Subgroups — #31806002

Order 17792.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581349	C₂	1	2	yes	maximal
2	#22581333	C₂	1	1	yes	maximal, characteristic
3	#22581303	C₂	1	2	yes	maximal
4	C₈×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#500×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₅₆	—	2	2	no	—
7	#17984842	C₂²	1	2	yes	—
8	#17984773	—	2	4	no	—
9	#17984766	—	2	4	no	—
10	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₃₉	C₂²	1	2	yes	—
13	#107×C₁₃₉	—	2	2	no	—
14	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₇₈	—	2	2	no	—
16	Dic₅₅₆	—	4	4	no	—
17	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
18	C₂×C₁₃₉⋊C₈	D₄	1	2	yes	—
19	C₄⋊Dic₁₃₉	—	2	2	no	—
20	C₄⋊Dic₁₃₉	—	2	2	no	—
21	C₂×Q₈×C₁₃₉	—	2	2	no	—
22	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₃₉	—	2	2	no	—
24	C₄⋊C₄×C₁₃₉	—	2	2	no	—
25	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
26	Dic₂₇₈	—	4	4	no	—
27	C₂×Dic₁₃₉	—	4	2	no	—
28	C₁₃₉⋊C₈	—	2	2	no	—
29	C₁₃₉⋊C₈	Q₁₆	1	4	yes	complemented
30	Q₈×C₁₃₉	—	4	4	no	—
31	C₂×C₄×C₁₃₉	—	4	2	no	—
32	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₃₉	SD₁₆	1	4	yes	—
36	C₂²×C₁₃₉	#127	1	1	yes	characteristic
37	C₄×C₁₃₉	—	8	2	no	—
38	C₄×C₁₃₉	C₂×SD₁₆	1	2	yes	—
39	C₄×C₁₃₉	C₂×Q₁₆	1	2	yes	—
40	C₄×C₁₃₉	#135	1	2	yes	—
41	Dic₁₃₉	—	8	2	no	—
42	C₂×C₁₃₉	#492	1	1	yes	characteristic
43	C₂×C₁₃₉	#494	1	1	yes	characteristic
44	C₂×C₁₃₉	#495	1	1	yes	characteristic
45	C₁₃₉	#1707	1	1	yes	characteristic, complemented, Sylow
46	#1707	—	139	1	no	maximal, Sylow
47	#500	D₁₃₉	1	1	yes	characteristic
48	#498	—	139	2	no	—
49	#494	—	139	1	no	—
50	#487	—	139	2	no	—
51	C₈²	—	139	1	no	—
52	C₂×Q₁₆	—	278	2	no	—
53	C₄⋊Q₈	—	139	2	no	—
54	C₄⋊Q₈	C₂×D₁₃₉	1	2	yes	—
55	#107	—	2	2	no	—
56	#106	—	278	4	no	—
57	#103	—	278	4	no	—
58	C₄×C₈	—	139	1	no	—
59	C₄×C₈	—	139	1	no	—
60	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	278	2	no	—
63	Q₁₆	—	556	4	no	—
64	C₂×C₈	—	139	2	no	—
65	C₂×C₈	—	139	2	no	—
66	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	278	2	no	—
70	C₄⋊C₄	—	278	2	no	—
71	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	556	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	556	2	no	—
76	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
77	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
79	C₈	—	278	2	no	—
80	C₈	—	139	4	no	—
81	C₈	#14555772	1	4	yes	—
82	C₂²	#17984838	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	1112	2	no	—
85	C₄	C₂×#14555772	1	2	yes	—
86	C₄	Q₁₆×D₁₃₉	1	2	yes	—
87	C₄	#17984861	1	2	yes	—
88	C₂	#22581327	1	1	yes	characteristic
89	C₂	#22581331	1	1	yes	characteristic
90	C₂	#22581336	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17984838
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#500×C₁₃₉	C₂
socle, prime radical	C₂²×C₁₃₉	#127
hypercenter	#500	D₁₃₉
hypocenter	C₁₃₉	#1707
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
29. C₁₃₉⋊C₈	Q₁₆	semidirect
45. C₁₃₉	#1707	semidirect

Maximal subgroups

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

subgroup	index
#22581349	2
#22581333	2
#22581303	2
C₈×C₁₃₉⋊C₈	2
#500×C₁₃₉	2
#1707	139