Group #45664315 — subgroups

Subgroups — #45664315

Order 32896.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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×#22551557	C₂	1	1	yes	maximal, characteristic
2	#30656760	C₂	1	2	yes	maximal
3	C₂×#22551510	C₂	1	2	yes	maximal
4	#30656506	C₂	1	1	yes	maximal, characteristic
5	#343×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₁₄	C₂²	1	1	yes	characteristic
7	C₂×#16861602	C₂²	1	2	yes	—
8	#22551557	—	2	2	no	—
9	#22551557	—	2	2	no	—
10	C₂×#16861597	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₅₇	C₄	1	2	yes	—
12	C₂×C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
13	#22551521	—	2	2	no	—
14	#22551521	—	2	2	no	—
15	#22551510	—	2	4	no	—
16	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₅₇	C₂²	1	1	yes	characteristic
18	C₂×Dic₅₁₄	—	2	2	no	—
19	C₂×Dic₅₁₄	—	2	2	no	—
20	C₂×Dic₅₁₄	D₄	1	2	yes	—
21	C₂×Dic₅₁₄	Q₈	1	2	yes	—
22	C₂²×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₂₅₇	C₂×C₄	1	2	yes	—
24	C₂²×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
25	#16861602	—	2	2	no	—
26	#16861602	—	2	2	no	—
27	C₂×#14523339	—	2	2	no	—
28	#16861597	—	2	4	no	—
29	C₄×Dic₂₅₇	—	2	2	no	—
30	C₄×Dic₂₅₇	—	2	2	no	—
31	C₄×Dic₂₅₇	D₄	1	2	yes	—
32	C₄×Dic₂₅₇	D₄	1	2	yes	—
33	C₂²×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₂₅₇	—	2	2	no	—
36	M₁₆×C₂₅₇	—	2	2	no	—
37	C₂×C₈×C₂₅₇	—	2	2	no	—
38	C₄⋊C₄×C₂₅₇	—	2	2	no	—
39	Dic₅₁₄	—	2	4	no	—
40	Dic₅₁₄	—	2	4	no	—
41	C₂×Dic₂₅₇	—	2	2	no	—
42	C₂×Dic₂₅₇	—	2	2	no	—
43	C₂×Dic₂₅₇	—	2	2	no	—
44	C₂×Dic₂₅₇	—	4	1	no	—
45	C₂×Dic₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×Dic₂₅₇	C₂²⋊C₄	1	2	yes	—
47	C₂×Dic₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₂₅₇	C₄⋊C₄	1	4	yes	—
49	#14523339	—	4	2	no	—
50	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₅₇	—	2	2	no	—
52	C₂×C₄×C₂₅₇	—	4	2	no	—
53	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₅₇	#41	1	2	yes	—
57	C₈×C₂₅₇	—	4	2	no	—
58	C₂²×C₂₅₇	—	2	1	no	—
59	C₂²×C₂₅₇	—	2	1	no	—
60	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₂₅₇	#117	1	1	yes	characteristic
63	C₄×C₂₅₇	—	4	2	no	—
64	C₄×C₂₅₇	#121	1	2	yes	—
65	C₄×C₂₅₇	C₂²⋊Q₈	1	2	yes	—
66	Dic₂₅₇	—	2	4	no	—
67	Dic₂₅₇	—	2	1	no	—
68	Dic₂₅₇	—	2	2	no	—
69	Dic₂₅₇	#101	1	2	yes	complemented
70	C₂×C₂₅₇	—	2	1	no	—
