Subgroups — C₃²×C₄²×C₉₇ #72192166

Order 13968.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₉₇	C₂	1	3	yes	maximal
2	C₃×C₄²×C₉₇	C₃	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₄×C₉₇	C₂×C₃	1	12	yes	—
4	C₂²×C₃²×C₉₇	C₂²	1	1	yes	characteristic
5	C₃²×C₄×C₉₇	C₄	1	6	yes	direct product factor
6	C₂×C₃²×C₉₇	C₂×C₄	1	3	yes	—
7	C₄²×C₉₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₂²×C₃×C₉₇	C₂²×C₃	1	4	yes	—
9	C₃×C₄×C₉₇	C₃×C₄	1	24	yes	direct product factor
10	C₃²×C₉₇	C₄²	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₄×C₉₇	C₂×C₃²	1	3	yes	—
12	C₂×C₃×C₉₇	C₂×C₃×C₄	1	12	yes	—
13	C₂²×C₉₇	C₂²×C₃²	1	1	yes	characteristic
14	C₄×C₉₇	C₃²×C₄	1	6	yes	direct product factor
15	C₃×C₉₇	C₃×C₄²	1	4	yes	direct product factor
16	C₂×C₉₇	C₂×C₃²×C₄	1	3	yes	—
17	C₃²×C₄²	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₉₇	C₃²×C₄²	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₃²×C₄	C₂×C₉₇	1	3	yes	—
20	C₃×C₄²	C₃×C₉₇	1	4	yes	direct product factor
21	C₂²×C₃²	C₂²×C₉₇	1	1	yes	characteristic
22	C₃²×C₄	C₄×C₉₇	1	6	yes	direct product factor
23	C₂×C₃×C₄	C₂×C₃×C₉₇	1	12	yes	—
24	C₂×C₃²	C₂×C₄×C₉₇	1	3	yes	—
25	C₄²	C₃²×C₉₇	1	1	yes	characteristic, direct product factor, Sylow
26	C₂²×C₃	C₂²×C₃×C₉₇	1	4	yes	—
27	C₃×C₄	C₃×C₄×C₉₇	1	24	yes	direct product factor
28	C₃²	C₄²×C₉₇	1	1	yes	characteristic, direct product factor, Sylow
29	C₂×C₄	C₂×C₃²×C₉₇	1	3	yes	—
30	C₂×C₃	C₂×C₃×C₄×C₉₇	1	12	yes	—
31	C₂²	C₂²×C₃²×C₉₇	1	1	yes	characteristic
32	C₄	C₃²×C₄×C₉₇	1	6	yes	direct product factor
33	C₃	C₃×C₄²×C₉₇	1	4	yes	direct product factor
34	C₂	C₂×C₃²×C₄×C₉₇	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²	C₂²×C₃²×C₉₇
socle, prime radical	C₂²×C₃²×C₉₇	C₂²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄²×C₉₇	C₃	direct (N × H)
5. C₃²×C₄×C₉₇	C₄	direct (N × H)
7. C₄²×C₉₇	C₃²	direct (N × H)
9. C₃×C₄×C₉₇	C₃×C₄	direct (N × H)
10. C₃²×C₉₇	C₄²	direct (N × H)
14. C₄×C₉₇	C₃²×C₄	direct (N × H)
15. C₃×C₉₇	C₃×C₄²	direct (N × H)
17. C₃²×C₄²	C₉₇	direct (N × H)
18. C₉₇	C₃²×C₄²	direct (N × H)
20. C₃×C₄²	C₃×C₉₇	direct (N × H)
22. C₃²×C₄	C₄×C₉₇	direct (N × H)
25. C₄²	C₃²×C₉₇	direct (N × H)
27. C₃×C₄	C₃×C₄×C₉₇	direct (N × H)
28. C₃²	C₄²×C₉₇	direct (N × H)
32. C₄	C₃²×C₄×C₉₇	direct (N × H)
33. C₃	C₃×C₄²×C₉₇	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×C₃²×C₄×C₉₇	2
C₃×C₄²×C₉₇	3
C₃²×C₄²	97

Subgroups — C32×C42×C97 #72192166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄²×C₉₇ #72192166