Group #44550293 — subgroups

Subgroups — #44550293

Order 32128.

584 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 125 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×#22548634	C₂	1	2	yes	maximal, complemented
2	C₂×#22548632	C₂	1	2	yes	maximal, complemented
3	#30631651	C₂	1	2	yes	maximal, complemented
4	C₂²×#41×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16860794	C₂²	1	4	yes	—
6	#22548634	—	2	4	no	—
7	#22548632	—	2	4	no	—
8	C₂²⋊(C₂₅₁⋊C₈)	—	2	4	no	—
9	C₂×#41×C₂₅₁	—	2	4	no	—
10	C₂×#41×C₂₅₁	—	2	4	no	—
11	C₂×#41×C₂₅₁	—	2	4	no	—
12	C₂²×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₂₅₁	C₄	1	2	yes	—
14	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₅₁	C₄	1	2	yes	—
16	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
17	#16860794	—	2	4	no	—
18	#16860794	—	2	4	no	—
19	C₂×C₂₅₁⋊C₈	—	2	4	no	—
20	C₂⁴×C₂₅₁	C₂×C₄	1	2	yes	—
21	C₂⁴×C₂₅₁	C₂×C₄	1	1	yes	characteristic
22	#41×C₂₅₁	—	2	32	no	—
23	C₂×Q₈×C₂₅₁	—	2	2	no	—
24	C₂×Q₈×C₂₅₁	—	2	2	no	—
25	C₂×Q₈×C₂₅₁	D₄	1	4	yes	—
26	C₂×D₄×C₂₅₁	—	2	4	no	—
27	C₂×D₄×C₂₅₁	—	2	4	no	—
28	C₂×D₄×C₂₅₁	—	2	4	no	—
29	C₂×D₄×C₂₅₁	—	2	2	no	—
30	C₂×D₄×C₂₅₁	—	2	2	no	—
31	C₂×D₄×C₂₅₁	D₄	1	4	yes	—
32	C₂²×C₄×C₂₅₁	—	2	4	no	—
33	C₂²×C₄×C₂₅₁	—	2	4	no	—
34	C₂²×C₄×C₂₅₁	—	2	4	no	—
35	C₂²×C₄×C₂₅₁	—	2	2	no	—
36	C₂²×C₄×C₂₅₁	—	2	2	no	—
37	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₅₁	D₄	1	4	yes	—
40	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
41	C₂₅₁⋊C₈	—	4	4	no	—
42	C₂³×C₂₅₁	—	2	1	no	—
43	C₂³×C₂₅₁	—	2	2	no	—
44	C₂³×C₂₅₁	—	2	1	no	—
45	C₂³×C₂₅₁	—	2	2	no	—
46	C₂³×C₂₅₁	—	4	2	no	—
47	C₂³×C₂₅₁	—	4	4	no	—
48	C₂³×C₂₅₁	C₂²⋊C₄	1	2	yes	—
49	C₂³×C₂₅₁	C₂²⋊C₄	1	4	yes	—
50	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
51	Q₈×C₂₅₁	—	2	8	no	—
52	D₄×C₂₅₁	—	2	16	no	—
53	D₄×C₂₅₁	—	2	8	no	—
54	C₂×C₄×C₂₅₁	—	2	16	no	—
55	C₂×C₄×C₂₅₁	—	2	1	no	—
56	C₂×C₄×C₂₅₁	—	2	1	no	—
57	C₂×C₄×C₂₅₁	—	2	2	no	—
58	C₂×C₄×C₂₅₁	—	2	2	no	—
59	C₂×C₄×C₂₅₁	—	2	8	no	—
60	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	4	yes	—
62	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
65	C₂²×C₂₅₁	—	2	2	no	—
66	C₂²×C₂₅₁	—	2	2	no	—
67	C₂²×C₂₅₁	—	2	4	no	—
68	C₂²×C₂₅₁	—	2	4	no	—
69	C₂²×C₂₅₁	—	4	4	no	—
70	C₂²×C₂₅₁	—	4	2	no	—
71	C₂²×C₂₅₁	—	4	2	no	—
72	C₂²×C₂₅₁	—	4	4	no	—
73	C₂²×C₂₅₁	—	4	4	no	—
74	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₂₅₁	—	2	2	no	—
78	C₄×C₂₅₁	—	2	4	no	—
79	C₄×C₂₅₁	#120	1	4	yes	—
