Group #44550449 — subgroups

Subgroups — #44550449

Order 32128.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₄⋊Dic₂₅₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₂₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#30631449	C₂	1	4	yes	maximal, complemented
4	#380×C₂₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₅₁	—	2	1	no	—
6	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₅₁	—	2	1	no	—
8	C₂×C₄⋊Dic₂₅₁	—	2	2	no	—
9	C₂×C₄⋊Dic₂₅₁	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
11	C₂²⋊(C₂₅₁⋊C₈)	—	2	2	no	—
12	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₄×C₂₅₁	—	2	1	no	—
15	#95×C₂₅₁	—	2	1	no	—
16	C₂²×Dic₂₅₁	—	2	4	no	—
17	C₂²×Dic₂₅₁	—	2	2	no	—
18	C₂²×Dic₂₅₁	—	4	2	no	—
19	C₂²×Dic₂₅₁	—	4	2	no	—
20	C₂×C₂₅₁⋊C₈	—	2	2	no	—
21	C₂×C₂₅₁⋊C₈	—	2	4	no	—
22	C₄⋊Dic₂₅₁	—	2	4	no	—
23	C₄⋊Dic₂₅₁	—	2	2	no	—
24	C₄⋊Dic₂₅₁	C₂×C₄	1	4	yes	—
25	C₂⁴×C₂₅₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₅₁	—	2	4	no	—
27	C₂²×C₄×C₂₅₁	—	2	2	no	—
28	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₅₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₅₁	D₄	1	2	yes	—
31	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₂₅₁	—	2	4	no	—
33	C₂²⋊C₄×C₂₅₁	—	4	2	no	—
34	C₂×Dic₂₅₁	—	2	4	no	—
35	C₂×Dic₂₅₁	—	4	8	no	—
36	C₂×Dic₂₅₁	—	4	2	no	—
37	C₂×Dic₂₅₁	—	4	2	no	—
38	C₂₅₁⋊C₈	—	4	2	no	—
39	C₂³×C₂₅₁	—	2	2	no	—
40	C₂³×C₂₅₁	—	2	2	no	—
41	C₂³×C₂₅₁	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₂₅₁	D₈	1	2	yes	—
43	C₂³×C₂₅₁	SD₁₆	1	2	yes	—
44	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₅₁	—	2	2	no	—
46	C₂×C₄×C₂₅₁	—	2	4	no	—
47	C₂×C₄×C₂₅₁	—	2	4	no	—
48	C₂×C₄×C₂₅₁	—	4	2	no	—
49	C₂×C₄×C₂₅₁	—	4	2	no	—
50	C₂×C₄×C₂₅₁	—	4	2	no	—
51	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₁	#41	1	4	yes	—
55	C₂²×C₂₅₁	—	2	2	no	—
56	C₂²×C₂₅₁	—	2	2	no	—
57	C₂²×C₂₅₁	—	2	4	no	—
58	C₂²×C₂₅₁	—	2	4	no	—
59	C₂²×C₂₅₁	#102	1	4	yes	—
60	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₅₁	#123	1	2	yes	—
62	C₂²×C₂₅₁	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₂₅₁	#137	1	2	yes	—
65	C₄×C₂₅₁	—	2	2	no	—
66	C₄×C₂₅₁	—	8	2	no	—
67	C₄×C₂₅₁	#117	1	2	yes	—
68	C₄×C₂₅₁	#123	1	2	yes	—
69	Dic₂₅₁	—	4	4	no	—
70	C₂×C₂₅₁	—	2	4	no	—
71	C₂×C₂₅₁	#381	1	1	yes	characteristic
72	C₂×C₂₅₁	C₂×#102	1	1	yes	characteristic
73	C₂×C₂₅₁	#419	1	1	yes	characteristic
