Subgroups — C₃×#20301292 #36347958

Order 22050.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅²⋊#3862	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₅⋊(C₅⋊(C₇⋊D₇))	C₃	1	1	yes	maximal, characteristic, complemented
3	#20301292	C₃	1	3	yes	maximal, direct product factor
4	C₅²⋊#3862	C₂×C₃	1	3	yes	complemented
5	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×#11760386	—	7	2	no	maximal
7	C₅⋊(C₅⋊(C₇⋊D₇))	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇⋊(C₅²⋊C₃)	—	7	2	no	—
9	C₃×C₅⋊(C₇⋊D₇)	—	15	2	no	—
10	C₅²×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
11	C₃×C₅⋊D₃₅	—	7	2	no	—
12	C₃×C₅⋊D₃₅	—	21	2	no	—
13	#11760386	—	7	6	no	—
14	C₃×#64425	—	25	1	no	maximal, Hall
15	C₃×C₅×C₇²	—	3	2	no	—
16	C₃×C₅²×C₇	—	3	2	no	—
17	C₃×C₅²×C₇	AGL(1,𝔽₇)	1	2	yes	complemented
18	C₇⋊(C₅²⋊C₃)	—	7	6	no	—
19	C₅⋊(C₇⋊D₇)	—	15	2	no	—
20	C₃×(C₅⋊D₅)⋊C₃	—	49	1	no	Hall
21	C₃×#3862	—	25	1	no	Hall
22	C₅⋊D₃₅	—	7	2	no	—
23	C₅⋊D₃₅	—	21	2	no	—
24	C₃×C₇⋊D₇	—	25	1	no	—
25	#64425	—	25	3	no	—
26	C₅×C₇²	—	3	2	no	—
27	C₃×C₅²⋊C₃	—	49	1	no	Hall
28	C₃×D₃₅	—	105	12	no	—
29	C₃×D₃₅	—	105	4	no	—
30	C₅²×C₇	—	3	2	no	—
31	C₅²×C₇	C₃×AGL(1,𝔽₇)	1	2	yes	complemented
32	C₃×C₅⋊D₅	—	49	1	no	—
33	(C₅⋊D₅)⋊C₃	—	49	3	no	—
34	C₃×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
35	#3862	—	25	3	no	—
36	C₃×AGL(1,𝔽₇)	—	175	2	no	—
37	C₃×C₅×C₇	—	3	12	no	—
38	C₃×C₅×C₇	—	3	4	no	—
39	C₇⋊D₇	—	25	1	no	Hall
40	C₃×C₅²	#64425	1	1	yes	characteristic, complemented
41	C₅²⋊C₃	—	49	3	no	—
42	D₃₅	—	105	12	no	—
43	D₃₅	—	105	4	no	—
44	C₃×ΓL(1,𝔽₈)	—	175	2	no	—
45	C₅⋊D₅	—	49	1	no	Hall
46	C₇²	C₃×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
47	C₃×D₇	—	525	2	no	—
48	C₃×D₇	—	175	2	no	—
49	AGL(1,𝔽₇)	—	175	6	no	—
50	C₅×C₇	—	3	12	no	—
51	C₅×C₇	—	3	4	no	—
52	C₃×D₅	—	735	2	no	—
53	C₅²	C₃×#64425	1	1	yes	characteristic, complemented, Sylow
54	C₃×C₇	—	3	2	no	—
55	C₃×C₇	#11760386	1	2	yes	complemented
56	ΓL(1,𝔽₈)	—	175	6	no	—
57	C₂×C₃²	—	1225	1	no	Hall
58	C₃×C₅	—	3	2	no	—
59	D₇	—	525	2	no	—
60	D₇	—	175	2	no	—
61	D₅	—	735	2	no	—
62	C₃²	—	1225	1	no	Sylow
63	C₇	—	3	2	no	—
64	C₇	C₃×#11760386	1	2	yes	complemented
65	C₂×C₃	—	1225	3	no	—
66	C₂×C₃	—	1225	1	no	—
67	C₅	—	3	2	no	—
68	C₃	—	1225	3	no	—
69	C₃	#20301292	1	1	yes	characteristic, direct product factor
70	C₂	—	1225	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	#20301292
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×C₅²⋊#3862	C₂	semidirect
2. C₃×C₅⋊(C₅⋊(C₇⋊D₇))	C₃	semidirect
3. #20301292	C₃	direct (N × H)
4. C₅²⋊#3862	C₂×C₃	semidirect
5. C₃×C₅²×C₇²	C₂×C₃	semidirect
7. C₅⋊(C₅⋊(C₇⋊D₇))	C₃²	semidirect
10. C₅²×C₇²	C₂×C₃²	semidirect
17. C₃×C₅²×C₇	AGL(1,𝔽₇)	semidirect
31. C₅²×C₇	C₃×AGL(1,𝔽₇)	semidirect
34. C₃×C₇²	(C₅⋊D₅)⋊C₃	semidirect
40. C₃×C₅²	#64425	semidirect
46. C₇²	C₃×(C₅⋊D₅)⋊C₃	semidirect
53. C₅²	C₃×#64425	semidirect
55. C₃×C₇	#11760386	semidirect
64. C₇	C₃×#11760386	semidirect
69. C₃	#20301292	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×C₅²⋊#3862	2
C₃×C₅⋊(C₅⋊(C₇⋊D₇))	3
#20301292	3
C₃×#11760386	7
C₃×#64425	25

Subgroups — C3×#20301292 #36347958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20301292 #36347958