Subgroups — C₃×#20301299 #36347959

Order 22050.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×C₅²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅⋊(C₅⋊(C₇⋊D₇))	C₃	1	1	yes	maximal, characteristic, complemented
3	#20301299	C₃	1	3	yes	maximal, direct product factor
4	C₇²×C₅²⋊C₃	C₂×C₃	1	3	yes	complemented
5	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×#11760384	—	7	8	no	maximal
7	C₅⋊(C₅⋊(C₇⋊D₇))	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇×C₅²⋊C₃	D₇	1	8	yes	complemented
9	C₃×C₅⋊(C₇⋊D₇)	—	15	2	no	—
10	C₅²×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
11	C₃×C₅⋊D₃₅	—	7	8	no	—
12	#11760384	—	7	24	no	—
13	C₃²×C₇⋊D₇	—	25	1	no	maximal, Hall
14	C₃×C₅×C₇²	—	3	2	no	—
15	C₃×C₅²×C₇	C₃×D₇	1	8	yes	complemented
16	C₇×C₅²⋊C₃	C₃×D₇	1	24	yes	complemented
17	C₅⋊(C₇⋊D₇)	—	15	2	no	—
18	C₃×(C₅⋊D₅)⋊C₃	—	49	1	no	Hall
19	C₃²×C₇²	—	25	1	no	Hall
20	C₅⋊D₃₅	—	7	8	no	—
21	C₃×C₇⋊D₇	—	25	3	no	—
22	C₃×C₇⋊D₇	—	25	1	no	—
23	C₅×C₇²	—	3	2	no	—
24	C₃×C₅²⋊C₃	C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
25	C₃×D₃₅	—	105	16	no	—
26	C₅²×C₇	C₃²×D₇	1	8	yes	complemented
27	C₃×C₅⋊D₅	—	49	1	no	—
28	(C₅⋊D₅)⋊C₃	—	49	3	no	—
29	C₃×C₇²	—	25	3	no	—
30	C₃×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
31	C₃²×D₇	—	175	8	no	—
32	C₃×C₅×C₇	—	3	16	no	—
33	C₇⋊D₇	—	25	1	no	Hall
34	C₃×C₅²	C₃×C₇⋊D₇	1	1	yes	characteristic, complemented
35	C₅²⋊C₃	C₃×C₇⋊D₇	1	3	yes	complemented
36	D₃₅	—	105	16	no	—
37	C₃²×C₇	—	25	8	no	—
38	C₅⋊D₅	—	49	1	no	Hall
39	C₇²	C₃×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
40	C₃×D₇	—	175	24	no	—
41	C₃×D₇	—	175	8	no	—
42	C₅×C₇	—	3	16	no	—
43	C₃×D₅	—	735	2	no	—
44	C₅²	C₃²×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
45	C₃×C₇	—	25	24	no	—
46	C₃×C₇	#11760384	1	8	yes	complemented
47	C₂×C₃²	—	1225	1	no	Hall
48	C₃×C₅	—	3	2	no	—
49	D₇	—	175	8	no	—
50	D₅	—	735	2	no	—
51	C₃²	—	25	1	no	Sylow
52	C₇	C₃×#11760384	1	8	yes	complemented
53	C₂×C₃	—	1225	3	no	—
54	C₂×C₃	—	1225	1	no	—
55	C₅	—	3	2	no	—
56	C₃	—	25	3	no	—
57	C₃	#20301299	1	1	yes	characteristic, direct product factor
58	C₂	—	1225	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	#20301299
commutator, cosocle, hypocenter	C₅²×C₇²	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃×C₇²×C₅²⋊C₃	C₂	semidirect
2. C₃×C₅⋊(C₅⋊(C₇⋊D₇))	C₃	semidirect
3. #20301299	C₃	direct (N × H)
4. C₇²×C₅²⋊C₃	C₂×C₃	semidirect
5. C₃×C₅²×C₇²	C₂×C₃	semidirect
7. C₅⋊(C₅⋊(C₇⋊D₇))	C₃²	semidirect
8. C₃×C₇×C₅²⋊C₃	D₇	semidirect
10. C₅²×C₇²	C₂×C₃²	semidirect
15. C₃×C₅²×C₇	C₃×D₇	semidirect
16. C₇×C₅²⋊C₃	C₃×D₇	semidirect
24. C₃×C₅²⋊C₃	C₇⋊D₇	semidirect
26. C₅²×C₇	C₃²×D₇	semidirect
30. C₃×C₇²	(C₅⋊D₅)⋊C₃	semidirect
34. C₃×C₅²	C₃×C₇⋊D₇	semidirect
35. C₅²⋊C₃	C₃×C₇⋊D₇	semidirect
39. C₇²	C₃×(C₅⋊D₅)⋊C₃	semidirect
44. C₅²	C₃²×C₇⋊D₇	semidirect
46. C₃×C₇	#11760384	semidirect
52. C₇	C₃×#11760384	semidirect
57. C₃	#20301299	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×C₇²×C₅²⋊C₃	2
C₃×C₅⋊(C₅⋊(C₇⋊D₇))	3
#20301299	3
C₃×#11760384	7
C₃²×C₇⋊D₇	25

Subgroups — C3×#20301299 #36347959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20301299 #36347959