Subgroups — C₂₃×#2840 #15670459

Order 2944.

580 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#393	C₂	1	12	yes	maximal, complemented
2	C₂₃×#379	C₂	1	3	yes	maximal
3	C₂⁵×C₂₃	C₂²	1	1	yes	characteristic
4	C₂²×Q₈×C₂₃	C₂²	1	4	yes	complemented
5	C₂×C₂²⋊C₄×C₂₃	C₂²	1	18	yes	—
6	C₂₃×#95	C₂²	1	12	yes	complemented
7	C₂⁴×C₂₃	—	4	6	no	—
8	C₂⁴×C₂₃	Q₈	1	4	yes	complemented
9	C₂⁴×C₂₃	C₂³	1	3	yes	—
10	C₂×Q₈×C₂₃	—	4	12	no	—
11	C₂²×C₄×C₂₃	C₂³	1	12	yes	—
12	C₂²⋊C₄×C₂₃	—	4	18	no	—
13	C₂³×C₂₃	—	2	18	no	—
14	C₂³×C₂₃	—	4	12	no	—
15	C₂³×C₂₃	—	4	16	no	—
16	C₂³×C₂₃	C₂×Q₈	1	6	yes	—
17	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
18	Q₈×C₂₃	—	4	16	no	—
19	C₂×C₄×C₂₃	—	2	36	no	—
20	#2840	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₂₃	—	2	18	no	—
22	C₂²×C₂₃	—	4	16	no	—
23	C₂²×C₂₃	—	4	12	no	—
24	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
25	C₂²×C₂₃	H(5,𝔽₂)	1	6	yes	—
26	C₄×C₂₃	—	4	12	no	—
27	#393	C₂×C₂₃	1	12	yes	complemented
28	#379	C₂×C₂₃	1	3	yes	—
29	C₂×C₂₃	—	4	6	no	—
30	C₂×C₂₃	#543	1	3	yes	—
31	C₂×C₂₃	H(3,𝔽₄)	1	4	yes	—
32	C₂⁵	C₂²×C₂₃	1	1	yes	characteristic
33	C₂²×Q₈	C₂²×C₂₃	1	4	yes	complemented
34	C₂×C₂²⋊C₄	C₂²×C₂₃	1	18	yes	—
35	#95	C₂²×C₂₃	1	12	yes	complemented
36	C₂₃	#2840	1	1	yes	characteristic, direct product factor, Sylow
37	C₂⁴	—	4	6	no	—
38	C₂⁴	Q₈×C₂₃	1	4	yes	complemented
39	C₂⁴	C₂³×C₂₃	1	3	yes	—
40	C₂×Q₈	—	4	12	no	—
41	C₂²×C₄	C₂³×C₂₃	1	12	yes	—
42	C₂²⋊C₄	—	4	18	no	—
43	C₂³	—	2	18	no	—
44	C₂³	—	4	12	no	—
45	C₂³	—	4	16	no	—
46	C₂³	C₂×Q₈×C₂₃	1	6	yes	—
47	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
48	Q₈	—	4	16	no	—
49	C₂×C₄	—	2	36	no	—
50	C₂²	—	2	18	no	—
51	C₂²	—	4	16	no	—
52	C₂²	—	4	12	no	—
53	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
54	C₂²	C₂₃×H(5,𝔽₂)	1	6	yes	—
55	C₄	—	4	12	no	—
56	C₂	—	4	6	no	—
57	C₂	C₂₃×#543	1	3	yes	—
58	C₂	C₂₃×H(3,𝔽₄)	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂⁵×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#393	C₂	semidirect
4. C₂²×Q₈×C₂₃	C₂²	semidirect
6. C₂₃×#95	C₂²	semidirect
8. C₂⁴×C₂₃	Q₈	semidirect
20. #2840	C₂₃	direct (N × H)
27. #393	C₂×C₂₃	semidirect
33. C₂²×Q₈	C₂²×C₂₃	semidirect
35. #95	C₂²×C₂₃	semidirect
36. C₂₃	#2840	direct (N × H)
38. C₂⁴	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#393	2
C₂₃×#379	2
#2840	23

Subgroups — C23×#2840 #15670459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2840 #15670459