Subgroups — #30668271

Order 16768.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553745	C₂	1	2	yes	maximal, complemented
2	#22553149	C₂	1	2	yes	maximal, complemented
3	D₄₁₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₆₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16862269	—	2	2	no	—
7	#16862189	—	2	2	no	—
8	D₂₀₉₆	C₂²	1	2	yes	—
9	C₁₆×D₁₃₁	C₂²	1	1	yes	characteristic
10	C₁₃₁⋊C₃₂	C₂²	1	1	yes	characteristic
11	Q₃₂×C₁₃₁	C₂²	1	2	yes	—
12	C₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
13	#14523514	—	4	2	no	—
14	#14523500	—	4	2	no	—
15	D₁₀₄₈	—	2	2	no	—
16	C₈×D₁₃₁	D₄	1	1	yes	characteristic
17	C₁₃₁⋊C₁₆	D₄	1	1	yes	characteristic
18	Q₁₆×C₁₃₁	—	2	2	no	—
19	C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
20	D₅₂₄	—	4	2	no	—
21	D₅₂₄	—	8	2	no	—
22	C₄×D₁₃₁	—	8	2	no	—
23	C₄×D₁₃₁	D₈	1	1	yes	characteristic
24	C₁₃₁⋊C₈	D₈	1	1	yes	characteristic
25	Q₈×C₁₃₁	—	4	2	no	—
26	C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₃₁	—	8	2	no	—
28	C₂×D₁₃₁	D₁₆	1	1	yes	characteristic
29	C₄×C₁₃₁	—	8	2	no	—
30	C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
31	Dic₁₃₁	D₁₆	1	1	yes	characteristic
32	C₂×C₁₃₁	C₂×D₁₆	1	1	yes	characteristic
33	D₁₃₁	—	2	1	no	—
34	D₁₃₁	—	16	2	no	—
35	C₁₃₁	#2262	1	1	yes	characteristic, complemented, Sylow
36	#2262	—	131	1	no	maximal, Sylow
37	#508	—	131	2	no	—
38	Q₆₄	D₁₃₁	1	1	yes	characteristic, complemented
39	SD₆₄	—	131	2	no	—
40	D₃₂	—	131	1	no	—
41	C₂×C₃₂	—	131	1	no	—
42	#135	—	262	2	no	—
43	Q₃₂	C₂×D₁₃₁	1	2	yes	—
44	SD₃₂	—	262	2	no	—
45	D₁₆	—	131	2	no	—
46	C₂×C₁₆	—	131	1	no	—
47	C₃₂	—	131	1	no	—
48	C₃₂	C₂×D₁₃₁	1	1	yes	characteristic
49	#41	—	524	2	no	—
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	524	2	no	—
52	D₈	—	262	2	no	—
53	C₂×C₈	—	131	1	no	—
54	C₁₆	—	131	1	no	—
55	C₁₆	C₂²×D₁₃₁	1	1	yes	characteristic
56	Q₈	—	4	2	no	—
57	D₄	—	524	2	no	—
58	D₄	—	1048	2	no	—
59	C₂×C₄	—	1048	2	no	—
60	C₂×C₄	—	131	1	no	—
61	C₈	—	131	1	no	—
62	C₈	D₄×D₁₃₁	1	1	yes	characteristic
63	C₂²	—	1048	2	no	—
64	C₂²	—	131	1	no	—
65	C₄	—	8	2	no	—
66	C₄	—	131	1	no	—
67	C₄	D₈×D₁₃₁	1	1	yes	characteristic
68	C₂	—	262	1	no	—
69	C₂	—	2096	2	no	—
70	C₂	D₁₆×D₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₆×D₁₃₁
commutator, cosocle	C₁₆×C₁₃₁	C₂³
Frattini	C₁₆	C₂²×D₁₃₁
Fitting	Q₆₄×C₁₃₁	C₂
socle	C₂×C₁₃₁	C₂×D₁₆
hypercenter	Q₆₄	D₁₃₁
hypocenter	C₁₃₁	#2262
LCS	G ▷ C₁₆×C₁₃₁ ▷ C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
1. #22553745	C₂	semidirect
2. #22553149	C₂	semidirect
3. D₄₁₉₂	C₂	semidirect
4. C₃₂×D₁₃₁	C₂	semidirect
5. Q₆₄×C₁₃₁	C₂	semidirect
35. C₁₃₁	#2262	semidirect
38. Q₆₄	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553745	2
#22553149	2
D₄₁₉₂	2
C₃₂×D₁₃₁	2
Q₆₄×C₁₃₁	2
#2262	131