Subgroups — #31804547

Order 17792.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581369	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984848	C₄	1	1	yes	characteristic
5	#17984782	C₄	1	1	yes	characteristic
6	#17984781	—	2	1	no	—
7	C₁₆×D₁₃₉	—	2	1	no	—
8	C₁₃₉⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₃₉	—	2	1	no	—
11	#14555785	C₈	1	1	yes	characteristic
12	#14555769	C₈	1	1	yes	characteristic
13	#14555768	—	2	1	no	—
14	C₈×D₁₃₉	—	2	1	no	—
15	C₁₃₉⋊C₁₆	—	2	1	no	—
16	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₃₉	D₄	1	1	yes	characteristic
18	C₁₆×C₁₃₉	D₄	1	1	yes	characteristic
19	Dic₂₇₈	C₁₆	1	1	yes	characteristic
20	D₅₅₆	C₁₆	1	1	yes	characteristic
21	C₂²⋊D₁₃₉	—	2	1	no	—
22	C₄×D₁₃₉	—	2	1	no	—
23	C₁₃₉⋊C₈	—	2	1	no	—
24	C₂×C₄×C₁₃₉	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₃₉	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₃₉	M₃₂	1	1	yes	characteristic
28	C₂×D₁₃₉	—	2	1	no	—
29	C₄×C₁₃₉	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₁₃₉	C₂×C₁₆	1	1	yes	characteristic
31	Dic₁₃₉	—	2	1	no	—
32	C₂×C₁₃₉	—	2	1	no	—
33	C₂×C₁₃₉	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₁₃₉	—	4	1	no	—
35	C₁₃₉	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	139	1	no	maximal, Sylow
37	#504	—	139	1	no	—
38	M₆₄	D₁₃₉	1	1	yes	characteristic, complemented
39	C₂×C₃₂	—	139	1	no	—
40	#131	—	139	1	no	—
41	M₃₂	—	278	1	no	—
42	M₃₂	—	139	1	no	—
43	C₂×C₁₆	—	278	1	no	—
44	C₂×C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	278	1	no	—
47	#41	—	139	1	no	—
48	M₁₆	—	278	1	no	—
49	M₁₆	—	139	1	no	—
50	C₂×C₈	—	278	1	no	—
51	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
52	C₁₆	—	278	1	no	—
53	C₁₆	C₂²⋊D₁₃₉	1	1	yes	characteristic
54	C₁₆	D₅₅₆	1	1	yes	characteristic
55	Q₈	—	139	1	no	—
56	D₄	—	278	1	no	—
57	D₄	—	139	1	no	—
58	C₂×C₄	—	278	1	no	—
59	C₂×C₄	C₈×D₁₃₉	1	1	yes	characteristic
60	C₈	—	278	1	no	—
61	C₈	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
62	C₈	#14555768	1	1	yes	characteristic
63	C₂²	—	278	1	no	—
64	C₂²	#17984781	1	1	yes	characteristic
65	C₄	—	278	1	no	—
66	C₄	(C₂×D₁₃₉)⋊C₈	1	1	yes	characteristic
67	C₄	C₁₆×D₁₃₉	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	556	1	no	—
70	C₂	(C₂×D₁₃₉)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₁₃₉
commutator	C₄×C₁₃₉	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₃₉
Fitting	M₆₄×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₂²⋊C₁₆
prime radical	#14555785	C₈
cosocle	C₂×C₁₆×C₁₃₉	C₂²
hypercenter	M₆₄	D₁₃₉
hypocenter	C₁₃₉	#1401
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃₉⋊C₃₂	C₂	semidirect
3. M₆₄×C₁₃₉	C₂	semidirect
35. C₁₃₉	#1401	semidirect
38. M₆₄	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581369	2
C₂×C₁₃₉⋊C₃₂	2
M₆₄×C₁₃₉	2
#1401	139