Group #44549522 — subgroups

Subgroups — #44549522

Order 32128.

188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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	#30632416	C₂	1	1	yes	maximal, characteristic
2	#30632037	C₂	1	1	yes	maximal, characteristic
3	#30631979	C₂	1	1	yes	maximal, characteristic
4	#30631432	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	#477×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	#22548715	—	2	1	no	—
9	#22548715	C₂²	1	1	yes	characteristic
10	#22548706	—	2	2	no	—
11	#22548706	—	2	1	no	—
12	Q₈×Dic₂₅₁	—	2	1	no	—
13	C₄×Dic₅₀₂	—	2	1	no	—
14	C₄×Dic₅₀₂	C₂²	1	1	yes	characteristic
15	#22548648	—	2	1	no	—
16	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
17	#22548639	—	2	2	no	—
18	#22548638	—	2	1	no	—
19	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₄×Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
22	#107×C₂₅₁	—	2	1	no	—
23	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
24	#103×C₂₅₁	—	2	1	no	—
25	C₂×Dic₅₀₂	—	2	1	no	—
26	C₂×Dic₅₀₂	D₄	1	1	yes	characteristic
27	C₂×C₂₅₁⋊C₈	—	2	1	no	—
28	C₂×C₂₅₁⋊C₈	—	2	1	no	—
29	C₄⋊Dic₂₅₁	—	2	2	no	—
30	C₄⋊Dic₂₅₁	—	2	2	no	—
31	C₄⋊Dic₂₅₁	—	2	1	no	—
32	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
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	C₂×C₈×C₂₅₁	—	2	1	no	—
41	C₄⋊C₄×C₂₅₁	—	2	1	no	—
42	C₄⋊C₄×C₂₅₁	—	2	1	no	—
43	C₄⋊C₄×C₂₅₁	—	2	1	no	—
44	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
45	C₄²×C₂₅₁	—	2	1	no	—
46	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
47	Dic₅₀₂	—	2	1	no	—
48	Dic₅₀₂	—	2	1	no	—
49	Dic₅₀₂	—	4	1	no	—
50	Dic₅₀₂	—	4	1	no	—
51	C₂×Dic₂₅₁	—	2	1	no	—
52	C₂×Dic₂₅₁	—	2	1	no	—
53	C₂×Dic₂₅₁	—	2	2	no	—
54	C₂₅₁⋊C₈	—	4	1	no	—
55	C₂₅₁⋊C₈	—	4	1	no	—
56	Q₈×C₂₅₁	—	2	1	no	—
57	Q₈×C₂₅₁	—	4	2	no	—
58	Q₈×C₂₅₁	Q₁₆	1	2	yes	—
59	C₂×C₄×C₂₅₁	—	2	1	no	—
60	C₂×C₄×C₂₅₁	—	2	1	no	—
61	C₂×C₄×C₂₅₁	—	4	1	no	—
62	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
65	C₈×C₂₅₁	—	4	1	no	—
66	C₂²×C₂₅₁	#120	1	1	yes	characteristic
67	C₄×C₂₅₁	—	2	2	no	—
68	C₄×C₂₅₁	—	2	1	no	—
69	C₄×C₂₅₁	—	4	1	no	—
70	C₄×C₂₅₁	—	8	1	no	—
71	C₄×C₂₅₁	C₂×Q₁₆	1	1	yes	characteristic
72	C₄×C₂₅₁	#135	1	1	yes	characteristic
73	C₄×C₂₅₁	#137	1	1	yes	characteristic
74	C₄×C₂₅₁	#137	1	1	yes	characteristic
75	Dic₂₅₁	—	4	1	no	—
76	Dic₂₅₁	—	4	2	no	—
77	Dic₂₅₁	—	4	1	no	—
78	C₂×C₂₅₁	#451	1	1	yes	characteristic
79	C₂×C₂₅₁	#452	1	1	yes	characteristic
80	C₂×C₂₅₁	#454	1	1	yes	characteristic
81	C₂₅₁	#1636	1	1	yes	characteristic, complemented, Sylow
82	#1636	—	251	1	no	maximal, Sylow
83	#557	—	251	1	no	—
84	#487	—	251	1	no	—
85	#477	—	251	1	no	—
86	#477	D₂₅₁	1	1	yes	characteristic
87	#332	—	251	1	no	—
88	Q₈⋊C₈	—	251	1	no	—
89	Q₈⋊C₈	—	251	1	no	—
90	C₄⋊Q₈	—	502	1	no	—
91	C₄⋊Q₈	—	251	1	no	—
92	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
93	#125	—	502	2	no	—
94	#125	—	502	1	no	—
95	C₄×Q₈	—	502	1	no	—
96	C₄×Q₈	—	502	1	no	—
97	C₄×Q₈	—	251	1	no	—
98	C₄×Q₈	C₂×D₂₅₁	1	1	yes	characteristic
99	#107	—	2	1	no	—
100	#107	—	502	1	no	—
101	C₄⋊C₈	—	251	1	no	—
102	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
103	#103	—	2	1	no	—
104	#103	—	502	1	no	—
105	#103	—	502	2	no	—
106	C₄×C₈	—	251	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	502	1	no	—
109	C₂×Q₈	—	251	1	no	—
110	C₂×Q₈	C₂²⋊D₂₅₁	1	1	yes	characteristic
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	502	1	no	—
113	C₂×C₈	—	502	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	502	2	no	—
118	C₄⋊C₄	—	502	2	no	—
119	C₄⋊C₄	—	502	2	no	—
120	C₄⋊C₄	—	502	1	no	—
121	C₄⋊C₄	—	502	1	no	—
122	C₄⋊C₄	—	251	2	no	—
123	C₄⋊C₄	—	251	1	no	—
124	C₄⋊C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	—	502	2	no	—
127	C₄²	—	502	1	no	—
128	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	4	2	no	—
131	Q₈	—	1004	1	no	—
132	Q₈	—	1004	1	no	—
133	Q₈	—	502	1	no	—
134	Q₈	—	502	1	no	—
135	Q₈	#16860800	1	2	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	502	1	no	—
140	C₂×C₄	—	502	1	no	—
141	C₂×C₄	—	502	2	no	—
142	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
143	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	—	1004	1	no	—
147	C₈	—	1004	1	no	—
148	C₂²	#22548683	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	8	1	no	—
153	C₄	—	1004	2	no	—
154	C₄	—	1004	1	no	—
155	C₄	—	1004	1	no	—
156	C₄	C₂×#16860800	1	1	yes	characteristic
157	C₄	#22548738	1	1	yes	characteristic
158	C₄	#22548745	1	1	yes	characteristic
159	C₄	#22548747	1	1	yes	characteristic
160	C₂	#30631842	1	1	yes	characteristic
161	C₂	#30631848	1	1	yes	characteristic
162	C₂	#30631858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632416