71	C₂×C₂₅₇	—	2	1	no	—
72	C₂×C₂₅₇	#380	1	1	yes	characteristic
73	C₂×C₂₅₇	C₂×#101	1	1	yes	characteristic
74	C₂×C₂₅₇	#413	1	1	yes	characteristic
75	C₂₅₇	#1801	1	1	yes	characteristic, complemented, Sylow
76	#1801	—	257	1	no	maximal, Sylow
77	C₂×C₄×Q₈	—	257	1	no	—
78	#423	—	257	2	no	—
79	C₂×#101	—	257	2	no	—
80	#343	—	257	1	no	—
81	#343	D₂₅₇	1	1	yes	characteristic
82	C₂²×Q₈	—	257	1	no	—
83	C₂×M₁₆	—	257	2	no	—
84	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
85	C₄×Q₈	—	514	2	no	—
86	C₄×Q₈	—	514	2	no	—
87	C₂×C₄⋊C₄	—	257	2	no	—
88	C₂×C₄⋊C₄	C₂×D₂₅₇	1	1	yes	characteristic
89	C₂×C₄²	—	257	1	no	—
90	C₂×C₄²	—	257	2	no	—
91	C₄⋊C₈	—	514	2	no	—
92	C₄⋊C₈	—	514	2	no	—
93	#101	—	514	4	no	—
94	C₂×Q₈	—	257	2	no	—
95	C₂×Q₈	—	257	2	no	—
96	C₂×Q₈	—	514	2	no	—
97	C₂×Q₈	—	514	2	no	—
98	C₂²×C₄	—	257	1	no	—
99	C₂²×C₄	—	257	2	no	—
100	C₂²×C₄	—	257	1	no	—
101	C₂²×C₄	C₄×D₂₅₇	1	2	yes	—
102	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	514	2	no	—
106	M₁₆	—	514	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	514	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	514	4	no	—
111	C₄²	—	257	2	no	—
112	C₄²	—	257	2	no	—
113	C₄²	—	514	2	no	—
114	C₄²	—	514	2	no	—
115	C₂³	C₂×C₄×D₂₅₇	1	1	yes	characteristic
116	Q₈	—	514	4	no	—
117	Q₈	—	514	4	no	—
118	C₂×C₄	—	257	1	no	—
119	C₂×C₄	—	257	1	no	—
120	C₂×C₄	—	257	2	no	—
121	C₂×C₄	—	257	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	514	2	no	—
124	C₂×C₄	—	514	2	no	—
125	C₂×C₄	—	514	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	1028	1	no	—
128	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
129	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
130	C₂×C₄	Q₈×D₂₅₇	1	1	yes	characteristic
131	C₂×C₄	#16861618	1	2	yes	—
132	C₈	—	4	2	no	—
133	C₈	—	1028	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	C₂²⋊C₄×D₂₅₇	1	1	yes	characteristic
137	C₂²	C₄⋊C₄×D₂₅₇	1	1	yes	characteristic
138	C₂²	#22551549	1	1	yes	characteristic
139	C₄	—	257	2	no	—
140	C₄	—	514	4	no	—
141	C₄	—	514	1	no	—
142	C₄	—	514	2	no	—
143	C₄	—	4	2	no	—
144	C₄	#22551565	1	2	yes	—
145	C₄	#22551570	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#30656596	1	1	yes	characteristic
149	C₂	#101×D₂₅₇	1	1	yes	characteristic
150	C₂	#30656718	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551557

