Subgroups — C₂×C₃×C₄²×C₂₈₉ #72204792

Order 27744.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₂₈₉	C₂	1	3	yes	maximal
2	C₃×C₄²×C₂₈₉	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄²×C₂₈₉	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₃×C₂₈₉	C₂²	1	1	yes	characteristic
5	C₂×C₃×C₄×C₂₈₉	C₄	1	12	yes	direct product factor
6	C₂×C₃×C₄×C₂₈₉	C₂²	1	6	yes	—
7	C₂²×C₄×C₂₈₉	C₂×C₃	1	3	yes	—
8	C₄²×C₂₈₉	C₂×C₃	1	4	yes	direct product factor
9	C₂³×C₂₈₉	C₂²×C₃	1	1	yes	characteristic
10	C₂×C₄×C₂₈₉	C₃×C₄	1	12	yes	direct product factor
11	C₂×C₄×C₂₈₉	C₂²×C₃	1	6	yes	—
12	C₂²×C₃×C₂₈₉	C₂×C₄	1	6	yes	—
13	C₂²×C₃×C₂₈₉	C₂³	1	1	yes	characteristic
14	C₃×C₄×C₂₈₉	C₂×C₄	1	12	yes	direct product factor
15	C₂×C₃×C₂₈₉	C₄²	1	4	yes	direct product factor
16	C₂×C₃×C₂₈₉	C₂²×C₄	1	3	yes	—
17	C₂×C₃×C₄²×C₁₇	C₁₇	1	1	yes	maximal, characteristic
18	C₂²×C₂₈₉	C₂×C₃×C₄	1	6	yes	—
19	C₂²×C₂₈₉	C₂³×C₃	1	1	yes	characteristic
20	C₄×C₂₈₉	C₂×C₃×C₄	1	12	yes	direct product factor
21	C₃×C₂₈₉	C₂×C₄²	1	1	yes	characteristic, direct product factor, Hall
22	C₂²×C₃×C₄×C₁₇	C₂×C₁₇	1	3	yes	—
23	C₃×C₄²×C₁₇	C₂×C₁₇	1	4	yes	—
24	C₂×C₂₈₉	C₃×C₄²	1	4	yes	direct product factor
25	C₂×C₂₈₉	C₂²×C₃×C₄	1	3	yes	—
26	C₂×C₄²×C₁₇	C₃×C₁₇	1	1	yes	characteristic
27	C₂³×C₃×C₁₇	C₂²×C₁₇	1	1	yes	characteristic
28	C₂×C₃×C₄×C₁₇	C₄×C₁₇	1	12	yes	—
29	C₂×C₃×C₄×C₁₇	C₂²×C₁₇	1	6	yes	—
30	C₂₈₉	C₂×C₃×C₄²	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²×C₄×C₁₇	C₂×C₃×C₁₇	1	3	yes	—
32	C₄²×C₁₇	C₂×C₃×C₁₇	1	4	yes	—
33	C₂²×C₃×C₁₇	C₂×C₄×C₁₇	1	6	yes	—
34	C₂²×C₃×C₁₇	C₂³×C₁₇	1	1	yes	characteristic
35	C₃×C₄×C₁₇	C₂×C₄×C₁₇	1	12	yes	—
36	C₂³×C₁₇	C₂²×C₃×C₁₇	1	1	yes	characteristic
37	C₂×C₄×C₁₇	C₃×C₄×C₁₇	1	12	yes	—
38	C₂×C₄×C₁₇	C₂²×C₃×C₁₇	1	6	yes	—
39	C₂×C₃×C₁₇	C₄²×C₁₇	1	4	yes	—
40	C₂×C₃×C₁₇	C₂²×C₄×C₁₇	1	3	yes	—
41	C₂×C₃×C₄²	C₂₈₉	1	1	yes	characteristic, direct product factor, Hall
42	C₂²×C₁₇	C₂×C₃×C₄×C₁₇	1	6	yes	—
43	C₂²×C₁₇	C₂³×C₃×C₁₇	1	1	yes	characteristic
44	C₄×C₁₇	C₂×C₃×C₄×C₁₇	1	12	yes	—
45	C₃×C₁₇	C₂×C₄²×C₁₇	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₂×C₂₈₉	1	3	yes	—
47	C₃×C₄²	C₂×C₂₈₉	1	4	yes	direct product factor
