Group #45665192 — subgroups

Subgroups — #45665192

Order 32896.

304 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 113 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×#22551524	C₂	1	1	yes	maximal, characteristic
2	C₂×#22551516	C₂	1	2	yes	maximal
3	C₂×#22551502	C₂	1	1	yes	maximal, characteristic
4	#30656644	C₂	1	2	yes	maximal
5	C₂²×Q₁₆×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×C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
9	#22551524	—	2	2	no	—
10	#22551516	—	2	4	no	—
11	#22551502	—	2	2	no	—
12	#22551499	—	2	4	no	—
13	C₂²×Q₈×C₂₅₇	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₂₅₇	—	2	2	no	—
15	C₂×Q₁₆×C₂₅₇	—	2	2	no	—
16	C₂×Q₁₆×C₂₅₇	C₄	1	4	yes	complemented
17	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₂₅₇	—	2	2	no	—
19	C₂²×Dic₂₅₇	D₄	1	2	yes	—
20	C₂×#14523339	—	2	1	no	—
21	C₂×#14523339	—	2	1	no	—
22	C₂×#14523339	D₄	1	2	yes	—
23	C₄⋊Dic₂₅₇	—	2	2	no	—
24	C₄⋊Dic₂₅₇	—	2	2	no	—
25	C₄×Dic₂₅₇	—	2	1	no	—
26	C₄×Dic₂₅₇	—	2	1	no	—
27	C₂×Q₈×C₂₅₇	—	2	2	no	—
28	C₂×Q₈×C₂₅₇	—	2	2	no	—
29	C₂×Q₈×C₂₅₇	—	2	2	no	—
30	C₂×Q₈×C₂₅₇	—	2	2	no	—
31	C₂×Q₈×C₂₅₇	—	2	2	no	—
32	C₂×Q₈×C₂₅₇	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₂₅₇	—	2	2	no	—
34	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
35	Q₁₆×C₂₅₇	—	2	8	no	—
36	C₂×C₈×C₂₅₇	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
38	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
39	C₂×Dic₂₅₇	—	2	2	no	—
40	C₂×Dic₂₅₇	—	2	4	no	—
41	C₂×Dic₂₅₇	—	4	2	no	—
42	C₂×Dic₂₅₇	—	4	4	no	—
43	#14523339	—	2	2	no	—
44	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₂₅₇	—	2	4	no	—
46	Q₈×C₂₅₇	—	2	4	no	—
47	Q₈×C₂₅₇	—	4	2	no	—
48	Q₈×C₂₅₇	—	4	2	no	—
49	C₂×C₄×C₂₅₇	—	2	4	no	—
50	C₂×C₄×C₂₅₇	—	4	2	no	—
51	C₂×C₄×C₂₅₇	—	4	2	no	—
52	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
57	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
58	C₈×C₂₅₇	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₂₅₇	C₄×D₄	1	1	yes	characteristic
60	C₂²×C₂₅₇	#121	1	1	yes	characteristic
61	C₂²×C₂₅₇	#127	1	1	yes	characteristic
62	C₂²×C₂₅₇	#137	1	4	yes	—
63	C₄×C₂₅₇	—	4	4	no	—
64	C₄×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₂₅₇	C₄×D₄	1	1	yes	characteristic
66	C₄×C₂₅₇	#121	1	1	yes	characteristic
67	C₄×C₂₅₇	#124	1	1	yes	characteristic
