Subgroups — C₂×C₃×C₄₀₉₆ #72201813

Order 24576.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₂₀₄₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄₀₉₆	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₄₀₉₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂×C₃×C₁₀₂₄	C₄	1	1	yes	characteristic
5	C₃×C₂₀₄₈	C₄	1	1	yes	characteristic
6	C₃×C₂₀₄₈	C₂²	1	1	yes	characteristic
7	C₂×C₂₀₄₈	C₂×C₃	1	1	yes	characteristic
8	C₄₀₉₆	C₂×C₃	1	2	yes	direct product factor
9	C₂×C₃×C₅₁₂	C₈	1	1	yes	characteristic
10	C₃×C₁₀₂₄	C₈	1	1	yes	characteristic
11	C₃×C₁₀₂₄	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₀₂₄	C₃×C₄	1	1	yes	characteristic
13	C₂₀₄₈	C₃×C₄	1	1	yes	characteristic
14	C₂₀₄₈	C₂²×C₃	1	1	yes	characteristic
15	C₂×C₃×C₂₅₆	C₁₆	1	1	yes	characteristic
16	C₃×C₅₁₂	C₁₆	1	1	yes	characteristic
17	C₃×C₅₁₂	C₂×C₈	1	1	yes	characteristic
18	C₂×C₅₁₂	C₃×C₈	1	1	yes	characteristic
19	C₁₀₂₄	C₃×C₈	1	1	yes	characteristic
20	C₁₀₂₄	C₂×C₃×C₄	1	1	yes	characteristic
21	C₂×C₃×C₁₂₈	C₃₂	1	1	yes	characteristic
22	C₃×C₂₅₆	C₃₂	1	1	yes	characteristic
23	C₃×C₂₅₆	C₂×C₁₆	1	1	yes	characteristic
24	C₂×C₂₅₆	C₃×C₁₆	1	1	yes	characteristic
25	C₅₁₂	C₃×C₁₆	1	1	yes	characteristic
26	C₅₁₂	C₂×C₃×C₈	1	1	yes	characteristic
27	C₂×C₃×C₆₄	C₆₄	1	1	yes	characteristic
28	C₃×C₁₂₈	C₆₄	1	1	yes	characteristic
29	C₃×C₁₂₈	C₂×C₃₂	1	1	yes	characteristic
30	C₂×C₁₂₈	C₃×C₃₂	1	1	yes	characteristic
31	C₂₅₆	C₃×C₃₂	1	1	yes	characteristic
32	C₂₅₆	C₂×C₃×C₁₆	1	1	yes	characteristic
33	C₂×C₃×C₃₂	C₁₂₈	1	1	yes	characteristic
34	C₃×C₆₄	C₁₂₈	1	1	yes	characteristic
35	C₃×C₆₄	C₂×C₆₄	1	1	yes	characteristic
36	C₂×C₆₄	C₃×C₆₄	1	1	yes	characteristic
37	C₁₂₈	C₃×C₆₄	1	1	yes	characteristic
38	C₁₂₈	C₂×C₃×C₃₂	1	1	yes	characteristic
39	C₂×C₃×C₁₆	C₂₅₆	1	1	yes	characteristic
40	C₃×C₃₂	C₂₅₆	1	1	yes	characteristic
41	C₃×C₃₂	C₂×C₁₂₈	1	1	yes	characteristic
42	C₂×C₃₂	C₃×C₁₂₈	1	1	yes	characteristic
43	C₆₄	C₃×C₁₂₈	1	1	yes	characteristic
44	C₆₄	C₂×C₃×C₆₄	1	1	yes	characteristic
45	C₂×C₃×C₈	C₅₁₂	1	1	yes	characteristic
46	C₃×C₁₆	C₅₁₂	1	1	yes	characteristic
47	C₃×C₁₆	C₂×C₂₅₆	1	1	yes	characteristic
48	C₂×C₁₆	C₃×C₂₅₆	1	1	yes	characteristic
49	C₃₂	C₃×C₂₅₆	1	1	yes	characteristic
50	C₃₂	C₂×C₃×C₁₂₈	1	1	yes	characteristic
51	C₂×C₃×C₄	C₁₀₂₄	1	1	yes	characteristic
52	C₃×C₈	C₁₀₂₄	1	1	yes	characteristic
53	C₃×C₈	C₂×C₅₁₂	1	1	yes	characteristic
54	C₂×C₈	C₃×C₅₁₂	1	1	yes	characteristic
55	C₁₆	C₃×C₅₁₂	1	1	yes	characteristic
56	C₁₆	C₂×C₃×C₂₅₆	1	1	yes	characteristic
57	C₂²×C₃	C₂₀₄₈	1	1	yes	characteristic
58	C₃×C₄	C₂₀₄₈	1	1	yes	characteristic
59	C₃×C₄	C₂×C₁₀₂₄	1	1	yes	characteristic
60	C₂×C₄	C₃×C₁₀₂₄	1	1	yes	characteristic
61	C₈	C₃×C₁₀₂₄	1	1	yes	characteristic
62	C₈	C₂×C₃×C₅₁₂	1	1	yes	characteristic
63	C₂×C₃	C₄₀₉₆	1	2	yes	direct product factor
64	C₂×C₃	C₂×C₂₀₄₈	1	1	yes	characteristic
65	C₂²	C₃×C₂₀₄₈	1	1	yes	characteristic
66	C₄	C₃×C₂₀₄₈	1	1	yes	characteristic
67	C₄	C₂×C₃×C₁₀₂₄	1	1	yes	characteristic
68	C₃	C₂×C₄₀₉₆	1	1	yes	characteristic, direct product factor, Sylow
69	C₂	C₃×C₄₀₉₆	1	2	yes	direct product factor
70	C₂	C₂×C₃×C₂₀₄₈	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₀₄₈	C₂²×C₃
socle, prime radical	C₂²×C₃	C₂₀₄₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₄₀₉₆	C₂	direct (N × H)
3. C₂×C₄₀₉₆	C₃	direct (N × H)
8. C₄₀₉₆	C₂×C₃	direct (N × H)
63. C₂×C₃	C₄₀₉₆	direct (N × H)
68. C₃	C₂×C₄₀₉₆	direct (N × H)
69. C₂	C₃×C₄₀₉₆	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×C₃×C₂₀₄₈	2
C₃×C₄₀₉₆	2
C₂×C₄₀₉₆	3

Subgroups — C2×C3×C4096 #72201813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄₀₉₆ #72201813