Group #45665419 — subgroups

Subgroups — #45665419

Order 32896.

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

normal subgroups only

This group has 370 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	C₂²×#16861608	C₂	1	2	yes	maximal
2	C₂×#22551592	C₂	1	1	yes	maximal, characteristic
3	C₂×#22551515	C₂	1	2	yes	maximal
4	#30656492	C₂	1	1	yes	maximal, characteristic
5	#391×C₂₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₁₄	C₂²	1	2	yes	—
7	C₂×#16861608	—	2	4	no	—
8	C₂×#16861608	—	2	4	no	—
9	C₂×#16861608	—	2	4	no	—
10	C₂²×#14523339	C₂²	1	2	yes	—
11	#22551592	—	2	4	no	—
12	C₂×C₄⋊Dic₂₅₇	—	2	2	no	—
13	#22551515	—	2	4	no	—
14	C₂²×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₅₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
17	#95×C₂₅₇	—	2	2	no	—
18	C₂×Dic₅₁₄	—	2	2	no	—
19	C₂×Dic₅₁₄	—	2	2	no	—
20	C₂×Dic₅₁₄	—	2	2	no	—
21	C₂×Dic₅₁₄	—	2	2	no	—
22	C₂×Dic₅₁₄	—	2	2	no	—
23	C₂×Dic₅₁₄	D₄	1	4	yes	—
24	C₂²×Dic₂₅₇	—	2	2	no	—
25	#16861608	—	2	16	no	—
26	C₂×#14523339	—	2	2	no	—
27	C₂×#14523339	—	2	2	no	—
28	C₂×#14523339	D₄	1	4	yes	—
29	C₄⋊Dic₂₅₇	—	2	8	no	—
30	C₂×Q₈×C₂₅₇	—	2	1	no	—
31	C₂×Q₈×C₂₅₇	—	2	2	no	—
32	C₂×Q₈×C₂₅₇	—	2	2	no	—
33	C₂×Q₈×C₂₅₇	D₄	1	2	yes	—
34	C₂²×C₄×C₂₅₇	—	2	1	no	—
35	C₂²×C₄×C₂₅₇	—	2	1	no	—
36	C₂²×C₄×C₂₅₇	D₄	1	2	yes	—
37	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₅₇	—	2	2	no	—
39	C₄²×C₂₅₇	—	2	2	no	—
40	Dic₅₁₄	—	2	4	no	—
41	Dic₅₁₄	—	2	4	no	—
42	Dic₅₁₄	—	4	2	no	—
43	Dic₅₁₄	—	4	2	no	—
44	C₂×Dic₂₅₇	—	2	4	no	—
45	C₂×Dic₂₅₇	—	4	2	no	—
46	C₂×Dic₂₅₇	—	4	2	no	—
47	#14523339	—	2	4	no	—
48	C₂³×C₂₅₇	C₂×D₄	1	1	yes	characteristic
49	Q₈×C₂₅₇	—	2	4	no	—
50	Q₈×C₂₅₇	—	4	2	no	—
51	C₂×C₄×C₂₅₇	—	2	2	no	—
52	C₂×C₄×C₂₅₇	—	2	2	no	—
53	C₂×C₄×C₂₅₇	—	2	2	no	—
54	C₂×C₄×C₂₅₇	—	4	1	no	—
55	C₂×C₄×C₂₅₇	—	4	1	no	—
56	C₂×C₄×C₂₅₇	—	4	1	no	—
57	C₂×C₄×C₂₅₇	—	4	2	no	—
58	C₂×C₄×C₂₅₇	Q₁₆	1	4	yes	complemented
59	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₅₇	#41	1	1	yes	characteristic
63	C₂²×C₂₅₇	#120	1	1	yes	characteristic
64	C₂²×C₂₅₇	#121	1	1	yes	characteristic
65	C₂²×C₂₅₇	#127	1	1	yes	characteristic
66	C₂²×C₂₅₇	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₂₅₇	#137	1	2	yes	—
68	C₄×C₂₅₇	—	2	4	no	—
69	C₄×C₂₅₇	—	4	2	no	—
70	C₄×C₂₅₇	—	8	1	no	—
71	C₄×C₂₅₇	#120	1	2	yes	—
72	C₄×C₂₅₇	#121	1	2	yes	—
73	Dic₂₅₇	—	4	4	no	—
