Group #44550574 — subgroups

Subgroups — #44550574

Order 32128.

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

normal subgroups only

This group has 324 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	#30632162	C₂	1	1	yes	maximal, characteristic
2	#30631727	C₂	1	2	yes	maximal, complemented
3	C₂×#22548632	C₂	1	1	yes	maximal, characteristic
4	#30631447	C₂	1	2	yes	maximal, complemented
5	#390×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16860794	C₂²	1	2	yes	—
7	D₄×Dic₂₅₁	—	2	2	no	—
8	D₄×Dic₂₅₁	—	2	2	no	—
9	#22548674	C₂²	1	2	yes	—
10	C₂×C₂²⋊Dic₂₅₁	—	2	2	no	—
11	#22548648	—	2	2	no	—
12	#22548646	—	2	2	no	—
13	#22548632	C₄	1	4	yes	complemented
14	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₂₅₁	—	2	2	no	—
17	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
18	C₂²×Dic₂₅₁	—	2	2	no	—
19	C₂²×Dic₂₅₁	—	2	2	no	—
20	#16860794	—	2	2	no	—
21	#16860794	C₂×C₄	1	4	yes	complemented
22	C₂×C₂₅₁⋊C₈	—	2	2	no	—
23	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
24	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
25	C₂²⋊Dic₂₅₁	—	2	4	no	—
26	C₂²⋊Dic₂₅₁	—	2	2	no	—
27	C₄×Dic₂₅₁	—	2	2	no	—
28	C₂⁴×C₂₅₁	D₄	1	2	yes	—
29	C₂×D₄×C₂₅₁	—	2	2	no	—
30	C₂×D₄×C₂₅₁	—	2	2	no	—
31	C₂×D₄×C₂₅₁	C₂×C₄	1	2	yes	—
32	C₂×D₄×C₂₅₁	Q₈	1	2	yes	—
33	C₂²×C₄×C₂₅₁	—	2	1	no	—
34	C₂²×C₄×C₂₅₁	—	2	1	no	—
35	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₅₁	—	2	2	no	—
37	C₂²⋊C₄×C₂₅₁	—	4	4	no	—
38	C₄²×C₂₅₁	—	2	2	no	—
39	C₂×Dic₂₅₁	—	2	2	no	—
40	C₂×Dic₂₅₁	—	2	2	no	—
41	C₂×Dic₂₅₁	—	8	2	no	—
42	C₂×Dic₂₅₁	—	8	2	no	—
43	C₂₅₁⋊C₈	—	2	4	no	—
44	C₂³×C₂₅₁	—	2	2	no	—
45	C₂³×C₂₅₁	—	2	2	no	—
46	C₂³×C₂₅₁	—	4	2	no	—
47	C₂³×C₂₅₁	—	4	2	no	—
48	C₂³×C₂₅₁	C₄⋊C₄	1	4	yes	—
49	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
50	D₄×C₂₅₁	—	2	4	no	—
51	D₄×C₂₅₁	—	2	4	no	—
52	C₂×C₄×C₂₅₁	—	2	2	no	—
53	C₂×C₄×C₂₅₁	—	4	1	no	—
54	C₂×C₄×C₂₅₁	—	4	1	no	—
55	C₂×C₄×C₂₅₁	—	4	2	no	—
56	C₂×C₄×C₂₅₁	—	4	1	no	—
57	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₂₅₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₂₅₁	#41	1	2	yes	—
61	C₂²×C₂₅₁	—	2	1	no	—
62	C₂²×C₂₅₁	—	2	1	no	—
63	C₂²×C₂₅₁	—	2	4	no	—
64	C₂²×C₂₅₁	—	4	4	no	—
65	C₂²×C₂₅₁	—	4	2	no	—
66	C₂²×C₂₅₁	—	4	2	no	—
67	C₂²×C₂₅₁	—	8	2	no	—
68	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₅₁	#120	1	1	yes	characteristic
70	C₂²×C₂₅₁	#123	1	1	yes	characteristic
71	C₄×C₂₅₁	—	4	2	no	—
72	C₄×C₂₅₁	—	8	1	no	—
73	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
74	C₄×C₂₅₁	C₂²⋊Q₈	1	2	yes	—
75	Dic₂₅₁	—	4	2	no	—
