Subgroups — #19212092

Order 5888.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15675706	C₂	1	1	yes	maximal, characteristic
2	#15671608	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1424	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092849	—	2	1	no	—
5	Dic₃₆₈	C₄	1	1	yes	characteristic, complemented
6	#13092256	—	2	1	no	—
7	D₇₃₆	C₄	1	1	yes	characteristic, complemented
8	C₂₃⋊C₆₄	—	2	1	no	—
9	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
10	C₂₃×#366	—	2	1	no	—
11	#10625075	—	4	1	no	—
12	C₂×C₁₆×C₂₃	D₄	1	1	yes	characteristic
13	C₂₃×#107	—	4	1	no	—
14	Dic₁₈₄	—	2	1	no	—
15	#10624982	—	4	1	no	—
16	D₃₆₈	—	2	1	no	—
17	C₂₃×C₃₂	C₂×C₄	1	1	yes	characteristic
18	C₂₃×C₃₂	D₄	1	1	yes	characteristic
19	#67288	—	8	1	no	—
20	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
21	C₄⋊C₄×C₂₃	—	8	1	no	—
22	Dic₉₂	—	4	1	no	—
23	D₁₈₄	—	4	1	no	—
24	#67263	—	8	1	no	—
25	C₁₆×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₂₃	SD₁₆	1	1	yes	characteristic
27	#7540	—	23	1	no	maximal, Sylow
28	C₂×C₄×C₂₃	—	16	1	no	—
29	C₂×C₄×C₂₃	D₁₆	1	1	yes	characteristic
30	C₂²⋊D₂₃	—	16	1	no	—
31	D₉₂	—	8	1	no	—
32	C₄×D₂₃	—	16	1	no	—
33	Dic₄₆	—	8	1	no	—
34	C₈×C₂₃	#102	1	1	yes	characteristic
35	C₈×C₂₃	SD₃₂	1	1	yes	characteristic
36	#2262	—	23	1	no	—
37	M₁₂₈	—	23	1	no	—
38	#1424	D₂₃	1	1	yes	characteristic, complemented
39	C₂²×C₂₃	D₃₂	1	1	yes	characteristic
40	C₂×D₂₃	—	16	1	no	—
41	C₄×C₂₃	—	32	1	no	—
42	C₄×C₂₃	#357	1	1	yes	characteristic
43	C₄×C₂₃	SD₆₄	1	1	yes	characteristic
44	Dic₂₃	—	16	1	no	—
45	#508	—	46	1	no	—
46	Q₆₄	—	23	1	no	—
47	SD₆₄	—	46	1	no	—
48	D₃₂	—	23	1	no	—
49	C₂×C₃₂	C₂×D₂₃	1	1	yes	characteristic
50	#366	—	2	1	no	—
51	C₆₄	—	46	1	no	—
52	C₂×C₂₃	—	2	1	no	—
53	C₂×C₂₃	#1415	1	1	yes	characteristic
54	D₂₃	—	32	1	no	—
55	#135	—	92	1	no	—
56	Q₃₂	—	46	1	no	—
57	SD₃₂	—	92	1	no	—
58	D₁₆	—	46	1	no	—
59	C₂×C₁₆	C₂²⋊D₂₃	1	1	yes	characteristic
60	#107	—	4	1	no	—
61	C₃₂	C₄×D₂₃	1	1	yes	characteristic
62	C₃₂	D₉₂	1	1	yes	characteristic
63	C₂₃	#7540	1	1	yes	characteristic, complemented, Sylow
64	#41	—	184	1	no	—
65	Q₁₆	—	92	1	no	—
66	SD₁₆	—	184	1	no	—
67	D₈	—	92	1	no	—
68	C₂×C₈	#67272	1	1	yes	characteristic
69	C₄⋊C₄	—	8	1	no	—
70	C₁₆	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
71	C₁₆	#67274	1	1	yes	characteristic
72	Q₈	—	184	1	no	—
73	D₄	—	368	1	no	—
74	D₄	—	184	1	no	—
75	C₂×C₄	—	16	1	no	—
76	C₂×C₄	—	368	1	no	—
77	C₂×C₄	#10625008	1	1	yes	characteristic
78	C₈	#10624991	1	1	yes	characteristic
79	C₈	#10625010	1	1	yes	characteristic
80	C₂²	—	368	1	no	—
81	C₂²	#13092252	1	1	yes	characteristic
82	C₄	—	32	1	no	—
83	C₄	—	368	1	no	—
84	C₄	#13092217	1	1	yes	characteristic
85	C₄	#13092254	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	736	1	no	—
88	C₂	#15671574	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#15671574
commutator	C₂₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₂₃
Fitting	C₂₃×#1424	C₂
socle	C₂×C₂₃	#1415
hypercenter	#1424	D₂₃
hypocenter	C₂₃	#7540
LCS	G ▷ C₂₃×C₃₂ ▷ C₁₆×C₂₃ ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1424

Complements

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

N	H ≅ G/N	type
2. #15671608	C₂	semidirect
3. C₂₃×#1424	C₂	semidirect
5. Dic₃₆₈	C₄	semidirect
7. D₇₃₆	C₄	semidirect
38. #1424	D₂₃	semidirect
63. C₂₃	#7540	semidirect