Subgroups — C₆₇×Q₂₅₆ #30686804

Order 17152.

42 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×Q₁₂₈	C₂	1	2	yes	maximal
2	C₆₇×C₁₂₈	C₂	1	1	yes	maximal, characteristic
3	Q₆₄×C₆₇	—	2	2	no	—
4	C₆₄×C₆₇	C₂²	1	1	yes	characteristic
5	Q₃₂×C₆₇	—	4	2	no	—
6	C₃₂×C₆₇	D₄	1	1	yes	characteristic
7	Q₁₆×C₆₇	—	8	2	no	—
8	C₁₆×C₆₇	D₈	1	1	yes	characteristic
9	Q₈×C₆₇	—	16	2	no	—
10	C₈×C₆₇	D₁₆	1	1	yes	characteristic
11	C₄×C₆₇	—	32	2	no	—
12	C₄×C₆₇	D₃₂	1	1	yes	characteristic
13	Q₂₅₆	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂×C₆₇	D₆₄	1	1	yes	characteristic
15	Q₁₂₈	C₂×C₆₇	1	2	yes	—
16	C₁₂₈	C₂×C₆₇	1	1	yes	characteristic
17	C₆₇	Q₂₅₆	1	1	yes	characteristic, direct product factor, Sylow
18	Q₆₄	—	2	2	no	—
19	C₆₄	C₂²×C₆₇	1	1	yes	characteristic
20	Q₃₂	—	4	2	no	—
21	C₃₂	D₄×C₆₇	1	1	yes	characteristic
22	Q₁₆	—	8	2	no	—
23	C₁₆	D₈×C₆₇	1	1	yes	characteristic
24	Q₈	—	16	2	no	—
25	C₈	D₁₆×C₆₇	1	1	yes	characteristic
26	C₄	—	32	2	no	—
27	C₄	D₃₂×C₆₇	1	1	yes	characteristic
28	C₂	C₆₇×D₆₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₆₇	D₆₄
commutator, Frattini, cosocle	C₆₄	C₂²×C₆₇
LCS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₄×C₆₇ ◁ C₈×C₆₇ ◁ C₁₆×C₆₇ ◁ C₃₂×C₆₇ ◁ C₆₄×C₆₇ ◁ G

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

N	H ≅ G/N	type
13. Q₂₅₆	C₆₇	direct (N × H)
17. C₆₇	Q₂₅₆	direct (N × H)

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