Subgroups — C₂₃×#7461 #19155087

Order 5888.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1371	C₂	1	2	yes	maximal
3	C₂×C₂₃×C₃₂	C₄	1	1	yes	characteristic
4	C₂×C₂₃×C₃₂	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#335	—	2	2	no	—
7	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
8	C₂×C₁₆×C₂₃	Q₈	1	1	yes	characteristic
9	C₂₃×C₄⋊C₈	—	4	2	no	—
10	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
11	C₂₃×C₃₂	—	2	1	no	—
12	C₂₃×C₃₂	C₈	1	2	yes	complemented
13	C₂×C₈×C₂₃	—	8	2	no	—
14	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₂₃	Q₁₆	1	1	yes	characteristic
16	C₄²×C₂₃	D₈	1	1	yes	characteristic
17	C₁₆×C₂₃	—	2	1	no	—
18	C₁₆×C₂₃	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₂₃	M₁₆	1	1	yes	characteristic
20	#7461	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₄×C₂₃	#107	1	1	yes	characteristic
22	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
23	C₂×C₄×C₂₃	Q₃₂	1	1	yes	characteristic
24	C₈×C₂₃	—	2	1	no	—
25	C₈×C₂₃	—	16	2	no	—
26	C₈×C₂₃	C₄⋊C₈	1	1	yes	characteristic
27	C₈×C₂₃	#108	1	1	yes	characteristic
28	C₄×C₃₂	C₂×C₂₃	1	1	yes	characteristic
29	#1371	C₂×C₂₃	1	2	yes	—
30	C₂²×C₂₃	#366	1	1	yes	characteristic
31	C₄×C₂₃	—	2	1	no	—
32	C₄×C₂₃	#335	1	1	yes	characteristic
33	C₄×C₂₃	#368	1	1	yes	characteristic
34	C₄×C₂₃	SD₆₄	1	2	yes	—
35	C₂×C₃₂	C₄×C₂₃	1	1	yes	characteristic
36	C₂×C₃₂	C₄×C₂₃	1	1	yes	characteristic
37	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
38	#335	—	2	2	no	—
39	C₂×C₂₃	#1371	1	1	yes	characteristic
40	C₂×C₂₃	#1424	1	1	yes	characteristic
41	C₂×C₂₃	#1425	1	1	yes	characteristic
42	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
43	C₂×C₁₆	Q₈×C₂₃	1	1	yes	characteristic
44	C₄⋊C₈	—	4	2	no	—
45	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
46	C₃₂	—	2	1	no	—
47	C₃₂	C₈×C₂₃	1	2	yes	complemented
48	C₂₃	#7461	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×C₈	—	8	2	no	—
50	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
51	C₂×C₈	Q₁₆×C₂₃	1	1	yes	characteristic
52	C₄²	D₈×C₂₃	1	1	yes	characteristic
53	C₁₆	—	2	1	no	—
54	C₁₆	C₂×C₈×C₂₃	1	1	yes	characteristic
55	C₁₆	M₁₆×C₂₃	1	1	yes	characteristic
56	C₂×C₄	C₂₃×#107	1	1	yes	characteristic
57	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
58	C₂×C₄	C₂₃×Q₃₂	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	—	16	2	no	—
61	C₈	C₂₃×C₄⋊C₈	1	1	yes	characteristic
62	C₈	C₂₃×#108	1	1	yes	characteristic
63	C₂²	C₂₃×#366	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	C₂₃×#335	1	1	yes	characteristic
66	C₄	C₂₃×#368	1	1	yes	characteristic
67	C₄	C₂₃×SD₆₄	1	2	yes	—
68	C₂	C₂₃×#1371	1	1	yes	characteristic
69	C₂	C₂₃×#1424	1	1	yes	characteristic
70	C₂	C₂₃×#1425	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	D₁₆
commutator	C₁₆	C₂×C₈×C₂₃
Frattini, cosocle	C₄×C₁₆	C₂²×C₂₃
socle, prime radical	C₂²×C₂₃	#366
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄²×C₂₃ ◁ C₄×C₈×C₂₃ ◁ C₄×C₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂₃×C₃₂	C₈	semidirect
20. #7461	C₂₃	direct (N × H)
47. C₃₂	C₈×C₂₃	semidirect
48. C₂₃	#7461	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×C₃₂	2
C₂₃×#1371	2
#7461	23

Subgroups — C23×#7461 #19155087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7461 #19155087