Group #45665308 — subgroups

Subgroups — #45665308

Order 32896.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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×#22551592	C₂	1	1	yes	maximal, characteristic
2	C₄×#16861602	C₂	1	1	yes	maximal, characteristic
3	#30656505	C₂	1	4	yes	maximal
4	#423×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₁₄	C₄	1	2	yes	—
6	C₂×#16861602	C₂²	1	2	yes	—
7	#22551592	—	2	1	no	—
8	#22551592	—	2	1	no	—
9	#22551592	—	2	1	no	—
10	#22551592	—	2	1	no	—
11	C₂×C₄⋊Dic₂₅₇	C₄	1	2	yes	—
12	C₂×C₄⋊Dic₂₅₇	C₂²	1	2	yes	—
13	#22551503	—	2	1	no	—
14	C₄×#14523339	—	2	1	no	—
15	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₂₅₇	—	2	1	no	—
18	C₄⋊C₈×C₂₅₇	—	2	1	no	—
19	C₂×Dic₅₁₄	—	2	2	no	—
20	C₂×Dic₅₁₄	—	2	2	no	—
21	C₂×Dic₅₁₄	—	4	4	no	—
22	C₂²×Dic₂₅₇	C₂×C₄	1	4	yes	—
23	#16861602	—	2	2	no	—
24	#16861602	Q₈	1	4	yes	complemented
25	C₂×#14523339	—	2	2	no	—
26	C₄⋊Dic₂₅₇	—	2	2	no	—
27	C₄⋊Dic₂₅₇	—	2	2	no	—
28	C₄⋊Dic₂₅₇	—	2	2	no	—
29	C₄⋊Dic₂₅₇	—	2	2	no	—
30	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₂₅₇	—	2	4	no	—
34	C₂×C₈×C₂₅₇	—	2	2	no	—
35	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
36	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
37	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
38	C₄²×C₂₅₇	D₄	1	1	yes	characteristic
39	Dic₅₁₄	—	4	4	no	—
40	Dic₅₁₄	—	4	4	no	—
41	C₂×Dic₂₅₇	—	2	4	no	—
42	C₂×Dic₂₅₇	—	4	4	no	—
43	#14523339	—	2	4	no	—
44	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₅₇	—	2	2	no	—
46	C₂×C₄×C₂₅₇	—	2	2	no	—
47	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₅₇	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₂₅₇	#41	1	2	yes	—
55	C₈×C₂₅₇	—	4	2	no	—
56	C₂²×C₂₅₇	—	2	1	no	—
57	C₂²×C₂₅₇	—	2	1	no	—
58	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₂₅₇	C₄×Q₈	1	2	yes	—
60	C₄×C₂₅₇	—	2	1	no	—
61	C₄×C₂₅₇	—	2	1	no	—
62	C₄×C₂₅₇	#100	1	2	yes	—
63	C₄×C₂₅₇	#101	1	2	yes	—
64	C₄×C₂₅₇	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₂₅₇	#124	1	1	yes	characteristic
66	C₄×C₂₅₇	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₂₅₇	—	4	4	no	—
68	C₂×C₂₅₇	—	2	2	no	—
69	C₂×C₂₅₇	#391	1	1	yes	characteristic
70	C₂×C₂₅₇	C₂×#100	1	1	yes	characteristic
