Group #20275685 — subgroups

Subgroups — #20275685

Order 6784.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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	#16810493	C₂	1	1	yes	maximal, characteristic
3	#16810385	C₂	1	4	yes	maximal
4	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₅₃	—	2	2	no	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	#13133128	—	2	2	no	—
9	#125×C₅₃	—	2	2	no	—
10	#125×C₅₃	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
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	8	no	—
19	C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
21	C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	C₄²×C₅₃	D₄	1	2	yes	—
23	C₄⋊Dic₅₃	—	2	4	no	—
24	C₄⋊Dic₅₃	—	4	4	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	—	2	2	no	—
27	C₂×#88322	Q₈	1	4	yes	—
28	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	4	no	—
32	C₂×C₄×C₅₃	—	4	2	no	—
33	C₂×C₄×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₅₃	#41	1	2	yes	—
39	C₂×Dic₅₃	—	4	2	no	—
40	C₂×Dic₅₃	—	4	4	no	—
41	#88322	—	2	4	no	—
42	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₅₃	#124	1	1	yes	characteristic
44	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₅₃	#137	1	2	yes	—
47	C₄×C₅₃	—	2	4	no	—
48	C₄×C₅₃	—	4	4	no	—
49	C₄×C₅₃	C₄×Q₈	1	2	yes	—
50	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
51	Dic₅₃	—	8	2	no	—
52	#1992	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	#391	1	1	yes	characteristic
54	C₂×C₅₃	#417	1	2	yes	—
55	C₂×C₅₃	#490	1	1	yes	characteristic
56	C₂×C₅₃	#491	1	1	yes	characteristic
57	C₂×C₅₃	#501	1	2	yes	—
58	C₂×#125	D₅₃	1	1	yes	characteristic
59	C₂×#97	—	53	1	no	—
60	#384	—	53	1	no	—
61	#340	—	53	4	no	—
62	C₅₃	#1992	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	—	53	2	no	—
64	#125	—	2	2	no	—
65	#125	Dic₅₃	1	4	yes	complemented
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	53	2	no	—
68	C₂×C₄⋊C₄	—	106	2	no	—
69	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
70	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
71	#97	—	106	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	106	2	no	—
74	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
75	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
76	C₂×C₈	—	53	4	no	—
77	C₂×C₈	—	106	2	no	—
78	C₂×C₈	—	106	2	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄⋊C₄	—	2	8	no	—
81	C₄⋊C₄	—	106	4	no	—
82	C₄⋊C₄	—	212	4	no	—
83	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
84	C₄²	C₂×Dic₅₃	1	2	yes	—
85	C₄²	C₂²⋊D₅₃	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	212	2	no	—
93	C₂×C₄	—	212	4	no	—
94	C₂×C₄	C₂²⋊Dic₅₃	1	4	yes	—
95	C₂×C₄	Q₈×D₅₃	1	2	yes	—
96	C₂×C₄	#11719949	1	2	yes	—
97	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
99	C₈	—	106	4	no	—
100	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
101	C₂²	#13133206	1	1	yes	characteristic
102	C₂²	#13133215	1	1	yes	characteristic
103	C₂²	#13133242	1	2	yes	—
104	C₂²	#13133247	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	424	2	no	—
108	C₄	Q₈×Dic₅₃	1	2	yes	—
109	C₄	#13133193	1	2	yes	—
110	C₂	#16810530	1	1	yes	characteristic
111	C₂	#16810624	1	2	yes	—
112	C₂	#16810989	1	1	yes	characteristic
113	C₂	#16810993	1	1	yes	characteristic
114	C₂	#16811033	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133128

C₂

yes

maximal, characteristic

#16810493

C₂

yes

maximal, characteristic

#16810385

C₂

yes

maximal

C₂×#125×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133128

—

#125×C₅₃

—

#125×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

Q₈

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

AGL(1,ℤ/8)

yes

—

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×#125

D₅₃

yes

characteristic

C₂×#97

—

#384

—

#340

—

C₅₃

#1992

yes

characteristic, complemented, Sylow

C₂²×C₈

—

#125

—

#125

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#97

—

106

—

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

—

212

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂×C₄

Q₈×D₅₃

yes

—

C₂×C₄

#11719949

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

106

—

100

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

101

C₂²

#13133206

yes

characteristic

102

C₂²

#13133215

yes

characteristic

103

C₂²

#13133242

yes

—

104

C₂²

#13133247

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

424

—

108

C₄

Q₈×Dic₅₃

yes

—

109

C₄

#13133193

yes

—

110

C₂

#16810530

yes

characteristic

111

C₂

#16810624

yes

—

112

C₂

#16810989

yes

characteristic

113

C₂

#16810993

yes

characteristic

114

C₂

#16811033

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#125×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#125	D₅₃
hypocenter	C₅₃	#1992
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#125×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₅₃	C₄	semidirect
62. C₅₃	#1992	semidirect
65. #125	Dic₅₃	semidirect

H ≅ G/N

type

10. #125×C₅₃

C₄

semidirect

62. C₅₃

#1992

semidirect

65. #125

Dic₅₃

semidirect

Subgroups — #20275685

Conjugacy classes of subgroups

Complements