Subgroups — #30666113

Order 16768.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	1	6	yes	maximal
2	C₄³×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₃₁⋊C₈	C₂²	1	4	yes	—
4	C₄⋊(C₁₃₁⋊C₈)	—	2	12	no	—
5	C₂×C₄²×C₁₃₁	C₄	1	3	yes	—
6	C₂×C₄²×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄²×C₁₃₁	C₂²	1	3	yes	—
8	C₂×C₁₃₁⋊C₈	—	2	12	no	—
9	C₂²×C₄×C₁₃₁	C₂×C₄	1	3	yes	—
10	C₂²×C₄×C₁₃₁	C₂×C₄	1	3	yes	—
11	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
12	C₄²×C₁₃₁	—	2	6	no	—
13	C₄²×C₁₃₁	C₈	1	4	yes	complemented
14	C₄²×C₁₃₁	D₄	1	6	yes	—
15	C₄²×C₁₃₁	Q₈	1	6	yes	—
16	C₁₃₁⋊C₈	—	4	4	no	—
17	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₃₁	—	2	6	no	—
19	C₂×C₄×C₁₃₁	C₄⋊C₄	1	6	yes	—
20	C₂×C₄×C₁₃₁	C₄⋊C₄	1	6	yes	—
21	C₂×C₄×C₁₃₁	C₂×C₈	1	6	yes	—
22	C₂×C₄×C₁₃₁	M₁₆	1	6	yes	—
23	C₂×C₄×C₁₃₁	C₂×D₄	1	3	yes	—
24	C₂×C₄×C₁₃₁	C₂×Q₈	1	3	yes	—
25	C₂²×C₁₃₁	C₂×C₄⋊C₄	1	3	yes	—
26	C₂²×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
27	C₂²×C₁₃₁	C₂×M₁₆	1	3	yes	—
28	C₄×C₁₃₁	—	2	6	no	—
29	C₄×C₁₃₁	C₄⋊C₈	1	12	yes	complemented
30	C₄×C₁₃₁	#127	1	1	yes	characteristic
31	C₄×C₁₃₁	C₄⋊Q₈	1	3	yes	—
32	C₂×C₁₃₁	#384	1	1	yes	characteristic
33	C₂×C₁₃₁	C₂×C₄⋊C₈	1	3	yes	—
34	C₂×C₁₃₁	#423	1	3	yes	—
35	C₁₃₁	#1842	1	1	yes	characteristic, complemented, Sylow
36	#1842	—	131	1	no	maximal, Sylow
37	C₂×C₄⋊C₈	—	131	6	no	—
38	C₄³	D₁₃₁	1	1	yes	characteristic
39	C₂²×C₈	—	131	4	no	—
40	C₂×C₄²	Dic₁₃₁	1	3	yes	—
41	C₂×C₄²	Dic₁₃₁	1	1	yes	characteristic
42	C₂×C₄²	C₂×D₁₃₁	1	3	yes	—
43	C₄⋊C₈	—	262	12	no	—
44	C₂²×C₄	C₂×Dic₁₃₁	1	3	yes	—
45	C₂²×C₄	C₂×Dic₁₃₁	1	3	yes	—
46	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
47	C₂×C₈	—	262	12	no	—
48	C₄²	—	2	6	no	—
49	C₄²	C₁₃₁⋊C₈	1	4	yes	complemented
50	C₄²	D₅₂₄	1	6	yes	—
51	C₄²	Dic₂₆₂	1	6	yes	—
52	C₂³	C₂²×Dic₁₃₁	1	1	yes	characteristic
53	C₂×C₄	—	2	6	no	—
54	C₂×C₄	C₄⋊Dic₁₃₁	1	6	yes	—
55	C₂×C₄	C₄⋊Dic₁₃₁	1	6	yes	—
56	C₂×C₄	C₂×C₁₃₁⋊C₈	1	6	yes	—
57	C₂×C₄	#14523497	1	6	yes	—
58	C₂×C₄	C₂×D₅₂₄	1	3	yes	—
59	C₂×C₄	C₂×Dic₂₆₂	1	3	yes	—
60	C₈	—	524	4	no	—
61	C₂²	C₂×C₄⋊Dic₁₃₁	1	3	yes	—
62	C₂²	C₂²×C₁₃₁⋊C₈	1	1	yes	characteristic
63	C₂²	C₂×#14523497	1	3	yes	—
64	C₄	—	2	6	no	—
65	C₄	C₄⋊(C₁₃₁⋊C₈)	1	12	yes	complemented
66	C₄	#16862243	1	1	yes	characteristic
67	C₄	#16862246	1	3	yes	—
68	C₂	#22553182	1	1	yes	characteristic
69	C₂	C₂×C₄⋊(C₁₃₁⋊C₈)	1	3	yes	—
70	C₂	#22553336	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₃₁
commutator	C₂²×C₁₃₁	C₂²×C₈
Fitting	C₄³×C₁₃₁	C₂
socle, prime radical	C₂³×C₁₃₁	C₂²×C₄
cosocle	C₂²×C₄×C₁₃₁	C₂³
hypercenter	C₄³	D₁₃₁
hypocenter	C₁₃₁	#1842
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
13. C₄²×C₁₃₁	C₈	semidirect
29. C₄×C₁₃₁	C₄⋊C₈	semidirect
35. C₁₃₁	#1842	semidirect
49. C₄²	C₁₃₁⋊C₈	semidirect
65. C₄	C₄⋊(C₁₃₁⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₃₁⋊C₈)	2
C₄³×C₁₃₁	2
#1842	131