Group #44558437 — subgroups

Subgroups — #44558437

Order 32128.

456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22548709	C₂	1	1	yes	maximal, characteristic, complemented
2	#30631620	C₂	1	3	yes	maximal, complemented
3	#30631610	C₂	1	3	yes	maximal, complemented
4	#30631609	C₂	1	3	yes	maximal, complemented
5	#30631603	C₂	1	3	yes	maximal
6	#30631557	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#126×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊D₂₅₁	C₂²	1	1	yes	characteristic
9	#22548709	—	2	4	no	—
10	C₂×C₄⋊Dic₂₅₁	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₂₅₁	C₂²	1	3	yes	—
12	C₂×Dic₂₅₁⋊C₄	C₂²	1	3	yes	—
13	C₂×Dic₂₅₁⋊C₄	C₂²	1	3	yes	—
14	C₂×C₂²⋊Dic₂₅₁	C₂²	1	3	yes	—
15	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	3	yes	—
16	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	3	yes	—
17	C₂₅₁⋊#95	C₂²	1	3	yes	—
18	C₂₅₁⋊#95	C₂²	1	3	yes	—
19	#126×C₂₅₁	—	2	4	no	—
20	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	3	yes	—
21	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	3	yes	—
22	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
23	C₂³×D₂₅₁	C₂³	1	1	yes	characteristic
24	C₂×C₂²⋊D₂₅₁	—	4	3	no	—
25	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₂₅₁	C₂³	1	3	yes	—
27	C₂²×Dic₂₅₁	C₂³	1	3	yes	—
28	C₄⋊Dic₂₅₁	—	4	3	no	—
29	C₄⋊Dic₂₅₁	—	4	3	no	—
30	Dic₂₅₁⋊C₄	—	2	6	no	—
31	Dic₂₅₁⋊C₄	—	4	3	no	—
32	C₂²⋊Dic₂₅₁	—	4	3	no	—
33	(C₂×D₂₅₁)⋊C₄	—	2	6	no	—
34	(C₂×D₂₅₁)⋊C₄	—	4	3	no	—
35	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
37	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
38	C₄⋊C₄×C₂₅₁	—	2	6	no	—
39	C₂²⋊C₄×C₂₅₁	—	2	6	no	—
40	C₄²×C₂₅₁	D₄	1	4	yes	complemented
41	C₂²×D₂₅₁	—	2	1	no	—
42	C₂²×D₂₅₁	—	2	3	no	—
43	C₂²×D₂₅₁	—	2	3	no	—
44	C₂²⋊D₂₅₁	—	4	4	no	—
45	C₂×Dic₂₅₁	—	2	3	no	—
46	C₂×Dic₂₅₁	—	2	3	no	—
47	C₂×Dic₂₅₁	—	2	3	no	—
48	C₂×Dic₂₅₁	—	2	3	no	—
49	C₂×Dic₂₅₁	—	2	3	no	—
50	C₂×Dic₂₅₁	—	2	3	no	—
51	C₂×Dic₂₅₁	—	2	3	no	—
52	C₂³×C₂₅₁	—	2	1	no	—
53	C₂³×C₂₅₁	—	2	3	no	—
54	C₂³×C₂₅₁	—	2	3	no	—
55	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
56	C₂×C₄×C₂₅₁	—	2	3	no	—
57	C₂×C₄×C₂₅₁	—	2	3	no	—
58	C₂×C₄×C₂₅₁	—	2	3	no	—
59	C₂×C₄×C₂₅₁	—	2	3	no	—
60	C₂×C₄×C₂₅₁	—	2	3	no	—
61	C₂×C₄×C₂₅₁	C₂×D₄	1	6	yes	—
62	C₂²×C₂₅₁	—	4	3	no	—
63	C₂²×C₂₅₁	—	4	1	no	—
64	C₂²×C₂₅₁	—	4	3	no	—
65	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
66	C₂²×C₂₅₁	H(5,𝔽₂)	1	3	yes	—
67	C₂²×C₂₅₁	2₋¹⁺⁴	1	3	yes	—
68	C₂×D₂₅₁	—	4	3	no	—
69	C₂×D₂₅₁	—	4	1	no	—
70	C₂×D₂₅₁	—	4	3	no	—
71	C₄×C₂₅₁	—	2	6	no	—
72	C₄×C₂₅₁	—	4	3	no	—
73	Dic₂₅₁	—	4	3	no	—
74	Dic₂₅₁	—	4	3	no	—
75	Dic₂₅₁	—	4	1	no	—
76	C₂×C₂₅₁	—	8	1	no	—
77	C₂×C₂₅₁	#537	1	3	yes	—
78	C₂×C₂₅₁	#560	1	1	yes	characteristic
79	C₂×C₂₅₁	#563	1	3	yes	—
80	D₂₅₁	—	8	1	no	—
81	C₂₅₁	#2823	1	1	yes	characteristic, complemented, Sylow
82	#2823	—	251	1	no	maximal, Sylow
83	C₂×#126	—	251	1	no	—
84	C₂×#126	D₂₅₁	1	1	yes	characteristic, complemented
85	#398	—	251	3	no	—
86	#396	—	251	3	no	—
87	#396	—	251	3	no	—
88	#394	—	251	3	no	—
89	#384	—	251	1	no	—
90	C₂²×D₄	—	251	1	no	—
91	#126	—	2	4	no	—
92	#126	—	502	4	no	—
93	C₂×C₄⋊C₄	—	251	3	no	—
94	C₂×C₄⋊C₄	—	251	3	no	—
95	C₂×C₄⋊C₄	—	251	3	no	—
96	C₂×C₄⋊C₄	—	251	3	no	—
97	C₂×C₄⋊C₄	C₂×D₂₅₁	1	3	yes	—
98	C₂×C₂²⋊C₄	—	251	3	no	—
99	C₂×C₂²⋊C₄	—	251	3	no	—
100	C₂×C₂²⋊C₄	—	251	3	no	—
101	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	3	yes	—
102	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
103	#95	—	251	3	no	—
104	#95	—	251	3	no	—
105	C₂⁴	—	251	1	no	—
106	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
107	C₂×D₄	—	1004	3	no	—
108	C₂²×C₄	—	251	1	no	—
109	C₂²×C₄	—	251	3	no	—
110	C₂²×C₄	—	251	3	no	—
111	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
112	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
113	C₄⋊C₄	—	2	6	no	—
114	C₄⋊C₄	—	1004	3	no	—
115	C₄⋊C₄	—	1004	3	no	—
116	C₄⋊C₄	—	1004	3	no	—
117	C₄⋊C₄	—	502	6	no	—
118	C₂²⋊C₄	—	2	6	no	—
119	C₂²⋊C₄	—	1004	3	no	—
120	C₂²⋊C₄	—	1004	3	no	—
121	C₂²⋊C₄	—	502	6	no	—
122	C₄²	C₂²⋊D₂₅₁	1	4	yes	complemented
123	C₂³	—	2	1	no	—
124	C₂³	—	2	3	no	—
125	C₂³	—	2	3	no	—
126	C₂³	—	502	1	no	—
127	C₂³	—	502	3	no	—
128	C₂³	—	502	3	no	—
129	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
130	D₄	—	1004	4	no	—
131	C₂×C₄	—	2	3	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	2	3	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	502	3	no	—
137	C₂×C₄	—	502	3	no	—
138	C₂×C₄	—	502	3	no	—
139	C₂×C₄	—	502	3	no	—
140	C₂×C₄	—	502	3	no	—
141	C₂×C₄	—	502	3	no	—
142	C₂×C₄	—	502	3	no	—
143	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	6	yes	—
144	C₂²	—	4	3	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	4	3	no	—
147	C₂²	—	1004	3	no	—
148	C₂²	—	1004	1	no	—
149	C₂²	—	1004	3	no	—
150	C₂²	C₂²×C₂²⋊D₂₅₁	1	1	yes	characteristic
151	C₂²	#22548761	1	3	yes	—
152	C₂²	#22548762	1	3	yes	—
153	C₄	—	2	6	no	—
154	C₄	—	4	3	no	—
155	C₄	—	1004	3	no	—
156	C₄	—	1004	1	no	—
157	C₄	—	1004	3	no	—
158	C₂	—	8	1	no	—
159	C₂	—	2008	1	no	—
160	C₂	#30632265	1	3	yes	—
161	C₂	#30632428	1	1	yes	characteristic
162	C₂	#30632439	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548709

