Subgroups — D₃×C₃⋊(D₇₁⋊C₇) #31822057

Order 17892.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊(D₇₁⋊C₇))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₃×C₇₁⋊C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊(D₇₁⋊C₇)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃⋊(D₇₁⋊C₇)	—	3	1	no	maximal
5	D₃×D₇₁⋊C₇	—	3	1	no	maximal
6	C₃²×C₇₁⋊C₇	C₂²	1	1	yes	characteristic, complemented, Hall
7	C₃⋊(D₇₁⋊C₇)	—	3	1	no	—
8	C₃⋊(D₇₁⋊C₇)	—	3	1	no	—
9	C₃⋊(D₇₁⋊C₇)	—	6	1	no	—
10	C₃⋊(D₇₁⋊C₇)	D₃	1	1	yes	characteristic, direct product factor
11	D₃×C₇₁⋊C₇	D₃	1	1	yes	characteristic, complemented
12	C₃×D₇₁⋊C₇	—	3	1	no	—
13	C₂×C₃×C₇₁⋊C₇	—	3	1	no	—
14	D₃×D₂₁₃	C₇	1	1	yes	maximal, characteristic, complemented, Hall
15	C₂×D₇₁⋊C₇	—	9	1	no	Hall
16	C₃×C₇₁⋊C₇	—	2	1	no	—
17	C₃×C₇₁⋊C₇	C₂×D₃	1	1	yes	characteristic, complemented
18	C₃×C₇₁⋊C₇	C₂×D₃	1	1	yes	characteristic, complemented
19	C₃⋊D₂₁₃	C₂×C₇	1	1	yes	characteristic, complemented
20	C₃×D₂₁₃	C₂×C₇	1	1	yes	characteristic, complemented
21	C₃×D₃×C₇₁	C₂×C₇	1	1	yes	characteristic, complemented
22	C₂×C₇₁⋊C₇	—	3	1	no	—
23	D₇₁⋊C₇	—	3	1	no	—
24	D₇₁⋊C₇	—	9	1	no	—
25	C₂×D₂₁₃	—	3	1	no	—
26	D₃×D₇₁	—	3	1	no	—
27	C₃²×C₇₁	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
28	C₇₁⋊C₇	D₃²	1	1	yes	characteristic, complemented, Hall
29	C₂×C₃×C₇₁	—	3	1	no	—
30	D₂₁₃	—	3	1	no	—
31	D₂₁₃	—	3	1	no	—
32	D₂₁₃	—	6	1	no	—
33	D₂₁₃	D₃×C₇	1	1	yes	characteristic, complemented
34	C₃×D₇₁	—	3	1	no	—
35	D₃×C₇₁	D₃×C₇	1	1	yes	characteristic, complemented
36	C₂×D₇₁	—	9	1	no	Hall
37	D₃²×C₇	—	71	1	no	maximal, Hall
38	C₃×C₇₁	—	2	1	no	—
39	C₃×C₇₁	C₂×D₃×C₇	1	1	yes	characteristic, complemented
40	C₃×C₇₁	C₂×D₃×C₇	1	1	yes	characteristic, complemented
41	C₂×C₇₁	—	3	1	no	—
42	D₇₁	—	3	1	no	—
43	D₇₁	—	9	1	no	—
44	C₇×C₃⋊D₃	—	71	1	no	—
45	C₃×D₃×C₇	—	71	1	no	—
46	C₃×D₃×C₇	—	71	1	no	—
47	C₂×D₃×C₇	—	213	1	no	—
48	C₂×D₃×C₇	—	213	1	no	—
49	C₇₁	D₃²×C₇	1	1	yes	characteristic, complemented, Sylow
50	C₃²×C₇	—	71	1	no	Hall
51	C₂×C₃×C₇	—	213	1	no	—
52	C₂×C₃×C₇	—	213	1	no	—
