Group #44552609 — subgroups

Subgroups — #44552609

Order 32128.

132 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 47 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	#30632088	C₂	1	1	yes	maximal, characteristic
2	#30632065	C₂	1	1	yes	maximal, characteristic
3	#30631492	C₂	1	1	yes	maximal, characteristic
4	#30631491	C₂	1	1	yes	maximal, characteristic
5	#30631490	C₂	1	1	yes	maximal, characteristic
6	#30631462	C₂	1	1	yes	maximal, characteristic
7	#358×C₂₅₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₀₀₄	C₂²	1	1	yes	characteristic
9	C₂×#16860800	—	2	1	no	—
10	#22548713	—	2	1	no	—
11	#22548706	—	2	1	no	—
12	C₂×C₂₅₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	#22548649	C₂²	1	1	yes	characteristic
14	#22548646	—	2	1	no	—
15	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
18	#107×C₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₅₀₂	—	2	1	no	—
20	Dic₁₀₀₄	—	2	1	no	—
21	Dic₁₀₀₄	—	2	1	no	—
22	#16860800	—	4	2	no	—
23	C₂×C₂₅₁⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₂₅₁	—	2	1	no	—
25	C₄⋊Dic₂₅₁	—	4	1	no	—
26	Dic₂₅₁⋊C₄	—	4	1	no	—
27	Dic₂₅₁⋊C₄	—	4	1	no	—
28	Dic₂₅₁⋊C₄	—	4	1	no	—
29	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
30	C₂₅₁⋊C₁₆	—	2	1	no	—
31	C₂×Q₈×C₂₅₁	—	2	1	no	—
32	Q₁₆×C₂₅₁	—	2	1	no	—
33	Q₁₆×C₂₅₁	—	2	1	no	—
34	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₂₅₁	—	2	1	no	—
36	C₁₆×C₂₅₁	—	2	1	no	—
37	Dic₅₀₂	—	4	1	no	—
38	Dic₅₀₂	—	4	1	no	—
39	C₂×Dic₂₅₁	—	4	1	no	—
40	C₂×Dic₂₅₁	—	4	1	no	—
41	C₂×Dic₂₅₁	D₈	1	1	yes	characteristic
42	C₂×Dic₂₅₁	D₈	1	1	yes	characteristic
43	C₂₅₁⋊C₈	SD₁₆	1	2	yes	—
44	Q₈×C₂₅₁	—	4	1	no	—
45	Q₈×C₂₅₁	—	4	1	no	—
46	C₂×C₄×C₂₅₁	—	4	1	no	—
47	C₂×C₄×C₂₅₁	—	4	1	no	—
48	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₅₁	#41	1	1	yes	characteristic
50	C₈×C₂₅₁	#41	1	1	yes	characteristic
51	C₂²×C₂₅₁	C₂×D₈	1	1	yes	characteristic
52	C₄×C₂₅₁	—	8	1	no	—
53	C₄×C₂₅₁	—	8	1	no	—
54	C₄×C₂₅₁	#124	1	1	yes	characteristic
55	C₄×C₂₅₁	C₂×SD₁₆	1	1	yes	characteristic
56	Dic₂₅₁	—	2	1	no	—
57	Dic₂₅₁	—	2	1	no	—
58	Dic₂₅₁	—	8	1	no	—
59	Dic₂₅₁	—	8	1	no	—
60	C₂×C₂₅₁	#486	1	1	yes	characteristic
61	C₂×C₂₅₁	#510	1	1	yes	characteristic
62	C₂×C₂₅₁	#510	1	1	yes	characteristic
63	C₂₅₁	#2245	1	1	yes	characteristic, complemented, Sylow
64	#2245	—	251	1	no	maximal, Sylow
65	#499	—	251	1	no	—
66	#494	—	251	1	no	—
67	#358	—	251	1	no	—
68	#358	—	251	1	no	—
69	#358	—	251	1	no	—
70	#358	D₂₅₁	1	1	yes	characteristic
71	#346	—	251	1	no	—
72	C₂×Q₁₆	—	502	1	no	—
73	C₂×Q₁₆	—	251	1	no	—
74	C₂×Q₁₆	C₂×D₂₅₁	1	1	yes	characteristic
75	C₄⋊Q₈	—	502	1	no	—
76	#125	—	502	1	no	—
77	C₂×C₁₆	—	251	1	no	—
78	C₂×C₁₆	C₂×D₂₅₁	1	1	yes	characteristic
79	#107	—	251	1	no	—
80	#107	C₂×D₂₅₁	1	1	yes	characteristic
81	#106	—	502	1	no	—
82	C₄×C₈	—	251	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	502	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	1004	2	no	—
88	Q₁₆	—	502	1	no	—
89	Q₁₆	—	502	1	no	—
90	C₂×C₈	—	251	1	no	—
91	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	1004	1	no	—
94	C₄⋊C₄	—	1004	1	no	—
95	C₄⋊C₄	—	1004	1	no	—
96	C₄⋊C₄	—	1004	1	no	—
97	C₄⋊C₄	—	502	1	no	—
98	C₄²	—	251	1	no	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	502	1	no	—
101	Q₈	—	4	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	1004	1	no	—
104	Q₈	—	1004	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	1004	1	no	—
108	C₂×C₄	—	1004	1	no	—
109	C₂×C₄	—	251	1	no	—
110	C₂×C₄	—	251	1	no	—
111	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
112	C₈	—	251	2	no	—
113	C₈	#16860809	1	1	yes	characteristic
114	C₈	#16860810	1	1	yes	characteristic
115	C₂²	D₈×D₂₅₁	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	—	2008	1	no	—
119	C₄	—	2008	1	no	—
120	C₄	—	502	1	no	—
121	C₄	—	502	1	no	—
122	C₄	#22548701	1	1	yes	characteristic
123	C₄	SD₁₆×D₂₅₁	1	1	yes	characteristic
124	C₂	#30632032	1	1	yes	characteristic
125	C₂	#30632126	1	1	yes	characteristic
126	C₂	#30632129	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30632088

