Subgroups — #30664263

Order 16768.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523496	C₂	1	3	yes	maximal
2	#22553033	C₂	1	3	yes	maximal, complemented
3	C₈²×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14523496	C₄	1	6	yes	—
5	C₄²×D₁₃₁	C₂²	1	1	yes	characteristic
6	#16862156	C₄	1	6	yes	—
7	C₄×C₁₃₁⋊C₈	C₂²	1	3	yes	—
8	C₄×C₈×C₁₃₁	C₂²	1	3	yes	—
9	C₂×C₄×D₁₃₁	C₂×C₄	1	3	yes	—
10	#14523496	—	2	12	no	—
11	C₂×C₁₃₁⋊C₈	C₂×C₄	1	6	yes	—
12	C₄×Dic₁₃₁	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₁₃₁	C₂×C₄	1	6	yes	—
14	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
15	C₂²×D₁₃₁	C₄²	1	1	yes	characteristic
16	C₂×Dic₁₃₁	C₄²	1	3	yes	—
17	C₄×D₁₃₁	—	2	6	no	—
18	C₁₃₁⋊C₈	—	2	6	no	—
19	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
20	C₈×C₁₃₁	M₁₆	1	12	yes	complemented
21	C₂²×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
22	C₂×D₁₃₁	—	2	3	no	—
23	C₄×C₁₃₁	C₂×M₁₆	1	6	yes	—
24	Dic₁₃₁	—	2	3	no	—
25	C₂×C₁₃₁	C₄×M₁₆	1	3	yes	—
26	D₁₃₁	—	4	1	no	—
27	C₁₃₁	#1455	1	1	yes	characteristic, complemented, Sylow
28	#1455	—	131	1	no	maximal, Sylow
29	C₄×M₁₆	—	131	3	no	—
30	#322	—	131	3	no	—
31	C₈²	D₁₃₁	1	1	yes	characteristic, complemented
32	C₂×M₁₆	—	131	6	no	—
33	C₂×C₄²	—	131	1	no	—
34	#97	—	131	6	no	—
35	C₄×C₈	—	131	3	no	—
36	C₄×C₈	C₂×D₁₃₁	1	3	yes	—
37	C₂²×C₄	—	131	3	no	—
38	M₁₆	—	262	12	no	—
39	C₂×C₈	—	131	6	no	—
40	C₂×C₈	C₄×D₁₃₁	1	6	yes	—
41	C₄²	—	131	3	no	—
42	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
43	C₂³	—	131	1	no	—
44	C₂×C₄	—	262	6	no	—
45	C₂×C₄	—	131	3	no	—
46	C₂×C₄	C₂×C₄×D₁₃₁	1	3	yes	—
47	C₈	—	262	6	no	—
48	C₈	#14523496	1	12	yes	complemented
49	C₂²	—	262	3	no	—
50	C₂²	C₄²×D₁₃₁	1	1	yes	characteristic
51	C₄	—	262	3	no	—
52	C₄	C₂×#14523496	1	6	yes	—
53	C₂	—	524	1	no	—
54	C₂	C₄×#14523496	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂×C₄²
Fitting	C₈²×C₁₃₁	C₂
prime radical	C₂²×D₁₃₁	C₄²
cosocle	C₄²×C₁₃₁	C₂³
hypercenter	C₈²	D₁₃₁
hypocenter	C₁₃₁	#1455
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. #22553033	C₂	semidirect
3. C₈²×C₁₃₁	C₂	semidirect
20. C₈×C₁₃₁	M₁₆	semidirect
27. C₁₃₁	#1455	semidirect
31. C₈²	D₁₃₁	semidirect
48. C₈	#14523496	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523496	2
#22553033	2
C₈²×C₁₃₁	2
#1455	131