Group #20274401 — subgroups

Subgroups — #20274401

Order 6784.

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

normal subgroups only

This group has 328 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	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133127	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₅₃	C₂	1	2	yes	maximal
4	#16810469	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
8	#13133128	C₄	1	4	yes	complemented
9	#13133127	C₄	1	8	yes	complemented
10	C₈×Dic₅₃	C₄	1	8	yes	complemented
11	#13133124	C₄	1	4	yes	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₄	1	4	yes	complemented
16	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
21	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₄²×C₅₃	C₂×C₄	1	2	yes	—
24	C₄²×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
26	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
27	C₂×#88322	C₂×C₄	1	4	yes	—
28	C₂×#88322	C₂×C₄	1	4	yes	—
29	C₂×#88322	C₂×C₄	1	4	yes	—
30	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	4	no	—
32	C₂×C₄×C₅₃	C₄²	1	4	yes	—
33	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₅₃	—	2	8	no	—
38	C₂×Dic₅₃	C₄²	1	8	yes	—
39	C₈×C₅₃	C₄²	1	4	yes	—
40	C₈×C₅₃	C₄²	1	4	yes	—
41	#88322	C₄²	1	8	yes	—
42	C₂²×C₅₃	C₂×C₄²	1	2	yes	—
43	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₅₃	#131	1	4	yes	—
45	C₄×C₅₃	—	2	4	no	—
46	C₄×C₅₃	C₂×C₄²	1	2	yes	—
47	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
49	Dic₅₃	—	2	8	no	—
50	#1726	—	53	1	no	maximal, Sylow
51	C₂×C₅₃	C₄³	1	1	yes	characteristic
52	C₂×C₅₃	#405	1	4	yes	—
53	C₂×C₅₃	#405	1	2	yes	—
54	C₂×#97	—	53	2	no	—
55	C₂×#97	—	53	1	no	—
56	C₂×C₄×C₈	—	53	2	no	—
57	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
58	#376	—	53	1	no	—
59	C₅₃	#1726	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	53	2	no	—
61	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
62	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
63	C₂×C₄²	—	53	2	no	—
64	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
65	#97	—	53	8	no	—
66	#97	—	53	4	no	—
67	C₄×C₈	—	53	8	no	—
68	C₄×C₈	Dic₅₃	1	4	yes	complemented
69	#95	—	53	4	no	—
70	C₂²×C₄	—	53	4	no	—
71	C₂²×C₄	C₄×D₅₃	1	2	yes	—
72	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
73	C₂×C₈	—	53	4	no	—
74	C₂×C₈	—	53	4	no	—
75	C₂×C₈	—	53	4	no	—
76	C₂×C₈	C₄×D₅₃	1	4	yes	—
77	C₂×C₈	C₄×D₅₃	1	4	yes	—
78	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
79	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
80	C₄²	—	53	4	no	—
81	C₄²	—	53	4	no	—
82	C₄²	C₄×D₅₃	1	2	yes	—
83	C₄²	C₂×Dic₅₃	1	2	yes	—
84	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	53	8	no	—
87	C₂×C₄	—	106	8	no	—
88	C₂×C₄	C₄×Dic₅₃	1	4	yes	—
89	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
91	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
92	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
93	C₈	—	53	8	no	—
94	C₈	C₄×Dic₅₃	1	4	yes	—
95	C₈	C₄×Dic₅₃	1	4	yes	—
96	C₂²	C₄²×D₅₃	1	2	yes	—
97	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
98	C₂²	#13133223	1	4	yes	—
99	C₄	—	2	4	no	—
100	C₄	—	106	8	no	—
101	C₄	C₄²×D₅₃	1	2	yes	—
102	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
103	C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
104	C₂	C₄²×Dic₅₃	1	1	yes	characteristic
105	C₂	#16810577	1	4	yes	—
106	C₂	#16810578	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133128

C₂

yes

maximal, characteristic

C₂×#13133127

C₂

yes

maximal

C₂×C₈×Dic₅₃

C₂

yes

maximal

#16810469

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133128

C₄

yes

complemented

#13133127

C₄

yes

complemented

C₈×Dic₅₃

C₄

yes

complemented

#13133124

C₄

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄²

yes

—

C₈×C₅₃

C₄²

yes

—

C₈×C₅₃

C₄²

yes

—

#88322

C₄²

yes

—

C₂²×C₅₃

C₂×C₄²

yes

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

C₄×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

C₂×C₄²

yes

characteristic

Dic₅₃

—

#1726

—

maximal, Sylow

C₂×C₅₃

C₄³

yes

characteristic

C₂×C₅₃

#405

yes

—

C₂×C₅₃

#405

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₅₃

yes

characteristic

#376

—

C₅₃

#1726

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#97

—

#97

—

C₄×C₈

—

C₄×C₈

Dic₅₃

yes

complemented

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄²

—

C₄²

—

C₄²

C₄×D₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₂³

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₄×Dic₅₃

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

C₄×Dic₅₃

yes

—

C₈

C₄×Dic₅₃

yes

—

C₂²

C₄²×D₅₃

yes

—

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

C₂²

#13133223

yes

—

C₄

—

100

C₄

—

106

—

101

C₄

C₄²×D₅₃

yes

—

102

C₄

C₂×C₄×Dic₅₃

yes

characteristic

103

C₄

C₂×C₄×Dic₅₃

yes

characteristic

104

C₂

C₄²×Dic₅₃

yes

characteristic

105

C₂

#16810577

yes

—

106

C₂

#16810578

yes

—

	subgroup	G/H
center	C₂²×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₄³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1726
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₈

C₂×D₅₃

commutator

C₂×C₅₃

C₄³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂×C₄×C₈

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
8. #13133128	C₄	semidirect
9. #13133127	C₄	semidirect
10. C₈×Dic₅₃	C₄	semidirect
15. C₄×C₈×C₅₃	C₄	semidirect
59. C₅₃	#1726	semidirect
68. C₄×C₈	Dic₅₃	semidirect

H ≅ G/N

type

8. #13133128

C₄

semidirect

9. #13133127

C₄

semidirect

10. C₈×Dic₅₃

C₄

semidirect

15. C₄×C₈×C₅₃

C₄

semidirect

59. C₅₃

#1726

semidirect

68. C₄×C₈

Dic₅₃

semidirect

Subgroups — #20274401

Conjugacy classes of subgroups

Complements