Group #45664365 — subgroups

Subgroups — #45664365

Order 32896.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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×#22551600	C₂	1	1	yes	maximal, characteristic
2	#30656773	C₂	1	2	yes	maximal, complemented
3	#30656710	C₂	1	2	yes	maximal
4	#30656494	C₂	1	1	yes	maximal, characteristic, complemented
5	H(3,ℤ/4)×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16861618	C₂²	1	1	yes	characteristic
7	#22551600	—	2	2	no	—
8	#22551600	—	2	2	no	—
9	C₂×#16861601	C₄	1	2	yes	complemented
10	C₂²×#14523339	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₂₅₇	C₄	1	2	yes	complemented
12	#22551551	C₂²	1	2	yes	—
13	#22551525	—	2	2	no	—
14	#22551523	—	2	2	no	—
15	#22551506	—	2	4	no	—
16	C₂×M₁₆×C₂₅₇	C₂²	1	1	yes	characteristic
17	#117×C₂₅₇	C₂²	1	2	yes	—
18	#16861618	—	2	2	no	—
19	#16861618	—	2	2	no	—
20	C₂×Dic₅₁₄	Q₈	1	1	yes	characteristic
21	C₂×D₁₀₂₈	Q₈	1	1	yes	characteristic
22	C₂×C₂²⋊D₂₅₇	D₄	1	2	yes	—
23	C₂×C₄×D₂₅₇	C₂×C₄	1	2	yes	—
24	#16861601	—	2	4	no	—
25	C₂×#14523339	C₂×C₄	1	1	yes	characteristic
26	C₂×#14523339	C₂×C₄	1	1	yes	characteristic
27	C₂×#14523339	D₄	1	2	yes	—
28	C₂×#14523339	D₄	1	2	yes	—
29	C₈×D₂₅₇	—	2	2	no	—
30	C₈×D₂₅₇	—	2	2	no	—
31	C₄⋊Dic₂₅₇	—	2	2	no	—
32	C₂²⋊Dic₂₅₇	—	2	2	no	—
33	C₄×Dic₂₅₇	—	2	2	no	—
34	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
35	M₁₆×C₂₅₇	—	2	2	no	—
36	C₂×C₈×C₂₅₇	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₂₅₇	—	2	2	no	—
38	C₂²⋊C₄×C₂₅₇	—	2	2	no	—
39	C₄²×C₂₅₇	—	2	2	no	—
40	C₂²×D₂₅₇	C₄⋊C₄	1	2	yes	—
41	Dic₅₁₄	—	2	2	no	—
42	D₁₀₂₈	—	2	2	no	—
43	C₂²⋊D₂₅₇	—	2	4	no	—
44	C₂×Dic₂₅₇	—	4	2	no	—
45	C₂×Dic₂₅₇	C₄⋊C₄	1	2	yes	—
46	C₄×D₂₅₇	—	2	2	no	—
47	C₄×D₂₅₇	—	2	2	no	—
48	#14523339	C₂²⋊C₄	1	1	yes	characteristic
49	#14523339	C₂²⋊C₄	1	1	yes	characteristic
50	#14523339	C₂²⋊C₄	1	2	yes	—
51	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₅₇	—	4	2	no	—
53	C₂×C₄×C₂₅₇	C₂²×C₄	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	—	2	2	no	—
58	C₂²×C₂₅₇	—	4	1	no	—
59	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₅₇	#121	1	2	yes	—
61	C₂×D₂₅₇	—	2	2	no	—
62	C₂×D₂₅₇	—	4	2	no	—
63	C₄×C₂₅₇	—	8	2	no	—
64	C₄×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₂₅₇	#117	1	1	yes	characteristic
66	C₄×C₂₅₇	C₂²⋊Q₈	1	2	yes	—
67	Dic₂₅₇	—	2	2	no	—
68	Dic₂₅₇	—	8	2	no	—
69	C₂×C₂₅₇	—	2	2	no	—
70	C₂×C₂₅₇	—	2	1	no	—
71	C₂×C₂₅₇	#380	1	1	yes	characteristic
72	D₂₅₇	—	4	2	no	—
73	C₂₅₇	#1812	1	1	yes	characteristic, complemented, Sylow
74	#1812	—	257	1	no	maximal, Sylow
75	C₂×#131	—	257	1	no	—
76	#427	—	257	2	no	—
77	#410	—	257	2	no	—
78	H(3,ℤ/4)	—	257	1	no	—
79	H(3,ℤ/4)	D₂₅₇	1	1	yes	characteristic, complemented
80	C₂×#41	—	257	1	no	—
81	#131	—	514	2	no	—
82	#131	—	514	2	no	—
83	C₂×M₁₆	—	257	2	no	—
84	C₂×M₁₆	C₂×D₂₅₇	1	1	yes	characteristic
85	C₂²×C₈	—	257	1	no	—
86	C₂²×C₈	—	257	2	no	—
87	#117	—	257	2	no	—
88	#117	C₂×D₂₅₇	1	2	yes	—
89	#107	—	514	2	no	—
90	#106	—	514	2	no	—
91	#99	—	514	4	no	—
92	#41	—	514	2	no	—
93	#41	—	514	2	no	—
94	C₂×Q₈	—	257	1	no	—
95	C₂×D₄	—	257	2	no	—
96	C₂×D₄	—	257	1	no	—
97	C₂²×C₄	—	257	2	no	—
98	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	514	4	no	—
101	C₂×C₈	—	257	1	no	—
102	C₂×C₈	—	257	1	no	—
103	C₂×C₈	—	257	2	no	—
104	C₂×C₈	—	257	2	no	—
105	C₂×C₈	—	514	2	no	—
106	C₂×C₈	—	514	2	no	—
107	C₂×C₈	C₄×D₂₅₇	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	514	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	514	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	514	2	no	—
114	C₂³	—	257	2	no	—
115	C₂³	D₄×D₂₅₇	1	1	yes	characteristic
116	Q₈	—	514	2	no	—
117	D₄	—	514	4	no	—
118	D₄	—	514	2	no	—
119	C₂×C₄	—	257	2	no	—
120	C₂×C₄	—	514	2	no	—
121	C₂×C₄	—	514	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	1028	2	no	—
124	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
125	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
126	C₂×C₄	Q₈×D₂₅₇	1	1	yes	characteristic
127	C₂×C₄	#16861618	1	2	yes	—
128	C₈	—	257	1	no	—
129	C₈	—	257	1	no	—
130	C₈	—	257	2	no	—
131	C₈	—	2	2	no	—
132	C₂²	—	514	2	no	—
133	C₂²	—	4	1	no	—
134	C₂²	—	1028	2	no	—
135	C₂²	C₄⋊C₄×D₂₅₇	1	1	yes	characteristic
136	C₂²	#22551565	1	2	yes	—
137	C₄	—	514	2	no	—
138	C₄	—	8	2	no	—
139	C₄	—	2056	2	no	—
140	C₄	C₂²⋊C₄×D₂₅₇	1	1	yes	characteristic
141	C₄	#22551549	1	1	yes	characteristic
142	C₄	#22551570	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	1028	2	no	—
146	C₂	#30656596	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551600

