Group #20276868 — subgroups

Subgroups — #20276868

Order 6784.

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

normal subgroups only

This group has 130 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	#16811028	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#16810449	C₂	1	2	yes	maximal
4	#16810428	C₂	1	2	yes	maximal
5	C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	#13133217	—	2	1	no	—
8	C₄×Dic₁₀₆	—	2	1	no	—
9	#13133151	—	2	1	no	—
10	#13133151	C₂²	1	2	yes	—
11	#13133151	C₂²	1	1	yes	characteristic
12	#13133142	—	2	1	no	—
13	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₀₆	—	2	1	no	—
16	C₂×Dic₁₀₆	—	4	1	no	—
17	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₄²×C₅₃	D₄	1	1	yes	characteristic
20	C₄⋊Dic₅₃	—	2	1	no	—
21	C₄⋊Dic₅₃	—	2	2	no	—
22	C₄⋊Dic₅₃	—	4	1	no	—
23	#11719919	—	4	1	no	—
24	C₄×Dic₅₃	—	4	1	no	—
25	Dic₂₁₂	—	2	1	no	—
26	Dic₂₁₂	Q₈	1	2	yes	complemented
27	C₁₆×C₅₃	—	2	2	no	—
28	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₅₃	—	4	1	no	—
32	C₂×Dic₅₃	—	4	2	no	—
33	C₂×Dic₅₃	—	4	1	no	—
34	Dic₁₀₆	—	2	2	no	—
35	Dic₁₀₆	—	4	1	no	—
36	Dic₁₀₆	—	8	2	no	—
37	C₈×C₅₃	—	2	1	no	—
38	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₅₃	#41	1	1	yes	characteristic
40	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
41	C₄×C₅₃	—	2	1	no	—
42	C₄×C₅₃	Q₃₂	1	2	yes	—
43	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
44	C₄×C₅₃	#137	1	1	yes	characteristic
45	Dic₅₃	—	4	2	no	—
46	Dic₅₃	—	8	2	no	—
47	Dic₅₃	—	8	1	no	—
48	#2227	—	53	1	no	maximal, Sylow
49	C₂×C₅₃	#474	1	1	yes	characteristic
50	C₂×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
51	C₂×C₅₃	#509	1	1	yes	characteristic
52	#500	—	53	1	no	—
53	C₄×Q₁₆	—	53	1	no	—
54	#366	—	53	2	no	—
55	C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
56	#358	—	53	2	no	—
57	C₅₃	#2227	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₁₆	—	53	1	no	—
59	C₄⋊Q₈	—	106	1	no	—
60	C₄×Q₈	—	106	1	no	—
61	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
62	#107	—	53	2	no	—
63	#107	—	53	1	no	—
64	#107	—	106	1	no	—
65	#103	—	106	1	no	—
66	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
67	C₂×Q₈	—	106	1	no	—
68	C₂×Q₈	—	212	1	no	—
69	Q₁₆	—	53	2	no	—
70	Q₁₆	—	106	1	no	—
71	C₂×C₈	D₂₁₂	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
73	C₄⋊C₄	—	106	1	no	—
74	C₄⋊C₄	—	106	2	no	—
75	C₄⋊C₄	—	212	1	no	—
76	C₄⋊C₄	—	212	1	no	—
77	C₄²	—	212	1	no	—
78	C₄²	D₂₁₂	1	1	yes	characteristic
79	C₁₆	—	2	2	no	—
80	Q₈	—	106	2	no	—
81	Q₈	—	424	2	no	—
82	Q₈	—	212	1	no	—
83	C₂×C₄	—	212	1	no	—
84	C₂×C₄	—	212	2	no	—
85	C₂×C₄	—	212	1	no	—
86	C₂×C₄	D₄₂₄	1	1	yes	characteristic
87	C₂×C₄	D₄₂₄	1	1	yes	characteristic
88	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	#11719947	1	1	yes	characteristic
91	C₈	Q₈×D₅₃	1	1	yes	characteristic
92	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	424	1	no	—
95	C₄	—	424	2	no	—
96	C₄	—	212	2	no	—
97	C₄	Dic₄₂₄	1	2	yes	—
98	C₄	#13133197	1	1	yes	characteristic
99	C₄	#13133250	1	1	yes	characteristic
100	C₂	#16810904	1	1	yes	characteristic
101	C₂	C₂×Dic₄₂₄	1	1	yes	characteristic
102	C₂	#16811061	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811028

C₂

yes

maximal, characteristic

C₄×Dic₂₁₂

C₂

yes

maximal, characteristic

#16810449

C₂

yes

maximal

#16810428

C₂

yes

maximal

C₅₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

C₂²

yes

characteristic

#13133217

—

C₄×Dic₁₀₆

—

#13133151

—

#13133151

C₂²

yes

—

#13133151

C₂²

yes

characteristic

#13133142

—

C₂×C₁₆×C₅₃

C₂²

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

—

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

—

C₄×Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

Q₈

yes

complemented

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

—

Dic₁₀₆

—

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

Q₃₂

yes

—

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

#137

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×Q₃₂

yes

C₂×C₅₃

yes

—

C₄×Q₁₆

—

#366

—

C₄⋊C₁₆

D₅₃

yes

characteristic

#358

—

C₅₃

#2227

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

106

—

C₄×Q₈

—

106

—

C₂×C₁₆

C₂×D₅₃

yes

—

#107

—

#107

—

#107

—

106

—

#103

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

Q₁₆

—

Q₁₆

—

106

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

—

212

—

C₄²

D₂₁₂

yes

characteristic

C₁₆

—

Q₈

—

106

—

Q₈

—

424

—

Q₈

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

D₄₂₄

yes

characteristic

C₂×C₄

D₄₂₄

yes

characteristic

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₈

—

C₈

#11719947

yes

characteristic

C₈

Q₈×D₅₃

yes

characteristic

C₂²

C₂×D₄₂₄

yes

characteristic

C₄

—

C₄

—

424

—

C₄

—

424

—

C₄

—

212

—

C₄

Dic₄₂₄

yes

—

C₄

yes

C₄

yes

100

C₂

#16810904

yes

characteristic

101

C₂

C₂×Dic₄₂₄

yes

characteristic

102

C₂

#16811061

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	#2227
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

#2227

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
26. Dic₂₁₂	Q₈	semidirect
57. C₅₃	#2227	semidirect

H ≅ G/N

type

26. Dic₂₁₂

Q₈

semidirect

57. C₅₃

#2227

semidirect

Subgroups — #20276868

Conjugacy classes of subgroups

Complements