Group #44549487 — subgroups

Subgroups — #44549487

Order 32128.

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

normal subgroups only

This group has 218 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
2	#30632059	C₂	1	1	yes	maximal, characteristic
3	#30631983	C₂	1	2	yes	maximal, complemented
4	#30631417	C₂	1	2	yes	maximal, complemented
5	#331×C₂₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16860798	—	2	2	no	—
7	#22548715	C₂²	1	1	yes	characteristic
8	#22548702	—	2	2	no	—
9	#22548696	—	2	2	no	—
10	C₄×Dic₅₀₂	C₂²	1	2	yes	—
11	D₄×Dic₂₅₁	—	2	2	no	—
12	#22548649	—	2	2	no	—
13	#22548640	—	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	—	2	1	no	—
18	C₂×Dic₅₀₂	D₄	1	2	yes	—
19	C₂²×Dic₂₅₁	—	2	2	no	—
20	#16860798	—	4	4	no	—
21	C₂×C₂₅₁⋊C₈	—	2	1	no	—
22	C₄⋊Dic₂₅₁	—	2	2	no	—
23	C₄⋊Dic₂₅₁	D₄	1	2	yes	—
24	Dic₂₅₁⋊C₄	—	2	2	no	—
25	C₂²⋊Dic₂₅₁	—	2	4	no	—
26	C₂²⋊Dic₂₅₁	—	2	2	no	—
27	C₄×Dic₂₅₁	—	2	2	no	—
28	C₂×D₄×C₂₅₁	—	2	2	no	—
29	C₂×D₄×C₂₅₁	D₄	1	2	yes	—
30	C₂×C₈×C₂₅₁	—	2	2	no	—
31	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
32	Dic₅₀₂	—	2	2	no	—
33	Dic₅₀₂	—	2	2	no	—
34	Dic₅₀₂	—	4	2	no	—
35	C₂×Dic₂₅₁	—	2	2	no	—
36	C₂×Dic₂₅₁	—	2	2	no	—
37	C₂×Dic₂₅₁	—	4	1	no	—
38	C₂×Dic₂₅₁	—	8	2	no	—
39	C₂₅₁⋊C₈	—	2	2	no	—
40	C₂³×C₂₅₁	—	2	2	no	—
41	D₄×C₂₅₁	—	4	2	no	—
42	D₄×C₂₅₁	—	4	2	no	—
43	D₄×C₂₅₁	—	4	2	no	—
44	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
46	C₈×C₂₅₁	—	4	2	no	—
47	C₂²×C₂₅₁	—	4	4	no	—
48	C₂²×C₂₅₁	—	4	2	no	—
49	C₂²×C₂₅₁	#120	1	1	yes	characteristic
50	C₄×C₂₅₁	—	2	1	no	—
51	C₄×C₂₅₁	#135	1	2	yes	—
52	C₄×C₂₅₁	#137	1	2	yes	—
53	Dic₂₅₁	—	4	2	no	—
54	Dic₂₅₁	—	4	2	no	—
55	Dic₂₅₁	—	8	1	no	—
56	C₂×C₂₅₁	—	8	2	no	—
57	C₂×C₂₅₁	#452	1	2	yes	—
58	C₂×C₂₅₁	#453	1	1	yes	characteristic
59	C₂₅₁	#1630	1	1	yes	characteristic, complemented, Sylow
60	#1630	—	251	1	no	maximal, Sylow
61	#559	—	251	1	no	—
62	#492	—	251	1	no	—
63	#477	—	251	2	no	—
64	#331	D₂₅₁	1	1	yes	characteristic
65	Q₈⋊C₈	—	251	2	no	—
66	C₂×SD₁₆	—	502	2	no	—
67	C₄⋊Q₈	—	251	1	no	—
68	#127	C₂×D₂₅₁	1	1	yes	characteristic
69	#124	—	502	2	no	—
70	#123	—	502	2	no	—
71	C₄×Q₈	—	251	2	no	—
72	C₄×D₄	—	502	2	no	—
73	#107	—	502	2	no	—
74	C₄⋊C₈	C₂×D₂₅₁	1	2	yes	—
75	#103	—	502	2	no	—
76	C₄×C₈	—	251	1	no	—
77	C₂×Q₈	—	502	1	no	—
78	C₂×Q₈	—	251	2	no	—
79	C₂×D₄	—	2	2	no	—
80	C₂×D₄	D₁₀₀₄	1	2	yes	—
81	C₂²×C₄	—	502	2	no	—
82	SD₁₆	—	1004	4	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	502	1	no	—
85	C₄⋊C₄	—	502	2	no	—
86	C₄⋊C₄	—	502	2	no	—
87	C₄⋊C₄	—	251	2	no	—
88	C₂²⋊C₄	—	502	4	no	—
89	C₂²⋊C₄	—	502	2	no	—
90	C₄²	—	502	2	no	—
91	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
92	C₂³	—	2	2	no	—
93	Q₈	—	1004	2	no	—
94	Q₈	—	502	2	no	—
95	Q₈	—	502	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	—	2008	2	no	—
100	C₂×C₄	—	1004	1	no	—
101	C₂×C₄	—	502	2	no	—
102	C₂×C₄	—	502	2	no	—
103	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
104	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	502	2	no	—
107	C₂²	—	4	4	no	—
108	C₂²	—	4	2	no	—
109	C₂²	#22548685	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2008	1	no	—
112	C₄	—	1004	2	no	—
113	C₄	—	1004	2	no	—
114	C₄	#22548734	1	2	yes	—
115	C₄	#22548744	1	2	yes	—
116	C₂	—	8	2	no	—
117	C₂	#30631850	1	2	yes	—
118	C₂	#30631857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632423

