Subgroups — #31804461

Order 17792.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581358	C₂	1	1	yes	maximal, characteristic
2	#22580706	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#17984842	—	2	1	no	—
5	#17984776	—	2	1	no	—
6	#17984776	C₄	1	2	yes	—
7	C₄⋊(C₁₃₉⋊C₈)	—	2	1	no	—
8	C₂×C₁₆×C₁₃₉	—	2	1	no	—
9	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₇₈	—	4	1	no	—
11	C₂×C₁₃₉⋊C₈	—	4	1	no	—
12	C₄⋊Dic₁₃₉	—	2	2	no	—
13	C₄⋊Dic₁₃₉	—	4	1	no	—
14	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
16	C₄²×C₁₃₉	D₄	1	1	yes	characteristic
17	C₁₆×C₁₃₉	—	4	1	no	—
18	Dic₂₇₈	—	8	2	no	—
19	C₂×Dic₁₃₉	—	4	2	no	—
20	C₁₃₉⋊C₈	—	8	1	no	—
21	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₃₉	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₃₉	SD₁₆	1	1	yes	characteristic
24	C₈×C₁₃₉	—	2	1	no	—
25	C₈×C₁₃₉	SD₁₆	1	1	yes	characteristic
26	C₈×C₁₃₉	Q₁₆	1	1	yes	characteristic
27	C₂²×C₁₃₉	#102	1	1	yes	characteristic
28	C₄×C₁₃₉	—	2	1	no	—
29	C₄×C₁₃₉	#101	1	1	yes	characteristic
30	C₄×C₁₃₉	#103	1	1	yes	characteristic
31	C₄×C₁₃₉	SD₃₂	1	1	yes	characteristic
32	C₄×C₁₃₉	Q₃₂	1	1	yes	characteristic
33	Dic₁₃₉	—	8	2	no	—
34	C₂×C₁₃₉	#332	1	1	yes	characteristic
35	C₂×C₁₃₉	#358	1	1	yes	characteristic
36	C₂×C₁₃₉	#361	1	1	yes	characteristic
37	C₁₃₉	#1365	1	1	yes	characteristic, complemented, Sylow
38	#1365	—	139	1	no	maximal, Sylow
39	#500	—	139	1	no	—
40	C₄⋊C₁₆	D₁₃₉	1	1	yes	characteristic
41	#335	—	139	1	no	—
42	C₄⋊Q₈	—	278	1	no	—
43	C₂×C₁₆	—	2	1	no	—
44	#107	—	278	1	no	—
45	#107	—	139	2	no	—
46	C₄⋊C₈	—	278	1	no	—
47	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
48	C₂×Q₈	—	556	1	no	—
49	C₂×C₈	—	556	1	no	—
50	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
51	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
52	C₄⋊C₄	—	278	2	no	—
53	C₄⋊C₄	—	556	1	no	—
54	C₄²	D₅₅₆	1	1	yes	characteristic
55	C₁₆	—	4	1	no	—
56	Q₈	—	1112	2	no	—
57	C₂×C₄	—	556	2	no	—
58	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
59	C₂×C₄	D₁₁₁₂	1	1	yes	characteristic
60	C₂×C₄	#14555774	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	1112	1	no	—
63	C₈	#14555773	1	1	yes	characteristic
64	C₈	#14555775	1	1	yes	characteristic
65	C₂²	#17984764	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	1112	2	no	—
68	C₄	#17984760	1	1	yes	characteristic
69	C₄	#17984765	1	1	yes	characteristic
70	C₄	#17984787	1	1	yes	characteristic
71	C₄	Dic₁₁₁₂	1	1	yes	characteristic
72	C₂	#22580698	1	1	yes	characteristic
73	C₂	#22580758	1	1	yes	characteristic
74	C₂	#22580767	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984764
commutator	C₂×C₈×C₁₃₉	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₃₉
Fitting	C₄⋊C₁₆×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#102
cosocle	C₄×C₈×C₁₃₉	C₂²
hypercenter	C₄⋊C₁₆	D₁₃₉
hypocenter	C₁₃₉	#1365
LCS	G ▷ C₂×C₈×C₁₃₉ ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
37. C₁₃₉	#1365	semidirect

Maximal subgroups

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

subgroup	index
#22581358	2
#22580706	2
C₄⋊C₁₆×C₁₃₉	2
#1365	139