71	C₂×C₂₅₇	C₂×#101	1	1	yes	characteristic
72	C₂₅₇	#1997	1	1	yes	characteristic, complemented, Sylow
73	#1997	—	257	1	no	maximal, Sylow
74	C₂×C₄⋊Q₈	—	257	1	no	—
75	#423	D₂₅₇	1	1	yes	characteristic
76	C₄×M₁₆	—	257	1	no	—
77	#343	—	257	4	no	—
78	C₂²×Q₈	—	257	2	no	—
79	C₂×M₁₆	—	257	2	no	—
80	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
81	C₄⋊Q₈	—	514	1	no	—
82	C₄⋊Q₈	—	514	1	no	—
83	C₄⋊Q₈	—	514	1	no	—
84	C₄⋊Q₈	—	514	1	no	—
85	C₂×C₄⋊C₄	—	257	2	no	—
86	C₂×C₄⋊C₄	—	257	2	no	—
87	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	#97	—	514	1	no	—
91	C₄×C₈	—	514	1	no	—
92	C₂×Q₈	—	514	2	no	—
93	C₂×Q₈	—	514	2	no	—
94	C₂×Q₈	—	1028	4	no	—
95	C₂²×C₄	—	257	4	no	—
96	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
99	M₁₆	—	257	4	no	—
100	M₁₆	—	2	4	no	—
101	M₁₆	—	514	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	514	2	no	—
104	C₄⋊C₄	—	514	2	no	—
105	C₄⋊C₄	—	514	2	no	—
106	C₄⋊C₄	—	514	2	no	—
107	C₄⋊C₄	—	514	2	no	—
108	C₄²	C₂²⋊D₂₅₇	1	1	yes	characteristic
109	C₄²	C₂²⋊D₂₅₇	1	1	yes	characteristic
110	C₄²	D₁₀₂₈	1	1	yes	characteristic
111	C₄²	D₁₀₂₈	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₂₅₇	1	1	yes	characteristic
113	Q₈	—	1028	4	no	—
114	Q₈	—	1028	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	514	4	no	—
118	C₂×C₄	—	1028	4	no	—
119	C₂×C₄	#16861595	1	1	yes	characteristic
120	C₂×C₄	#16861595	1	1	yes	characteristic
121	C₂×C₄	#16861595	1	1	yes	characteristic
122	C₂×C₄	#16861595	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
124	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₂₅₇	1	2	yes	—
126	C₂×C₄	#16861617	1	2	yes	—
127	C₈	—	514	4	no	—
128	C₈	—	4	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×#16861595	1	1	yes	characteristic
132	C₂²	#22551557	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	1028	4	no	—
136	C₄	#22551508	1	2	yes	—
137	C₄	#22551510	1	2	yes	—
138	C₄	#22551572	1	2	yes	—
139	C₄	#22551579	1	1	yes	characteristic
140	C₄	#22551590	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#30656642	1	1	yes	characteristic
143	C₂	C₂×#22551508	1	1	yes	characteristic
144	C₂	C₂×#22551510	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22551592