C₂

yes

maximal, characteristic

#30632059

C₂

yes

maximal, characteristic

#30631983

C₂

yes

maximal, complemented

#30631417

C₂

yes

maximal, complemented

#331×C₂₅₁

C₂

yes

maximal, characteristic

C₂×#16860798

—

#22548715

C₂²

yes

characteristic

#22548702

—

#22548696

—

C₄×Dic₅₀₂

C₂²

yes

—

D₄×Dic₂₅₁

—

#22548649

—

#22548640

—

C₄×C₂₅₁⋊C₈

C₂²

yes

characteristic

#127×C₂₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₅₁

C₂²

yes

—

C₂×Dic₅₀₂

—

C₂×Dic₅₀₂

D₄

yes

—

C₂²×Dic₂₅₁

—

#16860798

—

C₂×C₂₅₁⋊C₈

—

C₄⋊Dic₂₅₁

—

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

Dic₅₀₂

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

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

C₄×C₂₅₁

—

C₄×C₂₅₁

#135

yes

—

C₄×C₂₅₁

#137

yes

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

—

C₂×C₂₅₁

#452

yes

—

C₂×C₂₅₁

#453

yes

characteristic

C₂₅₁

#1630

yes

characteristic, complemented, Sylow

#1630

—

251

maximal, Sylow

#559

—

251

—

#492

—

251

—

#477

—

251

—

#331

D₂₅₁

yes

characteristic

Q₈⋊C₈

—

251

—

C₂×SD₁₆

—

502

—

C₄⋊Q₈

—

251

—

#127

C₂×D₂₅₁

yes

characteristic

#124

—

502

—

#123

—

502

—

C₄×Q₈

—

251

—

C₄×D₄

—

502

—

#107

—

502

—

C₄⋊C₈

C₂×D₂₅₁

yes

—

#103

—

502

—

C₄×C₈

—

251

—

C₂×Q₈

—

502

—

C₂×Q₈

—

251

—

C₂×D₄

—

C₂×D₄

D₁₀₀₄

yes

—

C₂²×C₄

—

502

—

SD₁₆

—

1004

—

C₂×C₈

—

C₂×C₈

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

502

—

C₄⋊C₄

—

251

—

C₂²⋊C₄

—

502

—

C₂²⋊C₄

—

502

—

C₄²

—

502

—

C₄²

C₂²×D₂₅₁

yes

characteristic

C₂³

—

Q₈

—

1004

—

Q₈

—

502

—

Q₈

—

502

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

2008

—

100

C₂×C₄

—

1004

—

101

C₂×C₄

—

502

—

102

C₂×C₄

—

502

—

103

C₂×C₄

D₄×D₂₅₁

yes

—

104

C₂×C₄

C₂×D₁₀₀₄

yes

characteristic

105

C₈

—

106

C₈

—

502

—

107

C₂²

—

108

C₂²

—

109

C₂²

#22548685

yes

characteristic

110

C₄

—

111

C₄

—

2008

—

112

C₄

—

1004

—

113

C₄

—

1004

—

114

C₄

#22548734

yes

—

115

C₄

#22548744

yes

—

116

C₂

—

117

C₂

#30631850

yes

—

118

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	#127×C₂₅₁	C₂²
hypercenter	#331	D₂₅₁
hypocenter	C₂₅₁	#1630
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

socle

C₂²×C₂₅₁

#120

prime radical

#127×C₂₅₁

C₂²

D₂₅₁

C₂₅₁

G ▷ C₄²×C₂₅₁ ▷ C₂²×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
3. #30631983	C₂	semidirect
4. #30631417	C₂	semidirect
59. C₂₅₁	#1630	semidirect

H ≅ G/N

type

3. #30631983

C₂

semidirect

4. #30631417

C₂

semidirect

59. C₂₅₁

#1630

semidirect

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

subgroup	index
#30632423	2
#30632059	2
#30631983	2
#30631417	2
#331×C₂₅₁	2
#1630	251

subgroup

index

#331×C₂₅₁

251

Subgroups — #44549487

Conjugacy classes of subgroups

Complements

Maximal subgroups