Subgroups — #15671456

Order 2944.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092822	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#335	C₂	1	1	yes	maximal, characteristic
4	#10625040	—	2	1	no	—
5	C₂₃×C₄⋊C₈	—	2	1	no	—
6	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	#10625000	—	2	1	no	—
8	#10625000	C₄	1	2	yes	—
9	C₂×C₂₃⋊C₁₆	—	2	1	no	—
10	C₂×Dic₄₆	—	4	1	no	—
11	C₂×C₈×C₂₃	—	4	1	no	—
12	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
14	C₄²×C₂₃	D₄	1	1	yes	characteristic
15	C₄⋊Dic₂₃	—	2	2	no	—
16	C₄⋊Dic₂₃	—	4	1	no	—
17	C₂₃⋊C₁₆	—	4	1	no	—
18	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
20	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
21	C₂×Dic₂₃	—	4	2	no	—
22	Dic₄₆	—	8	2	no	—
23	C₈×C₂₃	—	2	1	no	—
24	C₈×C₂₃	—	8	1	no	—
25	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
26	C₈×C₂₃	Q₁₆	1	1	yes	characteristic
27	#1365	—	23	1	no	maximal, Sylow
28	C₂²×C₂₃	#102	1	1	yes	characteristic
29	C₄×C₂₃	—	2	1	no	—
30	C₄×C₂₃	#101	1	1	yes	characteristic
31	C₄×C₂₃	#103	1	1	yes	characteristic
32	C₄×C₂₃	SD₃₂	1	1	yes	characteristic
33	C₄×C₂₃	Q₃₂	1	1	yes	characteristic
34	Dic₂₃	—	8	2	no	—
35	#500	—	23	1	no	—
36	C₄⋊C₁₆	—	23	1	no	—
37	#335	D₂₃	1	1	yes	characteristic
38	C₂×C₂₃	#332	1	1	yes	characteristic
39	C₂×C₂₃	#358	1	1	yes	characteristic
40	C₂×C₂₃	#361	1	1	yes	characteristic
41	C₄⋊Q₈	—	46	1	no	—
42	C₂×C₁₆	—	46	1	no	—
43	#107	—	23	2	no	—
44	#107	—	46	1	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
47	C₂₃	#1365	1	1	yes	characteristic, complemented, Sylow
48	C₂×Q₈	—	92	1	no	—
49	C₂×C₈	—	4	1	no	—
50	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
51	C₂×C₈	D₉₂	1	1	yes	characteristic
52	C₄⋊C₄	—	46	2	no	—
53	C₄⋊C₄	—	92	1	no	—
54	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
55	C₁₆	—	92	1	no	—
56	Q₈	—	184	2	no	—
57	C₂×C₄	—	92	2	no	—
58	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
59	C₂×C₄	#67272	1	1	yes	characteristic
60	C₂×C₄	#67274	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	8	1	no	—
63	C₈	#67263	1	1	yes	characteristic
64	C₈	Dic₉₂	1	1	yes	characteristic
65	C₂²	#10624991	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	184	2	no	—
68	C₄	#10624998	1	1	yes	characteristic
69	C₄	#10625006	1	1	yes	characteristic
70	C₄	#10625009	1	1	yes	characteristic
71	C₄	#10625011	1	1	yes	characteristic
72	C₂	#13092164	1	1	yes	characteristic
73	C₂	#13092220	1	1	yes	characteristic
74	C₂	#13092229	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10624991
commutator	C₂×C₈×C₂₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₂₃×#335	C₂
socle, prime radical	C₂²×C₂₃	#102
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	#335	D₂₃
hypocenter	C₂₃	#1365
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
47. C₂₃	#1365	semidirect

Maximal subgroups

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

subgroup	index
#13092822	2
C₄⋊(C₂₃⋊C₁₆)	2
C₂₃×#335	2
#1365	23