74	C₂×C₂₅₁	#480	1	2	yes	—
75	C₂×C₂₅₁	#483	1	2	yes	—
76	C₂₅₁	#1823	1	1	yes	characteristic, complemented, Sylow
77	#1823	—	251	1	no	maximal, Sylow
78	C₂²×C₄⋊C₄	—	251	1	no	—
79	C₂×C₂²⋊C₈	—	251	1	no	—
80	#380	D₂₅₁	1	1	yes	characteristic
81	#340	—	251	4	no	—
82	C₂³×C₄	—	502	1	no	—
83	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
84	C₂²×C₈	—	251	2	no	—
85	C₂×C₄⋊C₄	—	502	1	no	—
86	C₂×C₄⋊C₄	—	502	2	no	—
87	C₂×C₄⋊C₄	—	251	4	no	—
88	C₂×C₄⋊C₄	—	251	2	no	—
89	C₂×C₄⋊C₄	C₂×D₂₅₁	1	2	yes	—
90	C₂×C₂²⋊C₄	—	2	1	no	—
91	C₂²⋊C₈	—	502	2	no	—
92	#95	—	2	1	no	—
93	C₂⁴	C₂²⋊D₂₅₁	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	1004	2	no	—
97	C₂²×C₄	—	1004	2	no	—
98	C₂²×C₄	—	502	4	no	—
99	C₂²×C₄	—	502	2	no	—
100	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
101	C₂²×C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
102	C₂²×C₄	D₁₀₀₄	1	2	yes	—
103	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
104	C₂×C₈	—	502	2	no	—
105	C₂×C₈	—	502	4	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	502	4	no	—
108	C₄⋊C₄	—	502	2	no	—
109	C₄⋊C₄	—	251	4	no	—
110	C₂²⋊C₄	—	4	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
114	C₂³	#16860795	1	2	yes	—
115	C₂³	#16860797	1	2	yes	—
116	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	1004	8	no	—
124	C₂×C₄	—	1004	2	no	—
125	C₂×C₄	—	1004	2	no	—
126	C₂×C₄	—	502	4	no	—
127	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
128	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
129	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
130	C₂×C₄	#16860809	1	4	yes	—
131	C₈	—	1004	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	4	no	—
134	C₂²	—	2	4	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#22548635	1	4	yes	—
137	C₂²	C₂×(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
138	C₂²	#22548696	1	2	yes	—
139	C₂²	C₂×#16860795	1	1	yes	characteristic
140	C₂²	C₂×#16860797	1	1	yes	characteristic
141	C₂²	#22548745	1	2	yes	—
142	C₄	—	2	2	no	—
143	C₄	—	8	2	no	—
144	C₄	—	1004	4	no	—
145	C₄	#22548671	1	2	yes	—
146	C₄	#22548699	1	2	yes	—
147	C₂	—	2	4	no	—
148	C₂	#30631548	1	1	yes	characteristic
149	C₂	C₂×#22548635	1	1	yes	characteristic
150	C₂	#30631695	1	1	yes	characteristic
151	C₂	#30631998	1	2	yes	—
152	C₂	#30632015	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₂₅₁

