Subgroups — C₂×Q₆₄×C₂₂₃ #43251276

Order 28544.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×C₂₂₃	C₂	1	2	yes	maximal
2	Q₆₄×C₂₂₃	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×C₂₂₃	—	2	2	no	—
5	Q₃₂×C₂₂₃	—	2	2	no	—
6	Q₃₂×C₂₂₃	C₂²	1	4	yes	—
7	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₃₂×C₂₂₃	C₂²	1	2	yes	—
9	C₂×Q₈×C₂₂₃	—	4	2	no	—
10	Q₁₆×C₂₂₃	—	2	4	no	—
11	Q₁₆×C₂₂₃	—	4	2	no	—
12	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
13	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
14	C₁₆×C₂₂₃	C₂³	1	1	yes	characteristic
15	Q₈×C₂₂₃	—	4	4	no	—
16	Q₈×C₂₂₃	—	8	2	no	—
17	C₂×C₄×C₂₂₃	—	8	2	no	—
18	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
19	C₈×C₂₂₃	D₈	1	1	yes	characteristic
20	C₈×C₂₂₃	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₂₂₃	D₁₆	1	1	yes	characteristic
22	C₄×C₂₂₃	—	8	4	no	—
23	C₄×C₂₂₃	D₁₆	1	1	yes	characteristic
24	C₄×C₂₂₃	C₂×D₈	1	1	yes	characteristic
25	C₂×C₂₂₃	Q₆₄	1	2	yes	direct product factor
26	C₂×C₂₂₃	C₂×D₁₆	1	1	yes	characteristic
27	C₂₂₃	C₂×Q₆₄	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×Q₆₄	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂×Q₃₂	C₂×C₂₂₃	1	2	yes	—
30	Q₆₄	C₂×C₂₂₃	1	4	yes	direct product factor
31	C₂×C₃₂	C₂×C₂₂₃	1	1	yes	characteristic
32	C₂×Q₁₆	—	2	2	no	—
33	Q₃₂	—	2	2	no	—
34	Q₃₂	C₂²×C₂₂₃	1	4	yes	—
35	C₂×C₁₆	C₂²×C₂₂₃	1	1	yes	characteristic
36	C₃₂	C₂²×C₂₂₃	1	2	yes	—
37	C₂×Q₈	—	4	2	no	—
38	Q₁₆	—	2	4	no	—
39	Q₁₆	—	4	2	no	—
40	C₂×C₈	D₄×C₂₂₃	1	1	yes	characteristic
41	C₁₆	D₄×C₂₂₃	1	1	yes	characteristic
42	C₁₆	C₂³×C₂₂₃	1	1	yes	characteristic
43	Q₈	—	4	4	no	—
44	Q₈	—	8	2	no	—
45	C₂×C₄	—	8	2	no	—
46	C₂×C₄	D₈×C₂₂₃	1	1	yes	characteristic
47	C₈	D₈×C₂₂₃	1	1	yes	characteristic
48	C₈	C₂×D₄×C₂₂₃	1	1	yes	characteristic
49	C₂²	D₁₆×C₂₂₃	1	1	yes	characteristic
50	C₄	—	8	4	no	—
51	C₄	D₁₆×C₂₂₃	1	1	yes	characteristic
52	C₄	C₂×D₈×C₂₂₃	1	1	yes	characteristic
53	C₂	Q₆₄×C₂₂₃	1	2	yes	direct product factor
54	C₂	C₂×D₁₆×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₂₃	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₂₂₃
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₂×C₄×C₂₂₃ ◁ C₂×C₈×C₂₂₃ ◁ C₂×C₁₆×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₆₄×C₂₂₃	C₂	direct (N × H)
25. C₂×C₂₂₃	Q₆₄	direct (N × H)
27. C₂₂₃	C₂×Q₆₄	direct (N × H)
28. C₂×Q₆₄	C₂₂₃	direct (N × H)
30. Q₆₄	C₂×C₂₂₃	direct (N × H)
53. C₂	Q₆₄×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×Q₃₂×C₂₂₃	2
Q₆₄×C₂₂₃	2
C₂×C₃₂×C₂₂₃	2
C₂×Q₆₄	223

Subgroups — C2×Q64×C223 #43251276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₆₄×C₂₂₃ #43251276