80	C₂×C₂₅₁	—	2	1	no	—
81	C₂×C₂₅₁	—	2	1	no	—
82	C₂×C₂₅₁	—	4	2	no	—
83	C₂×C₂₅₁	—	4	4	no	—
84	C₂×C₂₅₁	#379	1	1	yes	characteristic
85	C₂×C₂₅₁	#413	1	2	yes	—
86	C₂₅₁	#1785	1	1	yes	characteristic, complemented, Sylow
87	#1785	—	251	1	no	maximal, Sylow
88	C₂²×#41	D₂₅₁	1	1	yes	characteristic
89	C₂×#101	—	251	2	no	—
90	C₂×#100	—	251	2	no	—
91	#407	—	251	2	no	—
92	C₂×#41	—	2	4	no	—
93	C₂×#41	—	2	4	no	—
94	C₂×#41	—	2	4	no	—
95	C₂²×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
96	C₂²×D₄	Dic₂₅₁	1	2	yes	—
97	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
98	C₂³×C₄	Dic₂₅₁	1	2	yes	—
99	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
100	C₂×M₁₆	—	251	4	no	—
101	#101	—	502	4	no	—
102	#100	—	502	4	no	—
103	C₂²⋊C₈	—	502	4	no	—
104	C₂⁴	C₂×Dic₂₅₁	1	2	yes	—
105	C₂⁴	C₂×Dic₂₅₁	1	1	yes	characteristic
106	#41	—	2	32	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	C₂²⋊D₂₅₁	1	4	yes	—
110	C₂×D₄	—	2	4	no	—
111	C₂×D₄	—	2	4	no	—
112	C₂×D₄	—	2	4	no	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	C₂²⋊D₂₅₁	1	4	yes	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	2	no	—
121	C₂²×C₄	C₂×Dic₂₅₁	1	1	yes	characteristic
122	C₂²×C₄	C₂×Dic₂₅₁	1	2	yes	—
123	C₂²×C₄	C₂²⋊D₂₅₁	1	4	yes	—
124	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
125	M₁₆	—	502	4	no	—
126	M₁₆	—	502	4	no	—
127	C₂×C₈	—	502	4	no	—
128	C₂³	—	2	1	no	—
129	C₂³	—	2	2	no	—
130	C₂³	—	2	1	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	4	2	no	—
133	C₂³	—	4	4	no	—
134	C₂³	C₂²⋊Dic₂₅₁	1	2	yes	—
135	C₂³	C₂²⋊Dic₂₅₁	1	4	yes	—
136	C₂³	C₂²×Dic₂₅₁	1	1	yes	characteristic
137	Q₈	—	2	8	no	—
138	D₄	—	2	16	no	—
139	D₄	—	2	8	no	—
140	C₂×C₄	—	2	16	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	8	no	—
146	C₂×C₄	C₂²⋊Dic₂₅₁	1	4	yes	—
147	C₂×C₄	C₂²⋊Dic₂₅₁	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	2	yes	—
151	C₈	—	1004	4	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	4	4	no	—
160	C₂²	—	4	4	no	—
161	C₂²	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
162	C₂²	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
163	C₂²	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
164	C₄	—	2	2	no	—
165	C₄	—	2	4	no	—
166	C₄	#22548684	1	4	yes	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	4	2	no	—
170	C₂	—	4	4	no	—
171	C₂	#30631543	1	1	yes	characteristic
172	C₂	#30631671	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548634

