Subgroups — C₂²×Q₃₂×C₂₂₃ #43252425

Order 28544.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₁₆×C₂₂₃	C₂	1	2	yes	maximal
2	C₂×Q₃₂×C₂₂₃	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₂₂₃	—	2	2	no	—
5	C₂×Q₁₆×C₂₂₃	—	2	6	no	—
6	C₂×Q₁₆×C₂₂₃	C₂²	1	12	yes	—
7	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
8	Q₃₂×C₂₂₃	C₂²	1	16	yes	direct product factor
9	C₂×C₁₆×C₂₂₃	C₂²	1	6	yes	—
10	C₂×Q₈×C₂₂₃	—	2	12	no	—
11	C₂×Q₈×C₂₂₃	—	4	6	no	—
12	C₂²×C₄×C₂₂₃	—	4	2	no	—
13	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
14	Q₁₆×C₂₂₃	—	2	12	no	—
15	Q₁₆×C₂₂₃	C₂³	1	8	yes	—
16	C₂×C₈×C₂₂₃	D₄	1	3	yes	—
17	C₂×C₈×C₂₂₃	C₂³	1	3	yes	—
18	C₁₆×C₂₂₃	C₂³	1	4	yes	—
19	C₂³×C₂₂₃	D₈	1	1	yes	characteristic
20	Q₈×C₂₂₃	—	2	8	no	—
21	Q₈×C₂₂₃	—	4	12	no	—
22	C₂×C₄×C₂₂₃	—	4	12	no	—
23	C₂×C₄×C₂₂₃	D₈	1	3	yes	—
24	C₂×C₄×C₂₂₃	C₂×D₄	1	3	yes	—
25	C₈×C₂₂₃	C₂×D₄	1	3	yes	—
26	C₈×C₂₂₃	C₂⁴	1	1	yes	characteristic
27	C₂²×C₂₂₃	Q₃₂	1	4	yes	direct product factor
28	C₂²×C₂₂₃	C₂×D₈	1	3	yes	—
29	C₄×C₂₂₃	—	4	8	no	—
30	C₄×C₂₂₃	C₂×D₈	1	3	yes	—
31	C₄×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
32	C₂×C₂₂₃	C₂×Q₃₂	1	6	yes	direct product factor
33	C₂×C₂₂₃	C₂²×D₈	1	1	yes	characteristic
34	C₂₂₃	C₂²×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
35	C₂²×Q₃₂	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×Q₁₆	C₂×C₂₂₃	1	2	yes	—
37	C₂×Q₃₂	C₂×C₂₂₃	1	12	yes	direct product factor
38	C₂²×C₁₆	C₂×C₂₂₃	1	1	yes	characteristic
39	C₂²×Q₈	—	2	2	no	—
40	C₂×Q₁₆	—	2	6	no	—
41	C₂×Q₁₆	C₂²×C₂₂₃	1	12	yes	—
42	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
43	Q₃₂	C₂²×C₂₂₃	1	16	yes	direct product factor
44	C₂×C₁₆	C₂²×C₂₂₃	1	6	yes	—
45	C₂×Q₈	—	2	12	no	—
46	C₂×Q₈	—	4	6	no	—
47	C₂²×C₄	—	4	2	no	—
48	C₂²×C₄	D₄×C₂₂₃	1	1	yes	characteristic
49	Q₁₆	—	2	12	no	—
50	Q₁₆	C₂³×C₂₂₃	1	8	yes	—
51	C₂×C₈	D₄×C₂₂₃	1	3	yes	—
52	C₂×C₈	C₂³×C₂₂₃	1	3	yes	—
53	C₁₆	C₂³×C₂₂₃	1	4	yes	—
54	C₂³	D₈×C₂₂₃	1	1	yes	characteristic
55	Q₈	—	2	8	no	—
56	Q₈	—	4	12	no	—
57	C₂×C₄	—	4	12	no	—
58	C₂×C₄	D₈×C₂₂₃	1	3	yes	—
59	C₂×C₄	C₂×D₄×C₂₂₃	1	3	yes	—
60	C₈	C₂×D₄×C₂₂₃	1	3	yes	—
61	C₈	C₂⁴×C₂₂₃	1	1	yes	characteristic
62	C₂²	Q₃₂×C₂₂₃	1	4	yes	direct product factor
63	C₂²	C₂×D₈×C₂₂₃	1	3	yes	—
64	C₄	—	4	8	no	—
65	C₄	C₂×D₈×C₂₂₃	1	3	yes	—
66	C₄	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
67	C₂	C₂×Q₃₂×C₂₂₃	1	6	yes	direct product factor
68	C₂	C₂²×D₈×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₂₃	D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₂₂₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ C₂²×C₄×C₂₂₃ ◁ C₂²×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×Q₃₂×C₂₂₃	C₂	direct (N × H)
8. Q₃₂×C₂₂₃	C₂²	direct (N × H)
27. C₂²×C₂₂₃	Q₃₂	direct (N × H)
32. C₂×C₂₂₃	C₂×Q₃₂	direct (N × H)
34. C₂₂₃	C₂²×Q₃₂	direct (N × H)
35. C₂²×Q₃₂	C₂₂₃	direct (N × H)
37. C₂×Q₃₂	C₂×C₂₂₃	direct (N × H)
43. Q₃₂	C₂²×C₂₂₃	direct (N × H)
62. C₂²	Q₃₂×C₂₂₃	direct (N × H)
67. C₂	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
C₂²×Q₁₆×C₂₂₃	2
C₂×Q₃₂×C₂₂₃	2
C₂²×C₁₆×C₂₂₃	2
C₂²×Q₃₂	223

Subgroups — C22×Q32×C223 #43252425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₃₂×C₂₂₃ #43252425