Group #44550360 — subgroups

Subgroups — #44550360

Order 32128.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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×#22548680	C₂	1	1	yes	maximal, characteristic
2	#30631710	C₂	1	2	yes	maximal
3	C₂×#22548633	C₂	1	2	yes	maximal
4	#30631456	C₂	1	1	yes	maximal, characteristic
5	#343×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
7	C₂×#16860794	C₂²	1	2	yes	—
8	#22548680	—	2	2	no	—
9	#22548680	—	2	2	no	—
10	C₂×Dic₂₅₁⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₅₁	C₄	1	2	yes	—
12	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
13	Dic₂₅₁⋊C₈	—	2	2	no	—
14	Dic₂₅₁⋊C₈	—	2	2	no	—
15	#22548633	—	2	4	no	—
16	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₅₀₂	—	2	2	no	—
19	C₂×Dic₅₀₂	—	2	2	no	—
20	C₂×Dic₅₀₂	D₄	1	2	yes	—
21	C₂×Dic₅₀₂	Q₈	1	2	yes	—
22	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
24	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
25	#16860794	—	2	2	no	—
26	#16860794	—	2	2	no	—
27	C₂×C₂₅₁⋊C₈	—	2	2	no	—
28	Dic₂₅₁⋊C₄	—	2	4	no	—
29	C₄×Dic₂₅₁	—	2	2	no	—
30	C₄×Dic₂₅₁	—	2	2	no	—
31	C₄×Dic₂₅₁	D₄	1	2	yes	—
32	C₄×Dic₂₅₁	D₄	1	2	yes	—
33	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
35	M₁₆×C₂₅₁	—	2	2	no	—
36	M₁₆×C₂₅₁	—	2	2	no	—
37	C₂×C₈×C₂₅₁	—	2	2	no	—
38	C₄⋊C₄×C₂₅₁	—	2	2	no	—
39	Dic₅₀₂	—	2	4	no	—
40	Dic₅₀₂	—	2	4	no	—
41	C₂×Dic₂₅₁	—	2	2	no	—
42	C₂×Dic₂₅₁	—	2	2	no	—
43	C₂×Dic₂₅₁	—	2	2	no	—
44	C₂×Dic₂₅₁	—	4	1	no	—
45	C₂×Dic₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×Dic₂₅₁	C₂²⋊C₄	1	2	yes	—
47	C₂×Dic₂₅₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₂₅₁	C₄⋊C₄	1	4	yes	—
49	C₂₅₁⋊C₈	—	4	2	no	—
50	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₅₁	—	2	2	no	—
52	C₂×C₄×C₂₅₁	—	4	2	no	—
53	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₅₁	#41	1	2	yes	—
57	C₈×C₂₅₁	—	4	2	no	—
58	C₂²×C₂₅₁	—	2	1	no	—
59	C₂²×C₂₅₁	—	2	1	no	—
60	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₂₅₁	#117	1	1	yes	characteristic
63	C₄×C₂₅₁	—	4	2	no	—
64	C₄×C₂₅₁	#121	1	2	yes	—
65	C₄×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
66	Dic₂₅₁	—	2	4	no	—
67	Dic₂₅₁	—	2	1	no	—
68	Dic₂₅₁	—	2	2	no	—
69	Dic₂₅₁	#101	1	2	yes	complemented
70	C₂×C₂₅₁	—	2	1	no	—
71	C₂×C₂₅₁	—	2	1	no	—
72	C₂×C₂₅₁	#380	1	1	yes	characteristic
73	C₂×C₂₅₁	C₂×#101	1	1	yes	characteristic
74	C₂×C₂₅₁	#413	1	1	yes	characteristic
75	C₂₅₁	#1801	1	1	yes	characteristic, complemented, Sylow
76	#1801	—	251	1	no	maximal, Sylow
77	C₂×C₄×Q₈	—	251	1	no	—
78	#423	—	251	2	no	—
79	C₂×#101	—	251	2	no	—
80	#343	—	251	1	no	—
81	#343	D₂₅₁	1	1	yes	characteristic
82	C₂²×Q₈	—	251	1	no	—
83	C₂×M₁₆	—	251	2	no	—
84	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
85	C₄×Q₈	—	502	2	no	—
86	C₄×Q₈	—	502	2	no	—
87	C₂×C₄⋊C₄	—	251	2	no	—
88	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
89	C₂×C₄²	—	251	2	no	—
90	C₂×C₄²	—	251	1	no	—
91	C₄⋊C₈	—	502	2	no	—
92	C₄⋊C₈	—	502	2	no	—
93	#101	—	502	4	no	—
94	C₂×Q₈	—	502	2	no	—
95	C₂×Q₈	—	502	2	no	—
96	C₂×Q₈	—	251	2	no	—
97	C₂×Q₈	—	251	2	no	—
98	C₂²×C₄	—	251	1	no	—
99	C₂²×C₄	—	251	1	no	—
100	C₂²×C₄	—	251	2	no	—
101	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
102	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	502	2	no	—
106	M₁₆	—	502	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	502	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	502	4	no	—
111	C₄²	—	502	2	no	—
112	C₄²	—	502	2	no	—
113	C₄²	—	251	2	no	—
114	C₄²	—	251	2	no	—
115	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
116	Q₈	—	502	4	no	—
117	Q₈	—	502	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	1004	1	no	—
121	C₂×C₄	—	502	2	no	—
122	C₂×C₄	—	502	2	no	—
123	C₂×C₄	—	502	2	no	—
124	C₂×C₄	—	251	1	no	—
125	C₂×C₄	—	251	2	no	—
126	C₂×C₄	—	251	4	no	—
127	C₂×C₄	—	251	1	no	—
128	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
129	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
130	C₂×C₄	Q₈×D₂₅₁	1	1	yes	characteristic
131	C₂×C₄	#16860810	1	2	yes	—
132	C₈	—	4	2	no	—
133	C₈	—	1004	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	C₂²⋊C₄×D₂₅₁	1	1	yes	characteristic
137	C₂²	C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
138	C₂²	#22548672	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	502	4	no	—
141	C₄	—	502	1	no	—
142	C₄	—	502	2	no	—
143	C₄	—	251	2	no	—
144	C₄	#22548688	1	2	yes	—
145	C₄	#22548693	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#30631546	1	1	yes	characteristic
149	C₂	#101×D₂₅₁	1	1	yes	characteristic
150	C₂	#30631668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548680

