Group #20275503 — subgroups

Subgroups — #20275503

Order 6784.

184 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16810671	C₂	1	2	yes	maximal
2	C₂×#13133153	C₂	1	1	yes	maximal, characteristic
3	#16810578	C₂	1	1	yes	maximal, characteristic
4	#16810394	C₂	1	2	yes	maximal
5	C₂×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂×#11719917	C₂²	1	1	yes	characteristic
8	#13133176	C₂²	1	1	yes	characteristic
9	#13133153	C₄	1	4	yes	complemented
10	#13133152	—	2	4	no	—
11	#13133127	—	2	1	no	—
12	C₈×Dic₅₃	—	2	1	no	—
13	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	#108×C₅₃	—	2	1	no	—
16	#108×C₅₃	—	2	1	no	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	M₁₆×C₅₃	—	2	2	no	—
19	M₁₆×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	Q₈	1	2	yes	—
25	C₂²⋊Dic₅₃	Q₈	1	2	yes	—
26	C₄⋊Dic₅₃	—	2	1	no	—
27	C₄×Dic₅₃	—	2	1	no	—
28	#11719917	—	2	2	no	—
29	#11719917	—	2	1	no	—
30	#11719917	C₂×C₄	1	4	yes	—
31	#11719917	D₄	1	2	yes	—
32	C₂×#88322	—	2	1	no	—
33	C₂×#88322	—	2	2	no	—
34	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
41	C₂×Dic₅₃	—	2	2	no	—
42	C₈×C₅₃	—	4	2	no	—
43	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
45	#88322	—	2	4	no	—
46	#88322	—	2	2	no	—
47	C₂²×C₅₃	—	4	1	no	—
48	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
53	C₄×C₅₃	#121	1	1	yes	characteristic
54	C₄×C₅₃	#125	1	1	yes	characteristic
55	Dic₅₃	—	4	2	no	—
56	#1953	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	—	2	1	no	—
58	C₂×C₅₃	—	2	1	no	—
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	#389	1	1	yes	characteristic
61	#430	—	53	2	no	—
62	C₂×#108	—	53	1	no	—
63	C₂×#108	D₅₃	1	1	yes	characteristic
64	#405	—	53	1	no	—
65	#344	—	53	2	no	—
66	C₅₃	#1953	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₁₆	—	53	2	no	—
68	C₂×M₁₆	—	53	1	no	—
69	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
70	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
71	#117	—	53	1	no	—
72	#108	—	2	1	no	—
73	#108	—	2	1	no	—
74	#108	—	53	4	no	—
75	#108	—	106	4	no	—
76	#97	—	106	1	no	—
77	C₄×C₈	—	106	1	no	—
78	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	53	4	no	—
82	M₁₆	—	53	2	no	—
83	M₁₆	—	106	1	no	—
84	M₁₆	—	106	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	106	2	no	—
87	C₂×C₈	—	106	1	no	—
88	C₂×C₈	Dic₁₀₆	1	2	yes	—
89	C₂×C₈	C₄×D₅₃	1	2	yes	—
90	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
92	C₄⋊C₄	—	106	1	no	—
93	C₂²⋊C₄	—	53	2	no	—
94	C₄²	—	106	1	no	—
95	C₂³	Q₈×D₅₃	1	1	yes	characteristic
96	C₂×C₄	—	106	2	no	—
97	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
98	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
99	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
100	C₂×C₄	#11719947	1	1	yes	characteristic
101	C₂×C₄	#11719949	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	106	4	no	—
105	C₈	—	106	2	no	—
106	C₈	#11719919	1	2	yes	—
107	C₈	#11719919	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#13133182	1	1	yes	characteristic
110	C₂²	#13133193	1	1	yes	characteristic
111	C₂²	#13133216	1	1	yes	characteristic
112	C₄	—	212	2	no	—
113	C₄	C₂×#11719919	1	1	yes	characteristic
114	C₄	#13133179	1	1	yes	characteristic
115	C₄	#13133188	1	1	yes	characteristic
116	C₄	#13133208	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#16810519	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810671

C₂

yes

maximal

C₂×#13133153

C₂

yes

maximal, characteristic

#16810578

C₂

yes

maximal, characteristic

#16810394

C₂

yes

maximal

C₂×#108×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×#11719917

C₂²

yes

characteristic

#13133176

C₂²

yes

characteristic

#13133153

C₄

yes

complemented

#13133152

—

#13133127

—

C₈×Dic₅₃

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#108×C₅₃

—

#108×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

—

C₂²⋊Dic₅₃

Q₈

yes

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

#11719917

C₂×C₄

yes

—

#11719917

D₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×Q₈

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1953

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#405

—

#344

—

C₅₃

#1953

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

C₂×D₅₃

yes

—

—

—

—

—

106

—

#97

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

C₄²

—

106

—

C₂³

Q₈×D₅₃

yes

characteristic

C₂×C₄

—

106

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

D₄×D₅₃

yes

characteristic

100

C₂×C₄

#11719947

yes

characteristic

101

C₂×C₄

#11719949

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

103

C₈

—

104

C₈

—

106

—

105

C₈

—

106

—

106

C₈

#11719919

yes

—

107

C₈

#11719919

yes

—

108

C₂²

—

109

C₂²

#13133182

yes

characteristic

110

C₂²

#13133193

yes

characteristic

111

C₂²

#13133216

yes

characteristic

112

C₄

—

212

—

113

C₄

C₂×#11719919

yes

characteristic

114

C₄

#13133179

yes

characteristic

115

C₄

#13133188

yes

characteristic

116

C₄

#13133208

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#16810519

yes

characteristic

	subgroup	G/H
center	C₄	#13133188
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#108×C₅₃	C₂
socle	C₂×C₅₃	#389
hypercenter	C₂×#108	D₅₃
hypocenter	C₅₃	#1953
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#13133188

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#108×C₅₃

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #13133153	C₄	semidirect
66. C₅₃	#1953	semidirect

H ≅ G/N

type

9. #13133153

C₄

semidirect

66. C₅₃

#1953

semidirect

Subgroups — #20275503

Conjugacy classes of subgroups

Complements