Subgroups — C₂×C₃×C₇²×C₈₁ #72201122

Order 23814.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×C₈₁	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃×C₇²×C₂₇	C₃	1	1	yes	maximal, characteristic
3	C₂×C₇²×C₈₁	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₇²×C₂₇	C₂×C₃	1	1	yes	characteristic
5	C₇²×C₈₁	C₂×C₃	1	3	yes	direct product factor
6	C₂×C₃×C₇×C₈₁	C₇	1	8	yes	maximal, direct product factor
7	C₂×C₃×C₇²×C₉	C₉	1	1	yes	characteristic
8	C₂×C₇²×C₂₇	C₉	1	2	yes	—
9	C₂×C₇²×C₂₇	C₃²	1	1	yes	characteristic
10	C₃×C₇×C₈₁	C₂×C₇	1	8	yes	direct product factor
11	C₃×C₇²×C₉	C₂×C₉	1	1	yes	characteristic
12	C₇²×C₂₇	C₂×C₉	1	2	yes	—
13	C₇²×C₂₇	C₂×C₃²	1	1	yes	characteristic
14	C₂×C₃×C₇×C₂₇	C₃×C₇	1	8	yes	—
15	C₂×C₇×C₈₁	C₃×C₇	1	24	yes	direct product factor
16	C₂×C₃²×C₇²	C₂₇	1	1	yes	characteristic
17	C₂×C₇²×C₉	C₂₇	1	2	yes	—
18	C₂×C₇²×C₉	C₃×C₉	1	1	yes	characteristic
19	C₃×C₇×C₂₇	C₂×C₃×C₇	1	8	yes	—
20	C₇×C₈₁	C₂×C₃×C₇	1	24	yes	direct product factor
21	C₂×C₃×C₈₁	C₇²	1	1	yes	characteristic, direct product factor, Hall
22	C₃²×C₇²	C₂×C₂₇	1	1	yes	characteristic
23	C₇²×C₉	C₂×C₂₇	1	2	yes	—
24	C₇²×C₉	C₂×C₃×C₉	1	1	yes	characteristic
25	C₂×C₃×C₇×C₉	C₇×C₉	1	8	yes	—
26	C₂×C₇×C₂₇	C₇×C₉	1	16	yes	—
27	C₂×C₇×C₂₇	C₃²×C₇	1	8	yes	—
28	C₂×C₃×C₇²	C₈₁	1	3	yes	direct product factor
29	C₂×C₃×C₇²	C₃×C₂₇	1	1	yes	characteristic
30	C₃×C₈₁	C₂×C₇²	1	1	yes	characteristic, direct product factor, Sylow
31	C₃×C₇×C₉	C₂×C₇×C₉	1	8	yes	—
32	C₇×C₂₇	C₂×C₇×C₉	1	16	yes	—
33	C₇×C₂₇	C₂×C₃²×C₇	1	8	yes	—
34	C₂×C₃×C₂₇	C₃×C₇²	1	1	yes	characteristic
35	C₂×C₈₁	C₃×C₇²	1	3	yes	direct product factor
36	C₃×C₇²	C₂×C₈₁	1	3	yes	direct product factor
37	C₃×C₇²	C₂×C₃×C₂₇	1	1	yes	characteristic
38	C₂×C₃²×C₇	C₇×C₂₇	1	8	yes	—
39	C₂×C₇×C₉	C₇×C₂₇	1	16	yes	—
40	C₂×C₇×C₉	C₃×C₇×C₉	1	8	yes	—
41	C₂×C₇²	C₃×C₈₁	1	1	yes	characteristic, direct product factor, Hall
42	C₃×C₂₇	C₂×C₃×C₇²	1	1	yes	characteristic
43	C₈₁	C₂×C₃×C₇²	1	3	yes	direct product factor
44	C₃²×C₇	C₂×C₇×C₂₇	1	8	yes	—
45	C₇×C₉	C₂×C₇×C₂₇	1	16	yes	—
46	C₇×C₉	C₂×C₃×C₇×C₉	1	8	yes	—
47	C₂×C₃×C₉	C₇²×C₉	1	1	yes	characteristic
