Subgroups — C₄⋊Q₈×C₇₃₉ #37556549

Order 23648.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₇₃₉	C₂	1	2	yes	maximal
2	C₄⋊C₄×C₇₃₉	C₂	1	4	yes	maximal
3	C₄²×C₇₃₉	C₂	1	1	yes	maximal, characteristic
4	Q₈×C₇₃₉	—	2	4	no	—
5	C₂×C₄×C₇₃₉	C₂²	1	4	yes	—
6	C₂×C₄×C₇₃₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇₃₉	C₂²	1	2	yes	—
8	C₂²×C₇₃₉	C₂³	1	1	yes	characteristic
9	C₄×C₇₃₉	—	2	4	no	—
10	C₄×C₇₃₉	D₄	1	2	yes	—
11	C₄×C₇₃₉	Q₈	1	4	yes	complemented
12	C₂×C₇₃₉	C₂×D₄	1	1	yes	characteristic
13	C₂×C₇₃₉	C₂×Q₈	1	2	yes	—
14	C₇₃₉	C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
15	C₄⋊Q₈	C₇₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×Q₈	C₂×C₇₃₉	1	2	yes	—
17	C₄⋊C₄	C₂×C₇₃₉	1	4	yes	—
18	C₄²	C₂×C₇₃₉	1	1	yes	characteristic
19	Q₈	—	2	4	no	—
20	C₂×C₄	C₂²×C₇₃₉	1	4	yes	—
21	C₂×C₄	C₂²×C₇₃₉	1	1	yes	characteristic
22	C₂×C₄	C₂²×C₇₃₉	1	2	yes	—
23	C₂²	C₂³×C₇₃₉	1	1	yes	characteristic
24	C₄	—	2	4	no	—
25	C₄	D₄×C₇₃₉	1	2	yes	—
26	C₄	Q₈×C₇₃₉	1	4	yes	complemented
27	C₂	C₂×D₄×C₇₃₉	1	1	yes	characteristic
28	C₂	C₂×Q₈×C₇₃₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₇₃₉	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₇₃₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃₉ ◁ 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.