Group #20283143 — subgroups

Subgroups — #20283143

Order 6784.

556 conjugacy classes of subgroups; only the 373 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 373 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×#13133224	C₂	1	2	yes	maximal
2	#16810579	C₂	1	6	yes	maximal, complemented
3	C₄×#11719917	C₂	1	3	yes	maximal, complemented
4	C₂×#13133128	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#13133224	C₄	1	16	yes	complemented
7	C₂×#11719917	C₂²	1	6	yes	—
8	C₂²×#88322	C₂²	1	6	yes	—
9	#13133128	C₂²	1	1	yes	characteristic
10	#13133128	C₂²	1	3	yes	—
11	#13133128	C₂²	1	6	yes	—
12	C₄×#88322	C₂²	1	6	yes	—
13	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₅₃	C₄	1	2	yes	—
15	C₄×D₄×C₅₃	C₄	1	6	yes	—
16	#117×C₅₃	C₂²	1	3	yes	—
17	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
18	#41×C₅₃	C₂×C₄	1	8	yes	—
19	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₅₃	C₂×C₄	1	3	yes	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	6	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
23	C₄⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
24	C₂²⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
25	C₄²×C₅₃	C₂×C₄	1	6	yes	—
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	C₄²×C₅₃	C₂³	1	3	yes	—
28	#11719917	C₂×C₄	1	24	yes	complemented
29	C₂×#88322	C₂×C₄	1	24	yes	complemented
30	C₂×#88322	C₂³	1	6	yes	—
31	C₂×#88322	C₂³	1	2	yes	—
32	C₂³×C₅₃	C₂²×C₄	1	3	yes	—
33	Q₈×C₅₃	C₄²	1	4	yes	—
34	D₄×C₅₃	C₄²	1	12	yes	—
35	C₂×C₄×C₅₃	—	4	6	no	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	12	yes	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
40	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
41	#88322	C₂²×C₄	1	4	yes	—
42	#88322	C₂²×C₄	1	12	yes	—
43	C₂²×C₅₃	—	2	6	no	—
44	C₂²×C₅₃	C₂×C₄²	1	6	yes	—
45	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
46	C₄×C₅₃	—	2	2	no	—
47	C₄×C₅₃	—	2	6	no	—
48	C₄×C₅₃	C₂×C₄²	1	6	yes	—
49	C₄×C₅₃	C₂³×C₄	1	2	yes	—
50	#2875	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	—	2	6	no	—
52	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
53	C₂×C₅₃	#568	1	2	yes	—
54	C₂×#131	—	53	2	no	—
55	C₄×#41	D₅₃	1	1	yes	characteristic
56	#405	—	53	6	no	—
57	C₄×M₁₆	—	53	3	no	—
58	C₂×#97	—	53	3	no	—
59	C₅₃	#2875	1	1	yes	characteristic, complemented, Sylow
60	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
61	#131	—	53	16	no	—
62	C₂×M₁₆	—	53	6	no	—
63	C₂²×C₈	—	53	6	no	—
64	C₄×Q₈	Dic₅₃	1	2	yes	—
65	C₄×D₄	Dic₅₃	1	6	yes	—
66	#117	C₂×D₅₃	1	3	yes	—
67	C₂×C₄²	C₂×D₅₃	1	3	yes	—
68	#97	—	53	1	no	—
69	#97	—	53	3	no	—
70	#97	—	53	6	no	—
71	C₄×C₈	—	53	6	no	—
72	#41	C₄×D₅₃	1	8	yes	—
73	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
74	C₂×D₄	C₂×Dic₅₃	1	3	yes	—
75	C₂²×C₄	C₂×Dic₅₃	1	6	yes	—
76	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
77	M₁₆	—	53	24	no	—
78	C₂×C₈	—	53	24	no	—
79	C₂×C₈	—	53	6	no	—
80	C₂×C₈	—	53	2	no	—
81	C₄⋊C₄	C₂×Dic₅₃	1	6	yes	—
82	C₂²⋊C₄	C₂×Dic₅₃	1	6	yes	—
83	C₄²	C₂×Dic₅₃	1	6	yes	—
84	C₄²	C₂²×D₅₃	1	3	yes	—
85	C₄²	C₂²×D₅₃	1	1	yes	characteristic
86	C₂³	C₂²×Dic₅₃	1	3	yes	—
87	Q₈	C₄×Dic₅₃	1	4	yes	—
88	D₄	C₄×Dic₅₃	1	12	yes	—
89	C₂×C₄	—	4	6	no	—
90	C₂×C₄	C₂×C₄×D₅₃	1	12	yes	—
91	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
92	C₂×C₄	C₂²×Dic₅₃	1	6	yes	—
93	C₂×C₄	C₂²×Dic₅₃	1	3	yes	—
94	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
95	C₈	—	53	4	no	—
96	C₈	—	53	12	no	—
97	C₂²	—	2	6	no	—
98	C₂²	C₂×C₄×Dic₅₃	1	6	yes	—
99	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	2	6	no	—
102	C₄	C₂×C₄×Dic₅₃	1	6	yes	—
103	C₄	C₂²×C₄×D₅₃	1	2	yes	—
104	C₂	—	2	6	no	—
105	C₂	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
106	C₂	#16811390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133224

