Group #44558274 — subgroups

Subgroups — #44558274

Order 32128.

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

normal subgroups only

This group has 420 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	C₂₅₁⋊#401	C₂	1	1	yes	maximal, characteristic, complemented
2	#30631624	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631603	C₂	1	3	yes	maximal
4	#30631586	C₂	1	3	yes	maximal, complemented
5	#30631580	C₂	1	3	yes	maximal, complemented
6	#30631577	C₂	1	3	yes	maximal, complemented
7	#396×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊D₂₅₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₅₁	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₂₅₁	C₂²	1	3	yes	—
11	C₂×C₂²⋊Dic₂₅₁	C₂²	1	3	yes	—
12	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	3	yes	—
13	C₂×(C₂×D₂₅₁)⋊C₄	C₂²	1	3	yes	—
14	C₂×C₄×Dic₂₅₁	C₂²	1	3	yes	—
15	C₂₅₁⋊#95	C₂²	1	3	yes	—
16	C₂₅₁⋊#95	C₂²	1	3	yes	—
17	C₂₅₁⋊#95	C₂²	1	3	yes	—
18	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	3	yes	—
19	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	3	yes	—
20	#95×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₂³×D₂₅₁	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₂₅₁	—	4	3	no	—
23	C₂²×Dic₂₅₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₂₅₁	C₂³	1	3	yes	—
25	C₂²×Dic₂₅₁	C₂³	1	3	yes	—
26	C₄⋊Dic₂₅₁	—	4	3	no	—
27	C₄⋊Dic₂₅₁	—	4	3	no	—
28	C₂²⋊Dic₂₅₁	—	4	3	no	—
29	(C₂×D₂₅₁)⋊C₄	—	2	6	no	—
30	(C₂×D₂₅₁)⋊C₄	—	4	3	no	—
31	C₄×Dic₂₅₁	—	4	3	no	—
32	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
34	C₂²×C₄×C₂₅₁	C₂³	1	3	yes	—
35	C₄⋊C₄×C₂₅₁	—	2	6	no	—
36	C₂²⋊C₄×C₂₅₁	—	2	6	no	—
37	C₂²×D₂₅₁	—	2	3	no	—
38	C₂²×D₂₅₁	—	2	3	no	—
39	C₂²×D₂₅₁	—	2	1	no	—
40	C₂²⋊D₂₅₁	—	4	4	no	—
41	C₂×Dic₂₅₁	—	2	3	no	—
42	C₂×Dic₂₅₁	—	2	3	no	—
43	C₂×Dic₂₅₁	—	2	3	no	—
44	C₂×Dic₂₅₁	—	2	3	no	—
45	C₂×Dic₂₅₁	—	2	3	no	—
46	C₂×Dic₂₅₁	—	2	3	no	—
47	C₂×Dic₂₅₁	—	2	3	no	—
48	C₂³×C₂₅₁	—	2	3	no	—
49	C₂³×C₂₅₁	—	2	3	no	—
50	C₂³×C₂₅₁	—	2	1	no	—
51	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₄×C₂₅₁	—	2	3	no	—
53	C₂×C₄×C₂₅₁	—	2	3	no	—
54	C₂×C₄×C₂₅₁	—	2	3	no	—
55	C₂×C₄×C₂₅₁	—	2	3	no	—
56	C₂×C₄×C₂₅₁	—	2	3	no	—
57	C₂×C₄×C₂₅₁	#41	1	6	yes	—
58	C₂²×C₂₅₁	—	4	3	no	—
59	C₂²×C₂₅₁	—	4	3	no	—
60	C₂²×C₂₅₁	—	4	1	no	—
61	C₂²×C₂₅₁	C₂×#41	1	3	yes	—
62	C₂²×C₂₅₁	H(5,𝔽₂)	1	3	yes	—
63	C₂²×C₂₅₁	2₋¹⁺⁴	1	1	yes	characteristic
64	C₂×D₂₅₁	—	4	3	no	—
65	C₂×D₂₅₁	—	4	3	no	—
