Group #20272724 — subgroups

Subgroups — #20272724

Order 6784.

126 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 41 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	C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810386	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#466	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₅₃	—	2	1	no	—
5	C₂×#11719908	—	2	1	no	—
6	C₂×D₈×C₅₃	C₄	1	1	yes	characteristic, complemented
7	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	#121×C₅₃	—	2	1	no	—
9	#107×C₅₃	C₄	1	1	yes	characteristic, complemented
10	#102×C₅₃	—	2	1	no	—
11	C₂×D₄×C₅₃	—	2	1	no	—
12	C₂×D₄×C₅₃	—	4	1	no	—
13	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
14	C₂²×Dic₅₃	—	4	1	no	—
15	D₈×C₅₃	—	4	1	no	—
16	C₂×C₈×C₅₃	—	4	1	no	—
17	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₄⋊C₄×C₅₃	—	2	1	no	—
20	C₂²⋊C₄×C₅₃	—	4	1	no	—
21	C₄⋊Dic₅₃	—	2	2	no	—
22	C₄⋊Dic₅₃	—	4	1	no	—
23	#11719908	—	4	1	no	—
24	C₂³×C₅₃	—	4	1	no	—
25	C₂³×C₅₃	D₈	1	1	yes	characteristic
26	D₄×C₅₃	—	4	1	no	—
27	D₄×C₅₃	—	4	1	no	—
28	D₄×C₅₃	—	8	1	no	—
29	D₄×C₅₃	—	8	1	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	4	1	no	—
33	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
35	C₂×Dic₅₃	—	4	2	no	—
36	C₂×Dic₅₃	—	8	1	no	—
37	C₂×Dic₅₃	—	8	1	no	—
38	C₈×C₅₃	—	2	1	no	—
39	C₈×C₅₃	—	2	1	no	—
40	C₂²×C₅₃	—	2	1	no	—
41	C₂²×C₅₃	—	2	1	no	—
42	C₂²×C₅₃	—	8	1	no	—
43	C₂²×C₅₃	—	8	1	no	—
44	C₂²×C₅₃	—	8	1	no	—
45	C₂²×C₅₃	#102	1	1	yes	characteristic
46	C₂²×C₅₃	D₁₆	1	1	yes	characteristic
47	C₂²×C₅₃	SD₃₂	1	1	yes	characteristic
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	8	1	no	—
50	C₄×C₅₃	#99	1	1	yes	characteristic
51	C₄×C₅₃	#104	1	1	yes	characteristic
52	Dic₅₃	—	8	2	no	—
53	#1347	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	—	2	1	no	—
55	C₂×C₅₃	—	2	1	no	—
56	C₂×C₅₃	—	16	1	no	—
57	C₂×C₅₃	#327	1	1	yes	characteristic
58	C₂×C₅₃	#357	1	1	yes	characteristic
59	C₂×C₅₃	#360	1	1	yes	characteristic
60	#466	D₅₃	1	1	yes	characteristic
61	C₂²⋊C₁₆	—	53	1	no	—
62	#340	—	53	1	no	—
63	C₅₃	#1347	1	1	yes	characteristic, complemented, Sylow
64	C₂×D₈	Dic₅₃	1	1	yes	characteristic, complemented
65	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
66	#121	—	2	1	no	—
67	C₂×C₄⋊C₄	—	106	1	no	—
68	C₂×C₁₆	—	106	1	no	—
69	#107	Dic₅₃	1	1	yes	characteristic, complemented
70	#102	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	C₂×D₄	—	4	1	no	—
73	C₂²×C₄	—	212	1	no	—
74	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
75	D₈	—	4	1	no	—
76	C₂×C₈	—	4	1	no	—
77	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	106	2	no	—
81	C₄⋊C₄	—	212	1	no	—
82	C₂²⋊C₄	—	4	1	no	—
83	C₁₆	—	212	1	no	—
84	C₂³	—	4	1	no	—
85	C₂³	#11719922	1	1	yes	characteristic
86	D₄	—	4	1	no	—
87	D₄	—	4	1	no	—
88	D₄	—	8	1	no	—
89	D₄	—	8	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	424	1	no	—
94	C₂×C₄	—	424	1	no	—
95	C₂×C₄	—	212	2	no	—
96	C₂×C₄	#11719923	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
98	C₈	—	2	1	no	—
99	C₈	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	8	1	no	—
103	C₂²	—	8	1	no	—
104	C₂²	—	8	1	no	—
105	C₂²	#13133138	1	1	yes	characteristic
106	C₂²	#13133158	1	1	yes	characteristic
107	C₂²	#13133161	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	8	1	no	—
110	C₄	—	424	2	no	—
111	C₄	#13133132	1	1	yes	characteristic
112	C₄	#13133144	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	—	16	1	no	—
116	C₂	#16810357	1	1	yes	characteristic
117	C₂	#16810424	1	1	yes	characteristic
118	C₂	#16810433	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²⋊#11719908

