Subgroups — C₇×#10624931 #19100340

Order 5103.

978 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×3₋¹⁺⁴	C₃	1	2	yes	maximal, complemented
2	C₇×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₇×#737	C₃	1	1	yes	maximal, characteristic, complemented
4	C₇×#6910	C₃	1	24	yes	maximal, complemented
5	C₃×C₇×#731	C₃	1	12	yes	maximal, complemented
6	#10624931	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₇×#737	—	3	9	no	—
8	C₇×#737	—	3	18	no	—
9	C₃×C₇×3₋¹⁺²	C₃²	1	24	yes	complemented
10	C₃×C₇×3₋¹⁺²	C₃²	1	8	yes	complemented
11	C₃×C₇×H(3,𝔽₃)	—	3	9	no	—
12	C₃×C₇×H(3,𝔽₃)	C₃²	1	12	yes	—
13	C₃×C₇×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃²×C₇×C₉	C₃²	1	4	yes	complemented
15	C₇×#731	C₃²	1	9	yes	complemented
16	C₇×#731	C₃²	1	72	yes	complemented
17	3₋¹⁺⁴	C₃×C₇	1	2	yes	complemented
18	H(5,𝔽₃)	C₃×C₇	1	1	yes	characteristic
19	C₃×#737	C₃×C₇	1	1	yes	characteristic, complemented
20	#6910	C₃×C₇	1	24	yes	complemented
21	C₃×#731	C₃×C₇	1	12	yes	complemented
22	C₃³×C₇	—	3	12	no	—
23	C₃³×C₇	C₃³	1	4	yes	—
24	C₇×3₋¹⁺²	—	3	48	no	—
25	C₇×3₋¹⁺²	—	3	24	no	—
26	C₇×3₋¹⁺²	C₃³	1	18	yes	—
27	C₇×H(3,𝔽₃)	—	3	24	no	—
28	C₇×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
29	C₇×H(3,𝔽₃)	C₃³	1	9	yes	—
30	C₃×C₇×C₉	—	3	12	no	—
31	C₃×C₇×C₉	—	3	24	no	—
32	C₃×C₇×C₉	C₃³	1	1	yes	characteristic
33	C₃×C₇×C₉	C₃³	1	8	yes	—
34	#737	—	3	9	no	—
35	#737	—	3	18	no	—
36	C₃×3₋¹⁺²	C₃²×C₇	1	24	yes	complemented
37	C₃×3₋¹⁺²	C₃²×C₇	1	8	yes	complemented
38	C₃×H(3,𝔽₃)	—	3	9	no	—
39	C₃×H(3,𝔽₃)	C₃²×C₇	1	12	yes	—
40	C₃×H(3,𝔽₃)	C₃²×C₇	1	1	yes	characteristic
41	C₃²×C₉	C₃²×C₇	1	4	yes	complemented
42	#731	C₃²×C₇	1	9	yes	complemented
43	#731	C₃²×C₇	1	72	yes	complemented
44	C₃²×C₇	—	3	9	no	—
45	C₃²×C₇	—	9	4	no	—
46	C₃²×C₇	—	27	12	no	—
47	C₃²×C₇	C₃×H(3,𝔽₃)	1	12	yes	—
48	C₃²×C₇	C₃⁴	1	1	yes	characteristic
49	C₇×C₉	—	3	1	no	—
50	C₇×C₉	—	3	18	no	—
51	C₇×C₉	—	3	8	no	—
52	C₃³	—	3	12	no	—
53	C₃³	C₃³×C₇	1	4	yes	—
54	3₋¹⁺²	—	3	48	no	—
55	3₋¹⁺²	—	3	24	no	—
56	3₋¹⁺²	C₃³×C₇	1	18	yes	—
57	H(3,𝔽₃)	—	3	24	no	—
58	H(3,𝔽₃)	C₇×H(3,𝔽₃)	1	9	yes	—
59	H(3,𝔽₃)	C₃³×C₇	1	9	yes	—
60	C₃×C₉	—	3	12	no	—
61	C₃×C₉	—	3	24	no	—
62	C₃×C₉	C₃³×C₇	1	1	yes	characteristic
63	C₃×C₉	C₃³×C₇	1	8	yes	—
64	C₃×C₇	—	3	1	no	—
65	C₃×C₇	—	3	12	no	—
66	C₃×C₇	—	9	9	no	—
67	C₃×C₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
68	C₃²	—	3	9	no	—
69	C₃²	—	9	4	no	—
70	C₃²	—	27	12	no	—
71	C₃²	C₃×C₇×H(3,𝔽₃)	1	12	yes	—
72	C₃²	C₃⁴×C₇	1	1	yes	characteristic
73	C₉	—	3	1	no	—
74	C₉	—	3	18	no	—
75	C₉	—	3	8	no	—
76	C₇	#10624931	1	1	yes	characteristic, direct product factor, Sylow
77	C₃	—	3	1	no	—
78	C₃	—	3	12	no	—
79	C₃	—	9	9	no	—
80	C₃	C₃²×C₇×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₇	C₃²×H(3,𝔽₃)
commutator, Frattini	C₃²	C₃⁴×C₇
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₇ ◁ C₃×C₇×H(3,𝔽₃) ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×3₋¹⁺⁴	C₃	semidirect
3. C₃×C₇×#737	C₃	semidirect
4. C₇×#6910	C₃	semidirect
5. C₃×C₇×#731	C₃	semidirect
6. #10624931	C₇	direct (N × H)
9. C₃×C₇×3₋¹⁺²	C₃²	semidirect
10. C₃×C₇×3₋¹⁺²	C₃²	semidirect
14. C₃²×C₇×C₉	C₃²	semidirect
15. C₇×#731	C₃²	semidirect
16. C₇×#731	C₃²	semidirect
17. 3₋¹⁺⁴	C₃×C₇	semidirect
19. C₃×#737	C₃×C₇	semidirect
20. #6910	C₃×C₇	semidirect
21. C₃×#731	C₃×C₇	semidirect
36. C₃×3₋¹⁺²	C₃²×C₇	semidirect
37. C₃×3₋¹⁺²	C₃²×C₇	semidirect
41. C₃²×C₉	C₃²×C₇	semidirect
42. #731	C₃²×C₇	semidirect
43. #731	C₃²×C₇	semidirect
76. C₇	#10624931	direct (N × H)

Subgroups — C7×#10624931 #19100340

Conjugacy classes of subgroups

Complements

Subgroups — C₇×#10624931 #19100340