48	C₂×C₂₇	C₇²×C₉	1	2	yes	—
49	C₂×C₂₇	C₃²×C₇²	1	1	yes	characteristic
50	C₇²	C₂×C₃×C₈₁	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃×C₇	C₇×C₈₁	1	24	yes	direct product factor
52	C₂×C₃×C₇	C₃×C₇×C₂₇	1	8	yes	—
53	C₃×C₉	C₂×C₇²×C₉	1	1	yes	characteristic
54	C₂₇	C₂×C₇²×C₉	1	2	yes	—
55	C₂₇	C₂×C₃²×C₇²	1	1	yes	characteristic
56	C₃×C₇	C₂×C₇×C₈₁	1	24	yes	direct product factor
57	C₃×C₇	C₂×C₃×C₇×C₂₇	1	8	yes	—
58	C₂×C₃²	C₇²×C₂₇	1	1	yes	characteristic
59	C₂×C₉	C₇²×C₂₇	1	2	yes	—
60	C₂×C₉	C₃×C₇²×C₉	1	1	yes	characteristic
61	C₂×C₇	C₃×C₇×C₈₁	1	8	yes	direct product factor
62	C₃²	C₂×C₇²×C₂₇	1	1	yes	characteristic
63	C₉	C₂×C₇²×C₂₇	1	2	yes	—
64	C₉	C₂×C₃×C₇²×C₉	1	1	yes	characteristic
65	C₇	C₂×C₃×C₇×C₈₁	1	8	yes	direct product factor
66	C₂×C₃	C₇²×C₈₁	1	3	yes	direct product factor
67	C₂×C₃	C₃×C₇²×C₂₇	1	1	yes	characteristic
68	C₃	C₂×C₇²×C₈₁	1	3	yes	direct product factor
69	C₃	C₂×C₃×C₇²×C₂₇	1	1	yes	characteristic
70	C₂	C₃×C₇²×C₈₁	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₇	C₂×C₃²×C₇²
socle, prime radical	C₂×C₃²×C₇²	C₂₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₇²×C₈₁	C₂	direct (N × H)
3. C₂×C₇²×C₈₁	C₃	direct (N × H)
5. C₇²×C₈₁	C₂×C₃	direct (N × H)
6. C₂×C₃×C₇×C₈₁	C₇	direct (N × H)
10. C₃×C₇×C₈₁	C₂×C₇	direct (N × H)
15. C₂×C₇×C₈₁	C₃×C₇	direct (N × H)
20. C₇×C₈₁	C₂×C₃×C₇	direct (N × H)
21. C₂×C₃×C₈₁	C₇²	direct (N × H)
28. C₂×C₃×C₇²	C₈₁	direct (N × H)
30. C₃×C₈₁	C₂×C₇²	direct (N × H)
35. C₂×C₈₁	C₃×C₇²	direct (N × H)
36. C₃×C₇²	C₂×C₈₁	direct (N × H)
41. C₂×C₇²	C₃×C₈₁	direct (N × H)
43. C₈₁	C₂×C₃×C₇²	direct (N × H)
50. C₇²	C₂×C₃×C₈₁	direct (N × H)
51. C₂×C₃×C₇	C₇×C₈₁	direct (N × H)
56. C₃×C₇	C₂×C₇×C₈₁	direct (N × H)
61. C₂×C₇	C₃×C₇×C₈₁	direct (N × H)
65. C₇	C₂×C₃×C₇×C₈₁	direct (N × H)
66. C₂×C₃	C₇²×C₈₁	direct (N × H)
68. C₃	C₂×C₇²×C₈₁	direct (N × H)
70. C₂	C₃×C₇²×C₈₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇²×C₈₁	2
C₂×C₃×C₇²×C₂₇	3
C₂×C₇²×C₈₁	3
C₂×C₃×C₇×C₈₁	7

Subgroups — C2×C3×C72×C81 #72201122

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇²×C₈₁ #72201122