Subgroups — C₂×C₈²×C₂₂₃ #43250462

Order 28544.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×C₂₂₃	C₂	1	3	yes	maximal
2	C₈²×C₂₂₃	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₈×C₂₂₃	C₄	1	6	yes	—
4	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
5	C₄×C₈×C₂₂₃	C₄	1	6	yes	—
6	C₄×C₈×C₂₂₃	C₂²	1	6	yes	—
7	C₂²×C₄×C₂₂₃	C₂×C₄	1	3	yes	—
8	C₂×C₈×C₂₂₃	C₈	1	24	yes	direct product factor
9	C₂×C₈×C₂₂₃	C₂×C₄	1	12	yes	—
10	C₄²×C₂₂₃	C₂×C₄	1	3	yes	—
11	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
12	C₂³×C₂₂₃	C₄²	1	1	yes	characteristic
13	C₂×C₄×C₂₂₃	C₄²	1	3	yes	—
14	C₂×C₄×C₂₂₃	C₂×C₈	1	12	yes	—
15	C₂×C₄×C₂₂₃	C₂²×C₄	1	3	yes	—
16	C₈×C₂₂₃	C₂×C₈	1	24	yes	direct product factor
17	C₂²×C₂₂₃	C₄×C₈	1	6	yes	—
18	C₂²×C₂₂₃	C₂×C₄²	1	1	yes	characteristic
19	C₄×C₂₂₃	C₄×C₈	1	6	yes	—
20	C₄×C₂₂₃	C₂²×C₈	1	6	yes	—
21	C₂×C₂₂₃	C₈²	1	4	yes	direct product factor
22	C₂×C₂₂₃	C₂×C₄×C₈	1	3	yes	—
23	C₂₂₃	C₂×C₈²	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₈²	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₄×C₈	C₂×C₂₂₃	1	3	yes	—
26	C₈²	C₂×C₂₂₃	1	4	yes	direct product factor
27	C₂²×C₈	C₄×C₂₂₃	1	6	yes	—
28	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
29	C₄×C₈	C₄×C₂₂₃	1	6	yes	—
30	C₄×C₈	C₂²×C₂₂₃	1	6	yes	—
31	C₂²×C₄	C₂×C₄×C₂₂₃	1	3	yes	—
32	C₂×C₈	C₈×C₂₂₃	1	24	yes	direct product factor
33	C₂×C₈	C₂×C₄×C₂₂₃	1	12	yes	—
34	C₄²	C₂×C₄×C₂₂₃	1	3	yes	—
35	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
36	C₂³	C₄²×C₂₂₃	1	1	yes	characteristic
37	C₂×C₄	C₄²×C₂₂₃	1	3	yes	—
38	C₂×C₄	C₂×C₈×C₂₂₃	1	12	yes	—
39	C₂×C₄	C₂²×C₄×C₂₂₃	1	3	yes	—
40	C₈	C₂×C₈×C₂₂₃	1	24	yes	direct product factor
41	C₂²	C₄×C₈×C₂₂₃	1	6	yes	—
42	C₂²	C₂×C₄²×C₂₂₃	1	1	yes	characteristic
43	C₄	C₄×C₈×C₂₂₃	1	6	yes	—
44	C₄	C₂²×C₈×C₂₂₃	1	6	yes	—
45	C₂	C₈²×C₂₂₃	1	4	yes	direct product factor
46	C₂	C₂×C₄×C₈×C₂₂₃	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂³×C₂₂₃
socle, prime radical	C₂³×C₂₂₃	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈²×C₂₂₃	C₂	direct (N × H)
8. C₂×C₈×C₂₂₃	C₈	direct (N × H)
16. C₈×C₂₂₃	C₂×C₈	direct (N × H)
21. C₂×C₂₂₃	C₈²	direct (N × H)
23. C₂₂₃	C₂×C₈²	direct (N × H)
24. C₂×C₈²	C₂₂₃	direct (N × H)
26. C₈²	C₂×C₂₂₃	direct (N × H)
32. C₂×C₈	C₈×C₂₂₃	direct (N × H)
40. C₈	C₂×C₈×C₂₂₃	direct (N × H)
45. C₂	C₈²×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×C₄×C₈×C₂₂₃	2
C₈²×C₂₂₃	2
C₂×C₈²	223

Subgroups — C2×C82×C223 #43250462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈²×C₂₂₃ #43250462