Group #20276479 — subgroups

Subgroups — #20276479

Order 6784.

152 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 75 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	M₃₂×D₅₃	C₂	1	2	yes	maximal
2	#16810577	C₂	1	1	yes	maximal, characteristic
3	#16810444	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810443	C₂	1	2	yes	maximal, complemented
5	C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719912	C₄	1	1	yes	characteristic
7	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	#13133174	C₄	1	1	yes	characteristic
9	#13133157	C₂²	1	2	yes	—
10	#13133156	—	2	2	no	—
11	C₁₆×D₅₃	—	2	2	no	—
12	#13133128	C₄	1	1	yes	characteristic
13	#13133127	C₄	1	1	yes	characteristic
14	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	M₃₂×C₅₃	C₂²	1	2	yes	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	#11719921	—	2	1	no	—
22	#11719919	—	2	1	no	—
23	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×#88322	C₂×C₄	1	1	yes	characteristic
26	#11719912	C₈	1	4	yes	—
27	C₈×D₅₃	D₄	1	2	yes	—
28	C₈×D₅₃	Q₈	1	2	yes	—
29	C₁₆×C₅₃	—	2	2	no	—
30	#11719908	—	2	2	no	—
31	C₂²×D₅₃	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₅₃	—	2	1	no	—
35	C₂×Dic₅₃	M₁₆	1	1	yes	characteristic
36	C₄×D₅₃	C₄⋊C₄	1	2	yes	—
37	C₄×D₅₃	C₂×C₈	1	2	yes	—
38	C₈×C₅₃	C₂×C₈	1	2	yes	—
39	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	#88322	C₄⋊C₄	1	2	yes	—
42	#88322	C₂×C₈	1	2	yes	—
43	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
44	C₂×D₅₃	—	4	1	no	—
45	C₂×D₅₃	C₄⋊C₈	1	2	yes	—
46	C₄×C₅₃	—	4	1	no	—
47	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	Dic₅₃	—	4	1	no	—
50	Dic₅₃	C₄⋊C₈	1	2	yes	—
51	#2153	—	53	1	no	maximal, Sylow
52	C₂×C₅₃	—	2	1	no	—
53	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
54	D₅₃	—	2	2	no	—
55	C₂×M₃₂	—	53	2	no	—
56	#405	—	53	1	no	—
57	#364	—	53	2	no	—
58	#364	—	53	1	no	—
59	#364	D₅₃	1	1	yes	characteristic, complemented
60	C₅₃	#2153	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	53	1	no	—
62	C₂²×C₈	—	53	1	no	—
63	#117	—	53	1	no	—
64	M₃₂	—	53	2	no	—
65	M₃₂	—	106	2	no	—
66	M₃₂	C₂×D₅₃	1	2	yes	—
67	C₂×C₁₆	—	106	2	no	—
68	#97	—	53	1	no	—
69	#97	—	53	1	no	—
70	C₄×C₈	—	53	1	no	—
71	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
72	C₂²×C₄	—	53	1	no	—
73	M₁₆	—	53	4	no	—
74	C₂×C₈	—	53	1	no	—
75	C₂×C₈	—	53	1	no	—
76	C₂×C₈	—	53	2	no	—
77	C₂×C₈	—	53	2	no	—
78	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
79	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
80	C₄⋊C₄	—	106	1	no	—
81	C₂²⋊C₄	—	106	1	no	—
82	C₄²	—	53	1	no	—
83	C₄²	C₄×D₅₃	1	1	yes	characteristic
84	C₁₆	—	2	2	no	—
85	C₁₆	—	106	2	no	—
86	C₂³	—	53	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	53	1	no	—
89	C₂×C₄	—	53	2	no	—
90	C₂×C₄	—	53	2	no	—
91	C₂×C₄	—	106	1	no	—
92	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
93	C₈	—	53	2	no	—
94	C₈	—	53	2	no	—
95	C₈	C₈×D₅₃	1	2	yes	—
96	C₈	D₄×D₅₃	1	1	yes	characteristic
97	C₈	Q₈×D₅₃	1	1	yes	characteristic
98	C₂²	—	53	2	no	—
99	C₂²	—	212	1	no	—
100	C₂²	M₁₆×D₅₃	1	1	yes	characteristic
101	C₄	—	4	1	no	—
102	C₄	—	53	2	no	—
103	C₄	—	212	1	no	—
104	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
105	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	106	2	no	—
108	C₂	C₄⋊C₈×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×D₅₃

C₂

yes

maximal

#16810577

C₂

yes

maximal, characteristic

#16810444

C₂

yes

maximal, characteristic, complemented

#16810443

C₂

yes

maximal, complemented

C₅₃×#364

C₂

yes

maximal, characteristic, complemented

C₂×#11719912

C₄

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

#13133174

C₄

yes

characteristic

#13133157

C₂²

yes

—

#13133156

—

C₁₆×D₅₃

—

#13133128

C₄

yes

characteristic

#13133127

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

—

#11719919

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

#11719912

C₈

yes

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

Q₈

yes

—

C₁₆×C₅₃

—

#11719908

—

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

C₂×C₈

yes

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

#88322

C₄⋊C₄

yes

—

#88322

C₂×C₈

yes

—

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

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2153

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

M₃₂

—

106

—

M₃₂

C₂×D₅₃

yes

—

C₂×C₁₆

—

106

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

C₄×D₅₃

yes

characteristic

C₁₆

—

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₈

—

C₈

—

C₈

C₈×D₅₃

yes

—

C₈

D₄×D₅₃

yes

characteristic

C₈

Q₈×D₅₃

yes

characteristic

C₂²

—

C₂²

—

212

—

100

C₂²

M₁₆×D₅₃

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

212

—

104

C₄

C₄⋊C₄×D₅₃

yes

characteristic

105

C₄

C₂×C₈×D₅₃

yes

characteristic

106

C₂

—

107

C₂

—

106

—

108

C₂

C₄⋊C₈×D₅₃

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#364	C₂
socle	C₂×C₅₃	C₂×C₄⋊C₈
hypercenter	#364	D₅₃
hypocenter	C₅₃	#2153
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×D₅₃

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#364

C₂

socle

C₂×C₅₃

C₂×C₄⋊C₈

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
3. #16810444	C₂	semidirect
4. #16810443	C₂	semidirect
5. C₅₃×#364	C₂	semidirect
59. #364	D₅₃	semidirect
60. C₅₃	#2153	semidirect

H ≅ G/N

type

3. #16810444

C₂

semidirect

4. #16810443

C₂

semidirect

5. C₅₃×#364

C₂

semidirect

59. #364

D₅₃

semidirect

60. C₅₃

#2153

semidirect

Subgroups — #20276479

Conjugacy classes of subgroups

Complements