Subgroups — C₂³×Q₈×C₁₉₉ #27163731

Order 12736.

850 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₁₉₉	C₂	1	28	yes	maximal, direct product factor
2	C₂³×C₄×C₁₉₉	C₂	1	3	yes	maximal
3	C₂⁴×C₁₉₉	C₂²	1	1	yes	characteristic
4	C₂×Q₈×C₁₉₉	C₂²	1	112	yes	direct product factor
5	C₂²×C₄×C₁₉₉	C₂²	1	42	yes	—
6	C₂³×C₁₉₉	Q₈	1	8	yes	direct product factor
7	C₂³×C₁₉₉	C₂³	1	7	yes	—
8	Q₈×C₁₉₉	C₂³	1	64	yes	direct product factor
9	C₂×C₄×C₁₉₉	C₂³	1	84	yes	—
10	C₂²×C₁₉₉	C₂×Q₈	1	28	yes	direct product factor
11	C₂²×C₁₉₉	C₂⁴	1	7	yes	—
12	C₄×C₁₉₉	C₂⁴	1	24	yes	—
13	C₂×C₁₉₉	C₂²×Q₈	1	14	yes	direct product factor
14	C₂×C₁₉₉	C₂⁵	1	1	yes	characteristic
15	C₁₉₉	C₂³×Q₈	1	1	yes	characteristic, direct product factor, Sylow
16	C₂³×Q₈	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×Q₈	C₂×C₁₉₉	1	28	yes	direct product factor
18	C₂³×C₄	C₂×C₁₉₉	1	3	yes	—
19	C₂⁴	C₂²×C₁₉₉	1	1	yes	characteristic
20	C₂×Q₈	C₂²×C₁₉₉	1	112	yes	direct product factor
21	C₂²×C₄	C₂²×C₁₉₉	1	42	yes	—
22	C₂³	Q₈×C₁₉₉	1	8	yes	direct product factor
23	C₂³	C₂³×C₁₉₉	1	7	yes	—
24	Q₈	C₂³×C₁₉₉	1	64	yes	direct product factor
25	C₂×C₄	C₂³×C₁₉₉	1	84	yes	—
26	C₂²	C₂×Q₈×C₁₉₉	1	28	yes	direct product factor
27	C₂²	C₂⁴×C₁₉₉	1	7	yes	—
28	C₄	C₂⁴×C₁₉₉	1	24	yes	—
29	C₂	C₂²×Q₈×C₁₉₉	1	14	yes	direct product factor
30	C₂	C₂⁵×C₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₉₉	C₂²
commutator, Frattini, cosocle	C₂	C₂⁵×C₁₉₉
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉₉ ◁ G

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

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.