Subgroups — #31808721

Order 17792.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₁₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#22580793	C₂	1	2	yes	maximal, complemented
3	#22580791	C₂	1	2	yes	maximal, complemented
4	#22580787	C₂	1	1	yes	maximal, characteristic, complemented
5	#508×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₅₆	—	2	1	no	—
7	Dic₁₁₁₂	—	2	1	no	—
8	Dic₁₁₁₂	C₂²	1	2	yes	—
9	C₁₃₉⋊C₃₂	C₂²	1	2	yes	—
10	#135×C₁₃₉	—	2	1	no	—
11	Q₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
12	SD₃₂×C₁₃₉	—	2	1	no	—
13	D₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₇₈	—	4	1	no	—
16	Dic₅₅₆	—	2	2	no	—
17	Dic₅₅₆	—	4	1	no	—
18	#41×C₁₃₉	—	4	1	no	—
19	Q₁₆×C₁₃₉	—	2	1	no	—
20	SD₁₆×C₁₃₉	—	4	1	no	—
21	D₈×C₁₃₉	—	2	1	no	—
22	C₂×C₈×C₁₃₉	D₄	1	1	yes	characteristic
23	C₁₆×C₁₃₉	D₄	1	1	yes	characteristic
24	C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
25	Dic₂₇₈	—	4	2	no	—
26	Dic₂₇₈	—	8	1	no	—
27	C₂×Dic₁₃₉	—	8	1	no	—
28	Q₈×C₁₃₉	—	4	1	no	—
29	D₄×C₁₃₉	—	4	1	no	—
30	D₄×C₁₃₉	—	8	1	no	—
31	C₂×C₄×C₁₃₉	—	8	1	no	—
32	C₂×C₄×C₁₃₉	D₈	1	1	yes	characteristic
33	C₈×C₁₃₉	D₈	1	1	yes	characteristic
34	C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₃₉	—	8	1	no	—
36	C₂²×C₁₃₉	D₁₆	1	1	yes	characteristic
37	C₄×C₁₃₉	—	8	1	no	—
38	C₄×C₁₃₉	D₁₆	1	1	yes	characteristic
39	C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
40	Dic₁₃₉	—	8	2	no	—
41	C₂×C₁₃₉	—	2	1	no	—
42	C₂×C₁₃₉	—	16	1	no	—
43	C₂×C₁₃₉	C₂×D₁₆	1	1	yes	characteristic
44	C₁₃₉	#2264	1	1	yes	characteristic, complemented, Sylow
45	#2264	—	139	1	no	maximal, Sylow
46	#508	D₁₃₉	1	1	yes	characteristic
47	C₂×Q₃₂	—	139	1	no	—
48	Q₆₄	—	139	2	no	—
49	SD₆₄	—	139	2	no	—
50	M₆₄	—	139	1	no	—
51	#135	—	2	1	no	—
52	C₂×Q₁₆	—	278	1	no	—
53	Q₃₂	—	278	1	no	—
54	Q₃₂	—	139	2	no	—
55	Q₃₂	C₂×D₁₃₉	1	1	yes	characteristic
56	SD₃₂	—	2	1	no	—
57	D₁₆	C₂×D₁₃₉	1	1	yes	characteristic
58	C₂×C₁₆	C₂×D₁₃₉	1	1	yes	characteristic
59	C₃₂	—	139	2	no	—
60	#41	—	4	1	no	—
61	C₂×Q₈	—	556	1	no	—
62	Q₁₆	—	2	1	no	—
63	Q₁₆	—	278	2	no	—
64	Q₁₆	—	556	1	no	—
65	SD₁₆	—	4	1	no	—
66	D₈	—	2	1	no	—
67	C₂×C₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
68	C₁₆	C₂²⋊D₁₃₉	1	1	yes	characteristic
69	C₁₆	C₂²×D₁₃₉	1	1	yes	characteristic
70	Q₈	—	4	1	no	—
71	Q₈	—	556	2	no	—
72	Q₈	—	1112	1	no	—
73	D₄	—	4	1	no	—
74	D₄	—	8	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	1112	1	no	—
77	C₂×C₄	#14555770	1	1	yes	characteristic
78	C₈	#14555770	1	1	yes	characteristic
79	C₈	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
80	C₂²	—	8	1	no	—
81	C₂²	#17984783	1	1	yes	characteristic
82	C₄	—	8	1	no	—
83	C₄	—	1112	2	no	—
84	C₄	#17984783	1	1	yes	characteristic
85	C₄	C₂×#14555770	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	16	1	no	—
88	C₂	C₂×#17984783	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#17984783
commutator, cosocle	C₁₆×C₁₃₉	C₂³
Frattini	C₁₆	C₂²×D₁₃₉
Fitting, prime radical	#508×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₂×D₁₆
hypercenter	#508	D₁₃₉
hypocenter	C₁₃₉	#2264
LCS	G ▷ C₁₆×C₁₃₉ ▷ C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #508

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₁₁₂	C₂	semidirect
2. #22580793	C₂	semidirect
3. #22580791	C₂	semidirect
4. #22580787	C₂	semidirect
44. C₁₃₉	#2264	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₁₁₂	2
#22580793	2
#22580791	2
#22580787	2
#508×C₁₃₉	2
#2264	139