Group #20275679 — subgroups

Subgroups — #20275679

Order 6784.

252 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 105 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×#13133209	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#16810385	C₂	1	4	yes	maximal
4	C₅₃×#382	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	#13133209	—	2	2	no	—
7	#13133209	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂×#11719919	—	2	2	no	—
10	C₄×#88322	—	2	2	no	—
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	#95×C₅₃	—	2	2	no	—
14	C₂²×C₄×C₅₃	—	2	2	no	—
15	C₂²×C₄×C₅₃	D₄	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	2	2	no	—
18	C₄⋊C₄×C₅₃	—	2	4	no	—
19	C₄²×C₅₃	C₂×C₄	1	2	yes	—
20	C₄²×C₅₃	D₄	1	2	yes	—
21	C₄⋊Dic₅₃	—	2	2	no	—
22	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
23	#11719919	—	2	8	no	—
24	C₂×#88322	—	2	2	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	Q₈	1	4	yes	—
27	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	4	no	—
29	C₂×C₄×C₅₃	—	4	2	no	—
30	C₂×C₄×C₅₃	—	4	4	no	—
31	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
32	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₅₃	#41	1	2	yes	—
36	C₂×Dic₅₃	—	2	4	no	—
37	C₂×Dic₅₃	—	4	4	no	—
38	#88322	—	2	4	no	—
39	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₅₃	#124	1	1	yes	characteristic
41	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₅₃	#135	1	4	yes	—
43	C₄×C₅₃	—	2	4	no	—
44	C₄×C₅₃	—	8	2	no	—
45	C₄×C₅₃	C₄×Q₈	1	2	yes	—
46	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
47	Dic₅₃	—	4	4	no	—
48	#1991	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	#391	1	1	yes	characteristic
50	C₂×C₅₃	#416	1	2	yes	—
51	C₂×C₅₃	#488	1	2	yes	—
52	C₂×C₅₃	#499	1	2	yes	—
53	C₂×#125	—	53	1	no	—
54	C₂×C₄×C₈	—	53	1	no	—
55	#382	D₅₃	1	1	yes	characteristic
56	#340	—	53	4	no	—
57	C₅₃	#1991	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	—	53	2	no	—
59	#125	—	53	4	no	—
60	#125	—	106	2	no	—
61	C₂×C₄⋊C₄	—	53	2	no	—
62	C₂×C₄⋊C₄	—	106	2	no	—
63	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
64	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
65	C₄×C₈	—	106	2	no	—
66	#95	—	2	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	106	2	no	—
69	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
70	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
71	C₂×C₈	—	53	4	no	—
72	C₂×C₈	—	106	2	no	—
73	C₂×C₈	—	106	2	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	53	4	no	—
76	C₄⋊C₄	—	106	2	no	—
77	C₄⋊C₄	—	106	8	no	—
78	C₄²	C₄×D₅₃	1	2	yes	—
79	C₄²	D₂₁₂	1	2	yes	—
80	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	106	4	no	—
85	C₂×C₄	—	212	4	no	—
86	C₂×C₄	#11719921	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
88	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
89	C₂×C₄	#11719947	1	2	yes	—
90	C₂×C₄	Q₈×D₅₃	1	2	yes	—
91	C₈	—	106	4	no	—
92	C₂²	C₂×#11719921	1	1	yes	characteristic
93	C₂²	#13133204	1	1	yes	characteristic
94	C₂²	#13133215	1	1	yes	characteristic
95	C₂²	#13133237	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	8	2	no	—
98	C₄	—	212	4	no	—
99	C₄	#13133182	1	2	yes	—
100	C₄	#13133197	1	2	yes	—
101	C₂	#16810528	1	1	yes	characteristic
102	C₂	#16810617	1	2	yes	—
103	C₂	#16810977	1	2	yes	—
104	C₂	#16811022	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133209

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

C₅₃×#382

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

#13133209

—

#13133209

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

—

C₄×#88322

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

—

—

—

Q₈

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#124

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₂²×C₅₃

#135

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#488

yes

—

C₂×C₅₃

#499

yes

—

C₂×#125

—

C₂×C₄×C₈

—

#382

D₅₃

yes

characteristic

#340

—

C₅₃

#1991

yes

characteristic, complemented, Sylow

C₂²×C₈

—

#125

—

#125

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄×C₈

—

106

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

C₄×D₅₃

yes

—

C₄²

D₂₁₂

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

#11719921

yes

—

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

#11719947

yes

—

C₂×C₄

Q₈×D₅₃

yes

—

C₈

—

106

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

212

—

C₄

#13133182

yes

—

100

C₄

#13133197

yes

—

101

C₂

#16810528

yes

characteristic

102

C₂

#16810617

yes

—

103

C₂

#16810977

yes

—

104

C₂

#16811022

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#382	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	#382	D₅₃
hypocenter	C₅₃	#1991
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#382

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
7. #13133209	C₄	semidirect
57. C₅₃	#1991	semidirect

H ≅ G/N

type

7. #13133209

C₄

semidirect

57. C₅₃

#1991

semidirect

Subgroups — #20275679

Conjugacy classes of subgroups

Complements