Subgroups — C₄²×#88322 #20274397

Order 6784.

408 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#88322	C₂	1	6	yes	maximal
2	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88322	C₂²	1	4	yes	—
4	C₄×#88322	C₄	1	24	yes	direct product factor
5	C₂×C₄²×C₅₃	C₄	1	4	yes	—
6	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
7	C₂²×C₄×C₅₃	C₂×C₄	1	6	yes	—
8	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
9	C₄²×C₅₃	C₈	1	16	yes	complemented
10	C₄²×C₅₃	C₂×C₄	1	12	yes	—
11	C₂×#88322	C₂×C₄	1	24	yes	—
12	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
13	C₂×C₄×C₅₃	C₄²	1	12	yes	—
14	C₂×C₄×C₅₃	C₂×C₈	1	24	yes	—
15	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
16	#88322	C₄²	1	16	yes	direct product factor
17	C₂²×C₅₃	C₂×C₄²	1	3	yes	—
18	C₂²×C₅₃	C₂²×C₈	1	4	yes	—
19	C₄×C₅₃	C₄×C₈	1	24	yes	complemented
20	C₄×C₅₃	C₂×C₄²	1	4	yes	—
21	C₄²×C₈	—	53	1	no	maximal, Sylow
22	C₂×C₅₃	C₄³	1	1	yes	characteristic
23	C₂×C₅₃	C₂×C₄×C₈	1	6	yes	—
24	C₂×C₄×C₈	—	53	6	no	—
25	C₄³	D₅₃	1	1	yes	characteristic
26	C₅₃	C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
27	C₂²×C₈	—	53	4	no	—
28	C₂×C₄²	Dic₅₃	1	4	yes	—
29	C₂×C₄²	C₂×D₅₃	1	3	yes	—
30	C₄×C₈	—	53	24	no	—
31	C₂²×C₄	C₂×Dic₅₃	1	6	yes	—
32	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
33	C₂×C₈	—	53	24	no	—
34	C₄²	#88322	1	16	yes	direct product factor
35	C₄²	C₄×D₅₃	1	12	yes	—
36	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
37	C₂×C₄	C₂×#88322	1	24	yes	—
38	C₂×C₄	C₄×Dic₅₃	1	12	yes	—
39	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
40	C₈	—	53	16	no	—
41	C₂²	C₂×C₄×Dic₅₃	1	3	yes	—
42	C₂²	C₂²×#88322	1	4	yes	—
43	C₄	C₄×#88322	1	24	yes	direct product factor
44	C₄	C₄²×D₅₃	1	4	yes	—
45	C₂	C₄²×Dic₅₃	1	1	yes	characteristic
46	C₂	C₂×C₄×#88322	1	6	yes	—

	subgroup	G/H
center, hypercenter	C₄³	D₅₃
commutator, hypocenter	C₅₃	C₄²×C₈
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₄³×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₄³

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₄×#88322	C₄	direct (N × H)
9. C₄²×C₅₃	C₈	semidirect
16. #88322	C₄²	direct (N × H)
19. C₄×C₅₃	C₄×C₈	semidirect
26. C₅₃	C₄²×C₈	semidirect
34. C₄²	#88322	direct (N × H)
43. C₄	C₄×#88322	direct (N × H)

Subgroups — C42×#88322 #20274397

Conjugacy classes of subgroups

Complements

Subgroups — C₄²×#88322 #20274397