Group #44549425 — subgroups

Subgroups — #44549425

Order 32128.

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

normal subgroups only

This group has 240 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	#30632423	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632063	C₂	1	1	yes	maximal, characteristic
3	#30631888	C₂	1	2	yes	maximal, complemented
4	#30631417	C₂	1	2	yes	maximal, complemented
5	#331×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16860799	—	2	2	no	—
7	C₂×#16860795	—	2	2	no	—
8	#22548712	C₂²	1	1	yes	characteristic
9	#22548702	—	2	2	no	—
10	#22548696	—	2	2	no	—
11	C₄×Dic₅₀₂	C₂²	1	2	yes	—
12	D₄×Dic₂₅₁	—	2	2	no	—
13	#22548637	—	2	2	no	—
14	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
15	#127×C₂₅₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₂₅₁	C₂²	1	2	yes	—
17	C₂×Dic₅₀₂	D₄	1	2	yes	—
18	C₂×D₁₀₀₄	—	2	2	no	—
19	C₂×D₁₀₀₄	—	2	1	no	—
20	C₂²×Dic₂₅₁	—	2	2	no	—
21	#16860799	—	4	4	no	—
22	#16860795	—	4	4	no	—
23	C₂×C₂₅₁⋊C₈	—	2	1	no	—
24	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
25	Dic₂₅₁⋊C₄	—	2	2	no	—
26	C₂²⋊Dic₂₅₁	—	2	4	no	—
27	C₂²⋊Dic₂₅₁	—	2	2	no	—
28	C₄×Dic₂₅₁	—	2	2	no	—
29	C₂×D₄×C₂₅₁	—	2	2	no	—
30	C₂×D₄×C₂₅₁	D₄	1	2	yes	—
31	C₂×C₈×C₂₅₁	—	2	2	no	—
32	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
33	C₂²×D₂₅₁	—	4	1	no	—
34	Dic₅₀₂	—	2	2	no	—
35	Dic₅₀₂	—	2	2	no	—
36	D₁₀₀₄	—	4	2	no	—
37	D₁₀₀₄	—	4	2	no	—
38	D₁₀₀₄	—	4	2	no	—
39	C₂×Dic₂₅₁	—	2	2	no	—
40	C₂×Dic₂₅₁	—	2	2	no	—
41	C₂×Dic₂₅₁	—	8	2	no	—
42	C₂₅₁⋊C₈	—	2	2	no	—
43	C₂³×C₂₅₁	—	2	2	no	—
44	D₄×C₂₅₁	—	4	2	no	—
45	D₄×C₂₅₁	—	4	2	no	—
46	D₄×C₂₅₁	—	4	2	no	—
47	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₅₁	—	4	2	no	—
50	C₂²×C₂₅₁	—	4	4	no	—
51	C₂²×C₂₅₁	—	4	2	no	—
52	C₂²×C₂₅₁	#120	1	1	yes	characteristic
53	C₂×D₂₅₁	—	8	2	no	—
54	C₂×D₂₅₁	—	8	1	no	—
55	C₄×C₂₅₁	—	2	1	no	—
56	C₄×C₂₅₁	#135	1	2	yes	—
57	C₄×C₂₅₁	AGL(1,ℤ/8)	1	2	yes	—
58	Dic₂₅₁	—	4	2	no	—
59	Dic₂₅₁	—	4	2	no	—
60	C₂×C₂₅₁	—	8	2	no	—
61	C₂×C₂₅₁	#449	1	2	yes	—
62	C₂×C₂₅₁	#453	1	1	yes	characteristic
63	D₂₅₁	—	16	1	no	—
64	C₂₅₁	#1620	1	1	yes	characteristic, complemented, Sylow
65	#1620	—	251	1	no	maximal, Sylow
66	#559	—	251	1	no	—
67	#493	—	251	1	no	—
68	#460	—	251	2	no	—
69	#331	D₂₅₁	1	1	yes	characteristic, complemented
70	Q₈⋊C₈	—	251	2	no	—
71	C₂×SD₁₆	—	502	2	no	—
72	C₂×D₈	—	502	2	no	—
73	#127	—	251	1	no	—
74	#127	C₂×D₂₅₁	1	1	yes	characteristic
75	#124	—	502	2	no	—
76	#123	—	502	2	no	—
77	C₄×Q₈	—	251	2	no	—
78	C₄×D₄	—	502	2	no	—
79	C₄⋊C₈	C₂×D₂₅₁	1	2	yes	—
80	#102	—	502	2	no	—
81	C₄×C₈	—	251	1	no	—
82	C₂×Q₈	—	251	2	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	502	2	no	—
85	C₂×D₄	—	502	1	no	—
86	C₂×D₄	D₁₀₀₄	1	2	yes	—
87	C₂²×C₄	—	502	2	no	—
88	SD₁₆	—	1004	4	no	—
89	D₈	—	1004	4	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	502	1	no	—
92	C₄⋊C₄	—	502	2	no	—
93	C₄⋊C₄	—	251	2	no	—
94	C₂²⋊C₄	—	502	4	no	—
95	C₂²⋊C₄	—	502	2	no	—
96	C₄²	—	502	2	no	—
97	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	1004	1	no	—
100	Q₈	—	502	2	no	—
101	Q₈	—	502	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	1004	2	no	—
106	D₄	—	1004	2	no	—
107	D₄	—	1004	2	no	—
108	C₂×C₄	—	2008	2	no	—
109	C₂×C₄	—	502	2	no	—
110	C₂×C₄	—	502	2	no	—
111	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
112	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
113	C₈	—	4	2	no	—
114	C₈	—	502	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	2008	2	no	—
118	C₂²	—	2008	1	no	—
119	C₂²	#22548685	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	1004	2	no	—
122	C₄	—	1004	2	no	—
123	C₄	#22548736	1	2	yes	—
124	C₄	#22548739	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	4016	1	no	—
127	C₂	#30631833	1	2	yes	—
128	C₂	#30631857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632423

