Group #44567034 — subgroups

Subgroups — #44567034

Order 32192.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#30632863	C₂	1	2	yes	maximal, complemented
2	#30632857	C₂	1	2	yes	maximal, complemented
3	#30632857	C₂	1	2	yes	maximal, complemented
4	#30632854	C₂	1	2	yes	maximal, complemented
5	#30632851	C₂	1	1	yes	maximal, characteristic
6	#30632845	C₂	1	1	yes	maximal, characteristic
7	#30632843	C₂	1	2	yes	maximal, complemented
8	#30632841	C₂	1	2	yes	maximal, complemented
9	#123×C₅₀₃	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₁₀₀₆	C₂²	1	2	yes	—
11	C₂×Dic₁₀₀₆	C₂²	1	1	yes	characteristic
12	C₂²×Dic₅₀₃	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₅₀₃	C₂²	1	2	yes	—
14	C₄⋊Dic₅₀₃	C₂²	1	2	yes	—
15	Dic₅₀₃⋊C₄	C₂²	1	2	yes	—
16	Dic₅₀₃⋊C₄	C₂²	1	2	yes	—
17	Dic₅₀₃⋊C₄	C₂²	1	2	yes	—
18	Dic₅₀₃⋊C₄	C₂²	1	2	yes	—
19	Dic₅₀₃⋊C₄	C₂²	1	2	yes	—
20	C₂²⋊Dic₅₀₃	C₂²	1	2	yes	—
21	C₂²⋊Dic₅₀₃	C₂²	1	2	yes	—
22	C₂²⋊Dic₅₀₃	C₂²	1	2	yes	—
23	C₂²⋊Dic₅₀₃	C₂²	1	1	yes	characteristic
24	C₄×Dic₅₀₃	C₂²	1	1	yes	characteristic
25	C₄×Dic₅₀₃	C₂²	1	2	yes	—
26	C₂×D₄×C₅₀₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₅₀₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₅₀₃	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₅₀₃	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₅₀₃	C₂²	1	1	yes	characteristic
31	Dic₁₀₀₆	—	4	2	no	—
32	Dic₁₀₀₆	—	4	1	no	—
33	C₂×Dic₅₀₃	—	4	1	no	—
34	C₂×Dic₅₀₃	C₂³	1	2	yes	—
35	C₂×Dic₅₀₃	C₂³	1	2	yes	—
36	C₂×Dic₅₀₃	C₂³	1	2	yes	—
37	C₂×Dic₅₀₃	C₂³	1	2	yes	—
38	C₂³×C₅₀₃	C₂³	1	1	yes	characteristic
39	C₂³×C₅₀₃	C₂³	1	1	yes	characteristic
40	D₄×C₅₀₃	—	4	1	no	—
41	C₂×C₄×C₅₀₃	—	4	1	no	—
42	C₂×C₄×C₅₀₃	C₂³	1	2	yes	—
43	C₂×C₄×C₅₀₃	C₂³	1	2	yes	—
44	C₂×C₄×C₅₀₃	C₂³	1	1	yes	characteristic
45	C₂²×C₅₀₃	—	2	1	no	—
46	C₂²×C₅₀₃	—	2	1	no	—
47	C₂²×C₅₀₃	—	2	2	no	—
48	C₂²×C₅₀₃	—	2	2	no	—
49	C₂²×C₅₀₃	C₂⁴	1	1	yes	characteristic
