Group #20276480 — subgroups

Subgroups — #20276480

Order 6784.

152 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 89 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×#13133157	C₂	1	1	yes	maximal, characteristic
2	#16810578	C₂	1	1	yes	maximal, characteristic
3	#16810443	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₄	1	1	yes	characteristic
6	#13133176	C₄	1	1	yes	characteristic
7	#13133157	—	2	1	no	—
8	#13133157	C₂²	1	2	yes	—
9	C₂×#11719908	—	2	1	no	—
10	#13133127	C₄	1	2	yes	—
11	C₈×Dic₅₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	M₃₂×C₅₃	—	2	1	no	—
14	M₃₂×C₅₃	C₂²	1	2	yes	—
15	C₂×C₁₆×C₅₃	—	2	1	no	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²⋊Dic₅₃	—	2	1	no	—
24	C₄⋊Dic₅₃	—	2	1	no	—
25	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
26	#11719917	C₈	1	2	yes	—
27	#11719917	C₈	1	2	yes	—
28	C₂×#88322	C₂×C₄	1	2	yes	—
29	C₁₆×C₅₃	—	2	2	no	—
30	#11719908	—	2	2	no	—
31	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₅₃	—	2	2	no	—
39	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	#88322	C₂×C₈	1	4	yes	—
44	C₂²×C₅₃	—	4	1	no	—
45	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
46	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
47	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
48	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
49	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
50	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
52	Dic₅₃	—	4	2	no	—
53	#2153	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	—	2	1	no	—
57	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₂×M₃₂	—	53	1	no	—
59	C₂×M₃₂	D₅₃	1	1	yes	characteristic
60	#405	—	53	1	no	—
61	#364	—	53	4	no	—
62	C₅₃	#2153	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₁₆	—	53	1	no	—
64	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
65	#117	—	53	1	no	—
66	M₃₂	—	2	1	no	—
67	M₃₂	—	53	2	no	—
68	M₃₂	—	106	1	no	—
69	M₃₂	C₂×D₅₃	1	2	yes	—
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	—	106	1	no	—
72	#97	—	53	2	no	—
73	C₄×C₈	—	53	2	no	—
74	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
75	M₁₆	—	53	2	no	—
76	M₁₆	—	53	2	no	—
77	C₂×C₈	—	53	2	no	—
78	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
79	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
80	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
83	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
84	C₄⋊C₄	—	106	1	no	—
85	C₂²⋊C₄	—	106	1	no	—
86	C₄²	—	53	2	no	—
87	C₁₆	—	2	2	no	—
88	C₁₆	—	106	2	no	—
89	C₂³	#11719912	1	1	yes	characteristic
90	C₂×C₄	—	106	2	no	—
91	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
92	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
93	C₂×C₄	#11719912	1	1	yes	characteristic
94	C₂×C₄	#11719919	1	1	yes	characteristic
95	C₂×C₄	#11719919	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
97	C₈	—	53	4	no	—
98	C₈	#11719919	1	1	yes	characteristic
99	C₈	#11719919	1	1	yes	characteristic
100	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
101	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂²	—	4	1	no	—
103	C₂²	#13133146	1	1	yes	characteristic
104	C₂²	#13133146	1	1	yes	characteristic
105	C₂²	C₂×#11719912	1	1	yes	characteristic
106	C₄	—	212	2	no	—
107	C₄	#13133146	1	1	yes	characteristic
108	C₄	#13133146	1	1	yes	characteristic
109	C₄	C₂×#11719919	1	1	yes	characteristic
110	C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
111	C₂	—	2	1	no	—
112	C₂	—	2	1	no	—
113	C₂	—	2	1	no	—
114	C₂	C₂×#13133146	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133157

C₂

yes

maximal, characteristic

#16810578

C₂

yes

maximal, characteristic

#16810443

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

characteristic

#13133176

C₄

yes

characteristic

#13133157

—

#13133157

C₂²

yes

—

C₂×#11719908

—

#13133127

C₄

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

C₂²

yes

—

C₂×C₁₆×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

—

#11719917

C₈

yes

—

#11719917

C₈

yes

—

C₂×#88322

C₂×C₄

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

#88322

C₂×C₈

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₈

yes

characteristic

C₂²×C₅₃

C₄⋊C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂²×C₈

yes

characteristic

Dic₅₃

—

#2153

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

D₅₃

yes

characteristic

#405

—

#364

—

C₅₃

#2153

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

M₃₂

—

M₃₂

—

M₃₂

—

106

—

M₃₂

C₂×D₅₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

#97

—

C₄×C₈

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

Dic₁₀₆

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

C₁₆

—

C₁₆

—

106

—

C₂³

#11719912

yes

characteristic

C₂×C₄

—

106

—

C₂×C₄

C₈×D₅₃

yes

characteristic

C₂×C₄

C₈×D₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₈

—

C₈

yes

C₈

yes

100

C₈

C₂×Dic₁₀₆

yes

characteristic

101

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

102

C₂²

—

103

C₂²

#13133146

yes

characteristic

104

C₂²

#13133146

yes

characteristic

105

C₂²

C₂×#11719912

yes

characteristic

106

C₄

—

212

—

107

C₄

#13133146

yes

characteristic

108

C₄

#13133146

yes

characteristic

109

C₄

C₂×#11719919

yes

characteristic

110

C₄

C₂×C₈×D₅₃

yes

characteristic

111

C₂

—

112

C₂

—

113

C₂

—

114

C₂

C₂×#13133146

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×M₃₂×C₅₃	C₂
socle	C₂×C₅₃	C₂×C₄⋊C₈
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#2153
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₈

C₂×C₂²⋊D₅₃

commutator

C₄×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×M₃₂×C₅₃

C₂

socle

C₂×C₅₃

C₂×C₄⋊C₈

hypercenter

C₂×M₃₂

D₅₃

hypocenter

C₅₃

#2153

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #16810443	C₂	semidirect
62. C₅₃	#2153	semidirect

H ≅ G/N

type

3. #16810443

C₂

semidirect

62. C₅₃

#2153

semidirect

Subgroups — #20276480

Conjugacy classes of subgroups

Complements