68	Dic₂₅₇	—	4	2	no	—
69	Dic₂₅₇	—	8	2	no	—
70	C₂×C₂₅₇	#390	1	1	yes	characteristic
71	C₂×C₂₅₇	#441	1	2	yes	—
72	C₂×C₂₅₇	#470	1	2	yes	—
73	C₂×C₂₅₇	#497	1	2	yes	—
74	C₂₅₇	#1971	1	1	yes	characteristic, complemented, Sylow
75	#1971	—	257	1	no	maximal, Sylow
76	C₂²×Q₁₆	D₂₅₇	1	1	yes	characteristic
77	C₂×#106	—	257	1	no	—
78	C₂×#103	—	257	2	no	—
79	C₂×#97	—	257	1	no	—
80	#391	—	257	2	no	—
81	C₂²×Q₈	C₂×D₂₅₇	1	2	yes	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×Q₁₆	—	2	2	no	—
84	C₂×Q₁₆	Dic₂₅₇	1	4	yes	complemented
85	C₂²×C₈	—	257	1	no	—
86	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	257	2	no	—
88	C₂×C₄²	—	257	1	no	—
89	#106	—	514	2	no	—
90	#103	—	514	4	no	—
91	#97	—	514	2	no	—
92	#95	—	514	4	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂×Dic₂₅₇	1	4	yes	—
99	C₂²×C₄	—	257	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	514	2	no	—
102	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
103	Q₁₆	—	2	8	no	—
104	C₂×C₈	—	257	2	no	—
105	C₂×C₈	—	514	1	no	—
106	C₂×C₈	—	514	1	no	—
107	C₂×C₈	C₂×Dic₂₅₇	1	2	yes	—
108	C₂×C₈	C₂²⋊D₂₅₇	1	2	yes	—
109	C₂×C₈	C₂²⋊D₂₅₇	1	2	yes	—
110	C₄⋊C₄	—	514	2	no	—
111	C₄⋊C₄	—	514	2	no	—
112	C₄²	—	514	1	no	—
113	C₄²	—	514	1	no	—
114	C₂³	D₄×D₂₅₇	1	1	yes	characteristic
115	Q₈	—	2	4	no	—
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	514	2	no	—
121	C₂×C₄	—	514	4	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	1028	2	no	—
125	C₂×C₄	—	1028	4	no	—
126	C₂×C₄	C₂²×Dic₂₅₇	1	1	yes	characteristic
127	C₂×C₄	D₄×D₂₅₇	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
130	C₂×C₄	#16861617	1	1	yes	characteristic
131	C₂×C₄	#16861617	1	1	yes	characteristic
132	C₈	—	514	2	no	—
133	C₈	C₂²⋊Dic₂₅₇	1	4	yes	—
134	C₂²	D₄×Dic₂₅₇	1	1	yes	characteristic
135	C₂²	#22551563	1	1	yes	characteristic
136	C₂²	#22551588	1	1	yes	characteristic
137	C₂²	#22551624	1	4	yes	—
138	C₄	—	4	4	no	—
139	C₄	—	1028	2	no	—
140	C₄	—	2056	2	no	—
141	C₄	C₂×C₂²⋊Dic₂₅₇	1	1	yes	characteristic
142	C₄	D₄×Dic₂₅₇	1	1	yes	characteristic
143	C₄	#22551563	1	1	yes	characteristic
144	C₄	#22551579	1	1	yes	characteristic
145	C₂	#30656640	1	1	yes	characteristic
146	C₂	#30656844	1	2	yes	—
147	C₂	#30656992	1	2	yes	—
148	C₂	#30657130	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551524

