Subgroups — #30660606

Order 16758.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#11924777	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₇×D₁₃₃	C₃	1	1	yes	maximal, characteristic
3	C₃×C₇²×C₁₉	C₂×C₃	1	1	yes	characteristic
4	#14558781	—	7	1	no	maximal
5	#4294⋊D₇	—	7	1	no	maximal
6	C₇×D₁₃₃	C₉	1	1	yes	characteristic, complemented, Hall
7	#11924777	D₇	1	1	yes	characteristic, complemented
8	C₇×#4294	—	7	1	no	—
9	C₇²×C₁₉	C₂×C₉	1	1	yes	characteristic, complemented, Hall
10	D₇×C₇⋊C₉	—	19	1	no	maximal, Hall
11	C₃×D₁₃₃	ΓL(1,𝔽₈)	1	1	yes	characteristic
12	C₃×C₇×D₁₉	—	7	1	no	—
13	C₇×C₇⋊C₉	—	19	1	no	Hall
14	C₃×C₇×C₁₉	—	6	1	no	—
15	C₃×C₇×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
16	C₃×C₇×C₁₉	C₃×D₇	1	1	yes	characteristic
17	#4294⋊C₂	—	49	1	no	Hall
18	C₃×C₇×D₇	—	57	1	no	—
19	D₁₃₃	C₇⋊C₉	1	1	yes	characteristic, complemented
20	C₇×D₁₉	—	7	1	no	—
21	#4294	—	7	1	no	Hall
22	C₃×C₇²	D₁₉⋊C₃	1	1	yes	characteristic
23	C₇×C₁₉	—	6	1	no	—
24	C₇×C₁₉	C₂×C₇⋊C₉	1	1	yes	characteristic, complemented
25	C₇×C₁₉	C₉×D₇	1	1	yes	characteristic, complemented
26	C₉×D₇	—	133	1	no	—
27	C₂×C₇⋊C₉	—	133	1	no	—
28	C₃×D₁₉	—	7	1	no	—
29	C₇×D₇	—	57	1	no	Hall
30	C₇×C₉	—	133	1	no	—
31	C₇⋊C₉	—	19	1	no	—
32	C₃×C₁₉	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
33	C₇²	#4294⋊C₂	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃×C₇	—	399	1	no	—
35	C₃×D₇	—	57	1	no	—
36	D₁₉	—	7	1	no	Hall
37	C₃×C₇	—	6	1	no	—
38	C₃×C₇	(C₁₉⋊C₃)⋊D₇	1	1	yes	characteristic
39	C₃×C₇	#11716322	1	1	yes	characteristic
40	C₁₉	D₇×C₇⋊C₉	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₉	—	931	1	no	Hall
42	C₂×C₇	—	399	1	no	—
43	D₇	—	57	1	no	—
44	C₉	—	133	1	no	Sylow
45	C₇	—	6	1	no	—
46	C₇	#4294⋊D₇	1	1	yes	characteristic, complemented
47	C₇	#14558781	1	1	yes	characteristic, complemented
48	C₂×C₃	—	399	1	no	—
49	C₃	#19150318	1	1	yes	characteristic
50	C₂	—	399	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₃	#19150318
commutator, hypocenter	C₇²×C₁₉	C₂×C₉
Fitting, socle, cosocle	C₃×C₇²×C₁₉	C₂×C₃
prime radical	C₃×C₇×D₁₃₃	C₃
LCS	G ▷ C₇²×C₁₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₇×#11924777	C₂	semidirect
6. C₇×D₁₃₃	C₉	semidirect
7. #11924777	D₇	semidirect
9. C₇²×C₁₉	C₂×C₉	semidirect
19. D₁₃₃	C₇⋊C₉	semidirect
24. C₇×C₁₉	C₂×C₇⋊C₉	semidirect
25. C₇×C₁₉	C₉×D₇	semidirect
33. C₇²	#4294⋊C₂	semidirect
40. C₁₉	D₇×C₇⋊C₉	semidirect
46. C₇	#4294⋊D₇	semidirect
47. C₇	#14558781	semidirect

Maximal subgroups

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

subgroup	index
C₇×#11924777	2
C₃×C₇×D₁₃₃	3
#14558781	7
#4294⋊D₇	7
D₇×C₇⋊C₉	19