Group #44551212 — subgroups

Subgroups — #44551212

Order 32128.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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×#22548703	C₂	1	1	yes	maximal, characteristic
2	C₂×#22548633	C₂	1	2	yes	maximal, complemented
3	C₂×#22548631	C₂	1	2	yes	maximal
4	C₄×#16860794	C₂	1	1	yes	maximal, characteristic, complemented
5	#423×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₅₀₂	C₂²	1	1	yes	characteristic
7	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic
8	C₂×#16860794	C₂²	1	2	yes	—
9	#22548703	—	2	2	no	—
10	#22548703	—	2	1	no	—
11	#22548703	—	2	1	no	—
12	C₂×(C₂×D₂₅₁)⋊C₄	C₄	1	4	yes	—
13	#22548633	—	2	4	no	—
14	#22548631	—	2	4	no	—
15	#22548626	—	2	1	no	—
16	C₄×C₂₅₁⋊C₈	—	2	1	no	—
17	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₂₅₁	—	2	2	no	—
20	C₂³×D₂₅₁	C₂×C₄	1	2	yes	—
21	C₂×Dic₅₀₂	—	2	2	no	—
22	C₂×Dic₅₀₂	—	2	1	no	—
23	C₂×Dic₅₀₂	—	2	1	no	—
24	C₂×Dic₅₀₂	—	2	2	no	—
25	C₂×D₁₀₀₄	—	2	2	no	—
26	C₂×D₁₀₀₄	—	2	1	no	—
27	C₂×D₁₀₀₄	—	2	1	no	—
28	C₂×D₁₀₀₄	—	2	2	no	—
29	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
30	#16860794	—	2	2	no	—
31	#16860794	D₄	1	4	yes	—
32	C₂×C₂₅₁⋊C₈	—	2	2	no	—
33	(C₂×D₂₅₁)⋊C₄	—	2	4	no	—
34	(C₂×D₂₅₁)⋊C₄	—	2	4	no	—
35	C₂²×C₄×C₂₅₁	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
37	M₁₆×C₂₅₁	—	2	2	no	—
38	M₁₆×C₂₅₁	—	2	2	no	—
39	C₂×C₈×C₂₅₁	—	2	2	no	—
40	C₄²×C₂₅₁	D₄	1	2	yes	—
41	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
42	C₄²×C₂₅₁	D₄	1	1	yes	characteristic
43	C₂²×D₂₅₁	—	2	2	no	—
44	C₂²×D₂₅₁	—	2	4	no	—
45	C₂²×D₂₅₁	—	4	4	no	—
46	Dic₅₀₂	—	4	2	no	—
47	Dic₅₀₂	—	4	1	no	—
48	Dic₅₀₂	—	4	1	no	—
49	D₁₀₀₄	—	4	2	no	—
50	D₁₀₀₄	—	4	1	no	—
51	D₁₀₀₄	—	4	1	no	—
52	C₂×Dic₂₅₁	—	2	2	no	—
53	C₂×Dic₂₅₁	—	2	4	no	—
54	C₂₅₁⋊C₈	—	2	4	no	—
55	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₅₁	—	4	2	no	—
57	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₂₅₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₅₁	#41	1	2	yes	—
63	C₈×C₂₅₁	—	4	2	no	—
64	C₂²×C₂₅₁	—	2	2	no	—
65	C₂²×C₂₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
67	C₂×D₂₅₁	—	4	2	no	—
68	C₂×D₂₅₁	—	4	4	no	—
69	C₂×D₂₅₁	—	8	4	no	—
70	C₄×C₂₅₁	—	2	2	no	—
71	C₄×C₂₅₁	—	2	2	no	—
72	C₄×C₂₅₁	#121	1	2	yes	—
73	C₄×C₂₅₁	#124	1	1	yes	characteristic
74	C₄×C₂₅₁	#127	1	1	yes	characteristic
75	Dic₂₅₁	—	4	2	no	—
76	C₂×C₂₅₁	—	2	2	no	—
77	C₂×C₂₅₁	#390	1	1	yes	characteristic
