Subgroups — #44482614

Order 30592.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29504523	C₂	1	2	yes	maximal
2	#346×C₂₃₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₃₉⋊C₈)	—	2	2	no	—
4	C₂×C₁₆×C₂₃₉	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₂₃₉	C₄	1	1	yes	characteristic
6	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
7	C₂×C₂₃₉⋊C₈	—	4	2	no	—
8	C₂×C₈×C₂₃₉	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₂₃₉	Q₈	1	1	yes	characteristic
10	C₄²×C₂₃₉	D₄	1	1	yes	characteristic
11	C₁₆×C₂₃₉	—	2	1	no	—
12	C₁₆×C₂₃₉	C₈	1	2	yes	complemented
13	C₂₃₉⋊C₈	—	8	2	no	—
14	C₂×C₄×C₂₃₉	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₃₉	SD₁₆	1	2	yes	—
16	C₈×C₂₃₉	—	2	1	no	—
17	C₈×C₂₃₉	C₂×C₈	1	1	yes	characteristic
18	C₈×C₂₃₉	M₁₆	1	1	yes	characteristic
19	C₂²×C₂₃₉	#106	1	1	yes	characteristic
20	C₄×C₂₃₉	—	2	2	no	—
21	C₄×C₂₃₉	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₂₃₉	#108	1	1	yes	characteristic
23	C₂×C₂₃₉	#334	1	1	yes	characteristic
24	C₂×C₂₃₉	#365	1	2	yes	—
25	C₂₃₉	#1368	1	1	yes	characteristic, complemented, Sylow
26	#1368	—	239	1	no	maximal, Sylow
27	#346	D₂₃₉	1	1	yes	characteristic
28	#334	—	239	2	no	—
29	C₂×C₁₆	Dic₂₃₉	1	1	yes	characteristic
30	C₂×C₁₆	Dic₂₃₉	1	1	yes	characteristic
31	C₄⋊C₈	—	478	2	no	—
32	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
33	C₂×C₈	—	956	2	no	—
34	C₂×C₈	C₂×Dic₂₃₉	1	1	yes	characteristic
35	C₂×C₈	Dic₄₇₈	1	1	yes	characteristic
36	C₄²	D₉₅₆	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	C₂₃₉⋊C₈	1	2	yes	complemented
39	C₂×C₄	C₄⋊Dic₂₃₉	1	1	yes	characteristic
40	C₂×C₄	#16838537	1	2	yes	—
41	C₈	—	2	1	no	—
42	C₈	—	1912	2	no	—
43	C₈	C₂×C₂₃₉⋊C₈	1	1	yes	characteristic
44	C₈	#16838532	1	1	yes	characteristic
45	C₂²	#21439892	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₄⋊(C₂₃₉⋊C₈)	1	1	yes	characteristic
48	C₄	#21439896	1	1	yes	characteristic
49	C₂	#29504523	1	1	yes	characteristic
50	C₂	#29504595	1	2	yes	—

	subgroup	G/H
center	C₂²	#21439892
commutator	C₈×C₂₃₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₃₉
Fitting	#346×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	#106
cosocle	C₄×C₈×C₂₃₉	C₂²
hypercenter	#346	D₂₃₉
hypocenter	C₂₃₉	#1368
LCS	G ▷ C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₂₃₉	C₈	semidirect
25. C₂₃₉	#1368	semidirect
38. C₁₆	C₂₃₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#29504523	2
#346×C₂₃₉	2
#1368	239