Subgroups — #44482750

Order 30592.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29504598	C₂	1	2	yes	maximal
2	C₂×C₃₂×C₂₃₉	C₂	1	1	yes	maximal, characteristic
3	#21439894	—	2	2	no	—
4	C₂×C₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
5	C₃₂×C₂₃₉	C₄	1	2	yes	complemented
6	C₄⋊Dic₂₃₉	—	4	2	no	—
7	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
8	C₁₆×C₂₃₉	C₂×C₄	1	1	yes	characteristic
9	C₁₆×C₂₃₉	Q₈	1	1	yes	characteristic
10	C₂×Dic₂₃₉	—	8	2	no	—
11	C₂×C₄×C₂₃₉	D₈	1	1	yes	characteristic
12	C₈×C₂₃₉	C₄⋊C₄	1	1	yes	characteristic
13	C₈×C₂₃₉	Q₁₆	1	1	yes	characteristic
14	C₂²×C₂₃₉	D₁₆	1	1	yes	characteristic
15	C₄×C₂₃₉	#107	1	1	yes	characteristic
16	C₄×C₂₃₉	Q₃₂	1	1	yes	characteristic
17	Dic₂₃₉	—	16	2	no	—
18	C₂×C₂₃₉	#366	1	1	yes	characteristic
19	C₂×C₂₃₉	SD₆₄	1	2	yes	—
20	C₂₃₉	#1424	1	1	yes	characteristic, complemented, Sylow
21	#1424	—	239	1	no	maximal, Sylow
22	C₂×C₃₂	D₂₃₉	1	1	yes	characteristic
23	#366	—	239	2	no	—
24	C₂×C₁₆	C₂×D₂₃₉	1	1	yes	characteristic
25	#107	—	478	2	no	—
26	C₃₂	Dic₂₃₉	1	2	yes	complemented
27	C₂×C₈	D₉₅₆	1	1	yes	characteristic
28	C₄⋊C₄	—	956	2	no	—
29	C₁₆	C₂×Dic₂₃₉	1	1	yes	characteristic
30	C₁₆	Dic₄₇₈	1	1	yes	characteristic
31	C₂×C₄	—	1912	2	no	—
32	C₂×C₄	D₁₉₁₂	1	1	yes	characteristic
33	C₈	C₄⋊Dic₂₃₉	1	1	yes	characteristic
34	C₈	Dic₉₅₆	1	1	yes	characteristic
35	C₂²	D₃₈₂₄	1	1	yes	characteristic
36	C₄	—	3824	2	no	—
37	C₄	#21439894	1	1	yes	characteristic
38	C₄	Dic₁₉₁₂	1	1	yes	characteristic
39	C₂	#29504598	1	1	yes	characteristic
40	C₂	#29504611	1	2	yes	—

	subgroup	G/H
center	C₂²	D₃₈₂₄
commutator	C₁₆×C₂₃₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₃₉
Fitting	C₂×C₃₂×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	D₁₆
cosocle	C₂×C₁₆×C₂₃₉	C₂²
hypercenter	C₂×C₃₂	D₂₃₉
hypocenter	C₂₃₉	#1424
LCS	G ▷ C₁₆×C₂₃₉ ▷ C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
5. C₃₂×C₂₃₉	C₄	semidirect
20. C₂₃₉	#1424	semidirect
26. C₃₂	Dic₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#29504598	2
C₂×C₃₂×C₂₃₉	2
#1424	239