Subgroups — C₂×C₈×C₇₂₉ #72190098

Order 11664.

77 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇₂₉	C₂	1	1	yes	maximal, characteristic
2	C₈×C₇₂₉	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₈×C₂₄₃	C₃	1	1	yes	maximal, characteristic
4	C₂²×C₇₂₉	C₄	1	1	yes	characteristic
5	C₄×C₇₂₉	C₄	1	1	yes	characteristic
6	C₄×C₇₂₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₂₄₃	C₂×C₃	1	1	yes	characteristic
8	C₈×C₂₄₃	C₂×C₃	1	2	yes	—
9	C₂×C₇₂₉	C₈	1	2	yes	direct product factor
10	C₂×C₇₂₉	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₈₁	C₉	1	1	yes	characteristic
12	C₂²×C₂₄₃	C₃×C₄	1	1	yes	characteristic
13	C₄×C₂₄₃	C₃×C₄	1	1	yes	characteristic
14	C₄×C₂₄₃	C₂²×C₃	1	1	yes	characteristic
15	C₇₂₉	C₂×C₈	1	1	yes	characteristic, direct product factor, Sylow
16	C₂×C₄×C₈₁	C₂×C₉	1	1	yes	characteristic
17	C₈×C₈₁	C₂×C₉	1	2	yes	—
18	C₂×C₂₄₃	C₃×C₈	1	2	yes	—
19	C₂×C₂₄₃	C₂×C₃×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₇	C₂₇	1	1	yes	characteristic
21	C₂²×C₈₁	C₄×C₉	1	1	yes	characteristic
22	C₄×C₈₁	C₄×C₉	1	1	yes	characteristic
23	C₄×C₈₁	C₂²×C₉	1	1	yes	characteristic
24	C₂₄₃	C₂×C₃×C₈	1	1	yes	characteristic
25	C₂×C₄×C₂₇	C₂×C₂₇	1	1	yes	characteristic
26	C₈×C₂₇	C₂×C₂₇	1	2	yes	—
27	C₂×C₈₁	C₈×C₉	1	2	yes	—
28	C₂×C₈₁	C₂×C₄×C₉	1	1	yes	characteristic
29	C₂×C₈×C₉	C₈₁	1	1	yes	characteristic
30	C₂²×C₂₇	C₄×C₂₇	1	1	yes	characteristic
31	C₄×C₂₇	C₄×C₂₇	1	1	yes	characteristic
32	C₄×C₂₇	C₂²×C₂₇	1	1	yes	characteristic
33	C₈₁	C₂×C₈×C₉	1	1	yes	characteristic
34	C₂×C₄×C₉	C₂×C₈₁	1	1	yes	characteristic
35	C₈×C₉	C₂×C₈₁	1	2	yes	—
36	C₂×C₂₇	C₈×C₂₇	1	2	yes	—
37	C₂×C₂₇	C₂×C₄×C₂₇	1	1	yes	characteristic
38	C₂×C₃×C₈	C₂₄₃	1	1	yes	characteristic
39	C₂²×C₉	C₄×C₈₁	1	1	yes	characteristic
40	C₄×C₉	C₄×C₈₁	1	1	yes	characteristic
41	C₄×C₉	C₂²×C₈₁	1	1	yes	characteristic
42	C₂₇	C₂×C₈×C₂₇	1	1	yes	characteristic
43	C₂×C₃×C₄	C₂×C₂₄₃	1	1	yes	characteristic
44	C₃×C₈	C₂×C₂₄₃	1	2	yes	—
45	C₂×C₉	C₈×C₈₁	1	2	yes	—
46	C₂×C₉	C₂×C₄×C₈₁	1	1	yes	characteristic
47	C₂×C₈	C₇₂₉	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²×C₃	C₄×C₂₄₃	1	1	yes	characteristic
49	C₃×C₄	C₄×C₂₄₃	1	1	yes	characteristic
50	C₃×C₄	C₂²×C₂₄₃	1	1	yes	characteristic
51	C₉	C₂×C₈×C₈₁	1	1	yes	characteristic
52	C₂×C₄	C₂×C₇₂₉	1	1	yes	characteristic
53	C₈	C₂×C₇₂₉	1	2	yes	direct product factor
54	C₂×C₃	C₈×C₂₄₃	1	2	yes	—
55	C₂×C₃	C₂×C₄×C₂₄₃	1	1	yes	characteristic
56	C₂²	C₄×C₇₂₉	1	1	yes	characteristic
57	C₄	C₄×C₇₂₉	1	1	yes	characteristic
58	C₄	C₂²×C₇₂₉	1	1	yes	characteristic
59	C₃	C₂×C₈×C₂₄₃	1	1	yes	characteristic
60	C₂	C₈×C₇₂₉	1	2	yes	direct product factor
61	C₂	C₂×C₄×C₇₂₉	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₂₄₃	C₂²×C₃
socle, prime radical	C₂²×C₃	C₄×C₂₄₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×C₇₂₉	C₂	direct (N × H)
9. C₂×C₇₂₉	C₈	direct (N × H)
15. C₇₂₉	C₂×C₈	direct (N × H)
47. C₂×C₈	C₇₂₉	direct (N × H)
53. C₈	C₂×C₇₂₉	direct (N × H)
60. C₂	C₈×C₇₂₉	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×C₄×C₇₂₉	2
C₈×C₇₂₉	2
C₂×C₈×C₂₄₃	3

Subgroups — C2×C8×C729 #72190098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×C₇₂₉ #72190098