Group #45688185 — subgroups

Subgroups — #45688185

Order 33344.

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	#30659707	C₂	1	2	yes	maximal, complemented
2	#30659701	C₂	1	2	yes	maximal, complemented
3	#30659701	C₂	1	2	yes	maximal, complemented
4	#30659698	C₂	1	2	yes	maximal, complemented
5	#30659695	C₂	1	1	yes	maximal, characteristic
6	#30659689	C₂	1	1	yes	maximal, characteristic
7	#30659687	C₂	1	2	yes	maximal, complemented
8	#30659685	C₂	1	2	yes	maximal, complemented
9	#123×C₅₂₁	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₁₀₄₂	C₂²	1	2	yes	—
11	C₂×Dic₁₀₄₂	C₂²	1	1	yes	characteristic
12	C₂²×Dic₅₂₁	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₅₂₁	C₂²	1	2	yes	—
14	C₄⋊Dic₅₂₁	C₂²	1	2	yes	—
15	#22552343	C₂²	1	2	yes	—
16	#22552343	C₂²	1	2	yes	—
17	#22552343	C₂²	1	2	yes	—
18	#22552343	C₂²	1	2	yes	—
19	#22552343	C₂²	1	2	yes	—
20	C₂²⋊Dic₅₂₁	C₂²	1	2	yes	—
21	C₂²⋊Dic₅₂₁	C₂²	1	2	yes	—
22	C₂²⋊Dic₅₂₁	C₂²	1	2	yes	—
23	C₂²⋊Dic₅₂₁	C₂²	1	1	yes	characteristic
24	C₄×Dic₅₂₁	C₂²	1	1	yes	characteristic
25	C₄×Dic₅₂₁	C₂²	1	2	yes	—
26	C₂×D₄×C₅₂₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₅₂₁	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₅₂₁	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₅₂₁	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₅₂₁	C₂²	1	1	yes	characteristic
31	Dic₁₀₄₂	—	4	2	no	—
32	Dic₁₀₄₂	—	4	1	no	—
33	C₂×Dic₅₂₁	—	4	1	no	—
34	C₂×Dic₅₂₁	C₂³	1	2	yes	—
35	C₂×Dic₅₂₁	C₂³	1	2	yes	—
36	C₂×Dic₅₂₁	C₂³	1	2	yes	—
37	C₂×Dic₅₂₁	C₂³	1	2	yes	—
38	C₂³×C₅₂₁	C₂³	1	1	yes	characteristic
39	C₂³×C₅₂₁	C₂³	1	1	yes	characteristic
40	D₄×C₅₂₁	—	4	1	no	—
41	C₂×C₄×C₅₂₁	—	4	1	no	—
42	C₂×C₄×C₅₂₁	C₂³	1	2	yes	—
43	C₂×C₄×C₅₂₁	C₂³	1	2	yes	—
44	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
45	C₂²×C₅₂₁	—	2	1	no	—
46	C₂²×C₅₂₁	—	2	1	no	—
47	C₂²×C₅₂₁	—	2	2	no	—
48	C₂²×C₅₂₁	—	2	2	no	—
49	C₂²×C₅₂₁	C₂⁴	1	1	yes	characteristic
50	C₄×C₅₂₁	—	2	2	no	—
51	C₄×C₅₂₁	—	2	1	no	—
52	C₄×C₅₂₁	—	2	2	no	—
53	Dic₅₂₁	—	2	2	no	—
54	Dic₅₂₁	—	2	2	no	—
55	Dic₅₂₁	—	2	2	no	—
56	Dic₅₂₁	—	2	2	no	—
57	C₂×C₅₂₁	—	4	1	no	—
58	C₂×C₅₂₁	—	4	1	no	—
59	C₂×C₅₂₁	H(5,𝔽₂)	1	1	yes	characteristic
60	C₂×C₅₂₁	2₋¹⁺⁴	1	2	yes	—
61	C₅₂₁	#563	1	1	yes	characteristic, complemented, Sylow
62	#563	—	521	1	no	maximal, Sylow
63	C₄⋊Q₈	—	521	2	no	—
64	#126	—	521	2	no	—
65	#126	—	521	2	no	—
66	#125	—	521	2	no	—
67	#124	—	521	1	no	—
68	#123	—	521	1	no	—
69	#123	D₅₂₁	1	1	yes	characteristic
70	C₂²⋊Q₈	—	521	2	no	—
71	C₂²⋊Q₈	—	521	2	no	—
72	C₂×Q₈	—	521	2	no	—
73	C₂×Q₈	—	521	1	no	—
74	C₂×D₄	C₂×D₅₂₁	1	1	yes	characteristic
75	C₂²×C₄	—	521	1	no	—
76	C₂²×C₄	C₂×D₅₂₁	1	1	yes	characteristic
77	C₄⋊C₄	—	521	2	no	—
78	C₄⋊C₄	—	521	2	no	—
79	C₄⋊C₄	—	521	2	no	—
80	C₄⋊C₄	—	521	2	no	—
81	C₄⋊C₄	—	521	2	no	—
82	C₄⋊C₄	—	521	2	no	—
83	C₄⋊C₄	—	521	2	no	—
84	C₄⋊C₄	C₂×D₅₂₁	1	2	yes	—
85	C₂²⋊C₄	—	521	2	no	—
86	C₂²⋊C₄	—	521	2	no	—
87	C₂²⋊C₄	—	521	2	no	—
88	C₂²⋊C₄	—	521	1	no	—
89	C₂²⋊C₄	C₂×D₅₂₁	1	2	yes	—
90	C₂²⋊C₄	C₂×D₅₂₁	1	1	yes	characteristic
91	C₄²	—	521	1	no	—
92	C₄²	—	521	2	no	—
93	C₂³	C₂²×D₅₂₁	1	1	yes	characteristic
94	C₂³	C₂²×D₅₂₁	1	1	yes	characteristic
95	Q₈	—	2084	2	no	—
96	Q₈	—	2084	1	no	—
97	D₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	521	2	no	—
100	C₂×C₄	—	521	2	no	—
101	C₂×C₄	—	521	2	no	—
102	C₂×C₄	—	521	2	no	—
103	C₂×C₄	—	2084	1	no	—
104	C₂×C₄	C₂²×D₅₂₁	1	2	yes	—
105	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
106	C₂×C₄	C₂²×D₅₂₁	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×D₅₂₁	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	2	no	—
115	C₄	—	1042	2	no	—
116	C₄	—	1042	2	no	—
117	C₄	—	1042	2	no	—
118	C₄	—	1042	2	no	—
119	C₂	—	4	1	no	—
120	C₂	—	4	1	no	—
121	C₂	#30659753	1	1	yes	characteristic
122	C₂	#30659755	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30659707

