Group #20272597 — subgroups

Subgroups — #20272597

Order 6784.

180 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 73 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×#13133128	C₂	1	1	yes	maximal, characteristic
2	#16810489	C₂	1	1	yes	maximal, characteristic
3	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	—	2	1	no	—
5	C₂×C₄⋊Dic₅₃	C₄	1	2	yes	—
6	#13133128	—	2	1	no	—
7	#13133128	C₄	1	2	yes	complemented
8	#13133124	—	2	1	no	—
9	#13133124	—	2	1	no	—
10	C₂²×C₈×C₅₃	—	2	1	no	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	#97×C₅₃	—	2	1	no	—
13	#97×C₅₃	C₄	1	2	yes	complemented
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	—	2	4	no	—
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	D₄	1	1	yes	characteristic
23	C₄²×C₅₃	Q₈	1	1	yes	characteristic
24	C₄⋊Dic₅₃	—	4	2	no	—
25	C₂×#88322	—	2	4	no	—
26	C₂×#88322	—	2	2	no	—
27	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	1	no	—
29	C₂×C₄×C₅₃	—	2	1	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×Dic₅₃	—	4	2	no	—
39	C₂×Dic₅₃	—	4	2	no	—
40	C₂×Dic₅₃	—	4	2	no	—
41	C₈×C₅₃	—	2	4	no	—
42	#88322	—	2	4	no	—
43	C₂²×C₅₃	#95	1	1	yes	characteristic
44	C₂²×C₅₃	#100	1	1	yes	characteristic
45	C₂²×C₅₃	#101	1	1	yes	characteristic
46	C₂²×C₅₃	#104	1	2	yes	—
47	C₂²×C₅₃	#104	1	2	yes	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	2	1	no	—
54	Dic₅₃	—	8	2	no	—
55	#1295	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	#329	1	2	yes	—
57	C₂×C₅₃	#330	1	2	yes	—
58	C₂×C₅₃	#339	1	1	yes	characteristic
59	C₂×C₅₃	#339	1	1	yes	characteristic
60	C₂×C₅₃	#343	1	1	yes	characteristic
61	C₂×#97	—	53	1	no	—
62	C₂×#97	D₅₃	1	1	yes	characteristic
63	#382	—	53	1	no	—
64	C₅₃	#1295	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	106	1	no	—
67	C₂×C₄⋊C₄	—	53	2	no	—
68	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
69	#97	—	2	1	no	—
70	#97	—	53	2	no	—
71	#97	—	106	1	no	—
72	#97	Dic₅₃	1	2	yes	complemented
73	#95	—	106	1	no	—
74	#95	—	106	1	no	—
75	C₂²×C₄	—	106	2	no	—
76	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
77	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	106	4	no	—
82	C₂×C₈	—	106	2	no	—
83	C₄⋊C₄	—	212	2	no	—
84	C₄²	Dic₁₀₆	1	1	yes	characteristic
85	C₄²	C₄×D₅₃	1	1	yes	characteristic
86	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
87	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
88	C₂³	#11719921	1	1	yes	characteristic
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	212	2	no	—
94	C₂×C₄	—	212	2	no	—
95	C₂×C₄	—	212	2	no	—
96	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
97	C₂×C₄	#11719919	1	1	yes	characteristic
98	C₂×C₄	#11719919	1	1	yes	characteristic
99	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
100	C₂×C₄	#11719921	1	1	yes	characteristic
101	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
102	C₈	—	2	4	no	—
103	C₈	—	106	4	no	—
104	C₂²	#13133124	1	1	yes	characteristic
105	C₂²	#13133133	1	1	yes	characteristic
106	C₂²	#13133135	1	1	yes	characteristic
107	C₂²	#13133143	1	2	yes	—
108	C₂²	#13133145	1	2	yes	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	424	2	no	—
116	C₂	#16810362	1	2	yes	—
117	C₂	#16810364	1	2	yes	—
118	C₂	#16810383	1	1	yes	characteristic
119	C₂	#16810384	1	1	yes	characteristic
120	C₂	#16810391	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133128

C₂

yes

maximal, characteristic

#16810489

C₂

yes

maximal, characteristic

C₂×#97×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

—

C₂×C₄⋊Dic₅₃

C₄

yes

—

#13133128

—

#13133128

C₄

yes

complemented

#13133124

—

#13133124

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#97×C₅₃

—

#97×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

Q₈

yes

characteristic

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#104

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#1295

—

maximal, Sylow

C₂×C₅₃

#329

yes

—

C₂×C₅₃

#330

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×#97

—

C₂×#97

D₅₃

yes

characteristic

#382

—

C₅₃

#1295

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

—

—

—

106

—

#97

Dic₅₃

yes

complemented

#95

—

106

—

#95

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

212

—

C₄²

Dic₁₀₆

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₂³

#11719921

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

C₄×Dic₅₃

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Dic₅₃

yes

characteristic

100

C₂×C₄

#11719921

yes

characteristic

101

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

102

C₈

—

103

C₈

—

106

—

104

C₂²

#13133124

yes

characteristic

105

C₂²

#13133133

yes

characteristic

106

C₂²

#13133135

yes

characteristic

107

C₂²

#13133143

yes

—

108

C₂²

#13133145

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

424

—

116

C₂

#16810362

yes

—

117

C₂

#16810364

yes

—

118

C₂

#16810383

yes

characteristic

119

C₂

#16810384

yes

characteristic

120

C₂

#16810391

yes

characteristic

	subgroup	G/H
center	C₂³	#11719921
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₂×#97×C₅₃	C₂
socle	C₂³×C₅₃	C₂²⋊C₄
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	#1295
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

#11719921

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₂×C₄²

C₂×D₅₃

Fitting

C₂×#97×C₅₃

C₂

socle

C₂³×C₅₃

C₂²⋊C₄

hypercenter

C₂×#97

D₅₃

hypocenter

C₅₃

#1295

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #13133128	C₄	semidirect
13. #97×C₅₃	C₄	semidirect
64. C₅₃	#1295	semidirect
72. #97	Dic₅₃	semidirect

H ≅ G/N

type

7. #13133128

C₄

semidirect

13. #97×C₅₃

C₄

semidirect

64. C₅₃

#1295

semidirect

72. #97

Dic₅₃

semidirect

Subgroups — #20272597

Conjugacy classes of subgroups

Complements