Subgroups — C₂²×C₄×C₂₃×C₇₁ #72203293

Order 26128.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂₃×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂₃×C₇₁	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₂₃×C₇₁	C₄	1	4	yes	direct product factor
4	C₂²×C₂₃×C₇₁	C₂²	1	3	yes	—
5	C₄×C₂₃×C₇₁	C₂²	1	4	yes	direct product factor
6	C₂×C₂₃×C₇₁	C₂×C₄	1	6	yes	direct product factor
7	C₂×C₂₃×C₇₁	C₂³	1	1	yes	characteristic
8	C₂₃×C₇₁	C₂²×C₄	1	1	yes	characteristic, direct product factor, Hall
9	C₂²×C₄×C₇₁	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂³×C₇₁	C₂×C₂₃	1	1	yes	characteristic
11	C₂×C₄×C₇₁	C₂×C₂₃	1	6	yes	direct product factor
12	C₂²×C₄×C₂₃	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂²×C₇₁	C₄×C₂₃	1	4	yes	direct product factor
14	C₂²×C₇₁	C₂²×C₂₃	1	3	yes	—
15	C₄×C₇₁	C₂²×C₂₃	1	4	yes	direct product factor
16	C₂³×C₂₃	C₂×C₇₁	1	1	yes	characteristic
17	C₂×C₄×C₂₃	C₂×C₇₁	1	6	yes	direct product factor
18	C₂×C₇₁	C₂×C₄×C₂₃	1	6	yes	direct product factor
19	C₂×C₇₁	C₂³×C₂₃	1	1	yes	characteristic
20	C₂²×C₂₃	C₄×C₇₁	1	4	yes	direct product factor
21	C₂²×C₂₃	C₂²×C₇₁	1	3	yes	—
22	C₄×C₂₃	C₂²×C₇₁	1	4	yes	direct product factor
23	C₇₁	C₂²×C₄×C₂₃	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₂₃	C₂×C₄×C₇₁	1	6	yes	direct product factor
25	C₂×C₂₃	C₂³×C₇₁	1	1	yes	characteristic
26	C₂₃	C₂²×C₄×C₇₁	1	1	yes	characteristic, direct product factor, Sylow
27	C₂²×C₄	C₂₃×C₇₁	1	1	yes	characteristic, direct product factor, Sylow
28	C₂³	C₂×C₂₃×C₇₁	1	1	yes	characteristic
29	C₂×C₄	C₂×C₂₃×C₇₁	1	6	yes	direct product factor
30	C₂²	C₄×C₂₃×C₇₁	1	4	yes	direct product factor
31	C₂²	C₂²×C₂₃×C₇₁	1	3	yes	—
32	C₄	C₂²×C₂₃×C₇₁	1	4	yes	direct product factor
33	C₂	C₂×C₄×C₂₃×C₇₁	1	6	yes	direct product factor
34	C₂	C₂³×C₂₃×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂³×C₂₃×C₇₁
socle, prime radical	C₂³×C₂₃×C₇₁	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₂₃×C₇₁	C₂	direct (N × H)
3. C₂²×C₂₃×C₇₁	C₄	direct (N × H)
5. C₄×C₂₃×C₇₁	C₂²	direct (N × H)
6. C₂×C₂₃×C₇₁	C₂×C₄	direct (N × H)
8. C₂₃×C₇₁	C₂²×C₄	direct (N × H)
9. C₂²×C₄×C₇₁	C₂₃	direct (N × H)
11. C₂×C₄×C₇₁	C₂×C₂₃	direct (N × H)
12. C₂²×C₄×C₂₃	C₇₁	direct (N × H)
13. C₂²×C₇₁	C₄×C₂₃	direct (N × H)
15. C₄×C₇₁	C₂²×C₂₃	direct (N × H)
17. C₂×C₄×C₂₃	C₂×C₇₁	direct (N × H)
18. C₂×C₇₁	C₂×C₄×C₂₃	direct (N × H)
20. C₂²×C₂₃	C₄×C₇₁	direct (N × H)
22. C₄×C₂₃	C₂²×C₇₁	direct (N × H)
23. C₇₁	C₂²×C₄×C₂₃	direct (N × H)
24. C₂×C₂₃	C₂×C₄×C₇₁	direct (N × H)
26. C₂₃	C₂²×C₄×C₇₁	direct (N × H)
27. C₂²×C₄	C₂₃×C₇₁	direct (N × H)
29. C₂×C₄	C₂×C₂₃×C₇₁	direct (N × H)
30. C₂²	C₄×C₂₃×C₇₁	direct (N × H)
32. C₄	C₂²×C₂₃×C₇₁	direct (N × H)
33. C₂	C₂×C₄×C₂₃×C₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂₃×C₇₁	2
C₂×C₄×C₂₃×C₇₁	2
C₂²×C₄×C₇₁	23
C₂²×C₄×C₂₃	71

Subgroups — C22×C4×C23×C71 #72203293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₂₃×C₇₁ #72203293