Subgroups — #13093222

Order 1472.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10625160	C₂	1	2	yes	maximal, complemented
2	C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	#10625079	C₂	1	6	yes	maximal, complemented
4	#10625075	C₂	1	3	yes	maximal, complemented
5	C₂×D₁₈₄	C₂	1	3	yes	maximal, complemented
6	#41×C₂₃	C₂²	1	1	yes	characteristic
7	#67292	—	2	2	no	—
8	D₄×D₂₃	—	2	6	no	—
9	#67288	C₂²	1	6	yes	complemented
10	C₂×D₉₂	C₂²	1	6	yes	complemented
11	M₁₆×C₂₃	C₂²	1	3	yes	complemented
12	C₂×C₈×C₂₃	C₂²	1	3	yes	complemented
13	Dic₉₂	C₂²	1	1	yes	characteristic, complemented
14	D₁₈₄	C₂²	1	3	yes	complemented
15	D₁₈₄	C₂²	1	6	yes	complemented
16	#67263	C₂²	1	6	yes	complemented
17	C₂²×D₂₃	—	2	6	no	—
18	Q₈×C₂₃	D₄	1	1	yes	characteristic
19	D₄×C₂₃	D₄	1	3	yes	—
20	C₂×C₄×C₂₃	C₂³	1	3	yes	—
21	C₂²⋊D₂₃	—	2	6	no	—
22	D₉₂	—	2	6	no	—
23	D₉₂	C₂³	1	6	yes	—
24	C₄×D₂₃	—	2	6	no	—
25	Dic₄₆	C₂³	1	2	yes	—
26	C₈×C₂₃	C₂³	1	3	yes	—
27	C₈×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₂₃	C₂×D₄	1	3	yes	—
29	C₂×D₂₃	—	2	6	no	—
30	C₂×D₂₃	—	8	6	no	—
31	C₄×C₂₃	C₂×D₄	1	3	yes	—
32	C₄×C₂₃	C₂⁴	1	1	yes	characteristic
33	Dic₂₃	—	2	2	no	—
34	#576	—	23	1	no	maximal, Sylow
35	C₂×C₂₃	—	2	3	no	—
36	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
37	D₂₃	—	4	6	no	—
38	H(5,𝔽₂)	—	23	2	no	—
39	AGL(1,ℤ/8)	—	23	6	no	—
40	#135	—	23	3	no	—
41	C₂×D₈	—	23	3	no	—
42	#131	D₂₃	1	1	yes	characteristic, complemented
43	C₂₃	#576	1	1	yes	characteristic, complemented, Sylow
44	#41	—	23	6	no	—
45	#41	—	46	2	no	—
46	#41	C₂×D₂₃	1	1	yes	characteristic
47	C₂×D₄	—	23	6	no	—
48	C₂×D₄	—	46	6	no	—
49	Q₁₆	—	23	1	no	—
50	SD₁₆	—	23	6	no	—
51	D₈	—	23	3	no	—
52	D₈	—	23	6	no	—
53	M₁₆	C₂×D₂₃	1	3	yes	complemented
54	C₂×C₈	C₂×D₂₃	1	3	yes	complemented
55	C₂³	—	46	6	no	—
56	Q₈	—	23	2	no	—
57	Q₈	D₉₂	1	1	yes	characteristic
58	D₄	—	23	6	no	—
59	D₄	—	46	6	no	—
60	D₄	—	46	6	no	—
61	D₄	D₉₂	1	3	yes	—
62	C₂×C₄	—	46	6	no	—
63	C₂×C₄	C₂²×D₂₃	1	3	yes	—
64	C₈	C₂²×D₂₃	1	1	yes	characteristic
65	C₈	C₂²×D₂₃	1	3	yes	—
66	C₂²	—	46	6	no	—
67	C₂²	—	184	6	no	—
68	C₂²	C₂×D₉₂	1	3	yes	—
69	C₄	—	46	2	no	—
70	C₄	C₂×D₉₂	1	3	yes	—
71	C₄	C₂³×D₂₃	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	92	6	no	—
74	C₂	C₂²×D₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₉₂
commutator, cosocle	C₄×C₂₃	C₂⁴
Frattini	C₄	C₂³×D₂₃
Fitting	C₂₃×#131	C₂
socle	C₂×C₂₃	C₂²×D₄
hypercenter	#131	D₂₃
hypocenter	C₂₃	#576
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #10625160	C₂	semidirect
2. C₂₃×#131	C₂	semidirect
3. #10625079	C₂	semidirect
4. #10625075	C₂	semidirect
5. C₂×D₁₈₄	C₂	semidirect
9. #67288	C₂²	semidirect
10. C₂×D₉₂	C₂²	semidirect
11. M₁₆×C₂₃	C₂²	semidirect
12. C₂×C₈×C₂₃	C₂²	semidirect
13. Dic₉₂	C₂²	semidirect
14. D₁₈₄	C₂²	semidirect
15. D₁₈₄	C₂²	semidirect
16. #67263	C₂²	semidirect
42. #131	D₂₃	semidirect
43. C₂₃	#576	semidirect
53. M₁₆	C₂×D₂₃	semidirect
54. C₂×C₈	C₂×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
#10625160	2
C₂₃×#131	2
#10625079	2
#10625075	2
C₂×D₁₈₄	2
#576	23