Group #44549417 — subgroups

Subgroups — #44549417

Order 32128.

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

normal subgroups only

This group has 336 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	#30632322	C₂	1	1	yes	maximal, characteristic, complemented
2	#30631879	C₂	1	2	yes	maximal, complemented
3	#30631431	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631416	C₂	1	2	yes	maximal, complemented
5	#493×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₂₅₁	—	2	2	no	—
7	C₂×#16860795	—	2	2	no	—
8	#22548686	—	2	2	no	—
9	#22548683	—	2	2	no	—
10	C₄×D₁₀₀₄	C₂²	1	2	yes	—
11	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	complemented
12	#22548636	—	2	2	no	—
13	C₂×D₈×C₂₅₁	—	2	2	no	—
14	#127×C₂₅₁	C₂²	1	1	yes	characteristic
15	#127×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
16	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₂₅₁	—	2	2	no	—
18	C₂×D₁₀₀₄	D₄	1	2	yes	—
19	C₂×C₂²⋊D₂₅₁	—	2	4	no	—
20	D₄×D₂₅₁	—	4	4	no	—
21	D₄×D₂₅₁	—	4	4	no	—
22	C₂×C₄×D₂₅₁	—	2	2	no	—
23	#16860795	—	4	4	no	—
24	C₂×C₂₅₁⋊C₈	—	2	2	no	—
25	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
26	C₂²⋊Dic₂₅₁	—	2	2	no	—
27	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
28	C₂×D₄×C₂₅₁	—	2	2	no	—
29	C₂×D₄×C₂₅₁	—	2	1	no	—
30	C₂×D₄×C₂₅₁	—	2	2	no	—
31	C₂×D₄×C₂₅₁	D₄	1	2	yes	—
32	D₈×C₂₅₁	—	4	4	no	—
33	C₂×C₈×C₂₅₁	—	2	1	no	—
34	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
35	C₂²×D₂₅₁	—	2	2	no	—
36	C₂²×D₂₅₁	—	4	8	no	—
37	C₂²×D₂₅₁	—	8	2	no	—
38	D₁₀₀₄	—	2	2	no	—
39	D₁₀₀₄	D₈	1	4	yes	complemented
40	C₂²⋊D₂₅₁	—	4	8	no	—
41	C₂×Dic₂₅₁	—	2	2	no	—
42	C₄×D₂₅₁	—	4	2	no	—
43	C₄×D₂₅₁	—	4	2	no	—
44	C₂₅₁⋊C₈	—	4	2	no	—
45	C₂³×C₂₅₁	—	2	2	no	—
46	C₂³×C₂₅₁	—	4	1	no	—
47	D₄×C₂₅₁	—	4	2	no	—
48	D₄×C₂₅₁	—	4	2	no	—
49	D₄×C₂₅₁	—	4	2	no	—
50	D₄×C₂₅₁	—	4	2	no	—
51	D₄×C₂₅₁	—	4	2	no	—
52	D₄×C₂₅₁	—	4	2	no	—
53	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
55	C₈×C₂₅₁	—	2	2	no	—
56	C₂²×C₂₅₁	—	4	4	no	—
57	C₂²×C₂₅₁	—	4	2	no	—
58	C₂²×C₂₅₁	—	8	2	no	—
59	C₂²×C₂₅₁	—	8	1	no	—
60	C₂²×C₂₅₁	#120	1	1	yes	characteristic
61	C₂×D₂₅₁	—	2	4	no	—
62	C₂×D₂₅₁	—	4	2	no	—
63	C₂×D₂₅₁	—	4	2	no	—
64	C₂×D₂₅₁	—	8	8	no	—
65	C₄×C₂₅₁	—	2	1	no	—
66	C₄×C₂₅₁	C₂×D₈	1	2	yes	—
67	C₄×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
68	Dic₂₅₁	—	4	2	no	—
69	C₂×C₂₅₁	—	8	2	no	—
70	C₂×C₂₅₁	—	16	1	no	—
71	C₂×C₂₅₁	#447	1	2	yes	—
72	C₂×C₂₅₁	#453	1	1	yes	characteristic
73	D₂₅₁	—	4	4	no	—
74	C₂₅₁	#1619	1	1	yes	characteristic, complemented, Sylow
75	#1619	—	251	1	no	maximal, Sylow
76	D₄²	—	251	1	no	—
77	#493	D₂₅₁	1	1	yes	characteristic, complemented
78	#459	—	251	2	no	—
79	#331	—	251	1	no	—
80	D₄⋊C₈	—	251	2	no	—
81	C₂²×D₄	—	502	2	no	—
82	C₂×D₈	—	2	2	no	—
83	C₂×D₈	—	502	2	no	—
84	#127	C₂×D₂₅₁	1	1	yes	characteristic
85	#127	C₂×D₂₅₁	1	1	yes	characteristic, complemented
86	#121	—	502	2	no	—
87	#120	—	502	2	no	—
88	C₄×D₄	—	251	2	no	—
89	C₄⋊C₈	—	251	2	no	—
90	#102	—	502	2	no	—
91	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic, complemented
92	C₂⁴	—	502	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	1004	4	no	—
97	C₂×D₄	—	1004	4	no	—
98	C₂×D₄	—	502	4	no	—
99	C₂×D₄	—	251	2	no	—
100	C₂×D₄	C₂²⋊D₂₅₁	1	2	yes	—
101	C₂²×C₄	—	502	2	no	—
102	D₈	—	4	4	no	—
103	D₈	—	1004	4	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	502	2	no	—
106	C₄⋊C₄	—	251	2	no	—
107	C₂²⋊C₄	—	502	2	no	—
108	C₂²⋊C₄	—	502	2	no	—
109	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
110	C₂³	—	2	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	2008	2	no	—
113	C₂³	—	1004	8	no	—
114	C₂³	—	502	2	no	—
115	D₄	—	4	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	4	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	1004	8	no	—
122	D₄	—	502	2	no	—
123	D₄	—	251	4	no	—
124	C₂×C₄	—	1004	2	no	—
125	C₂×C₄	—	1004	2	no	—
126	C₂×C₄	—	502	2	no	—
127	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
128	C₂×C₄	C₂×C₂²⋊D₂₅₁	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	1004	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	8	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	2008	8	no	—
136	C₂²	—	1004	2	no	—
137	C₂²	—	1004	2	no	—
138	C₂²	—	502	4	no	—
139	C₂²	#22548683	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	1004	2	no	—
142	C₄	D₈×D₂₅₁	1	2	yes	—
143	C₄	#22548739	1	2	yes	—
144	C₂	—	8	2	no	—
145	C₂	—	16	1	no	—
146	C₂	—	1004	4	no	—
147	C₂	#30631819	1	2	yes	—
148	C₂	#30631854	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632322

