Subgroups — C₄²×#88324 #20290367

Order 6784.

408 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#88324	C₂	1	6	yes	maximal
2	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88324	C₂²	1	4	yes	—
4	C₄×#88324	C₄	1	24	yes	direct product factor
5	C₂×C₄×Dic₅₃	C₄	1	3	yes	—
6	C₂×C₄×Dic₅₃	C₂²	1	3	yes	—
7	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
9	C₂²×Dic₅₃	C₂×C₄	1	3	yes	—
10	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
11	C₂×#88324	C₂×C₄	1	24	yes	—
12	C₄²×C₅₃	C₈	1	4	yes	complemented
13	C₄×Dic₅₃	C₈	1	12	yes	complemented
14	C₄×Dic₅₃	C₂×C₄	1	12	yes	—
15	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅₃	C₄²	1	6	yes	—
17	C₂×C₄×C₅₃	C₂×C₈	1	6	yes	—
18	C₂×C₄×C₅₃	C₂×C₈	1	6	yes	—
19	C₂×Dic₅₃	C₄²	1	6	yes	—
20	C₂×Dic₅₃	C₂×C₈	1	12	yes	—
21	C₂×Dic₅₃	C₂²×C₄	1	6	yes	—
22	#88324	C₄²	1	16	yes	direct product factor
23	C₂²×C₅₃	C₂×C₄²	1	3	yes	—
24	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
25	C₂²×C₅₃	C₂²×C₈	1	3	yes	—
26	C₄×C₅₃	C₄×C₈	1	12	yes	complemented
27	Dic₅₃	C₄×C₈	1	12	yes	complemented
28	Dic₅₃	C₂×C₄²	1	4	yes	—
29	C₄²×C₈	—	53	1	no	maximal, Sylow
30	C₂×C₅₃	C₄³	1	1	yes	characteristic
31	C₂×C₅₃	C₂×C₄×C₈	1	3	yes	—
32	C₂×C₅₃	C₂×C₄×C₈	1	3	yes	—
33	C₂×C₄×C₈	—	53	6	no	—
34	C₄³	—	53	1	no	—
35	C₅₃	C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₈	—	53	4	no	—
37	C₂×C₄²	—	53	3	no	—
38	C₂×C₄²	—	53	3	no	—
39	C₂×C₄²	#6157	1	1	yes	characteristic
40	C₄×C₈	—	53	24	no	—
41	C₂²×C₄	—	53	3	no	—
42	C₂²×C₄	—	53	1	no	—
43	C₂²×C₄	C₂×#6157	1	3	yes	—
44	C₂×C₈	—	53	24	no	—
45	C₄²	—	53	12	no	—
46	C₄²	—	53	12	no	—
47	C₄²	#88324	1	4	yes	direct product factor
48	C₂³	C₂²×#6157	1	1	yes	characteristic
49	C₂×C₄	—	53	6	no	—
50	C₂×C₄	—	53	6	no	—
51	C₂×C₄	—	53	12	no	—
52	C₂×C₄	D₅₃⋊C₈	1	6	yes	—
53	C₂×C₄	C₄×#6157	1	6	yes	—
54	C₂×C₄	C₂×#88324	1	6	yes	—
55	C₈	—	53	16	no	—
56	C₂²	C₂×C₄×#6157	1	3	yes	—
57	C₂²	C₂²×#88324	1	1	yes	characteristic
58	C₂²	C₂×D₅₃⋊C₈	1	3	yes	—
59	C₄	—	53	4	no	—
60	C₄	—	53	12	no	—
61	C₄	C₄×#88324	1	12	yes	direct product factor
62	C₂	C₄²×#6157	1	1	yes	characteristic
63	C₂	C₂×C₄×#88324	1	3	yes	—
64	C₂	C₄×D₅₃⋊C₈	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄²	#6157
commutator, hypocenter	C₅₃	C₄²×C₈
Frattini	C₂³	C₂²×#6157
Fitting	C₂×C₄²×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₄
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. C₄×#88324	C₄	direct (N × H)
12. C₄²×C₅₃	C₈	semidirect
13. C₄×Dic₅₃	C₈	semidirect
22. #88324	C₄²	direct (N × H)
26. C₄×C₅₃	C₄×C₈	semidirect
27. Dic₅₃	C₄×C₈	semidirect
35. C₅₃	C₄²×C₈	semidirect
47. C₄²	#88324	direct (N × H)
61. C₄	C₄×#88324	direct (N × H)

Subgroups — C42×#88324 #20290367

Conjugacy classes of subgroups

Complements

Subgroups — C₄²×#88324 #20290367