Subgroups — #31804326

Order 17792.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22580729	C₂	1	1	yes	maximal, characteristic
2	#22580674	C₂	1	1	yes	maximal, characteristic
3	#346×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃₉⋊C₁₆	C₄	1	2	yes	—
5	C₄×C₁₃₉⋊C₈	C₄	1	2	yes	—
6	C₂×C₁₆×C₁₃₉	C₄	1	2	yes	—
7	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₃₉⋊C₈	—	2	2	no	—
9	C₁₃₉⋊C₁₆	C₈	1	4	yes	complemented
10	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
12	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₃₉	—	2	2	no	—
14	C₁₃₉⋊C₈	—	4	2	no	—
15	C₂×C₄×C₁₃₉	C₄²	1	1	yes	characteristic
16	C₂×C₄×C₁₃₉	M₁₆	1	2	yes	—
17	C₈×C₁₃₉	C₂×C₈	1	2	yes	—
18	C₈×C₁₃₉	M₁₆	1	2	yes	—
19	C₂²×C₁₃₉	#97	1	1	yes	characteristic
20	C₄×C₁₃₉	—	2	2	no	—
21	C₄×C₁₃₉	C₄×C₈	1	1	yes	characteristic
22	C₄×C₁₃₉	#97	1	1	yes	characteristic
23	C₂×C₁₃₉	#322	1	1	yes	characteristic
24	C₂×C₁₃₉	#347	1	2	yes	—
25	C₁₃₉	#1313	1	1	yes	characteristic, complemented, Sylow
26	#1313	—	139	1	no	maximal, Sylow
27	#346	—	139	1	no	—
28	#346	D₁₃₉	1	1	yes	characteristic
29	#322	—	139	1	no	—
30	C₂×C₁₆	—	139	2	no	—
31	C₂×C₁₆	Dic₁₃₉	1	2	yes	—
32	C₄×C₈	—	139	2	no	—
33	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
34	C₂×C₈	—	278	2	no	—
35	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
36	C₂×C₈	C₂×Dic₁₃₉	1	1	yes	characteristic
37	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
38	C₁₆	—	2	2	no	—
39	C₁₆	—	139	4	no	—
40	C₂×C₄	C₄×Dic₁₃₉	1	1	yes	characteristic
41	C₂×C₄	#14555768	1	2	yes	—
42	C₈	—	556	2	no	—
43	C₈	C₈×D₁₃₉	1	2	yes	—
44	C₈	#14555769	1	2	yes	—
45	C₂²	#17984752	1	1	yes	characteristic
46	C₄	—	2	2	no	—
47	C₄	C₈×Dic₁₃₉	1	1	yes	characteristic
48	C₄	#17984753	1	1	yes	characteristic
49	C₂	#22580675	1	1	yes	characteristic
50	C₂	#22580730	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₃₉
commutator	C₄×C₁₃₉	C₄×C₈
Frattini	C₄×C₈	C₂×D₁₃₉
Fitting	#346×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#97
cosocle	C₄×C₈×C₁₃₉	C₂²
hypercenter	#346	D₁₃₉
hypocenter	C₁₃₉	#1313
LCS	G ▷ C₄×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ #346

Complements

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

N	H ≅ G/N	type
9. C₁₃₉⋊C₁₆	C₈	semidirect
25. C₁₃₉	#1313	semidirect

Maximal subgroups

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

subgroup	index
#22580729	2
#22580674	2
#346×C₁₃₉	2
#1313	139