C₂

yes

maximal

#16810579

C₂

yes

maximal, complemented

C₄×#11719917

C₂

yes

maximal, complemented

C₂×#13133128

C₂

yes

maximal, complemented

C₄×#41×C₅₃

C₂

yes

maximal, characteristic

#13133224

C₄

yes

complemented

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

#13133128

C₂²

yes

characteristic

#13133128

C₂²

yes

—

#13133128

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₄

yes

—

C₄×D₄×C₅₃

C₄

yes

—

#117×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

#41×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

#11719917

C₂×C₄

yes

complemented

C₂×#88322

C₂×C₄

yes

complemented

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

Q₈×C₅₃

C₄²

yes

—

D₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

#88322

C₂²×C₄

yes

—

#88322

C₂²×C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄²

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

C₄×C₅₃

C₂³×C₄

yes

—

#2875

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

#568

yes

—

C₂×#131

—

C₄×#41

D₅₃

yes

characteristic

#405

—

C₄×M₁₆

—

C₂×#97

—

C₅₃

#2875

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

Dic₅₃

yes

—

C₄×D₄

Dic₅₃

yes

—

#117

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

#97

—

#97

—

#97

—

C₄×C₈

—

#41

C₄×D₅₃

yes

—

C₂×Q₈

C₂×Dic₅₃

yes

characteristic

C₂×D₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

C₂²×Dic₅₃

yes

—

Q₈

C₄×Dic₅₃

yes

—

D₄

C₄×Dic₅₃

yes

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

—

C₂×C₄

C₂³×D₅₃

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

C₂×C₄×Dic₅₃

yes

—

C₂²

C₂³×Dic₅₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

C₂×C₄×Dic₅₃

yes

—

103

C₄

C₂²×C₄×D₅₃

yes

—

104

C₂

—

105

C₂

C₂²×C₄×Dic₅₃

yes

characteristic

106

C₂

#16811390

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Fitting	C₄×#41×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	C₄×#41	D₅₃
hypocenter	C₅₃	#2875
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator

C₂×C₅₃

C₂²×C₄²

Fitting

C₄×#41×C₅₃

C₂

socle

C₂²×C₅₃

C₂³×C₄

hypercenter

C₄×#41

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
2. #16810579	C₂	semidirect
3. C₄×#11719917	C₂	semidirect
4. C₂×#13133128	C₂	semidirect
6. #13133224	C₄	semidirect
28. #11719917	C₂×C₄	semidirect
29. C₂×#88322	C₂×C₄	semidirect
59. C₅₃	#2875	semidirect

H ≅ G/N

type

2. #16810579

C₂

semidirect

3. C₄×#11719917

C₂

semidirect

4. C₂×#13133128

C₂

semidirect

6. #13133224

C₄

semidirect

28. #11719917

C₂×C₄

semidirect

29. C₂×#88322

C₂×C₄

semidirect

59. C₅₃

#2875

semidirect

Subgroups — #20283143

Conjugacy classes of subgroups

Complements