Subgroups — C₃×(C₆₇⋊C₃)⋊#261 #44535705

Order 31356.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₁₃×C₆₇⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₆₇⋊#261	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₆₇⋊C₃)⋊#261	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₁₃×C₆₇⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
5	D₁₃×C₆₇⋊C₃	C₂×C₃	1	3	yes	—
6	C₃×D₁₃×C₆₇	C₂×C₃	1	1	yes	characteristic
7	C₆₇⋊#261	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₁₃×C₆₇	C₃×C₄	1	1	yes	characteristic, complemented
9	C₁₃×C₆₇⋊C₃	C₃×C₄	1	3	yes	complemented
10	C₃×(C₆₇⋊C₃)⋊C₄	—	13	1	no	maximal, Hall
11	D₁₃×C₆₇	C₂×C₃²	1	1	yes	characteristic
12	C₂×C₃×C₆₇⋊C₃	—	13	1	no	—
13	C₁₃×C₆₇	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
14	C₃×Dic₆₇	—	13	1	no	—
15	(C₆₇⋊C₃)⋊C₄	—	13	3	no	—
16	C₃×C₆₇⋊C₃	#261	1	1	yes	characteristic, complemented, Hall
17	C₃²×#261	—	67	1	no	maximal, Hall
18	C₂×C₃×C₆₇	—	13	1	no	—
19	C₂×C₆₇⋊C₃	—	13	3	no	—
20	Dic₆₇	—	13	1	no	Hall
21	C₃²×D₁₃	—	67	1	no	—
22	C₃×C₆₇	C₃×#261	1	1	yes	characteristic, complemented
23	C₆₇⋊C₃	C₃×#261	1	3	yes	complemented
24	C₃×#261	—	67	3	no	—
25	C₃×#261	—	67	1	no	—
26	C₂×C₆₇	—	13	1	no	—
27	C₃²×C₁₃	—	67	1	no	Hall
28	C₃×D₁₃	—	67	3	no	—
29	C₃×D₁₃	D₆₇⋊C₃	1	1	yes	characteristic
30	C₆₇	C₃²×#261	1	1	yes	characteristic, complemented, Sylow
31	#261	—	67	1	no	Hall
32	C₃×C₁₃	—	67	3	no	—
33	C₃×C₁₃	(C₆₇⋊C₃)⋊C₄	1	1	yes	characteristic, complemented
34	C₃²×C₄	—	871	1	no	Hall
35	D₁₃	C₃×D₆₇⋊C₃	1	1	yes	characteristic
36	C₂×C₃²	—	871	1	no	—
37	C₁₃	C₃×(C₆₇⋊C₃)⋊C₄	1	1	yes	characteristic, complemented, Sylow
38	C₃×C₄	—	871	1	no	—
39	C₃×C₄	—	871	3	no	—
40	C₃²	—	67	1	no	Sylow
41	C₂×C₃	—	13	1	no	—
42	C₂×C₃	—	871	3	no	—
43	C₄	—	871	1	no	Sylow
44	C₃	—	67	3	no	—
45	C₃	(C₆₇⋊C₃)⋊#261	1	1	yes	characteristic, direct product factor
46	C₂	—	13	1	no	—

	subgroup	G/H
center, hypercenter	C₃	(C₆₇⋊C₃)⋊#261
commutator, hypocenter	C₁₃×C₆₇	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₁₃×C₆₇	C₃×C₄
prime radical	C₃×D₁₃×C₆₇⋊C₃	C₂
cosocle	D₁₃×C₆₇	C₂×C₃²
LCS	G ▷ C₁₃×C₆₇
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. C₃×C₆₇⋊#261	C₃	semidirect
3. (C₆₇⋊C₃)⋊#261	C₃	direct (N × H)
4. C₃×C₁₃×C₆₇⋊C₃	C₄	semidirect
7. C₆₇⋊#261	C₃²	semidirect
8. C₃×C₁₃×C₆₇	C₃×C₄	semidirect
9. C₁₃×C₆₇⋊C₃	C₃×C₄	semidirect
13. C₁₃×C₆₇	C₃²×C₄	semidirect
16. C₃×C₆₇⋊C₃	#261	semidirect
22. C₃×C₆₇	C₃×#261	semidirect
23. C₆₇⋊C₃	C₃×#261	semidirect
30. C₆₇	C₃²×#261	semidirect
33. C₃×C₁₃	(C₆₇⋊C₃)⋊C₄	semidirect
37. C₁₃	C₃×(C₆₇⋊C₃)⋊C₄	semidirect
45. C₃	(C₆₇⋊C₃)⋊#261	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×D₁₃×C₆₇⋊C₃	2
C₃×C₆₇⋊#261	3
(C₆₇⋊C₃)⋊#261	3
C₃×(C₆₇⋊C₃)⋊C₄	13
C₃²×#261	67

Subgroups — C3×(C67⋊C3)⋊#261 #44535705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×(C₆₇⋊C₃)⋊#261 #44535705