Subgroups — #28384385

Order 14272.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297350	C₂	1	2	yes	maximal, complemented
2	#20297332	C₂	1	6	yes	maximal, complemented
3	#20297326	C₂	1	3	yes	maximal, complemented
4	C₂×D₁₇₈₄	C₂	1	3	yes	maximal, complemented
5	#131×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813171	—	2	2	no	—
7	#16813170	C₂²	1	6	yes	complemented
8	C₂×D₈₉₂	C₂²	1	6	yes	complemented
9	D₄×D₂₂₃	—	2	6	no	—
10	Dic₈₉₂	C₂²	1	1	yes	characteristic, complemented
11	#16813159	C₂²	1	6	yes	complemented
12	D₁₇₈₄	C₂²	1	3	yes	complemented
13	D₁₇₈₄	C₂²	1	6	yes	complemented
14	#41×C₂₂₃	C₂²	1	1	yes	characteristic
15	M₁₆×C₂₂₃	C₂²	1	3	yes	complemented
16	C₂×C₈×C₂₂₃	C₂²	1	3	yes	complemented
17	C₂²×D₂₂₃	—	2	6	no	—
18	Dic₄₄₆	C₂³	1	2	yes	—
19	D₈₉₂	—	2	6	no	—
20	D₈₉₂	C₂³	1	6	yes	—
21	C₂²⋊D₂₂₃	—	2	6	no	—
22	C₄×D₂₂₃	—	2	6	no	—
23	Q₈×C₂₂₃	D₄	1	1	yes	characteristic
24	D₄×C₂₂₃	D₄	1	3	yes	—
25	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
26	C₈×C₂₂₃	C₂³	1	1	yes	characteristic
27	C₈×C₂₂₃	C₂³	1	3	yes	—
28	C₂²×C₂₂₃	C₂×D₄	1	3	yes	—
29	C₂×D₂₂₃	—	2	6	no	—
30	C₂×D₂₂₃	—	8	6	no	—
31	C₄×C₂₂₃	C₂×D₄	1	3	yes	—
32	C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
33	Dic₂₂₃	—	2	2	no	—
34	C₂×C₂₂₃	—	2	3	no	—
35	C₂×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
36	D₂₂₃	—	4	6	no	—
37	C₂₂₃	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	223	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	223	2	no	—
40	AGL(1,ℤ/8)	—	223	6	no	—
41	#135	—	223	3	no	—
42	C₂×D₈	—	223	3	no	—
43	#131	D₂₂₃	1	1	yes	characteristic, complemented
44	#41	—	446	2	no	—
45	#41	—	223	6	no	—
46	#41	C₂×D₂₂₃	1	1	yes	characteristic
47	C₂×D₄	—	446	6	no	—
48	C₂×D₄	—	223	6	no	—
49	Q₁₆	—	223	1	no	—
50	SD₁₆	—	223	6	no	—
51	D₈	—	223	3	no	—
52	D₈	—	223	6	no	—
53	M₁₆	C₂×D₂₂₃	1	3	yes	complemented
54	C₂×C₈	C₂×D₂₂₃	1	3	yes	complemented
55	C₂³	—	446	6	no	—
56	Q₈	—	223	2	no	—
57	Q₈	D₈₉₂	1	1	yes	characteristic
58	D₄	—	446	6	no	—
59	D₄	—	446	6	no	—
60	D₄	—	223	6	no	—
61	D₄	D₈₉₂	1	3	yes	—
62	C₂×C₄	—	446	6	no	—
63	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
64	C₈	C₂²×D₂₂₃	1	1	yes	characteristic
65	C₈	C₂²×D₂₂₃	1	3	yes	—
66	C₂²	—	446	6	no	—
67	C₂²	—	1784	6	no	—
68	C₂²	C₂×D₈₉₂	1	3	yes	—
69	C₄	—	446	2	no	—
70	C₄	C₂×D₈₉₂	1	3	yes	—
71	C₄	C₂³×D₂₂₃	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	892	6	no	—
74	C₂	C₂²×D₈₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₈₉₂
commutator, cosocle	C₄×C₂₂₃	C₂⁴
Frattini	C₄	C₂³×D₂₂₃
Fitting	#131×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₂²×D₄
hypercenter	#131	D₂₂₃
hypocenter	C₂₂₃	#576
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #20297350	C₂	semidirect
2. #20297332	C₂	semidirect
3. #20297326	C₂	semidirect
4. C₂×D₁₇₈₄	C₂	semidirect
5. #131×C₂₂₃	C₂	semidirect
7. #16813170	C₂²	semidirect
8. C₂×D₈₉₂	C₂²	semidirect
10. Dic₈₉₂	C₂²	semidirect
11. #16813159	C₂²	semidirect
12. D₁₇₈₄	C₂²	semidirect
13. D₁₇₈₄	C₂²	semidirect
15. M₁₆×C₂₂₃	C₂²	semidirect
16. C₂×C₈×C₂₂₃	C₂²	semidirect
37. C₂₂₃	#576	semidirect
43. #131	D₂₂₃	semidirect
53. M₁₆	C₂×D₂₂₃	semidirect
54. C₂×C₈	C₂×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297350	2
#20297332	2
#20297326	2
C₂×D₁₇₈₄	2
#131×C₂₂₃	2
#576	223