C₂

yes

maximal, characteristic, complemented

#30632063

C₂

yes

maximal, characteristic

#30631888

C₂

yes

maximal, complemented

#30631417

C₂

yes

maximal, complemented

#331×C₂₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#16860799

—

C₂×#16860795

—

#22548712

C₂²

yes

characteristic

#22548702

—

#22548696

—

C₄×Dic₅₀₂

C₂²

yes

—

D₄×Dic₂₅₁

—

#22548637

—

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

#127×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

D₄

yes

—

C₂×D₁₀₀₄

—

C₂×D₁₀₀₄

—

C₂²×Dic₂₅₁

—

#16860799

—

#16860795

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

D₄

yes

—

Dic₂₅₁⋊C₄

—

C₂²⋊Dic₂₅₁

—

C₂²⋊Dic₂₅₁

—

C₄×Dic₂₅₁

—

C₂×D₄×C₂₅₁

—

C₂×D₄×C₂₅₁

D₄

yes

—

C₂×C₈×C₂₅₁

—

C₄²×C₂₅₁

C₂³

yes

characteristic

C₂²×D₂₅₁

—

Dic₅₀₂

—

Dic₅₀₂

—

D₁₀₀₄

—

D₁₀₀₄

—

D₁₀₀₄

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂₅₁⋊C₈

—

C₂³×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

#120

yes

characteristic

C₂×D₂₅₁

—

C₂×D₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#135

yes

—

C₄×C₂₅₁

AGL(1,ℤ/8)

yes

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#449

yes

—

C₂×C₂₅₁

#453

yes

characteristic

D₂₅₁

—

C₂₅₁

#1620

yes

characteristic, complemented, Sylow

#1620

—

251

maximal, Sylow

#559

—

251

—

#493

—

251

—

#460

—

251

—

#331

D₂₅₁

yes

characteristic, complemented

Q₈⋊C₈

—

251

—

C₂×SD₁₆

—

502

—

C₂×D₈

—

502

—

#127

—

251

—

#127

C₂×D₂₅₁

yes

characteristic

#124

—

502

—

#123

—

502

—

C₄×Q₈

—

251

—

C₄×D₄

—

502

—

C₄⋊C₈

C₂×D₂₅₁

yes

—

#102

—

502

—

C₄×C₈

—

251

—

C₂×Q₈

—

251

—

C₂×D₄

—

C₂×D₄

—

502

—

C₂×D₄

—

502

—

C₂×D₄

D₁₀₀₄

yes

—

C₂²×C₄

—

502

—

SD₁₆

—

1004

—

D₈

—

1004

—

C₂×C₈

—

C₂×C₈

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

251

—

C₂²⋊C₄

—

502

—

C₂²⋊C₄

—

502

—

C₄²

—

502

—

C₄²

C₂²×D₂₅₁

yes

characteristic

C₂³

—

C₂³

—

1004

—

100

Q₈

—

502

—

101

Q₈

—

502

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

1004

—

106

D₄

—

1004

—

107

D₄

—

1004

—

108

C₂×C₄

—

2008

—

109

C₂×C₄

—

502

—

110

C₂×C₄

—

502

—

111

C₂×C₄

D₄×D₂₅₁

yes

—

112

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

113

C₈

—

114

C₈

—

502

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

2008

—

118

C₂²

—

2008

—

119

C₂²

#22548685

yes

characteristic

120

C₄

—

121

C₄

—

1004

—

122

C₄

—

1004

—

123

C₄

#22548736

yes

—

124

C₄

#22548739

yes

—

125

C₂

—

126

C₂

—

4016

—

127

C₂

#30631833

yes

—

128

C₂

#30631857

yes

characteristic

	subgroup	G/H
center	C₂²	#22548685
commutator, cosocle	C₄²×C₂₅₁	C₂³
Frattini	C₄²	C₂²×D₂₅₁
Fitting	#331×C₂₅₁	C₂
socle	C₂²×C₂₅₁	#120
prime radical	#30632063	C₂
hypercenter	#331	D₂₅₁
hypocenter	C₂₅₁	#1620
LCS	G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#22548685

commutator, cosocle

C₄²×C₂₅₁

C₂³

Frattini

C₄²

C₂²×D₂₅₁

Fitting

#331×C₂₅₁

C₂

C₂²×C₂₅₁

C₂

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #30632423	C₂	semidirect
3. #30631888	C₂	semidirect
4. #30631417	C₂	semidirect
5. #331×C₂₅₁	C₂	semidirect
64. C₂₅₁	#1620	semidirect
69. #331	D₂₅₁	semidirect

H ≅ G/N

type

1. #30632423

C₂

semidirect

3. #30631888

C₂

semidirect

4. #30631417

C₂

semidirect

5. #331×C₂₅₁

C₂

semidirect

64. C₂₅₁

#1620

semidirect

69. #331

D₂₅₁

semidirect

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

subgroup	index
#30632423	2
#30632063	2
#30631888	2
#30631417	2
#331×C₂₅₁	2
#1620	251

subgroup

index

#331×C₂₅₁

251

Subgroups — #44549425

Conjugacy classes of subgroups

Complements

Maximal subgroups