Group #44550355 — subgroups

Subgroups — #44550355

Order 32128.

336 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 95 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×D₄×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
2	#30631710	C₂	1	2	yes	maximal, complemented
3	C₂×#22548632	C₂	1	1	yes	maximal, characteristic
4	#30631455	C₂	1	2	yes	maximal, complemented
5	C₂×#100×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₂₅₁	C₄	1	2	yes	—
7	C₂×#16860794	C₂²	1	2	yes	—
8	D₄×Dic₂₅₁	—	2	2	no	—
9	D₄×Dic₂₅₁	—	2	2	no	—
10	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₂₅₁	C₄	1	2	yes	—
12	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
13	Dic₂₅₁⋊C₈	—	2	2	no	—
14	Dic₂₅₁⋊C₈	—	2	2	no	—
15	#22548632	—	2	2	no	—
16	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
18	#100×C₂₅₁	—	2	2	no	—
19	C₂²×Dic₂₅₁	—	2	2	no	—
20	C₂²×Dic₂₅₁	—	2	2	no	—
21	C₂²×Dic₂₅₁	—	4	4	no	—
22	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
24	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
25	#16860794	—	2	2	no	—
26	#16860794	—	2	2	no	—
27	C₂×C₂₅₁⋊C₈	—	2	2	no	—
28	C₄⋊Dic₂₅₁	—	2	2	no	—
29	C₂²⋊Dic₂₅₁	—	2	4	no	—
30	C₄×Dic₂₅₁	D₄	1	2	yes	—
31	C₄×Dic₂₅₁	D₄	1	2	yes	—
32	C₂⁴×C₂₅₁	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₂₅₁	—	2	2	no	—
34	C₂×D₄×C₂₅₁	—	2	2	no	—
35	C₂×D₄×C₂₅₁	D₄	1	2	yes	—
36	C₂×D₄×C₂₅₁	Q₈	1	2	yes	—
37	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
38	M₁₆×C₂₅₁	—	2	2	no	—
39	M₁₆×C₂₅₁	—	2	2	no	—
40	C₂×C₈×C₂₅₁	—	2	2	no	—
41	C₂×Dic₂₅₁	—	2	2	no	—
42	C₂×Dic₂₅₁	—	2	2	no	—
43	C₂×Dic₂₅₁	—	4	1	no	—
44	C₂×Dic₂₅₁	—	4	4	no	—
45	C₂×Dic₂₅₁	—	4	4	no	—
46	C₂×Dic₂₅₁	—	4	2	no	—
47	C₂×Dic₂₅₁	C₂²⋊C₄	1	2	yes	—
48	C₂×Dic₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂₅₁⋊C₈	—	4	2	no	—
51	C₂³×C₂₅₁	—	2	2	no	—
52	C₂³×C₂₅₁	—	2	2	no	—
53	C₂³×C₂₅₁	—	4	2	no	—
54	C₂³×C₂₅₁	—	4	2	no	—
55	C₂³×C₂₅₁	C₄⋊C₄	1	4	yes	—
56	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
57	D₄×C₂₅₁	—	2	4	no	—
58	D₄×C₂₅₁	—	2	4	no	—
59	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₂₅₁	#41	1	2	yes	—
63	C₈×C₂₅₁	—	4	2	no	—
64	C₂²×C₂₅₁	—	2	1	no	—
65	C₂²×C₂₅₁	—	2	1	no	—
66	C₂²×C₂₅₁	—	2	4	no	—
67	C₂²×C₂₅₁	—	4	4	no	—
68	C₂²×C₂₅₁	—	4	4	no	—
69	C₂²×C₂₅₁	—	4	2	no	—
70	C₂²×C₂₅₁	—	4	2	no	—
71	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₂₅₁	#117	1	1	yes	characteristic
74	C₄×C₂₅₁	#121	1	2	yes	—
75	C₄×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
76	Dic₂₅₁	—	2	1	no	—
77	Dic₂₅₁	—	2	2	no	—
78	Dic₂₅₁	—	4	2	no	—
79	Dic₂₅₁	#100	1	2	yes	complemented
80	C₂×C₂₅₁	—	2	1	no	—
81	C₂×C₂₅₁	—	2	1	no	—
82	C₂×C₂₅₁	—	4	4	no	—
83	C₂×C₂₅₁	#380	1	1	yes	characteristic
84	C₂×C₂₅₁	C₂×#100	1	1	yes	characteristic
85	C₂×C₂₅₁	#413	1	1	yes	characteristic
86	C₂₅₁	#1800	1	1	yes	characteristic, complemented, Sylow
87	#1800	—	251	1	no	maximal, Sylow
88	C₂×C₄×D₄	—	251	1	no	—
89	#423	—	251	2	no	—
90	C₂×#100	—	251	1	no	—
91	C₂×#100	D₂₅₁	1	1	yes	characteristic
92	#343	—	251	2	no	—
93	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
94	C₂³×C₄	—	251	2	no	—
95	C₂×M₁₆	—	251	2	no	—
96	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
97	C₄×D₄	—	502	2	no	—
98	C₄×D₄	—	502	2	no	—
99	C₂×C₄⋊C₄	—	251	1	no	—
100	C₂×C₂²⋊C₄	—	251	2	no	—
101	C₂×C₄²	—	251	1	no	—
102	C₄⋊C₈	—	502	2	no	—
103	C₄⋊C₈	—	502	2	no	—
104	#100	—	2	2	no	—
105	#100	—	502	2	no	—
106	C₂⁴	C₄×D₂₅₁	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	C₂²⋊D₂₅₁	1	2	yes	—
110	C₂×D₄	Dic₅₀₂	1	2	yes	—
111	C₂²×C₄	—	1004	4	no	—
112	C₂²×C₄	—	502	2	no	—
113	C₂²×C₄	—	502	2	no	—
114	C₂²×C₄	—	251	2	no	—
115	C₂²×C₄	—	251	1	no	—
116	C₂²×C₄	—	251	1	no	—
117	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
118	M₁₆	—	2	2	no	—
119	M₁₆	—	2	2	no	—
120	M₁₆	—	502	2	no	—
121	M₁₆	—	502	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	502	2	no	—
124	C₄⋊C₄	—	502	2	no	—
125	C₂²⋊C₄	—	502	4	no	—
126	C₄²	—	251	2	no	—
127	C₄²	—	251	2	no	—
128	C₂³	—	2	2	no	—
129	C₂³	—	2	2	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	4	2	no	—
132	C₂³	Dic₂₅₁⋊C₄	1	4	yes	—
133	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
134	D₄	—	2	4	no	—
135	D₄	—	2	4	no	—
136	C₂×C₄	—	1004	4	no	—
137	C₂×C₄	—	1004	4	no	—
138	C₂×C₄	—	1004	2	no	—
139	C₂×C₄	—	1004	1	no	—
140	C₂×C₄	—	502	2	no	—
141	C₂×C₄	—	502	2	no	—
142	C₂×C₄	—	251	2	no	—
143	C₂×C₄	—	251	1	no	—
144	C₂×C₄	—	251	1	no	—
145	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
147	C₂×C₄	C₂×Dic₅₀₂	1	1	yes	characteristic
148	C₂×C₄	#16860809	1	2	yes	—
149	C₈	—	4	2	no	—
150	C₈	—	1004	2	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	4	4	no	—
155	C₂²	—	4	4	no	—
156	C₂²	—	4	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	C₂²⋊C₄×D₂₅₁	1	1	yes	characteristic
159	C₂²	C₂×Dic₂₅₁⋊C₄	1	1	yes	characteristic
160	C₂²	#22548671	1	1	yes	characteristic
161	C₄	—	1004	2	no	—
162	C₄	—	502	1	no	—
163	C₄	—	502	2	no	—
164	C₄	—	251	2	no	—
165	C₄	#22548687	1	2	yes	—
166	C₄	#22548692	1	2	yes	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	4	4	no	—
170	C₂	#30631545	1	1	yes	characteristic
171	C₂	#100×D₂₅₁	1	1	yes	characteristic
172	C₂	#30631668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×Dic₂₅₁