C₂

yes

maximal, characteristic, complemented

#30631879

C₂

yes

maximal, complemented

#30631431

C₂

yes

maximal, characteristic, complemented

#30631416

C₂

yes

maximal, complemented

#493×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₂₅₁

—

C₂×#16860795

—

#22548686

—

#22548683

—

C₄×D₁₀₀₄

C₂²

yes

—

C₄⋊(C₂₅₁⋊C₈)

C₂²

yes

complemented

#22548636

—

C₂×D₈×C₂₅₁

—

#127×C₂₅₁

C₂²

yes

characteristic

#127×C₂₅₁

C₂²

yes

characteristic, complemented

C₄×C₈×C₂₅₁

C₂²

yes

characteristic, complemented

C₂³×D₂₅₁

—

C₂×D₁₀₀₄

D₄

yes

—

C₂×C₂²⋊D₂₅₁

—

D₄×D₂₅₁

—

D₄×D₂₅₁

—

C₂×C₄×D₂₅₁

—

#16860795

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

D₄

yes

—

C₂²⋊Dic₂₅₁

—

(C₂×D₂₅₁)⋊C₄

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

D₄

yes

—

D₈×C₂₅₁

—

C₂×C₈×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

C₂²×D₂₅₁

—

D₁₀₀₄

—

D₁₀₀₄

D₈

yes

complemented

C₂²⋊D₂₅₁

—

C₂×Dic₂₅₁

—

C₄×D₂₅₁

—

C₄×D₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×C₂₅₁

—

C₂³×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

D₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

characteristic

C₈×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

—

C₂²×C₂₅₁

#120

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

C₂×D₈

yes

—

C₄×C₂₅₁

AGL(1,ℤ/8)