C₂

yes

maximal, characteristic

#30631710

C₂

yes

maximal

C₂×#22548633

C₂

yes

maximal

#30631456

C₂

yes

maximal, characteristic

#343×C₂₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₀₂

C₂²

yes

characteristic

C₂×#16860794

C₂²

yes

—

#22548680

—

#22548680

—

C₂×Dic₂₅₁⋊C₄

C₄

yes

—

C₂×C₄×Dic₂₅₁

C₄

yes

—

C₂×C₄×Dic₂₅₁

C₂²

yes

characteristic

Dic₂₅₁⋊C₈

—

Dic₂₅₁⋊C₈

—

#22548633

—

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

D₄

yes

—

C₂×Dic₅₀₂

Q₈

yes

—

C₂²×Dic₂₅₁

C₂×C₄

yes

characteristic

C₂²×Dic₂₅₁

C₂×C₄

yes

—

C₂²×Dic₂₅₁

C₂×C₄

yes

characteristic

#16860794

—

#16860794

—

C₂×C₂₅₁⋊C₈

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

D₄

yes

—

C₄×Dic₂₅₁

D₄

yes

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₅₁

C₂²⋊C₄

yes

—

C₂×Dic₂₅₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₅₁

C₄⋊C₄

yes

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₅₁

#117

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

#121

yes

—

C₄×C₂₅₁

C₂²⋊Q₈

yes

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

#101

yes

complemented

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#1801

—

251

maximal, Sylow

C₂×C₄×Q₈

—

251

—

#423

—

251

—

C₂×#101

—

251

—

#343

—

251

—

#343

D₂₅₁

yes

characteristic

C₂²×Q₈

—

251

—

C₂×M₁₆

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

C₄×Q₈

—

502

—

C₄×Q₈

—

502

—

C₂×C₄⋊C₄

—

251

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

characteristic

C₂×C₄²

—

251

—

C₂×C₄²

—

251

—

C₄⋊C₈

—

502

—

C₄⋊C₈

—

502

—

#101

—

502

—

C₂×Q₈

—

502

—

C₂×Q₈

—

502

—

C₂×Q₈

—

251

—

C₂×Q₈

—

251

—

C₂²×C₄

—

251

—

C₂²×C₄

—

251

—

100

C₂²×C₄

—

251

—

101

C₂²×C₄

C₄×D₂₅₁

yes

—

102

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

502

—

106

M₁₆

—

502

—

107

C₂×C₈

—

108

C₂×C₈

—

502

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

502

—

111

C₄²

—

502

—

112

C₄²

—

502

—

113

C₄²

—

251

—

114

C₄²

—

251

—

115

C₂³

C₂×C₄×D₂₅₁

yes

characteristic

116

Q₈

—

502

—

117

Q₈

—

502

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

1004

—

121

C₂×C₄

—

502

—

122

C₂×C₄

—

502

—

123

C₂×C₄

—

502

—

124

C₂×C₄

—

251

—

125

C₂×C₄

—

251

—

126

C₂×C₄

—

251

—

127

C₂×C₄

—

251

—

128

C₂×C₄

D₄×D₂₅₁

yes

characteristic

129

C₂×C₄

D₄×D₂₅₁

yes

—

130

C₂×C₄

Q₈×D₂₅₁

yes

characteristic

131

C₂×C₄

#16860810

yes

—

132

C₈

—

133

C₈

—

1004

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂²⋊C₄×D₂₅₁

yes

characteristic

137

C₂²

C₄⋊C₄×D₂₅₁

yes

characteristic

138

C₂²

#22548672

yes

characteristic

139

C₄

—

140

C₄

—

502

—

141

C₄

—

502

—

142

C₄

—

502

—

143

C₄

—

251

—

144

C₄

#22548688

yes

—

145

C₄

#22548693

yes

—

146

C₂

—

147

C₂

—

148

C₂

#30631546

yes

characteristic

149

C₂

#101×D₂₅₁

yes

characteristic

150

C₂

#30631668

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₂₅₁
commutator, prime radical	C₂³×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	#343×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#343	D₂₅₁
hypocenter	C₂₅₁	#1801
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₂₅₁

commutator, prime radical

C₂³×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

#343×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
69. Dic₂₅₁	#101	semidirect
75. C₂₅₁	#1801	semidirect

H ≅ G/N

type

69. Dic₂₅₁

#101

semidirect

75. C₂₅₁

#1801

semidirect

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

subgroup	index
C₂×#22548680	2
#30631710	2
C₂×#22548633	2
#30631456	2
#343×C₂₅₁	2
#1801	251

subgroup

index

#343×C₂₅₁

251

Subgroups — #44550360

Conjugacy classes of subgroups

Complements

Maximal subgroups