C₂

yes

maximal, characteristic

C₂×#22551516

C₂

yes

maximal

C₂×#22551502

C₂

yes

maximal, characteristic

#30656644

C₂

yes

maximal

C₂²×Q₁₆×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×#14523339

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₅₇

C₂²

yes

—

C₂×C₄×Dic₂₅₇

C₂²

yes

—

—

—

—

C₂²×Q₈×C₂₅₇

C₂²

yes

—

C₂×Q₁₆×C₂₅₇

—

C₂×Q₁₆×C₂₅₇

—

C₂×Q₁₆×C₂₅₇

C₄

yes

complemented

C₂²×C₈×C₂₅₇

C₂²

yes

characteristic

C₂²×Dic₂₅₇

—

C₂²×Dic₂₅₇

D₄

yes

—

C₂×#14523339

—

C₂×#14523339

—

C₂×#14523339

D₄

yes

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

C₄×Dic₂₅₇

—

C₄×Dic₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

C₂×C₄

yes

—

C₂²×C₄×C₂₅₇

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

Q₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

C₂×C₄

yes

—

C₂×C₈×C₂₅₇

D₄

yes

—

C₂×C₈×C₂₅₇

D₄

yes

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

C₂×D₄

yes

characteristic

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂×D₄

yes

C₂×C₄×C₂₅₇

yes

C₂×C₄×C₂₅₇

yes

C₈×C₂₅₇

C₂²⋊C₄

yes

—

C₂²×C₂₅₇

C₄×D₄

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₅₇

C₄×D₄

yes

C₄×C₂₅₇

yes

C₄×C₂₅₇

yes

Dic₂₅₇

—

Dic₂₅₇

—

C₂×C₂₅₇

#390

yes

characteristic

C₂×C₂₅₇

#441

yes

—

C₂×C₂₅₇

#470

yes

—

C₂×C₂₅₇

#497

yes

—

C₂₅₇

#1971

yes

characteristic, complemented, Sylow

#1971

—

257

maximal, Sylow

C₂²×Q₁₆

D₂₅₇

yes

characteristic

C₂×#106

—

257

—

C₂×#103

—

257

—

C₂×#97

—

257

—

#391

—

257

—

C₂²×Q₈

C₂×D₂₅₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₂₅₇

yes

complemented

C₂²×C₈

—

257

—

C₂²×C₈

C₂×D₂₅₇

yes

characteristic

C₂×C₄⋊C₄

—

257

—

C₂×C₄²

—

257

—

#106

—

514

—

#103

—

514

—

#97

—

514

—

#95

—

514

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₅₇

yes

—

C₂²×C₄

—

257

—

100

C₂²×C₄

—

101

C₂²×C₄

—

514

—

102

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

103

Q₁₆

—

104

C₂×C₈

—

257

—

105

C₂×C₈

—

514

—

106

C₂×C₈

—

514

—

107

C₂×C₈

C₂×Dic₂₅₇

yes

—

108

C₂×C₈

C₂²⋊D₂₅₇

yes

—

109

C₂×C₈

C₂²⋊D₂₅₇

yes

—

110

C₄⋊C₄

—

514

—

111

C₄⋊C₄

—

514

—

112

C₄²

—

514

—

113

C₄²

—

514

—

114

C₂³

D₄×D₂₅₇

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

514

—

121

C₂×C₄

—

514

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

1028

—

125

C₂×C₄

—

1028

—

126

C₂×C₄

C₂²×Dic₂₅₇

yes

characteristic

127

C₂×C₄

D₄×D₂₅₇

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

130

C₂×C₄

#16861617

yes

characteristic

131

C₂×C₄

#16861617

yes

characteristic

132

C₈

—

514

—

133

C₈

C₂²⋊Dic₂₅₇

yes

—

134

C₂²

D₄×Dic₂₅₇

yes

characteristic

135

C₂²

#22551563

yes

characteristic

136

C₂²

#22551588

yes

characteristic

137

C₂²

#22551624

yes

—

138

C₄

—

139

C₄

—

1028

—

140

C₄

—

2056

—

141

C₄

C₂×C₂²⋊Dic₂₅₇

yes

characteristic

142

C₄

D₄×Dic₂₅₇

yes

characteristic

143

C₄

#22551563

yes

characteristic

144

C₄

#22551579

yes

characteristic

145

C₂

#30656640

yes

characteristic

146

C₂

#30656844

yes

—

147

C₂

#30656992

yes

—

148

C₂

#30657130

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₅₇
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	C₂²×Q₁₆×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂×D₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₂²×Q₁₆	D₂₅₇
hypocenter	C₂₅₇	#1971
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₂₅₇

commutator

C₂×C₄×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

C₂²×Q₁₆×C₂₅₇

C₂

socle, prime radical

C₂³×C₂₅₇

C₂×D₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

hypercenter

C₂²×Q₁₆

D₂₅₇

hypocenter

C₂₅₇

#1971

LCS

G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
16. C₂×Q₁₆×C₂₅₇	C₄	semidirect
74. C₂₅₇	#1971	semidirect
84. C₂×Q₁₆	Dic₂₅₇	semidirect

H ≅ G/N

type

16. C₂×Q₁₆×C₂₅₇

C₄

semidirect

74. C₂₅₇

#1971

semidirect

84. C₂×Q₁₆

Dic₂₅₇

semidirect

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

subgroup	index
C₂×#22551524	2
C₂×#22551516	2
C₂×#22551502	2
#30656644	2
C₂²×Q₁₆×C₂₅₇	2
#1971	257

subgroup

index

C₂²×Q₁₆×C₂₅₇

257

Subgroups — #45665192

Conjugacy classes of subgroups

Complements

Maximal subgroups