Subgroups — C₂×#20301299 #28413446

Order 14700.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇²×C₅²⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#20301299	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅⋊(C₅⋊(C₇⋊D₇))	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₇²×C₅²⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₅⋊(C₅⋊(C₇⋊D₇))	C₂×C₃	1	2	yes	complemented
6	C₂×C₅²×C₇²	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×#11760384	—	7	8	no	maximal
8	C₅²×C₇²	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₇×C₅²⋊C₃	D₇	1	8	yes	complemented
10	#11760384	—	7	16	no	—
11	C₂×C₅⋊(C₇⋊D₇)	—	15	2	no	—
12	C₂×C₅⋊D₃₅	—	7	8	no	—
13	C₂×C₃×C₇⋊D₇	—	25	1	no	maximal, Hall
14	C₇×C₅²⋊C₃	C₂×D₇	1	8	yes	complemented
15	C₂×C₅×C₇²	—	3	2	no	—
16	C₅⋊(C₇⋊D₇)	—	15	4	no	—
17	C₂×C₅²×C₇	C₃×D₇	1	8	yes	complemented
18	C₅⋊D₃₅	—	7	16	no	—
19	C₂×(C₅⋊D₅)⋊C₃	—	49	1	no	Hall
20	C₂×C₃×C₇²	—	25	1	no	—
21	C₃×C₇⋊D₇	—	25	2	no	—
22	C₅×C₇²	—	3	2	no	—
23	C₂×C₇⋊D₇	—	25	1	no	Hall
24	C₅²×C₇	C₂×C₃×D₇	1	8	yes	complemented
25	C₂×C₅²⋊C₃	C₇⋊D₇	1	1	yes	characteristic, complemented
26	(C₅⋊D₅)⋊C₃	—	49	2	no	—
27	C₃×C₇²	—	25	1	no	Hall
28	C₂×D₃₅	—	105	16	no	—
29	C₂×C₅⋊D₅	—	49	1	no	Hall
30	C₂×C₇²	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
31	C₇⋊D₇	—	25	2	no	—
32	C₂×C₃×D₇	—	175	8	no	—
33	C₅²⋊C₃	C₂×C₇⋊D₇	1	1	yes	characteristic, complemented, Hall
34	C₂×C₅×C₇	—	3	16	no	—
35	D₃₅	—	105	32	no	—
36	C₂×C₅²	C₃×C₇⋊D₇	1	1	yes	characteristic, complemented
37	C₅⋊D₅	—	49	2	no	—
38	C₇²	C₂×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃×C₇	—	25	8	no	—
40	C₃×D₇	—	175	16	no	—
41	C₅×C₇	—	3	16	no	—
42	C₂×D₇	—	175	8	no	—
43	C₅²	C₂×C₃×C₇⋊D₇	1	1	yes	characteristic, complemented, Sylow
44	C₃×C₇	—	25	8	no	—
45	C₂×D₅	—	735	2	no	—
46	C₂×C₇	#11760384	1	8	yes	complemented
47	D₇	—	175	16	no	—
48	C₂²×C₃	—	1225	1	no	Hall
49	C₂×C₅	—	3	2	no	—
50	D₅	—	735	4	no	—
51	C₇	C₂×#11760384	1	8	yes	complemented
52	C₂×C₃	—	25	1	no	—
53	C₂×C₃	—	1225	2	no	—
54	C₅	—	3	2	no	—
55	C₂²	—	1225	1	no	Sylow
56	C₃	—	25	1	no	Sylow
57	C₂	—	1225	2	no	—
58	C₂	#20301299	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	#20301299
commutator, cosocle, hypocenter	C₅²×C₇²	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₇²×C₅²⋊C₃	C₂	semidirect
2. #20301299	C₂	direct (N × H)
3. C₂×C₅⋊(C₅⋊(C₇⋊D₇))	C₃	semidirect
4. C₇²×C₅²⋊C₃	C₂²	semidirect
5. C₅⋊(C₅⋊(C₇⋊D₇))	C₂×C₃	semidirect
6. C₂×C₅²×C₇²	C₂×C₃	semidirect
8. C₅²×C₇²	C₂²×C₃	semidirect
9. C₂×C₇×C₅²⋊C₃	D₇	semidirect
14. C₇×C₅²⋊C₃	C₂×D₇	semidirect
17. C₂×C₅²×C₇	C₃×D₇	semidirect
24. C₅²×C₇	C₂×C₃×D₇	semidirect
25. C₂×C₅²⋊C₃	C₇⋊D₇	semidirect
30. C₂×C₇²	(C₅⋊D₅)⋊C₃	semidirect
33. C₅²⋊C₃	C₂×C₇⋊D₇	semidirect
36. C₂×C₅²	C₃×C₇⋊D₇	semidirect
38. C₇²	C₂×(C₅⋊D₅)⋊C₃	semidirect
43. C₅²	C₂×C₃×C₇⋊D₇	semidirect
46. C₂×C₇	#11760384	semidirect
51. C₇	C₂×#11760384	semidirect
58. C₂	#20301299	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇²×C₅²⋊C₃	2
#20301299	2
C₂×C₅⋊(C₅⋊(C₇⋊D₇))	3
C₂×#11760384	7
C₂×C₃×C₇⋊D₇	25

Subgroups — C2×#20301299 #28413446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20301299 #28413446