Subgroups — C₂×C₃²×C₈×C₁₃₇ #72197402

Order 19728.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₈×C₁₃₇	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₈×C₁₃₇	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₄×C₁₃₇	C₂×C₃	1	4	yes	—
5	C₃×C₈×C₁₃₇	C₂×C₃	1	8	yes	direct product factor
6	C₂²×C₃²×C₁₃₇	C₄	1	1	yes	characteristic
7	C₃²×C₄×C₁₃₇	C₄	1	1	yes	characteristic
8	C₃²×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₁₃₇	C₈	1	2	yes	direct product factor
10	C₂×C₃²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₁₃₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
12	C₂²×C₃×C₁₃₇	C₃×C₄	1	4	yes	—
13	C₃×C₄×C₁₃₇	C₃×C₄	1	4	yes	—
14	C₃×C₄×C₁₃₇	C₂²×C₃	1	4	yes	—
15	C₃²×C₁₃₇	C₂×C₈	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₄×C₁₃₇	C₂×C₃²	1	1	yes	characteristic
17	C₈×C₁₃₇	C₂×C₃²	1	2	yes	direct product factor
18	C₂×C₃×C₁₃₇	C₃×C₈	1	8	yes	direct product factor
19	C₂×C₃×C₁₃₇	C₂×C₃×C₄	1	4	yes	—
20	C₂²×C₁₃₇	C₃²×C₄	1	1	yes	characteristic
21	C₄×C₁₃₇	C₃²×C₄	1	1	yes	characteristic
22	C₄×C₁₃₇	C₂²×C₃²	1	1	yes	characteristic
23	C₃×C₁₃₇	C₂×C₃×C₈	1	4	yes	direct product factor
24	C₂×C₁₃₇	C₃²×C₈	1	2	yes	direct product factor
25	C₂×C₁₃₇	C₂×C₃²×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₁₃₇	C₂×C₃²×C₈	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×C₃²×C₄	C₂×C₁₃₇	1	1	yes	characteristic
29	C₃²×C₈	C₂×C₁₃₇	1	2	yes	direct product factor
30	C₂×C₃×C₈	C₃×C₁₃₇	1	4	yes	direct product factor
31	C₂²×C₃²	C₄×C₁₃₇	1	1	yes	characteristic
32	C₃²×C₄	C₄×C₁₃₇	1	1	yes	characteristic
33	C₃²×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
34	C₂×C₃×C₄	C₂×C₃×C₁₃₇	1	4	yes	—
35	C₃×C₈	C₂×C₃×C₁₃₇	1	8	yes	direct product factor
36	C₂×C₃²	C₈×C₁₃₇	1	2	yes	direct product factor
37	C₂×C₃²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
38	C₂×C₈	C₃²×C₁₃₇	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₃	C₃×C₄×C₁₃₇	1	4	yes	—
40	C₃×C₄	C₃×C₄×C₁₃₇	1	4	yes	—
41	C₃×C₄	C₂²×C₃×C₁₃₇	1	4	yes	—
42	C₃²	C₂×C₈×C₁₃₇	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₄	C₂×C₃²×C₁₃₇	1	1	yes	characteristic
44	C₈	C₂×C₃²×C₁₃₇	1	2	yes	direct product factor
45	C₂×C₃	C₃×C₈×C₁₃₇	1	8	yes	direct product factor
46	C₂×C₃	C₂×C₃×C₄×C₁₃₇	1	4	yes	—
47	C₂²	C₃²×C₄×C₁₃₇	1	1	yes	characteristic
48	C₄	C₃²×C₄×C₁₃₇	1	1	yes	characteristic
49	C₄	C₂²×C₃²×C₁₃₇	1	1	yes	characteristic
50	C₃	C₂×C₃×C₈×C₁₃₇	1	4	yes	direct product factor
51	C₂	C₃²×C₈×C₁₃₇	1	2	yes	direct product factor
52	C₂	C₂×C₃²×C₄×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂²×C₃²×C₁₃₇
socle, prime radical	C₂²×C₃²×C₁₃₇	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₈×C₁₃₇	C₂	direct (N × H)
3. C₂×C₃×C₈×C₁₃₇	C₃	direct (N × H)
5. C₃×C₈×C₁₃₇	C₂×C₃	direct (N × H)
9. C₂×C₃²×C₁₃₇	C₈	direct (N × H)
11. C₂×C₈×C₁₃₇	C₃²	direct (N × H)
15. C₃²×C₁₃₇	C₂×C₈	direct (N × H)
17. C₈×C₁₃₇	C₂×C₃²	direct (N × H)
18. C₂×C₃×C₁₃₇	C₃×C₈	direct (N × H)
23. C₃×C₁₃₇	C₂×C₃×C₈	direct (N × H)
24. C₂×C₁₃₇	C₃²×C₈	direct (N × H)
26. C₂×C₃²×C₈	C₁₃₇	direct (N × H)
27. C₁₃₇	C₂×C₃²×C₈	direct (N × H)
29. C₃²×C₈	C₂×C₁₃₇	direct (N × H)
30. C₂×C₃×C₈	C₃×C₁₃₇	direct (N × H)
35. C₃×C₈	C₂×C₃×C₁₃₇	direct (N × H)
36. C₂×C₃²	C₈×C₁₃₇	direct (N × H)
38. C₂×C₈	C₃²×C₁₃₇	direct (N × H)
42. C₃²	C₂×C₈×C₁₃₇	direct (N × H)
44. C₈	C₂×C₃²×C₁₃₇	direct (N × H)
45. C₂×C₃	C₃×C₈×C₁₃₇	direct (N × H)
50. C₃	C₂×C₃×C₈×C₁₃₇	direct (N × H)
51. C₂	C₃²×C₈×C₁₃₇	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×C₁₃₇	2
C₃²×C₈×C₁₃₇	2
C₂×C₃×C₈×C₁₃₇	3
C₂×C₃²×C₈	137

Subgroups — C2×C32×C8×C137 #72197402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₈×C₁₃₇ #72197402