50	C₄×C₅₀₃	—	2	2	no	—
51	C₄×C₅₀₃	—	2	1	no	—
52	C₄×C₅₀₃	—	2	2	no	—
53	Dic₅₀₃	—	2	2	no	—
54	Dic₅₀₃	—	2	2	no	—
55	Dic₅₀₃	—	2	2	no	—
56	Dic₅₀₃	—	2	2	no	—
57	C₂×C₅₀₃	—	4	1	no	—
58	C₂×C₅₀₃	—	4	1	no	—
59	C₂×C₅₀₃	H(5,𝔽₂)	1	1	yes	characteristic
60	C₂×C₅₀₃	2₋¹⁺⁴	1	2	yes	—
61	C₅₀₃	#563	1	1	yes	characteristic, complemented, Sylow
62	#563	—	503	1	no	maximal, Sylow
63	C₄⋊Q₈	—	503	2	no	—
64	#126	—	503	2	no	—
65	#126	—	503	2	no	—
66	#125	—	503	2	no	—
67	#124	—	503	1	no	—
68	#123	—	503	1	no	—
69	#123	D₅₀₃	1	1	yes	characteristic
70	C₂²⋊Q₈	—	503	2	no	—
71	C₂²⋊Q₈	—	503	2	no	—
72	C₂×Q₈	—	503	2	no	—
73	C₂×Q₈	—	503	1	no	—
74	C₂×D₄	C₂×D₅₀₃	1	1	yes	characteristic
75	C₂²×C₄	—	503	1	no	—
76	C₂²×C₄	C₂×D₅₀₃	1	1	yes	characteristic
77	C₄⋊C₄	—	503	2	no	—
78	C₄⋊C₄	—	503	2	no	—
79	C₄⋊C₄	—	503	2	no	—
80	C₄⋊C₄	—	503	2	no	—
81	C₄⋊C₄	—	503	2	no	—
82	C₄⋊C₄	—	503	2	no	—
83	C₄⋊C₄	—	503	2	no	—
84	C₄⋊C₄	C₂×D₅₀₃	1	2	yes	—
85	C₂²⋊C₄	—	503	2	no	—
86	C₂²⋊C₄	—	503	2	no	—
87	C₂²⋊C₄	—	503	2	no	—
88	C₂²⋊C₄	—	503	1	no	—
89	C₂²⋊C₄	C₂×D₅₀₃	1	2	yes	—
90	C₂²⋊C₄	C₂×D₅₀₃	1	1	yes	characteristic
91	C₄²	—	503	1	no	—
92	C₄²	—	503	2	no	—
93	C₂³	C₂²×D₅₀₃	1	1	yes	characteristic
94	C₂³	C₂²×D₅₀₃	1	1	yes	characteristic
95	Q₈	—	2012	2	no	—
96	Q₈	—	2012	1	no	—
97	D₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	2012	1	no	—
100	C₂×C₄	—	503	2	no	—
101	C₂×C₄	—	503	2	no	—
102	C₂×C₄	—	503	2	no	—
103	C₂×C₄	—	503	2	no	—
104	C₂×C₄	C₂²×D₅₀₃	1	2	yes	—
105	C₂×C₄	C₂²×D₅₀₃	1	1	yes	characteristic
106	C₂×C₄	C₂²×D₅₀₃	1	2	yes	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	2	no	—
111	C₂²	C₂³×D₅₀₃	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	2	no	—
115	C₄	—	1006	2	no	—
116	C₄	—	1006	2	no	—
117	C₄	—	1006	2	no	—
118	C₄	—	1006	2	no	—
119	C₂	—	4	1	no	—
120	C₂	—	4	1	no	—
121	C₂	#30632909	1	1	yes	characteristic
122	C₂	#30632911	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30632863

