Group #20291074 — subgroups

Subgroups — #20291074

Order 6784.

160 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 78 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	#16811544	C₂	1	2	yes	maximal, complemented
2	#16811518	C₂	1	2	yes	maximal
3	C₄×#11719910	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719910	C₂²	1	4	yes	—
6	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
7	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₄²×D₅₃	C₄	1	1	yes	characteristic
9	C₄×#88322	C₂²	1	1	yes	characteristic
10	C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×D₅₃	C₈	1	1	yes	characteristic
14	C₂×C₄×D₅₃	C₈	1	1	yes	characteristic
15	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₅₃	C₈	1	1	yes	characteristic
20	C₄×Dic₅₃	C₈	1	1	yes	characteristic
21	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×#88322	C₂³	1	1	yes	characteristic
24	C₈×D₅₃	—	2	2	no	—
25	C₈×D₅₃	—	2	2	no	—
26	#11719910	—	2	4	no	—
27	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
34	C₄×D₅₃	—	2	2	no	—
35	C₄×D₅₃	—	2	1	no	—
36	C₄×D₅₃	—	2	1	no	—
37	C₄×D₅₃	C₁₆	1	4	yes	complemented
38	C₈×C₅₃	—	2	1	no	—
39	C₈×C₅₃	—	2	1	no	—
40	#88322	M₁₆	1	2	yes	—
41	#88322	#41	1	2	yes	—
42	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
43	C₂×D₅₃	—	2	1	no	—
44	C₂×D₅₃	—	2	1	no	—
45	C₂×D₅₃	C₂×C₁₆	1	2	yes	—
46	C₄×C₅₃	C₂×C₁₆	1	2	yes	—
47	C₄×C₅₃	M₃₂	1	2	yes	—
48	C₄×C₅₃	#117	1	1	yes	characteristic
49	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
50	Dic₅₃	—	2	1	no	—
51	Dic₅₃	—	2	1	no	—
52	Dic₅₃	C₂×C₁₆	1	2	yes	—
53	#2162	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	#431	1	1	yes	characteristic
55	C₂×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
56	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
57	D₅₃	—	2	2	no	—
58	C₂×C₄×C₈	—	53	1	no	—
59	C₄⋊C₁₆	—	53	2	no	—
60	C₂²⋊C₁₆	—	53	2	no	—
61	C₄×C₁₆	—	53	2	no	—
62	C₅₃	#2162	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	—	53	1	no	—
64	C₂²×C₈	—	53	1	no	—
65	C₂×C₄²	—	53	1	no	—
66	C₂×C₁₆	—	53	4	no	—
67	C₄×C₈	—	53	1	no	—
68	C₄×C₈	—	53	1	no	—
69	C₄×C₈	—	53	1	no	—
70	C₄×C₈	#6157	1	1	yes	characteristic
71	C₂²×C₄	—	53	1	no	—
72	C₂²×C₄	—	53	1	no	—
73	C₂²×C₄	—	53	1	no	—
74	C₂×C₈	—	53	1	no	—
75	C₂×C₈	—	53	1	no	—
76	C₂×C₈	—	106	2	no	—
77	C₂×C₈	—	106	2	no	—
78	C₂×C₈	C₂×#6157	1	1	yes	characteristic
79	C₂×C₈	C₂×#6157	1	1	yes	characteristic
80	C₄²	—	53	1	no	—
81	C₄²	—	53	1	no	—
82	C₄²	—	53	1	no	—
83	C₄²	C₂×#6157	1	1	yes	characteristic
84	C₁₆	—	106	4	no	—
85	C₂³	—	53	1	no	—
86	C₂×C₄	—	53	1	no	—
87	C₂×C₄	—	53	4	no	—
88	C₂×C₄	—	53	1	no	—
89	C₂×C₄	—	53	1	no	—
90	C₂×C₄	—	106	2	no	—
91	C₂×C₄	—	106	1	no	—
92	C₂×C₄	—	106	1	no	—
93	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
94	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
95	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	2	1	no	—
98	C₈	—	53	2	no	—
99	C₈	—	53	2	no	—
100	C₂²	—	53	2	no	—
101	C₂²	—	106	1	no	—
102	C₂²	—	106	1	no	—
103	C₂²	C₂×D₅₃⋊C₈	1	1	yes	characteristic
104	C₄	—	53	2	no	—
105	C₄	—	106	1	no	—
106	C₄	—	106	1	no	—
107	C₄	D₅₃⋊C₁₆	1	2	yes	—
108	C₄	#13133292	1	2	yes	—
109	C₄	#13133296	1	1	yes	characteristic
110	C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
111	C₂	—	106	2	no	—
112	C₂	#16811641	1	1	yes	characteristic
113	C₂	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
114	C₂	C₂×#13133292	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811544

C₂

yes

maximal, complemented

#16811518

C₂

yes

maximal

C₄×#11719910

C₂

yes

maximal, complemented

C₄×C₈×D₅₃

C₂

yes

maximal, characteristic

C₂×#11719910

C₂²

yes

—

C₂×C₈×D₅₃

C₄

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₄²×D₅₃

C₄

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₈×Dic₅₃

C₄

yes

characteristic

C₈×Dic₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₈

yes

characteristic

C₂×C₄×D₅₃

C₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₈

yes

characteristic

C₄×Dic₅₃

C₈

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

#11719910

—

C₂²×D₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₁₆

yes

complemented

C₈×C₅₃

—

C₈×C₅₃

—

#88322

M₁₆

yes

—

#88322

#41

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂×C₁₆

yes

—

C₄×C₅₃

C₂×C₁₆

yes

—

C₄×C₅₃

M₃₂

yes

—

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×C₁₆

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂²×C₁₆

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

D₅₃

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₅₃

#2162

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

D₅₃⋊C₈

yes

characteristic

C₂×C₄

D₅₃⋊C₈

yes

characteristic

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

100

C₂²

—

101

C₂²

—

106

—

102

C₂²

—

106

—

103

C₂²

C₂×D₅₃⋊C₈

yes

characteristic

104

C₄

—

105

C₄

—

106

—

106

C₄

—

106

—

107

C₄

D₅₃⋊C₁₆

yes

—

108

C₄

#13133292

yes

—

109

C₄

#13133296

yes

characteristic

110

C₄

D₅₃⋊M₁₆

yes

characteristic

111

C₂

—

106

—

112

C₂

#16811641

yes

characteristic

113

C₂

C₂×D₅₃⋊C₁₆

yes

characteristic

114

C₂

C₂×#13133292

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×#6157
commutator	C₂×C₅₃	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₈
hypercenter	C₄×C₈	#6157
hypocenter	C₅₃	#2162
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₄²

C₂×#6157

commutator

C₂×C₅₃

C₂²×C₁₆

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄×C₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂²×C₈

C₄×C₈

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
1. #16811544	C₂	semidirect
3. C₄×#11719910	C₂	semidirect
37. C₄×D₅₃	C₁₆	semidirect
62. C₅₃	#2162	semidirect

H ≅ G/N

type

1. #16811544

C₂

semidirect

3. C₄×#11719910

C₂

semidirect

37. C₄×D₅₃

C₁₆

semidirect

62. C₅₃

#2162

semidirect

Subgroups — #20291074

Conjugacy classes of subgroups

Complements