C₂

yes

maximal, characteristic, complemented

#30631620

C₂

yes

maximal, complemented

#30631610

C₂

yes

maximal, complemented

#30631609

C₂

yes

maximal, complemented

#30631603

C₂

yes

maximal

#30631557

C₂

yes

maximal, characteristic, complemented

C₂×#126×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₂₅₁

C₂²

yes

characteristic

#22548709

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

C₂²

yes

—

C₂×Dic₂₅₁⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₂₅₁

C₂²

yes

—

C₂×(C₂×D₂₅₁)⋊C₄

C₂²

yes

—

C₂×(C₂×D₂₅₁)⋊C₄

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

#126×C₂₅₁

—

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₂³×D₂₅₁

C₂³

yes

characteristic

C₂×C₂²⋊D₂₅₁

—

C₂²×Dic₂₅₁

C₂³

yes

characteristic

C₂²×Dic₂₅₁

C₂³

yes

—

C₂²×Dic₂₅₁

C₂³

yes

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₂²⋊Dic₂₅₁

—

(C₂×D₂₅₁)⋊C₄

—

(C₂×D₂₅₁)⋊C₄

—

C₂⁴×C₂₅₁

C₂³

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₄⋊C₄×C₂₅₁

—

C₂²⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

D₄

yes

complemented

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

C₂²⋊D₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₂⁴

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂²×D₄

yes

characteristic

C₂²×C₂₅₁

H(5,𝔽₂)

