Group #44559448 — subgroups

Subgroups — #44559448

Order 32128.

356 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 177 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×C₄×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic
2	#30631817	C₂	1	4	yes	maximal, complemented
3	#30631814	C₂	1	4	yes	maximal, complemented
4	#30631741	C₂	1	4	yes	maximal
5	#30631740	C₂	1	1	yes	maximal, characteristic
6	#431×C₂₅₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₅₀₂	C₄	1	1	yes	characteristic
8	C₄×Dic₅₀₂	C₄	1	2	yes	—
9	C₄×Dic₅₀₂	C₄	1	2	yes	—
10	C₄×Dic₅₀₂	C₂²	1	2	yes	—
11	C₄×Dic₅₀₂	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₂₅₁	C₄	1	1	yes	characteristic
13	C₂×Dic₂₅₁⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
15	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
16	Dic₂₅₁⋊C₈	C₂²	1	8	yes	—
17	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
18	#22548625	C₂²	1	8	yes	—
19	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
22	C₂²⋊C₈×C₂₅₁	C₂²	1	2	yes	—
23	C₄×C₈×C₂₅₁	C₂²	1	2	yes	—
24	C₂×Dic₅₀₂	—	2	4	no	—
25	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
26	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
27	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
28	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
29	C₂×C₂₅₁⋊C₈	C₂³	1	4	yes	—
30	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
32	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
33	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
34	C₄×Dic₂₅₁	C₂×C₄	1	4	yes	—
35	C₄×Dic₂₅₁	C₂³	1	4	yes	—
36	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
40	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
43	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
44	Dic₅₀₂	—	2	4	no	—
45	Dic₅₀₂	—	2	4	no	—
46	C₂×Dic₂₅₁	—	2	4	no	—
47	C₂×Dic₂₅₁	—	4	2	no	—
48	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
49	C₂×Dic₂₅₁	C₂²×C₄	1	4	yes	—
50	C₂₅₁⋊C₈	—	2	4	no	—
51	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₅₁	—	2	2	no	—
53	C₂×C₄×C₂₅₁	—	4	1	no	—
54	C₂×C₄×C₂₅₁	M₁₆	1	4	yes	—
55	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
61	C₈×C₂₅₁	—	2	4	no	—
62	C₂²×C₂₅₁	—	2	1	no	—
63	C₂²×C₂₅₁	—	2	1	no	—
64	C₂²×C₂₅₁	C₂×M₁₆	1	2	yes	—
65	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₂₅₁	—	2	1	no	—
67	C₄×C₂₅₁	—	2	1	no	—
68	C₄×C₂₅₁	—	2	1	no	—
69	C₄×C₂₅₁	C₂×M₁₆	1	2	yes	—
70	C₄×C₂₅₁	C₂×M₁₆	1	2	yes	—
71	C₄×C₂₅₁	2₋¹⁺⁴	1	2	yes	—
72	Dic₂₅₁	—	2	4	no	—
73	Dic₂₅₁	—	2	4	no	—
74	C₂×C₂₅₁	—	2	2	no	—
75	C₂×C₂₅₁	#519	1	1	yes	characteristic
76	C₂×C₂₅₁	C₂²×M₁₆	1	1	yes	characteristic
77	C₂×C₂₅₁	#568	1	1	yes	characteristic
78	C₂₅₁	#2983	1	1	yes	characteristic, complemented, Sylow
79	#2983	—	251	1	no	maximal, Sylow
80	C₂×C₄×Q₈	—	251	1	no	—
81	#446	—	251	4	no	—
82	#446	—	251	4	no	—
83	#432	—	251	4	no	—
84	#431	—	251	1	no	—
85	#431	D₂₅₁	1	1	yes	characteristic
86	C₂²×Q₈	—	251	1	no	—
87	C₄×Q₈	—	251	2	no	—
88	C₄×Q₈	—	251	2	no	—
89	C₄×Q₈	—	251	2	no	—
90	C₄×Q₈	—	251	2	no	—
91	C₂×C₄⋊C₄	—	251	2	no	—
92	C₂×C₄⋊C₄	—	251	1	no	—
93	C₂×C₄²	—	251	2	no	—
94	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
95	C₄⋊C₈	—	251	8	no	—
96	C₄⋊C₈	—	251	2	no	—
97	C₄⋊C₈	C₂×D₂₅₁	1	2	yes	—
98	C₂²⋊C₈	—	251	2	no	—
99	C₂²⋊C₈	C₂×D₂₅₁	1	2	yes	—
100	#97	—	251	8	no	—
101	C₄×C₈	—	251	2	no	—
102	C₄×C₈	C₂×D₂₅₁	1	2	yes	—
103	C₂×Q₈	—	502	4	no	—
104	C₂×Q₈	—	251	2	no	—
105	C₂×Q₈	—	251	2	no	—
106	C₂²×C₄	—	251	2	no	—
107	C₂²×C₄	—	251	2	no	—
108	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
111	C₂×C₈	—	251	4	no	—
112	C₂×C₈	C₂²×D₂₅₁	1	4	yes	—
113	C₄⋊C₄	—	251	4	no	—
114	C₄⋊C₄	—	251	4	no	—
115	C₄⋊C₄	—	251	2	no	—
116	C₄⋊C₄	—	251	2	no	—
117	C₄²	—	251	4	no	—
118	C₄²	—	251	4	no	—
119	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
120	C₄²	C₄×D₂₅₁	1	1	yes	characteristic
121	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
122	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
123	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
124	Q₈	—	502	4	no	—
125	Q₈	—	502	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	1004	2	no	—
129	C₂×C₄	—	502	4	no	—
130	C₂×C₄	—	251	4	no	—
131	C₂×C₄	—	251	4	no	—
132	C₂×C₄	#16860793	1	4	yes	—
133	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	502	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#16860793	1	2	yes	—
144	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	502	4	no	—
149	C₄	—	502	4	no	—
150	C₄	C₂×#16860793	1	2	yes	—
151	C₄	C₂×#16860793	1	2	yes	—
152	C₄	#22548762	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#30632164	1	1	yes	characteristic
155	C₂	C₂²×#16860793	1	1	yes	characteristic
156	C₂	#30632453	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₅₀₂

