Subgroups — C₂²×C₄×C₂₇×C₂₉ #72190875

Order 12528.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂₇×C₂₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₂₇×C₂₉	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₄×C₉×C₂₉	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₂₇×C₂₉	C₄	1	4	yes	direct product factor
5	C₂²×C₂₇×C₂₉	C₂²	1	3	yes	—
6	C₄×C₂₇×C₂₉	C₂²	1	4	yes	direct product factor
7	C₂³×C₉×C₂₉	C₂×C₃	1	1	yes	characteristic
8	C₂×C₄×C₉×C₂₉	C₂×C₃	1	6	yes	—
9	C₂×C₂₇×C₂₉	C₂×C₄	1	6	yes	direct product factor
10	C₂×C₂₇×C₂₉	C₂³	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₂₉	C₉	1	1	yes	characteristic
12	C₂²×C₉×C₂₉	C₃×C₄	1	4	yes	—
13	C₂²×C₉×C₂₉	C₂²×C₃	1	3	yes	—
14	C₄×C₉×C₂₉	C₂²×C₃	1	4	yes	—
15	C₂₇×C₂₉	C₂²×C₄	1	1	yes	characteristic, direct product factor, Hall
16	C₂³×C₃×C₂₉	C₂×C₉	1	1	yes	characteristic
17	C₂×C₃×C₄×C₂₉	C₂×C₉	1	6	yes	—
18	C₂×C₉×C₂₉	C₂×C₃×C₄	1	6	yes	—
19	C₂×C₉×C₂₉	C₂³×C₃	1	1	yes	characteristic
20	C₂²×C₄×C₂₉	C₂₇	1	1	yes	characteristic, direct product factor, Hall
21	C₂²×C₄×C₂₇	C₂₉	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂²×C₃×C₂₉	C₄×C₉	1	4	yes	—
23	C₂²×C₃×C₂₉	C₂²×C₉	1	3	yes	—
24	C₃×C₄×C₂₉	C₂²×C₉	1	4	yes	—
25	C₉×C₂₉	C₂²×C₃×C₄	1	1	yes	characteristic
26	C₂³×C₂₉	C₂×C₂₇	1	1	yes	characteristic
27	C₂×C₄×C₂₉	C₂×C₂₇	1	6	yes	direct product factor
28	C₂³×C₂₇	C₂×C₂₉	1	1	yes	characteristic
29	C₂×C₄×C₂₇	C₂×C₂₉	1	6	yes	direct product factor
30	C₂×C₃×C₂₉	C₂×C₄×C₉	1	6	yes	—
31	C₂×C₃×C₂₉	C₂³×C₉	1	1	yes	characteristic
32	C₂²×C₄×C₉	C₃×C₂₉	1	1	yes	characteristic
33	C₂²×C₂₉	C₄×C₂₇	1	4	yes	direct product factor
34	C₂²×C₂₉	C₂²×C₂₇	1	3	yes	—
35	C₄×C₂₉	C₂²×C₂₇	1	4	yes	direct product factor
36	C₂²×C₂₇	C₄×C₂₉	1	4	yes	direct product factor
37	C₂²×C₂₇	C₂²×C₂₉	1	3	yes	—
38	C₄×C₂₇	C₂²×C₂₉	1	4	yes	direct product factor
39	C₃×C₂₉	C₂²×C₄×C₉	1	1	yes	characteristic
40	C₂³×C₉	C₂×C₃×C₂₉	1	1	yes	characteristic
41	C₂×C₄×C₉	C₂×C₃×C₂₉	1	6	yes	—
42	C₂×C₂₉	C₂×C₄×C₂₇	1	6	yes	direct product factor
43	C₂×C₂₉	C₂³×C₂₇	1	1	yes	characteristic
44	C₂×C₂₇	C₂×C₄×C₂₉	1	6	yes	direct product factor
45	C₂×C₂₇	C₂³×C₂₉	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₉×C₂₉	1	1	yes	characteristic
