Subgroups — C₆₇×(C₃⋊D₃)⋊D₇ #30683234

Order 16884.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₃⋊D₃×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₂₁×C₆₇	C₂	1	2	yes	maximal, complemented
3	D₃×D₇×C₆₇	—	3	2	no	maximal
4	C₃²×C₇×C₆₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₂₁×C₆₇	D₃	1	2	yes	complemented
6	D₃×C₇×C₆₇	—	3	2	no	—
7	D₃×C₇×C₆₇	—	6	1	no	—
8	C₃×D₇×C₆₇	—	3	2	no	—
9	D₃²×C₆₇	—	7	1	no	maximal, Hall
10	C₂×D₇×C₆₇	—	9	1	no	Hall
11	C₃×C₇×C₆₇	—	2	1	no	—
12	C₃×C₇×C₆₇	C₂×D₃	1	2	yes	complemented
13	C₃⋊D₃×C₆₇	D₇	1	1	yes	characteristic, complemented
14	C₃×D₃×C₆₇	—	7	2	no	—
15	C₂×C₇×C₆₇	—	9	1	no	—
16	D₇×C₆₇	—	3	2	no	—
17	C₂×D₃×C₆₇	—	21	2	no	—
18	C₃²×C₆₇	C₂×D₇	1	1	yes	characteristic, complemented, Hall
19	C₇×C₆₇	D₃²	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃×C₆₇	—	21	2	no	—
21	D₃×C₆₇	—	3	2	no	—
22	D₃×C₆₇	—	6	1	no	—
23	D₃×C₆₇	—	7	2	no	—
24	C₂²×C₆₇	—	63	1	no	Hall
25	(C₃⋊D₃)⋊D₇	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₃×C₆₇	—	2	1	no	—
27	C₃×C₆₇	D₃×D₇	1	2	yes	complemented
28	C₂×C₆₇	—	9	1	no	—
29	C₂×C₆₇	—	21	2	no	—
30	C₇×C₃⋊D₃	C₂×C₆₇	1	1	yes	characteristic, complemented
31	C₃×D₂₁	C₂×C₆₇	1	2	yes	complemented
32	D₃×D₇	—	3	2	no	—
33	C₆₇	(C₃⋊D₃)⋊D₇	1	1	yes	characteristic, direct product factor, Sylow
34	C₃²×C₇	C₂²×C₆₇	1	1	yes	characteristic, complemented, Hall
35	D₂₁	D₃×C₆₇	1	2	yes	complemented
36	C₃×D₇	—	3	2	no	—
37	D₃×C₇	—	3	2	no	—
38	D₃×C₇	—	6	1	no	—
39	D₃²	—	7	1	no	Hall
40	C₂×D₇	—	9	1	no	Hall
41	C₃×C₇	—	2	1	no	—
42	C₃×C₇	C₂×D₃×C₆₇	1	2	yes	complemented
43	C₃⋊D₃	D₇×C₆₇	1	1	yes	characteristic, complemented
44	C₃×D₃	—	7	2	no	—
45	C₂×C₇	—	9	1	no	—
46	D₇	—	3	2	no	—
47	C₂×D₃	—	21	2	no	—
48	C₃²	C₂×D₇×C₆₇	1	1	yes	characteristic, complemented, Sylow
49	C₇	D₃²×C₆₇	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃	—	21	2	no	—
51	D₃	—	3	2	no	—
52	D₃	—	6	1	no	—
53	D₃	—	7	2	no	—
54	C₂²	—	63	1	no	Sylow
55	C₃	—	2	1	no	—
56	C₃	D₃×D₇×C₆₇	1	2	yes	complemented
57	C₂	—	9	1	no	—
58	C₂	—	21	2	no	—

	subgroup	G/H
center, hypercenter	C₆₇	(C₃⋊D₃)⋊D₇
commutator, cosocle, hypocenter	C₃²×C₇	C₂²×C₆₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₆₇	C₂²
LCS	G ▷ C₃²×C₇
UCS	C₁ ◁ C₆₇

Complements

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

N	H ≅ G/N	type
1. C₇×C₃⋊D₃×C₆₇	C₂	semidirect
2. C₃×D₂₁×C₆₇	C₂	semidirect
4. C₃²×C₇×C₆₇	C₂²	semidirect
5. D₂₁×C₆₇	D₃	semidirect
12. C₃×C₇×C₆₇	C₂×D₃	semidirect
13. C₃⋊D₃×C₆₇	D₇	semidirect
18. C₃²×C₆₇	C₂×D₇	semidirect
19. C₇×C₆₇	D₃²	semidirect
25. (C₃⋊D₃)⋊D₇	C₆₇	direct (N × H)
27. C₃×C₆₇	D₃×D₇	semidirect
30. C₇×C₃⋊D₃	C₂×C₆₇	semidirect
31. C₃×D₂₁	C₂×C₆₇	semidirect
33. C₆₇	(C₃⋊D₃)⋊D₇	direct (N × H)
34. C₃²×C₇	C₂²×C₆₇	semidirect
35. D₂₁	D₃×C₆₇	semidirect
42. C₃×C₇	C₂×D₃×C₆₇	semidirect
43. C₃⋊D₃	D₇×C₆₇	semidirect
48. C₃²	C₂×D₇×C₆₇	semidirect
49. C₇	D₃²×C₆₇	semidirect
56. C₃	D₃×D₇×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₃⋊D₃×C₆₇	2
C₃×D₂₁×C₆₇	2
D₃×D₇×C₆₇	3
D₃²×C₆₇	7
(C₃⋊D₃)⋊D₇	67

Subgroups — C67×(C3⋊D3)⋊D7 #30683234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×(C₃⋊D₃)⋊D₇ #30683234