Group #20283111 — subgroups

Subgroups — #20283111

Order 6784.

648 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 245 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133215	C₂	1	4	yes	maximal
3	#16810492	C₂	1	8	yes	maximal
4	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
7	#13133217	—	2	4	no	—
8	#13133215	—	2	16	no	—
9	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
10	C₂×#11719919	C₂²	1	16	yes	—
11	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
12	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
13	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₅₃	—	2	4	no	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	4	no	—
18	C₂×Q₈×C₅₃	—	2	8	no	—
19	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
22	C₂²×Dic₅₃	C₂³	1	4	yes	—
23	C₂²×Dic₅₃	C₂³	1	4	yes	—
24	C₂×Dic₁₀₆	—	2	8	no	—
25	C₂×Dic₁₀₆	—	2	4	no	—
26	C₄⋊C₄×C₅₃	—	2	8	no	—
27	C₄²×C₅₃	D₄	1	4	yes	—
28	C₄⋊Dic₅₃	—	2	8	no	—
29	#11719919	—	2	32	no	—
30	C₄×Dic₅₃	D₄	1	8	yes	—
31	C₄×Dic₅₃	Q₈	1	16	yes	complemented
32	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
33	Q₈×C₅₃	—	4	8	no	—
34	C₂×C₄×C₅₃	—	2	8	no	—
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
37	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
39	C₂×C₄×C₅₃	C₂×Q₈	1	8	yes	—
40	C₂×Dic₅₃	—	2	8	no	—
41	C₂×Dic₅₃	—	2	4	no	—
42	C₂×Dic₅₃	C₂×D₄	1	8	yes	—
43	C₂×Dic₅₃	C₂×Q₈	1	16	yes	—
44	Dic₁₀₆	—	4	8	no	—
45	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
46	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
47	C₂²×C₅₃	C₂²×Q₈	1	4	yes	—
48	C₄×C₅₃	—	4	4	no	—
49	C₄×C₅₃	#127	1	4	yes	—
50	C₄×C₅₃	C₄⋊Q₈	1	8	yes	complemented
51	Dic₅₃	—	4	4	no	—
52	Dic₅₃	C₄⋊Q₈	1	16	yes	complemented
53	#2868	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
55	C₂×C₅₃	C₂×C₄⋊Q₈	1	4	yes	—
56	C₂×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
57	C₂×C₄⋊Q₈	—	53	4	no	—
58	C₂×C₄⋊Q₈	—	53	1	no	—
59	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
60	#384	—	53	8	no	—
61	C₄³	—	53	1	no	—
62	C₅₃	#2868	1	1	yes	characteristic, complemented, Sylow
63	C₂²×Q₈	—	53	2	no	—
64	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
65	C₄⋊Q₈	—	2	4	no	—
66	C₄⋊Q₈	—	106	16	no	—
67	C₄⋊Q₈	—	106	4	no	—
68	C₂×C₄⋊C₄	—	53	16	no	—
69	C₂×C₄⋊C₄	—	53	4	no	—
70	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
71	C₂×C₄²	—	53	2	no	—
72	C₂×C₄²	—	53	4	no	—
73	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
74	C₂×Q₈	—	2	8	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	106	8	no	—
77	C₂×Q₈	—	106	4	no	—
78	C₂²×C₄	—	53	4	no	—
79	C₂²×C₄	—	53	4	no	—
80	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
81	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
83	C₄⋊C₄	—	2	8	no	—
84	C₄⋊C₄	—	106	32	no	—
85	C₄⋊C₄	—	106	8	no	—
86	C₄²	—	53	8	no	—
87	C₄²	—	53	16	no	—
88	C₄²	C₂²⋊D₅₃	1	4	yes	—
89	C₂³	C₂³×D₅₃	1	1	yes	characteristic
90	Q₈	—	4	8	no	—
91	Q₈	—	212	8	no	—
92	C₂×C₄	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	53	8	no	—
95	C₂×C₄	—	53	16	no	—
96	C₂×C₄	—	106	8	no	—
97	C₂×C₄	—	106	4	no	—
98	C₂×C₄	D₄×D₅₃	1	4	yes	—
99	C₂×C₄	Q₈×D₅₃	1	8	yes	—
100	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
102	C₂²	C₂×D₄×D₅₃	1	2	yes	—
103	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂²	C₂×Q₈×D₅₃	1	4	yes	—
105	C₄	—	4	4	no	—
106	C₄	—	53	16	no	—
107	C₄	—	212	4	no	—
108	C₄	#13133213	1	4	yes	—
109	C₄	#13133215	1	8	yes	complemented
110	C₂	C₂×#13133213	1	1	yes	characteristic
111	C₂	C₄⋊Q₈×D₅₃	1	4	yes	—
112	C₂	C₂×#13133215	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

C₂×#13133215

C₂

yes

maximal

#16810492

C₂

yes

maximal

C₄²×Dic₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

#13133217

—

#13133215

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

Q₈

yes

complemented

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×Q₈

yes

—

Dic₁₀₆

—

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

#127

yes

—

C₄×C₅₃

C₄⋊Q₈

yes

complemented

Dic₅₃

—

Dic₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄⋊Q₈

yes

—

C₂×C₅₃

C₂×C₄⋊Q₈

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

#384

—

C₄³

—

C₅₃

#2868

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

D₄×D₅₃

yes

—

C₂×C₄

Q₈×D₅₃

yes

—

100

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

101

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

102

C₂²

C₂×D₄×D₅₃

yes

—

103

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

104

C₂²

C₂×Q₈×D₅₃

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

212

—

108

C₄

#13133213

yes

—

109

C₄

#13133215

yes

complemented

110

C₂

C₂×#13133213

yes

characteristic

111

C₂

C₄⋊Q₈×D₅₃

yes

—

112

C₂

C₂×#13133215

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#2868
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

hypercenter

C₂×C₄⋊Q₈

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
31. C₄×Dic₅₃	Q₈	semidirect
50. C₄×C₅₃	C₄⋊Q₈	semidirect
52. Dic₅₃	C₄⋊Q₈	semidirect
62. C₅₃	#2868	semidirect
109. C₄	#13133215	semidirect

H ≅ G/N

type

31. C₄×Dic₅₃

Q₈

semidirect

50. C₄×C₅₃

C₄⋊Q₈

semidirect

52. Dic₅₃

C₄⋊Q₈

semidirect

62. C₅₃

#2868

semidirect

109. C₄

#13133215

semidirect

Subgroups — #20283111

Conjugacy classes of subgroups

Complements