Subgroups — C₂³×#11719908 #20287808

Order 6784.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂³×C₈	D₅₃
commutator, hypocenter	C₅₃	C₂³×C₁₆
Frattini	C₈	C₂³×D₅₃
Fitting	C₂³×C₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₈
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#11719908	C₂	direct (N × H)
3. C₂×#11719908	C₂²	direct (N × H)
12. #11719908	C₂³	direct (N × H)
13. C₂³×C₅₃	C₁₆	semidirect
19. C₂²×C₅₃	C₂×C₁₆	semidirect
24. C₂×C₅₃	C₂²×C₁₆	semidirect
28. C₅₃	C₂³×C₁₆	semidirect
39. C₂³	#11719908	direct (N × H)
45. C₂²	C₂×#11719908	direct (N × H)
49. C₂	C₂²×#11719908	direct (N × H)

Subgroups — C23×#11719908 #20287808

Conjugacy classes of subgroups

Complements

Subgroups — C₂³×#11719908 #20287808