Group #20283265 — subgroups

Subgroups — C₂²×#13133146 #20283265

Order 6784.

640 conjugacy classes of subgroups; only the 389 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 389 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	C₂³×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133146	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₅₃	C₄	1	1	yes	characteristic
6	C₂³×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂²×#88322	—	2	4	no	—
8	C₂²×#88322	C₂²	1	6	yes	—
9	C₂×C₄×Dic₅₃	C₄	1	6	yes	—
10	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
11	#13133146	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	—	2	4	no	—
14	C₂²×C₈×C₅₃	C₂²	1	6	yes	—
15	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
17	C₂²×C₄×C₅₃	D₄	1	4	yes	—
18	C₂²×C₄×C₅₃	Q₈	1	4	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
20	C₂²×Dic₅₃	—	2	4	no	—
21	C₂²×Dic₅₃	C₈	1	8	yes	complemented
22	C₂²×Dic₅₃	C₂×C₄	1	6	yes	—
23	C₂²×Dic₅₃	C₂×C₄	1	6	yes	—
24	C₂×C₈×C₅₃	—	2	12	no	—
25	C₂×C₈×C₅₃	C₂³	1	4	yes	—
26	C₄×Dic₅₃	C₂×C₄	1	12	yes	—
27	C₄×Dic₅₃	C₂³	1	4	yes	—
28	C₂×#88322	—	2	12	no	—
29	C₂×#88322	C₂³	1	4	yes	—
30	C₂³×C₅₃	C₄⋊C₄	1	4	yes	—
31	C₂³×C₅₃	C₂×C₈	1	4	yes	—
32	C₂³×C₅₃	M₁₆	1	4	yes	—
33	C₂³×C₅₃	C₂²×C₄	1	3	yes	—
34	C₂×C₄×C₅₃	C₄⋊C₄	1	12	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
36	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
37	C₂×C₄×C₅₃	C₂×Q₈	1	6	yes	—
38	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
39	C₂×Dic₅₃	—	2	12	no	—
40	C₂×Dic₅₃	C₂×C₈	1	24	yes	complemented
41	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
42	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
43	C₈×C₅₃	—	2	4	no	—
44	#88322	—	2	4	no	—
45	C₂²×C₅₃	C₄⋊C₈	1	16	yes	complemented
46	C₂²×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
47	C₂²×C₅₃	C₂²×C₈	1	6	yes	—
48	C₂²×C₅₃	C₂×M₁₆	1	6	yes	—
49	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
50	C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
51	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
52	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
53	Dic₅₃	—	2	4	no	—
54	Dic₅₃	C₂²×C₈	1	8	yes	complemented
55	C₂²×C₄⋊C₈	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	C₂×C₄⋊C₈	1	12	yes	complemented
57	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
59	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
60	C₂³×C₈	—	53	1	no	—
61	C₂³×C₈	D₅₃	1	1	yes	characteristic
62	C₂²×C₄²	—	53	1	no	—
63	C₂×C₄⋊C₈	—	53	12	no	—
64	C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
65	C₂³×C₄	—	53	1	no	—
66	C₂³×C₄	—	53	1	no	—
67	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
68	C₂²×C₈	—	2	4	no	—
69	C₂²×C₈	—	53	6	no	—
70	C₂²×C₈	—	106	4	no	—
71	C₂²×C₈	C₂×D₅₃	1	6	yes	—
72	C₂×C₄²	—	53	6	no	—
73	C₂×C₄²	—	53	6	no	—
74	C₄⋊C₈	—	53	16	no	—
75	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
76	C₂²×C₄	—	53	6	no	—
77	C₂²×C₄	—	53	8	no	—
78	C₂²×C₄	—	53	6	no	—
79	C₂²×C₄	—	106	4	no	—
80	C₂²×C₄	Dic₁₀₆	1	4	yes	—
81	C₂²×C₄	C₄×D₅₃	1	3	yes	—
82	C₂²×C₄	C₂²⋊D₅₃	1	4	yes	—
83	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
84	C₂×C₈	—	2	12	no	—
85	C₂×C₈	—	53	4	no	—
86	C₂×C₈	—	106	12	no	—
87	C₂×C₈	C₂²×D₅₃	1	4	yes	—
88	C₄²	—	53	12	no	—
89	C₄²	—	53	4	no	—
90	C₂³	C₈×D₅₃	1	4	yes	—
91	C₂³	#11719912	1	4	yes	—
92	C₂³	#11719919	1	4	yes	—
93	C₂³	C₂×C₄×D₅₃	1	3	yes	—
94	C₂×C₄	—	53	24	no	—
95	C₂×C₄	—	53	4	no	—
96	C₂×C₄	—	53	4	no	—
97	C₂×C₄	—	106	12	no	—
98	C₂×C₄	#11719919	1	12	yes	—
99	C₂×C₄	C₂×Dic₁₀₆	1	6	yes	—
100	C₂×C₄	C₂×C₄×D₅₃	1	3	yes	—
101	C₂×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
102	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₈	—	106	4	no	—
105	C₂²	#13133146	1	16	yes	direct product factor
106	C₂²	C₂×#11719919	1	6	yes	—
107	C₂²	C₂×C₈×D₅₃	1	6	yes	—
108	C₂²	C₂×#11719912	1	6	yes	—
109	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
110	C₄	—	53	8	no	—
111	C₄	—	106	4	no	—
112	C₄	C₂×#11719919	1	6	yes	—
113	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
114	C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
115	C₂	C₂×#13133146	1	12	yes	direct product factor
116	C₂	C₂²×#11719919	1	1	yes	characteristic
117	C₂	C₂²×C₈×D₅₃	1	1	yes	characteristic
118	C₂	C₂²×#11719912	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#88322