yes

—

C₂²×C₂₅₁

2₋¹⁺⁴

yes

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#537

yes

—

C₂×C₂₅₁

#560

yes

characteristic

C₂×C₂₅₁

#563

yes

—

D₂₅₁

—

C₂₅₁

#2823

yes

characteristic, complemented, Sylow

#2823

—

251

maximal, Sylow

C₂×#126

—

251

—

C₂×#126

D₂₅₁

yes

characteristic, complemented

#398

—

251

—

#396

—

251

—

#396

—

251

—

#394

—

251

—

#384

—

251

—

C₂²×D₄

—

251

—

#126

—

#126

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₂²⋊C₄

—

251

—

C₂×C₂²⋊C₄

—

251

—

100

C₂×C₂²⋊C₄

—

251

—

101

C₂×C₂²⋊C₄

C₂×D₂₅₁

yes

—

102

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

103

#95

—

251

—

104

#95

—

251

—

105

C₂⁴

—

251

—

106

C₂⁴

C₂²×D₂₅₁

yes

characteristic

107

C₂×D₄

—

1004

—

108

C₂²×C₄

—

251

—

109

C₂²×C₄

—

251

—

110

C₂²×C₄

—

251

—

111

C₂²×C₄

C₂²×D₂₅₁

yes

—

112

C₂²×C₄

C₂²×D₂₅₁

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

1004

—

115

C₄⋊C₄

—

1004

—

116

C₄⋊C₄

—

1004

—

117

C₄⋊C₄

—

502

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

1004

—

120

C₂²⋊C₄

—

1004

—

121

C₂²⋊C₄

—

502

—

122

C₄²

C₂²⋊D₂₅₁

yes

complemented

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

502

—

127

C₂³

—

502

—

128

C₂³

—

502

—

129

C₂³

C₂³×D₂₅₁

yes

characteristic

130

D₄

—

1004

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

502

—

137

C₂×C₄

—

502

—

138

C₂×C₄

—

502

—

139

C₂×C₄

—

502

—

140

C₂×C₄

—

502

—

141

C₂×C₄

—

502

—

142

C₂×C₄

—

502

—

143

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

1004

—

148

C₂²

—

1004

—

149

C₂²

—

1004

—

150

C₂²

C₂²×C₂²⋊D₂₅₁

yes

characteristic

151

C₂²

#22548761

yes

—

152

C₂²

#22548762

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

1004

—

156

C₄

—

1004

—

157

C₄

—

1004

—

158

C₂

—

159

C₂

—

2008

—

160

C₂

#30632265

yes

—

161

C₂

#30632428

yes

characteristic

162

C₂

#30632439

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	C₂×#126×C₂₅₁	C₂
prime radical	C₂²×C₂²⋊D₂₅₁	C₂²
hypercenter	C₂×#126	D₂₅₁
hypocenter	C₂₅₁	#2823
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×#126

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₅₁

commutator, socle, cosocle

C₂³×C₂₅₁

C₂⁴

Fitting

C₂×#126×C₂₅₁

C₂

prime radical

C₂²×C₂²⋊D₂₅₁

C₂²

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂³ ◁ C₂×#126

N	H ≅ G/N	type
1. C₂×#22548709	C₂	semidirect
2. #30631620	C₂	semidirect
3. #30631610	C₂	semidirect
4. #30631609	C₂	semidirect
6. #30631557	C₂	semidirect
7. C₂×#126×C₂₅₁	C₂	semidirect
40. C₄²×C₂₅₁	D₄	semidirect
81. C₂₅₁	#2823	semidirect
84. C₂×#126	D₂₅₁	semidirect
122. C₄²	C₂²⋊D₂₅₁	semidirect

H ≅ G/N

type

1. C₂×#22548709

C₂

semidirect

2. #30631620

C₂

semidirect

3. #30631610

C₂

semidirect

4. #30631609

C₂

semidirect

6. #30631557

C₂

semidirect

7. C₂×#126×C₂₅₁

C₂

semidirect

40. C₄²×C₂₅₁

D₄

semidirect

81. C₂₅₁

#2823

semidirect

84. C₂×#126

D₂₅₁

semidirect

122. C₄²

C₂²⋊D₂₅₁

semidirect

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

subgroup	index
C₂×#22548709	2
#30631620	2
#30631610	2
#30631609	2
#30631603	2
#30631557	2
C₂×#126×C₂₅₁	2
#2823	251

subgroup

index

C₂×#126×C₂₅₁

251

Subgroups — #44558437

Conjugacy classes of subgroups

Complements

Maximal subgroups