C₂

yes

maximal, characteristic, complemented

#16810386

C₂

yes

maximal, characteristic, complemented

C₅₃×#466

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

—

C₂×#11719908

—

C₂×D₈×C₅₃

C₄

yes

characteristic, complemented

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#121×C₅₃

—

#107×C₅₃

C₄

yes

characteristic, complemented

#102×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

D₈

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

characteristic

C₂²×C₅₃

D₁₆

yes

characteristic

C₂²×C₅₃

SD₃₂

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1347

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

C₂²⋊C₁₆

—

#340

—

C₅₃

#1347

yes

characteristic, complemented, Sylow

C₂×D₈

Dic₅₃

yes

characteristic, complemented

C₂²×C₈

C₂×D₅₃

yes

characteristic

#121

—

C₂×C₄⋊C₄

—

106

—

C₂×C₁₆

—

106

—

#107

Dic₅₃

yes

characteristic, complemented

#102

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

D₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

C₁₆

—

212

—

C₂³

—

C₂³

#11719922

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

424

—

C₂×C₄

—

424

—

C₂×C₄

—

212

—

C₂×C₄

#11719923

yes

characteristic

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₈

—

C₈

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

#13133138

yes

characteristic

106

C₂²

#13133158

yes

characteristic

107

C₂²

#13133161

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

424

—

111

C₄

#13133132

yes

characteristic

112

C₄

#13133144

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

—

116

C₂

#16810357

yes

characteristic

117

C₂

#16810424

yes

characteristic

118

C₂

#16810433

yes

characteristic

	subgroup	G/H
center	C₂²	#13133138
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₅₃×#466	C₂
socle	C₂²×C₅₃	#102
hypercenter	#466	D₅₃
hypocenter	C₅₃	#1347
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

subgroup

G/H

center

C₂²

#13133138

commutator

C₂×C₈×C₅₃

C₂×C₄

Frattini

C₂²×C₈

C₂×D₅₃

Fitting

C₅₃×#466

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

N	H ≅ G/N	type
1. C₂²⋊#11719908	C₂	semidirect
2. #16810386	C₂	semidirect
6. C₂×D₈×C₅₃	C₄	semidirect
9. #107×C₅₃	C₄	semidirect
63. C₅₃	#1347	semidirect
64. C₂×D₈	Dic₅₃	semidirect
69. #107	Dic₅₃	semidirect

H ≅ G/N

type

1. C₂²⋊#11719908

C₂

semidirect

2. #16810386

C₂

semidirect

6. C₂×D₈×C₅₃

C₄

semidirect

9. #107×C₅₃

C₄

semidirect

63. C₅₃

#1347

semidirect

64. C₂×D₈

Dic₅₃

semidirect

69. #107

Dic₅₃

semidirect

Subgroups — #20272724

Conjugacy classes of subgroups

Complements