Subgroups — C₂₃×#7542 #19155168

Order 5888.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×Q₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1425	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×Q₃₂	—	2	1	no	—
5	C₂₃×Q₆₄	—	2	1	no	—
6	C₂₃×Q₆₄	C₄	1	2	yes	—
7	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
8	C₂₃×#368	—	2	1	no	—
9	C₂₃×C₆₄	—	2	1	no	—
10	C₂×Q₁₆×C₂₃	—	4	1	no	—
11	C₂₃×Q₃₂	—	2	2	no	—
12	C₂₃×Q₃₂	—	4	1	no	—
13	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
14	C₂₃×#108	—	4	1	no	—
15	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
16	C₂₃×C₃₂	D₄	1	1	yes	characteristic
17	C₂×Q₈×C₂₃	—	8	1	no	—
18	Q₁₆×C₂₃	—	4	2	no	—
19	Q₁₆×C₂₃	—	8	1	no	—
20	M₁₆×C₂₃	—	8	1	no	—
21	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
22	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₂₃	SD₁₆	1	1	yes	characteristic
24	#7542	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	Q₈×C₂₃	—	8	2	no	—
26	Q₈×C₂₃	—	16	1	no	—
27	C₂×C₄×C₂₃	—	16	1	no	—
28	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
29	C₈×C₂₃	—	16	1	no	—
30	C₈×C₂₃	#102	1	1	yes	characteristic
31	C₈×C₂₃	SD₃₂	1	1	yes	characteristic
32	C₂×Q₆₄	C₂×C₂₃	1	1	yes	characteristic
33	M₁₂₈	C₂×C₂₃	1	1	yes	characteristic
34	#1425	C₂×C₂₃	1	1	yes	characteristic
35	C₂²×C₂₃	SD₆₄	1	1	yes	characteristic
36	C₄×C₂₃	—	16	2	no	—
37	C₄×C₂₃	#357	1	1	yes	characteristic
38	C₄×C₂₃	D₃₂	1	1	yes	characteristic
39	C₂×Q₃₂	—	2	1	no	—
40	Q₆₄	—	2	1	no	—
41	Q₆₄	C₄×C₂₃	1	2	yes	—
42	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
43	#368	—	2	1	no	—
44	C₆₄	—	2	1	no	—
45	C₂×C₂₃	—	2	1	no	—
46	C₂×C₂₃	#1415	1	1	yes	characteristic
47	C₂×Q₁₆	—	4	1	no	—
48	Q₃₂	—	2	2	no	—
49	Q₃₂	—	4	1	no	—
50	C₂×C₁₆	D₄×C₂₃	1	1	yes	characteristic
51	#108	—	4	1	no	—
52	C₃₂	C₂×C₄×C₂₃	1	1	yes	characteristic
53	C₃₂	D₄×C₂₃	1	1	yes	characteristic
54	C₂₃	#7542	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×Q₈	—	8	1	no	—
56	Q₁₆	—	4	2	no	—
57	Q₁₆	—	8	1	no	—
58	M₁₆	—	8	1	no	—
59	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
60	C₁₆	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
61	C₁₆	SD₁₆×C₂₃	1	1	yes	characteristic
62	Q₈	—	8	2	no	—
63	Q₈	—	16	1	no	—
64	C₂×C₄	—	16	1	no	—
65	C₂×C₄	C₂₃×D₁₆	1	1	yes	characteristic
66	C₈	—	16	1	no	—
67	C₈	C₂₃×#102	1	1	yes	characteristic
68	C₈	C₂₃×SD₃₂	1	1	yes	characteristic
69	C₂²	C₂₃×SD₆₄	1	1	yes	characteristic
70	C₄	—	16	2	no	—
71	C₄	C₂₃×#357	1	1	yes	characteristic
72	C₄	C₂₃×D₃₂	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	C₂₃×#1415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	#1415
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
24. #7542	C₂₃	direct (N × H)
54. C₂₃	#7542	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×Q₆₄	2
C₂₃×M₁₂₈	2
C₂₃×#1425	2
#7542	23

Subgroups — C23×#7542 #19155168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7542 #19155168