Group #44549862 — subgroups

Subgroups — #44549862

Order 32128.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 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	#30632083	C₂	1	1	yes	maximal, characteristic, complemented
2	#30631977	C₂	1	1	yes	maximal, characteristic
3	#30631765	C₂	1	1	yes	maximal, characteristic
4	#30631439	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631435	C₂	1	1	yes	maximal, characteristic, complemented
6	#30631433	C₂	1	1	yes	maximal, characteristic, complemented
7	#461×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#16860794	—	2	1	no	—
9	#22548715	C₂²	1	1	yes	characteristic
10	#22548646	—	2	1	no	—
11	#22548646	—	2	1	no	—
12	#22548646	—	2	1	no	—
13	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
15	#22548636	—	2	1	no	—
16	C₂²⋊(C₂₅₁⋊C₈)	—	2	1	no	—
17	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×SD₁₆×C₂₅₁	—	2	1	no	—
19	C₄⋊Q₈×C₂₅₁	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
22	#103×C₂₅₁	—	2	1	no	—
23	C₂×Dic₅₀₂	—	2	1	no	—
24	#16860794	—	4	2	no	—
25	C₂×C₂₅₁⋊C₈	—	2	1	no	—
26	C₂×C₂₅₁⋊C₈	—	2	1	no	—
27	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
29	C₄⋊Dic₂₅₁	—	2	1	no	—
30	C₄⋊Dic₂₅₁	—	2	1	no	—
31	C₂×Q₈×C₂₅₁	—	2	1	no	—
32	C₂×D₄×C₂₅₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₅₁	—	2	1	no	—
34	SD₁₆×C₂₅₁	—	4	2	no	—
35	C₂×C₈×C₂₅₁	—	2	1	no	—
36	C₄⋊C₄×C₂₅₁	—	2	1	no	—
37	C₄⋊C₄×C₂₅₁	—	2	1	no	—
38	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₂₅₁	—	2	1	no	—
40	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
41	Dic₅₀₂	—	8	1	no	—
42	C₂×Dic₂₅₁	—	4	1	no	—
43	C₂₅₁⋊C₈	—	2	1	no	—
44	C₂₅₁⋊C₈	—	4	1	no	—
45	C₂₅₁⋊C₈	—	4	1	no	—
46	C₂₅₁⋊C₈	SD₁₆	1	2	yes	complemented
47	C₂³×C₂₅₁	—	2	1	no	—
48	Q₈×C₂₅₁	—	4	1	no	—
49	Q₈×C₂₅₁	—	4	1	no	—
50	D₄×C₂₅₁	—	2	1	no	—
51	D₄×C₂₅₁	—	2	1	no	—
52	C₂×C₄×C₂₅₁	—	2	1	no	—
53	C₂×C₄×C₂₅₁	—	4	1	no	—
54	C₂×C₄×C₂₅₁	—	4	1	no	—
55	C₂×C₄×C₂₅₁	—	4	1	no	—
56	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₅₁	#41	1	1	yes	characteristic
59	C₈×C₂₅₁	—	4	1	no	—
60	C₂²×C₂₅₁	—	4	1	no	—
61	C₂²×C₂₅₁	—	4	1	no	—
62	C₂²×C₂₅₁	—	4	1	no	—
63	C₂²×C₂₅₁	#121	1	1	yes	characteristic
64	C₄×C₂₅₁	—	2	1	no	—
65	C₄×C₂₅₁	—	4	1	no	—
66	C₄×C₂₅₁	—	8	1	no	—
67	C₄×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₂₅₁	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₂₅₁	#137	1	1	yes	characteristic
70	C₄×C₂₅₁	#137	1	1	yes	characteristic
71	Dic₂₅₁	—	8	1	no	—
72	C₂×C₂₅₁	—	8	1	no	—
73	C₂×C₂₅₁	#461	1	1	yes	characteristic
74	C₂×C₂₅₁	#468	1	1	yes	characteristic
75	C₂×C₂₅₁	#471	1	1	yes	characteristic
76	C₂₅₁	#1691	1	1	yes	characteristic, complemented, Sylow
77	#1691	—	251	1	no	maximal, Sylow
78	#498	—	251	1	no	—
79	#476	—	251	1	no	—
80	#461	D₂₅₁	1	1	yes	characteristic
81	#436	—	251	1	no	—
82	#335	—	251	1	no	—
83	#333	—	251	1	no	—
84	#332	—	251	1	no	—
85	C₂×SD₁₆	—	2	1	no	—
86	C₂×M₁₆	—	502	1	no	—
87	C₄⋊Q₈	—	251	1	no	—
88	C₄⋊Q₈	C₂×D₂₅₁	1	1	yes	characteristic
89	C₄×D₄	C₂×D₂₅₁	1	1	yes	characteristic
90	#106	—	502	1	no	—
91	#106	—	502	1	no	—
92	#106	—	502	1	no	—
93	C₄⋊C₈	—	251	1	no	—
94	C₄⋊C₈	—	251	1	no	—
95	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
96	#103	—	2	1	no	—
97	#102	—	502	1	no	—
98	C₂²⋊C₈	—	502	1	no	—
99	C₄×C₈	—	251	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	502	1	no	—
102	C₂×D₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	—	4	2	no	—
105	M₁₆	—	1004	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	502	1	no	—
108	C₂×C₈	—	502	1	no	—
109	C₂×C₈	—	251	1	no	—
110	C₂×C₈	—	251	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	502	1	no	—
114	C₄⋊C₄	—	502	1	no	—
115	C₄⋊C₄	C₂²⋊D₂₅₁	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
118	C₂³	—	2	1	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	2008	1	no	—
122	D₄	—	2	1	no	—
123	D₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	1004	1	no	—
129	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
131	C₂×C₄	#16860809	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	1004	1	no	—
134	C₈	—	1004	1	no	—
135	C₈	—	502	1	no	—
136	C₈	—	251	2	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	#22548687	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	—	8	1	no	—
144	C₄	—	2008	1	no	—
145	C₄	SD₁₆×D₂₅₁	1	1	yes	characteristic
146	C₄	#22548741	1	1	yes	characteristic
147	C₄	#22548744	1	1	yes	characteristic
148	C₄	#22548745	1	1	yes	characteristic
149	C₂	—	8	1	no	—
150	C₂	#30631890	1	1	yes	characteristic
151	C₂	#30631931	1	1	yes	characteristic
152	C₂	#30631952	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632083

