Group #20275697 — subgroups

Subgroups — #20275697

Order 6784.

244 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16810647	C₂	1	1	yes	maximal, characteristic
2	#16810469	C₂	1	1	yes	maximal, characteristic
3	#16810382	C₂	1	4	yes	maximal
4	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	2	yes	—
6	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
7	C₄⋊#88322	—	2	2	no	—
8	#13133124	—	2	2	no	—
9	C₄⋊Q₈×C₅₃	C₄	1	2	yes	complemented
10	#125×C₅₃	C₄	1	2	yes	complemented
11	C₂²⋊Q₈×C₅₃	—	2	2	no	—
12	C₄×Q₈×C₅₃	—	2	2	no	—
13	#117×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	2	2	no	—
16	C₂²×C₄×C₅₃	D₄	1	2	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	2	2	no	—
19	C₄⋊C₄×C₅₃	—	2	4	no	—
20	C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
22	C₂²⋊C₄×C₅₃	—	2	2	no	—
23	C₄²×C₅₃	—	2	2	no	—
24	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₅₃	D₄	1	2	yes	—
27	C₄×Dic₅₃	—	2	2	no	—
28	C₄×Dic₅₃	Q₈	1	4	yes	complemented
29	#11719917	—	2	4	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₅₃	—	4	4	no	—
33	C₂×C₄×C₅₃	—	2	4	no	—
34	C₂×C₄×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₅₃	#41	1	2	yes	—
41	C₂×Dic₅₃	—	2	4	no	—
42	C₂×Dic₅₃	—	4	4	no	—
43	#88322	—	4	2	no	—
44	C₂²×C₅₃	—	2	2	no	—
45	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₅₃	—	2	2	no	—
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	—	4	4	no	—
50	C₄×C₅₃	C₄×Q₈	1	2	yes	—
51	C₄×C₅₃	#124	1	1	yes	characteristic
52	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₅₃	—	4	4	no	—
54	#1995	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	2	2	no	—
56	C₂×C₅₃	#391	1	1	yes	characteristic
57	C₂×C₅₃	#421	1	2	yes	—
58	#533	D₅₃	1	1	yes	characteristic
59	#423	—	53	1	no	—
60	#376	—	53	1	no	—
61	#339	—	53	4	no	—
62	C₅₃	#1995	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₁₆	—	53	2	no	—
64	C₄⋊Q₈	Dic₅₃	1	2	yes	complemented
65	#125	Dic₅₃	1	2	yes	complemented
66	C₂²⋊Q₈	—	2	2	no	—
67	C₄×Q₈	—	2	2	no	—
68	#117	C₂×D₅₃	1	2	yes	—
69	C₂×C₄²	—	53	2	no	—
70	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
71	C₄⋊C₈	—	106	2	no	—
72	#95	—	106	2	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂²×C₄	—	106	2	no	—
75	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
76	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
77	M₁₆	—	106	4	no	—
78	C₂×C₈	—	106	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	53	4	no	—
85	C₄²	—	106	2	no	—
86	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
87	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
88	C₄²	C₂²⋊D₅₃	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	106	4	no	—
94	C₂×C₄	—	212	4	no	—
95	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
96	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
97	C₂×C₄	Q₈×D₅₃	1	2	yes	—
98	C₂×C₄	#11719949	1	2	yes	—
99	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
101	C₈	—	212	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
104	C₂²	#13133193	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	212	4	no	—
109	C₄	Q₈×Dic₅₃	1	2	yes	—
110	C₄	#13133206	1	1	yes	characteristic
111	C₄	#13133215	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#16810530	1	1	yes	characteristic
114	C₂	#16810640	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal, characteristic

#16810469

C₂

yes

maximal, characteristic

#16810382

C₂

yes

maximal

C₅₃×#533

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄⋊#88322

—

#13133124

—

C₄⋊Q₈×C₅₃

C₄

yes

complemented

#125×C₅₃

C₄

yes

complemented

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×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

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

Q₈

yes

complemented

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

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

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

#124

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

#1995

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#391

yes

characteristic

C₂×C₅₃

#421

yes

—

#533

D₅₃

yes

—

—

—

C₅₃

#1995

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

Dic₅₃

yes

complemented

#125

Dic₅₃

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

#95

—

106

—

C₂×Q₈

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

106

—

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂×C₄

C₂²⋊Dic₅₃

yes

—

C₂×C₄

Q₈×D₅₃

yes

—

C₂×C₄

#11719949

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

101

C₈

—

212

—

102

C₂²

—

103

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

104

C₂²

#13133193

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

212

—

109

C₄

Q₈×Dic₅₃

yes

—

110

C₄

#13133206

yes

characteristic

111

C₄

#13133215

yes

characteristic

112

C₂

—

113

C₂

#16810530

yes

characteristic

114

C₂

#16810640

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#533	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#533	D₅₃
hypocenter	C₅₃	#1995
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#533

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
9. C₄⋊Q₈×C₅₃	C₄	semidirect
10. #125×C₅₃	C₄	semidirect
28. C₄×Dic₅₃	Q₈	semidirect
62. C₅₃	#1995	semidirect
64. C₄⋊Q₈	Dic₅₃	semidirect
65. #125	Dic₅₃	semidirect

H ≅ G/N

type

9. C₄⋊Q₈×C₅₃

C₄

semidirect

10. #125×C₅₃

C₄

semidirect

28. C₄×Dic₅₃

Q₈

semidirect

62. C₅₃

#1995

semidirect

64. C₄⋊Q₈

Dic₅₃

semidirect

65. #125

Dic₅₃

semidirect

Subgroups — #20275697

Conjugacy classes of subgroups

Complements