Group #44549584 — subgroups

Subgroups — #44549584

Order 32128.

200 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₅₀₂	C₂	1	1	yes	maximal, characteristic
2	#30632039	C₂	1	1	yes	maximal, characteristic
3	#30631969	C₂	1	1	yes	maximal, characteristic
4	#30631435	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₂₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#30631417	C₂	1	1	yes	maximal, characteristic
7	#462×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	#22548715	—	2	1	no	—
9	#22548715	C₂²	1	1	yes	characteristic
10	#22548714	—	2	2	no	—
11	#22548714	—	2	1	no	—
12	Q₈×Dic₂₅₁	—	2	1	no	—
13	C₄×Dic₅₀₂	—	2	1	no	—
14	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
15	#22548646	—	2	1	no	—
16	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
17	#22548638	—	2	1	no	—
18	#22548638	—	2	2	no	—
19	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×Q₁₆×C₂₅₁	—	2	1	no	—
21	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
24	#103×C₂₅₁	—	2	1	no	—
25	C₂×Dic₅₀₂	—	2	2	no	—
26	C₂×Dic₅₀₂	D₄	1	1	yes	characteristic
27	C₂×Dic₅₀₂	D₄	1	2	yes	—
28	C₂×C₂₅₁⋊C₈	—	2	1	no	—
29	C₂×C₂₅₁⋊C₈	—	2	1	no	—
30	C₄⋊Dic₂₅₁	—	2	2	no	—
31	C₄⋊Dic₂₅₁	—	2	1	no	—
32	C₄⋊Dic₂₅₁	—	2	1	no	—
33	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
34	Dic₂₅₁⋊C₄	—	2	2	no	—
35	Dic₂₅₁⋊C₄	—	2	1	no	—
36	C₄×Dic₂₅₁	—	2	2	no	—
37	C₄×Dic₂₅₁	—	2	1	no	—
38	C₂×Q₈×C₂₅₁	—	2	1	no	—
39	C₂×Q₈×C₂₅₁	D₄	1	1	yes	characteristic
40	Q₁₆×C₂₅₁	—	4	2	no	—
41	C₂×C₈×C₂₅₁	—	2	1	no	—
42	C₄⋊C₄×C₂₅₁	—	2	1	no	—
43	C₄⋊C₄×C₂₅₁	—	2	1	no	—
44	C₄⋊C₄×C₂₅₁	—	2	1	no	—
45	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
46	C₄²×C₂₅₁	—	2	1	no	—
47	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
48	Dic₅₀₂	—	2	1	no	—
49	Dic₅₀₂	—	4	2	no	—
50	Dic₅₀₂	—	8	2	no	—
51	Dic₅₀₂	Q₁₆	1	2	yes	complemented
52	C₂×Dic₂₅₁	—	2	1	no	—
53	C₂×Dic₂₅₁	—	2	2	no	—
54	C₂×Dic₂₅₁	—	2	1	no	—
55	C₂₅₁⋊C₈	—	4	1	no	—
56	C₂₅₁⋊C₈	—	4	1	no	—
57	Q₈×C₂₅₁	—	2	1	no	—
58	Q₈×C₂₅₁	—	4	1	no	—
59	Q₈×C₂₅₁	—	4	1	no	—
60	Q₈×C₂₅₁	SD₁₆	1	2	yes	—
61	C₂×C₄×C₂₅₁	—	2	1	no	—
62	C₂×C₄×C₂₅₁	—	2	1	no	—
63	C₂×C₄×C₂₅₁	—	4	1	no	—
64	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
67	C₈×C₂₅₁	—	4	1	no	—
68	C₂²×C₂₅₁	#120	1	1	yes	characteristic
69	C₄×C₂₅₁	—	2	2	no	—
70	C₄×C₂₅₁	—	2	1	no	—
71	C₄×C₂₅₁	—	4	1	no	—
72	C₄×C₂₅₁	—	8	1	no	—
73	C₄×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
74	C₄×C₂₅₁	C₂×Q₁₆	1	1	yes	characteristic
75	C₄×C₂₅₁	#137	1	1	yes	characteristic
76	C₄×C₂₅₁	#137	1	1	yes	characteristic
77	Dic₂₅₁	—	2	2	no	—
