Subgroups — C₂³×#14582115 #36342428

Order 21568.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14582115	C₂	1	14	yes	maximal, direct product factor
2	C₂³×#13062438	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582115	C₂²	1	28	yes	direct product factor
4	C₂²×#13062438	C₄	1	7	yes	—
5	C₂²×#13062438	C₂²	1	7	yes	—
6	C₂³×D₃₃₇	C₄	1	1	yes	characteristic
7	#14582115	C₂³	1	8	yes	direct product factor
8	C₂×#13062438	C₂×C₄	1	21	yes	—
9	C₂×#13062438	C₂³	1	7	yes	—
10	C₂²×D₃₃₇	C₈	1	7	yes	complemented
11	C₂²×D₃₃₇	C₂×C₄	1	7	yes	—
12	C₂³×C₃₃₇	C₈	1	1	yes	characteristic, complemented
13	C₂²×C₃₃₇	C₂×C₈	1	7	yes	complemented
14	C₂×D₃₃₇	C₂×C₈	1	21	yes	complemented
15	C₂×D₃₃₇	C₂²×C₄	1	7	yes	—
16	#13062438	C₂²×C₄	1	7	yes	—
17	#13062438	C₂⁴	1	1	yes	characteristic
18	C₂×C₃₃₇	C₂²×C₈	1	7	yes	complemented
19	D₃₃₇	C₂²×C₈	1	7	yes	complemented
20	D₃₃₇	C₂³×C₄	1	1	yes	characteristic
21	C₃₃₇	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
22	C₂³×C₈	—	337	1	no	maximal, Sylow
23	C₂³×C₄	—	337	1	no	—
24	C₂²×C₈	—	337	14	no	—
25	C₂⁴	—	337	1	no	—
26	C₂²×C₄	—	337	7	no	—
27	C₂²×C₄	—	337	7	no	—
28	C₂×C₈	—	337	28	no	—
29	C₂³	—	337	7	no	—
30	C₂³	—	337	7	no	—
31	C₂³	#14582115	1	1	yes	characteristic, direct product factor
32	C₂×C₄	—	337	7	no	—
33	C₂×C₄	—	337	21	no	—
34	C₈	—	337	8	no	—
35	C₂²	—	337	7	no	—
36	C₂²	—	337	21	no	—
37	C₂²	C₂×#14582115	1	7	yes	direct product factor
38	C₄	—	337	1	no	—
39	C₄	—	337	7	no	—
40	C₂	—	337	1	no	—
41	C₂	—	337	7	no	—
42	C₂	C₂²×#14582115	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	#14582115
commutator, hypocenter	C₃₃₇	C₂³×C₈
Frattini	C₁	G
Fitting, socle	C₂³×C₃₃₇	C₈
prime radical	C₂³×D₃₃₇	C₄
cosocle	#13062438	C₂⁴
LCS	G ▷ C₃₃₇
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂²×#14582115	C₂	direct (N × H)
3. C₂×#14582115	C₂²	direct (N × H)
7. #14582115	C₂³	direct (N × H)
10. C₂²×D₃₃₇	C₈	semidirect
12. C₂³×C₃₃₇	C₈	semidirect
13. C₂²×C₃₃₇	C₂×C₈	semidirect
14. C₂×D₃₃₇	C₂×C₈	semidirect
18. C₂×C₃₃₇	C₂²×C₈	semidirect
19. D₃₃₇	C₂²×C₈	semidirect
21. C₃₃₇	C₂³×C₈	semidirect
31. C₂³	#14582115	direct (N × H)
37. C₂²	C₂×#14582115	direct (N × H)
42. C₂	C₂²×#14582115	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×#14582115	2
C₂³×#13062438	2
C₂³×C₈	337

Subgroups — C23×#14582115 #36342428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14582115 #36342428