Subgroups — C₂³×#13133123 #27542195

Order 13568.

542 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133123	C₂	1	14	yes	maximal, direct product factor
2	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133123	C₂²	1	28	yes	direct product factor
4	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₅₃	C₄	1	7	yes	—
6	C₂²×C₁₆×C₅₃	C₂²	1	7	yes	—
7	#13133123	C₂³	1	8	yes	direct product factor
8	C₂³×C₄×C₅₃	C₈	1	1	yes	characteristic
9	C₂²×C₈×C₅₃	C₈	1	7	yes	—
10	C₂²×C₈×C₅₃	C₂×C₄	1	7	yes	—
11	C₂×C₁₆×C₅₃	C₂×C₄	1	21	yes	—
12	C₂×C₁₆×C₅₃	C₂³	1	7	yes	—
13	C₂⁴×C₅₃	C₁₆	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₁₆	1	7	yes	—
15	C₂²×C₄×C₅₃	C₂×C₈	1	7	yes	—
16	C₂×C₈×C₅₃	C₂×C₈	1	21	yes	—
17	C₂×C₈×C₅₃	C₂²×C₄	1	7	yes	—
18	C₁₆×C₅₃	C₂²×C₄	1	7	yes	—
19	C₁₆×C₅₃	C₂⁴	1	1	yes	characteristic
20	C₂³×C₅₃	C₃₂	1	8	yes	complemented
21	C₂³×C₅₃	C₂×C₁₆	1	7	yes	—
22	C₂×C₄×C₅₃	C₂×C₁₆	1	21	yes	—
23	C₂×C₄×C₅₃	C₂²×C₈	1	7	yes	—
24	C₈×C₅₃	C₂²×C₈	1	7	yes	—
25	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
26	C₂³×C₃₂	—	53	1	no	maximal, Sylow
27	C₂²×C₅₃	C₂×C₃₂	1	28	yes	complemented
28	C₂²×C₅₃	C₂²×C₁₆	1	7	yes	—
29	C₄×C₅₃	C₂²×C₁₆	1	7	yes	—
30	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
31	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
32	C₂²×C₃₂	—	53	14	no	—
33	C₂×C₅₃	C₂²×C₃₂	1	14	yes	complemented
34	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
35	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
36	C₂²×C₁₆	Dic₅₃	1	7	yes	—
37	C₂²×C₁₆	C₂×D₅₃	1	7	yes	—
38	C₂×C₃₂	—	53	28	no	—
39	C₅₃	C₂³×C₃₂	1	1	yes	characteristic, complemented, Sylow
40	C₂³×C₄	#88322	1	1	yes	characteristic
41	C₂²×C₈	#88322	1	7	yes	—
42	C₂²×C₈	C₂×Dic₅₃	1	7	yes	—
43	C₂×C₁₆	C₂×Dic₅₃	1	21	yes	—
44	C₂×C₁₆	C₂²×D₅₃	1	7	yes	—
45	C₃₂	—	53	8	no	—
46	C₂⁴	#11719908	1	1	yes	characteristic
47	C₂²×C₄	#11719908	1	7	yes	—
48	C₂²×C₄	C₂×#88322	1	7	yes	—
49	C₂×C₈	C₂×#88322	1	21	yes	—
50	C₂×C₈	C₂²×Dic₅₃	1	7	yes	—
51	C₁₆	C₂²×Dic₅₃	1	7	yes	—
52	C₁₆	C₂³×D₅₃	1	1	yes	characteristic
53	C₂³	#13133123	1	8	yes	direct product factor
54	C₂³	C₂×#11719908	1	7	yes	—
55	C₂×C₄	C₂×#11719908	1	21	yes	—
56	C₂×C₄	C₂²×#88322	1	7	yes	—
57	C₈	C₂²×#88322	1	7	yes	—
58	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
59	C₂²	C₂×#13133123	1	28	yes	direct product factor
60	C₂²	C₂²×#11719908	1	7	yes	—
61	C₄	C₂²×#11719908	1	7	yes	—
62	C₄	C₂³×#88322	1	1	yes	characteristic
63	C₂	C₂²×#13133123	1	14	yes	direct product factor
64	C₂	C₂³×#11719908	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₁₆	D₅₃
commutator, hypocenter	C₅₃	C₂³×C₃₂
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₁₆
cosocle	C₁₆×C₅₃	C₂⁴
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133123	C₂	direct (N × H)
3. C₂×#13133123	C₂²	direct (N × H)
7. #13133123	C₂³	direct (N × H)
20. C₂³×C₅₃	C₃₂	semidirect
27. C₂²×C₅₃	C₂×C₃₂	semidirect
33. C₂×C₅₃	C₂²×C₃₂	semidirect
39. C₅₃	C₂³×C₃₂	semidirect
53. C₂³	#13133123	direct (N × H)
59. C₂²	C₂×#13133123	direct (N × H)
63. C₂	C₂²×#13133123	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133123	2
C₂³×C₁₆×C₅₃	2
C₂³×C₃₂	53

Subgroups — C23×#13133123 #27542195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133123 #27542195