C₂

yes

maximal, characteristic

#30632037

C₂

yes

maximal, characteristic

#30631979

C₂

yes

maximal, characteristic

#30631432

C₂

yes

maximal, characteristic

Q₈⋊(C₂₅₁⋊C₈)

C₂

yes

maximal, characteristic

#30631417

C₂

yes

maximal, characteristic

#477×C₂₅₁

C₂

yes

maximal, characteristic

#22548715

—

#22548715

C₂²

yes

characteristic

#22548706

—

#22548706

—

Q₈×Dic₂₅₁

—

C₄×Dic₅₀₂

—

C₄×Dic₅₀₂

C₂²

yes

characteristic

#22548648

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

characteristic

#22548639

—

#22548638

—

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄×Q₈×C₂₅₁

C₂²

yes

characteristic

#107×C₂₅₁

—

C₄⋊C₈×C₂₅₁

C₂²

yes

characteristic

#103×C₂₅₁

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

D₄

yes

characteristic

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

D₄

yes

—

C₄⋊Dic₂₅₁

D₄

yes

characteristic

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×Q₈×C₂₅₁

D₄

yes

characteristic

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

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

Q₁₆

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×Q₁₆

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#1636

—

251

maximal, Sylow

#557

—

251

—

#487

—

251

—

#477

—

251

—

#477

D₂₅₁

yes

characteristic

#332

—

251

—

Q₈⋊C₈

—

251

—

Q₈⋊C₈

—

251

—

C₄⋊Q₈

—

502

—

C₄⋊Q₈

—

251

—

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

#125

—

502

—

#125

—

502

—

C₄×Q₈

—

502

—

C₄×Q₈

—

502

—

C₄×Q₈

—

251

—

C₄×Q₈

C₂×D₂₅₁

yes

characteristic

#107

—

100

#107

—

502

—

101

C₄⋊C₈

—

251

—

102

C₄⋊C₈

C₂×D₂₅₁

yes

characteristic

103

#103

—

104

#103

—

502

—

105

#103

—

502

—

106

C₄×C₈

—

251

—

107

C₂×Q₈

—

108

C₂×Q₈

—

502

—

109

C₂×Q₈

—

251

—

110

C₂×Q₈

C₂²⋊D₂₅₁

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

502

—

113

C₂×C₈

—

502

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

502

—

118

C₄⋊C₄

—

502

—

119

C₄⋊C₄

—

502

—

120

C₄⋊C₄

—

502

—

121

C₄⋊C₄

—

502

—

122

C₄⋊C₄

—

251

—

123

C₄⋊C₄

—

251

—

124

C₄⋊C₄

C₂²⋊D₂₅₁

yes

characteristic

125

C₄²

—

126

C₄²

—

502

—

127

C₄²

—

502

—

128

C₄²

C₂²×D₂₅₁

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

1004

—

132

Q₈

—

1004

—

133

Q₈

—

502

—

134

Q₈

—

502

—

135

Q₈

#16860800

yes

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

502

—

140

C₂×C₄

—

502

—

141

C₂×C₄

—

502

—

142

C₂×C₄

D₄×D₂₅₁

yes

characteristic

143

C₂×C₄

D₄×D₂₅₁

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

145

C₈

—

146

C₈

—

1004

—

147

C₈

—

1004

—

148

C₂²

#22548683

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

1004

—

154

C₄

—

1004

—

155

C₄

—

1004

—

156

C₄

C₂×#16860800

yes

characteristic

157

C₄

#22548738

yes

characteristic

158

C₄

#22548745

yes

characteristic

159

C₄

#22548747

yes

characteristic

160

C₂

#30631842

yes

characteristic

161

C₂

#30631848

yes

characteristic

162

C₂

#30631858

yes

characteristic

	subgroup	G/H
center	C₂²	#22548683
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#477×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	#120
hypercenter	#477	D₂₅₁
hypocenter	C₂₅₁	#1636
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

subgroup

G/H

center

C₂²

#22548683

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#477×C₂₅₁

C₂

C₂²×C₂₅₁

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #477

N	H ≅ G/N	type
81. C₂₅₁	#1636	semidirect

H ≅ G/N

type

81. C₂₅₁

#1636

semidirect

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

subgroup	index
#30632416	2
#30632037	2
#30631979	2
#30631432	2
Q₈⋊(C₂₅₁⋊C₈)	2
#30631417	2
#477×C₂₅₁	2
#1636	251

subgroup

index

Q₈⋊(C₂₅₁⋊C₈)

#477×C₂₅₁

251

Subgroups — #44549522

Conjugacy classes of subgroups

Complements

Maximal subgroups