Group #44548269 — subgroups

Subgroups — C₂₅₁⋊#1304 #44548269

Order 32128.

396 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 65 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	#30631543	C₂	1	2	yes	maximal
2	#379×C₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊Dic₂₅₁	—	2	2	no	—
4	C₂×C₂²⋊Dic₂₅₁	—	2	2	no	—
5	C₂×C₂²⋊Dic₂₅₁	C₄	1	4	yes	complemented
6	C₂⁵×C₂₅₁	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₄×C₂₅₁	—	2	2	no	—
8	C₂×C₂²⋊C₄×C₂₅₁	C₄	1	2	yes	complemented
9	C₂²×Dic₂₅₁	—	2	4	no	—
10	C₂²⋊Dic₂₅₁	—	4	4	no	—
11	C₂²⋊Dic₂₅₁	—	4	4	no	—
12	C₂²⋊Dic₂₅₁	—	4	4	no	—
13	C₂⁴×C₂₅₁	—	4	2	no	—
14	C₂⁴×C₂₅₁	—	4	4	no	—
15	C₂⁴×C₂₅₁	C₂×C₄	1	2	yes	—
16	C₂⁴×C₂₅₁	C₂×C₄	1	1	yes	characteristic
17	C₂⁴×C₂₅₁	D₄	1	2	yes	—
18	C₂⁴×C₂₅₁	D₄	1	1	yes	characteristic
19	C₂⁴×C₂₅₁	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₂₅₁	—	2	2	no	—
21	C₂²⋊C₄×C₂₅₁	—	4	4	no	—
22	C₂²⋊C₄×C₂₅₁	—	4	2	no	—
23	C₂×Dic₂₅₁	—	4	4	no	—
24	C₂×Dic₂₅₁	—	4	4	no	—
25	C₂×Dic₂₅₁	—	4	4	no	—
26	C₂³×C₂₅₁	—	2	2	no	—
27	C₂³×C₂₅₁	—	2	4	no	—
28	C₂³×C₂₅₁	—	2	4	no	—
29	C₂³×C₂₅₁	—	2	4	no	—
30	C₂³×C₂₅₁	—	2	4	no	—
31	C₂³×C₂₅₁	—	4	2	no	—
32	C₂³×C₂₅₁	—	4	2	no	—
33	C₂³×C₂₅₁	—	4	2	no	—
34	C₂³×C₂₅₁	—	4	2	no	—
35	C₂³×C₂₅₁	—	4	1	no	—
36	C₂³×C₂₅₁	—	4	2	no	—
37	C₂³×C₂₅₁	—	4	1	no	—
38	C₂³×C₂₅₁	—	4	16	no	—
39	C₂³×C₂₅₁	C₄²	1	1	yes	characteristic
40	C₂³×C₂₅₁	C₂²⋊C₄	1	2	yes	—
41	C₂³×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂³×C₂₅₁	C₄⋊C₄	1	2	yes	—
43	C₂³×C₂₅₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₅₁	—	4	4	no	—
45	C₂×C₄×C₂₅₁	—	4	2	no	—
46	C₂²×C₂₅₁	—	2	2	no	—
47	C₂²×C₂₅₁	—	2	2	no	—
48	C₂²×C₂₅₁	—	2	2	no	—
49	C₂²×C₂₅₁	—	2	2	no	—
50	C₂²×C₂₅₁	—	2	1	no	—
51	C₂²×C₂₅₁	—	2	2	no	—
52	C₂²×C₂₅₁	—	2	2	no	—
53	C₂²×C₂₅₁	—	2	2	no	—
54	C₂²×C₂₅₁	—	2	2	no	—
55	C₂²×C₂₅₁	—	2	1	no	—
56	C₂²×C₂₅₁	—	4	4	no	—
57	C₂²×C₂₅₁	—	4	8	no	—
58	C₂²×C₂₅₁	—	4	8	no	—
59	C₂²×C₂₅₁	—	4	4	no	—
60	C₂²×C₂₅₁	—	4	4	no	—
61	C₂²×C₂₅₁	#95	1	1	yes	characteristic
62	C₂²×C₂₅₁	#99	1	4	yes	—
63	C₂²×C₂₅₁	#99	1	2	yes	—
64	C₄×C₂₅₁	—	8	2	no	—
65	Dic₂₅₁	—	8	4	no	—
