#22551578	1	2	yes	—
130	C₂	#30656650	1	1	yes	characteristic
131	C₂	#30656901	1	2	yes	—
132	C₂	#30657045	1	2	yes	—
133	C₂	#30657092	1	1	yes	characteristic
134	C₂	#30657106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551584

C₂

yes

maximal, characteristic

C₂×#22551517

C₂

yes

maximal

#30656643

C₂

yes

maximal, characteristic

#30656500

C₂

yes

maximal

#336×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₁₄

C₂²

yes

characteristic

#22551584

—

C₂×C₄⋊Dic₂₅₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₅₇

C₂²

yes

—

C₂×#16861597

—

#22551517

—

#22551499

—

C₂²×C₈×C₂₅₇

C₂²

yes

—

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

Q₈

yes

—

C₂²×Dic₂₅₇

—

C₂²×Dic₂₅₇

—

C₂²×Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

D₄

yes

—

#16861597

—

C₂²×C₄×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₄²×C₂₅₇

—

Dic₅₁₄

—

Dic₅₁₄

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂³×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×Q₈

yes

C₂×C₄×C₂₅₇

yes

C₂×C₄×C₂₅₇

yes

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

#120

yes

characteristic

C₂²×C₂₅₇

C₂²⋊Q₈

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

yes

—

C₂²×C₂₅₇

#137

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

C₂²⋊Q₈

yes

—

C₄×C₂₅₇

#124

yes

—

Dic₂₅₇

—

Dic₂₅₇

—

Dic₂₅₇

—

C₂×C₂₅₇

#393

yes

characteristic

C₂×C₂₅₇

#452

yes

—

C₂×C₂₅₇

#479

yes

—

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂₅₇

yes

characteristic, complemented, Sylow

#2026

—

257

maximal, Sylow

C₂×#125

—

257

—

C₂×#103

—

257

—

#391

—

257

—

#340

—

257

—

#336

D₂₅₇

yes

characteristic

C₂²×Q₈

—

257

—

C₂²×C₈

C₂×D₂₅₇

yes

—

#125

—

514

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄⋊C₄

—

514

—

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

#103

—

514

—

#95

—

514

—

C₂×Q₈

—

257

—

C₂×Q₈

—

514

—

C₂×Q₈

—

514

—

C₂×Q₈

—

514

—

C₂²×C₄

—

514

—

C₂²×C₄

—

514

—

C₂²×C₄

—

514

—

C₂²×C₄

D₁₀₂₈

yes

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

257

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

514

—

C₄⋊C₄

—

514

—

C₄²

—

100

C₂³

C₂×D₁₀₂₈

yes

characteristic

101

Q₈

—

514

—

102

Q₈

—

1028

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

514

—

107

C₂×C₄

—

514

—

108

C₂×C₄

—

1028

—

109

C₂×C₄

—

1028

—

110

C₂×C₄

—

1028

—

111

C₂×C₄

—

1028

—

112

C₂×C₄

—

1028

—

113

C₂×C₄

—

1028

—

114

C₂×C₄

D₄×D₂₅₇

yes

—

115

C₂×C₄

Q₈×D₂₅₇

yes

characteristic

116

C₂×C₄

#16861617

yes

characteristic

117

C₂×C₄

#16861618

yes

—

118

C₈

—

119

C₂²

#22551562

yes

characteristic

120

C₂²

#22551572

yes

characteristic

121

C₂²

#22551580

yes

characteristic

122

C₂²

#22551614

yes

—

123

C₂²

#22551625

yes

—

124

C₄

—

125

C₄

—

1028

—

126

C₄

—

1028

—

127

C₄

—

2056

—

128

C₄

#22551570

yes

—

129

C₄

#22551578

yes

—

130

C₂

#30656650

yes

characteristic

131

C₂

#30656901

yes

—

132

C₂

#30657045

yes

—

133

C₂

#30657092

yes

characteristic

134

C₂

#30657106

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₂₈
commutator, cosocle	C₂²×C₄×C₂₅₇	C₂³
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#336×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂×D₄
hypercenter	#336	D₂₅₇
hypocenter	C₂₅₇	#2026
LCS	G ▷ C₂²×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₁₀₂₈

commutator, cosocle

C₂²×C₄×C₂₅₇

C₂³

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#336×C₂₅₇

C₂

socle, prime radical

C₂³×C₂₅₇

C₂×D₄

D₂₅₇

C₂₅₇

G ▷ C₂²×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
67. C₂₅₇	#2026	semidirect

H ≅ G/N

type

67. C₂₅₇

#2026

semidirect

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

subgroup	index
C₂×#22551584	2
C₂×#22551517	2
#30656643	2
#30656500	2
#336×C₂₅₇	2
#2026	257

subgroup

index

#336×C₂₅₇

257

Subgroups — #45665472

Conjugacy classes of subgroups

Complements

Maximal subgroups