Subgroups — C₂²×C₇²×C₁₂₁ #37557359

Order 23716.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇²×C₁₂₁	C₂	1	3	yes	maximal, direct product factor
2	C₇²×C₁₂₁	C₂²	1	1	yes	characteristic, direct product factor, Hall
3	C₂²×C₇×C₁₂₁	C₇	1	8	yes	maximal, direct product factor
4	C₂²×C₇²×C₁₁	C₁₁	1	1	yes	maximal, characteristic
5	C₂×C₇×C₁₂₁	C₂×C₇	1	24	yes	direct product factor
6	C₂×C₇²×C₁₁	C₂×C₁₁	1	3	yes	—
7	C₇×C₁₂₁	C₂²×C₇	1	8	yes	direct product factor
8	C₇²×C₁₁	C₂²×C₁₁	1	1	yes	characteristic
9	C₂²×C₁₂₁	C₇²	1	1	yes	characteristic, direct product factor, Hall
10	C₂²×C₇×C₁₁	C₇×C₁₁	1	8	yes	—
11	C₂×C₁₂₁	C₂×C₇²	1	3	yes	direct product factor
12	C₂²×C₇²	C₁₂₁	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₇×C₁₁	C₂×C₇×C₁₁	1	24	yes	—
14	C₁₂₁	C₂²×C₇²	1	1	yes	characteristic, direct product factor, Sylow
15	C₂×C₇²	C₂×C₁₂₁	1	3	yes	direct product factor
16	C₇×C₁₁	C₂²×C₇×C₁₁	1	8	yes	—
17	C₇²	C₂²×C₁₂₁	1	1	yes	characteristic, direct product factor, Sylow
18	C₂²×C₁₁	C₇²×C₁₁	1	1	yes	characteristic
19	C₂²×C₇	C₇×C₁₂₁	1	8	yes	direct product factor
20	C₂×C₁₁	C₂×C₇²×C₁₁	1	3	yes	—
21	C₂×C₇	C₂×C₇×C₁₂₁	1	24	yes	direct product factor
22	C₁₁	C₂²×C₇²×C₁₁	1	1	yes	characteristic
23	C₇	C₂²×C₇×C₁₂₁	1	8	yes	direct product factor
24	C₂²	C₇²×C₁₂₁	1	1	yes	characteristic, direct product factor, Sylow
25	C₂	C₂×C₇²×C₁₂₁	1	3	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₁	C₂²×C₇²×C₁₁
socle, prime radical	C₂²×C₇²×C₁₁	C₁₁
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇²×C₁₂₁	C₂	direct (N × H)
2. C₇²×C₁₂₁	C₂²	direct (N × H)
3. C₂²×C₇×C₁₂₁	C₇	direct (N × H)
5. C₂×C₇×C₁₂₁	C₂×C₇	direct (N × H)
7. C₇×C₁₂₁	C₂²×C₇	direct (N × H)
9. C₂²×C₁₂₁	C₇²	direct (N × H)
11. C₂×C₁₂₁	C₂×C₇²	direct (N × H)
12. C₂²×C₇²	C₁₂₁	direct (N × H)
14. C₁₂₁	C₂²×C₇²	direct (N × H)
15. C₂×C₇²	C₂×C₁₂₁	direct (N × H)
17. C₇²	C₂²×C₁₂₁	direct (N × H)
19. C₂²×C₇	C₇×C₁₂₁	direct (N × H)
21. C₂×C₇	C₂×C₇×C₁₂₁	direct (N × H)
23. C₇	C₂²×C₇×C₁₂₁	direct (N × H)
24. C₂²	C₇²×C₁₂₁	direct (N × H)
25. C₂	C₂×C₇²×C₁₂₁	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×C₁₂₁	2
C₂²×C₇×C₁₂₁	7
C₂²×C₇²×C₁₁	11

Subgroups — C22×C72×C121 #37557359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇²×C₁₂₁ #37557359