Group #20272660 — subgroups

Subgroups — #20272660

Order 6784.

134 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 37 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	#16810492	C₂	1	1	yes	maximal, characteristic
2	#16810349	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#324	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
5	C₂×#11719919	—	2	1	no	—
6	C₂×#11719919	—	2	1	no	—
7	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
8	C₂²⋊#88322	—	2	1	no	—
9	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²⋊C₈×C₅₃	—	2	1	no	—
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
14	C₂²×Dic₅₃	—	2	1	no	—
15	C₂²×Dic₅₃	—	2	1	no	—
16	C₂×C₈×C₅₃	—	4	1	no	—
17	C₄⋊C₄×C₅₃	—	4	1	no	—
18	C₄⋊Dic₅₃	—	4	1	no	—
19	#11719919	—	4	2	no	—
20	#11719919	—	4	2	no	—
21	C₄×Dic₅₃	—	2	1	no	—
22	C₄×Dic₅₃	C₈	1	2	yes	complemented
23	C₂×#88322	—	4	1	no	—
24	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	—	2	1	no	—
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	—	2	1	no	—
28	C₂×C₄×C₅₃	—	2	1	no	—
29	C₂×C₄×C₅₃	—	4	1	no	—
30	C₂×C₄×C₅₃	—	4	1	no	—
31	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
33	C₂×Dic₅₃	—	2	2	no	—
34	C₂×Dic₅₃	—	2	2	no	—
35	C₂×Dic₅₃	—	2	2	no	—
36	C₂×Dic₅₃	—	4	1	no	—
37	C₂×Dic₅₃	—	4	1	no	—
38	C₂×Dic₅₃	—	4	1	no	—
39	C₈×C₅₃	—	8	1	no	—
40	#88322	—	8	1	no	—
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
44	C₂²×C₅₃	#99	1	1	yes	characteristic
45	C₂²×C₅₃	#101	1	1	yes	characteristic
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	4	1	no	—
49	C₄×C₅₃	—	4	1	no	—
50	Dic₅₃	—	2	4	no	—
51	Dic₅₃	—	8	1	no	—
52	#1325	—	53	1	no	maximal, Sylow
53	C₂×C₅₃	—	2	1	no	—
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	#324	1	1	yes	characteristic
56	C₂×C₅₃	#353	1	1	yes	characteristic
57	C₂×C₅₃	#356	1	1	yes	characteristic
58	#384	—	53	1	no	—
59	#324	—	53	1	no	—
60	#324	D₅₃	1	1	yes	characteristic
61	C₅₃	#1325	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₄⋊C₄	—	53	1	no	—
63	C₂×C₄⋊C₄	—	106	1	no	—
64	C₂×C₄⋊C₄	—	106	1	no	—
65	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
66	C₂×C₄²	—	53	1	no	—
67	C₂²⋊C₈	—	2	1	no	—
68	C₂²⋊C₈	—	106	1	no	—
69	C₂²×C₄	—	106	1	no	—
70	C₂²×C₄	—	106	1	no	—
71	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
72	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	—	212	1	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₄⋊C₄	—	212	1	no	—
78	C₄⋊C₄	—	212	2	no	—
79	C₄⋊C₄	—	212	2	no	—
80	C₄²	—	53	2	no	—
81	C₄²	—	106	1	no	—
82	C₂³	#11719921	1	1	yes	characteristic
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	106	2	no	—
90	C₂×C₄	—	106	2	no	—
91	C₂×C₄	—	106	2	no	—
92	C₂×C₄	—	212	1	no	—
93	C₂×C₄	—	212	1	no	—
94	C₂×C₄	—	212	1	no	—
95	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
96	C₂×C₄	#11719912	1	1	yes	characteristic
97	C₈	—	8	1	no	—
98	C₈	—	424	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#13133129	1	1	yes	characteristic
102	C₂²	#13133131	1	1	yes	characteristic
103	C₂²	#13133135	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	4	1	no	—
108	C₄	—	106	4	no	—
109	C₄	—	424	1	no	—
110	C₂	—	2	1	no	—
111	C₂	—	2	1	no	—
112	C₂	#16810348	1	1	yes	characteristic
113	C₂	#16810415	1	1	yes	characteristic
114	C₂	#16810421	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810492

C₂

yes

maximal, characteristic

#16810349

C₂

yes

maximal, characteristic

C₅₃×#324

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

C₄

yes

characteristic

C₂×#11719919

—

C₂×#11719919

—

C₂×C₄×Dic₅₃

C₄

yes

characteristic

C₂²⋊#88322

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₈

yes

complemented

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊C₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

#1325

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

C₅₃

#1325

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

212

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

—

C₄²

—

106

—

C₂³

#11719921

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₈×D₅₃

yes

characteristic

C₂×C₄

#11719912

yes

characteristic

C₈

—

C₈

—

424

—

C₂²

—

100

C₂²

—

101

C₂²

#13133129

yes

characteristic

102

C₂²

#13133131

yes

characteristic

103

C₂²

#13133135

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

106

—

109

C₄

—

424

—

110

C₂

—

111

C₂

—

112

C₂

#16810348

yes

characteristic

113

C₂

#16810415

yes

characteristic

114

C₂

#16810421

yes

characteristic

	subgroup	G/H
center	C₂²	#13133131
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₅₃
Fitting	C₅₃×#324	C₂
socle	C₂²×C₅₃	#99
hypercenter	#324	D₅₃
hypocenter	C₅₃	#1325
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #324

subgroup

G/H

center

C₂²

#13133131

commutator

C₂×C₄×C₅₃

C₂×C₈

Frattini

C₂×C₄⋊C₄

C₂×D₅₃

Fitting

C₅₃×#324

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #324

N	H ≅ G/N	type
22. C₄×Dic₅₃	C₈	semidirect
61. C₅₃	#1325	semidirect

H ≅ G/N

type

22. C₄×Dic₅₃

C₈

semidirect

61. C₅₃

#1325

semidirect

Subgroups — #20272660

Conjugacy classes of subgroups

Complements