Group #20275468 — subgroups

Subgroups — #20275468

Order 6784.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13133153	C₂	1	1	yes	maximal, characteristic
2	#16810661	C₂	1	1	yes	maximal, characteristic
3	#16810649	C₂	1	2	yes	maximal
4	#16810383	C₂	1	2	yes	maximal
5	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	#13133176	C₂²	1	2	yes	—
8	#13133153	—	2	2	no	—
9	#13133151	C₄	1	2	yes	complemented
10	#13133149	C₄	1	2	yes	complemented
11	#13133146	—	2	4	no	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	#97×C₅₃	—	2	2	no	—
16	C₂²×C₄×C₅₃	—	2	1	no	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₅₃	—	2	1	no	—
19	C₂×C₈×C₅₃	—	2	1	no	—
20	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	D₄	1	2	yes	—
23	C₂×C₈×C₅₃	D₄	1	2	yes	—
24	C₂×C₈×C₅₃	Q₈	1	2	yes	—
25	C₄²×C₅₃	—	2	1	no	—
26	C₄²×C₅₃	Q₈	1	2	yes	—
27	C₂²⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
29	C₄×Dic₅₃	—	2	2	no	—
30	#11719917	—	2	2	no	—
31	#11719917	—	2	2	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂³×C₅₃	#41	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	—	4	2	no	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
42	C₂×Dic₅₃	—	2	4	no	—
43	C₈×C₅₃	—	2	2	no	—
44	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	#88322	—	4	2	no	—
47	C₂²×C₅₃	—	2	2	no	—
48	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₅₃	#125	1	1	yes	characteristic
51	C₄×C₅₃	—	4	2	no	—
52	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₅₃	#121	1	1	yes	characteristic
55	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₅₃	—	4	4	no	—
57	#1946	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	—	2	1	no	—
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	#389	1	1	yes	characteristic
61	C₂×C₅₃	#444	1	2	yes	—
62	C₂×#108	—	53	1	no	—
63	#427	—	53	1	no	—
64	#424	—	53	2	no	—
65	C₂×#97	D₅₃	1	1	yes	characteristic
66	#339	—	53	2	no	—
67	C₅₃	#1946	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₁₆	—	53	2	no	—
69	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
71	#117	—	53	2	no	—
72	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
73	#108	—	106	2	no	—
74	#107	—	53	2	no	—
75	#106	—	53	2	no	—
76	C₄⋊C₈	—	106	4	no	—
77	#97	—	2	2	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
80	M₁₆	—	106	2	no	—
81	M₁₆	—	106	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	106	2	no	—
85	C₂×C₈	Dic₁₀₆	1	2	yes	—
86	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
87	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
88	C₂×C₈	D₂₁₂	1	2	yes	—
89	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
90	C₄⋊C₄	—	53	4	no	—
91	C₂²⋊C₄	—	106	2	no	—
92	C₄²	—	2	1	no	—
93	C₄²	—	106	2	no	—
94	C₄²	Dic₁₀₆	1	2	yes	—
95	C₂³	#11719945	1	1	yes	characteristic
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	106	4	no	—
99	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
102	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
103	C₂×C₄	#11719945	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	212	2	no	—
107	C₈	#11719919	1	2	yes	—
108	C₈	#11719919	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	C₄×D₂₁₂	1	1	yes	characteristic
111	C₂²	#13133196	1	1	yes	characteristic
112	C₂²	#13133209	1	1	yes	characteristic
113	C₄	—	4	2	no	—
114	C₄	—	212	4	no	—
115	C₄	C₂×#11719919	1	1	yes	characteristic
116	C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
117	C₄	#13133191	1	1	yes	characteristic
118	C₄	#13133217	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#16810521	1	1	yes	characteristic
122	C₂	#16810746	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133153

C₂

yes

maximal, characteristic

#16810661

C₂

yes

maximal, characteristic

#16810649

C₂

yes

maximal

#16810383

C₂

yes

maximal

C₂×#97×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

#13133176

C₂²

yes

—

#13133153

—

#13133151

C₄

yes

complemented

#13133149

C₄

yes

complemented

#13133146

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#97×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

Q₈

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

—

#11719917

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#41

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

#125

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

C₄×C₅₃

#121

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

#1946

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#389

yes

characteristic

C₂×C₅₃

#444

yes

—

C₂×#108

—

#427

—

#424

—

C₂×#97

D₅₃

yes

characteristic

#339

—

C₅₃

#1946

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#108

—

106

—

#107

—

#106

—

C₄⋊C₈

—

106

—

#97

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

C₄×D₅₃

yes

characteristic

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

C₂²⋊D₅₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

106

—

C₄²

—

C₄²

—

106

—

C₄²

Dic₁₀₆

yes

—

C₂³

#11719945

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

100

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

101

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

102

C₂×C₄

C₂×D₂₁₂

yes

characteristic

103

C₂×C₄

#11719945

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

105

C₈

—

106

C₈

—

212

—

107

C₈

#11719919

yes

—

108

C₈

#11719919

yes

—

109

C₂²

—

110

C₂²

C₄×D₂₁₂

yes

characteristic

111

C₂²

#13133196

yes

characteristic

112

C₂²

#13133209

yes

characteristic

113

C₄

—

114

C₄

—

212

—

115

C₄

C₂×#11719919

yes

characteristic

116

C₄

C₄×Dic₁₀₆

yes

characteristic

117

C₄

#13133191

yes

characteristic

118

C₄

#13133217

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#16810521

yes

characteristic

122

C₂

#16810746

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#97×C₅₃	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	#1946
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

subgroup

G/H

center

C₂×C₄

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#97×C₅₃

C₂

socle

C₂²×C₅₃

C₄×D₄

hypercenter

C₂×#97

D₅₃

hypocenter

C₅₃

#1946

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#97

N	H ≅ G/N	type
9. #13133151	C₄	semidirect
10. #13133149	C₄	semidirect
67. C₅₃	#1946	semidirect

H ≅ G/N

type

9. #13133151

C₄

semidirect

10. #13133149

C₄

semidirect

67. C₅₃

#1946

semidirect

Subgroups — #20275468

Conjugacy classes of subgroups

Complements