Group #44548834 — subgroups

Subgroups — #44548834

Order 32128.

246 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 81 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	#30631744	C₂	1	1	yes	maximal, characteristic, complemented
2	#30631740	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₂₅₁⋊C₈)	C₂	1	4	yes	maximal, complemented
4	#532×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₂₅₁⋊C₈)	—	2	1	no	—
6	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
7	C₂²⋊(C₂₅₁⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₂₅₁⋊C₈)	—	2	1	no	—
9	#22548626	—	2	1	no	—
10	C₄×C₂₅₁⋊C₈	C₂²	1	2	yes	—
11	C₂×#41×C₂₅₁	—	2	1	no	—
12	C₄⋊Q₈×C₂₅₁	C₄	1	1	yes	characteristic
13	#127×C₂₅₁	C₄	1	1	yes	characteristic
14	#124×C₂₅₁	—	2	1	no	—
15	#121×C₂₅₁	—	2	1	no	—
16	#121×C₂₅₁	C₄	1	2	yes	—
17	C₄×D₄×C₂₅₁	—	2	1	no	—
18	C₄×D₄×C₂₅₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
20	C₂×C₂₅₁⋊C₈	—	2	2	no	—
21	C₂×C₂₅₁⋊C₈	—	2	2	no	—
22	#41×C₂₅₁	—	4	4	no	—
23	C₂×Q₈×C₂₅₁	—	2	1	no	—
24	C₂×D₄×C₂₅₁	—	2	2	no	—
25	C₂×D₄×C₂₅₁	—	2	1	no	—
26	C₂×D₄×C₂₅₁	—	2	1	no	—
27	C₂×D₄×C₂₅₁	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₅₁	—	2	2	no	—
29	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₂₅₁	—	2	1	no	—
33	C₄⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
34	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
35	C₂²⋊C₄×C₂₅₁	—	2	2	no	—
36	C₄²×C₂₅₁	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
38	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
39	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
40	C₂₅₁⋊C₈	—	4	2	no	—
41	C₂₅₁⋊C₈	—	4	2	no	—
42	C₂³×C₂₅₁	—	2	2	no	—
43	C₂³×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
44	Q₈×C₂₅₁	—	4	2	no	—
45	D₄×C₂₅₁	—	2	4	no	—
46	D₄×C₂₅₁	—	4	2	no	—
47	D₄×C₂₅₁	—	4	2	no	—
48	D₄×C₂₅₁	—	4	4	no	—
49	C₂×C₄×C₂₅₁	—	2	2	no	—
50	C₂×C₄×C₂₅₁	—	2	2	no	—
51	C₂×C₄×C₂₅₁	—	2	2	no	—
52	C₂×C₄×C₂₅₁	—	4	1	no	—
53	C₂×C₄×C₂₅₁	—	4	4	no	—
54	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₂₅₁	—	2	1	no	—
61	C₂²×C₂₅₁	—	2	1	no	—
62	C₂²×C₂₅₁	—	2	1	no	—
63	C₂²×C₂₅₁	—	4	2	no	—
64	C₂²×C₂₅₁	—	4	2	no	—
65	C₂²×C₂₅₁	—	4	2	no	—
66	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₂₅₁	—	2	1	no	—
68	C₄×C₂₅₁	—	2	1	no	—
69	C₄×C₂₅₁	—	2	1	no	—
70	C₄×C₂₅₁	—	2	1	no	—
71	C₄×C₂₅₁	—	4	2	no	—
72	C₄×C₂₅₁	#131	1	2	yes	—
73	C₄×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₂₅₁	—	4	1	no	—
75	C₂×C₂₅₁	—	8	2	no	—
76	C₂×C₂₅₁	#408	1	1	yes	characteristic
77	C₂×C₂₅₁	#418	1	1	yes	characteristic
78	C₂×C₂₅₁	#421	1	1	yes	characteristic
79	C₂₅₁	#1496	1	1	yes	characteristic, complemented, Sylow
80	#1496	—	251	1	no	maximal, Sylow
81	#532	D₂₅₁	1	1	yes	characteristic
82	#432	—	251	1	no	—
83	#431	—	251	1	no	—
84	D₄⋊C₈	—	251	4	no	—
85	C₂×#41	—	2	1	no	—
86	C₄⋊Q₈	Dic₂₅₁	1	1	yes	characteristic
87	#127	Dic₂₅₁	1	1	yes	characteristic
88	#124	—	2	1	no	—
89	#121	—	2	1	no	—
90	#121	Dic₂₅₁	1	2	yes	—
91	C₄×D₄	—	2	1	no	—
92	C₄×D₄	C₂×D₂₅₁	1	2	yes	—
93	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
94	C₄⋊C₈	—	502	1	no	—
95	C₄⋊C₈	—	251	2	no	—
96	C₂²⋊C₈	—	502	1	no	—
97	C₂²⋊C₈	—	502	1	no	—
98	#97	—	502	1	no	—
99	C₄×C₈	—	251	2	no	—
100	#41	—	4	4	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	C₂×Dic₂₅₁	1	2	yes	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	C₂×Dic₂₅₁	1	1	yes	characteristic
108	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
110	C₂×C₈	—	502	2	no	—
111	C₂×C₈	—	502	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	C₂×Dic₂₅₁	1	2	yes	—
114	C₂²⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	2	2	no	—
116	C₄²	C₂×Dic₂₅₁	1	1	yes	characteristic
117	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
118	C₄²	C₂²⋊D₂₅₁	1	1	yes	characteristic
119	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
120	C₂³	—	2	2	no	—
121	C₂³	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
122	Q₈	—	4	2	no	—
123	D₄	—	2	4	no	—
124	D₄	—	4	2	no	—
125	D₄	—	4	2	no	—
126	D₄	—	4	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
133	C₂×C₄	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊Dic₂₅₁	1	1	yes	characteristic
135	C₂×C₄	C₂²×Dic₂₅₁	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
138	C₈	—	1004	2	no	—
139	C₈	—	1004	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	2	no	—
146	C₂²	C₂×C₂²⋊Dic₂₅₁	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	2	no	—
152	C₄	#22548722	1	2	yes	—
153	C₄	#22548741	1	2	yes	—
154	C₂	—	4	1	no	—
155	C₂	—	8	2	no	—
156	C₂	#30631654	1	1	yes	characteristic
157	C₂	#30631692	1	1	yes	characteristic
158	C₂	#30631704	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30631744

