Subgroups — C₂₃×#8817 #19156443

Order 5888.

390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1834	C₂	1	6	yes	maximal
2	C₂×C₈²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
4	C₂₃×#384	C₂²	1	4	yes	—
5	C₂₃×#340	—	2	12	no	—
6	C₈²×C₂₃	C₄	1	4	yes	complemented
7	C₂²×C₈×C₂₃	D₄	1	6	yes	—
8	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
9	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
10	C₄×C₈×C₂₃	C₂×C₄	1	6	yes	—
11	C₄×C₈×C₂₃	Q₈	1	6	yes	—
12	C₂²×C₄×C₂₃	—	4	4	no	—
13	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
14	C₂×C₈×C₂₃	—	2	12	no	—
15	C₂×C₈×C₂₃	C₄⋊C₄	1	12	yes	—
16	C₄⋊C₄×C₂₃	—	4	24	no	—
17	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
18	C₄²×C₂₃	C₂×Q₈	1	3	yes	—
19	#8817	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₂₃	#127	1	1	yes	characteristic
21	C₂×C₄×C₂₃	—	8	12	no	—
22	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
23	C₂×C₄×C₂₃	C₄⋊Q₈	1	3	yes	—
24	C₂×C₄×C₂₃	#135	1	12	yes	—
25	C₈×C₂₃	—	2	12	no	—
26	#1834	C₂×C₂₃	1	6	yes	—
27	C₂×C₈²	C₂×C₂₃	1	1	yes	characteristic
28	C₂²×C₂₃	#384	1	1	yes	characteristic
29	C₂²×C₂₃	#495	1	6	yes	—
30	C₄×C₂₃	—	16	4	no	—
31	C₄×C₂₃	#427	1	6	yes	—
32	C₄×C₂₃	#499	1	6	yes	—
33	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
34	#384	C₂²×C₂₃	1	4	yes	—
35	#340	—	2	12	no	—
36	C₈²	C₄×C₂₃	1	4	yes	complemented
37	C₂×C₂₃	#1711	1	4	yes	—
38	C₂×C₂₃	#1846	1	3	yes	—
39	C₂²×C₈	D₄×C₂₃	1	6	yes	—
40	C₂×C₄⋊C₄	—	2	12	no	—
41	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
42	C₄×C₈	C₂×C₄×C₂₃	1	6	yes	—
43	C₄×C₈	Q₈×C₂₃	1	6	yes	—
44	C₂₃	#8817	1	1	yes	characteristic, direct product factor, Sylow
45	C₂²×C₄	—	4	4	no	—
46	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
47	C₂×C₈	—	2	12	no	—
48	C₂×C₈	C₄⋊C₄×C₂₃	1	12	yes	—
49	C₄⋊C₄	—	4	24	no	—
50	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
51	C₄²	C₂×Q₈×C₂₃	1	3	yes	—
52	C₂³	C₂₃×#127	1	1	yes	characteristic
53	C₂×C₄	—	8	12	no	—
54	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
55	C₂×C₄	C₂₃×C₄⋊Q₈	1	3	yes	—
56	C₂×C₄	C₂₃×#135	1	12	yes	—
57	C₈	—	2	12	no	—
58	C₂²	C₂₃×#384	1	1	yes	characteristic
59	C₂²	C₂₃×#495	1	6	yes	—
60	C₄	—	16	4	no	—
61	C₄	C₂₃×#427	1	6	yes	—
62	C₄	C₂₃×#499	1	6	yes	—
63	C₂	C₂₃×#1711	1	4	yes	—
64	C₂	C₂₃×#1846	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₈²×C₂₃	C₄	semidirect
19. #8817	C₂₃	direct (N × H)
36. C₈²	C₄×C₂₃	semidirect
44. C₂₃	#8817	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#1834	2
C₂×C₈²×C₂₃	2
#8817	23

Subgroups — C23×#8817 #19156443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8817 #19156443