Subgroups — #1726×C₂₂₃ #43250741

Order 28544.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#97×C₂₂₃	C₂	1	3	yes	maximal
2	C₂×C₄×C₈×C₂₂₃	C₂	1	3	yes	maximal
3	#376×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×C₂₂₃	C₂²	1	3	yes	—
5	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₂₂₃	C₂²	1	3	yes	—
7	#97×C₂₂₃	C₄	1	12	yes	complemented
8	C₄×C₈×C₂₂₃	C₄	1	12	yes	complemented
9	#95×C₂₂₃	C₄	1	4	yes	—
10	C₂²×C₄×C₂₂₃	C₂×C₄	1	6	yes	—
11	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
12	C₂×C₈×C₂₂₃	C₂×C₄	1	6	yes	—
13	C₂×C₈×C₂₂₃	C₂×C₄	1	6	yes	—
14	C₂×C₈×C₂₂₃	C₂×C₄	1	12	yes	—
15	C₄²×C₂₂₃	C₂×C₄	1	6	yes	—
16	C₄²×C₂₂₃	C₂×C₄	1	6	yes	—
17	C₂³×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	—	2	12	no	—
19	C₂×C₄×C₂₂₃	C₄²	1	12	yes	—
20	C₂×C₄×C₂₂₃	C₂²×C₄	1	3	yes	—
21	C₂×C₄×C₂₂₃	C₂²×C₄	1	3	yes	—
22	C₈×C₂₂₃	C₄²	1	12	yes	—
23	C₈×C₂₂₃	C₄²	1	4	yes	—
24	C₂²×C₂₂₃	C₂×C₄²	1	3	yes	—
25	C₂²×C₂₂₃	#131	1	4	yes	—
26	C₄×C₂₂₃	—	2	12	no	—
27	C₄×C₂₂₃	C₂×C₄²	1	3	yes	—
28	C₄×C₂₂₃	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₂₂₃	C₄³	1	1	yes	characteristic
30	C₂×C₂₂₃	#405	1	6	yes	—
31	C₂₂₃	#1726	1	1	yes	characteristic, direct product factor, Sylow
32	#1726	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×#97	C₂×C₂₂₃	1	3	yes	—
34	C₂×C₄×C₈	C₂×C₂₂₃	1	3	yes	—
35	#376	C₂×C₂₂₃	1	1	yes	characteristic
36	C₂²×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
37	C₂²×C₈	C₂²×C₂₂₃	1	3	yes	—
38	C₂×C₄²	C₂²×C₂₂₃	1	3	yes	—
39	#97	C₄×C₂₂₃	1	12	yes	complemented
40	C₄×C₈	C₄×C₂₂₃	1	12	yes	complemented
41	#95	C₄×C₂₂₃	1	4	yes	—
42	C₂²×C₄	C₂×C₄×C₂₂₃	1	6	yes	—
43	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
44	C₂×C₈	C₂×C₄×C₂₂₃	1	6	yes	—
45	C₂×C₈	C₂×C₄×C₂₂₃	1	6	yes	—
46	C₂×C₈	C₂×C₄×C₂₂₃	1	12	yes	—
47	C₄²	C₂×C₄×C₂₂₃	1	6	yes	—
48	C₄²	C₂×C₄×C₂₂₃	1	6	yes	—
49	C₂³	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
50	C₂×C₄	—	2	12	no	—
51	C₂×C₄	C₄²×C₂₂₃	1	12	yes	—
52	C₂×C₄	C₂²×C₄×C₂₂₃	1	3	yes	—
53	C₂×C₄	C₂²×C₄×C₂₂₃	1	3	yes	—
54	C₈	C₄²×C₂₂₃	1	4	yes	—
55	C₈	C₄²×C₂₂₃	1	12	yes	—
56	C₂²	C₂×C₄²×C₂₂₃	1	3	yes	—
57	C₂²	#131×C₂₂₃	1	4	yes	—
58	C₄	—	2	12	no	—
59	C₄	C₂×C₄²×C₂₂₃	1	1	yes	characteristic
60	C₄	C₂×C₄²×C₂₂₃	1	3	yes	—
61	C₂	C₄³×C₂₂₃	1	1	yes	characteristic
62	C₂	#405×C₂₂₃	1	6	yes	—

	subgroup	G/H
center	C₂²×C₈×C₂₂₃	C₂²
commutator	C₂	C₄³×C₂₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₂₃
socle, prime radical	C₂³×C₂₂₃	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
7. #97×C₂₂₃	C₄	semidirect
8. C₄×C₈×C₂₂₃	C₄	semidirect
31. C₂₂₃	#1726	direct (N × H)
32. #1726	C₂₂₃	direct (N × H)
39. #97	C₄×C₂₂₃	semidirect
40. C₄×C₈	C₄×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#97×C₂₂₃	2
C₂×C₄×C₈×C₂₂₃	2
#376×C₂₂₃	2
#1726	223

Subgroups — #1726×C223 #43250741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1726×C₂₂₃ #43250741