78	Dic₂₅₁	—	4	2	no	—
79	Dic₂₅₁	—	4	1	no	—
80	C₂×C₂₅₁	#448	1	1	yes	characteristic
81	C₂×C₂₅₁	#451	1	1	yes	characteristic
82	C₂×C₂₅₁	#455	1	1	yes	characteristic
83	C₂₅₁	#1645	1	1	yes	characteristic, complemented, Sylow
84	#1645	—	251	1	no	maximal, Sylow
85	Q₈²	—	251	1	no	—
86	#487	—	251	1	no	—
87	#475	—	251	1	no	—
88	#462	D₂₅₁	1	1	yes	characteristic
89	#333	—	251	1	no	—
90	Q₈⋊C₈	—	251	1	no	—
91	Q₈⋊C₈	—	251	1	no	—
92	C₂×Q₁₆	—	2	1	no	—
93	C₄⋊Q₈	—	502	1	no	—
94	C₄⋊Q₈	—	502	2	no	—
95	C₄⋊Q₈	—	502	1	no	—
96	C₄⋊Q₈	—	251	1	no	—
97	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
98	C₄×Q₈	—	502	1	no	—
99	C₄×Q₈	—	502	1	no	—
100	C₄×Q₈	—	251	1	no	—
101	C₄×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
102	#106	—	502	1	no	—
103	C₄⋊C₈	—	251	1	no	—
104	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
105	#103	—	2	1	no	—
106	#103	—	502	1	no	—
107	#103	—	502	2	no	—
108	C₄×C₈	—	251	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	502	2	no	—
111	C₂×Q₈	—	251	2	no	—
112	C₂×Q₈	—	251	1	no	—
113	C₂×Q₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
114	Q₁₆	—	4	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	502	1	no	—
117	C₂×C₈	—	502	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	502	2	no	—
122	C₄⋊C₄	—	502	2	no	—
123	C₄⋊C₄	—	502	1	no	—
124	C₄⋊C₄	—	502	1	no	—
125	C₄⋊C₄	—	502	1	no	—
126	C₄⋊C₄	—	251	1	no	—
127	C₄⋊C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
128	C₄²	—	2	1	no	—
129	C₄²	—	502	2	no	—
130	C₄²	—	502	1	no	—
131	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	2008	2	no	—
136	Q₈	—	1004	2	no	—
137	Q₈	—	502	1	no	—
138	Q₈	—	251	2	no	—
139	Q₈	#16860798	1	2	yes	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	502	2	no	—
144	C₂×C₄	—	502	1	no	—
145	C₂×C₄	—	502	1	no	—
146	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
147	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
148	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
149	C₈	—	4	1	no	—
150	C₈	—	1004	1	no	—
151	C₈	—	1004	1	no	—
152	C₂²	#22548683	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	1	no	—
156	C₄	—	8	1	no	—
157	C₄	—	1004	1	no	—
158	C₄	—	1004	2	no	—
159	C₄	—	502	2	no	—
160	C₄	C₂×#16860798	1	1	yes	characteristic
161	C₄	Q₁₆×D₂₅₁	1	1	yes	characteristic
162	C₄	#22548745	1	1	yes	characteristic
163	C₄	#22548747	1	1	yes	characteristic
164	C₂	#30631825	1	1	yes	characteristic
165	C₂	#30631841	1	1	yes	characteristic
166	C₂	#30631863	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₅₀₂