66	C₂×C₂₅₁	—	2	4	no	—
67	C₂×C₂₅₁	—	2	2	no	—
68	C₂×C₂₅₁	—	4	1	no	—
69	C₂×C₂₅₁	—	4	2	no	—
70	C₂×C₂₅₁	—	4	1	no	—
71	C₂×C₂₅₁	#342	1	2	yes	—
72	C₂×C₂₅₁	#342	1	1	yes	characteristic
73	C₂₅₁	#1304	1	1	yes	characteristic, complemented, Sylow
74	#1304	—	251	1	no	maximal, Sylow
75	#379	—	251	2	no	—
76	#379	D₂₅₁	1	1	yes	characteristic
77	C₂⁵	C₂×D₂₅₁	1	1	yes	characteristic
78	C₂×C₂²⋊C₄	—	2	2	no	—
79	C₂×C₂²⋊C₄	—	502	2	no	—
80	C₂×C₂²⋊C₄	—	502	2	no	—
81	C₂×C₂²⋊C₄	—	251	4	no	—
82	C₂×C₂²⋊C₄	Dic₂₅₁	1	2	yes	complemented
83	C₂⁴	—	4	4	no	—
84	C₂⁴	—	4	2	no	—
85	C₂⁴	C₄×D₂₅₁	1	2	yes	—
86	C₂⁴	C₂×Dic₂₅₁	1	1	yes	characteristic
87	C₂⁴	C₂²⋊D₂₅₁	1	2	yes	—
88	C₂⁴	D₁₀₀₄	1	1	yes	characteristic
89	C₂⁴	Dic₅₀₂	1	1	yes	characteristic
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	502	4	no	—
92	C₂²⋊C₄	—	4	4	no	—
93	C₂²⋊C₄	—	4	2	no	—
94	C₂²⋊C₄	—	1004	4	no	—
95	C₂²⋊C₄	—	1004	4	no	—
96	C₂²⋊C₄	—	1004	4	no	—
97	C₂³	—	2	4	no	—
98	C₂³	—	2	4	no	—
99	C₂³	—	2	4	no	—
100	C₂³	—	2	2	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	4	2	no	—
103	C₂³	—	4	2	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	—	4	1	no	—
107	C₂³	—	4	2	no	—
108	C₂³	—	4	1	no	—
109	C₂³	—	4	16	no	—
110	C₂³	C₄×Dic₂₅₁	1	1	yes	characteristic
111	C₂³	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
112	C₂³	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
113	C₂³	Dic₂₅₁⋊C₄	1	2	yes	—
114	C₂³	C₄⋊Dic₂₅₁	1	1	yes	characteristic
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	1004	4	no	—
118	C₂×C₄	—	1004	4	no	—
119	C₂×C₄	—	1004	4	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	4	8	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	4	4	no	—
133	C₂²	—	4	8	no	—
134	C₂²	—	4	4	no	—
135	C₂²	C₂₅₁⋊#95	1	1	yes	characteristic
136	C₂²	#22548629	1	4	yes	—
137	C₂²	#22548630	1	2	yes	—
138	C₄	—	8	2	no	—
139	C₄	—	2008	4	no	—
140	C₂	—	2	2	no	—
141	C₂	—	2	4	no	—
142	C₂	—	4	1	no	—
143	C₂	—	4	2	no	—
144	C₂	—	4	1	no	—
145	C₂	#30631453	1	2	yes	—
146	C₂	#30631454	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30631543