76	Dic₂₅₁	—	4	2	no	—
77	C₂×C₂₅₁	—	2	1	no	—
78	C₂×C₂₅₁	—	2	1	no	—
79	C₂×C₂₅₁	—	4	4	no	—
80	C₂×C₂₅₁	#385	1	1	yes	characteristic
81	C₂×C₂₅₁	#453	1	1	yes	characteristic
82	C₂×C₂₅₁	#455	1	1	yes	characteristic
83	C₂₅₁	#1856	1	1	yes	characteristic, complemented, Sylow
84	#1856	—	251	1	no	maximal, Sylow
85	#518	—	251	1	no	—
86	#428	—	251	2	no	—
87	C₂×#100	—	251	1	no	—
88	#390	D₂₅₁	1	1	yes	characteristic
89	#339	—	251	2	no	—
90	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
91	C₂×M₁₆	—	251	2	no	—
92	C₄×D₄	—	502	2	no	—
93	C₄×D₄	—	502	2	no	—
94	#117	—	251	2	no	—
95	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
96	C₂×C₂²⋊C₄	—	2	2	no	—
97	C₂×C₂²⋊C₄	—	502	2	no	—
98	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
99	#107	—	502	2	no	—
100	#106	—	502	2	no	—
101	#100	—	251	4	no	—
102	C₂⁴	C₂²⋊D₂₅₁	1	2	yes	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	C₄×D₂₅₁	1	2	yes	—
106	C₂×D₄	Dic₅₀₂	1	2	yes	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	502	2	no	—
110	C₂²×C₄	—	502	2	no	—
111	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
112	M₁₆	—	502	2	no	—
113	M₁₆	—	251	4	no	—
114	C₂×C₈	—	502	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	251	2	no	—
117	C₄⋊C₄	—	251	2	no	—
118	C₂²⋊C₄	—	4	4	no	—
119	C₂²⋊C₄	—	502	4	no	—
120	C₂²⋊C₄	—	502	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	502	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	2	2	no	—
125	C₂³	—	4	2	no	—
126	C₂³	—	4	2	no	—
127	C₂³	Dic₂₅₁⋊C₄	1	4	yes	—
128	C₂³	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
129	D₄	—	2	4	no	—
130	D₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	2008	2	no	—
137	C₂×C₄	—	2008	2	no	—
138	C₂×C₄	—	502	2	no	—
139	C₂×C₄	—	502	2	no	—
140	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
141	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
142	C₂×C₄	C₂×Dic₅₀₂	1	1	yes	characteristic
143	C₂×C₄	#16860809	1	2	yes	—
144	C₈	—	502	4	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	8	2	no	—
152	C₂²	C₂×Dic₂₅₁⋊C₄	1	1	yes	characteristic
153	C₂²	#22548683	1	1	yes	characteristic
154	C₂²	#22548696	1	1	yes	characteristic
155	C₄	—	4	2	no	—
156	C₄	—	8	1	no	—
157	C₄	—	1004	2	no	—
158	C₄	—	1004	2	no	—
159	C₄	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
160	C₄	#22548692	1	2	yes	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	4	4	no	—
164	C₂	#30631559	1	1	yes	characteristic
165	C₂	#30631854	1	1	yes	characteristic
166	C₂	#30631864	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632162