C₂

yes

maximal, complemented

#30659701

C₂

yes

maximal, complemented

#30659701

C₂

yes

maximal, complemented

#30659698

C₂

yes

maximal, complemented

#30659695

C₂

yes

maximal, characteristic

#30659689

C₂

yes

maximal, characteristic

#30659687

C₂

yes

maximal, complemented

#30659685

C₂

yes

maximal, complemented

#123×C₅₂₁

C₂

yes

maximal, characteristic

C₂×Dic₁₀₄₂

C₂²

yes

—

C₂×Dic₁₀₄₂

C₂²

yes

characteristic

C₂²×Dic₅₂₁

C₂²

yes

characteristic

C₄⋊Dic₅₂₁

C₂²

yes

—

C₄⋊Dic₅₂₁

C₂²

yes

—

#22552343

C₂²

yes

—

#22552343

C₂²

yes

—

#22552343

C₂²

yes

—

#22552343

C₂²

yes

—

#22552343

C₂²

yes

—

C₂²⋊Dic₅₂₁

C₂²

yes

—

C₂²⋊Dic₅₂₁

C₂²

yes

—

C₂²⋊Dic₅₂₁

C₂²

yes

—

C₂²⋊Dic₅₂₁

C₂²

yes

characteristic

C₄×Dic₅₂₁

C₂²

yes

characteristic

C₄×Dic₅₂₁

C₂²

yes

—

C₂×D₄×C₅₂₁

C₂²

yes

characteristic

C₂²×C₄×C₅₂₁

C₂²

yes

characteristic

C₄⋊C₄×C₅₂₁

C₂²

yes

—

C₂²⋊C₄×C₅₂₁

C₂²

yes

—

C₂²⋊C₄×C₅₂₁

C₂²

yes

characteristic

Dic₁₀₄₂

—

Dic₁₀₄₂

—

C₂×Dic₅₂₁

—

C₂×Dic₅₂₁

C₂³

yes

—

C₂×Dic₅₂₁

C₂³

yes

—

C₂×Dic₅₂₁

C₂³

yes

—

C₂×Dic₅₂₁

C₂³

yes

—

C₂³×C₅₂₁

C₂³

yes

characteristic

C₂³×C₅₂₁

C₂³

yes

characteristic

D₄×C₅₂₁

—

C₂×C₄×C₅₂₁

—

C₂×C₄×C₅₂₁

C₂³

yes

—

C₂×C₄×C₅₂₁

C₂³

yes

—

C₂×C₄×C₅₂₁

C₂³

yes

characteristic

C₂²×C₅₂₁

—

C₂²×C₅₂₁

—

C₂²×C₅₂₁

—

C₂²×C₅₂₁

—

C₂²×C₅₂₁

C₂⁴

yes

characteristic

C₄×C₅₂₁

—

C₄×C₅₂₁

—

C₄×C₅₂₁

—

Dic₅₂₁

—

Dic₅₂₁

—

Dic₅₂₁

—

Dic₅₂₁

—

C₂×C₅₂₁

—

C₂×C₅₂₁

—

C₂×C₅₂₁

H(5,𝔽₂)

yes

characteristic

C₂×C₅₂₁

2₋¹⁺⁴

yes

—

C₅₂₁

#563

yes

characteristic, complemented, Sylow

#563

—

521

maximal, Sylow

C₄⋊Q₈

—

521

—

#126

—

521

—

#126

—

521

—

#125

—

521

—

#124

—

521

—

#123

—

521

—

#123

D₅₂₁

yes

characteristic

C₂²⋊Q₈

—

521

—

C₂²⋊Q₈

—

521

—

C₂×Q₈

—

521

—

C₂×Q₈

—

521

—

C₂×D₄

C₂×D₅₂₁

yes

characteristic

C₂²×C₄

—

521

—

C₂²×C₄

C₂×D₅₂₁

yes

characteristic

