Subgroups — C₃×C₁₃×C₂²⋊(C₆₇⋊C₃) #44535813

Order 31356.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×C₂²⋊(C₆₇⋊C₃)	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₁₃×C₆₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₁₃×C₆₇⋊C₃	—	4	1	no	maximal, Hall
4	C₂×C₃×C₁₃×C₆₇	—	3	1	no	—
5	C₂²×C₁₃×C₆₇	C₃²	1	1	yes	characteristic, complemented, Hall
6	C₃×C₁₃×C₆₇	A₄	1	1	yes	characteristic, complemented
7	C₁₃×C₆₇⋊C₃	—	4	3	no	—
8	C₃×C₂²⋊(C₆₇⋊C₃)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₁₃×C₆₇	—	3	1	no	—
10	C₁₃×C₆₇	C₃×A₄	1	1	yes	characteristic, complemented, Hall
11	C₂²×C₃×C₆₇	C₃×C₁₃	1	1	yes	characteristic, complemented
12	C₂²⋊(C₆₇⋊C₃)	C₃×C₁₃	1	3	yes	direct product factor
13	C₃×C₆₇⋊C₃	—	4	1	no	Hall
14	C₃×A₄×C₁₃	—	67	1	no	maximal, Hall
15	C₂×C₃×C₆₇	—	3	1	no	—
16	C₂²×C₆₇	C₃²×C₁₃	1	1	yes	characteristic, complemented, Hall
17	C₃×C₆₇	A₄×C₁₃	1	1	yes	characteristic, complemented
18	C₆₇⋊C₃	—	4	3	no	—
19	C₂²×C₃×C₁₃	C₆₇⋊C₃	1	1	yes	characteristic, complemented
20	A₄×C₁₃	—	67	3	no	—
21	C₂×C₆₇	—	3	1	no	—
22	C₃²×C₁₃	—	268	1	no	Hall
23	C₂×C₃×C₁₃	—	3	1	no	—
24	C₆₇	C₃×A₄×C₁₃	1	1	yes	characteristic, complemented, Sylow
25	C₂²×C₁₃	C₃×C₆₇⋊C₃	1	1	yes	characteristic, complemented, Hall
26	C₃×C₁₃	—	268	3	no	—
27	C₃×C₁₃	C₂²⋊(C₆₇⋊C₃)	1	1	yes	characteristic, direct product factor
28	C₃×A₄	—	67	1	no	Hall
29	C₂×C₁₃	—	3	1	no	—
30	C₁₃	C₃×C₂²⋊(C₆₇⋊C₃)	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²×C₃	C₁₃×C₆₇⋊C₃	1	1	yes	characteristic, complemented
32	A₄	—	67	3	no	—
33	C₃²	—	268	1	no	Sylow
34	C₂×C₃	—	3	1	no	—
35	C₂²	C₃×C₁₃×C₆₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
36	C₃	—	268	3	no	—
37	C₃	C₁₃×C₂²⋊(C₆₇⋊C₃)	1	1	yes	characteristic, direct product factor
38	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₁₃	C₂²⋊(C₆₇⋊C₃)
commutator, cosocle, hypocenter	C₂²×C₆₇	C₃²×C₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₁₃×C₆₇	C₃
LCS	G ▷ C₂²×C₆₇
UCS	C₁ ◁ C₃×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₁₃×C₂²⋊(C₆₇⋊C₃)	C₃	direct (N × H)
2. C₂²×C₃×C₁₃×C₆₇	C₃	semidirect
5. C₂²×C₁₃×C₆₇	C₃²	semidirect
6. C₃×C₁₃×C₆₇	A₄	semidirect
8. C₃×C₂²⋊(C₆₇⋊C₃)	C₁₃	direct (N × H)
10. C₁₃×C₆₇	C₃×A₄	semidirect
11. C₂²×C₃×C₆₇	C₃×C₁₃	semidirect
12. C₂²⋊(C₆₇⋊C₃)	C₃×C₁₃	direct (N × H)
16. C₂²×C₆₇	C₃²×C₁₃	semidirect
17. C₃×C₆₇	A₄×C₁₃	semidirect
19. C₂²×C₃×C₁₃	C₆₇⋊C₃	semidirect
24. C₆₇	C₃×A₄×C₁₃	semidirect
25. C₂²×C₁₃	C₃×C₆₇⋊C₃	semidirect
27. C₃×C₁₃	C₂²⋊(C₆₇⋊C₃)	direct (N × H)
30. C₁₃	C₃×C₂²⋊(C₆₇⋊C₃)	direct (N × H)
31. C₂²×C₃	C₁₃×C₆₇⋊C₃	semidirect
35. C₂²	C₃×C₁₃×C₆₇⋊C₃	semidirect
37. C₃	C₁₃×C₂²⋊(C₆₇⋊C₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×C₂²⋊(C₆₇⋊C₃)	3
C₂²×C₃×C₁₃×C₆₇	3
C₃×C₁₃×C₆₇⋊C₃	4
C₃×C₂²⋊(C₆₇⋊C₃)	13
C₃×A₄×C₁₃	67

Subgroups — C3×C13×C22⋊(C67⋊C3) #44535813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₁₃×C₂²⋊(C₆₇⋊C₃) #44535813