Subgroups — C₄²×C₁₆×C₁₀₉ #42130531

Order 27904.

558 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₁₀₉	C₂	1	6	yes	maximal
2	C₄²×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₁₀₉	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₁₀₉	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
6	C₄³×C₁₀₉	C₄	1	1	yes	characteristic
7	C₄×C₁₆×C₁₀₉	C₄	1	24	yes	direct product factor
8	C₂²×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₁₀₉	C₈	1	4	yes	—
11	C₂×C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
12	C₂×C₁₆×C₁₀₉	C₂×C₄	1	24	yes	—
13	C₄×C₈×C₁₀₉	C₈	1	12	yes	—
14	C₄×C₈×C₁₀₉	C₂×C₄	1	12	yes	—
15	C₂²×C₄×C₁₀₉	C₂×C₈	1	6	yes	—
16	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₀₉	C₄²	1	6	yes	—
18	C₂×C₈×C₁₀₉	C₂×C₈	1	12	yes	—
19	C₂×C₈×C₁₀₉	C₂²×C₄	1	6	yes	—
20	C₄²×C₁₀₉	C₁₆	1	16	yes	direct product factor
21	C₄²×C₁₀₉	C₄²	1	6	yes	—
22	C₄²×C₁₀₉	C₂×C₈	1	6	yes	—
23	C₁₆×C₁₀₉	C₄²	1	16	yes	direct product factor
24	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	C₄×C₈	1	12	yes	—
26	C₂×C₄×C₁₀₉	C₂×C₁₆	1	24	yes	—
27	C₂×C₄×C₁₀₉	C₂×C₄²	1	3	yes	—
28	C₂×C₄×C₁₀₉	C₂²×C₈	1	3	yes	—
29	C₈×C₁₀₉	C₄×C₈	1	12	yes	—
30	C₈×C₁₀₉	C₂×C₄²	1	4	yes	—
31	C₂²×C₁₀₉	C₂×C₄×C₈	1	3	yes	—
32	C₂²×C₁₀₉	C₂²×C₁₆	1	4	yes	—
33	C₄×C₁₀₉	C₄×C₁₆	1	24	yes	direct product factor
34	C₄×C₁₀₉	C₄³	1	1	yes	characteristic
35	C₄×C₁₀₉	C₂×C₄×C₈	1	3	yes	—
36	C₄²×C₁₆	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₁₀₉	C₄²×C₈	1	1	yes	characteristic
38	C₂×C₁₀₉	C₂×C₄×C₁₆	1	6	yes	—
39	C₂×C₄×C₁₆	C₂×C₁₀₉	1	6	yes	—
40	C₄²×C₈	C₂×C₁₀₉	1	1	yes	characteristic
41	C₁₀₉	C₄²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²×C₁₆	C₂²×C₁₀₉	1	4	yes	—
43	C₂×C₄×C₈	C₄×C₁₀₉	1	3	yes	—
44	C₂×C₄×C₈	C₂²×C₁₀₉	1	3	yes	—
45	C₄³	C₄×C₁₀₉	1	1	yes	characteristic
46	C₄×C₁₆	C₄×C₁₀₉	1	24	yes	direct product factor
47	C₂²×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
48	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
49	C₂×C₄²	C₈×C₁₀₉	1	4	yes	—
50	C₂×C₄²	C₂×C₄×C₁₀₉	1	3	yes	—
51	C₂×C₁₆	C₂×C₄×C₁₀₉	1	24	yes	—
52	C₄×C₈	C₈×C₁₀₉	1	12	yes	—
53	C₄×C₈	C₂×C₄×C₁₀₉	1	12	yes	—
54	C₂²×C₄	C₂×C₈×C₁₀₉	1	6	yes	—
55	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
56	C₂×C₈	C₄²×C₁₀₉	1	6	yes	—
57	C₂×C₈	C₂×C₈×C₁₀₉	1	12	yes	—
58	C₂×C₈	C₂²×C₄×C₁₀₉	1	6	yes	—
59	C₄²	C₁₆×C₁₀₉	1	16	yes	direct product factor
60	C₄²	C₄²×C₁₀₉	1	6	yes	—
61	C₄²	C₂×C₈×C₁₀₉	1	6	yes	—
62	C₁₆	C₄²×C₁₀₉	1	16	yes	direct product factor
63	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
64	C₂×C₄	C₄×C₈×C₁₀₉	1	12	yes	—
65	C₂×C₄	C₂×C₁₆×C₁₀₉	1	24	yes	—
66	C₂×C₄	C₂×C₄²×C₁₀₉	1	3	yes	—
67	C₂×C₄	C₂²×C₈×C₁₀₉	1	3	yes	—
68	C₈	C₄×C₈×C₁₀₉	1	12	yes	—
69	C₈	C₂×C₄²×C₁₀₉	1	4	yes	—
70	C₂²	C₂×C₄×C₈×C₁₀₉	1	3	yes	—
71	C₂²	C₂²×C₁₆×C₁₀₉	1	4	yes	—
72	C₄	C₄×C₁₆×C₁₀₉	1	24	yes	direct product factor
73	C₄	C₄³×C₁₀₉	1	1	yes	characteristic
74	C₄	C₂×C₄×C₈×C₁₀₉	1	3	yes	—
75	C₂	C₄²×C₈×C₁₀₉	1	1	yes	characteristic
76	C₂	C₂×C₄×C₁₆×C₁₀₉	1	6	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle, prime radical	C₂³×C₁₀₉	C₂²×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₄×C₁₆×C₁₀₉	C₄	direct (N × H)
20. C₄²×C₁₀₉	C₁₆	direct (N × H)
23. C₁₆×C₁₀₉	C₄²	direct (N × H)
33. C₄×C₁₀₉	C₄×C₁₆	direct (N × H)
36. C₄²×C₁₆	C₁₀₉	direct (N × H)
41. C₁₀₉	C₄²×C₁₆	direct (N × H)
46. C₄×C₁₆	C₄×C₁₀₉	direct (N × H)
59. C₄²	C₁₆×C₁₀₉	direct (N × H)
62. C₁₆	C₄²×C₁₀₉	direct (N × H)
72. C₄	C₄×C₁₆×C₁₀₉	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×C₁₆×C₁₀₉	2
C₄²×C₈×C₁₀₉	2
C₄²×C₁₆	109

Subgroups — C42×C16×C109 #42130531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₁₆×C₁₀₉ #42130531