Subgroups — C₃×#20301305 #36347944

Order 22050.

144 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×C₅²×C₇×D₇	C₃	1	1	yes	maximal, characteristic, complemented
3	#20301305	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	C₂×C₃×C₇⋊(C₅²⋊C₃)	—	7	1	no	maximal
7	C₃×D₇⋊(C₅²⋊C₃)	—	7	1	no	maximal
8	C₅²×C₇×D₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇⋊(C₅²⋊C₃)	—	7	1	no	—
10	C₃×C₇⋊(C₅²⋊C₃)	—	7	1	no	—
11	C₃×C₅×C₇×D₇	—	3	2	no	—
12	C₅²×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃×C₅²×C₇	—	21	1	no	—
14	C₃×C₅²×D₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
15	C₂×C₇⋊(C₅²⋊C₃)	—	7	3	no	—
16	D₇⋊(C₅²⋊C₃)	—	7	3	no	—
17	C₃×#64422	—	25	1	no	maximal, Hall
18	C₃×C₅×C₇²	—	3	2	no	—
19	C₃×C₅²×C₇	—	6	1	no	—
20	C₃×C₅²×C₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
21	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
22	C₇⋊(C₅²⋊C₃)	—	7	3	no	—
23	C₇⋊(C₅²⋊C₃)	—	7	3	no	—
24	C₅×C₇×D₇	—	3	2	no	—
25	C₂×C₃×C₅²⋊C₃	—	49	1	no	Hall
26	C₃×#3862	—	25	1	no	Hall
27	C₂×C₅²×C₇	—	21	1	no	—
28	C₅²×D₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
29	C₃×C₇×D₇	C₅²⋊C₃	1	1	yes	characteristic, complemented
30	#64422	—	25	3	no	—
31	C₅×C₇²	—	3	2	no	—
32	C₃×C₅²⋊C₃	—	49	1	no	Hall
33	C₂×C₃×C₅×C₇	—	21	2	no	—
34	C₃×C₅×D₇	—	3	2	no	—
35	C₅²×C₇	—	6	1	no	—
36	C₅²×C₇	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
37	C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
38	C₂×C₃×C₅²	—	21	1	no	—
39	C₂×C₅²⋊C₃	—	49	3	no	—
40	C₃×C₇²	C₂×C₅²⋊C₃	1	1	yes	characteristic, complemented
41	#3862	—	25	3	no	—
42	C₂×C₃×ΓL(1,𝔽₈)	—	175	1	no	—
43	C₃×AGL(1,𝔽₇)	—	175	1	no	—
44	C₃×C₅×C₇	—	3	2	no	—
45	C₃×C₅×C₇	—	3	2	no	—
46	C₃×C₅×C₇	—	6	6	no	—
47	C₇×D₇	C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Hall
48	C₃×C₅²	#64422	1	1	yes	characteristic, complemented
49	C₅²⋊C₃	—	49	3	no	—
50	C₂×C₅×C₇	—	21	2	no	—
51	C₅×D₇	—	3	2	no	—
52	C₃×ΓL(1,𝔽₈)	—	175	1	no	—
53	C₃×ΓL(1,𝔽₈)	—	175	1	no	—
54	C₂×C₅²	—	21	1	no	Hall
55	C₇²	C₂×C₃×C₅²⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₃×C₇	—	21	1	no	—
57	C₃×D₇	C₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
58	C₂×ΓL(1,𝔽₈)	—	175	3	no	—
59	AGL(1,𝔽₇)	—	175	3	no	—
60	C₅×C₇	—	3	2	no	—
61	C₅×C₇	—	3	2	no	—
62	C₅×C₇	—	6	6	no	—
63	C₂×C₃×C₅	—	21	2	no	—
64	C₅²	C₃×#64422	1	1	yes	characteristic, complemented, Sylow
65	C₃×C₇	—	6	1	no	—
66	C₃×C₇	D₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
67	C₃×C₇	C₂×C₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
68	ΓL(1,𝔽₈)	—	175	3	no	—
69	ΓL(1,𝔽₈)	—	175	3	no	—
70	C₂×C₃²	—	1225	1	no	Hall
71	C₃×C₅	—	3	2	no	—
72	C₂×C₇	—	21	1	no	—
73	D₇	C₃×C₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
74	C₂×C₅	—	21	2	no	—
75	C₃²	—	1225	1	no	Sylow
76	C₇	—	6	1	no	—
77	C₇	C₃×D₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
78	C₇	C₂×C₃×C₇⋊(C₅²⋊C₃)	1	1	yes	characteristic, complemented
79	C₂×C₃	—	21	1	no	—
80	C₂×C₃	—	1225	3	no	—
81	C₅	—	3	2	no	—
82	C₃	—	1225	3	no	—
83	C₃	#20301305	1	1	yes	characteristic, direct product factor
84	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	#20301305
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×C₅²×C₇×D₇	C₃	semidirect
3. #20301305	C₃	direct (N × H)
4. C₅²⋊#3862	C₂×C₃	semidirect
5. C₃×C₅²×C₇²	C₂×C₃	semidirect
8. C₅²×C₇×D₇	C₃²	semidirect
12. C₅²×C₇²	C₂×C₃²	semidirect
14. C₃×C₅²×D₇	ΓL(1,𝔽₈)	semidirect
20. C₃×C₅²×C₇	AGL(1,𝔽₇)	semidirect
21. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
28. C₅²×D₇	C₃×ΓL(1,𝔽₈)	semidirect
29. C₃×C₇×D₇	C₅²⋊C₃	semidirect
36. C₅²×C₇	C₃×AGL(1,𝔽₇)	semidirect
37. C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
40. C₃×C₇²	C₂×C₅²⋊C₃	semidirect
47. C₇×D₇	C₃×C₅²⋊C₃	semidirect
48. C₃×C₅²	#64422	semidirect
55. C₇²	C₂×C₃×C₅²⋊C₃	semidirect
57. C₃×D₇	C₇⋊(C₅²⋊C₃)	semidirect
64. C₅²	C₃×#64422	semidirect
66. C₃×C₇	D₇⋊(C₅²⋊C₃)	semidirect
67. C₃×C₇	C₂×C₇⋊(C₅²⋊C₃)	semidirect
73. D₇	C₃×C₇⋊(C₅²⋊C₃)	semidirect
77. C₇	C₃×D₇⋊(C₅²⋊C₃)	semidirect
78. C₇	C₂×C₃×C₇⋊(C₅²⋊C₃)	semidirect
83. C₃	#20301305	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×C₅²×C₇×D₇	3
#20301305	3
C₂×C₃×C₇⋊(C₅²⋊C₃)	7
C₃×D₇⋊(C₅²⋊C₃)	7
C₃×#64422	25

Subgroups — C3×#20301305 #36347944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20301305 #36347944