Group #20275469 — subgroups

Subgroups — #20275469

Order 6784.

200 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16810650	C₂	1	2	yes	maximal
3	C₂×#13133127	C₂	1	1	yes	maximal, characteristic
4	#16810382	C₂	1	2	yes	maximal
5	C₅₃×#427	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
9	#13133153	—	2	2	no	—
10	C₄⋊#88322	—	2	4	no	—
11	#13133127	—	2	2	no	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	#117×C₅₃	C₂²	1	2	yes	—
14	#107×C₅₃	C₄	1	2	yes	complemented
15	#106×C₅₃	C₄	1	2	yes	complemented
16	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	2	1	no	—
18	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	D₄	1	2	yes	—
21	C₂×C₈×C₅₃	Q₈	1	2	yes	—
22	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
23	C₂²⋊C₄×C₅₃	—	2	2	no	—
24	C₄²×C₅₃	—	2	2	no	—
25	C₄×Dic₅₃	—	2	1	no	—
26	C₄×Dic₅₃	Q₈	1	2	yes	—
27	#11719917	—	2	2	no	—
28	#11719917	—	2	2	no	—
29	C₂×#88322	—	2	1	no	—
30	C₂×#88322	—	2	1	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂×#88322	D₄	1	2	yes	—
33	C₂³×C₅₃	#41	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	4	no	—
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	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
41	C₂×Dic₅₃	—	2	2	no	—
42	C₂×Dic₅₃	—	4	2	no	—
43	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
45	#88322	—	2	2	no	—
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	4	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	2	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	D₅₃	1	1	yes	characteristic
64	#424	—	53	2	no	—
65	C₂×#97	—	53	1	no	—
66	#339	—	53	2	no	—
67	C₅₃	#1946	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₁₆	—	53	2	no	—
69	C₂²×C₈	—	53	1	no	—
70	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
71	#117	C₂×D₅₃	1	2	yes	—
72	C₂×C₄²	—	53	1	no	—
73	#108	—	106	2	no	—
74	#107	Dic₅₃	1	2	yes	complemented
75	#106	Dic₅₃	1	2	yes	complemented
76	C₄⋊C₈	—	106	4	no	—
77	#97	—	106	2	no	—
78	C₂²×C₄	—	106	1	no	—
79	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
80	M₁₆	—	106	2	no	—
81	M₁₆	—	106	2	no	—
82	C₂×C₈	—	53	2	no	—
83	C₂×C₈	—	106	1	no	—
84	C₂×C₈	—	106	1	no	—
85	C₂×C₈	—	106	2	no	—
86	C₂×C₈	Dic₁₀₆	1	2	yes	—
87	C₂×C₈	D₂₁₂	1	2	yes	—
88	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
89	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
90	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	53	2	no	—
94	C₄²	—	106	1	no	—
95	C₂³	#11719947	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	106	2	no	—
98	C₂×C₄	—	212	2	no	—
99	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
100	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
101	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
103	C₂×C₄	#11719949	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
105	C₈	—	106	2	no	—
106	C₈	—	212	2	no	—
107	C₈	C₄⋊Dic₅₃	1	2	yes	—
108	C₈	C₄⋊Dic₅₃	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
111	C₂²	#13133197	1	1	yes	characteristic
112	C₂²	#13133208	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	212	2	no	—
115	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
116	C₄	Q₈×Dic₅₃	1	1	yes	characteristic
117	C₄	#13133192	1	1	yes	characteristic
118	C₄	#13133216	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#16810522	1	1	yes	characteristic
122	C₂	#16810745	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133153

C₂

yes

maximal, characteristic

#16810650

C₂

yes

maximal

C₂×#13133127

C₂

yes

maximal, characteristic

#16810382

C₂

yes

maximal

C₅₃×#427

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

—

—

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

—

#107×C₅₃

C₄

yes

complemented

#106×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

Q₈

yes

—

—

—

—

—

—

D₄

yes

—

C₂³×C₅₃

#41

yes

characteristic

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

—

C₂×Dic₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

—

#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

D₅₃

yes

characteristic

#424

—

C₂×#97

—

#339

—

C₅₃

#1946

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

C₂×D₅₃

yes

—

C₂×C₄²

—

#108

—

106

—

#107

Dic₅₃

yes

complemented

#106

Dic₅₃

yes

complemented

C₄⋊C₈

—

106

—

#97

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

106

—

C₂³

#11719947

yes

characteristic

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

100

C₂×C₄

C₂×D₂₁₂

yes

characteristic

101

C₂×C₄

D₄×D₅₃

yes

characteristic

102

C₂×C₄

Q₈×D₅₃

yes

characteristic

103

C₂×C₄

#11719949

yes

characteristic

104

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

105

C₈

—

106

—

106

C₈

—

212

—

107

C₈

C₄⋊Dic₅₃

yes

—

108

C₈

C₄⋊Dic₅₃

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₅₃

yes

characteristic

111

C₂²

#13133197

yes

characteristic

112

C₂²

#13133208

yes

characteristic

113

C₄

—

114

C₄

—

212

—

115

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

116

C₄

Q₈×Dic₅₃

yes

characteristic

117

C₄

#13133192

yes

characteristic

118

C₄

#13133216

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#16810522

yes

characteristic

122

C₂

#16810745

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#427	C₂
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#427	D₅₃
hypocenter	C₅₃	#1946
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#427

C₂

socle

C₂²×C₅₃

C₄×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₅₃	C₄	semidirect
15. #106×C₅₃	C₄	semidirect
67. C₅₃	#1946	semidirect
74. #107	Dic₅₃	semidirect
75. #106	Dic₅₃	semidirect

H ≅ G/N

type

14. #107×C₅₃

C₄

semidirect

15. #106×C₅₃

C₄

semidirect

67. C₅₃

#1946

semidirect

74. #107

Dic₅₃

semidirect

75. #106

Dic₅₃

semidirect

Subgroups — #20275469

Conjugacy classes of subgroups

Complements