Subgroups — C₂₃×#7548 #19155174

Order 5888.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1425	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1424	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
5	C₂₃×#368	—	2	1	no	—
6	C₂₃×#366	—	2	1	no	—
7	C₂₃×C₆₄	C₄	1	2	yes	complemented
8	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
9	C₂₃×#108	—	4	1	no	—
10	C₂₃×#107	—	4	1	no	—
11	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
12	C₂₃×C₃₂	Q₈	1	1	yes	characteristic
13	M₁₆×C₂₃	—	8	1	no	—
14	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
15	C₄⋊C₄×C₂₃	—	8	1	no	—
16	C₁₆×C₂₃	C₄⋊C₄	1	1	yes	characteristic
17	C₁₆×C₂₃	Q₁₆	1	1	yes	characteristic
18	#7548	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₄×C₂₃	—	16	1	no	—
20	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
21	C₈×C₂₃	—	16	1	no	—
22	C₈×C₂₃	#107	1	1	yes	characteristic
23	C₈×C₂₃	Q₃₂	1	1	yes	characteristic
24	M₁₂₈	C₂×C₂₃	1	1	yes	characteristic
25	#1425	C₂×C₂₃	1	1	yes	characteristic
26	#1424	C₂×C₂₃	1	1	yes	characteristic
27	C₂²×C₂₃	SD₆₄	1	1	yes	characteristic
28	C₄×C₂₃	—	32	1	no	—
29	C₄×C₂₃	#366	1	1	yes	characteristic
30	C₄×C₂₃	SD₆₄	1	1	yes	characteristic
31	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
32	#368	—	2	1	no	—
33	#366	—	2	1	no	—
34	C₆₄	C₄×C₂₃	1	2	yes	complemented
35	C₂×C₂₃	—	2	1	no	—
36	C₂×C₂₃	#1424	1	1	yes	characteristic
37	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
38	#108	—	4	1	no	—
39	#107	—	4	1	no	—
40	C₃₂	C₂×C₄×C₂₃	1	1	yes	characteristic
41	C₃₂	Q₈×C₂₃	1	1	yes	characteristic
42	C₂₃	#7548	1	1	yes	characteristic, direct product factor, Sylow
43	M₁₆	—	8	1	no	—
44	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
45	C₄⋊C₄	—	8	1	no	—
46	C₁₆	C₄⋊C₄×C₂₃	1	1	yes	characteristic
47	C₁₆	Q₁₆×C₂₃	1	1	yes	characteristic
48	C₂×C₄	—	16	1	no	—
49	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
50	C₈	—	16	1	no	—
51	C₈	C₂₃×#107	1	1	yes	characteristic
52	C₈	C₂₃×Q₃₂	1	1	yes	characteristic
53	C₂²	C₂₃×SD₆₄	1	1	yes	characteristic
54	C₄	—	32	1	no	—
55	C₄	C₂₃×#366	1	1	yes	characteristic
56	C₄	C₂₃×SD₆₄	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	C₂₃×#1424	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#1424
commutator	C₃₂	C₂×C₄×C₂₃
Frattini, cosocle	C₂×C₃₂	C₂²×C₂₃
prime radical	C₂²×C₂₃	SD₆₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂×C₄×C₂₃ ◁ C₂×C₈×C₂₃ ◁ C₂×C₁₆×C₂₃ ◁ C₂×C₂₃×C₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₂₃×C₆₄	C₄	semidirect
18. #7548	C₂₃	direct (N × H)
34. C₆₄	C₄×C₂₃	semidirect
42. C₂₃	#7548	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×M₁₂₈	2
C₂₃×#1425	2
C₂₃×#1424	2
#7548	23

Subgroups — C23×#7548 #19155174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7548 #19155174