C₂

yes

maximal, characteristic

#30656760

C₂

yes

maximal

C₂×#22551510

C₂

yes

maximal

#30656506

C₂

yes

maximal, characteristic

#343×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₁₄

C₂²

yes

characteristic

C₂×#16861602

C₂²

yes

—

#22551557

—

#22551557

—

C₂×#16861597

C₄

yes

—

C₂×C₄×Dic₂₅₇

C₄

yes

—

C₂×C₄×Dic₂₅₇

C₂²

yes

—

—

—

C₂×M₁₆×C₂₅₇

C₂²

yes

—

C₂×C₄⋊C₄×C₂₅₇

C₂²

yes

characteristic

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

D₄

yes

—

C₂×Dic₅₁₄

Q₈

yes

—

C₂²×Dic₂₅₇

C₂×C₄

yes

characteristic

C₂²×Dic₂₅₇

C₂×C₄

yes

—

C₂²×Dic₂₅₇

C₂×C₄

yes

—

—

—

—

C₄×Dic₂₅₇

—

C₄×Dic₂₅₇

—

C₄×Dic₂₅₇

D₄

yes

—

C₄×Dic₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂×C₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

M₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₄⋊C₄×C₂₅₇

—

Dic₅₁₄

—

Dic₅₁₄

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₅₇

C₂²⋊C₄

yes

—

C₂×Dic₂₅₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₅₇

C₄⋊C₄

yes

—

#14523339

—

C₂³×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₇

#41

yes

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₇

#117

yes

characteristic

C₄×C₂₅₇

—

C₄×C₂₅₇

#121

yes

—

C₄×C₂₅₇

C₂²⋊Q₈

yes

—

Dic₂₅₇

—

Dic₂₅₇

—

Dic₂₅₇

—

Dic₂₅₇

#101

yes

complemented

C₂×C₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂₅₇

yes

characteristic, complemented, Sylow

#1801

—

257

maximal, Sylow

C₂×C₄×Q₈

—

257

—

#423

—

257

—

C₂×#101

—

257

—

#343

—

257

—

#343

D₂₅₇

yes

characteristic

C₂²×Q₈

—

257

—

C₂×M₁₆

—

257

—

C₂×M₁₆

C₂×D₂₅₇

yes

—

C₄×Q₈

—

514

—

C₄×Q₈

—

514

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄⋊C₄

C₂×D₂₅₇

yes

characteristic

C₂×C₄²

—

257

—

C₂×C₄²

—

257

—

C₄⋊C₈

—

514

—

C₄⋊C₈

—

514

—

#101

—

514

—

C₂×Q₈

—

257

—

C₂×Q₈

—

257

—

C₂×Q₈

—

514

—

C₂×Q₈

—

514

—

C₂²×C₄

—

257

—

C₂²×C₄

—

257

—

100

C₂²×C₄

—

257

—

101

C₂²×C₄

C₄×D₂₅₇

yes

—

102

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

514

—

106

M₁₆

—

514

—

107

C₂×C₈

—

108

C₂×C₈

—

514

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

514

—

111

C₄²

—

257

—

112

C₄²

—

257

—

113

C₄²

—

514

—

114

C₄²

—

514

—

115

C₂³

C₂×C₄×D₂₅₇

yes

characteristic

116

Q₈

—

514

—

117

Q₈

—

514

—

118

C₂×C₄

—

257

—

119

C₂×C₄

—

257

—

120

C₂×C₄

—

257

—

121

C₂×C₄

—

257

—

122

C₂×C₄

—

123

C₂×C₄

—

514

—

124

C₂×C₄

—

514

—

125

C₂×C₄

—

514

—

126

C₂×C₄

—

127

C₂×C₄

—

1028

—

128

C₂×C₄

D₄×D₂₅₇

yes

characteristic

129

C₂×C₄

D₄×D₂₅₇

yes

—

130

C₂×C₄

Q₈×D₂₅₇

yes

characteristic

131

C₂×C₄

#16861618

yes

—

132

C₈

—

133

C₈

—

1028

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂²⋊C₄×D₂₅₇

yes

characteristic

137

C₂²

C₄⋊C₄×D₂₅₇

yes

characteristic

138

C₂²

#22551549

yes

characteristic

139

C₄

—

257

—

140

C₄

—

514

—

141

C₄

—

514

—

142

C₄

—

514

—

143

C₄

—

144

C₄

#22551565

yes

—

145

C₄

#22551570

yes

—

146

C₂

—

147

C₂

—

148

C₂

#30656596

yes

characteristic

149

C₂

#101×D₂₅₇

yes

characteristic

150

C₂

#30656718

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₅₇
commutator, prime radical	C₂³×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#343×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#343	D₂₅₇
hypocenter	C₂₅₇	#1801
LCS	G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₅₇

commutator, prime radical

C₂³×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#343×C₂₅₇

C₂

socle

C₂²×C₂₅₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
69. Dic₂₅₇	#101	semidirect
75. C₂₅₇	#1801	semidirect

H ≅ G/N

type

69. Dic₂₅₇

#101

semidirect

75. C₂₅₇

#1801

semidirect

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

subgroup	index
C₂×#22551557	2
#30656760	2
C₂×#22551510	2
#30656506	2
#343×C₂₅₇	2
#1801	257

subgroup

index

#343×C₂₅₇

257

Subgroups — #45664315

Conjugacy classes of subgroups

Complements

Maximal subgroups