Subgroups — #1705×C₂₂₃ #43250720

Order 28544.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#498×C₂₂₃	C₂	1	3	yes	maximal, complemented
2	#492×C₂₂₃	C₂	1	3	yes	maximal
3	C₈²×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₁₆×C₂₂₃	—	2	6	no	—
5	C₄⋊Q₈×C₂₂₃	C₂²	1	3	yes	—
6	#127×C₂₂₃	C₂²	1	1	yes	characteristic
7	#106×C₂₂₃	—	2	6	no	—
8	C₄×C₈×C₂₂₃	C₂²	1	3	yes	—
9	C₂×Q₈×C₂₂₃	—	2	3	no	—
10	C₂×D₄×C₂₂₃	—	2	3	no	—
11	SD₁₆×C₂₂₃	—	4	12	no	—
12	C₂×C₈×C₂₂₃	D₄	1	6	yes	—
13	C₄⋊C₄×C₂₂₃	—	2	6	no	—
14	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
15	C₂³×C₂₂₃	—	4	1	no	—
16	Q₈×C₂₂₃	—	4	6	no	—
17	D₄×C₂₂₃	—	4	6	no	—
18	C₂×C₄×C₂₂₃	—	4	3	no	—
19	C₂×C₄×C₂₂₃	C₂×D₄	1	3	yes	—
20	C₈×C₂₂₃	SD₁₆	1	12	yes	complemented
21	C₂²×C₂₂₃	—	8	3	no	—
22	C₂²×C₂₂₃	#127	1	1	yes	characteristic
23	C₄×C₂₂₃	—	8	3	no	—
24	C₄×C₂₂₃	C₂×SD₁₆	1	6	yes	—
25	C₂×C₂₂₃	—	16	1	no	—
26	C₂×C₂₂₃	#492	1	3	yes	—
27	C₂₂₃	#1705	1	1	yes	characteristic, direct product factor, Sylow
28	#1705	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	#498	C₂×C₂₂₃	1	3	yes	complemented
30	#492	C₂×C₂₂₃	1	3	yes	—
31	C₈²	C₂×C₂₂₃	1	1	yes	characteristic, complemented
32	C₂×SD₁₆	—	2	6	no	—
33	C₄⋊Q₈	C₂²×C₂₂₃	1	3	yes	—
34	#127	C₂²×C₂₂₃	1	1	yes	characteristic
35	#106	—	2	6	no	—
36	C₄×C₈	C₂²×C₂₂₃	1	3	yes	—
37	C₂×Q₈	—	2	3	no	—
38	C₂×D₄	—	2	3	no	—
39	SD₁₆	—	4	12	no	—
40	C₂×C₈	D₄×C₂₂₃	1	6	yes	—
41	C₄⋊C₄	—	2	6	no	—
42	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
43	C₂³	—	4	1	no	—
44	Q₈	—	4	6	no	—
45	D₄	—	4	6	no	—
46	C₂×C₄	—	4	3	no	—
47	C₂×C₄	C₂×D₄×C₂₂₃	1	3	yes	—
48	C₈	SD₁₆×C₂₂₃	1	12	yes	complemented
49	C₂²	—	8	3	no	—
50	C₂²	#127×C₂₂₃	1	1	yes	characteristic
51	C₄	—	8	3	no	—
52	C₄	C₂×SD₁₆×C₂₂₃	1	6	yes	—
53	C₂	—	16	1	no	—
54	C₂	#492×C₂₂₃	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₂₂₃
prime radical	#127×C₂₂₃	C₂²
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
1. #498×C₂₂₃	C₂	semidirect
3. C₈²×C₂₂₃	C₂	semidirect
20. C₈×C₂₂₃	SD₁₆	semidirect
27. C₂₂₃	#1705	direct (N × H)
28. #1705	C₂₂₃	direct (N × H)
29. #498	C₂×C₂₂₃	semidirect
31. C₈²	C₂×C₂₂₃	semidirect
48. C₈	SD₁₆×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#498×C₂₂₃	2
#492×C₂₂₃	2
C₈²×C₂₂₃	2
#1705	223

Subgroups — #1705×C223 #43250720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1705×C₂₂₃ #43250720