Group #20274238 — subgroups

Subgroups — #20274238

Order 6784.

154 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16811028	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810913	C₂	1	2	yes	maximal
3	#16810374	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810371	C₂	1	2	yes	maximal, complemented
5	C₅₃×#436	C₂	1	1	yes	maximal, characteristic
6	#13133217	C₂²	1	2	yes	—
7	#13133151	—	2	1	no	—
8	#13133151	—	2	1	no	—
9	#13133148	—	2	2	no	—
10	C₄⋊#88322	C₂²	1	2	yes	—
11	#13133138	—	2	2	no	—
12	C₂×M₁₆×C₅₃	—	2	1	no	—
13	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₅₃	—	2	1	no	—
16	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	—	2	1	no	—
19	C₂×Dic₁₀₆	—	2	2	no	—
20	M₁₆×C₅₃	—	4	2	no	—
21	C₂×C₈×C₅₃	—	2	1	no	—
22	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²⋊C₄×C₅₃	—	2	1	no	—
26	C₄²×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×#88322	—	2	2	no	—
30	C₂³×C₅₃	—	2	1	no	—
31	D₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	4	1	no	—
34	C₂×C₄×C₅₃	—	4	1	no	—
35	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
38	C₂×Dic₅₃	—	4	2	no	—
39	Dic₁₀₆	—	8	2	no	—
40	C₈×C₅₃	—	2	1	no	—
41	C₈×C₅₃	—	4	1	no	—
42	C₈×C₅₃	SD₁₆	1	2	yes	—
43	#88322	—	4	2	no	—
44	C₂²×C₅₃	—	4	1	no	—
45	C₂²×C₅₃	—	4	1	no	—
46	C₂²×C₅₃	—	4	1	no	—
47	C₂²×C₅₃	#121	1	1	yes	characteristic
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	4	1	no	—
50	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₅₃	#137	1	1	yes	characteristic
52	C₄×C₅₃	#137	1	2	yes	—
53	Dic₅₃	—	8	2	no	—
54	#1695	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	—	8	1	no	—
56	C₂×C₅₃	#461	1	1	yes	characteristic
57	C₂×C₅₃	#470	1	1	yes	characteristic
58	C₂×C₅₃	#472	1	1	yes	characteristic
59	#500	—	53	1	no	—
60	#476	—	53	2	no	—
61	#436	D₅₃	1	1	yes	characteristic
62	#334	—	53	1	no	—
63	#333	—	53	2	no	—
64	C₅₃	#1695	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₁₆	—	2	1	no	—
66	C₄⋊Q₈	—	53	2	no	—
67	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
68	#107	—	106	1	no	—
69	#107	—	106	1	no	—
70	#106	—	106	2	no	—
71	C₄⋊C₈	—	53	2	no	—
72	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
73	#102	—	106	2	no	—
74	C₂²⋊C₈	—	2	1	no	—
75	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
76	C₂×Q₈	—	106	2	no	—
77	C₂×D₄	D₂₁₂	1	1	yes	characteristic
78	C₂²×C₄	—	2	1	no	—
79	M₁₆	—	4	2	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	106	2	no	—
82	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
84	C₄⋊C₄	—	106	2	no	—
85	C₄⋊C₄	—	106	2	no	—
86	C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
87	C₂²⋊C₄	—	2	1	no	—
88	C₄²	C₂²×D₅₃	1	1	yes	characteristic
89	C₂³	—	2	1	no	—
90	Q₈	—	424	2	no	—
91	D₄	—	2	2	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	212	2	no	—
96	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
97	C₂×C₄	#11719945	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	212	2	no	—
102	C₈	#11719923	1	2	yes	—
103	C₂²	—	4	1	no	—
104	C₂²	—	4	1	no	—
105	C₂²	—	4	1	no	—
106	C₂²	#13133191	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	4	1	no	—
109	C₄	—	424	2	no	—
110	C₄	C₂×#11719923	1	1	yes	characteristic
111	C₄	#13133247	1	1	yes	characteristic
112	C₄	#13133250	1	2	yes	—
113	C₂	—	8	1	no	—
114	C₂	#16810828	1	1	yes	characteristic
115	C₂	#16810881	1	1	yes	characteristic
116	C₂	#16810893	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811028

C₂

yes

maximal, characteristic, complemented

#16810913

C₂

yes

maximal

#16810374

C₂

yes

maximal, characteristic, complemented

#16810371

C₂

yes

maximal, complemented

C₅₃×#436

C₂

yes

maximal, characteristic

#13133217

C₂²

yes

—

—

—

—

C₂²

yes

—

#13133138

—

C₂×M₁₆×C₅₃

—

C₄×D₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂³×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

SD₁₆

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#121

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

#1695

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

characteristic

#334

—

#333

—

C₅₃

#1695

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

C₄×D₄

C₂×D₅₃

yes

characteristic

#107

—

106

—

#107

—

106

—

#106

—

106

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

characteristic

#102

—

106

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×D₄

D₂₁₂

yes

characteristic

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

D₂₁₂

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂²×D₅₃

yes

characteristic

C₂³

—

Q₈

—

424

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₂×C₄

#11719945

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

212

—

102

C₈

#11719923

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

#13133191

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

424

—

110

C₄

C₂×#11719923

yes

characteristic

111

C₄

#13133247

yes

characteristic

112

C₄

#13133250

yes

—

113

C₂

—

114

C₂

#16810828

yes

characteristic

115

C₂

#16810881

yes

characteristic

116

C₂

#16810893

yes

characteristic

	subgroup	G/H
center	C₂²	#13133191
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#436	C₂
socle	C₂²×C₅₃	#121
hypercenter	#436	D₅₃
hypocenter	C₅₃	#1695
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #436

subgroup

G/H

center

C₂²

#13133191

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#436

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #436

N	H ≅ G/N	type
1. #16811028	C₂	semidirect
3. #16810374	C₂	semidirect
4. #16810371	C₂	semidirect
64. C₅₃	#1695	semidirect

H ≅ G/N

type

1. #16811028

C₂

semidirect

3. #16810374

C₂

semidirect

4. #16810371

C₂

semidirect

64. C₅₃

#1695

semidirect

Subgroups — #20274238

Conjugacy classes of subgroups

Complements