Subgroups — #1710×C₂₂₃ #43250725

Order 28544.

168 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	#487×C₂₂₃	C₂	1	4	yes	maximal
3	C₈²×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₂₂₃	C₂²	1	4	yes	—
5	#107×C₂₂₃	—	2	4	no	—
6	#103×C₂₂₃	—	2	8	no	—
7	C₄×C₈×C₂₂₃	C₂²	1	2	yes	—
8	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₂₂₃	—	2	4	no	—
10	C₂×C₈×C₂₂₃	D₄	1	4	yes	—
11	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
12	C₄⋊C₄×C₂₂₃	—	2	4	no	—
13	C₄⋊C₄×C₂₂₃	—	2	4	no	—
14	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
15	Q₈×C₂₂₃	—	4	8	no	—
16	C₂×C₄×C₂₂₃	—	4	4	no	—
17	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
19	C₈×C₂₂₃	—	2	4	no	—
20	C₈×C₂₂₃	Q₁₆	1	4	yes	—
21	C₂²×C₂₂₃	#127	1	1	yes	characteristic
22	C₄×C₂₂₃	—	8	4	no	—
23	C₄×C₂₂₃	C₂×Q₁₆	1	2	yes	—
24	C₄×C₂₂₃	#135	1	4	yes	—
25	C₂×C₂₂₃	#494	1	1	yes	characteristic
26	C₂×C₂₂₃	#495	1	2	yes	—
27	C₂₂₃	#1710	1	1	yes	characteristic, direct product factor, Sylow
28	#1710	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	#500	C₂×C₂₂₃	1	2	yes	—
30	#487	C₂×C₂₂₃	1	4	yes	—
31	C₈²	C₂×C₂₂₃	1	1	yes	characteristic
32	C₄⋊Q₈	C₂²×C₂₂₃	1	4	yes	—
33	#107	—	2	4	no	—
34	#103	—	2	8	no	—
35	C₄×C₈	C₂²×C₂₂₃	1	2	yes	—
36	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
37	C₂×Q₈	—	2	4	no	—
38	C₂×C₈	D₄×C₂₂₃	1	4	yes	—
39	C₂×C₈	D₄×C₂₂₃	1	2	yes	—
40	C₄⋊C₄	—	2	4	no	—
41	C₄⋊C₄	—	2	4	no	—
42	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
43	Q₈	—	4	8	no	—
44	C₂×C₄	—	4	4	no	—
45	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
46	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
47	C₈	—	2	4	no	—
48	C₈	Q₁₆×C₂₂₃	1	4	yes	—
49	C₂²	#127×C₂₂₃	1	1	yes	characteristic
50	C₄	—	8	4	no	—
51	C₄	C₂×Q₁₆×C₂₂₃	1	2	yes	—
52	C₄	#135×C₂₂₃	1	4	yes	—
53	C₂	#494×C₂₂₃	1	1	yes	characteristic
54	C₂	#495×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₂₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
27. C₂₂₃	#1710	direct (N × H)
28. #1710	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
#500×C₂₂₃	2
#487×C₂₂₃	2
C₈²×C₂₂₃	2
#1710	223

Subgroups — #1710×C223 #43250725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1710×C₂₂₃ #43250725