Subgroups — C₄×Q₁₆×C₂₂₃ #28383151

Order 14272.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×C₂₂₃	C₂	1	2	yes	maximal
3	#107×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#103×C₂₂₃	C₂	1	2	yes	maximal
5	C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
7	Q₁₆×C₂₂₃	C₄	1	4	yes	direct product factor
8	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₂₂₃	—	2	2	no	—
11	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
12	C₄²×C₂₂₃	—	2	2	no	—
13	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
14	Q₈×C₂₂₃	—	2	2	no	—
15	Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₂₂₃	—	2	2	no	—
17	C₂×C₄×C₂₂₃	—	2	2	no	—
18	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
21	C₈×C₂₂₃	—	2	1	no	—
22	C₈×C₂₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₂₂₃	—	2	4	no	—
25	C₄×C₂₂₃	—	2	1	no	—
26	C₄×C₂₂₃	—	4	2	no	—
27	C₄×C₂₂₃	Q₁₆	1	2	yes	direct product factor
28	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₂₃	#41	1	1	yes	characteristic
30	C₂×C₂₂₃	C₄×D₄	1	1	yes	characteristic
31	C₂×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
32	C₂×C₂₂₃	#135	1	1	yes	characteristic
33	C₂₂₃	C₄×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
34	C₄×Q₁₆	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×Q₁₆	C₂×C₂₂₃	1	1	yes	characteristic
36	C₄×Q₈	C₂×C₂₂₃	1	2	yes	—
37	#107	C₂×C₂₂₃	1	1	yes	characteristic
38	#103	C₂×C₂₂₃	1	2	yes	—
39	C₄×C₈	C₂×C₂₂₃	1	1	yes	characteristic
40	C₂×Q₈	C₂²×C₂₂₃	1	2	yes	—
41	Q₁₆	C₄×C₂₂₃	1	4	yes	direct product factor
42	C₂×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
43	C₂×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
44	C₄⋊C₄	—	2	2	no	—
45	C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
46	C₄²	—	2	2	no	—
47	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
48	Q₈	—	2	2	no	—
49	Q₈	C₂×C₄×C₂₂₃	1	4	yes	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
53	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
54	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	C₂×C₄×C₂₂₃	1	2	yes	—
57	C₂²	C₂×D₄×C₂₂₃	1	1	yes	characteristic
58	C₄	—	2	4	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	2	no	—
61	C₄	Q₁₆×C₂₂₃	1	2	yes	direct product factor
62	C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
63	C₄	#41×C₂₂₃	1	1	yes	characteristic
64	C₂	C₄×D₄×C₂₂₃	1	1	yes	characteristic
65	C₂	C₂×Q₁₆×C₂₂₃	1	1	yes	characteristic
66	C₂	#135×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₂₃	D₄
commutator	C₄	C₂²×C₄×C₂₂₃
Frattini, cosocle	C₂×C₄	C₂³×C₂₂₃
socle, prime radical	C₂²×C₂₂₃	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
7. Q₁₆×C₂₂₃	C₄	direct (N × H)
27. C₄×C₂₂₃	Q₁₆	direct (N × H)
33. C₂₂₃	C₄×Q₁₆	direct (N × H)
34. C₄×Q₁₆	C₂₂₃	direct (N × H)
41. Q₁₆	C₄×C₂₂₃	direct (N × H)
61. C₄	Q₁₆×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
C₄×Q₈×C₂₂₃	2
#107×C₂₂₃	2
#103×C₂₂₃	2
C₄×C₈×C₂₂₃	2
C₄×Q₁₆	223

Subgroups — C4×Q16×C223 #28383151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₁₆×C₂₂₃ #28383151