C₂

yes

maximal, complemented

C₂×#22548632

C₂

yes

maximal, complemented

#30631651

C₂

yes

maximal, complemented

C₂²×#41×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860794

C₂²

yes

—

#22548634

—

#22548632

—

C₂²⋊(C₂₅₁⋊C₈)

—

C₂×#41×C₂₅₁

—

C₂×#41×C₂₅₁

—

C₂×#41×C₂₅₁

—

C₂²×Q₈×C₂₅₁

C₂²

yes

characteristic

C₂²×D₄×C₂₅₁

C₄

yes

—

C₂²×D₄×C₂₅₁

C₂²

yes

characteristic

C₂³×C₄×C₂₅₁

C₄

yes

—

C₂³×C₄×C₂₅₁

C₂²

yes

characteristic

#16860794

—

#16860794

—

C₂×C₂₅₁⋊C₈

—

C₂⁴×C₂₅₁

C₂×C₄

yes

—

C₂⁴×C₂₅₁

C₂×C₄

yes

characteristic

#41×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

D₄

yes

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₂²⋊C₄

yes

—

C₂³×C₂₅₁

C₂²⋊C₄

yes

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

Q₈×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#120

yes

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#379

yes

characteristic

C₂×C₂₅₁

#413

yes

—

C₂₅₁

#1785

yes

characteristic, complemented, Sylow

#1785

—

251

maximal, Sylow

C₂²×#41

D₂₅₁

yes

characteristic

C₂×#101

—

251

—

C₂×#100

—

251

—

#407

—

251

—

C₂×#41

—

C₂×#41

—

C₂×#41

—

C₂²×Q₈

C₂×D₂₅₁

yes

characteristic

C₂²×D₄

Dic₂₅₁

yes

—

C₂²×D₄

C₂×D₂₅₁

yes

characteristic

C₂³×C₄

Dic₂₅₁

yes

—

C₂³×C₄

C₂×D₂₅₁

yes

characteristic

100

C₂×M₁₆

—

251

—

101

#101

—

502

—

102

#100

—

502

—

103

C₂²⋊C₈

—

502

—

104

C₂⁴

C₂×Dic₂₅₁

yes

—

105

C₂⁴

C₂×Dic₂₅₁

yes

characteristic

106

#41

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

C₂²⋊D₂₅₁

yes

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

C₂²⋊D₂₅₁

yes

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

C₂×Dic₂₅₁

yes

characteristic

122

C₂²×C₄

C₂×Dic₂₅₁

yes

—

123

C₂²×C₄

C₂²⋊D₂₅₁

yes

—

124

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

125

M₁₆

—

502

—

126

M₁₆

—

502

—

127

C₂×C₈

—

502

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂²⋊Dic₂₅₁

yes

—

135

C₂³

C₂²⋊Dic₂₅₁

yes

—

136

C₂³

C₂²×Dic₂₅₁

yes

characteristic

137

Q₈

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊Dic₂₅₁

yes

—

147

C₂×C₄

C₂²⋊Dic₂₅₁

yes

—

148

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

149

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

150

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

151

C₈

—

1004

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

C₂×C₂²⋊Dic₂₅₁

yes

characteristic

162

C₂²

C₂×C₂²⋊Dic₂₅₁

yes

characteristic

163

C₂²

C₂×C₂²⋊Dic₂₅₁

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

#22548684

yes

—

167

C₂

—

168

C₂

—

169

C₂

—

170

C₂

—

171

C₂

#30631543

yes

characteristic

172

C₂

#30631671

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₅₁
commutator	C₂³×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting, prime radical	C₂²×#41×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂²×#41	D₂₅₁
hypocenter	C₂₅₁	#1785
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#41

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₅₁

commutator

C₂³×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting, prime radical

C₂²×#41×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

hypercenter

C₂²×#41

D₂₅₁

hypocenter

C₂₅₁

#1785

LCS

G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#41

N	H ≅ G/N	type
1. C₂×#22548634	C₂	semidirect
2. C₂×#22548632	C₂	semidirect
3. #30631651	C₂	semidirect
86. C₂₅₁	#1785	semidirect

H ≅ G/N

type

1. C₂×#22548634

C₂

semidirect

2. C₂×#22548632

C₂

semidirect

3. #30631651

C₂

semidirect

86. C₂₅₁

#1785

semidirect

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

subgroup	index
C₂×#22548634	2
C₂×#22548632	2
#30631651	2
C₂²×#41×C₂₅₁	2
#1785	251

subgroup

index

C₂²×#41×C₂₅₁

251

Subgroups — #44550293

Conjugacy classes of subgroups

Complements

Maximal subgroups