Subgroups — #19212068

Order 5888.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15671566	C₂	1	2	yes	maximal, complemented
2	C₂₃×#1709	C₂	1	1	yes	maximal, characteristic
3	#13092216	—	2	2	no	—
4	C₂₃×#494	C₂²	1	1	yes	characteristic
5	C₂₃×#493	C₄	1	1	yes	characteristic
6	C₂₃×#486	—	2	2	no	—
7	C₈²×C₂₃	C₄	1	1	yes	characteristic
8	C₂×Q₁₆×C₂₃	—	4	1	no	—
9	C₂×D₈×C₂₃	—	4	1	no	—
10	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
11	C₂₃×#127	—	2	1	no	—
12	C₂₃×#102	—	4	4	no	—
13	C₄×C₈×C₂₃	—	2	1	no	—
14	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
15	#10625018	—	4	2	no	—
16	C₂×Q₈×C₂₃	—	2	2	no	—
17	C₂×D₄×C₂₃	—	4	2	no	—
18	C₂×D₄×C₂₃	—	4	1	no	—
19	Q₁₆×C₂₃	—	8	2	no	—
20	D₈×C₂₃	—	8	2	no	—
21	C₂×C₈×C₂₃	—	2	1	no	—
22	C₂×C₈×C₂₃	—	2	2	no	—
23	C₄⋊C₄×C₂₃	—	4	2	no	—
24	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	#67267	—	8	2	no	—
26	#7531	—	23	1	no	maximal, Sylow
27	C₂³×C₂₃	—	8	1	no	—
28	Q₈×C₂₃	—	8	2	no	—
29	D₄×C₂₃	—	8	2	no	—
30	D₄×C₂₃	—	8	4	no	—
31	C₂×C₄×C₂₃	—	2	1	no	—
32	C₂×C₄×C₂₃	—	4	2	no	—
33	C₂×C₄×C₂₃	#99	1	1	yes	characteristic
34	C₈×C₂₃	—	2	2	no	—
35	C₈×C₂₃	—	4	2	no	—
36	C₂₃⋊C₈	—	16	2	no	—
37	#1709	D₂₃	1	1	yes	characteristic
38	#1411	—	23	2	no	—
39	C₂²×C₂₃	—	16	2	no	—
40	C₂²×C₂₃	—	16	1	no	—
41	C₂²×C₂₃	#353	1	1	yes	characteristic
42	C₄×C₂₃	—	2	2	no	—
43	C₄×C₂₃	—	2	1	no	—
44	C₄×C₂₃	—	8	2	no	—
45	#494	C₂×D₂₃	1	1	yes	characteristic
46	#493	Dic₂₃	1	1	yes	characteristic
47	#486	—	2	2	no	—
48	#356	—	46	2	no	—
49	C₈²	Dic₂₃	1	1	yes	characteristic
50	C₂×C₂₃	—	2	1	no	—
51	C₂×C₂₃	—	32	1	no	—
52	C₂×C₂₃	#1408	1	1	yes	characteristic
53	C₂×Q₁₆	—	4	1	no	—
54	C₂×D₈	—	4	1	no	—
55	C₄⋊Q₈	C₂²⋊D₂₃	1	2	yes	—
56	#127	—	2	1	no	—
57	#102	—	4	4	no	—
58	#101	—	92	2	no	—
59	C₄×C₈	—	2	1	no	—
60	C₄×C₈	C₂×Dic₂₃	1	1	yes	characteristic
61	C₂₃	#7531	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₈	—	2	2	no	—
63	C₂×D₄	—	4	2	no	—
64	C₂×D₄	—	4	1	no	—
65	Q₁₆	—	8	2	no	—
66	D₈	—	8	2	no	—
67	M₁₆	—	184	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	C₂²⋊Dic₂₃	1	1	yes	characteristic
72	C₂³	—	8	1	no	—
73	Q₈	—	8	2	no	—
74	D₄	—	8	2	no	—
75	D₄	—	8	4	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	#10625016	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	4	2	no	—
81	C₈	—	368	2	no	—
82	C₂²	—	16	2	no	—
83	C₂²	—	16	1	no	—
84	C₂²	#13092210	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	2	1	no	—
87	C₄	—	8	2	no	—
88	C₂	—	2	1	no	—
89	C₂	—	32	1	no	—
90	C₂	#15671560	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#15671560
commutator	C₄×C₈×C₂₃	C₂×C₄
Frattini	#494	C₂×D₂₃
Fitting	C₂₃×#1709	C₂
socle	C₂×C₂₃	#1408
hypercenter	#1709	D₂₃
hypocenter	C₂₃	#7531
LCS	G ▷ C₄×C₈×C₂₃ ▷ C₄²×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #494 ◁ #1709

Complements

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

N	H ≅ G/N	type
1. #15671566	C₂	semidirect
61. C₂₃	#7531	semidirect