78	C₂×C₂₅₁	#413	1	2	yes	—
79	D₂₅₁	—	8	2	no	—
80	C₂₅₁	#1967	1	1	yes	characteristic, complemented, Sylow
81	#1967	—	251	1	no	maximal, Sylow
82	C₂×#124	—	251	1	no	—
83	#423	D₂₅₁	1	1	yes	characteristic, complemented
84	C₂×#101	—	251	2	no	—
85	C₂×#100	—	251	2	no	—
86	C₄×M₁₆	—	251	1	no	—
87	C₂²×Q₈	—	251	1	no	—
88	C₂²×D₄	—	251	1	no	—
89	C₂×M₁₆	—	251	2	no	—
90	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
91	#124	—	502	2	no	—
92	#124	—	502	1	no	—
93	#124	—	502	1	no	—
94	C₂×C₂²⋊C₄	—	251	4	no	—
95	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
96	C₄⋊C₈	—	2	2	no	—
97	#101	—	502	4	no	—
98	#100	—	502	4	no	—
99	#97	—	502	1	no	—
100	C₄×C₈	—	502	1	no	—
101	C₂⁴	—	251	2	no	—
102	C₂×Q₈	—	502	2	no	—
103	C₂×Q₈	—	502	1	no	—
104	C₂×Q₈	—	502	1	no	—
105	C₂×Q₈	—	502	2	no	—
106	C₂×D₄	—	502	2	no	—
107	C₂×D₄	—	502	1	no	—
108	C₂×D₄	—	502	1	no	—
109	C₂×D₄	—	502	2	no	—
110	C₂²×C₄	—	251	2	no	—
111	C₂²×C₄	C₄×D₂₅₁	1	2	yes	—
112	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	502	2	no	—
116	M₁₆	—	251	4	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	502	2	no	—
119	C₂²⋊C₄	—	502	4	no	—
120	C₂²⋊C₄	—	502	4	no	—
121	C₄²	C₂²⋊D₂₅₁	1	2	yes	—
122	C₄²	D₁₀₀₄	1	1	yes	characteristic
123	C₄²	D₁₀₀₄	1	1	yes	characteristic
124	C₂³	—	1004	4	no	—
125	C₂³	—	502	4	no	—
126	C₂³	—	502	2	no	—
127	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
128	Q₈	—	1004	2	no	—
129	Q₈	—	1004	1	no	—
130	Q₈	—	1004	1	no	—
131	D₄	—	1004	2	no	—
132	D₄	—	1004	1	no	—
133	D₄	—	1004	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	502	2	no	—
136	C₂×C₄	—	502	4	no	—
137	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
138	C₂×C₄	(C₂×D₂₅₁)⋊C₄	1	2	yes	—
139	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
141	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
142	C₂×C₄	#16860811	1	2	yes	—
143	C₈	—	4	2	no	—
144	C₈	—	502	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	2008	4	no	—
147	C₂²	—	1004	2	no	—
148	C₂²	—	1004	4	no	—
149	C₂²	C₂×(C₂×D₂₅₁)⋊C₄	1	1	yes	characteristic
150	C₂²	#22548677	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	1004	2	no	—
154	C₄	#22548690	1	2	yes	—
155	C₄	#22548704	1	1	yes	characteristic
156	C₄	#22548711	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	2008	2	no	—
159	C₂	#30631588	1	1	yes	characteristic
160	C₂	#30631669	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22548703