C₂

yes

maximal, characteristic

#30631710

C₂

yes

maximal, complemented

C₂×#22548632

C₂

yes

maximal, characteristic

#30631455

C₂

yes

maximal, complemented

C₂×#100×C₂₅₁

C₂

yes

maximal, characteristic

C₂³×Dic₂₅₁

C₄

yes

—

C₂×#16860794

C₂²

yes

—

D₄×Dic₂₅₁

—

D₄×Dic₂₅₁

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

characteristic

C₂×C₂²⋊Dic₂₅₁

C₄

yes

—

C₂×C₄×Dic₂₅₁

C₂²

yes

characteristic

Dic₂₅₁⋊C₈

—

Dic₂₅₁⋊C₈

—

#22548632

—

C₂²×D₄×C₂₅₁

C₂²

yes

characteristic

C₂×M₁₆×C₂₅₁

C₂²

yes

—

#100×C₂₅₁

—

C₂²×Dic₂₅₁

—

C₂²×Dic₂₅₁

—

C₂²×Dic₂₅₁

—

C₂²×Dic₂₅₁

C₂×C₄

yes

characteristic

C₂²×Dic₂₅₁

C₂×C₄

yes

—

C₂²×Dic₂₅₁

C₂×C₄

yes

characteristic

#16860794

—

#16860794

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

C₂²⋊Dic₂₅₁

—

C₄×Dic₂₅₁

D₄

yes

—

C₄×Dic₂₅₁

D₄

yes

—

C₂⁴×C₂₅₁

C₂×C₄

yes

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

D₄

yes

—

C₂×D₄×C₂₅₁

Q₈

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

C₂²⋊C₄

yes

—

C₂×Dic₂₅₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₅₁

C₂²⋊C₄

yes

characteristic

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₄⋊C₄

yes

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

D₄×C₂₅₁

—

D₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₈×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×C₄⋊C₄

