Group #20291057 — subgroups

Subgroups — #20291057

Order 6784.

152 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 72 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	D₅₃⋊M₃₂	C₂	1	2	yes	maximal
2	#16811546	C₂	1	2	yes	maximal, complemented
3	#16811545	C₂	1	2	yes	maximal, complemented
4	#16810580	C₂	1	1	yes	maximal, characteristic
5	#13133292	—	2	2	no	—
6	#13133291	C₂²	1	4	yes	—
7	D₅₃⋊C₁₆	—	2	2	no	—
8	C₂×#11719912	C₄	1	1	yes	characteristic
9	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	#13133174	C₄	1	1	yes	characteristic
11	#13133127	C₄	1	1	yes	characteristic
12	C₄×#88322	C₂²	1	1	yes	characteristic
13	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	#97×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	#11719921	—	2	1	no	—
20	#11719919	—	2	1	no	—
21	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×#88322	C₂³	1	1	yes	characteristic
24	#11719912	C₈	1	4	yes	—
25	C₈×D₅₃	D₄	1	2	yes	—
26	C₈×D₅₃	Q₈	1	2	yes	—
27	#11719910	—	2	4	no	—
28	C₂²×D₅₃	M₁₆	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₅₃	—	2	1	no	—
32	C₂×Dic₅₃	M₁₆	1	1	yes	characteristic
33	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
34	C₄×D₅₃	C₂×C₈	1	2	yes	—
35	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₈×C₅₃	C₂×C₈	1	2	yes	—
37	#88322	C₂×C₈	1	2	yes	—
38	#88322	C₂×D₄	1	1	yes	characteristic
39	#88322	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
41	C₂×D₅₃	—	4	1	no	—
42	C₂×D₅₃	C₄⋊C₈	1	2	yes	—
43	C₄×C₅₃	—	4	1	no	—
44	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
46	Dic₅₃	—	4	1	no	—
47	Dic₅₃	C₄⋊C₈	1	2	yes	—
48	#2153	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	—	2	1	no	—
50	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
51	D₅₃	—	2	2	no	—
52	C₂×M₃₂	—	53	2	no	—
53	#405	—	53	1	no	—
54	#364	—	53	2	no	—
55	#364	—	53	2	no	—
56	C₅₃	#2153	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₁₆	—	53	1	no	—
58	C₂²×C₈	—	53	1	no	—
59	#117	—	53	1	no	—
60	M₃₂	—	53	4	no	—
61	M₃₂	—	106	2	no	—
62	C₂×C₁₆	—	106	2	no	—
63	#97	—	53	1	no	—
64	#97	#6157	1	1	yes	characteristic
65	C₄×C₈	—	53	1	no	—
66	C₄×C₈	—	53	1	no	—
67	C₂²×C₄	—	53	1	no	—
68	M₁₆	—	53	4	no	—
69	C₂×C₈	—	53	1	no	—
70	C₂×C₈	—	53	1	no	—
71	C₂×C₈	—	53	2	no	—
72	C₂×C₈	—	53	2	no	—
73	C₂×C₈	C₂×#6157	1	1	yes	characteristic
74	C₂×C₈	C₂×#6157	1	1	yes	characteristic
75	C₄⋊C₄	—	106	1	no	—
76	C₂²⋊C₄	—	106	1	no	—
77	C₄²	—	53	1	no	—
78	C₄²	C₂×#6157	1	1	yes	characteristic
79	C₁₆	—	106	4	no	—
80	C₂³	—	53	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	53	1	no	—
83	C₂×C₄	—	53	2	no	—
84	C₂×C₄	—	53	2	no	—
85	C₂×C₄	—	106	1	no	—
86	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
87	C₈	—	53	1	no	—
88	C₈	—	53	1	no	—
89	C₈	—	53	2	no	—
90	C₈	D₅₃⋊C₈	1	2	yes	—
91	C₈	C₄⋊#6157	1	2	yes	—
92	C₂²	—	53	2	no	—
93	C₂²	—	212	1	no	—
94	C₂²	D₅₃⋊M₁₆	1	1	yes	characteristic
95	C₄	—	4	1	no	—
96	C₄	—	53	2	no	—
97	C₄	—	212	1	no	—
98	C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
99	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	106	2	no	—
102	C₂	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊M₃₂

C₂

yes

maximal

#16811546

C₂

yes

maximal, complemented

#16811545

C₂

yes

maximal, complemented

#16810580

C₂

yes

maximal, characteristic

#13133292

—

#13133291

C₂²

yes

—

D₅₃⋊C₁₆

—

C₂×#11719912

C₄

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133174

C₄

yes

characteristic

#13133127

C₄

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

#97×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

—

#11719919

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

#11719912

C₈

yes

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

#11719910

—

C₂²×D₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

M₁₆

yes

characteristic

C₄×D₅₃

C₄⋊C₄

yes

—

C₄×D₅₃

C₂×C₈

yes

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₂×C₈

yes

—

#88322

C₂×C₈

yes

—

#88322

C₂×D₄

yes

characteristic

#88322

C₂×Q₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

characteristic

Dic₅₃

—

Dic₅₃

C₄⋊C₈

yes

—

#2153

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

D₅₃

—

C₂×M₃₂

—

#405

—

#364

—

#364

—

C₅₃

#2153

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

—

yes

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

D₅₃⋊C₈

yes

—

C₈

C₄⋊#6157

yes

—

C₂²

—

C₂²

—

212

—

C₂²

D₅₃⋊M₁₆

yes

characteristic

C₄

—

C₄

—

C₄

—

212

—

C₄

C₂×C₄⋊#6157

yes

characteristic

C₄

C₂×D₅₃⋊C₈

yes

characteristic

100

C₂

—

101

C₂

—

106

—

102

C₂

D₅₃⋊(C₄⋊C₈)

yes

characteristic

	subgroup	G/H
center	C₄	C₂×C₄⋊#6157
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	#97×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₄⋊C₈
hypercenter	#97	#6157
hypocenter	C₅₃	#2153
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #97

subgroup

G/H

center

C₄

C₂×C₄⋊#6157

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6157

Fitting

#97×C₅₃

C₄

socle

C₂×C₅₃

C₂×C₄⋊C₈

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #97

N	H ≅ G/N	type
2. #16811546	C₂	semidirect
3. #16811545	C₂	semidirect
56. C₅₃	#2153	semidirect

H ≅ G/N

type

2. #16811546

C₂

semidirect

3. #16811545

C₂

semidirect

56. C₅₃

#2153

semidirect

Subgroups — #20291057

Conjugacy classes of subgroups

Complements