C₂

yes

maximal, characteristic, complemented

#30631977

C₂

yes

maximal, characteristic

#30631765

C₂

yes

maximal, characteristic

#30631439

C₂

yes

maximal, characteristic, complemented

#30631435

C₂

yes

maximal, characteristic, complemented

#30631433

C₂

yes

maximal, characteristic, complemented

#461×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860794

—

#22548715

C₂²

yes

—

—

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

characteristic

#22548636

—

C₂²⋊(C₂₅₁⋊C₈)

—

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₂₅₁

—

C₄⋊Q₈×C₂₅₁

C₂²

yes

characteristic

C₄×D₄×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

C₂²

yes

characteristic

#103×C₂₅₁

—

C₂×Dic₅₀₂

—

#16860794

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

—

C₂×C₂₅₁⋊C₈

D₄

yes

characteristic

C₂×C₂₅₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

C₂×Q₈×C₂₅₁

—

C₂×D₄×C₂₅₁

D₄

yes

characteristic

C₂²×C₄×C₂₅₁

—

SD₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

—

C₄⋊C₄×C₂₅₁

D₄

yes

characteristic

C₂²⋊C₄×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

—

C₂₅₁⋊C₈

SD₁₆

yes

complemented

C₂³×C₂₅₁

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

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

#41

yes

characteristic

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#121

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×SD₁₆

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

C₄×C₂₅₁

yes

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#1691

—

251

maximal, Sylow

#498

—

251

—

#476

—

251

—

#461

D₂₅₁

yes

characteristic

#436

—

251

—

#335

—

251

—

#333

—

251

—

#332

—

251

—

C₂×SD₁₆

—

C₂×M₁₆

—

502

—

C₄⋊Q₈

—

251

—

C₄⋊Q₈

C₂×D₂₅₁

yes

characteristic

C₄×D₄

C₂×D₂₅₁

yes

characteristic

#106

—

502

—

#106

—

502

—

#106

—

502

—

C₄⋊C₈

—

251

—

C₄⋊C₈

—

251

—

C₄⋊C₈

C₂×D₂₅₁

yes

characteristic

#103

—

#102

—

502

—

C₂²⋊C₈

—

502

—

C₄×C₈

—

251

—

100

C₂×Q₈

—

101

C₂×Q₈

—

502

—

102

C₂×D₄

C₂²⋊D₂₅₁

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

—

105

M₁₆

—

1004

—

106

C₂×C₈

—

107

C₂×C₈

—

502

—

108

C₂×C₈

—

502

—

109

C₂×C₈

—

251

—

110

C₂×C₈

—

251

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

502

—

114

C₄⋊C₄

—

502

—

115

C₄⋊C₄

C₂²⋊D₂₅₁

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

C₂²×D₂₅₁

yes

characteristic

118

C₂³

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

2008

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

1004

—

129

C₂×C₄

D₄×D₂₅₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

131

C₂×C₄

#16860809

yes

characteristic

132

C₈

—

133

C₈

—

1004

—

134

C₈

—

1004

—

135

C₈

—

502

—

136

C₈

—

251

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#22548687

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

2008

—

145

C₄

SD₁₆×D₂₅₁

yes

characteristic

146

C₄

#22548741

yes

characteristic

147

C₄

#22548744

yes

characteristic

148

C₄

#22548745

yes

characteristic

149

C₂

—

150

C₂

#30631890

yes

characteristic

151

C₂

#30631931

yes

characteristic

152

C₂

#30631952

yes

characteristic

	subgroup	G/H
center	C₂²	#22548687
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#461×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#121
prime radical	C₂×D₄×C₂₅₁	D₄
hypercenter	#461	D₂₅₁
hypocenter	C₂₅₁	#1691
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #461

subgroup

G/H

center

C₂²

#22548687

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#461×C₂₅₁

C₂

socle

C₂²×C₂₅₁

#121

prime radical

C₂×D₄×C₂₅₁

D₄

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #461

N	H ≅ G/N	type
1. #30632083	C₂	semidirect
4. #30631439	C₂	semidirect
5. #30631435	C₂	semidirect
6. #30631433	C₂	semidirect
46. C₂₅₁⋊C₈	SD₁₆	semidirect
76. C₂₅₁	#1691	semidirect

H ≅ G/N

type

1. #30632083

C₂

semidirect

4. #30631439

C₂

semidirect

5. #30631435

C₂

semidirect

6. #30631433

C₂

semidirect

46. C₂₅₁⋊C₈

SD₁₆

semidirect

76. C₂₅₁

#1691

semidirect

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

subgroup	index
#30632083	2
#30631977	2
#30631765	2
#30631439	2
#30631435	2
#30631433	2
#461×C₂₅₁	2
#1691	251

subgroup

index

#461×C₂₅₁

251

Subgroups — #44549862

Conjugacy classes of subgroups

Complements

Maximal subgroups