Subgroups — C₃×#23782980 #44535652

Order 31356.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₁₃⋊C₃×C₆₇	C₂	1	1	yes	maximal, characteristic
2	#23782980	C₃	1	3	yes	maximal, direct product factor
3	C₃×Dic₈₇₁	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₁₃⋊C₃×C₆₇	C₄	1	1	yes	characteristic, complemented, Hall
5	C₂×C₁₃⋊C₃×C₆₇	C₂×C₃	1	3	yes	—
6	C₂×C₃×C₁₃×C₆₇	C₂×C₃	1	1	yes	characteristic
7	Dic₈₇₁	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₁₃×C₆₇	C₃×C₄	1	1	yes	characteristic, complemented
9	C₁₃⋊C₃×C₆₇	C₃×C₄	1	3	yes	complemented
10	C₃²×Dic₆₇	—	13	1	no	maximal, Hall
11	C₂×C₁₃×C₆₇	C₂×C₃²	1	1	yes	characteristic
12	C₂×C₃²×C₆₇	—	13	1	no	—
13	C₁₃×C₆₇	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
14	C₃×Dic₆₇	—	13	3	no	—
15	C₃×Dic₆₇	—	13	1	no	—
16	C₃²×C₆₇	—	13	1	no	Hall
17	C₃×Dic₁₃⋊C₃	—	67	1	no	maximal, Hall
18	C₂×C₃×C₆₇	—	13	3	no	—
19	C₂×C₃×C₆₇	D₁₃⋊C₃	1	1	yes	characteristic
20	Dic₆₇	—	13	1	no	Hall
21	C₂×C₃×C₁₃⋊C₃	D₆₇	1	1	yes	characteristic
22	C₃×C₆₇	—	13	3	no	—
23	C₃×C₆₇	Dic₁₃⋊C₃	1	1	yes	characteristic, complemented
24	C₃×Dic₁₃	—	67	1	no	—
25	Dic₁₃⋊C₃	—	67	3	no	—
26	C₂×C₆₇	C₃×D₁₃⋊C₃	1	1	yes	characteristic
27	C₃×C₁₃⋊C₃	Dic₆₇	1	1	yes	characteristic, complemented, Hall
28	C₂×C₃×C₁₃	C₃×D₆₇	1	1	yes	characteristic
29	C₂×C₁₃⋊C₃	C₃×D₆₇	1	3	yes	—
30	C₆₇	C₃×Dic₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
31	Dic₁₃	—	67	1	no	Hall
32	C₃×C₁₃	C₃×Dic₆₇	1	1	yes	characteristic, complemented
33	C₁₃⋊C₃	C₃×Dic₆₇	1	3	yes	complemented
34	C₃²×C₄	—	871	1	no	Hall
35	C₂×C₁₃	C₃²×D₆₇	1	1	yes	characteristic
36	C₂×C₃²	—	13	1	no	—
37	C₁₃	C₃²×Dic₆₇	1	1	yes	characteristic, complemented, Sylow
38	C₃×C₄	—	871	3	no	—
39	C₃×C₄	—	871	1	no	—
40	C₃²	—	13	1	no	Sylow
41	C₂×C₃	—	13	3	no	—
42	C₂×C₃	(C₁₃⋊C₃)⋊D₆₇	1	1	yes	characteristic
43	C₄	—	871	1	no	Sylow
44	C₃	—	13	3	no	—
45	C₃	#23782980	1	1	yes	characteristic, direct product factor
46	C₂	C₃×(C₁₃⋊C₃)⋊D₆₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₁₃⋊C₃)⋊D₆₇
commutator, hypocenter	C₁₃×C₆₇	C₃²×C₄
Frattini	C₂	C₃×(C₁₃⋊C₃)⋊D₆₇
Fitting, socle	C₂×C₃×C₁₃×C₆₇	C₂×C₃
prime radical	C₂×C₃×C₁₃⋊C₃×C₆₇	C₂
cosocle	C₂×C₁₃×C₆₇	C₂×C₃²
LCS	G ▷ C₁₃×C₆₇
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
2. #23782980	C₃	direct (N × H)
3. C₃×Dic₈₇₁	C₃	semidirect
4. C₃×C₁₃⋊C₃×C₆₇	C₄	semidirect
7. Dic₈₇₁	C₃²	semidirect
8. C₃×C₁₃×C₆₇	C₃×C₄	semidirect
9. C₁₃⋊C₃×C₆₇	C₃×C₄	semidirect
13. C₁₃×C₆₇	C₃²×C₄	semidirect
23. C₃×C₆₇	Dic₁₃⋊C₃	semidirect
27. C₃×C₁₃⋊C₃	Dic₆₇	semidirect
30. C₆₇	C₃×Dic₁₃⋊C₃	semidirect
32. C₃×C₁₃	C₃×Dic₆₇	semidirect
33. C₁₃⋊C₃	C₃×Dic₆₇	semidirect
37. C₁₃	C₃²×Dic₆₇	semidirect
45. C₃	#23782980	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₁₃⋊C₃×C₆₇	2
#23782980	3
C₃×Dic₈₇₁	3
C₃²×Dic₆₇	13
C₃×Dic₁₃⋊C₃	67

Subgroups — C3×#23782980 #44535652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#23782980 #44535652