Group #20289727 — subgroups

Subgroups — #20289727

Order 6784.

102 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 102 conjugacy classes of subgroups — too many to draw. Only its 32 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	#16811544	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811485	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811006	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719910	—	2	1	no	—
5	C₄⋊#88324	—	2	1	no	—
6	C₂×Dic₂₁₂	C₄	1	1	yes	characteristic
7	C₂×#11719923	—	2	1	no	—
8	#13133204	—	2	1	no	—
9	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×D₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×D₄×C₅₃	—	2	1	no	—
12	C₂×Dic₁₀₆	—	2	1	no	—
13	C₂×#88324	—	4	1	no	—
14	D₈×C₅₃	—	4	1	no	—
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²⋊Dic₅₃	—	4	1	no	—
17	C₂²⋊Dic₅₃	—	4	1	no	—
18	#11719923	—	4	1	no	—
19	#11719923	—	4	1	no	—
20	C₄×Dic₅₃	D₄	1	1	yes	characteristic
21	C₂×#88322	D₄	1	1	yes	characteristic
22	Dic₂₁₂	—	4	1	no	—
23	#11719910	—	4	1	no	—
24	C₂³×C₅₃	—	4	1	no	—
25	D₄×C₅₃	—	4	1	no	—
26	D₄×C₅₃	—	4	1	no	—
27	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₅₃	—	4	1	no	—
29	C₂×Dic₅₃	D₈	1	1	yes	characteristic
30	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
31	Dic₁₀₆	—	4	1	no	—
32	Dic₁₀₆	—	4	1	no	—
33	#88324	—	8	1	no	—
34	C₈×C₅₃	—	2	1	no	—
35	#88322	D₈	1	1	yes	characteristic
36	#88322	SD₁₆	1	1	yes	characteristic
37	C₂²×C₅₃	—	8	1	no	—
38	C₂²×C₅₃	—	8	1	no	—
39	C₂²×C₅₃	—	8	1	no	—
40	C₂²×C₅₃	#102	1	1	yes	characteristic
41	C₄×C₅₃	#100	1	1	yes	characteristic
42	C₄×C₅₃	#102	1	1	yes	characteristic
43	Dic₅₃	—	2	1	no	—
44	Dic₅₃	—	2	1	no	—
45	Dic₅₃	—	8	1	no	—
46	#1360	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	—	16	1	no	—
48	C₂×C₅₃	#331	1	1	yes	characteristic
49	C₂×C₅₃	#359	1	1	yes	characteristic
50	C₂×C₅₃	#360	1	1	yes	characteristic
51	#495	—	53	1	no	—
52	C₄⋊C₁₆	—	53	1	no	—
53	#334	—	53	1	no	—
54	C₅₃	#1360	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	—	53	1	no	—
56	C₂×SD₁₆	—	106	1	no	—
57	C₂×D₈	#6157	1	1	yes	characteristic
58	#124	—	106	1	no	—
59	C₂×C₁₆	—	106	1	no	—
60	C₄⋊C₈	—	106	1	no	—
61	C₄×C₈	—	53	1	no	—
62	C₂×Q₈	—	106	1	no	—
63	C₂×D₄	—	2	1	no	—
64	Q₁₆	—	212	1	no	—
65	SD₁₆	—	212	1	no	—
66	SD₁₆	—	212	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	—	53	1	no	—
69	C₂×C₈	—	212	1	no	—
70	C₂×C₈	C₂×#6157	1	1	yes	characteristic
71	C₂²⋊C₄	—	212	1	no	—
72	C₂²⋊C₄	—	212	1	no	—
73	C₄²	—	53	1	no	—
74	C₁₆	—	212	1	no	—
75	C₂³	—	4	1	no	—
76	Q₈	—	212	1	no	—
77	Q₈	—	212	1	no	—
78	D₄	—	4	1	no	—
79	D₄	—	4	1	no	—
80	C₂×C₄	—	53	1	no	—
81	C₂×C₄	—	53	1	no	—
82	C₂×C₄	—	212	1	no	—
83	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	53	1	no	—
86	C₈	—	53	1	no	—
87	C₈	—	424	1	no	—
88	C₂²	—	8	1	no	—
89	C₂²	—	8	1	no	—
90	C₂²	—	8	1	no	—
91	C₂²	#13133279	1	1	yes	characteristic
92	C₄	—	106	1	no	—
93	C₄	—	106	1	no	—
94	C₄	—	424	1	no	—
95	C₄	#13133277	1	1	yes	characteristic
96	C₄	#13133279	1	1	yes	characteristic
97	C₂	—	16	1	no	—
98	C₂	#16811481	1	1	yes	characteristic
99	C₂	#16811537	1	1	yes	characteristic
100	C₂	#16811539	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811544

C₂

yes

maximal, characteristic, complemented

#16811485

C₂

yes

maximal, characteristic, complemented

#16811006

C₂

yes

maximal, characteristic

C₂×#11719910

—

C₄⋊#88324

—

C₂×Dic₂₁₂

C₄

yes

characteristic

C₂×#11719923

—

#13133204

—

C₈×Dic₅₃

C₂²

yes

characteristic

C₂×D₈×C₅₃

C₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

C₂×#88324

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

—

#11719923

—

#11719923

—

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

—

#11719910

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₂×Dic₅₃

SD₁₆

yes

characteristic

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

C₈×C₅₃

—

#88322

D₈

yes

characteristic

#88322

SD₁₆

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

#1360

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₄⋊C₁₆

—

#334

—

C₅₃

#1360

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×SD₁₆

—

106

—

C₂×D₈

#6157

yes

characteristic

#124

—

106

—

C₂×C₁₆

—

106

—

C₄⋊C₈

—

106

—

C₄×C₈

—

C₂×Q₈

—

106

—

C₂×D₄

—

Q₁₆

—

212

—

SD₁₆

—

212

—

SD₁₆

—

212

—

D₈

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₂×#6157

yes

characteristic

C₂²⋊C₄

—

212

—

C₂²⋊C₄

—

212

—

C₄²

—

C₁₆

—

212

—

C₂³

—

Q₈

—

212

—

Q₈

—

212

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

C₂²⋊#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

424

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#13133279

yes

characteristic

C₄

—

106

—

C₄

—

106

—

C₄

—

424

—

C₄

yes

C₄

yes

C₂

—

C₂

yes

C₂

yes

100

C₂

#16811539

yes

characteristic

	subgroup	G/H
center	C₂²	#13133279
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₂×C₈	C₂×#6157
Fitting	C₂×D₈×C₅₃	C₄
socle	C₂²×C₅₃	#102
hypercenter	C₂×D₈	#6157
hypocenter	C₅₃	#1360
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

subgroup

G/H

center

C₂²

#13133279

commutator

C₂×C₈×C₅₃

C₂×C₄

Frattini

C₂×C₈

C₂×#6157

Fitting

C₂×D₈×C₅₃

C₄

C₂²×C₅₃

C₂×D₈

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

N	H ≅ G/N	type
1. #16811544	C₂	semidirect
2. #16811485	C₂	semidirect
54. C₅₃	#1360	semidirect

H ≅ G/N

type

1. #16811544

C₂

semidirect

2. #16811485

C₂

semidirect

54. C₅₃

#1360

semidirect

Subgroups — #20289727

Conjugacy classes of subgroups

Complements