Subgroups — #20297336

Order 7136.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813171	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#16813160	C₂	1	1	yes	maximal, characteristic, complemented
4	#16813159	C₂	1	1	yes	maximal, characteristic, complemented
5	#16813157	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813153	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₄₄₆	—	2	1	no	—
9	Dic₄₄₆	C₂²	1	1	yes	characteristic
10	D₈₉₂	—	2	1	no	—
11	D₈₉₂	C₂²	1	1	yes	characteristic
12	C₄×D₂₂₃	—	2	1	no	—
13	C₄×D₂₂₃	—	2	1	no	—
14	C₄×D₂₂₃	C₂²	1	1	yes	characteristic
15	C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
17	Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₈×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₂×D₂₂₃	—	2	1	no	—
20	C₂×D₂₂₃	D₄	1	1	yes	characteristic
21	C₄×C₂₂₃	—	2	1	no	—
22	C₄×C₂₂₃	—	2	1	no	—
23	C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	Dic₂₂₃	—	2	1	no	—
25	Dic₂₂₃	D₄	1	1	yes	characteristic
26	C₂×C₂₂₃	C₂×D₄	1	1	yes	characteristic
27	D₂₂₃	—	2	1	no	—
28	D₂₂₃	—	4	1	no	—
29	C₂₂₃	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	223	1	no	maximal, Sylow
31	#41	—	223	1	no	—
32	C₂×Q₈	—	223	1	no	—
33	Q₁₆	—	223	1	no	—
34	Q₁₆	D₂₂₃	1	1	yes	characteristic, complemented
35	SD₁₆	—	223	1	no	—
36	SD₁₆	—	223	1	no	—
37	M₁₆	—	223	1	no	—
38	Q₈	—	446	1	no	—
39	Q₈	—	223	1	no	—
40	Q₈	C₂×D₂₂₃	1	1	yes	characteristic
41	Q₈	C₂×D₂₂₃	1	1	yes	characteristic
42	D₄	—	446	1	no	—
43	D₄	—	223	1	no	—
44	C₂×C₄	—	446	1	no	—
45	C₂×C₄	—	446	1	no	—
46	C₂×C₄	—	223	1	no	—
47	C₈	—	223	1	no	—
48	C₈	C₂×D₂₂₃	1	1	yes	characteristic
49	C₂²	—	446	1	no	—
50	C₂²	—	223	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	446	1	no	—
54	C₄	—	223	1	no	—
55	C₄	C₂²×D₂₂₃	1	1	yes	characteristic
56	C₂	—	892	1	no	—
57	C₂	—	446	1	no	—
58	C₂	D₄×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₂₂₃
commutator	C₄×C₂₂₃	C₂³
Frattini	C₄	C₂²×D₂₂₃
Fitting	Q₁₆×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₂×D₄
hypercenter	Q₁₆	D₂₂₃
hypocenter	C₂₂₃	#137
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₂₂₃	C₂	semidirect
3. #16813160	C₂	semidirect
4. #16813159	C₂	semidirect
5. #16813157	C₂	semidirect
6. #16813153	C₂	semidirect
7. Q₁₆×C₂₂₃	C₂	semidirect
29. C₂₂₃	#137	semidirect
34. Q₁₆	D₂₂₃	semidirect