C₂

yes

maximal, characteristic

C₂×#22548633

C₂

yes

maximal, complemented

C₂×#22548631

C₂

yes

maximal

C₄×#16860794

C₂

yes

maximal, characteristic, complemented

#423×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₅₀₂

C₂²

yes

characteristic

C₂²×D₁₀₀₄

C₂²

yes

characteristic

C₂×#16860794

C₂²

yes

—

#22548703

—

#22548703

—

#22548703

—

C₂×(C₂×D₂₅₁)⋊C₄

C₄

yes

—

#22548633

—

#22548631

—

#22548626

—

C₄×C₂₅₁⋊C₈

—

C₂×M₁₆×C₂₅₁

C₂²

yes

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

—

C₂³×D₂₅₁

C₂×C₄

yes

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

C₂²×Dic₂₅₁

C₂×C₄

yes

—

#16860794

—

#16860794

D₄

yes

—

C₂×C₂₅₁⋊C₈

—

(C₂×D₂₅₁)⋊C₄

—

(C₂×D₂₅₁)⋊C₄

—

C₂²×C₄×C₂₅₁

C₂×C₄

yes

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

M₁₆×C₂₅₁

—

M₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄²×C₂₅₁

D₄

yes

—

C₄²×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

D₄

yes

characteristic

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

D₁₀₀₄

—

D₁₀₀₄

—

D₁₀₀₄

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₅₁

C₄×D₄

yes

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#121

yes

—

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#390

yes

characteristic

C₂×C₂₅₁

#413

yes

—

D₂₅₁

—

C₂₅₁

#1967

yes

characteristic, complemented, Sylow

#1967

—

251

maximal, Sylow

C₂×#124

—

251

—

#423

D₂₅₁

yes

characteristic, complemented

C₂×#101

—

251

—

C₂×#100

—

251

—

C₄×M₁₆

—

251

—

C₂²×Q₈

—

251

—

C₂²×D₄

—

251

—

C₂×M₁₆

—

251

—

C₂×M₁₆

C₂×D₂₅₁

yes

—

#124

—

502

—

#124

—

502

—

#124

—

502

—

C₂×C₂²⋊C₄

—

251

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

C₄⋊C₈

—

#101

—

502

—

#100

—

502

—

#97

—

502

—

100

C₄×C₈

—

502

—

101

C₂⁴

—

251

—

102

C₂×Q₈

—

502

—

103

C₂×Q₈

—

502

—

104

C₂×Q₈

—

502

—

105

C₂×Q₈

—

502

—

106

C₂×D₄

—

502

—

107

C₂×D₄

—

502

—

108

C₂×D₄

—

502

—

109

C₂×D₄

—

502

—

110

C₂²×C₄

—

251

—

111

C₂²×C₄

C₄×D₂₅₁

yes

—

112

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

502

—

116

M₁₆

—

251

—

117

C₂×C₈

—

118

C₂×C₈

—

502

—

119

C₂²⋊C₄

—

502

—

120

C₂²⋊C₄

—

502

—

121

C₄²

C₂²⋊D₂₅₁

yes

—

122

C₄²

D₁₀₀₄

yes

characteristic

123

C₄²

D₁₀₀₄

yes

characteristic

124

C₂³

—

1004

—

125

C₂³

—

502

—

126

C₂³

—

502

—

127

C₂³

C₂×C₄×D₂₅₁

yes

characteristic

128

Q₈

—

1004

—

129

Q₈

—

1004

—

130

Q₈

—

1004

—

131

D₄

—

1004

—

132

D₄

—

1004

—

133

D₄

—

1004

—

134

C₂×C₄

—

135

C₂×C₄

—

502

—

136

C₂×C₄

—

502

—

137

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

138

C₂×C₄

(C₂×D₂₅₁)⋊C₄

yes

—

139

C₂×C₄

D₄×D₂₅₁

yes

—

140

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

141

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

142

C₂×C₄

#16860811

yes

—

143

C₈

—

144

C₈

—

502

—

145

C₂²

—

146

C₂²

—

2008

—

147

C₂²

—

1004

—

148

C₂²

—

1004

—

149

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

yes

characteristic

150

C₂²

#22548677

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

1004

—

154

C₄

#22548690

yes

—

155

C₄

#22548704

yes

characteristic

156

C₄

#22548711

yes

characteristic

157

C₂

—

158

C₂

—

2008

—

159

C₂

#30631588

yes

characteristic

160

C₂

#30631669

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₂₅₁)⋊C₄
commutator	C₂³×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	#423×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₀₄	C₂²
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#423	D₂₅₁
hypocenter	C₂₅₁	#1967
LCS	G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₂₅₁)⋊C₄

commutator

C₂³×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

#423×C₂₅₁

C₂

socle

C₂²×C₂₅₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₀₄

C₂²

cosocle

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂³×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#22548633	C₂	semidirect
4. C₄×#16860794	C₂	semidirect
5. #423×C₂₅₁	C₂	semidirect
80. C₂₅₁	#1967	semidirect
83. #423	D₂₅₁	semidirect

H ≅ G/N

type

2. C₂×#22548633

C₂

semidirect

4. C₄×#16860794

C₂

semidirect

5. #423×C₂₅₁

C₂

semidirect

80. C₂₅₁

#1967

semidirect

83. #423

D₂₅₁

semidirect

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

subgroup	index
C₂×#22548703	2
C₂×#22548633	2
C₂×#22548631	2
C₄×#16860794	2
#423×C₂₅₁	2
#1967	251

subgroup

index

#423×C₂₅₁

251

Subgroups — #44551212

Conjugacy classes of subgroups

Complements

Maximal subgroups