Subgroups — #30664017

Order 16768.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553717	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₃₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#335×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#16862246	—	2	1	no	—
5	C₂×C₁₃₁⋊C₁₆	—	2	1	no	—
6	#16862180	—	2	1	no	—
7	#16862180	C₄	1	2	yes	—
8	C₄⋊C₈×C₁₃₁	—	2	1	no	—
9	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₆₂	—	4	1	no	—
11	C₄⋊Dic₁₃₁	—	2	2	no	—
12	C₄⋊Dic₁₃₁	—	4	1	no	—
13	C₁₃₁⋊C₁₆	—	4	1	no	—
14	C₂×C₈×C₁₃₁	—	4	1	no	—
15	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
17	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
18	Dic₂₆₂	—	8	1	no	—
19	Dic₂₆₂	—	8	1	no	—
20	C₂×Dic₁₃₁	—	4	2	no	—
21	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₃₁	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₃₁	SD₁₆	1	1	yes	characteristic
24	C₈×C₁₃₁	—	2	1	no	—
25	C₈×C₁₃₁	—	8	1	no	—
26	C₈×C₁₃₁	SD₁₆	1	1	yes	characteristic
27	C₈×C₁₃₁	Q₁₆	1	1	yes	characteristic
28	C₂²×C₁₃₁	#102	1	1	yes	characteristic
29	C₄×C₁₃₁	—	2	1	no	—
30	C₄×C₁₃₁	#101	1	1	yes	characteristic
31	C₄×C₁₃₁	#103	1	1	yes	characteristic
32	C₄×C₁₃₁	D₁₆	1	1	yes	characteristic
33	C₄×C₁₃₁	SD₃₂	1	1	yes	characteristic
34	Dic₁₃₁	—	8	2	no	—
35	C₂×C₁₃₁	#332	1	1	yes	characteristic
36	C₂×C₁₃₁	#357	1	1	yes	characteristic
37	C₂×C₁₃₁	#362	1	1	yes	characteristic
38	C₁₃₁	#1364	1	1	yes	characteristic, complemented, Sylow
39	#1364	—	131	1	no	maximal, Sylow
40	#500	—	131	1	no	—
41	C₄⋊C₁₆	—	131	1	no	—
42	#335	D₁₃₁	1	1	yes	characteristic
43	C₄⋊Q₈	—	262	1	no	—
44	C₂×C₁₆	—	262	1	no	—
45	#107	—	262	1	no	—
46	#107	—	131	2	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
49	C₂×Q₈	—	524	1	no	—
50	C₂×C₈	—	4	1	no	—
51	C₂×C₈	C₄×D₁₃₁	1	1	yes	characteristic
52	C₂×C₈	D₅₂₄	1	1	yes	characteristic
53	C₄⋊C₄	—	262	2	no	—
54	C₄⋊C₄	—	524	1	no	—
55	C₄²	C₂²⋊D₁₃₁	1	1	yes	characteristic
56	C₁₆	—	524	1	no	—
57	Q₈	—	1048	1	no	—
58	Q₈	—	1048	1	no	—
59	C₂×C₄	—	524	2	no	—
60	C₂×C₄	(C₂×D₁₃₁)⋊C₄	1	1	yes	characteristic
61	C₂×C₄	#14523498	1	1	yes	characteristic
62	C₂×C₄	#14523500	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	8	1	no	—
65	C₈	#14523502	1	1	yes	characteristic
66	C₈	Dic₅₂₄	1	1	yes	characteristic
67	C₂²	#16862166	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	1048	2	no	—
70	C₄	#16862164	1	1	yes	characteristic
71	C₄	#16862171	1	1	yes	characteristic
72	C₄	#16862187	1	1	yes	characteristic
73	C₄	#16862189	1	1	yes	characteristic
74	C₂	#22553059	1	1	yes	characteristic
75	C₂	#22553112	1	1	yes	characteristic
76	C₂	#22553127	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16862166
commutator	C₂×C₈×C₁₃₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₃₁
Fitting	#335×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#102
cosocle	C₄×C₈×C₁₃₁	C₂²
hypercenter	#335	D₁₃₁
hypocenter	C₁₃₁	#1364
LCS	G ▷ C₂×C₈×C₁₃₁ ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
38. C₁₃₁	#1364	semidirect

Maximal subgroups

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

subgroup	index
#22553717	2
C₄⋊(C₁₃₁⋊C₁₆)	2
#335×C₁₃₁	2
#1364	131