C₂

yes

maximal, characteristic

C₄×#16861602

C₂

yes

maximal, characteristic

#30656505

C₂

yes

maximal

#423×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₁₄

C₄

yes

—

C₂×#16861602

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₂₅₇

C₄

yes

—

C₂×C₄⋊Dic₂₅₇

C₂²

yes

—

#22551503

—

C₄×#14523339

—

C₂×M₁₆×C₂₅₇

C₂²

yes

—

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₇

—

C₄⋊C₈×C₂₅₇

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂²×Dic₂₅₇

C₂×C₄

yes

—

#16861602

—

#16861602

Q₈

yes

complemented

C₂×#14523339

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

C₄⋊Dic₂₅₇

—

C₂²×C₄×C₂₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

M₁₆×C₂₅₇

—

C₂×C₈×C₂₅₇

—

C₄²×C₂₅₇

D₄

yes

characteristic

C₄²×C₂₅₇

D₄

yes

characteristic

C₄²×C₂₅₇

D₄

yes

characteristic

C₄²×C₂₅₇

D₄

yes

characteristic

Dic₅₁₄

—

Dic₅₁₄

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₅₇

C₂²⋊C₄

yes

characteristic

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×Q₈

yes

—

C₂×C₄×C₂₅₇

#41

yes

—

C₈×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

—

C₂²×C₂₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₇

C₄×Q₈

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#100

yes

—

C₄×C₂₅₇

#101

yes

—

C₄×C₂₅₇

C₂²⋊Q₈

yes

—

C₄×C₂₅₇

#124

yes

characteristic

C₄×C₂₅₇

C₄⋊Q₈

yes

characteristic

Dic₂₅₇

—

C₂×C₂₅₇

—

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂₅₇

yes

characteristic, complemented, Sylow

#1997

—

257

maximal, Sylow

C₂×C₄⋊Q₈

—

257

—

#423

D₂₅₇

yes

characteristic

C₄×M₁₆

—

257

—

#343

—

257

—

C₂²×Q₈

—

257

—

C₂×M₁₆

—

257

—

C₂×M₁₆

C₂×D₂₅₇

yes

—

C₄⋊Q₈

—

514

—

C₄⋊Q₈

—

514

—

C₄⋊Q₈

—

514

—

C₄⋊Q₈

—

514

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄⋊C₄

—

257

—

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

514

—

C₄×C₈

—

514

—

C₂×Q₈

—

514

—

C₂×Q₈

—

514

—

C₂×Q₈

—

1028

—

C₂²×C₄

—

257

—

C₂²×C₄

C₄×D₂₅₇

yes

characteristic

C₂²×C₄

C₄×D₂₅₇

yes

characteristic

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

M₁₆

—

257

—

100

M₁₆

—

101

M₁₆

—

514

—

102

C₂×C₈

—

103

C₂×C₈

—

514

—

104

C₄⋊C₄

—

514

—

105

C₄⋊C₄

—

514

—

106

C₄⋊C₄

—

514

—

107

C₄⋊C₄

—

514

—

108

C₄²

C₂²⋊D₂₅₇

yes

characteristic

109

C₄²

C₂²⋊D₂₅₇

yes

characteristic

110

C₄²

D₁₀₂₈

yes

characteristic

111

C₄²

D₁₀₂₈

yes

characteristic

112

C₂³

C₂×C₄×D₂₅₇

yes

characteristic

113

Q₈

—

1028

—

114

Q₈

—

1028

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

514

—

118

C₂×C₄

—

1028

—

119

C₂×C₄

#16861595

yes

characteristic

120

C₂×C₄

#16861595

yes

characteristic

121

C₂×C₄

#16861595

yes

characteristic

122

C₂×C₄

#16861595

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₂₅₇

yes

characteristic

124

C₂×C₄

C₂×D₁₀₂₈

yes

characteristic

125

C₂×C₄

Q₈×D₂₅₇

yes

—

126

C₂×C₄

#16861617

yes

—

127

C₈

—

514

—

128

C₈

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×#16861595

yes

characteristic

132

C₂²

#22551557

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

1028

—

136

C₄

#22551508

yes

—

137

C₄

#22551510

yes

—

138

C₄

#22551572

yes

—

139

C₄

#22551579

yes

characteristic

140

C₄

#22551590

yes

characteristic

141

C₂

—

142

C₂

#30656642

yes

characteristic

143

C₂

C₂×#22551508

yes

characteristic

144

C₂

C₂×#22551510

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#16861595
commutator, prime radical	C₂³×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#423×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#423	D₂₅₇
hypocenter	C₂₅₇	#1997
LCS	G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#16861595

commutator, prime radical

C₂³×C₂₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#423×C₂₅₇

C₂

socle

C₂²×C₂₅₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₅₇

C₂³

D₂₅₇

C₂₅₇

G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
24. #16861602	Q₈	semidirect
72. C₂₅₇	#1997	semidirect

H ≅ G/N

type

24. #16861602

Q₈

semidirect

72. C₂₅₇

#1997

semidirect

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

subgroup	index
C₂×#22551592	2
C₄×#16861602	2
#30656505	2
#423×C₂₅₇	2
#1997	257

subgroup

index

#423×C₂₅₇

257

Subgroups — #45665308

Conjugacy classes of subgroups

Complements

Maximal subgroups