Subgroups — #21439989

Order 7648.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838547	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₉₅₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#16838538	C₂	1	1	yes	maximal, characteristic, complemented
5	#16838536	C₂	1	1	yes	maximal, characteristic, complemented
6	#16838531	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₄₇₈	—	2	1	no	—
9	Dic₄₇₈	C₂²	1	1	yes	characteristic
10	Dic₄₇₈	C₂²	1	1	yes	characteristic
11	C₂²⋊D₂₃₉	—	2	1	no	—
12	C₂×Dic₂₃₉	—	2	1	no	—
13	C₄×D₂₃₉	—	2	1	no	—
14	C₄×D₂₃₉	C₂²	1	1	yes	characteristic
15	C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
17	D₄×C₂₃₉	C₂²	1	1	yes	characteristic
18	C₈×C₂₃₉	C₂²	1	1	yes	characteristic
19	C₂²×C₂₃₉	—	2	1	no	—
20	C₂×D₂₃₉	D₄	1	1	yes	characteristic
21	C₄×C₂₃₉	—	2	1	no	—
22	C₄×C₂₃₉	C₂³	1	1	yes	characteristic
23	Dic₂₃₉	—	2	1	no	—
24	Dic₂₃₉	—	2	1	no	—
25	Dic₂₃₉	D₄	1	1	yes	characteristic
26	C₂×C₂₃₉	—	4	1	no	—
27	C₂×C₂₃₉	C₂×D₄	1	1	yes	characteristic
28	D₂₃₉	—	2	1	no	—
29	C₂₃₉	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	239	1	no	maximal, Sylow
31	#41	—	239	1	no	—
32	C₂×Q₈	—	239	1	no	—
33	Q₁₆	—	239	1	no	—
34	Q₁₆	—	239	1	no	—
35	SD₁₆	—	239	1	no	—
36	SD₁₆	D₂₃₉	1	1	yes	characteristic, complemented
37	M₁₆	—	239	1	no	—
38	Q₈	—	478	1	no	—
39	Q₈	—	239	1	no	—
40	Q₈	—	239	1	no	—
41	Q₈	C₂×D₂₃₉	1	1	yes	characteristic
42	D₄	—	478	1	no	—
43	D₄	C₂×D₂₃₉	1	1	yes	characteristic
44	C₂×C₄	—	478	1	no	—
45	C₂×C₄	—	478	1	no	—
46	C₂×C₄	—	239	1	no	—
47	C₈	—	239	1	no	—
48	C₈	C₂×D₂₃₉	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	239	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	478	1	no	—
53	C₄	—	478	1	no	—
54	C₄	—	239	1	no	—
55	C₄	C₂²×D₂₃₉	1	1	yes	characteristic
56	C₂	—	4	1	no	—
57	C₂	—	478	1	no	—
58	C₂	D₄×D₂₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₂₃₉
commutator	C₄×C₂₃₉	C₂³
Frattini	C₄	C₂²×D₂₃₉
Fitting	SD₁₆×C₂₃₉	C₂
socle	C₂×C₂₃₉	C₂×D₄
hypercenter	SD₁₆	D₂₃₉
hypocenter	C₂₃₉	#137
LCS	G ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₂₃₉	C₂	semidirect
3. Dic₉₅₆	C₂	semidirect
4. #16838538	C₂	semidirect
5. #16838536	C₂	semidirect
6. #16838531	C₂	semidirect
7. SD₁₆×C₂₃₉	C₂	semidirect
29. C₂₃₉	#137	semidirect
36. SD₁₆	D₂₃₉	semidirect