Group #20283078 — subgroups

Subgroups — #20283078

Order 6784.

472 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 213 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	C₂×#13133209	C₂	1	2	yes	maximal
2	#16810521	C₂	1	8	yes	maximal
3	#16810493	C₂	1	2	yes	maximal
4	#16810489	C₂	1	2	yes	maximal
5	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#13133209	—	2	8	no	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
9	C₂×#11719919	C₂²	1	8	yes	—
10	#13133124	C₂²	1	8	yes	—
11	C₂×C₄²×C₅₃	C₂²	1	4	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
19	C₂²×Dic₅₃	C₂³	1	8	yes	—
20	C₄²×C₅₃	—	2	8	no	—
21	C₄²×C₅₃	D₄	1	4	yes	—
22	C₄²×C₅₃	Q₈	1	4	yes	—
23	C₄²×C₅₃	Q₈	1	4	yes	—
24	C₄⋊Dic₅₃	—	2	8	no	—
25	C₄⋊Dic₅₃	—	2	16	no	—
26	#11719919	—	2	16	no	—
27	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	4	no	—
29	C₂×C₄×C₅₃	—	2	4	no	—
30	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
31	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
33	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₅₃	#41	1	8	yes	—
37	C₂×Dic₅₃	—	2	8	no	—
38	C₂×Dic₅₃	—	2	16	no	—
39	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
40	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
41	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
42	C₂²×C₅₃	C₂×#41	1	4	yes	—
43	C₄×C₅₃	—	2	4	no	—
44	C₄×C₅₃	—	2	4	no	—
45	C₄×C₅₃	#125	1	8	yes	complemented
46	C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
47	Dic₅₃	—	4	8	no	—
48	#2859	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	C₂×#125	1	2	yes	—
50	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₅₃	#532	1	2	yes	—
52	C₂×C₅₃	#533	1	2	yes	—
53	C₂×#125	—	53	2	no	—
54	#389	—	53	8	no	—
55	#384	—	53	2	no	—
56	#382	—	53	2	no	—
57	C₄³	D₅₃	1	1	yes	characteristic
58	C₅₃	#2859	1	1	yes	characteristic, complemented, Sylow
59	#125	—	106	8	no	—
60	C₂×C₄⋊C₄	—	53	8	no	—
61	C₂×C₄⋊C₄	—	53	4	no	—
62	C₂×C₄⋊C₄	—	53	8	no	—
63	C₂×C₄²	C₂×D₅₃	1	4	yes	—
64	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
67	#95	—	53	8	no	—
68	C₂²×C₄	—	53	8	no	—
69	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
70	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
71	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
73	C₄⋊C₄	—	106	16	no	—
74	C₄⋊C₄	—	106	8	no	—
75	C₄⋊C₄	—	106	16	no	—
76	C₄²	—	2	8	no	—
77	C₄²	Dic₁₀₆	1	4	yes	—
78	C₄²	Dic₁₀₆	1	4	yes	—
79	C₄²	D₂₁₂	1	4	yes	—
80	C₂³	C₂³×D₅₃	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	106	8	no	—
84	C₂×C₄	—	106	16	no	—
85	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
86	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
87	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
88	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
89	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
90	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
91	C₂×C₄	#11719945	1	8	yes	—
92	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
93	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
94	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
95	C₂²	C₂×#11719945	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	2	4	no	—
98	C₄	—	212	8	no	—
99	C₄	#13133209	1	8	yes	complemented
100	C₄	#13133217	1	4	yes	—
101	C₂	C₂×#13133209	1	2	yes	—
102	C₂	C₂×#13133217	1	1	yes	characteristic
103	C₂	#16811173	1	2	yes	—
104	C₂	#16811180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133209

C₂

yes

maximal

#16810521

C₂

yes

maximal

#16810493

C₂

yes

maximal

#16810489

C₂

yes

maximal

C₄³×C₅₃

C₂

yes

maximal, characteristic

#13133209

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

#13133124

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#125

yes

complemented

C₄×C₅₃

C₄⋊Q₈

yes

—

Dic₅₃

—

#2859

—

maximal, Sylow

C₂×C₅₃

C₂×#125

yes

—

C₂×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₃

#532

yes

—

C₂×C₅₃

#533

yes

—

—

—

—

—

C₄³

D₅₃

yes

characteristic

C₅₃

#2859

yes

characteristic, complemented, Sylow

#125

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

C₄²

Dic₁₀₆

yes

—

C₄²

Dic₁₀₆

yes

—

C₄²

D₂₁₂

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

—

C₂×C₄

C₂×D₂₁₂

yes

—

C₂×C₄

C₂×D₂₁₂

yes

—

C₂×C₄

#11719945

yes

—

C₂²

C₂²×D₂₁₂

yes

characteristic

C₂²

C₂²×Dic₁₀₆

yes

characteristic

C₂²

C₂²×Dic₁₀₆

yes

characteristic

C₂²

C₂×#11719945

yes

—

C₄

—

C₄

—

C₄

—

212

—

C₄

#13133209

yes

complemented

100

C₄

#13133217

yes

—

101

C₂

C₂×#13133209

yes

—

102

C₂

C₂×#13133217

yes

characteristic

103

C₂

#16811173

yes

—

104

C₂

#16811180

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₄³×C₅₃	C₂
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	#2859
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₄³×C₅₃

C₂

hypercenter

C₄³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
45. C₄×C₅₃	#125	semidirect
58. C₅₃	#2859	semidirect
99. C₄	#13133209	semidirect

H ≅ G/N

type

45. C₄×C₅₃

#125

semidirect

58. C₅₃

#2859

semidirect

99. C₄

#13133209

semidirect

Subgroups — #20283078

Conjugacy classes of subgroups

Complements