Subgroups — C₂²×Q₁₆×C₁₉₉ #27163721

Order 12736.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₁₉₉	C₂	1	2	yes	maximal
2	C₂×Q₁₆×C₁₉₉	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₁₉₉	—	2	6	no	—
5	C₂×Q₈×C₁₉₉	C₂²	1	12	yes	—
6	C₂²×C₄×C₁₉₉	—	2	2	no	—
7	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
8	Q₁₆×C₁₉₉	C₂²	1	16	yes	direct product factor
9	C₂×C₈×C₁₉₉	C₂²	1	6	yes	—
10	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
11	Q₈×C₁₉₉	—	2	12	no	—
12	Q₈×C₁₉₉	C₂³	1	8	yes	—
13	C₂×C₄×C₁₉₉	—	2	12	no	—
14	C₂×C₄×C₁₉₉	D₄	1	3	yes	—
15	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
16	C₈×C₁₉₉	C₂³	1	4	yes	—
17	C₂²×C₁₉₉	Q₁₆	1	4	yes	direct product factor
18	C₂²×C₁₉₉	C₂×D₄	1	3	yes	—
19	C₄×C₁₉₉	—	2	8	no	—
20	C₄×C₁₉₉	C₂×D₄	1	3	yes	—
21	C₄×C₁₉₉	C₂⁴	1	1	yes	characteristic
22	C₂×C₁₉₉	C₂×Q₁₆	1	6	yes	direct product factor
23	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
24	C₁₉₉	C₂²×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
25	C₂²×Q₁₆	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×Q₈	C₂×C₁₉₉	1	2	yes	—
27	C₂×Q₁₆	C₂×C₁₉₉	1	12	yes	direct product factor
28	C₂²×C₈	C₂×C₁₉₉	1	1	yes	characteristic
29	C₂×Q₈	—	2	6	no	—
30	C₂×Q₈	C₂²×C₁₉₉	1	12	yes	—
31	C₂²×C₄	—	2	2	no	—
32	C₂²×C₄	C₂²×C₁₉₉	1	1	yes	characteristic
33	Q₁₆	C₂²×C₁₉₉	1	16	yes	direct product factor
34	C₂×C₈	C₂²×C₁₉₉	1	6	yes	—
35	C₂³	D₄×C₁₉₉	1	1	yes	characteristic
36	Q₈	—	2	12	no	—
37	Q₈	C₂³×C₁₉₉	1	8	yes	—
38	C₂×C₄	—	2	12	no	—
39	C₂×C₄	D₄×C₁₉₉	1	3	yes	—
40	C₂×C₄	C₂³×C₁₉₉	1	3	yes	—
41	C₈	C₂³×C₁₉₉	1	4	yes	—
42	C₂²	Q₁₆×C₁₉₉	1	4	yes	direct product factor
43	C₂²	C₂×D₄×C₁₉₉	1	3	yes	—
44	C₄	—	2	8	no	—
45	C₄	C₂×D₄×C₁₉₉	1	3	yes	—
46	C₄	C₂⁴×C₁₉₉	1	1	yes	characteristic
47	C₂	C₂×Q₁₆×C₁₉₉	1	6	yes	direct product factor
48	C₂	C₂²×D₄×C₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉₉	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₁₉₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉₉ ◁ C₂²×C₄×C₁₉₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×C₁₉₉	C₂	direct (N × H)
8. Q₁₆×C₁₉₉	C₂²	direct (N × H)
17. C₂²×C₁₉₉	Q₁₆	direct (N × H)
22. C₂×C₁₉₉	C₂×Q₁₆	direct (N × H)
24. C₁₉₉	C₂²×Q₁₆	direct (N × H)
25. C₂²×Q₁₆	C₁₉₉	direct (N × H)
27. C₂×Q₁₆	C₂×C₁₉₉	direct (N × H)
33. Q₁₆	C₂²×C₁₉₉	direct (N × H)
42. C₂²	Q₁₆×C₁₉₉	direct (N × H)
47. C₂	C₂×Q₁₆×C₁₉₉	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×C₁₉₉	2
C₂×Q₁₆×C₁₉₉	2
C₂²×C₈×C₁₉₉	2
C₂²×Q₁₆	199

Subgroups — C22×Q16×C199 #27163721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₁₆×C₁₉₉ #27163721