C₂

yes

maximal

#379×C₂₅₁

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₂₅₁

—

C₂×C₂²⋊Dic₂₅₁

—

C₂×C₂²⋊Dic₂₅₁

C₄

yes

complemented

C₂⁵×C₂₅₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₂₅₁

—

C₂×C₂²⋊C₄×C₂₅₁

C₄

yes

complemented

C₂²×Dic₂₅₁

—

C₂²⋊Dic₂₅₁

—

C₂²⋊Dic₂₅₁

—

C₂²⋊Dic₂₅₁

—

C₂⁴×C₂₅₁

—

C₂⁴×C₂₅₁

—

C₂⁴×C₂₅₁

C₂×C₄

yes

—

C₂⁴×C₂₅₁

C₂×C₄

yes

characteristic

C₂⁴×C₂₅₁

D₄

yes

—

C₂⁴×C₂₅₁

D₄

yes

characteristic

C₂⁴×C₂₅₁

Q₈

yes

characteristic

C₂²×C₄×C₂₅₁

—

C₂²⋊C₄×C₂₅₁

—

C₂²⋊C₄×C₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₄²

yes

characteristic

C₂³×C₂₅₁

C₂²⋊C₄

yes

—

C₂³×C₂₅₁

C₂²⋊C₄

yes

characteristic

C₂³×C₂₅₁

C₄⋊C₄

yes

—

C₂³×C₂₅₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#95

yes

characteristic

C₂²×C₂₅₁

#99

yes

—

C₂²×C₂₅₁

#99

yes

—

C₄×C₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#342

yes

—

C₂×C₂₅₁

#342

yes

characteristic

C₂₅₁

#1304

yes

characteristic, complemented, Sylow

#1304

—

251

maximal, Sylow

#379

—

251

—

#379

D₂₅₁

yes

characteristic

C₂⁵

C₂×D₂₅₁

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

502

—

C₂×C₂²⋊C₄

—

502

—

C₂×C₂²⋊C₄

—

251

—

C₂×C₂²⋊C₄

Dic₂₅₁

yes

complemented

C₂⁴

—

C₂⁴

—

C₂⁴

C₄×D₂₅₁

yes

—

C₂⁴

C₂×Dic₂₅₁

yes

characteristic

C₂⁴

C₂²⋊D₂₅₁

yes

—

C₂⁴

D₁₀₀₄

yes

characteristic

C₂⁴

Dic₅₀₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

502

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

1004

—

C₂²⋊C₄

—

1004

—

C₂²⋊C₄

—

1004

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₄×Dic₂₅₁

yes

characteristic

111

C₂³

(C₂×D₂₅₁)⋊C₄

yes

—

112

C₂³

C₂²⋊Dic₂₅₁

yes

characteristic

113

C₂³

Dic₂₅₁⋊C₄

yes

—

114

C₂³

C₄⋊Dic₂₅₁

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

1004

—

118

C₂×C₄

—

1004

—

119

C₂×C₄

—

1004

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₂₅₁⋊#95

yes

characteristic

136

C₂²

#22548629

yes

—

137

C₂²

#22548630

yes

—

138

C₄

—

139

C₄

—

2008

—

140

C₂

—

141

C₂

—

142

C₂

—

143

C₂

—

144

C₂

—

145

C₂

#30631453

yes

—

146

C₂

#30631454

yes

characteristic

	subgroup	G/H
center	C₂²	C₂₅₁⋊#95
commutator	C₂³×C₂₅₁	C₄²
Frattini	C₂⁵	C₂×D₂₅₁
Fitting	#379×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#95
prime radical, cosocle	C₂⁵×C₂₅₁	C₂²
hypercenter	#379	D₂₅₁
hypocenter	C₂₅₁	#1304
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #379

subgroup

G/H

center

C₂²

C₂₅₁⋊#95

commutator

C₂³×C₂₅₁

C₄²

Frattini

C₂⁵

C₂×D₂₅₁

Fitting

#379×C₂₅₁

C₂

socle

C₂²×C₂₅₁

#95

prime radical, cosocle

C₂⁵×C₂₅₁

C₂²

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂⁵ ◁ #379

N	H ≅ G/N	type
5. C₂×C₂²⋊Dic₂₅₁	C₄	semidirect
8. C₂×C₂²⋊C₄×C₂₅₁	C₄	semidirect
73. C₂₅₁	#1304	semidirect
82. C₂×C₂²⋊C₄	Dic₂₅₁	semidirect

H ≅ G/N

type

5. C₂×C₂²⋊Dic₂₅₁

C₄

semidirect

8. C₂×C₂²⋊C₄×C₂₅₁

C₄

semidirect

73. C₂₅₁

#1304

semidirect

82. C₂×C₂²⋊C₄

Dic₂₅₁

semidirect

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

subgroup	index
#30631543	2
#379×C₂₅₁	2
#1304	251

subgroup

index

#30631543

#379×C₂₅₁

#1304

251

Subgroups — C₂₅₁⋊#1304 #44548269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C251⋊#1304 #44548269

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₅₁⋊#1304 #44548269