Group #20276762 — subgroups

Subgroups — #20276762

Order 6784.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₂×#13133163	C₂	1	2	yes	maximal
2	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
4	#16810427	C₂	1	2	yes	maximal
5	C₅₃×#494	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₀₆	—	2	1	no	—
8	#13133163	—	2	4	no	—
9	C₂×#11719908	C₂²	1	2	yes	—
10	#13133151	C₂²	1	1	yes	characteristic
11	#13133142	—	2	1	no	—
12	C₂×Q₁₆×C₅₃	—	2	1	no	—
13	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₅₃	—	2	1	no	—
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	2	2	no	—
17	C₂×Dic₁₀₆	—	2	1	no	—
18	Q₁₆×C₅₃	—	2	2	no	—
19	Q₁₆×C₅₃	—	2	2	no	—
20	Q₁₆×C₅₃	—	4	2	no	—
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₅₃	—	4	1	no	—
24	C₄⋊C₄×C₅₃	—	4	1	no	—
25	C₄²×C₅₃	D₄	1	1	yes	characteristic
26	C₄⋊Dic₅₃	—	2	1	no	—
27	#11719919	—	4	1	no	—
28	C₄×Dic₅₃	—	4	1	no	—
29	Dic₂₁₂	—	2	1	no	—
30	Dic₂₁₂	D₄	1	2	yes	—
31	#11719908	—	2	2	no	—
32	Q₈×C₅₃	—	4	2	no	—
33	Q₈×C₅₃	—	4	2	no	—
34	C₂×C₄×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₅₃	—	4	1	no	—
39	C₂×Dic₅₃	—	4	1	no	—
40	Dic₁₀₆	—	2	2	no	—
41	Dic₁₀₆	—	4	1	no	—
42	C₈×C₅₃	—	2	1	no	—
43	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₈×C₅₃	#41	1	1	yes	characteristic
45	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	8	2	no	—
48	C₄×C₅₃	Q₃₂	1	2	yes	complemented
49	C₄×C₅₃	#121	1	1	yes	characteristic
50	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
51	Dic₅₃	—	4	2	no	—
52	Dic₅₃	—	8	1	no	—
53	#2209	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	#459	1	1	yes	characteristic
55	C₂×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
56	C₂×C₅₃	#510	1	1	yes	characteristic
57	C₂×Q₃₂	—	53	2	no	—
58	#494	D₅₃	1	1	yes	characteristic
59	C₄×Q₁₆	—	53	1	no	—
60	C₄⋊C₁₆	—	53	1	no	—
61	#358	—	53	2	no	—
62	C₅₃	#2209	1	1	yes	characteristic, complemented, Sylow
63	C₂×Q₁₆	—	2	1	no	—
64	C₂×Q₁₆	—	53	1	no	—
65	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
66	C₄⋊Q₈	—	2	1	no	—
67	C₄×Q₈	—	106	1	no	—
68	Q₃₂	—	106	4	no	—
69	C₂×C₁₆	—	53	2	no	—
70	#107	—	53	1	no	—
71	#103	—	106	1	no	—
72	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	106	1	no	—
75	Q₁₆	—	2	2	no	—
76	Q₁₆	—	2	2	no	—
77	Q₁₆	—	4	2	no	—
78	Q₁₆	—	53	2	no	—
79	Q₁₆	—	106	1	no	—
80	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
82	C₄⋊C₄	—	4	1	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₄⋊C₄	—	106	1	no	—
85	C₄⋊C₄	—	212	1	no	—
86	C₄²	—	212	1	no	—
87	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
88	C₁₆	—	106	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	4	2	no	—
91	Q₈	—	106	2	no	—
92	Q₈	—	212	1	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	212	1	no	—
95	C₂×C₄	—	212	1	no	—
96	C₂×C₄	#11719922	1	1	yes	characteristic
97	C₂×C₄	#11719922	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₅₃	1	1	yes	characteristic
101	C₈	#11719947	1	1	yes	characteristic
102	C₂²	C₂×#11719922	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	8	2	no	—
105	C₄	—	424	1	no	—
106	C₄	—	212	2	no	—
107	C₄	#13133163	1	2	yes	complemented
108	C₄	#13133188	1	1	yes	characteristic
109	C₄	#13133243	1	1	yes	characteristic
110	C₂	#16810815	1	1	yes	characteristic
111	C₂	C₂×#13133163	1	1	yes	characteristic
112	C₂	#16811064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133163

C₂

yes

maximal

C₄×Dic₂₁₂

C₂

yes

maximal, characteristic

C₄⋊#11719908

C₂

yes

maximal, characteristic

#16810427

C₂

yes

maximal

C₅₃×#494

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₄×Dic₁₀₆

—

#13133163

—

C₂×#11719908

C₂²

yes

—

#13133151

C₂²

yes

characteristic

#13133142

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

D₄

yes

—

#11719908

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

Q₃₂

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×Q₃₂

yes

characteristic

C₂×C₅₃

#510

yes

characteristic

C₂×Q₃₂

—

#494

D₅₃

yes

characteristic

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

C₅₃

#2209

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₅₃

yes

—

C₄⋊Q₈

—

C₄×Q₈

—

106

—

Q₃₂

—

106

—

C₂×C₁₆

—

#107

—

#103

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

106

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

106

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

—

212

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

Q₈

—

Q₈

—

Q₈

—

106

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

C₈

—

100

C₈

D₄×D₅₃

yes

characteristic

101

C₈

#11719947

yes

characteristic

102

C₂²

C₂×#11719922

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

424

—

106

C₄

—

212

—

107

C₄

#13133163

yes

complemented

108

C₄

#13133188

yes

characteristic

109

C₄

#13133243

yes

characteristic

110

C₂

#16810815

yes

characteristic

111

C₂

C₂×#13133163

yes

characteristic

112

C₂

#16811064

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#494	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	#494	D₅₃
hypocenter	C₅₃	#2209
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#494

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

N	H ≅ G/N	type
48. C₄×C₅₃	Q₃₂	semidirect
62. C₅₃	#2209	semidirect
107. C₄	#13133163	semidirect

H ≅ G/N

type

48. C₄×C₅₃

Q₃₂

semidirect

62. C₅₃

#2209

semidirect

107. C₄

#13133163

semidirect

Subgroups — #20276762

Conjugacy classes of subgroups

Complements