Subgroups — C₂³×#15667734 #36378910

Order 22592.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667734	C₂	1	14	yes	maximal, direct product factor
2	C₂³×#13083443	C₂	1	1	yes	maximal, characteristic
3	C₂×#15667734	C₂²	1	28	yes	direct product factor
4	C₂²×#13083443	C₄	1	7	yes	—
5	C₂²×#13083443	C₂²	1	7	yes	—
6	C₂³×D₃₅₃	C₄	1	1	yes	characteristic
7	#15667734	C₂³	1	8	yes	direct product factor
8	C₂×#13083443	C₂×C₄	1	21	yes	—
9	C₂×#13083443	C₂³	1	7	yes	—
10	C₂²×D₃₅₃	C₈	1	7	yes	complemented
11	C₂²×D₃₅₃	C₂×C₄	1	7	yes	—
12	C₂³×C₃₅₃	C₈	1	1	yes	characteristic, complemented
13	C₂²×C₃₅₃	C₂×C₈	1	7	yes	complemented
14	C₂×D₃₅₃	C₂×C₈	1	21	yes	complemented
15	C₂×D₃₅₃	C₂²×C₄	1	7	yes	—
16	#13083443	C₂²×C₄	1	7	yes	—
17	#13083443	C₂⁴	1	1	yes	characteristic
18	C₂×C₃₅₃	C₂²×C₈	1	7	yes	complemented
19	D₃₅₃	C₂²×C₈	1	7	yes	complemented
20	D₃₅₃	C₂³×C₄	1	1	yes	characteristic
21	C₃₅₃	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
22	C₂³×C₈	—	353	1	no	maximal, Sylow
23	C₂³×C₄	—	353	1	no	—
24	C₂²×C₈	—	353	14	no	—
25	C₂⁴	—	353	1	no	—
26	C₂²×C₄	—	353	7	no	—
27	C₂²×C₄	—	353	7	no	—
28	C₂×C₈	—	353	28	no	—
29	C₂³	—	353	7	no	—
30	C₂³	—	353	7	no	—
31	C₂³	#15667734	1	1	yes	characteristic, direct product factor
32	C₂×C₄	—	353	21	no	—
33	C₂×C₄	—	353	7	no	—
34	C₈	—	353	8	no	—
35	C₂²	—	353	7	no	—
36	C₂²	—	353	21	no	—
37	C₂²	C₂×#15667734	1	7	yes	direct product factor
38	C₄	—	353	1	no	—
39	C₄	—	353	7	no	—
40	C₂	—	353	1	no	—
41	C₂	—	353	7	no	—
42	C₂	C₂²×#15667734	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	#15667734
commutator, hypocenter	C₃₅₃	C₂³×C₈
Frattini	C₁	G
Fitting, socle	C₂³×C₃₅₃	C₈
prime radical	C₂³×D₃₅₃	C₄
cosocle	#13083443	C₂⁴
LCS	G ▷ C₃₅₃
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂²×#15667734	C₂	direct (N × H)
3. C₂×#15667734	C₂²	direct (N × H)
7. #15667734	C₂³	direct (N × H)
10. C₂²×D₃₅₃	C₈	semidirect
12. C₂³×C₃₅₃	C₈	semidirect
13. C₂²×C₃₅₃	C₂×C₈	semidirect
14. C₂×D₃₅₃	C₂×C₈	semidirect
18. C₂×C₃₅₃	C₂²×C₈	semidirect
19. D₃₅₃	C₂²×C₈	semidirect
21. C₃₅₃	C₂³×C₈	semidirect
31. C₂³	#15667734	direct (N × H)
37. C₂²	C₂×#15667734	direct (N × H)
42. C₂	C₂²×#15667734	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×#15667734	2
C₂³×#13083443	2
C₂³×C₈	353

Subgroups — C23×#15667734 #36378910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#15667734 #36378910