C₂

yes

maximal, characteristic

#30632065

C₂

yes

maximal, characteristic

#30631492

C₂

yes

maximal, characteristic

#30631491

C₂

yes

maximal, characteristic

#30631490

C₂

yes

maximal, characteristic

#30631462

C₂

yes

maximal, characteristic

#358×C₂₅₁

C₂

yes

maximal, characteristic

C₂×Dic₁₀₀₄

C₂²

yes

—

—

—

C₂×C₂₅₁⋊C₁₆

C₂²

yes

characteristic

#22548649

C₂²

yes

characteristic

#22548646

—

C₈×Dic₂₅₁

C₂²

yes

characteristic

C₂×Q₁₆×C₂₅₁

C₂²

yes

characteristic

C₂×C₁₆×C₂₅₁

C₂²

yes

characteristic

#107×C₂₅₁

C₂²

yes

characteristic

C₂×Dic₅₀₂

—

Dic₁₀₀₄

—

Dic₁₀₀₄

—

#16860800

—

C₂×C₂₅₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₂₅₁

—

C₄⋊Dic₂₅₁

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

Dic₂₅₁⋊C₄

—

C₄×Dic₂₅₁

D₄

yes

characteristic

C₂₅₁⋊C₁₆

—

C₂×Q₈×C₂₅₁

—

Q₁₆×C₂₅₁

—

Q₁₆×C₂₅₁

—

C₂×C₈×C₂₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₅₁

—

C₁₆×C₂₅₁

—

Dic₅₀₂

—

Dic₅₀₂

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

—

C₂×Dic₂₅₁

D₈

yes

characteristic

C₂×Dic₂₅₁

D₈

yes

characteristic

C₂₅₁⋊C₈

SD₁₆

yes

—

Q₈×C₂₅₁

—

Q₈×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

—

C₂×C₄×C₂₅₁

C₂×D₄

yes

C₈×C₂₅₁

yes

C₈×C₂₅₁

yes

C₂²×C₂₅₁

C₂×D₈

yes

characteristic

C₄×C₂₅₁

—

C₄×C₂₅₁

—

C₄×C₂₅₁

#124

yes

characteristic

C₄×C₂₅₁

C₂×SD₁₆

yes

characteristic

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

Dic₂₅₁

—

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂×C₂₅₁

yes

C₂₅₁

yes

characteristic, complemented, Sylow

#2245

—

251

maximal, Sylow

#499

—

251

—

#494

—

251

—

#358

—

251

—

#358

—

251

—

#358

—

251

—

#358

D₂₅₁

yes

characteristic

#346

—

251

—

C₂×Q₁₆

—

502

—

C₂×Q₁₆

—

251

—

C₂×Q₁₆

C₂×D₂₅₁

yes

characteristic

C₄⋊Q₈

—

502

—

#125

—

502

—

C₂×C₁₆

—

251

—

C₂×C₁₆

C₂×D₂₅₁

yes

characteristic

#107

—

251

—

#107

C₂×D₂₅₁

yes

characteristic

#106

—

502

—

C₄×C₈

—

251

—

C₂×Q₈

—

C₂×Q₈

—

502

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

1004

—

Q₁₆

—

502

—

Q₁₆

—

502

—

C₂×C₈

—

251

—

C₂×C₈

C₂²×D₂₅₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

1004

—

C₄⋊C₄

—

502

—

C₄²

—

251

—

C₁₆

—

100

C₁₆

—

502

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

1004

—

104

Q₈

—

1004

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

1004

—

108

C₂×C₄

—

1004

—

109

C₂×C₄

—

251

—

110

C₂×C₄

—

251

—

111

C₂×C₄

D₄×D₂₅₁

yes

characteristic

112

C₈

—

251

—

113

C₈

#16860809

yes

characteristic

114

C₈

#16860810

yes

characteristic

115

C₂²

D₈×D₂₅₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

2008

—

119

C₄

—

2008

—

120

C₄

—

502

—

121

C₄

—

502

—

122

C₄

#22548701

yes

characteristic

123

C₄

SD₁₆×D₂₅₁

yes

characteristic

124

C₂

#30632032

yes

characteristic

125

C₂

#30632126

yes

characteristic

126

C₂

#30632129

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₅₁
commutator, cosocle	C₂×C₈×C₂₅₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₅₁
Fitting	#358×C₂₅₁	C₂
socle, prime radical	C₂²×C₂₅₁	C₂×D₈
hypercenter	#358	D₂₅₁
hypocenter	C₂₅₁	#2245
LCS	G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₂₅₁

commutator, cosocle

C₂×C₈×C₂₅₁

C₂³

Frattini

C₂×C₈

C₂²×D₂₅₁

Fitting

#358×C₂₅₁

C₂

socle, prime radical

C₂²×C₂₅₁

C₂×D₈

D₂₅₁

C₂₅₁

G ▷ C₂×C₈×C₂₅₁ ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
63. C₂₅₁	#2245	semidirect

H ≅ G/N

type

63. C₂₅₁

#2245

semidirect

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

subgroup	index
#30632088	2
#30632065	2
#30631492	2
#30631491	2
#30631490	2
#30631462	2
#358×C₂₅₁	2
#2245	251

subgroup

index

#358×C₂₅₁

251

Subgroups — #44552609

Conjugacy classes of subgroups

Complements

Maximal subgroups