Group #20276880 — subgroups

Subgroups — #20276880

Order 6784.

124 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16811025	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#16810451	C₂	1	1	yes	maximal, characteristic
4	#16810449	C₂	1	1	yes	maximal, characteristic
5	#16810428	C₂	1	1	yes	maximal, characteristic
6	#16810428	C₂	1	1	yes	maximal, characteristic
7	C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
9	#13133209	—	2	1	no	—
10	C₄×Dic₁₀₆	—	2	1	no	—
11	#13133151	C₂²	1	1	yes	characteristic
12	#13133151	C₂²	1	1	yes	characteristic
13	#13133151	C₂²	1	1	yes	characteristic
14	#13133149	—	2	1	no	—
15	#13133142	—	2	1	no	—
16	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₀₆	—	2	1	no	—
20	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₄²×C₅₃	D₄	1	1	yes	characteristic
23	C₄⋊Dic₅₃	—	2	1	no	—
24	C₄⋊Dic₅₃	—	2	1	no	—
25	C₄⋊Dic₅₃	—	2	1	no	—
26	#11719919	—	4	1	no	—
27	#11719919	—	4	1	no	—
28	#11719919	—	4	1	no	—
29	C₄×Dic₅₃	—	4	1	no	—
30	Dic₂₁₂	—	2	1	no	—
31	Dic₂₁₂	Q₈	1	2	yes	—
32	C₁₆×C₅₃	—	2	1	no	—
33	C₁₆×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
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	—	4	1	no	—
40	C₂×Dic₅₃	—	4	1	no	—
41	Dic₁₀₆	—	2	2	no	—
42	Dic₁₀₆	—	4	1	no	—
43	C₈×C₅₃	—	2	1	no	—
44	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₅₃	#41	1	1	yes	characteristic
46	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₄×C₅₃	#137	1	1	yes	characteristic
51	Dic₅₃	—	4	2	no	—
52	Dic₅₃	—	8	1	no	—
53	Dic₅₃	—	8	1	no	—
54	Dic₅₃	—	8	1	no	—
55	#2229	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	#474	1	1	yes	characteristic
57	C₂×C₅₃	#508	1	1	yes	characteristic
58	C₂×C₅₃	#510	1	1	yes	characteristic
59	#499	—	53	1	no	—
60	C₄×Q₁₆	—	53	1	no	—
61	#367	—	53	1	no	—
62	#366	—	53	1	no	—
63	C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
64	#358	—	53	1	no	—
65	#358	—	53	1	no	—
66	C₅₃	#2229	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₁₆	—	53	1	no	—
68	#125	—	106	1	no	—
69	C₄×Q₈	—	106	1	no	—
70	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
71	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
72	#107	—	53	1	no	—
73	#107	—	53	1	no	—
74	#107	—	53	1	no	—
75	#106	—	106	1	no	—
76	#103	—	106	1	no	—
77	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
78	C₂×Q₈	—	106	1	no	—
79	Q₁₆	—	53	2	no	—
80	Q₁₆	—	106	1	no	—
81	C₂×C₈	D₂₁₂	1	1	yes	characteristic
82	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
83	C₄⋊C₄	—	106	1	no	—
84	C₄⋊C₄	—	106	1	no	—
85	C₄⋊C₄	—	106	1	no	—
86	C₄⋊C₄	—	212	1	no	—
87	C₄⋊C₄	—	212	1	no	—
88	C₄⋊C₄	—	212	1	no	—
89	C₄²	—	212	1	no	—
90	C₄²	D₂₁₂	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	2	1	no	—
93	Q₈	—	106	2	no	—
94	Q₈	—	212	1	no	—
95	C₂×C₄	—	212	1	no	—
96	C₂×C₄	—	212	1	no	—
97	C₂×C₄	—	212	1	no	—
98	C₂×C₄	—	212	1	no	—
99	C₂×C₄	D₄₂₄	1	1	yes	characteristic
100	C₂×C₄	D₄₂₄	1	1	yes	characteristic
101	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	#11719947	1	1	yes	characteristic
104	C₈	Q₈×D₅₃	1	1	yes	characteristic
105	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	424	1	no	—
109	C₄	—	424	1	no	—
110	C₄	—	424	1	no	—
111	C₄	—	212	2	no	—
112	C₄	#13133197	1	1	yes	characteristic
113	C₄	#13133250	1	1	yes	characteristic
114	C₂	#16810904	1	1	yes	characteristic
115	C₂	#16811055	1	1	yes	characteristic
116	C₂	#16811066	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811025

C₂

yes

maximal, characteristic

C₄×Dic₂₁₂

C₂

yes

maximal, characteristic

#16810451

C₂

yes

maximal, characteristic

#16810449

C₂

yes

maximal, characteristic

#16810428

C₂

yes

maximal, characteristic

#16810428

C₂

yes

maximal, characteristic

C₅₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

characteristic

#13133209

—

C₄×Dic₁₀₆

—

#13133151

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133151

C₂²

yes

characteristic

#13133149

—

#13133142

—

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

#11719919

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

Q₈

yes

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₈

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

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

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₄×Q₁₆

—

#367

—

#366

—

C₄⋊C₁₆

D₅₃

yes

characteristic

#358

—

#358

—

C₅₃

#2229

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

#125

—

106

—

C₄×Q₈

—

106

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

C₂×C₁₆

C₂×D₅₃

yes

—

—

—

—

106

—

#103

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

Q₁₆

—

Q₁₆

—

106

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

—

212

—

C₄²

D₂₁₂

yes

characteristic

C₁₆

—

C₁₆

—

Q₈

—

106

—

Q₈

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

D₄₂₄

yes

characteristic

100

C₂×C₄

D₄₂₄

yes

characteristic

101

C₂×C₄

C₂×D₂₁₂

yes

characteristic

102

C₈

—

103

C₈

#11719947

yes

characteristic

104

C₈

Q₈×D₅₃

yes

characteristic

105

C₂²

C₂×D₄₂₄

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

424

—

109

C₄

—

424

—

110

C₄

—

424

—

111

C₄

—

212

—

112

C₄

#13133197

yes

characteristic

113

C₄

#13133250

yes

characteristic

114

C₂

#16810904

yes

characteristic

115

C₂

#16811055

yes

characteristic

116

C₂

#16811066

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#2229
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×C₄⋊C₁₆

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₄⋊C₁₆

D₅₃

hypocenter

C₅₃

#2229

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
66. C₅₃	#2229	semidirect

H ≅ G/N

type

66. C₅₃

#2229

semidirect

Subgroups — #20276880

Conjugacy classes of subgroups

Complements