Subgroups — C₂³×#15692552 #39910370

Order 26176.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15692552	C₂	1	14	yes	maximal, direct product factor
2	C₂³×#13109888	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692552	C₂²	1	28	yes	direct product factor
4	C₂²×#13109888	C₄	1	7	yes	—
5	C₂²×#13109888	C₂²	1	7	yes	—
6	C₂³×D₄₀₉	C₄	1	1	yes	characteristic
7	#15692552	C₂³	1	8	yes	direct product factor
8	C₂×#13109888	C₂×C₄	1	21	yes	—
9	C₂×#13109888	C₂³	1	7	yes	—
10	C₂²×D₄₀₉	C₈	1	7	yes	complemented
11	C₂²×D₄₀₉	C₂×C₄	1	7	yes	—
12	C₂³×C₄₀₉	C₈	1	1	yes	characteristic, complemented
13	C₂²×C₄₀₉	C₂×C₈	1	7	yes	complemented
14	C₂×D₄₀₉	C₂×C₈	1	21	yes	complemented
15	C₂×D₄₀₉	C₂²×C₄	1	7	yes	—
16	#13109888	C₂²×C₄	1	7	yes	—
17	#13109888	C₂⁴	1	1	yes	characteristic
18	C₂×C₄₀₉	C₂²×C₈	1	7	yes	complemented
19	D₄₀₉	C₂²×C₈	1	7	yes	complemented
20	D₄₀₉	C₂³×C₄	1	1	yes	characteristic
21	C₄₀₉	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
22	C₂³×C₈	—	409	1	no	maximal, Sylow
23	C₂³×C₄	—	409	1	no	—
24	C₂²×C₈	—	409	14	no	—
25	C₂⁴	—	409	1	no	—
26	C₂²×C₄	—	409	7	no	—
27	C₂²×C₄	—	409	7	no	—
28	C₂×C₈	—	409	28	no	—
29	C₂³	—	409	7	no	—
30	C₂³	—	409	7	no	—
31	C₂³	#15692552	1	1	yes	characteristic, direct product factor
32	C₂×C₄	—	409	21	no	—
33	C₂×C₄	—	409	7	no	—
34	C₈	—	409	8	no	—
35	C₂²	—	409	7	no	—
36	C₂²	—	409	21	no	—
37	C₂²	C₂×#15692552	1	7	yes	direct product factor
38	C₄	—	409	1	no	—
39	C₄	—	409	7	no	—
40	C₂	—	409	1	no	—
41	C₂	—	409	7	no	—
42	C₂	C₂²×#15692552	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	#15692552
commutator, hypocenter	C₄₀₉	C₂³×C₈
Frattini	C₁	G
Fitting, socle	C₂³×C₄₀₉	C₈
prime radical	C₂³×D₄₀₉	C₄
cosocle	#13109888	C₂⁴
LCS	G ▷ C₄₀₉
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂²×#15692552	C₂	direct (N × H)
3. C₂×#15692552	C₂²	direct (N × H)
7. #15692552	C₂³	direct (N × H)
10. C₂²×D₄₀₉	C₈	semidirect
12. C₂³×C₄₀₉	C₈	semidirect
13. C₂²×C₄₀₉	C₂×C₈	semidirect
14. C₂×D₄₀₉	C₂×C₈	semidirect
18. C₂×C₄₀₉	C₂²×C₈	semidirect
19. D₄₀₉	C₂²×C₈	semidirect
21. C₄₀₉	C₂³×C₈	semidirect
31. C₂³	#15692552	direct (N × H)
37. C₂²	C₂×#15692552	direct (N × H)
42. C₂	C₂²×#15692552	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×#15692552	2
C₂³×#13109888	2
C₂³×C₈	409

Subgroups — C23×#15692552 #39910370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#15692552 #39910370