C₂

yes

maximal, characteristic

#30656773

C₂

yes

maximal, complemented

#30656710

C₂

yes

maximal

#30656494

C₂

yes

maximal, characteristic, complemented

H(3,ℤ/4)×C₂₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#16861618

C₂²

yes

—

—

C₄

yes

complemented

C₂²×#14523339

C₂²

yes

characteristic

C₂×C₈×D₂₅₇

C₄

yes

complemented

#22551551

C₂²

yes

—

#22551525

—

#22551523

—

#22551506

—

C₂×M₁₆×C₂₅₇

C₂²

yes

characteristic

#117×C₂₅₇

C₂²

yes

—

#16861618

—

#16861618

—

C₂×Dic₅₁₄

Q₈

yes

characteristic

C₂×D₁₀₂₈

Q₈

yes

characteristic

C₂×C₂²⋊D₂₅₇

D₄

yes

—

C₂×C₄×D₂₅₇

C₂×C₄

yes

—

#16861601

—

C₂×#14523339

C₂×C₄

yes

characteristic

C₂×#14523339

C₂×C₄

yes

characteristic

C₂×#14523339

D₄

yes

—

C₂×#14523339

D₄

yes

—

C₈×D₂₅₇

—

C₈×D₂₅₇

—

C₄⋊Dic₂₅₇

—

C₂²⋊Dic₂₅₇

—

C₄×Dic₂₅₇

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

C₂×C₄

yes

—

C₄⋊C₄×C₂₅₇

—

C₂²⋊C₄×C₂₅₇

—

C₄²×C₂₅₇

—

C₂²×D₂₅₇

C₄⋊C₄

yes

—

Dic₅₁₄

—

D₁₀₂₈

—

C₂²⋊D₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

C₄⋊C₄

yes

—

C₄×D₂₅₇

—

C₄×D₂₅₇

—

#14523339

C₂²⋊C₄

yes

characteristic

#14523339

C₂²⋊C₄

yes

characteristic

#14523339

C₂²⋊C₄

yes

—

C₂³×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²×C₄

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

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₇

#121

yes

—

C₂×D₂₅₇

—

C₂×D₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₅₇

#117

yes

characteristic

C₄×C₂₅₇

C₂²⋊Q₈

yes

—

Dic₂₅₇

—

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

#380

yes

characteristic

D₂₅₇

—

C₂₅₇

#1812

yes

characteristic, complemented, Sylow

#1812

—

257

maximal, Sylow

C₂×#131

—

257

—

#427

—

257

—

#410

—

257

—

H(3,ℤ/4)

