Subgroups — #31808714

Order 17792.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581386	C₂	1	2	yes	maximal, complemented
2	#22580790	C₂	1	2	yes	maximal, complemented
3	D₄₄₄₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984865	—	2	2	no	—
7	#17984785	—	2	2	no	—
8	D₂₂₂₄	C₂²	1	2	yes	—
9	C₁₆×D₁₃₉	C₂²	1	1	yes	characteristic
10	C₁₃₉⋊C₃₂	C₂²	1	1	yes	characteristic
11	Q₃₂×C₁₃₉	C₂²	1	2	yes	—
12	C₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
13	#14555786	—	4	2	no	—
14	#14555772	—	4	2	no	—
15	D₁₁₁₂	—	2	2	no	—
16	C₈×D₁₃₉	D₄	1	1	yes	characteristic
17	C₁₃₉⋊C₁₆	D₄	1	1	yes	characteristic
18	Q₁₆×C₁₃₉	—	2	2	no	—
19	C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
20	D₅₅₆	—	4	2	no	—
21	D₅₅₆	—	8	2	no	—
22	C₄×D₁₃₉	—	8	2	no	—
23	C₄×D₁₃₉	D₈	1	1	yes	characteristic
24	C₁₃₉⋊C₈	D₈	1	1	yes	characteristic
25	Q₈×C₁₃₉	—	4	2	no	—
26	C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₃₉	—	8	2	no	—
28	C₂×D₁₃₉	D₁₆	1	1	yes	characteristic
29	C₄×C₁₃₉	—	8	2	no	—
30	C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
31	Dic₁₃₉	D₁₆	1	1	yes	characteristic
32	C₂×C₁₃₉	C₂×D₁₆	1	1	yes	characteristic
33	D₁₃₉	—	2	1	no	—
34	D₁₃₉	—	16	2	no	—
35	C₁₃₉	#2262	1	1	yes	characteristic, complemented, Sylow
36	#2262	—	139	1	no	maximal, Sylow
37	#508	—	139	2	no	—
38	Q₆₄	D₁₃₉	1	1	yes	characteristic, complemented
39	SD₆₄	—	139	2	no	—
40	D₃₂	—	139	1	no	—
41	C₂×C₃₂	—	139	1	no	—
42	#135	—	278	2	no	—
43	Q₃₂	C₂×D₁₃₉	1	2	yes	—
44	SD₃₂	—	278	2	no	—
45	D₁₆	—	139	2	no	—
46	C₂×C₁₆	—	139	1	no	—
47	C₃₂	—	139	1	no	—
48	C₃₂	C₂×D₁₃₉	1	1	yes	characteristic
49	#41	—	556	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	556	2	no	—
52	D₈	—	278	2	no	—
53	C₂×C₈	—	139	1	no	—
54	C₁₆	—	139	1	no	—
55	C₁₆	C₂²×D₁₃₉	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	556	2	no	—
58	D₄	—	1112	2	no	—
59	C₂×C₄	—	1112	2	no	—
60	C₂×C₄	—	139	1	no	—
61	C₈	—	139	1	no	—
62	C₈	D₄×D₁₃₉	1	1	yes	characteristic
63	C₂²	—	1112	2	no	—
64	C₂²	—	139	1	no	—
65	C₄	—	8	2	no	—
66	C₄	—	139	1	no	—
67	C₄	D₈×D₁₃₉	1	1	yes	characteristic
68	C₂	—	278	1	no	—
69	C₂	—	2224	2	no	—
70	C₂	D₁₆×D₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₁₃₉
commutator, cosocle	C₁₆×C₁₃₉	C₂³
Frattini	C₁₆	C₂²×D₁₃₉
Fitting	Q₆₄×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₂×D₁₆
hypercenter	Q₆₄	D₁₃₉
hypocenter	C₁₃₉	#2262
LCS	G ▷ C₁₆×C₁₃₉ ▷ C₈×C₁₃₉ ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #22581386	C₂	semidirect
2. #22580790	C₂	semidirect
3. D₄₄₄₈	C₂	semidirect
4. C₃₂×D₁₃₉	C₂	semidirect
5. Q₆₄×C₁₃₉	C₂	semidirect
35. C₁₃₉	#2262	semidirect
38. Q₆₄	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581386	2
#22580790	2
D₄₄₄₈	2
C₃₂×D₁₃₉	2
Q₆₄×C₁₃₉	2
#2262	139