Group #20276434 — subgroups

Subgroups — #20276434

Order 6784.

220 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 85 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×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133157	C₂	1	1	yes	maximal, characteristic
3	#16810426	C₂	1	4	yes	maximal, complemented
4	C₅₃×#427	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₆	—	2	1	no	—
6	C₂×Dic₂₁₂	—	2	2	no	—
7	C₂×Dic₂₁₂	—	2	1	no	—
8	C₂×Dic₂₁₂	C₄	1	4	yes	complemented
9	C₂×Dic₂₁₂	C₂²	1	2	yes	—
10	#13133157	—	2	2	no	—
11	C₂×#11719908	C₂²	1	2	yes	—
12	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	#117×C₅₃	—	2	1	no	—
14	#107×C₅₃	C₂²	1	2	yes	—
15	#106×C₅₃	—	2	1	no	—
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	4	1	no	—
18	C₂×Dic₁₀₆	—	2	4	no	—
19	C₂×Dic₁₀₆	—	2	2	no	—
20	C₂×Dic₁₀₆	—	4	1	no	—
21	C₂×Dic₁₀₆	—	4	2	no	—
22	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	D₄	1	2	yes	—
25	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂²⋊C₄×C₅₃	—	4	1	no	—
28	C₄²×C₅₃	—	4	1	no	—
29	Dic₂₁₂	—	2	4	no	—
30	Dic₂₁₂	—	2	2	no	—
31	Dic₂₁₂	C₂×C₄	1	4	yes	—
32	#11719908	—	2	2	no	—
33	C₂³×C₅₃	D₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
37	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₅₃	—	4	4	no	—
40	C₂×Dic₅₃	—	4	2	no	—
41	Dic₁₀₆	—	2	4	no	—
42	Dic₁₀₆	—	4	4	no	—
43	Dic₁₀₆	—	4	2	no	—
44	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₅₃	—	2	2	no	—
48	C₂²×C₅₃	#102	1	2	yes	—
49	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
50	C₄×C₅₃	—	8	2	no	—
51	C₄×C₅₃	#102	1	2	yes	—
52	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₅₃	—	4	4	no	—
55	#2141	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	—	2	2	no	—
57	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
58	C₂×C₅₃	#510	1	2	yes	—
59	C₂²×Q₁₆	—	53	1	no	—
60	C₂×M₃₂	—	53	1	no	—
61	#427	D₅₃	1	1	yes	characteristic
62	#358	—	53	4	no	—
63	C₅₃	#2141	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	—	106	1	no	—
65	C₂×Q₁₆	—	53	4	no	—
66	C₂×Q₁₆	—	53	2	no	—
67	C₂×Q₁₆	—	106	2	no	—
68	C₂×Q₁₆	—	106	1	no	—
69	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
70	#117	—	2	1	no	—
71	M₃₂	—	106	2	no	—
72	C₂×C₁₆	—	53	2	no	—
73	#107	C₂×D₅₃	1	2	yes	—
74	#106	—	2	1	no	—
75	C₂×Q₈	—	106	2	no	—
76	C₂×Q₈	—	106	4	no	—
77	C₂×Q₈	—	212	1	no	—
78	C₂×Q₈	—	212	2	no	—
79	C₂²×C₄	—	212	1	no	—
80	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
81	Q₁₆	—	53	4	no	—
82	Q₁₆	—	106	2	no	—
83	Q₁₆	—	106	4	no	—
84	C₂×C₈	C₄×D₅₃	1	2	yes	—
85	C₂×C₈	D₂₁₂	1	2	yes	—
86	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
88	C₄⋊C₄	—	2	2	no	—
89	C₂²⋊C₄	—	4	1	no	—
90	C₄²	—	4	1	no	—
91	C₁₆	—	106	2	no	—
92	C₂³	#11719922	1	1	yes	characteristic
93	Q₈	—	106	4	no	—
94	Q₈	—	212	4	no	—
95	Q₈	—	212	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	212	4	no	—
98	C₂×C₄	—	212	2	no	—
99	C₂×C₄	#11719921	1	2	yes	—
100	C₂×C₄	#11719922	1	1	yes	characteristic
101	C₂×C₄	#11719924	1	2	yes	—
102	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
103	C₈	#11719921	1	2	yes	—
104	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
105	C₈	C₂×D₂₁₂	1	1	yes	characteristic
106	C₂²	—	2	2	no	—
107	C₂²	#13133137	1	2	yes	—
108	C₂²	C₂×#11719922	1	1	yes	characteristic
109	C₄	—	8	2	no	—
110	C₄	—	212	4	no	—
111	C₄	#13133137	1	2	yes	—
112	C₄	C₂×#11719921	1	1	yes	characteristic
113	C₄	C₂×#11719924	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	C₂×#13133137	1	1	yes	characteristic
116	C₂	#16811063	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

C₂×#13133157

C₂

yes

maximal, characteristic

#16810426

C₂

yes

maximal, complemented

C₅₃×#427

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₄

yes

complemented

C₂×Dic₂₁₂

C₂²

yes

—

#13133157

—

C₂×#11719908

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#117×C₅₃

—

#107×C₅₃

C₂²

yes

—

#106×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

C₂×C₄

yes

—

#11719908

—

C₂³×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

#2141

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×#102

yes

characteristic

C₂×C₅₃

#510

yes

—

C₂²×Q₁₆

—

C₂×M₃₂

—

#427

D₅₃

yes

characteristic

#358

—

C₅₃

#2141

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

M₃₂

—

106

—

C₂×C₁₆

—

#107

C₂×D₅₃

yes

—

#106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

Q₁₆

—

Q₁₆

—

106

—

Q₁₆

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

106

—

C₂³

#11719922

yes

characteristic

Q₈

—

106

—

Q₈

—

212

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

#11719921

yes

—

100

C₂×C₄

#11719922

yes

characteristic

101

C₂×C₄

#11719924

yes

—

102

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

103

C₈

#11719921

yes

—

104

C₈

C₂×C₄×D₅₃

yes

characteristic

105

C₈

C₂×D₂₁₂

yes

characteristic

106

C₂²

—

107

C₂²

#13133137

yes

—

108

C₂²

C₂×#11719922

yes

characteristic

109

C₄

—

110

C₄

—

212

—

111

C₄

#13133137

yes

—

112

C₄

C₂×#11719921

yes

characteristic

113

C₄

C₂×#11719924

yes

characteristic

114

C₂

—

115

C₂

C₂×#13133137

yes

characteristic

116

C₂

#16811063

yes

—

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#427	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#427	D₅₃
hypocenter	C₅₃	#2141
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#427

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
3. #16810426	C₂	semidirect
8. C₂×Dic₂₁₂	C₄	semidirect
63. C₅₃	#2141	semidirect

H ≅ G/N

type

3. #16810426

C₂

semidirect

8. C₂×Dic₂₁₂

C₄

semidirect

63. C₅₃

#2141

semidirect

Subgroups — #20276434

Conjugacy classes of subgroups

Complements