Subgroups — Q₈²×C₁₉₉ #27163708

Order 12736.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₁₉₉	C₂	1	9	yes	maximal
2	C₄×Q₈×C₁₉₉	C₂	1	6	yes	maximal
3	C₂×Q₈×C₁₉₉	C₂²	1	6	yes	—
4	C₂×Q₈×C₁₉₉	C₂²	1	2	yes	—
5	C₄⋊C₄×C₁₉₉	C₂²	1	18	yes	—
6	C₄²×C₁₉₉	C₂²	1	9	yes	—
7	Q₈×C₁₉₉	—	4	6	no	—
8	Q₈×C₁₉₉	Q₈	1	8	yes	direct product factor
9	C₂×C₄×C₁₉₉	C₂³	1	9	yes	—
10	C₂×C₄×C₁₉₉	C₂³	1	6	yes	—
11	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
12	C₄×C₁₉₉	—	2	9	no	—
13	C₄×C₁₉₉	C₂×Q₈	1	12	yes	—
14	C₂×C₁₉₉	C₂²×Q₈	1	2	yes	—
15	C₂×C₁₉₉	H(5,𝔽₂)	1	1	yes	characteristic
16	C₁₉₉	Q₈²	1	1	yes	characteristic, direct product factor, Sylow
17	Q₈²	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₄⋊Q₈	C₂×C₁₉₉	1	9	yes	—
19	C₄×Q₈	C₂×C₁₉₉	1	6	yes	—
20	C₂×Q₈	C₂²×C₁₉₉	1	6	yes	—
21	C₂×Q₈	C₂²×C₁₉₉	1	2	yes	—
22	C₄⋊C₄	C₂²×C₁₉₉	1	18	yes	—
23	C₄²	C₂²×C₁₉₉	1	9	yes	—
24	Q₈	—	4	6	no	—
25	Q₈	Q₈×C₁₉₉	1	8	yes	direct product factor
26	C₂×C₄	C₂³×C₁₉₉	1	9	yes	—
27	C₂×C₄	C₂³×C₁₉₉	1	6	yes	—
28	C₂²	C₂⁴×C₁₉₉	1	1	yes	characteristic
29	C₄	—	2	9	no	—
30	C₄	C₂×Q₈×C₁₉₉	1	12	yes	—
31	C₂	C₂²×Q₈×C₁₉₉	1	2	yes	—
32	C₂	H(5,𝔽₂)×C₁₉₉	1	1	yes	characteristic

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