C₂

yes

maximal, characteristic

#30631817

C₂

yes

maximal, complemented

#30631814

C₂

yes

maximal, complemented

#30631741

C₂

yes

maximal

#30631740

C₂

yes

maximal, characteristic

#431×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₀₂

C₄

yes

characteristic

C₄×Dic₅₀₂

C₄

yes

—

C₄×Dic₅₀₂

C₄

yes

—

C₄×Dic₅₀₂

C₂²

yes

—

C₄×Dic₅₀₂

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

C₄

yes

characteristic

C₂×Dic₂₅₁⋊C₄

C₄

yes

—

C₂×C₄×Dic₂₅₁

C₂²

yes

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

—

Dic₂₅₁⋊C₈

C₂²

yes

—

C₂²⋊(C₂₅₁⋊C₈)

C₂²

yes

—

#22548625

C₂²

yes

—

C₄×C₂₅₁⋊C₈

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

C₂²

yes

—

C₂²⋊C₈×C₂₅₁

C₂²

yes

—

C₄×C₈×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

C₂×C₄

yes

—

C₂×Dic₅₀₂

C₂×C₄

yes

—

C₂²×Dic₂₅₁

C₂×C₄

yes

—

C₂²×Dic₂₅₁

C₂×C₄

yes

—

C₂×C₂₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₂₅₁

C₂×C₄

yes

—

C₄⋊Dic₂₅₁

C₂×C₄

yes

—

Dic₂₅₁⋊C₄

C₂×C₄

yes

—

Dic₂₅₁⋊C₄

C₂×C₄

yes

—

C₄×Dic₂₅₁

C₂×C₄

yes

—

C₄×Dic₂₅₁

C₂³

yes

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₂×C₈×C₂₅₁

C₂³

yes

—

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

characteristic

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

C₂²×C₄

yes

—

C₂×Dic₂₅₁

C₂²×C₄

yes

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

M₁₆

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂⁴

yes

characteristic

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×M₁₆

yes

—

C₂²×C₂₅₁

C₂³×C₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×M₁₆

yes

—

C₄×C₂₅₁

C₂×M₁₆

yes

—

C₄×C₂₅₁

2₋¹⁺⁴

yes

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#519

yes

characteristic

C₂×C₂₅₁

C₂²×M₁₆

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#2983

—

251

maximal, Sylow

C₂×C₄×Q₈

—

251

—

#446

—

251

—

#446

—

251

—

#432

—

251

—

#431

—

251

—

#431

D₂₅₁

yes

characteristic

C₂²×Q₈

—

251

—

C₄×Q₈

—

251

—

C₄×Q₈

—

251

—

C₄×Q₈

—

251

—

C₄×Q₈

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄²

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

251

—

C₄⋊C₈

—

251

—

C₄⋊C₈

C₂×D₂₅₁

yes

—

C₂²⋊C₈

—

251

—

C₂²⋊C₈

C₂×D₂₅₁

yes

—

100

#97

—

251

—

101

C₄×C₈

—

251

—

102

C₄×C₈

C₂×D₂₅₁

yes

—

103

C₂×Q₈

—

502

—

104

C₂×Q₈

—

251

—

105

C₂×Q₈

—

251

—

106

C₂²×C₄

—

251

—

107