47	C₂²×C₉	C₃×C₄×C₂₉	1	4	yes	—
48	C₂²×C₉	C₂²×C₃×C₂₉	1	3	yes	—
49	C₄×C₉	C₂²×C₃×C₂₉	1	4	yes	—
50	C₂₉	C₂²×C₄×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
51	C₂₇	C₂²×C₄×C₂₉	1	1	yes	characteristic, direct product factor, Sylow
52	C₂³×C₃	C₂×C₉×C₂₉	1	1	yes	characteristic
53	C₂×C₃×C₄	C₂×C₉×C₂₉	1	6	yes	—
54	C₂×C₉	C₂×C₃×C₄×C₂₉	1	6	yes	—
55	C₂×C₉	C₂³×C₃×C₂₉	1	1	yes	characteristic
56	C₂²×C₄	C₂₇×C₂₉	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²×C₃	C₄×C₉×C₂₉	1	4	yes	—
58	C₂²×C₃	C₂²×C₉×C₂₉	1	3	yes	—
59	C₃×C₄	C₂²×C₉×C₂₉	1	4	yes	—
60	C₉	C₂²×C₃×C₄×C₂₉	1	1	yes	characteristic
61	C₂³	C₂×C₂₇×C₂₉	1	1	yes	characteristic
62	C₂×C₄	C₂×C₂₇×C₂₉	1	6	yes	direct product factor
63	C₂×C₃	C₂×C₄×C₉×C₂₉	1	6	yes	—
64	C₂×C₃	C₂³×C₉×C₂₉	1	1	yes	characteristic
65	C₂²	C₄×C₂₇×C₂₉	1	4	yes	direct product factor
66	C₂²	C₂²×C₂₇×C₂₉	1	3	yes	—
67	C₄	C₂²×C₂₇×C₂₉	1	4	yes	direct product factor
68	C₃	C₂²×C₄×C₉×C₂₉	1	1	yes	characteristic
69	C₂	C₂×C₄×C₂₇×C₂₉	1	6	yes	direct product factor
70	C₂	C₂³×C₂₇×C₂₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₉	C₂³×C₃×C₂₉
socle, prime radical	C₂³×C₃×C₂₉	C₂×C₉
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×C₂₇×C₂₉	C₂	direct (N × H)
4. C₂²×C₂₇×C₂₉	C₄	direct (N × H)
6. C₄×C₂₇×C₂₉	C₂²	direct (N × H)
9. C₂×C₂₇×C₂₉	C₂×C₄	direct (N × H)
15. C₂₇×C₂₉	C₂²×C₄	direct (N × H)
20. C₂²×C₄×C₂₉	C₂₇	direct (N × H)
21. C₂²×C₄×C₂₇	C₂₉	direct (N × H)
27. C₂×C₄×C₂₉	C₂×C₂₇	direct (N × H)
29. C₂×C₄×C₂₇	C₂×C₂₉	direct (N × H)
33. C₂²×C₂₉	C₄×C₂₇	direct (N × H)
35. C₄×C₂₉	C₂²×C₂₇	direct (N × H)
36. C₂²×C₂₇	C₄×C₂₉	direct (N × H)
38. C₄×C₂₇	C₂²×C₂₉	direct (N × H)
42. C₂×C₂₉	C₂×C₄×C₂₇	direct (N × H)
44. C₂×C₂₇	C₂×C₄×C₂₉	direct (N × H)
50. C₂₉	C₂²×C₄×C₂₇	direct (N × H)
51. C₂₇	C₂²×C₄×C₂₉	direct (N × H)
56. C₂²×C₄	C₂₇×C₂₉	direct (N × H)
62. C₂×C₄	C₂×C₂₇×C₂₉	direct (N × H)
65. C₂²	C₄×C₂₇×C₂₉	direct (N × H)
67. C₄	C₂²×C₂₇×C₂₉	direct (N × H)
69. C₂	C₂×C₄×C₂₇×C₂₉	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×C₂₇×C₂₉	2
C₂×C₄×C₂₇×C₂₉	2
C₂²×C₄×C₉×C₂₉	3
C₂²×C₄×C₂₇	29

Subgroups — C22×C4×C27×C29 #72190875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₂₇×C₂₉ #72190875