Group #20276856 — subgroups

Subgroups — #20276856

Order 6784.

130 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 49 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	#16811027	C₂	1	1	yes	maximal, characteristic
2	#16810717	C₂	1	1	yes	maximal, characteristic
3	#16810450	C₂	1	1	yes	maximal, characteristic
4	#16810441	C₂	1	1	yes	maximal, characteristic
5	#16810428	C₂	1	1	yes	maximal, characteristic
6	#16810426	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#367	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
9	#13133216	—	2	1	no	—
10	#13133182	—	2	1	no	—
11	C₂×#11719908	C₂²	1	1	yes	characteristic
12	#13133151	C₂²	1	1	yes	characteristic
13	#13133150	—	2	1	no	—
14	#13133140	—	2	1	no	—
15	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	#107×C₅₃	C₂²	1	1	yes	characteristic
18	#107×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₀₆	—	2	1	no	—
20	C₂×Dic₁₀₆	—	4	1	no	—
21	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₅₃	—	2	1	no	—
23	C₄⋊C₄×C₅₃	—	2	1	no	—
24	C₄⋊Dic₅₃	—	2	1	no	—
25	#11719919	—	4	1	no	—
26	#11719919	—	4	1	no	—
27	C₄×Dic₅₃	—	4	1	no	—
28	C₄×Dic₅₃	D₄	1	1	yes	characteristic
29	C₂×#88322	D₄	1	1	yes	characteristic
30	Dic₂₁₂	—	2	1	no	—
31	Dic₂₁₂	Q₈	1	2	yes	complemented
32	C₁₆×C₅₃	—	2	1	no	—
33	#11719908	—	2	1	no	—
34	C₂×C₄×C₅₃	—	4	1	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₅₃	—	4	1	no	—
38	C₂×Dic₅₃	—	4	1	no	—
39	C₂×Dic₅₃	D₈	1	1	yes	characteristic
40	C₂×Dic₅₃	D₈	1	1	yes	characteristic
41	Dic₁₀₆	—	2	2	no	—
42	Dic₁₀₆	—	4	1	no	—
43	Dic₁₀₆	—	8	2	no	—
44	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₅₃	#41	1	1	yes	characteristic
46	#88322	—	2	1	no	—
47	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
48	C₄×C₅₃	—	8	1	no	—
49	C₄×C₅₃	—	8	1	no	—
50	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
51	C₄×C₅₃	#137	1	1	yes	characteristic
52	Dic₅₃	—	2	1	no	—
53	Dic₅₃	—	4	2	no	—
54	Dic₅₃	—	8	1	no	—
55	Dic₅₃	SD₃₂	1	2	yes	—
56	#2225	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	#474	1	1	yes	characteristic
58	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
59	C₂×C₅₃	#510	1	1	yes	characteristic
60	#500	—	53	1	no	—
61	C₄×Q₁₆	—	53	1	no	—
62	#367	—	53	1	no	—
63	#367	D₅₃	1	1	yes	characteristic
64	C₄⋊C₁₆	—	53	1	no	—
65	#358	—	53	1	no	—
66	#358	—	53	1	no	—
67	C₅₃	#2225	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₁₆	—	53	1	no	—
69	C₄⋊Q₈	—	106	1	no	—
70	C₄×Q₈	—	106	1	no	—
71	C₂×C₁₆	—	53	1	no	—
72	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
73	#107	—	53	1	no	—
74	#107	—	106	1	no	—
75	#107	C₂×D₅₃	1	1	yes	characteristic
76	#107	C₂×D₅₃	1	1	yes	characteristic
77	#103	—	106	1	no	—
78	C₄×C₈	—	53	1	no	—
79	C₂×Q₈	—	106	1	no	—
80	C₂×Q₈	—	212	1	no	—
81	Q₁₆	—	53	2	no	—
82	Q₁₆	—	106	1	no	—
83	C₂×C₈	—	53	1	no	—
84	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	2	1	no	—
87	C₄⋊C₄	—	106	1	no	—
88	C₄⋊C₄	—	212	1	no	—
89	C₄⋊C₄	—	212	1	no	—
90	C₄²	—	53	1	no	—
91	C₄²	—	212	1	no	—
92	C₁₆	—	2	1	no	—
93	C₁₆	—	106	1	no	—
94	Q₈	—	106	2	no	—
95	Q₈	—	424	2	no	—
96	Q₈	—	212	1	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	212	1	no	—
102	C₂×C₄	—	212	1	no	—
103	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
104	C₈	—	106	1	no	—
105	C₈	#11719945	1	1	yes	characteristic
106	C₈	Q₈×D₅₃	1	1	yes	characteristic
107	C₂²	D₈×D₅₃	1	1	yes	characteristic
108	C₄	—	8	1	no	—
109	C₄	—	8	1	no	—
110	C₄	—	53	2	no	—
111	C₄	—	106	1	no	—
112	C₄	—	424	1	no	—
113	C₄	—	212	2	no	—
114	C₄	#13133195	1	1	yes	characteristic
115	C₄	#13133249	1	1	yes	characteristic
116	C₂	#16810902	1	1	yes	characteristic
117	C₂	SD₃₂×D₅₃	1	1	yes	characteristic
118	C₂	#16811062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811027

C₂

yes

maximal, characteristic

#16810717

C₂

yes

maximal, characteristic

#16810450

C₂

yes

maximal, characteristic

#16810441

C₂

yes

maximal, characteristic

#16810428

C₂

yes

maximal, characteristic

#16810426

C₂

yes

maximal, characteristic

C₅₃×#367

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

—

—

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133150

—

#13133140

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

—

Dic₂₁₂

Q₈

yes

complemented

C₁₆×C₅₃

—

#11719908

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₂×Dic₅₃

D₈

yes

characteristic

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂×Q₈

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#137

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

SD₃₂

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×SD₃₂

yes

C₂×C₅₃

yes

—

C₄×Q₁₆

—

#367

—

#367

D₅₃

yes

characteristic

C₄⋊C₁₆

—

#358

—

#358

—

C₅₃

#2225

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

106

—

C₄×Q₈

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#107

—

#107

—

106

—

#107

C₂×D₅₃

yes

characteristic

#107

C₂×D₅₃

yes

characteristic

#103

—

106

—

C₄×C₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

Q₁₆

—

Q₁₆

—

106

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

—

C₄²

—

212

—

C₁₆

—

C₁₆

—

106

—

Q₈

—

106

—

Q₈

—

424

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

212

—

102

C₂×C₄

—

212

—

103

C₂×C₄

D₄×D₅₃

yes

characteristic

104

C₈

—

106

—

105

C₈

#11719945

yes

characteristic

106

C₈

Q₈×D₅₃

yes

characteristic

107

C₂²

D₈×D₅₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

106

—

112

C₄

—

424

—

113

C₄

—

212

—

114

C₄

#13133195

yes

characteristic

115

C₄

#13133249

yes

characteristic

116

C₂

#16810902

yes

characteristic

117

C₂

SD₃₂×D₅₃

yes

characteristic

118

C₂

#16811062

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#367	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#367	D₅₃
hypocenter	C₅₃	#2225
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#367

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
31. Dic₂₁₂	Q₈	semidirect
67. C₅₃	#2225	semidirect

H ≅ G/N

type

31. Dic₂₁₂

Q₈

semidirect

67. C₅₃

#2225

semidirect

Subgroups — #20276856

Conjugacy classes of subgroups

Complements