C₂²×C₄

—

251

—

108

C₂²×C₄

C₄×D₂₅₁

yes

characteristic

109

C₂²×C₄

C₄×D₂₅₁

yes

characteristic

110

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

111

C₂×C₈

—

251

—

112

C₂×C₈

C₂²×D₂₅₁

yes

—

113

C₄⋊C₄

—

251

—

114

C₄⋊C₄

—

251

—

115

C₄⋊C₄

—

251

—

116

C₄⋊C₄

—

251

—

117

C₄²

—

251

—

118

C₄²

—

251

—

119

C₄²

C₄×D₂₅₁

yes

characteristic

120

C₄²

C₄×D₂₅₁

yes

characteristic

121

C₄²

C₂²×D₂₅₁

yes

characteristic

122

C₄²

C₂²×D₂₅₁

yes

characteristic

123

C₂³

C₂×C₄×D₂₅₁

yes

characteristic

124

Q₈

—

502

—

125

Q₈

—

502

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

1004

—

129

C₂×C₄

—

502

—

130

C₂×C₄

—

251

—

131

C₂×C₄

—

251

—

132

C₂×C₄

#16860793

yes

—

133

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

134

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

135

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

136

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

137

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

138

C₂×C₄

C₂³×D₂₅₁

yes

characteristic

139

C₈

—

140

C₈

—

502

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#16860793

yes

—

144

C₂²

C₂²×C₄×D₂₅₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

502

—

149

C₄

—

502

—

150

C₄

C₂×#16860793

yes

—

151

C₄

C₂×#16860793

yes

—

152

C₄

#22548762

yes

—

153

C₂

—

154

C₂

#30632164

yes

characteristic

155

C₂

C₂²×#16860793

yes

characteristic

156

C₂

#30632453

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting	#431×C₂₅₁	C₂
prime radical	C₂³×C₂₅₁	C₂²×C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#431	D₂₅₁
hypocenter	C₂₅₁	#2983
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₅₁

commutator, socle

C₂²×C₂₅₁

C₂³×C₄

Fitting

#431×C₂₅₁

C₂

prime radical

C₂³×C₂₅₁

C₂²×C₄

cosocle

C₂×C₄×C₂₅₁

C₂⁴

D₂₅₁

C₂₅₁

G ▷ C₂²×C₂₅₁ ▷ C₂₅₁

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #30631817	C₂	semidirect
3. #30631814	C₂	semidirect
78. C₂₅₁	#2983	semidirect

H ≅ G/N

type

2. #30631817

C₂

semidirect

3. #30631814

C₂

semidirect

78. C₂₅₁

#2983

semidirect

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

subgroup	index
C₂×C₄×Dic₅₀₂	2
#30631817	2
#30631814	2
#30631741	2
#30631740	2
#431×C₂₅₁	2
#2983	251

subgroup

index

C₂×C₄×Dic₅₀₂

#431×C₂₅₁

251

Subgroups — #44559448

Conjugacy classes of subgroups

Complements

Maximal subgroups