53	D₃×C₇	—	71	1	no	—
54	D₃×C₇	—	71	1	no	—
55	D₃×C₇	—	426	1	no	—
56	D₃×C₇	—	213	1	no	—
57	D₃×C₇	—	213	1	no	—
58	D₃²	—	497	1	no	Hall
59	C₂²×C₇	—	639	1	no	Hall
60	C₃×C₇	—	71	1	no	—
61	C₃×C₇	—	71	1	no	—
62	C₃×C₇	—	142	1	no	—
63	C₃⋊D₃	—	497	1	no	—
64	C₃×D₃	—	497	1	no	—
65	C₃×D₃	D₇₁⋊C₇	1	1	yes	characteristic, complemented
66	C₂×C₇	—	639	1	no	—
67	C₂×C₇	—	213	1	no	—
68	C₂×C₇	—	213	1	no	—
69	C₂×D₃	—	1491	1	no	—
70	C₂×D₃	—	1491	1	no	—
71	C₃²	C₂×D₇₁⋊C₇	1	1	yes	characteristic, complemented, Sylow
72	C₇	—	71	1	no	Sylow
73	C₂×C₃	—	3	1	no	—
74	C₂×C₃	—	1491	1	no	—
75	D₃	—	2982	1	no	—
76	D₃	—	1491	1	no	—
77	D₃	—	1491	1	no	—
78	D₃	—	497	1	no	—
79	D₃	C₃⋊(D₇₁⋊C₇)	1	1	yes	characteristic, direct product factor
80	C₂²	—	4473	1	no	Sylow
81	C₃	—	2	1	no	—
82	C₃	D₃×D₇₁⋊C₇	1	1	yes	characteristic, complemented
83	C₃	C₂×C₃⋊(D₇₁⋊C₇)	1	1	yes	characteristic, complemented
84	C₂	—	3	1	no	—
85	C₂	—	639	1	no	—
86	C₂	—	1491	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₇₁	C₂²×C₇
LCS	G ▷ C₃²×C₇₁
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃⋊(C₃⋊(D₇₁⋊C₇))	C₂	semidirect
2. C₃×D₃×C₇₁⋊C₇	C₂	semidirect
3. C₃×C₃⋊(D₇₁⋊C₇)	C₂	semidirect
6. C₃²×C₇₁⋊C₇	C₂²	semidirect
10. C₃⋊(D₇₁⋊C₇)	D₃	direct (N × H)
11. D₃×C₇₁⋊C₇	D₃	semidirect
14. D₃×D₂₁₃	C₇	semidirect
17. C₃×C₇₁⋊C₇	C₂×D₃	semidirect
18. C₃×C₇₁⋊C₇	C₂×D₃	semidirect
19. C₃⋊D₂₁₃	C₂×C₇	semidirect
20. C₃×D₂₁₃	C₂×C₇	semidirect
21. C₃×D₃×C₇₁	C₂×C₇	semidirect
27. C₃²×C₇₁	C₂²×C₇	semidirect
28. C₇₁⋊C₇	D₃²	semidirect
33. D₂₁₃	D₃×C₇	semidirect
35. D₃×C₇₁	D₃×C₇	semidirect
39. C₃×C₇₁	C₂×D₃×C₇	semidirect
40. C₃×C₇₁	C₂×D₃×C₇	semidirect
49. C₇₁	D₃²×C₇	semidirect
65. C₃×D₃	D₇₁⋊C₇	semidirect
71. C₃²	C₂×D₇₁⋊C₇	semidirect
79. D₃	C₃⋊(D₇₁⋊C₇)	direct (N × H)
82. C₃	D₃×D₇₁⋊C₇	semidirect
83. C₃	C₂×C₃⋊(D₇₁⋊C₇)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃⋊(C₃⋊(D₇₁⋊C₇))	2
C₃×D₃×C₇₁⋊C₇	2
C₃×C₃⋊(D₇₁⋊C₇)	2
C₂×C₃⋊(D₇₁⋊C₇)	3
D₃×D₇₁⋊C₇	3
D₃×D₂₁₃	7
D₃²×C₇	71

Subgroups — D3×C3⋊(D71⋊C7) #31822057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₃⋊(D₇₁⋊C₇) #31822057