Group #20283817 — subgroups

Subgroups — #20283817

Order 6784.

330 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 205 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	#16810754	C₂	1	4	yes	maximal, complemented
2	Q₈×#88322	C₂	1	4	yes	maximal, complemented
3	#16810684	C₂	1	4	yes	maximal
4	#16810676	C₂	1	2	yes	maximal
5	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#88322	C₂²	1	2	yes	—
7	C₄⋊#88322	C₂²	1	8	yes	—
8	C₄⋊#88322	C₂²	1	2	yes	—
9	C₂²⋊#88322	C₂²	1	4	yes	—
10	#13133128	C₂²	1	4	yes	—
11	C₄×#88322	C₂²	1	4	yes	—
12	C₄×#88322	C₂²	1	4	yes	—
13	C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
14	#125×C₅₃	C₄	1	2	yes	—
15	C₂²⋊Q₈×C₅₃	C₄	1	4	yes	—
16	C₄×Q₈×C₅₃	C₂²	1	4	yes	—
17	#117×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
23	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
24	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₂³	1	2	yes	—
29	C₄²×C₅₃	C₂³	1	4	yes	—
30	C₂×#88322	C₂³	1	4	yes	—
31	C₂×#88322	C₂³	1	4	yes	—
32	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	Q₈×C₅₃	—	2	8	no	—
34	C₂×C₄×C₅₃	—	2	4	no	—
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
41	#88322	—	2	4	no	—
42	#88322	—	2	4	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	—	2	2	no	—
45	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	2	2	no	—
48	C₄×C₅₃	—	2	8	no	—
49	C₄×C₅₃	#131	1	4	yes	—
50	C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
51	#2986	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	—	4	1	no	—
53	C₂×C₅₃	#519	1	1	yes	characteristic
54	C₂×C₅₃	C₂×#131	1	2	yes	—
55	#533	D₅₃	1	1	yes	characteristic
56	#446	—	53	4	no	—
57	C₈×Q₈	—	53	4	no	—
58	#433	—	53	4	no	—
59	#431	—	53	2	no	—
60	C₅₃	#2986	1	1	yes	characteristic, complemented, Sylow
61	C₄⋊Q₈	Dic₅₃	1	2	yes	—
62	#125	Dic₅₃	1	2	yes	—
63	C₂²⋊Q₈	Dic₅₃	1	4	yes	—
64	C₄×Q₈	C₂×D₅₃	1	4	yes	—
65	#117	C₂×D₅₃	1	2	yes	—
66	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
67	C₄⋊C₈	—	53	2	no	—
68	C₄⋊C₈	—	53	2	no	—
69	C₄⋊C₈	—	53	8	no	—
70	C₂²⋊C₈	—	53	4	no	—
71	#97	—	53	4	no	—
72	C₄×C₈	—	53	4	no	—
73	C₄×C₈	—	53	4	no	—
74	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
75	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
76	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
77	C₂×C₈	—	53	4	no	—
78	C₂×C₈	—	53	4	no	—
79	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
80	C₄⋊C₄	C₂×Dic₅₃	1	8	yes	—
81	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
82	C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
83	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
84	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
85	C₄²	C₂²×D₅₃	1	2	yes	—
86	C₄²	C₂²×D₅₃	1	4	yes	—
87	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
88	Q₈	—	2	8	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
92	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
93	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
94	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
95	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
96	C₈	—	106	4	no	—
97	C₈	—	106	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	2	2	no	—
103	C₄	—	2	8	no	—
104	C₄	#13133224	1	4	yes	—
105	C₄	#13133265	1	2	yes	—
106	C₂	—	4	1	no	—
107	C₂	#16811101	1	1	yes	characteristic
108	C₂	C₂×#13133224	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810754

C₂

yes

maximal, complemented

Q₈×#88322

C₂

yes

maximal, complemented

#16810684

C₂

yes

maximal

#16810676

C₂

yes

maximal

C₅₃×#533

C₂

yes

maximal, characteristic

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

C₂²

yes

—

#13133128

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₄

yes

—

#125×C₅₃

C₄

yes

—

C₂²⋊Q₈×C₅₃

C₄

yes

—

C₄×Q₈×C₅₃

C₂²

yes

—

#117×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

2₋¹⁺⁴

yes

—

#2986

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#519

yes

characteristic

C₂×C₅₃

C₂×#131

yes

—

#533

D₅₃

yes

characteristic

#446

—

C₈×Q₈

—

#433

—

#431

—

C₅₃

#2986

yes

characteristic, complemented, Sylow

C₄⋊Q₈

Dic₅₃

yes

—

#125

Dic₅₃

yes

—

C₂²⋊Q₈

Dic₅₃

yes

—

C₄×Q₈

C₂×D₅₃

yes

—

#117

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²×D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

C₂²×Dic₅₃

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

—

C₂×C₄

C₂³×D₅₃

yes

characteristic

C₈

—

106

—

C₈

—

106

—

C₂²

—

C₂²

—

100

C₂²

C₂³×Dic₅₃

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#13133224

yes

—

105

C₄

#13133265

yes

—

106

C₂

—

107

C₂

#16811101

yes

characteristic

108

C₂

C₂×#13133224

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#533	C₂
hypercenter	#533	D₅₃
hypocenter	C₅₃	#2986
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#533

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #16810754	C₂	semidirect
2. Q₈×#88322	C₂	semidirect
60. C₅₃	#2986	semidirect

H ≅ G/N

type

1. #16810754

C₂

semidirect

2. Q₈×#88322

C₂

semidirect

60. C₅₃

#2986

semidirect

Subgroups — #20283817

Conjugacy classes of subgroups

Complements