C₂

yes

maximal, characteristic

C₂²×C₄×Dic₅₃

C₂

yes

maximal, characteristic

C₂×#13133146

C₂

yes

maximal, direct product factor

C₂³×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

C₄

yes

characteristic

C₂³×Dic₅₃

C₄

yes

characteristic

C₂²×#88322

—

C₂²×#88322

C₂²

yes

—

C₂×C₄×Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133146

C₂²

yes

direct product factor

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₈

yes

complemented

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂×C₈

yes

—

C₂³×C₅₃

M₁₆

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

complemented

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₄⋊C₈

yes

complemented

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂²×C₈

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

C₂²×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

C₂²×C₈

yes

complemented

C₂²×C₄⋊C₈

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊C₈

yes

complemented

C₂×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₅₃

C₂³×C₈

yes

characteristic

C₂×C₅₃

C₂²×M₁₆

yes

characteristic

C₂³×C₈

—

C₂³×C₈

D₅₃

yes

characteristic

C₂²×C₄²

—

C₂×C₄⋊C₈

—

C₅₃

C₂²×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂⁴

C₄×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

Dic₁₀₆

yes

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂²×D₅₃

yes

—

C₄²

—

C₄²

—

C₂³

C₈×D₅₃

yes

—

C₂³

#11719912

yes

—

C₂³

#11719919

yes

—

C₂³

C₂×C₄×D₅₃

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

#11719919

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

100

C₂×C₄

C₂×C₄×D₅₃

yes

—

101

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

102

C₂×C₄

C₂³×D₅₃

yes

characteristic

103

C₈

—

104

C₈

—

106

—

105

C₂²

#13133146

yes

direct product factor

106

C₂²

C₂×#11719919

yes

—

107

C₂²

C₂×C₈×D₅₃

yes

—

108

C₂²

C₂×#11719912

yes

—

109

C₂²

C₂²×C₄×D₅₃

yes

characteristic

110

C₄

—

111

C₄

—

106

—

112

C₄

C₂×#11719919

yes

—

113

C₄

C₂²×C₂²⋊D₅₃

yes

characteristic

114

C₄

C₂²×Dic₁₀₆

yes

characteristic

115

C₂

C₂×#13133146

yes

direct product factor

116

C₂

C₂²×#11719919

yes

characteristic

117

C₂

C₂²×C₈×D₅₃

yes

characteristic

118

C₂

C₂²×#11719912

yes

characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂³×C₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂×C₄
hypercenter	C₂³×C₈	D₅₃
hypocenter	C₅₃	C₂²×C₄⋊C₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂³×C₄

C₂×D₅₃

commutator

C₂×C₅₃

C₂³×C₈

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂³×C₈×C₅₃

C₂

socle

C₂⁴×C₅₃

C₂×C₄

hypercenter

C₂³×C₈

D₅₃

hypocenter

C₅₃

C₂²×C₄⋊C₈

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
3. C₂×#13133146	C₂	direct (N × H)
11. #13133146	C₂²	direct (N × H)
21. C₂²×Dic₅₃	C₈	semidirect
40. C₂×Dic₅₃	C₂×C₈	semidirect
45. C₂²×C₅₃	C₄⋊C₈	semidirect
54. Dic₅₃	C₂²×C₈	semidirect
56. C₂×C₅₃	C₂×C₄⋊C₈	semidirect
64. C₅₃	C₂²×C₄⋊C₈	semidirect
105. C₂²	#13133146	direct (N × H)
115. C₂	C₂×#13133146	direct (N × H)

H ≅ G/N

type

3. C₂×#13133146

C₂

direct (N × H)

11. #13133146

C₂²

direct (N × H)

21. C₂²×Dic₅₃

C₈

semidirect

40. C₂×Dic₅₃

C₂×C₈

semidirect

45. C₂²×C₅₃

C₄⋊C₈

semidirect

54. Dic₅₃

C₂²×C₈

semidirect

56. C₂×C₅₃

C₂×C₄⋊C₈

semidirect

64. C₅₃

C₂²×C₄⋊C₈

semidirect

105. C₂²

#13133146

direct (N × H)

115. C₂

C₂×#13133146

direct (N × H)

Subgroups — C₂²×#13133146 #20283265

Conjugacy classes of subgroups

Complements

Subgroups — C22×#13133146 #20283265

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#13133146 #20283265