—

257

—

H(3,ℤ/4)

D₂₅₇

yes

characteristic, complemented

C₂×#41

—

257

—

#131

—

514

—

#131

—

514

—

C₂×M₁₆

—

257

—

C₂×M₁₆

C₂×D₂₅₇

yes

characteristic

C₂²×C₈

—

257

—

C₂²×C₈

—

257

—

#117

—

257

—

#117

C₂×D₂₅₇

yes

—

#107

—

514

—

#106

—

514

—

#99

—

514

—

#41

—

514

—

#41

—

514

—

C₂×Q₈

—

257

—

C₂×D₄

—

257

—

C₂×D₄

—

257

—

C₂²×C₄

—

257

—

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

M₁₆

—

100

M₁₆

—

514

—

101

C₂×C₈

—

257

—

102

C₂×C₈

—

257

—

103

C₂×C₈

—

257

—

104

C₂×C₈

—

257

—

105

C₂×C₈

—

514

—

106

C₂×C₈

—

514

—

107

C₂×C₈

C₄×D₂₅₇

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

514

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

514

—

112

C₄²

—

113

C₄²

—

514

—

114

C₂³

—

257

—

115

C₂³

D₄×D₂₅₇

yes

characteristic

116

Q₈

—

514

—

117

D₄

—

514

—

118

D₄

—

514

—

119

C₂×C₄

—

257

—

120

C₂×C₄

—

514

—

121

C₂×C₄

—

514

—

122

C₂×C₄

—

123

C₂×C₄

—

1028

—

124

C₂×C₄

C₂×C₄×D₂₅₇

yes

characteristic

125

C₂×C₄

D₄×D₂₅₇

yes

—

126

C₂×C₄

Q₈×D₂₅₇

yes

characteristic

127

C₂×C₄

#16861618

yes

—

128

C₈

—

257

—

129

C₈

—

257

—

130

C₈

—

257

—

131

C₈

—

132

C₂²

—

514

—

133

C₂²

—

134

C₂²

—

1028

—

135

C₂²

C₄⋊C₄×D₂₅₇

yes

characteristic

136

C₂²

#22551565

yes

—

137

C₄

—

514

—

138

C₄

—

139

C₄

—

2056

—

140

C₄

C₂²⋊C₄×D₂₅₇

yes

characteristic

141

C₄

#22551549

yes

characteristic

142

C₄

#22551570

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

1028

—

146

C₂

#30656596

yes

characteristic

	subgroup	G/H
center	C₄	#22551549
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	H(3,ℤ/4)×C₂₅₇	C₂
socle	C₂×C₂₅₇	#380
prime radical	C₂×#16861618	C₂²
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	H(3,ℤ/4)	D₂₅₇
hypocenter	C₂₅₇	#1812
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#22551549

commutator

C₂×C₄×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

H(3,ℤ/4)×C₂₅₇

C₂

C₂×C₂₅₇

C₂²

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #30656773	C₂	semidirect
4. #30656494	C₂	semidirect
5. H(3,ℤ/4)×C₂₅₇	C₂	semidirect
9. C₂×#16861601	C₄	semidirect
11. C₂×C₈×D₂₅₇	C₄	semidirect
73. C₂₅₇	#1812	semidirect
79. H(3,ℤ/4)	D₂₅₇	semidirect

H ≅ G/N

type

2. #30656773

C₂

semidirect

4. #30656494

C₂

semidirect

5. H(3,ℤ/4)×C₂₅₇

C₂

semidirect

9. C₂×#16861601

C₄

semidirect

11. C₂×C₈×D₂₅₇

C₄

semidirect

73. C₂₅₇

#1812

semidirect

79. H(3,ℤ/4)

D₂₅₇

semidirect

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

subgroup	index
C₂×#22551600	2
#30656773	2
#30656710	2
#30656494	2
H(3,ℤ/4)×C₂₅₇	2
#1812	257

subgroup

index

H(3,ℤ/4)×C₂₅₇

257

Subgroups — #45664365

Conjugacy classes of subgroups

Complements

Maximal subgroups