Subgroups — C₄²×C₈×C₂₁₁ #41009344

Order 27008.

408 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×C₂₁₁	C₂	1	6	yes	maximal
2	C₄³×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×C₂₁₁	C₂²	1	4	yes	—
4	C₂×C₄²×C₂₁₁	C₄	1	4	yes	—
5	C₂×C₄²×C₂₁₁	C₂²	1	3	yes	—
6	C₄×C₈×C₂₁₁	C₄	1	24	yes	direct product factor
7	C₂²×C₄×C₂₁₁	C₂×C₄	1	6	yes	—
8	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
9	C₂×C₈×C₂₁₁	C₂×C₄	1	24	yes	—
10	C₄²×C₂₁₁	C₈	1	16	yes	direct product factor
11	C₄²×C₂₁₁	C₂×C₄	1	12	yes	—
12	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
13	C₂×C₄×C₂₁₁	C₄²	1	12	yes	—
14	C₂×C₄×C₂₁₁	C₂×C₈	1	24	yes	—
15	C₂×C₄×C₂₁₁	C₂²×C₄	1	6	yes	—
16	C₈×C₂₁₁	C₄²	1	16	yes	direct product factor
17	C₂²×C₂₁₁	C₂×C₄²	1	3	yes	—
18	C₂²×C₂₁₁	C₂²×C₈	1	4	yes	—
19	C₄×C₂₁₁	C₄×C₈	1	24	yes	direct product factor
20	C₄×C₂₁₁	C₂×C₄²	1	4	yes	—
21	C₂×C₂₁₁	C₄³	1	1	yes	characteristic
22	C₂×C₂₁₁	C₂×C₄×C₈	1	6	yes	—
23	C₂₁₁	C₄²×C₈	1	1	yes	characteristic, direct product factor, Sylow
24	C₄²×C₈	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₄×C₈	C₂×C₂₁₁	1	6	yes	—
26	C₄³	C₂×C₂₁₁	1	1	yes	characteristic
27	C₂²×C₈	C₂²×C₂₁₁	1	4	yes	—
28	C₂×C₄²	C₄×C₂₁₁	1	4	yes	—
29	C₂×C₄²	C₂²×C₂₁₁	1	3	yes	—
30	C₄×C₈	C₄×C₂₁₁	1	24	yes	direct product factor
31	C₂²×C₄	C₂×C₄×C₂₁₁	1	6	yes	—
32	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
33	C₂×C₈	C₂×C₄×C₂₁₁	1	24	yes	—
34	C₄²	C₈×C₂₁₁	1	16	yes	direct product factor
35	C₄²	C₂×C₄×C₂₁₁	1	12	yes	—
36	C₂³	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
37	C₂×C₄	C₄²×C₂₁₁	1	12	yes	—
38	C₂×C₄	C₂×C₈×C₂₁₁	1	24	yes	—
39	C₂×C₄	C₂²×C₄×C₂₁₁	1	6	yes	—
40	C₈	C₄²×C₂₁₁	1	16	yes	direct product factor
41	C₂²	C₂×C₄²×C₂₁₁	1	3	yes	—
42	C₂²	C₂²×C₈×C₂₁₁	1	4	yes	—
43	C₄	C₄×C₈×C₂₁₁	1	24	yes	direct product factor
44	C₄	C₂×C₄²×C₂₁₁	1	4	yes	—
45	C₂	C₄³×C₂₁₁	1	1	yes	characteristic
46	C₂	C₂×C₄×C₈×C₂₁₁	1	6	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₄×C₈×C₂₁₁	C₄	direct (N × H)
10. C₄²×C₂₁₁	C₈	direct (N × H)
16. C₈×C₂₁₁	C₄²	direct (N × H)
19. C₄×C₂₁₁	C₄×C₈	direct (N × H)
23. C₂₁₁	C₄²×C₈	direct (N × H)
24. C₄²×C₈	C₂₁₁	direct (N × H)
30. C₄×C₈	C₄×C₂₁₁	direct (N × H)
34. C₄²	C₈×C₂₁₁	direct (N × H)
40. C₈	C₄²×C₂₁₁	direct (N × H)
43. C₄	C₄×C₈×C₂₁₁	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×C₈×C₂₁₁	2
C₄³×C₂₁₁	2
C₄²×C₈	211

Subgroups — C42×C8×C211 #41009344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₈×C₂₁₁ #41009344