C₂

yes

maximal, characteristic, complemented

#30631740

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₂₅₁⋊C₈)

C₂

yes

maximal, complemented

#532×C₂₅₁

C₂

yes

maximal, characteristic

C₄⋊(C₂₅₁⋊C₈)

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₂₅₁⋊C₈)

—

C₂²⋊(C₂₅₁⋊C₈)

—

#22548626

—

C₄×C₂₅₁⋊C₈

C₂²

yes

—

C₂×#41×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₄

yes

characteristic

#127×C₂₅₁

C₄

yes

characteristic

#124×C₂₅₁

—

#121×C₂₅₁

—

#121×C₂₅₁

C₄

yes

—

C₄×D₄×C₂₅₁

—

C₄×D₄×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

#41×C₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₅₁

D₄

yes

characteristic

C₂²×C₄×C₂₅₁

D₄

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

C₂×C₄

yes

—

C₂²⋊C₄×C₂₅₁

—

C₂²⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂×C₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₂²⋊C₄

yes

characteristic

Q₈×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#131

yes

—

C₄×C₂₅₁

AGL(1,ℤ/8)

yes

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#1496

—

251

maximal, Sylow

#532

D₂₅₁

yes

characteristic

#432

—

251

—

#431

—

251

—

D₄⋊C₈

—

251

—

C₂×#41

—

C₄⋊Q₈

Dic₂₅₁

yes

characteristic

#127

Dic₂₅₁

yes

—

—

Dic₂₅₁

yes

—

C₄×D₄

—

C₄×D₄

C₂×D₂₅₁

yes

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

502

—

C₄⋊C₈

—

251

—

C₂²⋊C₈

—

502

—

C₂²⋊C₈

—

502

—

#97

—

502

—

C₄×C₈

—

251

—

100

#41

—

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₂×Dic₂₅₁

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

C₂×Dic₂₅₁

yes

characteristic

108

C₂²×C₄

C₂²⋊D₂₅₁

yes

characteristic

109

C₂²×C₄

C₂²⋊D₂₅₁

yes

characteristic

110

C₂×C₈

—

502

—

111

C₂×C₈

—

502

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₂×Dic₂₅₁

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

C₂×Dic₂₅₁

yes

characteristic

117

C₄²

C₂²⋊D₂₅₁

yes

characteristic

118

C₄²

C₂²⋊D₂₅₁

yes

characteristic

119

C₄²

C₂²×D₂₅₁

yes

characteristic

120

C₂³

—

121

C₂³

C₂²⋊Dic₂₅₁

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₂²⋊Dic₂₅₁

yes

characteristic

133

C₂×C₄

C₂²⋊Dic₂₅₁

yes

characteristic

134

C₂×C₄

C₂²⋊Dic₂₅₁

yes

characteristic

135

C₂×C₄

C₂²×Dic₂₅₁

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

138

C₈

—

1004

—

139

C₈

—

1004

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊Dic₂₅₁

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#22548722

yes

—

153

C₄

#22548741

yes

—

154

C₂

—

155

C₂

—

156

C₂

#30631654

yes

characteristic

157

C₂

#30631692

yes

characteristic

158

C₂

#30631704

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₁
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₄²	C₂²×D₂₅₁
Fitting, prime radical	#532×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
cosocle	C₄²×C₂₅₁	C₂³
hypercenter	#532	D₂₅₁
hypocenter	C₂₅₁	#1496
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₅₁

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₄²

C₂²×D₂₅₁

Fitting, prime radical

#532×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₂²⋊C₄

cosocle

C₄²×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #30631744	C₂	semidirect
2. #30631740	C₂	semidirect
3. D₄⋊(C₂₅₁⋊C₈)	C₂	semidirect
79. C₂₅₁	#1496	semidirect

H ≅ G/N

type

1. #30631744

C₂

semidirect

2. #30631740

C₂

semidirect

3. D₄⋊(C₂₅₁⋊C₈)

C₂

semidirect

79. C₂₅₁

#1496

semidirect

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

subgroup	index
#30631744	2
#30631740	2
D₄⋊(C₂₅₁⋊C₈)	2
#532×C₂₅₁	2
#1496	251

subgroup

index

#30631744

#30631740

D₄⋊(C₂₅₁⋊C₈)

#532×C₂₅₁

#1496

251

Subgroups — #44548834

Conjugacy classes of subgroups

Complements

Maximal subgroups