Subgroups — #501×C₂₂₃ #28383213

Order 14272.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#125×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#107×C₂₂₃	C₂	1	2	yes	maximal
4	#106×C₂₂₃	C₂	1	2	yes	maximal
5	#97×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₂₂₃	—	2	1	no	—
7	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
8	C₄⋊C₄×C₂₂₃	—	2	1	no	—
9	C₄⋊C₄×C₂₂₃	—	2	1	no	—
10	C₄⋊C₄×C₂₂₃	—	2	1	no	—
11	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
13	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
15	Q₈×C₂₂₃	—	4	2	no	—
16	C₂×C₄×C₂₂₃	—	2	1	no	—
17	C₂×C₄×C₂₂₃	—	2	1	no	—
18	C₂×C₄×C₂₂₃	—	2	2	no	—
19	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
22	C₈×C₂₂₃	Q₈	1	4	yes	complemented
23	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
24	C₄×C₂₂₃	—	2	1	no	—
25	C₄×C₂₂₃	—	2	1	no	—
26	C₄×C₂₂₃	—	4	1	no	—
27	C₄×C₂₂₃	—	4	2	no	—
28	C₄×C₂₂₃	—	4	1	no	—
29	C₄×C₂₂₃	C₂×Q₈	1	2	yes	—
30	C₂×C₂₂₃	C₄⋊Q₈	1	1	yes	characteristic
31	C₂×C₂₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
32	C₂×C₂₂₃	#137	1	1	yes	characteristic
33	C₂₂₃	#501	1	1	yes	characteristic, direct product factor, Sylow
34	#501	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₄⋊Q₈	C₂×C₂₂₃	1	1	yes	characteristic
36	#125	C₂×C₂₂₃	1	1	yes	characteristic
37	#107	C₂×C₂₂₃	1	2	yes	—
38	#106	C₂×C₂₂₃	1	2	yes	—
39	#97	C₂×C₂₂₃	1	1	yes	characteristic
40	C₂×Q₈	—	2	1	no	—
41	C₂×C₈	C₂²×C₂₂₃	1	2	yes	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	2	1	no	—
45	C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
46	C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
47	C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
48	C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
49	Q₈	—	4	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
54	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
55	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
56	C₈	Q₈×C₂₂₃	1	4	yes	complemented
57	C₂²	C₂×D₄×C₂₂₃	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	2	no	—
61	C₄	—	4	1	no	—
62	C₄	—	4	1	no	—
63	C₄	C₂×Q₈×C₂₂₃	1	2	yes	—
64	C₂	C₄⋊Q₈×C₂₂₃	1	1	yes	characteristic
65	C₂	AGL(1,ℤ/8)×C₂₂₃	1	1	yes	characteristic
66	C₂	#137×C₂₂₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
22. C₈×C₂₂₃	Q₈	semidirect
33. C₂₂₃	#501	direct (N × H)
34. #501	C₂₂₃	direct (N × H)
56. C₈	Q₈×C₂₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄⋊Q₈×C₂₂₃	2
#125×C₂₂₃	2
#107×C₂₂₃	2
#106×C₂₂₃	2
#97×C₂₂₃	2
#501	223

Subgroups — #501×C223 #28383213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #501×C₂₂₃ #28383213