Subgroups — C₇×D₅×D₂₂₃ #44530240

Order 31220.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×D₁₁₁₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₅×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×C₇×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₇×C₂₂₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇×D₂₂₃	—	5	1	no	maximal, Hall
6	D₅×D₂₂₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₇×C₂₂₃	—	5	1	no	—
8	C₇×D₂₂₃	—	5	1	no	—
9	C₇×D₂₂₃	D₅	1	1	yes	characteristic, direct product factor
10	D₁₁₁₅	C₂×C₇	1	1	yes	characteristic, complemented
11	C₅×D₂₂₃	C₂×C₇	1	1	yes	characteristic, complemented
12	D₅×C₂₂₃	C₂×C₇	1	1	yes	characteristic, complemented
13	C₇×C₂₂₃	C₂×D₅	1	1	yes	characteristic, complemented, Hall
14	C₅×C₂₂₃	C₂²×C₇	1	1	yes	characteristic, complemented, Hall
15	C₂×D₂₂₃	—	5	1	no	Hall
16	C₂×C₂₂₃	—	5	1	no	—
17	D₂₂₃	—	5	1	no	—
18	D₂₂₃	C₇×D₅	1	1	yes	characteristic, direct product factor
19	C₂₂₃	C₂×C₇×D₅	1	1	yes	characteristic, complemented, Sylow
20	C₂×C₇×D₅	—	223	1	no	maximal, Hall
21	C₂×C₅×C₇	—	223	1	no	—
22	C₇×D₅	—	223	1	no	—
23	C₇×D₅	D₂₂₃	1	1	yes	characteristic, direct product factor
24	C₅×C₇	C₂×D₂₂₃	1	1	yes	characteristic, complemented, Hall
25	C₂²×C₇	—	1115	1	no	Hall
26	C₂×D₅	—	223	1	no	Hall
27	C₂×C₇	—	5	1	no	—
28	C₂×C₇	—	1115	1	no	—
29	C₂×C₇	—	223	1	no	—
30	C₂×C₅	—	223	1	no	—
31	D₅	—	223	1	no	—
32	D₅	C₇×D₂₂₃	1	1	yes	characteristic, direct product factor
33	C₇	D₅×D₂₂₃	1	1	yes	characteristic, direct product factor, Sylow
34	C₅	C₂×C₇×D₂₂₃	1	1	yes	characteristic, complemented, Sylow
35	C₂²	—	1115	1	no	Sylow
36	C₂	—	5	1	no	—
37	C₂	—	1115	1	no	—
38	C₂	—	223	1	no	—

	subgroup	G/H
center, hypercenter	C₇	D₅×D₂₂₃
commutator, cosocle, hypocenter	C₅×C₂₂₃	C₂²×C₇
Frattini	C₁	G
Fitting, socle	C₅×C₇×C₂₂₃	C₂²
LCS	G ▷ C₅×C₂₂₃
UCS	C₁ ◁ C₇

Complements

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

N	H ≅ G/N	type
1. C₇×D₁₁₁₅	C₂	semidirect
2. C₇×D₅×C₂₂₃	C₂	semidirect
3. C₅×C₇×D₂₂₃	C₂	semidirect
4. C₅×C₇×C₂₂₃	C₂²	semidirect
6. D₅×D₂₂₃	C₇	direct (N × H)
9. C₇×D₂₂₃	D₅	direct (N × H)
10. D₁₁₁₅	C₂×C₇	semidirect
11. C₅×D₂₂₃	C₂×C₇	semidirect
12. D₅×C₂₂₃	C₂×C₇	semidirect
13. C₇×C₂₂₃	C₂×D₅	semidirect
14. C₅×C₂₂₃	C₂²×C₇	semidirect
18. D₂₂₃	C₇×D₅	direct (N × H)
19. C₂₂₃	C₂×C₇×D₅	semidirect
23. C₇×D₅	D₂₂₃	direct (N × H)
24. C₅×C₇	C₂×D₂₂₃	semidirect
32. D₅	C₇×D₂₂₃	direct (N × H)
33. C₇	D₅×D₂₂₃	direct (N × H)
34. C₅	C₂×C₇×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₇×D₁₁₁₅	2
C₇×D₅×C₂₂₃	2
C₅×C₇×D₂₂₃	2
C₂×C₇×D₂₂₃	5
D₅×D₂₂₃	7
C₂×C₇×D₅	223

Subgroups — C7×D5×D223 #44530240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×D₅×D₂₂₃ #44530240