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

C₂×D₅₂₁

yes

—

C₂²⋊C₄

—

521

—

C₂²⋊C₄

—

521

—

C₂²⋊C₄

—

521

—

C₂²⋊C₄

—

521

—

C₂²⋊C₄

C₂×D₅₂₁

yes

—

C₂²⋊C₄

C₂×D₅₂₁

yes

characteristic

C₄²

—

521

—

C₄²

—

521

—

C₂³

C₂²×D₅₂₁

yes

characteristic

C₂³

C₂²×D₅₂₁

yes

characteristic

Q₈

—

2084

—

Q₈

—

2084

—

D₄

—

C₂×C₄

—

C₂×C₄

—

521

—

100

C₂×C₄

—

521

—

101

C₂×C₄

—

521

—

102

C₂×C₄

—

521

—

103

C₂×C₄

—

2084

—

104

C₂×C₄

C₂²×D₅₂₁

yes

—

105

C₂×C₄

C₂²×D₅₂₁

yes

characteristic

106

C₂×C₄

C₂²×D₅₂₁

yes

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂³×D₅₂₁

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

1042

—

116

C₄

—

1042

—

117

C₄

—

1042

—

118

C₄

—

1042

—

119

C₂

—

120

C₂

—

121

C₂

#30659753

yes

characteristic

122

C₂

#30659755

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₂₁
commutator, socle, cosocle	C₂²×C₅₂₁	C₂⁴
Fitting	#123×C₅₂₁	C₂
prime radical	C₂×D₄×C₅₂₁	C₂²
hypercenter	#123	D₅₂₁
hypocenter	C₅₂₁	#563
LCS	G ▷ C₂²×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₂₁

commutator, socle, cosocle

C₂²×C₅₂₁

C₂⁴

Fitting

#123×C₅₂₁

C₂

prime radical

C₂×D₄×C₅₂₁

C₂²

D₅₂₁

C₅₂₁

G ▷ C₂²×C₅₂₁ ▷ C₅₂₁

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
1. #30659707	C₂	semidirect
2. #30659701	C₂	semidirect
3. #30659701	C₂	semidirect
4. #30659698	C₂	semidirect
7. #30659687	C₂	semidirect
8. #30659685	C₂	semidirect
61. C₅₂₁	#563	semidirect

H ≅ G/N

type

1. #30659707

C₂

semidirect

2. #30659701

C₂

semidirect

3. #30659701

C₂

semidirect

4. #30659698

C₂

semidirect

7. #30659687

C₂

semidirect

8. #30659685

C₂

semidirect

61. C₅₂₁

#563

semidirect

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

subgroup	index
#30659707	2
#30659701	2
#30659698	2
#30659695	2
#30659689	2
#30659687	2
#30659685	2
#123×C₅₂₁	2
#563	521

subgroup

index

#123×C₅₂₁

521

Subgroups — #45688185

Conjugacy classes of subgroups

Complements

Maximal subgroups