Subgroups — #30674353

Order 16768.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553203	C₂	1	7	yes	maximal
2	#22553196	C₂	1	7	yes	maximal, complemented
3	C₄³×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₁₃₁⋊C₄	C₂²	1	7	yes	—
5	C₂×(C₂×D₁₃₁)⋊C₄	C₂²	1	7	yes	—
6	C₁₃₁⋊#95	C₂²	1	14	yes	—
7	C₂×C₄²×C₁₃₁	C₂²	1	7	yes	—
8	C₂³×D₁₃₁	C₂³	1	1	yes	characteristic
9	C₂²×Dic₁₃₁	C₂³	1	7	yes	—
10	Dic₁₃₁⋊C₄	—	2	14	no	—
11	(C₂×D₁₃₁)⋊C₄	—	2	14	no	—
12	C₂²×C₄×C₁₃₁	C₂³	1	7	yes	—
13	C₄²×C₁₃₁	—	2	14	no	—
14	C₂²×D₁₃₁	—	2	7	no	—
15	C₂×Dic₁₃₁	—	2	7	no	—
16	C₂×Dic₁₃₁	—	2	14	no	—
17	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
18	C₂×C₄×C₁₃₁	—	2	14	no	—
19	C₂×C₄×C₁₃₁	#41	1	14	yes	—
20	C₂²×C₁₃₁	C₂×#41	1	7	yes	—
21	C₂×D₁₃₁	—	4	7	no	—
22	C₄×C₁₃₁	—	2	14	no	—
23	Dic₁₃₁	—	4	7	no	—
24	C₂×C₁₃₁	#529	1	7	yes	—
25	D₁₃₁	—	8	1	no	—
26	C₁₃₁	#2866	1	1	yes	characteristic, complemented, Sylow
27	#2866	—	131	1	no	maximal, Sylow
28	#388	—	131	7	no	—
29	#387	—	131	7	no	—
30	C₄³	D₁₃₁	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄	—	131	7	no	—
32	C₂×C₂²⋊C₄	—	131	7	no	—
33	C₂×C₄²	C₂×D₁₃₁	1	7	yes	—
34	#95	—	131	14	no	—
35	C₂⁴	—	131	1	no	—
36	C₂²×C₄	—	131	7	no	—
37	C₂²×C₄	C₂²×D₁₃₁	1	7	yes	—
38	C₄⋊C₄	—	262	14	no	—
39	C₂²⋊C₄	—	262	14	no	—
40	C₄²	—	2	14	no	—
41	C₂³	—	262	7	no	—
42	C₂³	C₂³×D₁₃₁	1	1	yes	characteristic
43	C₂×C₄	—	2	14	no	—
44	C₂×C₄	—	262	7	no	—
45	C₂×C₄	—	262	14	no	—
46	C₂×C₄	#14523513	1	14	yes	—
47	C₂²	—	524	7	no	—
48	C₂²	C₂×#14523513	1	7	yes	—
49	C₄	—	2	14	no	—
50	C₄	—	524	7	no	—
51	C₂	—	1048	1	no	—
52	C₂	#22553846	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₁
commutator, socle, cosocle	C₂³×C₁₃₁	C₂⁴
Fitting	C₄³×C₁₃₁	C₂
prime radical	C₂³×D₁₃₁	C₂³
hypercenter	C₄³	D₁₃₁
hypocenter	C₁₃₁	#2866
LCS	G ▷ C₂³×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
2. #22553196	C₂	semidirect
3. C₄³×C₁₃₁	C₂	semidirect
26. C₁₃₁	#2866	semidirect
30. C₄³	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553203	2
#22553196	2
C₄³×C₁₃₁	2
#2866	131