Subgroups — #15671580

Order 2944.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092849	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625075	—	2	1	no	—
5	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#108	—	2	1	no	—
7	Dic₁₈₄	C₄	1	1	yes	characteristic
8	#10624982	—	2	1	no	—
9	D₃₆₈	C₄	1	1	yes	characteristic
10	C₂₃⋊C₃₂	—	2	1	no	—
11	#67288	—	4	1	no	—
12	M₁₆×C₂₃	—	4	1	no	—
13	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
14	Dic₉₂	—	2	1	no	—
15	D₁₈₄	—	2	1	no	—
16	#67263	—	4	1	no	—
17	C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₂₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
20	C₂²⋊D₂₃	—	8	1	no	—
21	D₉₂	—	4	1	no	—
22	C₄×D₂₃	—	8	1	no	—
23	Dic₄₆	—	4	1	no	—
24	C₈×C₂₃	—	8	1	no	—
25	C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
27	#1417	—	23	1	no	maximal, Sylow
28	C₂²×C₂₃	SD₃₂	1	1	yes	characteristic
29	C₂×D₂₃	—	8	1	no	—
30	C₄×C₂₃	#102	1	1	yes	characteristic
31	C₄×C₂₃	D₁₆	1	1	yes	characteristic
32	Dic₂₃	—	8	1	no	—
33	#508	—	23	1	no	—
34	C₂×C₃₂	—	23	1	no	—
35	#368	D₂₃	1	1	yes	characteristic, complemented
36	C₂×C₂₃	—	2	1	no	—
37	C₂×C₂₃	#357	1	1	yes	characteristic
38	D₂₃	—	16	1	no	—
39	#135	—	46	1	no	—
40	Q₃₂	—	23	1	no	—
41	SD₃₂	—	46	1	no	—
42	D₁₆	—	23	1	no	—
43	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic
44	#108	—	2	1	no	—
45	C₃₂	—	46	1	no	—
46	C₂₃	#1417	1	1	yes	characteristic, complemented, Sylow
47	#41	—	92	1	no	—
48	Q₁₆	—	46	1	no	—
49	SD₁₆	—	92	1	no	—
50	D₈	—	46	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
53	C₁₆	C₄×D₂₃	1	1	yes	characteristic
54	C₁₆	D₉₂	1	1	yes	characteristic
55	Q₈	—	92	1	no	—
56	D₄	—	92	1	no	—
57	D₄	—	184	1	no	—
58	C₂×C₄	—	184	1	no	—
59	C₂×C₄	#67272	1	1	yes	characteristic
60	C₈	—	8	1	no	—
61	C₈	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
62	C₈	#67274	1	1	yes	characteristic
63	C₂²	—	184	1	no	—
64	C₂²	#10625010	1	1	yes	characteristic
65	C₄	—	184	1	no	—
66	C₄	#10624991	1	1	yes	characteristic
67	C₄	#10625008	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	368	1	no	—
70	C₂	#13092217	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10625008
commutator	C₁₆×C₂₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₃
Fitting	C₂₃×#368	C₂
socle	C₂×C₂₃	#357
prime radical	#13092849	C₂
cosocle	C₂×C₁₆×C₂₃	C₂²
hypercenter	#368	D₂₃
hypocenter	C₂₃	#1417
LCS	G ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃⋊C₃₂	C₂	semidirect
3. C₂₃×#368	C₂	semidirect
35. #368	D₂₃	semidirect
46. C₂₃	#1417	semidirect

Maximal subgroups

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

subgroup	index
#13092849	2
C₂×C₂₃⋊C₃₂	2
C₂₃×#368	2
#1417	23