Subgroups — C₂₃×#7456 #19155082

Order 5888.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1422	C₂	1	2	yes	maximal
2	C₂₃×#1311	C₂	1	1	yes	maximal, characteristic
3	C₂₃×M₆₄	—	2	2	no	—
4	C₄×C₁₆×C₂₃	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
6	C₈²×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₂₃	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₂₃	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
10	C₂×C₁₆×C₂₃	Q₈	1	1	yes	characteristic
11	C₄×C₈×C₂₃	—	2	1	no	—
12	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂₃×C₃₂	—	4	2	no	—
14	C₂×C₈×C₂₃	—	2	2	no	—
15	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
16	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₂₃	M₁₆	1	1	yes	characteristic
18	C₁₆×C₂₃	—	2	1	no	—
19	C₁₆×C₂₃	—	2	1	no	—
20	C₁₆×C₂₃	C₁₆	1	2	yes	—
21	C₁₆×C₂₃	SD₁₆	1	2	yes	—
22	#7456	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂×C₄×C₂₃	—	2	1	no	—
24	C₂×C₄×C₂₃	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₂₃	—	4	2	no	—
26	C₈×C₂₃	#106	1	1	yes	characteristic
27	C₈×C₂₃	#108	1	1	yes	characteristic
28	C₈×C₂₃	C₂×C₁₆	1	1	yes	characteristic
29	C₈×C₂₃	M₃₂	1	1	yes	characteristic
30	#1422	C₂×C₂₃	1	2	yes	—
31	#1311	C₂×C₂₃	1	1	yes	characteristic
32	C₂²×C₂₃	#364	1	1	yes	characteristic
33	C₄×C₂₃	—	2	2	no	—
34	C₄×C₂₃	#334	1	1	yes	characteristic
35	C₄×C₂₃	C₄⋊C₁₆	1	1	yes	characteristic
36	M₆₄	—	2	2	no	—
37	C₄×C₁₆	C₄×C₂₃	1	1	yes	characteristic
38	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
39	C₈²	C₄×C₂₃	1	1	yes	characteristic
40	C₂×C₂₃	—	2	1	no	—
41	C₂×C₂₃	#1367	1	1	yes	characteristic
42	C₂×C₁₆	C₈×C₂₃	1	1	yes	characteristic
43	C₂×C₁₆	C₈×C₂₃	1	1	yes	characteristic
44	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
45	C₂×C₁₆	Q₈×C₂₃	1	1	yes	characteristic
46	C₄×C₈	—	2	1	no	—
47	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
48	C₃₂	—	4	2	no	—
49	C₂₃	#7456	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
52	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
53	C₄²	M₁₆×C₂₃	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	2	1	no	—
56	C₁₆	C₁₆×C₂₃	1	2	yes	—
57	C₁₆	SD₁₆×C₂₃	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₂₃×C₄⋊C₈	1	1	yes	characteristic
60	C₈	—	4	2	no	—
61	C₈	C₂₃×#106	1	1	yes	characteristic
62	C₈	C₂₃×#108	1	1	yes	characteristic
63	C₈	C₂×C₁₆×C₂₃	1	1	yes	characteristic
64	C₈	C₂₃×M₃₂	1	1	yes	characteristic
65	C₂²	C₂₃×#364	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	C₂₃×#334	1	1	yes	characteristic
68	C₄	C₂₃×C₄⋊C₁₆	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	C₂₃×#1367	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#106
commutator	C₈	C₂×C₁₆×C₂₃
Frattini, cosocle	C₄×C₁₆	C₂²×C₂₃
socle	C₂×C₂₃	#1367
prime radical	C₂²×C₂₃	#364
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂×C₁₆×C₂₃ ◁ C₄×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
22. #7456	C₂₃	direct (N × H)
49. C₂₃	#7456	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×#1422	2
C₂₃×#1311	2
#7456	23

Subgroups — C23×#7456 #19155082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7456 #19155082