yes

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#447

yes

—

C₂×C₂₅₁

#453

yes

characteristic

D₂₅₁

—

C₂₅₁

#1619

yes

characteristic, complemented, Sylow

#1619

—

251

maximal, Sylow

D₄²

—

251

—

#493

D₂₅₁

yes

characteristic, complemented

#459

—

251

—

#331

—

251

—

D₄⋊C₈

—

251

—

C₂²×D₄

—

502

—

C₂×D₈

—

C₂×D₈

—

502

—

#127

C₂×D₂₅₁

yes

characteristic

#127

C₂×D₂₅₁

yes

characteristic, complemented

#121

—

502

—

#120

—

502

—

C₄×D₄

—

251

—

C₄⋊C₈

—

251

—

#102

—

502

—

C₄×C₈

C₂×D₂₅₁

yes

characteristic, complemented

C₂⁴

—

502

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

1004

—

C₂×D₄

—

1004

—

C₂×D₄

—

502

—

C₂×D₄

—

251

—

100

C₂×D₄

C₂²⋊D₂₅₁

yes

—

101

C₂²×C₄

—

502

—

102

D₈

—

103

D₈

—

1004

—

104

C₂×C₈

—

105

C₂×C₈

—

502

—

106

C₄⋊C₄

—

251

—

107

C₂²⋊C₄

—

502

—

108

C₂²⋊C₄

—

502

—

109

C₄²

C₂²×D₂₅₁

yes

characteristic

110

C₂³

—

111

C₂³

—

112

C₂³

—

2008

—

113

C₂³

—

1004

—

114

C₂³

—

502

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

1004

—

122

D₄

—

502

—

123

D₄

—

251

—

124

C₂×C₄

—

1004

—

125

C₂×C₄

—

1004

—

126

C₂×C₄

—

502

—

127

C₂×C₄

D₄×D₂₅₁

yes

—

128

C₂×C₄

C₂×C₂²⋊D₂₅₁

yes

characteristic

129

C₈

—

130

C₈

—

1004

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

2008

—

136

C₂²

—

1004

—

137

C₂²

—

1004

—

138

C₂²

—

502

—

139

C₂²

#22548683

yes

characteristic

140

C₄

—

141

C₄

—

1004

—

142

C₄

D₈×D₂₅₁

yes

—

143

C₄

#22548739

yes

—

144

C₂

—

145

C₂

—

146

C₂

—

1004

—

147

C₂

#30631819

yes

—

148

C₂

#30631854

yes

characteristic

	subgroup	G/H
center	C₂²	#22548683
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#493×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#120
hypercenter	#493	D₂₅₁
hypocenter	C₂₅₁	#1619
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#22548683

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#493×C₂₅₁

C₂

C₂²×C₂₅₁

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #30632322	C₂	semidirect
2. #30631879	C₂	semidirect
3. #30631431	C₂	semidirect
4. #30631416	C₂	semidirect
5. #493×C₂₅₁	C₂	semidirect
11. C₄⋊(C₂₅₁⋊C₈)	C₂²	semidirect
15. #127×C₂₅₁	C₂²	semidirect
16. C₄×C₈×C₂₅₁	C₂²	semidirect
39. D₁₀₀₄	D₈	semidirect
74. C₂₅₁	#1619	semidirect
77. #493	D₂₅₁	semidirect
85. #127	C₂×D₂₅₁	semidirect
91. C₄×C₈	C₂×D₂₅₁	semidirect

H ≅ G/N

type

1. #30632322

C₂

semidirect

2. #30631879

C₂

semidirect

3. #30631431

C₂

semidirect

4. #30631416

C₂

semidirect

5. #493×C₂₅₁

C₂

semidirect

11. C₄⋊(C₂₅₁⋊C₈)

C₂²

semidirect

15. #127×C₂₅₁

C₂²

semidirect

16. C₄×C₈×C₂₅₁

C₂²

semidirect

39. D₁₀₀₄

D₈

semidirect

74. C₂₅₁

#1619

semidirect

77. #493

D₂₅₁

semidirect

85. #127

C₂×D₂₅₁

semidirect

91. C₄×C₈

C₂×D₂₅₁

semidirect

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

subgroup	index
#30632322	2
#30631879	2
#30631431	2
#30631416	2
#493×C₂₅₁	2
#1619	251

subgroup

index

#493×C₂₅₁

251

Subgroups — #44549417

Conjugacy classes of subgroups

Complements

Maximal subgroups