Subgroups — #2252×C₂₂₃ #43251267

Order 28544.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×C₂₂₃	C₂	1	2	yes	maximal
2	#366×C₂₂₃	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₂₂₃	—	2	2	no	—
5	C₂×C₁₆×C₂₂₃	C₂²	1	2	yes	—
6	#107×C₂₂₃	—	2	2	no	—
7	#107×C₂₂₃	C₂²	1	4	yes	—
8	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₂₂₃	—	4	2	no	—
10	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
11	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
12	C₄⋊C₄×C₂₂₃	—	2	4	no	—
13	C₄⋊C₄×C₂₂₃	—	4	2	no	—
14	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
15	C₁₆×C₂₂₃	Q₈	1	4	yes	complemented
16	Q₈×C₂₂₃	—	8	4	no	—
17	C₂×C₄×C₂₂₃	—	4	4	no	—
18	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	D₈	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
21	C₈×C₂₂₃	Q₁₆	1	2	yes	—
22	C₈×C₂₂₃	C₂×Q₈	1	2	yes	—
23	C₂²×C₂₂₃	C₂×D₈	1	1	yes	characteristic
24	C₄×C₂₂₃	—	8	4	no	—
25	C₄×C₂₂₃	D₁₆	1	2	yes	—
26	C₄×C₂₂₃	Q₃₂	1	2	yes	—
27	C₄×C₂₂₃	C₄⋊Q₈	1	1	yes	characteristic
28	C₄×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
29	C₂×C₂₂₃	#500	1	1	yes	characteristic
30	C₂×C₂₂₃	C₂×D₁₆	1	1	yes	characteristic
31	C₂×C₂₂₃	C₂×Q₃₂	1	1	yes	characteristic
32	C₂₂₃	#2252	1	1	yes	characteristic, direct product factor, Sylow
33	#2252	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	#500	C₂×C₂₂₃	1	2	yes	—
35	#366	C₂×C₂₂₃	1	4	yes	—
36	C₄×C₁₆	C₂×C₂₂₃	1	1	yes	characteristic
37	C₄⋊Q₈	—	2	2	no	—
38	C₂×C₁₆	C₂²×C₂₂₃	1	2	yes	—
39	#107	—	2	2	no	—
40	#107	C₂²×C₂₂₃	1	4	yes	—
41	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
42	C₂×Q₈	—	4	2	no	—
43	C₂×C₈	D₄×C₂₂₃	1	1	yes	characteristic
44	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
45	C₄⋊C₄	—	2	4	no	—
46	C₄⋊C₄	—	4	2	no	—
47	C₄²	D₄×C₂₂₃	1	1	yes	characteristic
48	C₁₆	Q₈×C₂₂₃	1	4	yes	complemented
49	Q₈	—	8	4	no	—
50	C₂×C₄	—	4	4	no	—
51	C₂×C₄	D₈×C₂₂₃	1	1	yes	characteristic
52	C₂×C₄	D₈×C₂₂₃	1	1	yes	characteristic
53	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
54	C₈	Q₁₆×C₂₂₃	1	2	yes	—
55	C₈	C₂×Q₈×C₂₂₃	1	2	yes	—
56	C₂²	C₂×D₈×C₂₂₃	1	1	yes	characteristic
57	C₄	—	8	4	no	—
58	C₄	D₁₆×C₂₂₃	1	2	yes	—
59	C₄	Q₃₂×C₂₂₃	1	2	yes	—
60	C₄	C₄⋊Q₈×C₂₂₃	1	1	yes	characteristic
61	C₄	C₂×Q₁₆×C₂₂₃	1	1	yes	characteristic
62	C₂	#500×C₂₂₃	1	1	yes	characteristic
63	C₂	C₂×D₁₆×C₂₂₃	1	1	yes	characteristic
64	C₂	C₂×Q₃₂×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ C₄×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₂₂₃	Q₈	semidirect
32. C₂₂₃	#2252	direct (N × H)
33. #2252	C₂₂₃	direct (N × H)
48. C₁₆	Q₈×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#500×C₂₂₃	2
#366×C₂₂₃	2
C₄×C₁₆×C₂₂₃	2
#2252	223

Subgroups — #2252×C223 #43251267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2252×C₂₂₃ #43251267