74	C₂×C₂₅₇	#392	1	1	yes	characteristic
75	C₂×C₂₅₇	#451	1	2	yes	—
76	C₂×C₂₅₇	#462	1	2	yes	—
77	C₂×C₂₅₇	#494	1	1	yes	characteristic
78	C₂×C₂₅₇	#497	1	1	yes	characteristic
79	C₂₅₇	#2016	1	1	yes	characteristic, complemented, Sylow
80	#2016	—	257	1	no	maximal, Sylow
81	C₂²×Q₁₆	—	257	2	no	—
82	C₂×C₄⋊Q₈	—	257	1	no	—
83	C₂×#103	—	257	2	no	—
84	#391	D₂₅₇	1	1	yes	characteristic
85	#336	—	257	1	no	—
86	C₂²×Q₈	—	257	2	no	—
87	C₂²×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
88	C₂×Q₁₆	—	514	4	no	—
89	C₂×Q₁₆	—	514	4	no	—
90	C₂×Q₁₆	—	514	4	no	—
91	C₂²×C₈	—	257	2	no	—
92	C₄⋊Q₈	—	514	4	no	—
93	C₂×C₄⋊C₄	—	514	2	no	—
94	C₂×C₄⋊C₄	C₂×D₂₅₇	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
96	#103	—	514	4	no	—
97	#95	—	2	2	no	—
98	C₂×Q₈	—	257	4	no	—
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	514	2	no	—
103	C₂×Q₈	—	514	2	no	—
104	C₂×Q₈	—	514	2	no	—
105	C₂×Q₈	—	514	2	no	—
106	C₂×Q₈	—	514	2	no	—
107	C₂×Q₈	D₁₀₂₈	1	2	yes	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	514	2	no	—
111	C₂²×C₄	C₂²⋊D₂₅₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
113	Q₁₆	—	514	16	no	—
114	C₂×C₈	—	257	4	no	—
115	C₂×C₈	—	514	2	no	—
116	C₂×C₈	—	514	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	514	8	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	514	4	no	—
123	Q₈	—	514	4	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	1028	2	no	—
126	Q₈	—	1028	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	514	4	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	1028	2	no	—
136	C₂×C₄	—	1028	2	no	—
137	C₂×C₄	#16861608	1	4	yes	complemented
138	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
139	C₂×C₄	D₄×D₂₅₇	1	2	yes	—
140	C₂×C₄	C₂×D₁₀₂₈	1	1	yes	characteristic
141	C₂×C₄	#16861618	1	1	yes	characteristic
142	C₈	—	514	4	no	—
143	C₂²	#22551560	1	1	yes	characteristic
144	C₂²	#22551566	1	1	yes	characteristic
145	C₂²	#22551588	1	1	yes	characteristic
146	C₂²	C₂×#16861608	1	2	yes	—
147	C₂²	#22551622	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	2	no	—
150	C₄	—	1028	4	no	—
151	C₄	—	8	1	no	—
152	C₄	#22551562	1	2	yes	—
153	C₄	#22551565	1	2	yes	—
154	C₂	#30656646	1	1	yes	characteristic
155	C₂	#30656892	1	2	yes	—
156	C₂	#30656947	1	2	yes	—
157	C₂	#30657116	1	1	yes	characteristic
158	C₂	#30657132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#16861608

