Subgroups — C₄³×#6157 #28312675

Order 13568.

3966 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×#6157	C₂	1	14	yes	maximal
2	C₄³×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×#6157	C₂²	1	28	yes	—
4	C₄²×#6157	C₄	1	112	yes	direct product factor
5	C₂×C₄²×D₅₃	C₂²	1	7	yes	—
6	C₄²×Dic₅₃	C₄	1	7	yes	complemented
7	C₄³×C₅₃	C₄	1	1	yes	characteristic, complemented
8	C₂³×#6157	C₂³	1	8	yes	—
9	C₂×C₄×#6157	C₂×C₄	1	336	yes	—
10	C₂²×C₄×D₅₃	C₂³	1	7	yes	—
11	C₂×C₄×Dic₅₃	C₂×C₄	1	21	yes	—
12	C₄²×D₅₃	C₂×C₄	1	56	yes	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	7	yes	—
14	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
15	C₂²×#6157	C₂²×C₄	1	112	yes	—
16	C₂²×C₄×C₅₃	C₂²×C₄	1	7	yes	—
17	C₂²×Dic₅₃	C₂²×C₄	1	7	yes	—
18	C₂×C₄×D₅₃	C₂²×C₄	1	84	yes	—
19	C₄×#6157	C₄²	1	448	yes	direct product factor
20	C₄²×C₅₃	C₄²	1	28	yes	complemented
21	C₄×Dic₅₃	C₄²	1	84	yes	complemented
22	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
23	C₂²×D₅₃	C₂³×C₄	1	14	yes	—
24	C₂×#6157	C₂×C₄²	1	224	yes	—
25	C₂×C₄×C₅₃	C₂×C₄²	1	42	yes	—
26	C₂×Dic₅₃	C₂×C₄²	1	42	yes	—
27	C₄×D₅₃	C₂×C₄²	1	112	yes	—
28	C₄⁴	—	53	1	no	maximal, Sylow
29	C₂²×C₅₃	C₂²×C₄²	1	7	yes	—
30	C₂×D₅₃	C₂²×C₄²	1	28	yes	—
31	#6157	C₄³	1	64	yes	direct product factor
32	C₄×C₅₃	C₄³	1	28	yes	complemented
33	Dic₅₃	C₄³	1	28	yes	complemented
34	C₂×C₄³	—	53	14	no	—
35	C₂×C₄³	—	53	1	no	—
36	C₂×C₅₃	C₂×C₄³	1	7	yes	—
37	D₅₃	C₂×C₄³	1	8	yes	—
38	C₂²×C₄²	—	53	28	no	—
39	C₂²×C₄²	—	53	7	no	—
40	C₄³	—	53	112	no	—
41	C₄³	—	53	7	no	—
42	C₄³	#6157	1	1	yes	characteristic, direct product factor
43	C₅₃	C₄⁴	1	1	yes	characteristic, complemented, Sylow
44	C₂³×C₄	—	53	8	no	—
45	C₂³×C₄	—	53	7	no	—
46	C₂×C₄²	—	53	336	no	—
47	C₂×C₄²	—	53	21	no	—
48	C₂×C₄²	—	53	56	no	—
49	C₂×C₄²	C₂×#6157	1	7	yes	—
50	C₂⁴	—	53	1	no	—
51	C₂²×C₄	—	53	112	no	—
52	C₂²×C₄	—	53	7	no	—
53	C₂²×C₄	—	53	84	no	—
54	C₂²×C₄	C₂²×#6157	1	7	yes	—
55	C₄²	—	53	448	no	—
56	C₄²	—	53	84	no	—
57	C₄²	C₄×#6157	1	28	yes	direct product factor
58	C₂³	—	53	14	no	—
59	C₂³	C₂³×#6157	1	1	yes	characteristic
60	C₂×C₄	—	53	224	no	—
61	C₂×C₄	—	53	42	no	—
62	C₂×C₄	—	53	112	no	—
63	C₂×C₄	C₂×C₄×#6157	1	42	yes	—
64	C₂²	—	53	28	no	—
65	C₂²	C₂²×C₄×#6157	1	7	yes	—
66	C₄	—	53	64	no	—
67	C₄	—	53	28	no	—
68	C₄	C₄²×#6157	1	28	yes	direct product factor
69	C₂	—	53	8	no	—
70	C₂	C₂×C₄²×#6157	1	7	yes	—

	subgroup	G/H
center, hypercenter	C₄³	#6157
commutator, hypocenter	C₅₃	C₄⁴
Frattini	C₂³	C₂³×#6157
Fitting	C₄³×C₅₃	C₄
socle	C₂³×C₅₃	C₂³×C₄
prime radical, cosocle	C₂³×D₅₃	C₂⁴
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₄³

Complements

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

N	H ≅ G/N	type
4. C₄²×#6157	C₄	direct (N × H)
6. C₄²×Dic₅₃	C₄	semidirect
7. C₄³×C₅₃	C₄	semidirect
19. C₄×#6157	C₄²	direct (N × H)
20. C₄²×C₅₃	C₄²	semidirect
21. C₄×Dic₅₃	C₄²	semidirect
31. #6157	C₄³	direct (N × H)
32. C₄×C₅₃	C₄³	semidirect
33. Dic₅₃	C₄³	semidirect
42. C₄³	#6157	direct (N × H)
43. C₅₃	C₄⁴	semidirect
57. C₄²	C₄×#6157	direct (N × H)
68. C₄	C₄²×#6157	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄²×#6157	2
C₄³×D₅₃	2
C₄⁴	53

Subgroups — C43×#6157 #28312675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄³×#6157 #28312675