Subgroups — C₂×C₄×Q₈×C₂₂₃ #28383228

Order 14272.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×C₂₂₃	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₄×C₂₂₃	C₂	1	3	yes	maximal
4	C₂×C₄²×C₂₂₃	C₂	1	3	yes	maximal
5	C₂×Q₈×C₂₂₃	C₄	1	8	yes	direct product factor
6	C₂×Q₈×C₂₂₃	C₂²	1	4	yes	—
7	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
8	C₂²×C₄×C₂₂₃	C₂²	1	3	yes	—
9	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₂₂₃	C₂²	1	12	yes	—
11	C₄²×C₂₂₃	C₂²	1	12	yes	—
12	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
13	Q₈×C₂₂₃	C₂×C₄	1	16	yes	direct product factor
14	C₂×C₄×C₂₂₃	—	2	6	no	—
15	C₂×C₄×C₂₂₃	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₂₂₃	Q₈	1	4	yes	direct product factor
17	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
18	C₂×C₄×C₂₂₃	C₂³	1	6	yes	—
19	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
20	C₂²×C₂₂₃	C₂²×C₄	1	2	yes	—
21	C₂²×C₂₂₃	C₂×Q₈	1	2	yes	—
22	C₂²×C₂₂₃	#41	1	2	yes	—
23	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
24	C₄×C₂₂₃	—	2	6	no	—
25	C₄×C₂₂₃	C₂²×C₄	1	12	yes	—
26	C₄×C₂₂₃	C₂×Q₈	1	4	yes	direct product factor
27	C₂×C₂₂₃	C₄×Q₈	1	4	yes	direct product factor
28	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
29	C₂×C₂₂₃	C₂²×Q₈	1	1	yes	characteristic
30	C₂×C₂₂₃	C₂×#41	1	1	yes	characteristic
31	C₂₂₃	C₂×C₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
32	C₂×C₄×Q₈	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂²×Q₈	C₂×C₂₂₃	1	1	yes	characteristic
34	C₄×Q₈	C₂×C₂₂₃	1	8	yes	direct product factor
35	C₂×C₄⋊C₄	C₂×C₂₂₃	1	3	yes	—
36	C₂×C₄²	C₂×C₂₂₃	1	3	yes	—
37	C₂×Q₈	C₄×C₂₂₃	1	8	yes	direct product factor
38	C₂×Q₈	C₂²×C₂₂₃	1	4	yes	—
39	C₂²×C₄	C₂²×C₂₂₃	1	3	yes	—
40	C₂²×C₄	C₂²×C₂₂₃	1	3	yes	—
41	C₂²×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
42	C₄⋊C₄	C₂²×C₂₂₃	1	12	yes	—
43	C₄²	C₂²×C₂₂₃	1	12	yes	—
44	C₂³	C₂³×C₂₂₃	1	1	yes	characteristic
45	Q₈	C₂×C₄×C₂₂₃	1	16	yes	direct product factor
46	C₂×C₄	—	2	6	no	—
47	C₂×C₄	C₂×C₄×C₂₂₃	1	12	yes	—
48	C₂×C₄	Q₈×C₂₂₃	1	4	yes	direct product factor
49	C₂×C₄	C₂³×C₂₂₃	1	6	yes	—
50	C₂×C₄	C₂³×C₂₂₃	1	6	yes	—
51	C₂×C₄	C₂³×C₂₂₃	1	2	yes	—
52	C₂²	C₂²×C₄×C₂₂₃	1	2	yes	—
53	C₂²	C₂×Q₈×C₂₂₃	1	2	yes	—
54	C₂²	#41×C₂₂₃	1	2	yes	—
55	C₂²	C₂⁴×C₂₂₃	1	1	yes	characteristic
56	C₄	—	2	6	no	—
57	C₄	C₂²×C₄×C₂₂₃	1	12	yes	—
58	C₄	C₂×Q₈×C₂₂₃	1	4	yes	direct product factor
59	C₂	C₄×Q₈×C₂₂₃	1	4	yes	direct product factor
60	C₂	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
61	C₂	C₂²×Q₈×C₂₂₃	1	1	yes	characteristic
62	C₂	C₂×#41×C₂₂₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×C₂₂₃	C₂	direct (N × H)
5. C₂×Q₈×C₂₂₃	C₄	direct (N × H)
13. Q₈×C₂₂₃	C₂×C₄	direct (N × H)
16. C₂×C₄×C₂₂₃	Q₈	direct (N × H)
26. C₄×C₂₂₃	C₂×Q₈	direct (N × H)
27. C₂×C₂₂₃	C₄×Q₈	direct (N × H)
31. C₂₂₃	C₂×C₄×Q₈	direct (N × H)
32. C₂×C₄×Q₈	C₂₂₃	direct (N × H)
34. C₄×Q₈	C₂×C₂₂₃	direct (N × H)
37. C₂×Q₈	C₄×C₂₂₃	direct (N × H)
45. Q₈	C₂×C₄×C₂₂₃	direct (N × H)
48. C₂×C₄	Q₈×C₂₂₃	direct (N × H)
58. C₄	C₂×Q₈×C₂₂₃	direct (N × H)
59. C₂	C₄×Q₈×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×Q₈×C₂₂₃	2
C₄×Q₈×C₂₂₃	2
C₂×C₄⋊C₄×C₂₂₃	2
C₂×C₄²×C₂₂₃	2
C₂×C₄×Q₈	223

Subgroups — C2×C4×Q8×C223 #28383228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×Q₈×C₂₂₃ #28383228