Subgroups — C₂²×Q₁₆×C₂₁₁ #27212849

Order 13504.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₂₁₁	C₂	1	2	yes	maximal
2	C₂×Q₁₆×C₂₁₁	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₂₁₁	—	2	6	no	—
5	C₂×Q₈×C₂₁₁	C₂²	1	12	yes	—
6	C₂²×C₄×C₂₁₁	—	2	2	no	—
7	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
8	Q₁₆×C₂₁₁	C₂²	1	16	yes	direct product factor
9	C₂×C₈×C₂₁₁	C₂²	1	6	yes	—
10	C₂³×C₂₁₁	D₄	1	1	yes	characteristic
11	Q₈×C₂₁₁	—	2	12	no	—
12	Q₈×C₂₁₁	C₂³	1	8	yes	—
13	C₂×C₄×C₂₁₁	—	2	12	no	—
14	C₂×C₄×C₂₁₁	D₄	1	3	yes	—
15	C₂×C₄×C₂₁₁	C₂³	1	3	yes	—
16	C₈×C₂₁₁	C₂³	1	4	yes	—
17	C₂²×C₂₁₁	Q₁₆	1	4	yes	direct product factor
18	C₂²×C₂₁₁	C₂×D₄	1	3	yes	—
19	C₄×C₂₁₁	—	2	8	no	—
20	C₄×C₂₁₁	C₂×D₄	1	3	yes	—
21	C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
22	C₂×C₂₁₁	C₂×Q₁₆	1	6	yes	direct product factor
23	C₂×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
24	C₂₁₁	C₂²×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
25	C₂²×Q₁₆	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×Q₈	C₂×C₂₁₁	1	2	yes	—
27	C₂×Q₁₆	C₂×C₂₁₁	1	12	yes	direct product factor
28	C₂²×C₈	C₂×C₂₁₁	1	1	yes	characteristic
29	C₂×Q₈	—	2	6	no	—
30	C₂×Q₈	C₂²×C₂₁₁	1	12	yes	—
31	C₂²×C₄	—	2	2	no	—
32	C₂²×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
33	Q₁₆	C₂²×C₂₁₁	1	16	yes	direct product factor
34	C₂×C₈	C₂²×C₂₁₁	1	6	yes	—
35	C₂³	D₄×C₂₁₁	1	1	yes	characteristic
36	Q₈	—	2	12	no	—
37	Q₈	C₂³×C₂₁₁	1	8	yes	—
38	C₂×C₄	—	2	12	no	—
39	C₂×C₄	D₄×C₂₁₁	1	3	yes	—
40	C₂×C₄	C₂³×C₂₁₁	1	3	yes	—
41	C₈	C₂³×C₂₁₁	1	4	yes	—
42	C₂²	Q₁₆×C₂₁₁	1	4	yes	direct product factor
43	C₂²	C₂×D₄×C₂₁₁	1	3	yes	—
44	C₄	—	2	8	no	—
45	C₄	C₂×D₄×C₂₁₁	1	3	yes	—
46	C₄	C₂⁴×C₂₁₁	1	1	yes	characteristic
47	C₂	C₂×Q₁₆×C₂₁₁	1	6	yes	direct product factor
48	C₂	C₂²×D₄×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₁₁	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₂₁₁
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×C₂₁₁	C₂	direct (N × H)
8. Q₁₆×C₂₁₁	C₂²	direct (N × H)
17. C₂²×C₂₁₁	Q₁₆	direct (N × H)
22. C₂×C₂₁₁	C₂×Q₁₆	direct (N × H)
24. C₂₁₁	C₂²×Q₁₆	direct (N × H)
25. C₂²×Q₁₆	C₂₁₁	direct (N × H)
27. C₂×Q₁₆	C₂×C₂₁₁	direct (N × H)
33. Q₁₆	C₂²×C₂₁₁	direct (N × H)
42. C₂²	Q₁₆×C₂₁₁	direct (N × H)
47. C₂	C₂×Q₁₆×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
C₂²×Q₈×C₂₁₁	2
C₂×Q₁₆×C₂₁₁	2
C₂²×C₈×C₂₁₁	2
C₂²×Q₁₆	211

Subgroups — C22×Q16×C211 #27212849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₁₆×C₂₁₁ #27212849