Subgroups — C₂×Q₃₂×C₂₂₃ #28383219

Order 14272.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×C₂₂₃	C₂	1	2	yes	maximal
2	Q₃₂×C₂₂₃	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₂₂₃	—	2	2	no	—
5	Q₁₆×C₂₂₃	—	2	2	no	—
6	Q₁₆×C₂₂₃	C₂²	1	4	yes	—
7	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₁₆×C₂₂₃	C₂²	1	2	yes	—
9	Q₈×C₂₂₃	—	2	4	no	—
10	Q₈×C₂₂₃	—	4	2	no	—
11	C₂×C₄×C₂₂₃	—	4	2	no	—
12	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
13	C₈×C₂₂₃	D₄	1	1	yes	characteristic
14	C₈×C₂₂₃	C₂³	1	1	yes	characteristic
15	C₂²×C₂₂₃	D₈	1	1	yes	characteristic
16	C₄×C₂₂₃	—	4	4	no	—
17	C₄×C₂₂₃	D₈	1	1	yes	characteristic
18	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
19	C₂×C₂₂₃	Q₃₂	1	2	yes	direct product factor
20	C₂×C₂₂₃	C₂×D₈	1	1	yes	characteristic
21	C₂₂₃	C₂×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
22	C₂×Q₃₂	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂×Q₁₆	C₂×C₂₂₃	1	2	yes	—
24	Q₃₂	C₂×C₂₂₃	1	4	yes	direct product factor
25	C₂×C₁₆	C₂×C₂₂₃	1	1	yes	characteristic
26	C₂×Q₈	—	2	2	no	—
27	Q₁₆	—	2	2	no	—
28	Q₁₆	C₂²×C₂₂₃	1	4	yes	—
29	C₂×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
30	C₁₆	C₂²×C₂₂₃	1	2	yes	—
31	Q₈	—	2	4	no	—
32	Q₈	—	4	2	no	—
33	C₂×C₄	—	4	2	no	—
34	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
35	C₈	D₄×C₂₂₃	1	1	yes	characteristic
36	C₈	C₂³×C₂₂₃	1	1	yes	characteristic
37	C₂²	D₈×C₂₂₃	1	1	yes	characteristic
38	C₄	—	4	4	no	—
39	C₄	D₈×C₂₂₃	1	1	yes	characteristic
40	C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
41	C₂	Q₃₂×C₂₂₃	1	2	yes	direct product factor
42	C₂	C₂×D₈×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₂₃	D₈
commutator, Frattini, cosocle	C₈	C₂³×C₂₂₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₂×C₄×C₂₂₃ ◁ C₂×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₃₂×C₂₂₃	C₂	direct (N × H)
19. C₂×C₂₂₃	Q₃₂	direct (N × H)
21. C₂₂₃	C₂×Q₃₂	direct (N × H)
22. C₂×Q₃₂	C₂₂₃	direct (N × H)
24. Q₃₂	C₂×C₂₂₃	direct (N × H)
41. C₂	Q₃₂×C₂₂₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×C₂₂₃	2
Q₃₂×C₂₂₃	2
C₂×C₁₆×C₂₂₃	2
C₂×Q₃₂	223

Subgroups — C2×Q32×C223 #28383219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₃₂×C₂₂₃ #28383219