Subgroups — C₇×C₃⋊#11716326 #30660808

Order 16758.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#87705	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇×C₁₉×D₂₁	C₃	1	1	yes	maximal, characteristic, complemented
3	C₇×#11716326	—	3	1	no	maximal
4	C₇×#87705	—	2	1	no	—
5	C₇×#87705	D₃	1	1	yes	characteristic, complemented
6	C₃×C₇²×C₁₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃⋊#11716326	C₇	1	1	yes	maximal, characteristic, direct product factor
8	D₃×C₇×C₁₉⋊C₃	—	7	1	no	maximal
9	C₇×D₇×C₁₉	—	3	1	no	Hall
10	C₃×#87705	C₂×C₇	1	1	yes	characteristic, complemented
11	C₃×C₇×C₁₉⋊C₃	—	14	1	no	—
12	C₇²×C₁₉	C₃×D₃	1	1	yes	characteristic, complemented, Hall
13	C₇×C₃⋊AGL(1,𝔽₇)	—	19	1	no	maximal, Hall
14	C₁₉×D₂₁	C₃×C₇	1	1	yes	characteristic, complemented
15	D₃×C₇×C₁₉	—	7	1	no	—
16	C₂×C₇×C₁₉⋊C₃	—	21	1	no	—
17	#11716326	—	3	1	no	—
18	C₃×C₇×ΓL(1,𝔽₈)	—	19	1	no	Hall
19	C₃×C₇×C₁₉	—	6	1	no	—
20	C₃×C₇×C₁₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
21	C₃×C₇×C₁₉	C₂×C₃×C₇	1	1	yes	characteristic, complemented
22	#87705	—	2	1	no	—
23	#87705	D₃×C₇	1	1	yes	characteristic, complemented
24	C₇×C₁₉⋊C₃	—	14	1	no	—
25	C₇×C₁₉⋊C₃	—	14	1	no	—
26	D₃×C₁₉⋊C₃	—	7	1	no	Hall
27	C₇×D₂₁	C₁₉⋊C₃	1	1	yes	characteristic, complemented
28	C₇×AGL(1,𝔽₇)	—	57	1	no	—
29	C₂×C₇×C₁₉	—	21	1	no	—
30	D₇×C₁₉	—	3	1	no	—
31	C₃×C₁₉⋊C₃	—	14	1	no	Hall
32	C₃×C₇²	C₂×C₁₉⋊C₃	1	1	yes	characteristic, complemented
33	C₇×ΓL(1,𝔽₈)	—	19	1	no	—
34	C₇×ΓL(1,𝔽₈)	—	38	1	no	—
35	C₇×C₁₉	—	6	1	no	—
36	C₇×C₁₉	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
37	C₇×C₁₉	C₃×D₃×C₇	1	1	yes	characteristic, complemented
38	C₃×D₃×C₇	—	133	1	no	—
39	C₃⋊AGL(1,𝔽₇)	—	19	1	no	—
40	D₃×C₁₉	—	7	1	no	—
41	C₂×C₁₉⋊C₃	—	21	1	no	—
42	C₇×D₇	—	3	1	no	Hall
43	C₃²×C₇	—	266	1	no	—
44	C₃×ΓL(1,𝔽₈)	—	19	1	no	—
45	C₃×C₁₉	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
46	C₁₉⋊C₃	—	14	1	no	—
47	C₁₉⋊C₃	—	14	1	no	—
48	C₇²	D₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃×C₇	—	399	1	no	—
50	D₂₁	C₇×C₁₉⋊C₃	1	1	yes	characteristic, complemented
51	D₃×C₇	—	7	1	no	—
52	AGL(1,𝔽₇)	—	57	1	no	—
53	C₂×C₁₉	—	21	1	no	Hall
54	C₃×C₇	—	6	1	no	—
55	C₃×C₇	—	266	1	no	—
56	C₃×C₇	—	266	1	no	—
57	C₃×C₇	#11716326	1	1	yes	characteristic, complemented
58	C₃×C₇	C₂×C₇×C₁₉⋊C₃	1	1	yes	characteristic, complemented
59	ΓL(1,𝔽₈)	—	19	1	no	—
60	ΓL(1,𝔽₈)	—	38	1	no	—
61	C₁₉	C₇×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
62	C₃×D₃	—	133	1	no	Hall
63	C₂×C₇	—	21	1	no	—
64	D₇	—	3	1	no	—
65	C₃²	—	266	1	no	Sylow
66	C₇	—	6	1	no	—
67	C₇	D₃×C₇×C₁₉⋊C₃	1	1	yes	characteristic, complemented
68	C₇	C₃⋊#11716326	1	1	yes	characteristic, direct product factor
69	C₂×C₃	—	399	1	no	—
70	D₃	—	7	1	no	—
71	C₃	—	266	1	no	—
72	C₃	—	266	1	no	—
73	C₃	C₇×#11716326	1	1	yes	characteristic, complemented
74	C₂	—	21	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	C₃⋊#11716326
commutator, cosocle, hypocenter	C₃×C₇×C₁₉	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₁₉	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₉
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×#87705	C₂	semidirect
2. C₇×C₁₉×D₂₁	C₃	semidirect
5. C₇×#87705	D₃	semidirect
6. C₃×C₇²×C₁₉	C₂×C₃	semidirect
7. C₃⋊#11716326	C₇	direct (N × H)
10. C₃×#87705	C₂×C₇	semidirect
12. C₇²×C₁₉	C₃×D₃	semidirect
14. C₁₉×D₂₁	C₃×C₇	semidirect
20. C₃×C₇×C₁₉	AGL(1,𝔽₇)	semidirect
21. C₃×C₇×C₁₉	C₂×C₃×C₇	semidirect
23. #87705	D₃×C₇	semidirect
27. C₇×D₂₁	C₁₉⋊C₃	semidirect
32. C₃×C₇²	C₂×C₁₉⋊C₃	semidirect
36. C₇×C₁₉	C₃⋊AGL(1,𝔽₇)	semidirect
37. C₇×C₁₉	C₃×D₃×C₇	semidirect
45. C₃×C₁₉	C₇×AGL(1,𝔽₇)	semidirect
48. C₇²	D₃×C₁₉⋊C₃	semidirect
50. D₂₁	C₇×C₁₉⋊C₃	semidirect
57. C₃×C₇	#11716326	semidirect
58. C₃×C₇	C₂×C₇×C₁₉⋊C₃	semidirect
61. C₁₉	C₇×C₃⋊AGL(1,𝔽₇)	semidirect
67. C₇	D₃×C₇×C₁₉⋊C₃	semidirect
68. C₇	C₃⋊#11716326	direct (N × H)
73. C₃	C₇×#11716326	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×#87705	2
C₇×C₁₉×D₂₁	3
C₇×#11716326	3
C₃⋊#11716326	7
D₃×C₇×C₁₉⋊C₃	7
C₇×C₃⋊AGL(1,𝔽₇)	19

Subgroups — C7×C3⋊#11716326 #30660808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₃⋊#11716326 #30660808