Subgroups — #15671578

Order 2944.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#13092243	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×C₂₃	—	2	1	no	—
5	C₂₃×Q₃₂	—	2	1	no	—
6	C₂₃×Q₃₂	C₄	1	2	yes	complemented
7	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	#10625000	—	2	1	no	—
9	C₂₃⋊C₃₂	—	2	1	no	—
10	C₂×Q₈×C₂₃	—	4	1	no	—
11	Q₁₆×C₂₃	—	2	2	no	—
12	Q₁₆×C₂₃	—	4	1	no	—
13	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
14	C₄⋊Dic₂₃	—	4	1	no	—
15	C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₃	D₄	1	1	yes	characteristic
17	Q₈×C₂₃	—	4	2	no	—
18	Q₈×C₂₃	—	8	1	no	—
19	C₂×C₄×C₂₃	—	8	1	no	—
20	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
21	C₂×Dic₂₃	—	8	1	no	—
22	C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
24	#1416	—	23	1	no	maximal, Sylow
25	C₂²×C₂₃	D₁₆	1	1	yes	characteristic
26	C₄×C₂₃	—	8	2	no	—
27	C₄×C₂₃	#102	1	1	yes	characteristic
28	C₄×C₂₃	SD₃₂	1	1	yes	characteristic
29	Dic₂₃	—	16	1	no	—
30	C₂×Q₃₂	D₂₃	1	1	yes	characteristic
31	C₂×C₃₂	—	23	1	no	—
32	#366	—	23	1	no	—
33	C₂×C₂₃	#357	1	1	yes	characteristic
34	C₂×C₂₃	SD₆₄	1	1	yes	characteristic
35	C₂×C₂₃	Q₆₄	1	1	yes	characteristic
36	C₂×Q₁₆	—	2	1	no	—
37	Q₃₂	—	2	1	no	—
38	Q₃₂	Dic₂₃	1	2	yes	complemented
39	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
40	#107	—	46	1	no	—
41	C₃₂	—	46	1	no	—
42	C₂₃	#1416	1	1	yes	characteristic, complemented, Sylow
43	C₂×Q₈	—	4	1	no	—
44	Q₁₆	—	2	2	no	—
45	Q₁₆	—	4	1	no	—
46	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
47	C₄⋊C₄	—	92	1	no	—
48	C₁₆	C₂×Dic₂₃	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₂₃	1	1	yes	characteristic
50	Q₈	—	4	2	no	—
51	Q₈	—	8	1	no	—
52	C₂×C₄	—	8	1	no	—
53	C₂×C₄	—	184	1	no	—
54	C₂×C₄	#67272	1	1	yes	characteristic
55	C₈	#67273	1	1	yes	characteristic
56	C₈	C₂²⋊Dic₂₃	1	1	yes	characteristic
57	C₂²	#10625008	1	1	yes	characteristic
58	C₄	—	8	2	no	—
59	C₄	—	368	1	no	—
60	C₄	#10625009	1	1	yes	characteristic
61	C₄	#10625014	1	1	yes	characteristic
62	C₂	#13092218	1	1	yes	characteristic
63	C₂	#13092254	1	1	yes	characteristic
64	C₂	#13092257	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10625008
commutator	C₁₆×C₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₃
Fitting	C₂×C₂₃×Q₃₂	C₂
socle, prime radical	C₂²×C₂₃	D₁₆
cosocle	C₂×C₁₆×C₂₃	C₂²
hypercenter	C₂×Q₃₂	D₂₃
hypocenter	C₂₃	#1416
LCS	G ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
6. C₂₃×Q₃₂	C₄	semidirect
38. Q₃₂	Dic₂₃	semidirect
42. C₂₃	#1416	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃⋊C₃₂	2
#13092243	2
C₂×C₂₃×Q₃₂	2
#1416	23