C₂

yes

maximal, characteristic

#30631727

C₂

yes

maximal, complemented

C₂×#22548632

C₂

yes

maximal, characteristic

#30631447

C₂

yes

maximal, complemented

#390×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860794

C₂²

yes

—

D₄×Dic₂₅₁

—

D₄×Dic₂₅₁

—

#22548674

C₂²

yes

—

C₂×C₂²⋊Dic₂₅₁

—

#22548648

—

#22548646

—

#22548632

C₄

yes

complemented

C₂²×D₄×C₂₅₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₅₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₂₅₁

—

C₂×C₄²×C₂₅₁

C₂²

yes

characteristic

C₂²×Dic₂₅₁

—

C₂²×Dic₂₅₁

—

#16860794

—

#16860794

C₂×C₄

yes

complemented

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

D₄

yes

—

C₄⋊Dic₂₅₁

D₄

yes

—

C₂²⋊Dic₂₅₁

—

C₂²⋊Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₂⁴×C₂₅₁

D₄

yes

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

C₂×C₄

yes

—

C₂×D₄×C₂₅₁

Q₈

yes

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

—

C₂²×C₄×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₂²⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

C₄⋊C₄

yes

—

C₂³×C₂₅₁

C₂×D₄

yes

characteristic

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

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₅₁

#41

yes

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

C₂×C₄⋊C₄

yes

C₂²×C₂₅₁

yes

C₂²×C₂₅₁

yes

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₄×D₄

yes

—

C₄×C₂₅₁

C₂²⋊Q₈

yes

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#1856

—

251

maximal, Sylow

#518

—

251

—

#428

—

251

—

C₂×#100

—

251

—

#390

D₂₅₁

yes

characteristic

#339

—

251

—

C₂²×D₄

C₂×D₂₅₁

yes

characteristic

C₂×M₁₆

—

251

—

C₄×D₄

—

502

—

C₄×D₄

—

502

—

#117

—

251

—

C₂×C₄⋊C₄

C₂×D₂₅₁

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

502

—

C₂×C₄²

C₂×D₂₅₁

yes

characteristic

#107

—

502

—

100

#106

—

502

—

101

#100

—

251

—

102

C₂⁴

C₂²⋊D₂₅₁

yes

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₄×D₂₅₁

yes

—

106

C₂×D₄

Dic₅₀₂

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

502

—

110

C₂²×C₄

—

502

—

111

C₂²×C₄

C₂²×D₂₅₁

yes

characteristic

112

M₁₆

—

502

—

113

M₁₆

—

251

—

114

C₂×C₈

—

502

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

251

—

117

C₄⋊C₄

—

251

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

502

—

120

C₂²⋊C₄

—

502

—

121

C₄²

—

122

C₄²

—

502

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

Dic₂₅₁⋊C₄

yes

—

128

C₂³

C₂×C₂²⋊D₂₅₁

yes

characteristic

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

2008

—

137

C₂×C₄

—

2008

—

138

C₂×C₄

—

502

—

139

C₂×C₄

—

502

—

140

C₂×C₄

C₂×C₄×D₂₅₁

yes

characteristic

141

C₂×C₄

D₄×D₂₅₁

yes

—

142

C₂×C₄

C₂×Dic₅₀₂

yes

characteristic

143

C₂×C₄

#16860809

yes

—

144

C₈

—

502

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×Dic₂₅₁⋊C₄

yes

characteristic

153

C₂²

#22548683

yes

characteristic

154

C₂²

#22548696

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

1004

—

158

C₄

—

1004

—

159

C₄

(C₂²⋊D₂₅₁)⋊C₄

yes

—

160

C₄

#22548692

yes

—

161

C₂

—

162

C₂

—

163

C₂

—

164

C₂

#30631559

yes

characteristic

165

C₂

#30631854

yes

characteristic

166

C₂

#30631864

yes

characteristic

	subgroup	G/H
center	C₂²	#22548683
commutator	C₂×C₄×C₂₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₁
Fitting	#390×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#120
prime radical	C₂²×D₄×C₂₅₁	C₂²
cosocle	C₂²×C₄×C₂₅₁	C₂³
hypercenter	#390	D₂₅₁
hypocenter	C₂₅₁	#1856
LCS	G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #390

subgroup

G/H

center

C₂²

#22548683

commutator

C₂×C₄×C₂₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₅₁

Fitting

#390×C₂₅₁

C₂

socle

C₂²×C₂₅₁

#120

prime radical

C₂²×D₄×C₂₅₁

C₂²

cosocle

C₂²×C₄×C₂₅₁

C₂³

D₂₅₁

C₂₅₁

G ▷ C₂×C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #390

N	H ≅ G/N	type
2. #30631727	C₂	semidirect
4. #30631447	C₂	semidirect
13. #22548632	C₄	semidirect
21. #16860794	C₂×C₄	semidirect
83. C₂₅₁	#1856	semidirect

H ≅ G/N

type

2. #30631727

C₂

semidirect

4. #30631447

C₂

semidirect

13. #22548632

C₄

semidirect

21. #16860794

C₂×C₄

semidirect

83. C₂₅₁

#1856

semidirect

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

subgroup	index
#30632162	2
#30631727	2
C₂×#22548632	2
#30631447	2
#390×C₂₅₁	2
#1856	251

subgroup

index

#390×C₂₅₁

251

Subgroups — #44550574

Conjugacy classes of subgroups

Complements

Maximal subgroups