yes

C₂²×C₂₅₁

yes

C₄×C₂₅₁

yes

—

C₄×C₂₅₁

C₂²⋊Q₈

yes

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

#100

yes

complemented

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#1800

—

251

maximal, Sylow

C₂×C₄×D₄

—

251

—

#423

—

251

—

C₂×#100

—

251

—

C₂×#100

D₂₅₁

yes

characteristic

#343

—

251

—

C₂²×D₄

C₂×D₂₅₁

yes

characteristic

C₂³×C₄

—

251

—

C₂×M₁₆

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

C₄×D₄

—

502

—

C₄×D₄

—

502

—

C₂×C₄⋊C₄

—

251

—

100

C₂×C₂²⋊C₄

—

251

—

101

C₂×C₄²

—

251

—

102

C₄⋊C₈

—

502

—

103

C₄⋊C₈

—

502

—

104

#100

—

105

#100

—

502

—

106

C₂⁴

C₄×D₂₅₁

yes

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂²⋊D₂₅₁

yes

—

110

C₂×D₄

Dic₅₀₂

yes

—

111

C₂²×C₄

—

1004

—

112

C₂²×C₄

—

502

—

113

C₂²×C₄

—

502

—

114

C₂²×C₄

—

251

—

115

C₂²×C₄

—

251

—

116

C₂²×C₄

—

251

—

117

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

M₁₆

—

502

—

121

M₁₆

—

502

—

122

C₂×C₈

—

123

C₂×C₈

—

502

—

124

C₄⋊C₄

—

502

—

125

C₂²⋊C₄

—

502

—

126

C₄²

—

251

—

127

C₄²

—

251

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

Dic₂₅₁⋊C₄

yes

—

133

C₂³

C₂×C₄×D₂₅₁

yes

characteristic

134

D₄

—

135

D₄

—

136

C₂×C₄

—

1004

—

137

C₂×C₄

—

1004

—

138

C₂×C₄

—

1004

—

139

C₂×C₄

—

1004

—

140

C₂×C₄

—

502

—

141

C₂×C₄

—

502

—

142

C₂×C₄

—

251

—

143

C₂×C₄

—

251

—

144

C₂×C₄

—

251

—

145

C₂×C₄

D₄×D₂₅₁

yes

—

146

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

147

C₂×C₄

C₂×Dic₅₀₂

yes

characteristic

148

C₂×C₄

#16860809

yes

—

149

C₈

—

150

C₈

—

1004

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂²⋊C₄×D₂₅₁

yes

characteristic

159

C₂²

C₂×Dic₂₅₁⋊C₄

yes

characteristic

160

C₂²

#22548671

yes

characteristic

161

C₄

—

1004

—

162

C₄

—

502

—

163

C₄

—

502

—

164

C₄

—

251

—

165

C₄

#22548687

yes

—

166

C₄

#22548692

yes

—

167

C₂

—

168

C₂

—

169

C₂

—

170

C₂

#30631545

yes

characteristic

171

C₂

#100×D₂₅₁

yes

characteristic

172

C₂

#30631668

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₅₁
commutator	C₂³×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	C₂×#100×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₂₅₁	C₂²
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	C₂×#100	D₂₅₁
hypocenter	C₂₅₁	#1800
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₅₁

commutator

C₂³×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

C₂×#100×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₂₅₁

C₂²

cosocle

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#100

N	H ≅ G/N	type
2. #30631710	C₂	semidirect
4. #30631455	C₂	semidirect
79. Dic₂₅₁	#100	semidirect
86. C₂₅₁	#1800	semidirect

H ≅ G/N

type

2. #30631710

C₂

semidirect

4. #30631455

C₂

semidirect

79. Dic₂₅₁

#100

semidirect

86. C₂₅₁

#1800

semidirect

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

subgroup	index
C₂×D₄×Dic₂₅₁	2
#30631710	2
C₂×#22548632	2
#30631455	2
C₂×#100×C₂₅₁	2
#1800	251

subgroup

index

C₂×D₄×Dic₂₅₁

C₂×#100×C₂₅₁

251

Subgroups — #44550355

Conjugacy classes of subgroups

Complements

Maximal subgroups