C₂

yes

maximal, complemented

#30632857

C₂

yes

maximal, complemented

#30632857

C₂

yes

maximal, complemented

#30632854

C₂

yes

maximal, complemented

#30632851

C₂

yes

maximal, characteristic

#30632845

C₂

yes

maximal, characteristic

#30632843

C₂

yes

maximal, complemented

#30632841

C₂

yes

maximal, complemented

#123×C₅₀₃

C₂

yes

maximal, characteristic

C₂×Dic₁₀₀₆

C₂²

yes

—

C₂×Dic₁₀₀₆

C₂²

yes

characteristic

C₂²×Dic₅₀₃

C₂²

yes

characteristic

C₄⋊Dic₅₀₃

C₂²

yes

—

C₄⋊Dic₅₀₃

C₂²

yes

—

Dic₅₀₃⋊C₄

C₂²

yes

—

Dic₅₀₃⋊C₄

C₂²

yes

—

Dic₅₀₃⋊C₄

C₂²

yes

—

Dic₅₀₃⋊C₄

C₂²

yes

—

Dic₅₀₃⋊C₄

C₂²

yes

—

C₂²⋊Dic₅₀₃

C₂²

yes

—

C₂²⋊Dic₅₀₃

C₂²

yes

—

C₂²⋊Dic₅₀₃

C₂²

yes

—

C₂²⋊Dic₅₀₃

C₂²

yes

characteristic

C₄×Dic₅₀₃

C₂²

yes

characteristic

C₄×Dic₅₀₃

C₂²

yes

—

C₂×D₄×C₅₀₃

C₂²

yes

characteristic

C₂²×C₄×C₅₀₃

C₂²

yes

characteristic

C₄⋊C₄×C₅₀₃

C₂²

yes

—

C₂²⋊C₄×C₅₀₃

C₂²

yes

—

C₂²⋊C₄×C₅₀₃

C₂²

yes

characteristic

Dic₁₀₀₆

—

Dic₁₀₀₆

—

C₂×Dic₅₀₃

—

C₂×Dic₅₀₃

C₂³

yes

—

C₂×Dic₅₀₃

C₂³

yes

—

C₂×Dic₅₀₃

C₂³

yes

—

C₂×Dic₅₀₃

C₂³

yes

—

C₂³×C₅₀₃

C₂³

yes

characteristic

C₂³×C₅₀₃

C₂³

yes

characteristic

D₄×C₅₀₃

—

C₂×C₄×C₅₀₃

—

C₂×C₄×C₅₀₃

C₂³

yes

—

C₂×C₄×C₅₀₃

C₂³

yes

—

C₂×C₄×C₅₀₃

C₂³

yes

characteristic

C₂²×C₅₀₃

—

C₂²×C₅₀₃

—

C₂²×C₅₀₃

—

C₂²×C₅₀₃

—

C₂²×C₅₀₃

C₂⁴

yes

characteristic

C₄×C₅₀₃

—

C₄×C₅₀₃

—

C₄×C₅₀₃

—

Dic₅₀₃

—

Dic₅₀₃

—

Dic₅₀₃

—

Dic₅₀₃

—

C₂×C₅₀₃

—

C₂×C₅₀₃

—

C₂×C₅₀₃

H(5,𝔽₂)

yes

characteristic

C₂×C₅₀₃

2₋¹⁺⁴

yes

—

C₅₀₃

#563

yes

characteristic, complemented, Sylow

#563

—

503

maximal, Sylow

C₄⋊Q₈

—

503

—

#126

—

503

—

#126

—

503

—

#125

—

503

—

#124

—

503

—

#123

—

503

—

#123

D₅₀₃

yes

characteristic

C₂²⋊Q₈

—

503

—

C₂²⋊Q₈

—

503

—

C₂×Q₈

—

503

—

C₂×Q₈

—

503

—

C₂×D₄

C₂×D₅₀₃

yes

characteristic

C₂²×C₄

—

503

—

C₂²×C₄

C₂×D₅₀₃

yes

characteristic

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

—

503

—

C₄⋊C₄

C₂×D₅₀₃

yes

—

C₂²⋊C₄

—

503

—

C₂²⋊C₄

—

503

—

C₂²⋊C₄

—

503

—

C₂²⋊C₄

—

503

—

C₂²⋊C₄

C₂×D₅₀₃

yes

—

C₂²⋊C₄

C₂×D₅₀₃

yes

characteristic

C₄²

—

503

—

C₄²

—

503

—

C₂³

C₂²×D₅₀₃

yes

characteristic

C₂³

C₂²×D₅₀₃

yes

characteristic

Q₈

—

2012

—

Q₈

—

2012

—

D₄

—

C₂×C₄

—

C₂×C₄

—

2012

—

100

C₂×C₄

—

503

—

101

C₂×C₄

—

503

—

102

C₂×C₄

—

503

—

103

C₂×C₄

—

503

—

104

C₂×C₄

C₂²×D₅₀₃

yes

—

105

C₂×C₄

C₂²×D₅₀₃

yes

characteristic

106

C₂×C₄

C₂²×D₅₀₃

yes

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂³×D₅₀₃

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

1006

—

116

C₄

—

1006

—

117

C₄

—

1006

—

118

C₄

—

1006

—

119

C₂

—

120

C₂

—

121

C₂

#30632909

yes

characteristic

122

C₂

#30632911

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₀₃
commutator, socle, cosocle	C₂²×C₅₀₃	C₂⁴
Fitting	#123×C₅₀₃	C₂
prime radical	C₂×D₄×C₅₀₃	C₂²
hypercenter	#123	D₅₀₃
hypocenter	C₅₀₃	#563
LCS	G ▷ C₂²×C₅₀₃ ▷ C₅₀₃
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₀₃

commutator, socle, cosocle

C₂²×C₅₀₃

C₂⁴

Fitting

#123×C₅₀₃

C₂

prime radical

C₂×D₄×C₅₀₃

C₂²

D₅₀₃

C₅₀₃

G ▷ C₂²×C₅₀₃ ▷ C₅₀₃

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
1. #30632863	C₂	semidirect
2. #30632857	C₂	semidirect
3. #30632857	C₂	semidirect
4. #30632854	C₂	semidirect
7. #30632843	C₂	semidirect
8. #30632841	C₂	semidirect
61. C₅₀₃	#563	semidirect

H ≅ G/N

type

1. #30632863

C₂

semidirect

2. #30632857

C₂

semidirect

3. #30632857

C₂

semidirect

4. #30632854

C₂

semidirect

7. #30632843

C₂

semidirect

8. #30632841

C₂

semidirect

61. C₅₀₃

#563

semidirect

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

subgroup	index
#30632863	2
#30632857	2
#30632854	2
#30632851	2
#30632845	2
#30632843	2
#30632841	2
#123×C₅₀₃	2
#563	503

subgroup

index

#123×C₅₀₃

503

Subgroups — #44567034

Conjugacy classes of subgroups

Complements

Maximal subgroups