48	C₂×C₁₇	C₃×C₄²×C₁₇	1	4	yes	—
49	C₂×C₁₇	C₂²×C₃×C₄×C₁₇	1	3	yes	—
50	C₂×C₄²	C₃×C₂₈₉	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³×C₃	C₂²×C₂₈₉	1	1	yes	characteristic
52	C₂×C₃×C₄	C₄×C₂₈₉	1	12	yes	direct product factor
53	C₂×C₃×C₄	C₂²×C₂₈₉	1	6	yes	—
54	C₁₇	C₂×C₃×C₄²×C₁₇	1	1	yes	characteristic
55	C₂²×C₄	C₂×C₃×C₂₈₉	1	3	yes	—
56	C₄²	C₂×C₃×C₂₈₉	1	4	yes	direct product factor
57	C₂²×C₃	C₂×C₄×C₂₈₉	1	6	yes	—
58	C₂²×C₃	C₂³×C₂₈₉	1	1	yes	characteristic
59	C₃×C₄	C₂×C₄×C₂₈₉	1	12	yes	direct product factor
60	C₂³	C₂²×C₃×C₂₈₉	1	1	yes	characteristic
61	C₂×C₄	C₃×C₄×C₂₈₉	1	12	yes	direct product factor
62	C₂×C₄	C₂²×C₃×C₂₈₉	1	6	yes	—
63	C₂×C₃	C₄²×C₂₈₉	1	4	yes	direct product factor
64	C₂×C₃	C₂²×C₄×C₂₈₉	1	3	yes	—
65	C₂²	C₂×C₃×C₄×C₂₈₉	1	6	yes	—
66	C₂²	C₂³×C₃×C₂₈₉	1	1	yes	characteristic
67	C₄	C₂×C₃×C₄×C₂₈₉	1	12	yes	direct product factor
68	C₃	C₂×C₄²×C₂₈₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂	C₃×C₄²×C₂₈₉	1	4	yes	direct product factor
70	C₂	C₂²×C₃×C₄×C₂₈₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₁₇	C₂³×C₃×C₁₇
socle, prime radical	C₂³×C₃×C₁₇	C₂²×C₁₇
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×C₂₈₉	C₂	direct (N × H)
3. C₂×C₄²×C₂₈₉	C₃	direct (N × H)
5. C₂×C₃×C₄×C₂₈₉	C₄	direct (N × H)
8. C₄²×C₂₈₉	C₂×C₃	direct (N × H)
10. C₂×C₄×C₂₈₉	C₃×C₄	direct (N × H)
14. C₃×C₄×C₂₈₉	C₂×C₄	direct (N × H)
15. C₂×C₃×C₂₈₉	C₄²	direct (N × H)
20. C₄×C₂₈₉	C₂×C₃×C₄	direct (N × H)
21. C₃×C₂₈₉	C₂×C₄²	direct (N × H)
24. C₂×C₂₈₉	C₃×C₄²	direct (N × H)
30. C₂₈₉	C₂×C₃×C₄²	direct (N × H)
41. C₂×C₃×C₄²	C₂₈₉	direct (N × H)
47. C₃×C₄²	C₂×C₂₈₉	direct (N × H)
50. C₂×C₄²	C₃×C₂₈₉	direct (N × H)
52. C₂×C₃×C₄	C₄×C₂₈₉	direct (N × H)
56. C₄²	C₂×C₃×C₂₈₉	direct (N × H)
59. C₃×C₄	C₂×C₄×C₂₈₉	direct (N × H)
61. C₂×C₄	C₃×C₄×C₂₈₉	direct (N × H)
63. C₂×C₃	C₄²×C₂₈₉	direct (N × H)
67. C₄	C₂×C₃×C₄×C₂₈₉	direct (N × H)
68. C₃	C₂×C₄²×C₂₈₉	direct (N × H)
69. C₂	C₃×C₄²×C₂₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×C₂₈₉	2
C₃×C₄²×C₂₈₉	2
C₂×C₄²×C₂₈₉	3
C₂×C₃×C₄²×C₁₇	17

Subgroups — C2×C3×C42×C289 #72204792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄²×C₂₈₉ #72204792