Subgroups — C₃×#20301298 #36347951

Order 22050.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₇⋊(C₅²⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₇×C₅⋊D₃₅	C₃	1	1	yes	maximal, characteristic, complemented
3	#20301298	C₃	1	3	yes	maximal, direct product factor
4	C₇×C₇⋊(C₅²⋊C₃)	C₂×C₃	1	3	yes	complemented
5	C₃×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×#11760384	—	7	1	no	maximal
7	C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	7	1	no	maximal
8	C₇×C₅⋊D₃₅	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇⋊(C₅²⋊C₃)	D₇	1	1	yes	characteristic, complemented
10	C₃×C₇×C₅²⋊C₃	—	7	1	no	—
11	C₃×C₇×D₃₅	—	15	2	no	—
12	C₅²×C₇²	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
13	C₃×C₅⋊D₃₅	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
14	C₃×C₇×C₅⋊D₅	—	7	1	no	—
15	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	—	7	3	no	—
16	#11760384	—	7	3	no	—
17	C₃×D₇×ΓL(1,𝔽₈)	—	25	1	no	maximal, Hall
18	C₃×C₅×C₇²	—	3	2	no	—
19	C₃×C₅²×C₇	—	6	1	no	—
20	C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₃×C₅²×C₇	C₃×D₇	1	1	yes	characteristic, complemented
22	C₇⋊(C₅²⋊C₃)	C₃×D₇	1	3	yes	complemented
23	C₇×C₅²⋊C₃	—	7	3	no	—
24	C₇×D₃₅	—	15	2	no	—
25	C₃×(C₅⋊D₅)⋊C₃	—	49	1	no	Hall
26	C₃×C₇×ΓL(1,𝔽₈)	—	25	1	no	Hall
27	C₅⋊D₃₅	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	C₇×C₅⋊D₅	—	7	1	no	—
29	C₃×C₇×D₇	—	75	1	no	—
30	D₇×ΓL(1,𝔽₈)	—	25	3	no	—
31	C₅×C₇²	—	3	2	no	—
32	C₃×C₅²⋊C₃	—	7	1	no	Hall
33	C₃×D₃₅	—	15	2	no	—
34	C₃×C₇×D₅	—	105	2	no	—
35	C₅²×C₇	—	6	1	no	—
36	C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
37	C₅²×C₇	C₃²×D₇	1	1	yes	characteristic, complemented
38	C₃×C₅⋊D₅	—	7	1	no	—
39	(C₅⋊D₅)⋊C₃	—	49	3	no	—
40	C₃×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
41	C₇×ΓL(1,𝔽₈)	—	25	3	no	—
42	C₃²×D₇	—	175	1	no	—
43	C₂×C₃×ΓL(1,𝔽₈)	—	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	—	75	1	no	Hall
48	C₃×C₅²	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
49	C₅²⋊C₃	—	7	3	no	—
50	D₃₅	—	15	2	no	—
51	C₇×D₅	—	105	2	no	—
52	C₃²×C₇	—	175	1	no	—
53	C₃×ΓL(1,𝔽₈)	—	25	1	no	—
54	C₅⋊D₅	—	7	1	no	Hall
55	C₇²	C₃×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₃×C₇	—	525	1	no	—
57	C₃×D₇	—	75	1	no	—
58	C₃×D₇	—	175	3	no	—
59	C₂×ΓL(1,𝔽₈)	—	175	3	no	—
60	C₅×C₇	—	3	2	no	—
61	C₅×C₇	—	3	2	no	—
62	C₅×C₇	—	6	6	no	—
63	C₃×D₅	—	105	2	no	—
64	C₅²	C₃×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
65	C₃×C₇	—	6	1	no	—
66	C₃×C₇	—	175	3	no	—
67	C₃×C₇	#11760384	1	1	yes	characteristic, complemented
68	C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
69	ΓL(1,𝔽₈)	—	25	3	no	—
70	C₂×C₃²	—	1225	1	no	Hall
71	C₃×C₅	—	3	2	no	—
72	C₂×C₇	—	525	1	no	—
73	D₇	—	75	1	no	—
74	D₅	—	105	2	no	—
75	C₃²	—	175	1	no	Sylow
76	C₇	—	6	1	no	—
77	C₇	C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
78	C₇	C₃×#11760384	1	1	yes	characteristic, complemented
79	C₂×C₃	—	525	1	no	—
80	C₂×C₃	—	1225	3	no	—
81	C₅	—	3	2	no	—
82	C₃	—	175	3	no	—
83	C₃	#20301298	1	1	yes	characteristic, direct product factor
84	C₂	—	525	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	#20301298
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×C₇×C₇⋊(C₅²⋊C₃)	C₂	semidirect
2. C₃×C₇×C₅⋊D₃₅	C₃	semidirect
3. #20301298	C₃	direct (N × H)
4. C₇×C₇⋊(C₅²⋊C₃)	C₂×C₃	semidirect
5. C₃×C₅²×C₇²	C₂×C₃	semidirect
8. C₇×C₅⋊D₃₅	C₃²	semidirect
9. C₃×C₇⋊(C₅²⋊C₃)	D₇	semidirect
12. C₅²×C₇²	C₂×C₃²	semidirect
13. C₃×C₅⋊D₃₅	ΓL(1,𝔽₈)	semidirect
20. C₃×C₅²×C₇	C₂×ΓL(1,𝔽₈)	semidirect
21. C₃×C₅²×C₇	C₃×D₇	semidirect
22. C₇⋊(C₅²⋊C₃)	C₃×D₇	semidirect
27. C₅⋊D₃₅	C₃×ΓL(1,𝔽₈)	semidirect
36. C₅²×C₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
37. C₅²×C₇	C₃²×D₇	semidirect
40. C₃×C₇²	(C₅⋊D₅)⋊C₃	semidirect
48. C₃×C₅²	D₇×ΓL(1,𝔽₈)	semidirect
55. C₇²	C₃×(C₅⋊D₅)⋊C₃	semidirect
64. C₅²	C₃×D₇×ΓL(1,𝔽₈)	semidirect
67. C₃×C₇	#11760384	semidirect
68. C₃×C₇	(C₅⋊D₅)⋊ΓL(1,𝔽₈)	semidirect
77. C₇	C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	semidirect
78. C₇	C₃×#11760384	semidirect
83. C₃	#20301298	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×C₇×C₇⋊(C₅²⋊C₃)	2
C₃×C₇×C₅⋊D₃₅	3
#20301298	3
C₃×#11760384	7
C₃×(C₅⋊D₅)⋊ΓL(1,𝔽₈)	7
C₃×D₇×ΓL(1,𝔽₈)	25

Subgroups — C3×#20301298 #36347951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20301298 #36347951