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₂₅₁⋊C₈)

C₂

yes

maximal, characteristic

#30631449

C₂

yes

maximal, complemented

#380×C₂₅₁

C₂

yes

maximal, characteristic

C₂³×Dic₂₅₁

—

C₂²×C₂₅₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₅₁

—

C₂×C₄⋊Dic₂₅₁

—

C₂×C₄⋊Dic₂₅₁

C₄

yes

complemented

C₂×C₄⋊Dic₂₅₁

C₂²

yes

—

C₂²⋊(C₂₅₁⋊C₈)

—

C₂³×C₄×C₂₅₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₅₁

—

#95×C₂₅₁

—

C₂²×Dic₂₅₁

—

C₂²×Dic₂₅₁

—

C₂²×Dic₂₅₁

—

C₂²×Dic₂₅₁

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

C₂×C₄

yes

—

C₂⁴×C₂₅₁

D₄

yes

characteristic

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

D₄

yes

characteristic

C₂²×C₄×C₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₂²⋊C₄×C₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₂²⋊C₄

yes

—

C₂³×C₂₅₁

D₈

yes

—

C₂³×C₂₅₁

SD₁₆

yes

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#102

yes

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

#123

yes

—

C₂²×C₂₅₁

C₂×D₈

yes

characteristic

C₂²×C₂₅₁

C₂×SD₁₆

yes

characteristic

C₂²×C₂₅₁

#137

yes

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#117

yes

—

C₄×C₂₅₁

#123

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

—

C₂×C₂₅₁

#483

yes

—

C₂₅₁

#1823

yes

characteristic, complemented, Sylow

#1823

—

251

maximal, Sylow

C₂²×C₄⋊C₄

—

251

—

C₂×C₂²⋊C₈

—

251

—

#380

D₂₅₁

yes

characteristic

#340

—

251

—

C₂³×C₄

—

502

—

C₂³×C₄

C₂×D₂₅₁

yes

characteristic

C₂²×C₈

—

251

—

C₂×C₄⋊C₄

—

502

—

C₂×C₄⋊C₄

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

502

—

#95

—

C₂⁴

C₂²⋊D₂₅₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

1004

—

C₂²×C₄

—

1004

—

C₂²×C₄

—

502

—

C₂²×C₄

—

502

—

100

C₂²×C₄

C₄×D₂₅₁

yes

—

101

C₂²×C₄

C₂²⋊D₂₅₁

yes

characteristic

102

C₂²×C₄

D₁₀₀₄

yes

—

103

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

104

C₂×C₈

—

502

—

105

C₂×C₈

—

502

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

502

—

108

C₄⋊C₄

—

502

—

109

C₄⋊C₄

—

251

—

110

C₂²⋊C₄

—

111

C₂³

—

112

C₂³

—

113

C₂³

(C₂×D₂₅₁)⋊C₄

yes

—

114

C₂³

#16860795

yes

—

115

C₂³

#16860797

yes

—

116

C₂³

C₂×C₂²⋊D₂₅₁

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

1004

—

124

C₂×C₄

—

1004

—

125

C₂×C₄

—

1004

—

126

C₂×C₄

—

502

—

127

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

128

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

129

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

130

C₂×C₄

#16860809

yes

—

131

C₈

—

1004

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#22548635

yes

—

137

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

yes

characteristic

138

C₂²

#22548696

yes

—

139

C₂²

C₂×#16860795

yes

characteristic

140

C₂²

C₂×#16860797

yes

characteristic

141

C₂²

#22548745

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

1004

—

145

C₄

#22548671

yes

—

146

C₄

#22548699

yes

—

147

C₂

—

148

C₂

#30631548

yes

characteristic

149

C₂

C₂×#22548635

yes

characteristic

150

C₂

#30631695

yes

characteristic

151

C₂

#30631998

yes

—

152

C₂

#30632015

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₅₁
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	#380×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂×D₄
prime radical	C₂⁴×C₂₅₁	D₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#380	D₂₅₁
hypocenter	C₂₅₁	#1823
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₅₁

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

#380×C₂₅₁

C₂

socle

C₂³×C₂₅₁

C₂×D₄

prime radical

C₂⁴×C₂₅₁

D₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
3. #30631449	C₂	semidirect
9. C₂×C₄⋊Dic₂₅₁	C₄	semidirect
76. C₂₅₁	#1823	semidirect

H ≅ G/N

type

3. #30631449

C₂

semidirect

9. C₂×C₄⋊Dic₂₅₁

C₄

semidirect

76. C₂₅₁

#1823

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₂₅₁	2
C₂×C₂²⋊(C₂₅₁⋊C₈)	2
#30631449	2
#380×C₂₅₁	2
#1823	251

subgroup

index

C₂²×C₄⋊Dic₂₅₁

C₂×C₂²⋊(C₂₅₁⋊C₈)

#30631449

#380×C₂₅₁

#1823

251

Subgroups — #44550449

Conjugacy classes of subgroups

Complements

Maximal subgroups