C₂

yes

maximal

C₂×#22551592

C₂

yes

maximal, characteristic

C₂×#22551515

C₂

yes

maximal

#30656492

C₂

yes

maximal, characteristic

#391×C₂₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₁₄

C₂²

yes

—

—

—

—

C₂²

yes

—

#22551592

—

C₂×C₄⋊Dic₂₅₇

—

#22551515

—

C₂²×Q₈×C₂₅₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₅₇

C₂²

yes

characteristic

C₂×C₄²×C₂₅₇

C₂²

yes

characteristic

#95×C₂₅₇

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

—

C₂×Dic₅₁₄

D₄

yes

—

C₂²×Dic₂₅₇

—

—

—

—

D₄

yes

—

C₄⋊Dic₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

—

C₂×Q₈×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

—

C₂²×C₄×C₂₅₇

—

C₂²×C₄×C₂₅₇

D₄

yes

—

C₂²×C₄×C₂₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₇

—

C₄²×C₂₅₇

—

Dic₅₁₄

—

Dic₅₁₄

—

Dic₅₁₄

—

Dic₅₁₄

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

C₂×Dic₂₅₇

—

#14523339

—

C₂³×C₂₅₇

C₂×D₄

yes

characteristic

Q₈×C₂₅₇

—

Q₈×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

—

C₂×C₄×C₂₅₇

Q₁₆

yes

complemented

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

—

C₂×C₄×C₂₅₇

C₂×D₄

yes

C₂×C₄×C₂₅₇

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

yes

C₂²×C₂₅₇

C₂×Q₁₆

yes

—

C₂²×C₂₅₇

#137

yes

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

—

C₄×C₂₅₇

#120

yes

—

C₄×C₂₅₇

#121

yes

—

Dic₂₅₇

—

C₂×C₂₅₇

#392

yes

characteristic

C₂×C₂₅₇

#451

yes

—

C₂×C₂₅₇

#462

yes

—

C₂×C₂₅₇

yes

C₂×C₂₅₇

yes

C₂₅₇

yes

characteristic, complemented, Sylow

#2016

—

257

maximal, Sylow

C₂²×Q₁₆

—

257

—

C₂×C₄⋊Q₈

—

257

—

C₂×#103

—

257

—

#391

D₂₅₇

yes

characteristic

#336

—

257

—

C₂²×Q₈

—

257

—

C₂²×Q₈

C₂×D₂₅₇

yes

characteristic

C₂×Q₁₆

—

514

—

C₂×Q₁₆

—

514

—

C₂×Q₁₆

—

514

—

C₂²×C₈

—

257

—

C₄⋊Q₈

—

514

—

C₂×C₄⋊C₄

—

514

—

C₂×C₄⋊C₄

C₂×D₂₅₇

yes

characteristic

C₂×C₄²

C₂×D₂₅₇

yes

characteristic

#103

—

514

—

#95

—

C₂×Q₈

—

257

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

514

—

103

C₂×Q₈

—

514

—

104

C₂×Q₈

—

514

—

105

C₂×Q₈

—

514

—

106

C₂×Q₈

—

514

—

107

C₂×Q₈

D₁₀₂₈

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

514

—

111

C₂²×C₄

C₂²⋊D₂₅₇

yes

—

112

C₂²×C₄

C₂²×D₂₅₇

yes

characteristic

113

Q₁₆

—

514

—

114

C₂×C₈

—

257

—

115

C₂×C₈

—

514

—

116

C₂×C₈

—

514

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

514

—

119

C₄²

—

120

C₂³

C₂×C₂²⋊D₂₅₇

yes

characteristic

121

Q₈

—

122

Q₈

—

514

—

123

Q₈

—

514

—

124

Q₈

—

125

Q₈

—

1028

—

126

Q₈

—

1028

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

514

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

1028

—

136

C₂×C₄

—

1028

—

137

C₂×C₄

#16861608

yes

complemented

138

C₂×C₄

D₄×D₂₅₇

yes

—

139

C₂×C₄

D₄×D₂₅₇

yes

—

140

C₂×C₄

C₂×D₁₀₂₈

yes

characteristic

141

C₂×C₄

#16861618

yes

characteristic

142

C₈

—

514

—

143

C₂²

#22551560

yes

characteristic

144

C₂²

#22551566

yes

characteristic

145

C₂²

#22551588

yes

characteristic

146

C₂²

C₂×#16861608

yes

—

147

C₂²

#22551622

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

1028

—

151

C₄

—

152

C₄

#22551562

yes

—

153

C₄

#22551565

yes

—

154

C₂

#30656646

yes

characteristic

155

C₂

#30656892

yes

—

156

C₂

#30656947

yes

—

157

C₂

#30657116

yes

characteristic

158

C₂

#30657132

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₅₇
commutator, cosocle	C₂²×C₄×C₂₅₇	C₂³
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#391×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂×D₄
hypercenter	#391	D₂₅₇
hypocenter	C₂₅₇	#2016
LCS	G ▷ C₂²×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₅₇

commutator, cosocle

C₂²×C₄×C₂₅₇

C₂³

Frattini

C₂²×C₄

C₂²×D₂₅₇

Fitting

#391×C₂₅₇

C₂

socle, prime radical

C₂³×C₂₅₇

C₂×D₄

D₂₅₇

C₂₅₇

G ▷ C₂²×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
58. C₂×C₄×C₂₅₇	Q₁₆	semidirect
79. C₂₅₇	#2016	semidirect
137. C₂×C₄	#16861608	semidirect

H ≅ G/N

type

58. C₂×C₄×C₂₅₇

Q₁₆

semidirect

79. C₂₅₇

#2016

semidirect

137. C₂×C₄

#16861608

semidirect

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

subgroup	index
C₂²×#16861608	2
C₂×#22551592	2
C₂×#22551515	2
#30656492	2
#391×C₂₅₇	2
#2016	257

subgroup

index

#391×C₂₅₇

257

Subgroups — #45665419

Conjugacy classes of subgroups

Complements

Maximal subgroups