Subgroups — #1712×C₂₂₃ #43250727

Order 28544.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#493×C₂₂₃	C₂	1	6	yes	maximal, complemented
2	C₈²×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₈×C₂₂₃	—	2	12	no	—
4	#127×C₂₂₃	C₂²	1	4	yes	—
5	C₄×C₈×C₂₂₃	C₂²	1	3	yes	—
6	C₂×D₄×C₂₂₃	—	2	12	no	—
7	D₈×C₂₂₃	—	4	24	no	—
8	C₂×C₈×C₂₂₃	D₄	1	6	yes	—
9	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
10	C₂³×C₂₂₃	—	4	4	no	—
11	D₄×C₂₂₃	—	4	24	no	—
12	C₂×C₄×C₂₂₃	C₂×D₄	1	3	yes	—
13	C₈×C₂₂₃	D₈	1	12	yes	complemented
14	C₂²×C₂₂₃	—	8	12	no	—
15	C₂²×C₂₂₃	#127	1	1	yes	characteristic
16	C₄×C₂₂₃	C₂×D₈	1	6	yes	—
17	C₂×C₂₂₃	—	16	4	no	—
18	C₂×C₂₂₃	#493	1	3	yes	—
19	C₂₂₃	#1712	1	1	yes	characteristic, direct product factor, Sylow
20	#1712	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	#493	C₂×C₂₂₃	1	6	yes	complemented
22	C₈²	C₂×C₂₂₃	1	1	yes	characteristic, complemented
23	C₂×D₈	—	2	12	no	—
24	#127	C₂²×C₂₂₃	1	4	yes	—
25	C₄×C₈	C₂²×C₂₂₃	1	3	yes	—
26	C₂×D₄	—	2	12	no	—
27	D₈	—	4	24	no	—
28	C₂×C₈	D₄×C₂₂₃	1	6	yes	—
29	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
30	C₂³	—	4	4	no	—
31	D₄	—	4	24	no	—
32	C₂×C₄	C₂×D₄×C₂₂₃	1	3	yes	—
33	C₈	D₈×C₂₂₃	1	12	yes	complemented
34	C₂²	—	8	12	no	—
35	C₂²	#127×C₂₂₃	1	1	yes	characteristic
36	C₄	C₂×D₈×C₂₂₃	1	6	yes	—
37	C₂	—	16	4	no	—
38	C₂	#493×C₂₂₃	1	3	yes	—

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

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

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