C₂

yes

maximal, characteristic

#30632039

C₂

yes

maximal, characteristic

#30631969

C₂

yes

maximal, characteristic

#30631435

C₂

yes

maximal, characteristic

Q₈⋊(C₂₅₁⋊C₈)

C₂

yes

maximal, characteristic

#30631417

C₂

yes

maximal, characteristic

#462×C₂₅₁

C₂

yes

maximal, characteristic

#22548715

—

#22548715

C₂²

yes

characteristic

#22548714

—

#22548714

—

Q₈×Dic₂₅₁

—

C₄×Dic₅₀₂

—

C₄×Dic₅₀₂

C₂²

yes

characteristic

#22548646

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

characteristic

#22548638

—

#22548638

—

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄×Q₈×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

C₂²

yes

characteristic

#103×C₂₅₁

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

D₄

yes

characteristic

C₂×Dic₅₀₂

D₄

yes

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

D₄

yes

characteristic

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

D₄

yes

characteristic

Q₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

D₄

yes

characteristic

C₄²×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

Q₁₆

yes

complemented

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

SD₁₆

yes

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

—

C₂²×C₂₅₁

#120

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×SD₁₆

yes

characteristic

C₄×C₂₅₁

C₂×Q₁₆

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#1645

—

251

maximal, Sylow

Q₈²

—

251

—

#487

—

251

—

#475

—

251

—

#462

D₂₅₁

yes

characteristic

#333

—

251

—

Q₈⋊C₈

—

251

—

Q₈⋊C₈

—

251

—

C₂×Q₁₆

—

C₄⋊Q₈

—

502

—

C₄⋊Q₈

—

502

—

C₄⋊Q₈

—

502

—

C₄⋊Q₈

—

251

—

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

C₄×Q₈

—

502

—

C₄×Q₈

—

502

—

100

C₄×Q₈

—

251

—

101

C₄×Q₈

C₂×D₂₅₁

yes

characteristic

102

#106

—

502

—

103

C₄⋊C₈

—

251

—

104

C₄⋊C₈

C₂×D₂₅₁

yes

characteristic

105

#103

—

106

#103

—

502

—

107

#103

—

502

—

108

C₄×C₈

—

251

—

109

C₂×Q₈

—

110

C₂×Q₈

—

502

—

111

C₂×Q₈

—

251

—

112

C₂×Q₈

—

251

—

113

C₂×Q₈

C₂²⋊D₂₅₁

yes

characteristic

114

Q₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

502

—

117

C₂×C₈

—

502

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

502

—

122

C₄⋊C₄

—

502

—

123

C₄⋊C₄

—

502

—

124

C₄⋊C₄

—

502

—

125

C₄⋊C₄

—

502

—

126

C₄⋊C₄

—

251

—

127

C₄⋊C₄

C₂²⋊D₂₅₁

yes

characteristic

128

C₄²

—

129

C₄²

—

502

—

130

C₄²

—

502

—

131

C₄²

C₂²×D₂₅₁

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

2008

—

136

Q₈

—

1004

—

137

Q₈

—

502

—

138

Q₈

—

251

—

139

Q₈

#16860798

yes

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

502

—

144

C₂×C₄

—

502

—

145

C₂×C₄

—

502

—

146

C₂×C₄

D₄×D₂₅₁

yes

characteristic

147

C₂×C₄

D₄×D₂₅₁

yes

characteristic

148

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

149

C₈

—

150

C₈

—

1004

—

151

C₈

—

1004

—

152

C₂²

#22548683

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

1004

—

158

C₄

—

1004

—

159

C₄

—

502

—

160

C₄

C₂×#16860798

yes

characteristic

161

C₄

Q₁₆×D₂₅₁

yes

characteristic

162

C₄

#22548745

yes

characteristic

163

C₄

#22548747

yes

characteristic

164

C₂

#30631825

yes

characteristic

165

C₂

#30631841

yes

characteristic

166

C₂

#30631863

yes

characteristic

	subgroup	G/H
center	C₂²	#22548683
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#462×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	#120
hypercenter	#462	D₂₅₁
hypocenter	C₂₅₁	#1645
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#22548683

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#462×C₂₅₁

C₂

C₂²×C₂₅₁

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
51. Dic₅₀₂	Q₁₆	semidirect
83. C₂₅₁	#1645	semidirect

H ≅ G/N

type

51. Dic₅₀₂

Q₁₆

semidirect

83. C₂₅₁

#1645

semidirect

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

subgroup	index
Q₈×Dic₅₀₂	2
#30632039	2
#30631969	2
#30631435	2
Q₈⋊(C₂₅₁⋊C₈)	2
#30631417	2
#462×C₂₅₁	2
#1645	251

subgroup

index

Q₈×Dic₅₀₂

Q₈⋊(C₂₅₁⋊C₈)

#462×C₂₅₁

251

Subgroups — #44549584

Conjugacy classes of subgroups

Complements

Maximal subgroups