66	C₂×D₂₅₁	—	4	1	no	—
67	C₄×C₂₅₁	—	2	6	no	—
68	C₄×C₂₅₁	—	4	3	no	—
69	Dic₂₅₁	—	4	3	no	—
70	Dic₂₅₁	—	4	3	no	—
71	Dic₂₅₁	—	4	1	no	—
72	C₂×C₂₅₁	—	8	1	no	—
73	C₂×C₂₅₁	#542	1	3	yes	—
74	C₂×C₂₅₁	#553	1	3	yes	—
75	C₂×C₂₅₁	#560	1	1	yes	characteristic
76	D₂₅₁	—	8	1	no	—
77	C₂₅₁	#2801	1	1	yes	characteristic, complemented, Sylow
78	#2801	—	251	1	no	maximal, Sylow
79	#401	—	251	1	no	—
80	#399	—	251	1	no	—
81	#396	D₂₅₁	1	1	yes	characteristic, complemented
82	#394	—	251	3	no	—
83	#389	—	251	3	no	—
84	#388	—	251	3	no	—
85	#388	—	251	3	no	—
86	C₂²×D₄	—	251	1	no	—
87	C₂×C₄⋊C₄	—	251	3	no	—
88	C₂×C₄⋊C₄	—	251	3	no	—
89	C₂×C₄⋊C₄	C₂×D₂₅₁	1	3	yes	—
90	C₂×C₂²⋊C₄	—	251	3	no	—
91	C₂×C₂²⋊C₄	—	251	3	no	—
92	C₂×C₂²⋊C₄	—	251	3	no	—
93	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	3	yes	—
94	C₂×C₄²	—	251	3	no	—
95	#95	—	251	3	no	—
96	#95	—	251	3	no	—
97	#95	—	251	3	no	—
98	#95	C₂×D₂₅₁	1	1	yes	characteristic
99	C₂⁴	—	251	1	no	—
100	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
101	C₂×D₄	—	1004	3	no	—
102	C₂²×C₄	—	251	3	no	—
103	C₂²×C₄	—	251	1	no	—
104	C₂²×C₄	—	251	3	no	—
105	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
106	C₂²×C₄	C₂²×D₂₅₁	1	3	yes	—
107	C₄⋊C₄	—	2	6	no	—
108	C₄⋊C₄	—	1004	3	no	—
109	C₄⋊C₄	—	1004	3	no	—
110	C₂²⋊C₄	—	2	6	no	—
111	C₂²⋊C₄	—	1004	3	no	—
112	C₂²⋊C₄	—	1004	3	no	—
113	C₂²⋊C₄	—	502	6	no	—
114	C₄²	—	1004	3	no	—
115	C₂³	—	2	3	no	—
116	C₂³	—	2	3	no	—
117	C₂³	—	2	1	no	—
118	C₂³	—	502	3	no	—
119	C₂³	—	502	3	no	—
120	C₂³	—	502	1	no	—
121	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
122	D₄	—	1004	4	no	—
123	C₂×C₄	—	2	3	no	—
124	C₂×C₄	—	2	3	no	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	502	3	no	—
129	C₂×C₄	—	502	3	no	—
130	C₂×C₄	—	502	3	no	—
131	C₂×C₄	—	502	3	no	—
132	C₂×C₄	—	502	3	no	—
133	C₂×C₄	—	502	3	no	—
134	C₂×C₄	—	502	3	no	—
135	C₂×C₄	#16860809	1	6	yes	—
136	C₂²	—	4	3	no	—
137	C₂²	—	4	3	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	1004	3	no	—
140	C₂²	—	1004	3	no	—
141	C₂²	—	1004	1	no	—
142	C₂²	C₂×#16860809	1	3	yes	—
143	C₂²	#22548761	1	3	yes	—
144	C₂²	#22548763	1	1	yes	characteristic
145	C₄	—	2	6	no	—
146	C₄	—	4	3	no	—
147	C₄	—	1004	3	no	—
148	C₄	—	1004	1	no	—
149	C₄	—	1004	3	no	—
150	C₂	—	8	1	no	—
151	C₂	—	2008	1	no	—
152	C₂	#30632310	1	3	yes	—
153	C₂	#30632389	1	3	yes	—
154	C₂	#30632428	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₅₁⋊#401

