Subgroups — C₂₃×#7333 #19154959

Order 5888.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1398	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1398	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1313	C₂	1	1	yes	maximal, characteristic
4	C₂₃×M₆₄	C₄	1	2	yes	—
5	C₂₃×M₆₄	C₄	1	2	yes	—
6	C₂₃×#346	C₄	1	1	yes	characteristic
7	C₂₃×#346	C₂²	1	1	yes	characteristic
8	C₂₃×#322	C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₂₃	—	2	1	no	—
10	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₂₃	—	2	1	no	—
13	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂₃×C₃₂	—	2	2	no	—
15	C₂₃×C₃₂	C₈	1	4	yes	complemented
16	C₂×C₈×C₂₃	—	4	1	no	—
17	C₂×C₈×C₂₃	C₄²	1	1	yes	characteristic
18	C₂×C₈×C₂₃	M₁₆	1	1	yes	characteristic
19	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
20	C₁₆×C₂₃	—	4	1	no	—
21	C₁₆×C₂₃	C₂×C₈	1	2	yes	—
22	C₁₆×C₂₃	M₁₆	1	2	yes	—
23	#7333	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₄×C₂₃	—	2	1	no	—
25	C₂×C₄×C₂₃	#97	1	1	yes	characteristic
26	C₈×C₂₃	—	2	1	no	—
27	C₈×C₂₃	—	8	1	no	—
28	C₈×C₂₃	C₄×C₈	1	1	yes	characteristic
29	C₈×C₂₃	#97	1	1	yes	characteristic
30	#1398	C₂×C₂₃	1	1	yes	characteristic
31	#1398	C₂×C₂₃	1	1	yes	characteristic
32	#1313	C₂×C₂₃	1	1	yes	characteristic
33	C₂²×C₂₃	#347	1	1	yes	characteristic
34	C₄×C₂₃	—	4	1	no	—
35	C₄×C₂₃	#322	1	1	yes	characteristic
36	C₄×C₂₃	#347	1	1	yes	characteristic
37	M₆₄	C₄×C₂₃	1	2	yes	—
38	M₆₄	C₄×C₂₃	1	2	yes	—
39	#346	C₄×C₂₃	1	1	yes	characteristic
40	#346	C₂²×C₂₃	1	1	yes	characteristic
41	#322	C₄×C₂₃	1	1	yes	characteristic
42	C₂×C₂₃	—	2	1	no	—
43	C₂×C₂₃	#1313	1	1	yes	characteristic
44	C₂×C₁₆	—	2	1	no	—
45	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
46	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
49	C₃₂	—	2	2	no	—
50	C₃₂	C₈×C₂₃	1	4	yes	complemented
51	C₂₃	#7333	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₈	—	4	1	no	—
53	C₂×C₈	C₄²×C₂₃	1	1	yes	characteristic
54	C₂×C₈	M₁₆×C₂₃	1	1	yes	characteristic
55	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
56	C₁₆	—	4	1	no	—
57	C₁₆	C₂×C₈×C₂₃	1	2	yes	—
58	C₁₆	M₁₆×C₂₃	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	C₂₃×#97	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	8	1	no	—
63	C₈	C₄×C₈×C₂₃	1	1	yes	characteristic
64	C₈	C₂₃×#97	1	1	yes	characteristic
65	C₂²	C₂₃×#347	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	C₂₃×#322	1	1	yes	characteristic
68	C₄	C₂₃×#347	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	C₂₃×#1313	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#322
commutator	C₈	C₄×C₈×C₂₃
Frattini, cosocle	#346	C₂²×C₂₃
socle	C₂×C₂₃	#1313
prime radical	C₂²×C₂₃	#347
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂₃×#346 ◁ G

Complements

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

N	H ≅ G/N	type
15. C₂₃×C₃₂	C₈	semidirect
23. #7333	C₂₃	direct (N × H)
50. C₃₂	C₈×C₂₃	semidirect
51. C₂₃	#7333	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×#1398	2
C₂₃×#1313	2
#7333	23

Subgroups — C23×#7333 #19154959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7333 #19154959