Group #20276976 — subgroups

Subgroups — #20276976

Order 6784.

184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 61 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×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133162	C₂	1	2	yes	maximal, complemented
3	C₂×D₈₄₈	C₂	1	1	yes	maximal, characteristic
4	#16811007	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₁₂	C₂²	1	2	yes	—
7	C₂×#11719913	—	2	2	no	—
8	C₂×D₄₂₄	C₂²	1	2	yes	—
9	#13133205	—	2	2	no	—
10	Dic₄₂₄	—	2	2	no	—
11	#13133162	—	2	4	no	—
12	D₈₄₈	—	2	2	no	—
13	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×D₂₁₂	—	2	2	no	—
17	C₂×Dic₁₀₆	—	2	2	no	—
18	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	C₄²×C₅₃	D₄	1	1	yes	characteristic
21	#11719921	—	4	4	no	—
22	Dic₂₁₂	—	2	2	no	—
23	Dic₂₁₂	—	2	2	no	—
24	D₄₂₄	—	2	2	no	—
25	D₄₂₄	—	2	2	no	—
26	#11719913	—	4	4	no	—
27	C₁₆×C₅₃	D₄	1	2	yes	—
28	C₁₆×C₅₃	D₄	1	2	yes	—
29	C₂²×D₅₃	—	4	2	no	—
30	C₂×C₄×C₅₃	D₈	1	2	yes	—
31	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×Dic₅₃	—	4	2	no	—
33	D₂₁₂	—	4	2	no	—
34	D₂₁₂	—	4	2	no	—
35	Dic₁₀₆	—	4	2	no	—
36	Dic₁₀₆	—	4	2	no	—
37	C₈×C₅₃	D₈	1	2	yes	—
38	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
41	C₂×D₅₃	—	8	2	no	—
42	C₂×D₅₃	—	8	4	no	—
43	C₄×C₅₃	—	2	2	no	—
44	C₄×C₅₃	#127	1	1	yes	characteristic
45	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
46	Dic₅₃	—	8	2	no	—
47	#2249	—	53	1	no	maximal, Sylow
48	C₂×C₅₃	#493	1	1	yes	characteristic
49	C₂×C₅₃	#508	1	2	yes	—
50	D₅₃	—	16	2	no	—
51	C₂×Q₃₂	—	53	1	no	—
52	C₂×SD₃₂	—	53	2	no	—
53	C₂×D₁₆	—	53	1	no	—
54	#495	—	53	2	no	—
55	C₄×C₁₆	D₅₃	1	1	yes	characteristic, complemented
56	C₅₃	#2249	1	1	yes	characteristic, complemented, Sylow
57	C₂×Q₁₆	—	53	2	no	—
58	C₂×SD₁₆	—	106	2	no	—
59	C₂×D₈	—	53	2	no	—
60	#124	—	106	2	no	—
61	Q₃₂	—	106	2	no	—
62	SD₃₂	—	106	4	no	—
63	D₁₆	—	106	2	no	—
64	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
65	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
66	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
67	C₂×Q₈	—	106	2	no	—
68	C₂×D₄	—	106	2	no	—
69	Q₁₆	—	106	2	no	—
70	Q₁₆	—	106	2	no	—
71	SD₁₆	—	212	4	no	—
72	D₈	—	106	2	no	—
73	D₈	—	106	2	no	—
74	C₂×C₈	D₂₁₂	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
76	C₂²⋊C₄	—	212	4	no	—
77	C₄²	D₂₁₂	1	1	yes	characteristic
78	C₁₆	D₂₁₂	1	2	yes	—
79	C₁₆	D₂₁₂	1	2	yes	—
80	C₂³	—	212	2	no	—
81	Q₈	—	212	2	no	—
82	Q₈	—	212	2	no	—
83	D₄	—	212	2	no	—
84	D₄	—	212	2	no	—
85	C₂×C₄	—	212	2	no	—
86	C₂×C₄	D₄₂₄	1	2	yes	—
87	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
88	C₈	D₄₂₄	1	2	yes	—
89	C₈	C₂×D₂₁₂	1	1	yes	characteristic
90	C₈	C₂×D₂₁₂	1	1	yes	characteristic
91	C₂²	—	424	4	no	—
92	C₂²	—	424	2	no	—
93	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	424	2	no	—
96	C₄	#13133214	1	1	yes	characteristic
97	C₄	C₂×D₄₂₄	1	1	yes	characteristic
98	C₂	—	848	2	no	—
99	C₂	#16811000	1	1	yes	characteristic
100	C₂	#16811055	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₄₂₄

C₂

yes

maximal, characteristic, complemented

C₂×#13133162

C₂

yes

maximal, complemented

C₂×D₈₄₈

C₂

yes

maximal, characteristic

#16811007

C₂

yes

maximal, complemented

C₄×C₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₂

C₂²

yes

—

C₂×#11719913

—

C₂×D₄₂₄

C₂²

yes

—

#13133205

—

Dic₄₂₄

—

#13133162

—

D₈₄₈

—

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

#11719921

—

Dic₂₁₂

—

Dic₂₁₂

—

D₄₂₄

—

D₄₂₄

—

#11719913

—

C₁₆×C₅₃

D₄

yes

—

C₁₆×C₅₃

D₄

yes

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

D₈

yes

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#127

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#495

—

C₄×C₁₆

D₅₃

yes

characteristic, complemented

C₅₃

#2249

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×SD₁₆

—

106

—

C₂×D₈

—

#124

—

106

—

Q₃₂

—

106

—

SD₃₂

—

106

—

D₁₆

—

106

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

C₂×C₁₆

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×D₄

—

106

—

Q₁₆

—

106

—

Q₁₆

—

106

—

SD₁₆

—

212

—

D₈

—

106

—

D₈

—

106

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₂²⋊C₄

—

212

—

C₄²

D₂₁₂

yes

characteristic

C₁₆

D₂₁₂

yes

—

C₁₆

D₂₁₂

yes

—

C₂³

—

212

—

Q₈

—

212

—

Q₈

—

212

—

D₄

—

212

—

D₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

D₄₂₄

yes

—

C₂×C₄

C₂×D₂₁₂

yes

characteristic

C₈

D₄₂₄

yes

—

C₈

C₂×D₂₁₂

yes

characteristic

C₈

C₂×D₂₁₂

yes

characteristic

C₂²

—

424

—

C₂²

—

424

—

C₂²

C₂×D₄₂₄

yes

characteristic

C₄

—

C₄

—

424

—

C₄

#13133214

yes

characteristic

C₄

C₂×D₄₂₄

yes

characteristic

C₂

—

848

—

C₂

#16811000

yes

characteristic

100

C₂

#16811055

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₄×C₁₆	D₅₃
hypocenter	C₅₃	#2249
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

C₄×C₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₄×C₁₆

D₅₃

hypocenter

C₅₃

#2249

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
1. C₂×Dic₄₂₄	C₂	semidirect
2. C₂×#13133162	C₂	semidirect
4. #16811007	C₂	semidirect
5. C₄×C₁₆×C₅₃	C₂	semidirect
55. C₄×C₁₆	D₅₃	semidirect
56. C₅₃	#2249	semidirect

H ≅ G/N

type

1. C₂×Dic₄₂₄

C₂

semidirect

2. C₂×#13133162

C₂

semidirect

4. #16811007

C₂

semidirect

5. C₄×C₁₆×C₅₃

C₂

semidirect

55. C₄×C₁₆

D₅₃

semidirect

56. C₅₃

#2249

semidirect

Subgroups — #20276976

Conjugacy classes of subgroups

Complements