C₂

yes

maximal, characteristic, complemented

#30631624

C₂

yes

maximal, characteristic, complemented

#30631603

C₂

yes

maximal

#30631586

C₂

yes

maximal, complemented

#30631580

C₂

yes

maximal, complemented

#30631577

C₂

yes

maximal, complemented

#396×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₂₅₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂×C₂²⋊Dic₂₅₁

C₂²

yes

—

C₂×(C₂×D₂₅₁)⋊C₄

C₂²

yes

—

C₂×(C₂×D₂₅₁)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₂₅₁

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂₅₁⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₅₁

C₂²

yes

—

#95×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

—

C₂²⋊Dic₂₅₁

—

(C₂×D₂₅₁)⋊C₄

—

(C₂×D₂₅₁)⋊C₄

—

C₄×Dic₂₅₁

—

C₂⁴×C₂₅₁

C₂³

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

—

C₄⋊C₄×C₂₅₁

—

C₂²⋊C₄×C₂₅₁

—

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

#41

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×#41

yes

—

C₂²×C₂₅₁

H(5,𝔽₂)

yes

—

C₂²×C₂₅₁

2₋¹⁺⁴

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#542

yes

—

C₂×C₂₅₁

#553

yes

—

C₂×C₂₅₁

#560

yes

characteristic

D₂₅₁

—

C₂₅₁

#2801

yes

characteristic, complemented, Sylow

#2801

—

251

maximal, Sylow

#401

—

251

—

#399

—

251

—

#396

D₂₅₁

yes

characteristic, complemented

#394

—

251

—

#389

—

251

—

#388

—

251

—

#388

—

251

—

C₂²×D₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₂²⋊C₄

—

251

—

C₂×C₂²⋊C₄

—

251

—

C₂×C₂²⋊C₄

—

251

—

C₂×C₂²⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₄²

—

251

—

#95

—

251

—

#95

—

251

—

#95

—

251

—

#95

C₂×D₂₅₁

yes

characteristic

C₂⁴

—

251

—

100

C₂⁴

C₂²×D₂₅₁

yes

characteristic

101

C₂×D₄

—

1004

—

102

C₂²×C₄

—

251

—

103

C₂²×C₄

—

251

—

104

C₂²×C₄

—

251

—

105

C₂²×C₄

C₂²×D₂₅₁

yes

—

106

C₂²×C₄

C₂²×D₂₅₁

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

1004

—

109

C₄⋊C₄

—

1004

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

1004

—

112

C₂²⋊C₄

—

1004

—

113

C₂²⋊C₄

—

502

—

114

C₄²

—

1004

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

502

—

119

C₂³

—

502

—

120

C₂³

—

502

—

121

C₂³

C₂³×D₂₅₁

yes

characteristic

122

D₄

—

1004

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

502

—

129

C₂×C₄

—

502

—

130

C₂×C₄

—

502

—

131

C₂×C₄

—

502

—

132

C₂×C₄

—

502

—

133

C₂×C₄

—

502

—

134

C₂×C₄

—

502

—

135

C₂×C₄

#16860809

yes

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

1004

—

140

C₂²

—

1004

—

141

C₂²

—

1004

—

142

C₂²

C₂×#16860809

yes

—

143

C₂²

#22548761

yes

—

144

C₂²

#22548763

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

1004

—

148

C₄

—

1004

—

149

C₄

—

1004

—

150

C₂

—

151

C₂

—

2008

—

152

C₂

#30632310

yes

—

153

C₂

#30632389

yes

—

154

C₂

#30632428

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator, socle, cosocle	C₂³×C₂₅₁	C₂⁴
Fitting	#396×C₂₅₁	C₂
prime radical	C₂²×C₂²⋊D₂₅₁	C₂²
hypercenter	#396	D₂₅₁
hypocenter	C₂₅₁	#2801
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ #396

subgroup

G/H

center, Frattini

C₂³

C₂³×D₂₅₁

commutator, socle, cosocle

C₂³×C₂₅₁

C₂⁴

Fitting

#396×C₂₅₁

C₂

prime radical

C₂²×C₂²⋊D₂₅₁

C₂²

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂³ ◁ #396

N	H ≅ G/N	type
1. C₂₅₁⋊#401	C₂	semidirect
2. #30631624	C₂	semidirect
4. #30631586	C₂	semidirect
5. #30631580	C₂	semidirect
6. #30631577	C₂	semidirect
7. #396×C₂₅₁	C₂	semidirect
77. C₂₅₁	#2801	semidirect
81. #396	D₂₅₁	semidirect

H ≅ G/N

type

1. C₂₅₁⋊#401

C₂

semidirect

2. #30631624

C₂

semidirect

4. #30631586

C₂

semidirect

5. #30631580

C₂

semidirect

6. #30631577

C₂

semidirect

7. #396×C₂₅₁

C₂

semidirect

77. C₂₅₁

#2801

semidirect

81. #396

D₂₅₁

semidirect

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

subgroup	index
C₂₅₁⋊#401	2
#30631624	2
#30631603	2
#30631586	2
#30631580	2
#30631577	2
#396×C₂₅₁	2
#2801	251

subgroup

index

C₂₅₁⋊#401

#396×C₂₅₁

251

Subgroups — #44558274

Conjugacy classes of subgroups

Complements

Maximal subgroups