Subgroups — #14580449

Order 2624.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₂₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₆₄	C₂	1	2	yes	maximal, complemented
3	#13050096	C₂	1	1	yes	maximal, characteristic, complemented
4	#13050086	C₂	1	2	yes	maximal, complemented
5	#97×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₄	C₂²	1	2	yes	—
7	C₂×C₄×D₄₁	—	2	2	no	—
8	C₂×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₄²×C₄₁	C₂²	1	1	yes	characteristic
11	#10620281	—	2	2	no	—
12	C₄⋊Dic₄₁	C₂²	1	2	yes	—
13	D₃₂₈	C₄	1	4	yes	complemented
14	C₂²×D₄₁	—	2	2	no	—
15	C₂×C₄×C₄₁	D₄	1	2	yes	—
16	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
17	C₂×Dic₄₁	—	2	2	no	—
18	D₁₆₄	—	2	2	no	—
19	D₁₆₄	C₂×C₄	1	4	yes	complemented
20	C₄×D₄₁	—	4	4	no	—
21	C₈×C₄₁	—	2	1	no	—
22	C₈×C₄₁	C₂×C₄	1	2	yes	complemented
23	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
24	C₂×D₄₁	—	2	4	no	—
25	C₂×D₄₁	—	4	2	no	—
26	C₂×D₄₁	—	4	2	no	—
27	C₄×C₄₁	—	2	2	no	—
28	C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₄₁	#41	1	1	yes	characteristic
30	Dic₄₁	—	4	2	no	—
31	C₂×C₄₁	C₄×D₄	1	1	yes	characteristic
32	C₂×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
33	D₄₁	—	4	4	no	—
34	#442	—	41	1	no	maximal, Sylow
35	C₄₁	#442	1	1	yes	characteristic, complemented, Sylow
36	C₂×D₈	—	41	1	no	—
37	C₄×D₄	—	41	2	no	—
38	#106	—	41	1	no	—
39	#102	—	41	2	no	—
40	#97	D₄₁	1	1	yes	characteristic, complemented
41	C₂×D₄	—	41	2	no	—
42	C₂²×C₄	—	82	2	no	—
43	D₈	—	41	4	no	—
44	C₂×C₈	C₂×D₄₁	1	1	yes	characteristic
45	C₂×C₈	C₂×D₄₁	1	1	yes	characteristic
46	C₄⋊C₄	—	41	2	no	—
47	C₂²⋊C₄	—	82	2	no	—
48	C₄²	C₂×D₄₁	1	1	yes	characteristic
49	C₂³	—	82	2	no	—
50	D₄	—	41	4	no	—
51	D₄	—	82	2	no	—
52	C₂×C₄	—	82	2	no	—
53	C₂×C₄	—	164	4	no	—
54	C₂×C₄	D₁₆₄	1	2	yes	—
55	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	C₄×D₄₁	1	2	yes	complemented
58	C₂²	—	82	4	no	—
59	C₂²	—	164	2	no	—
60	C₂²	—	164	2	no	—
61	C₂²	C₂×D₁₆₄	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	164	2	no	—
64	C₄	C₂×C₄×D₄₁	1	1	yes	characteristic
65	C₄	#10620306	1	1	yes	characteristic
66	C₂	—	164	4	no	—
67	C₂	C₄×D₁₆₄	1	1	yes	characteristic
68	C₂	#13050192	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₆₄
commutator	C₄×C₄₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	#97×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
prime radical	C₂×D₃₂₈	C₂
cosocle	C₂×C₄×C₄₁	C₂³
hypercenter	#97	D₄₁
hypocenter	C₄₁	#442
LCS	G ▷ C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. C₄×D₁₆₄	C₂	semidirect
3. #13050096	C₂	semidirect
4. #13050086	C₂	semidirect
5. #97×C₄₁	C₂	semidirect
13. D₃₂₈	C₄	semidirect
19. D₁₆₄	C₂×C₄	semidirect
22. C₈×C₄₁	C₂×C₄	semidirect
35. C₄₁	#442	semidirect
40. #97	D₄₁	semidirect
57. C₈	C₄×D₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₂₈	2
C₄×D₁₆₄	2
#13050096	2
#13050086	2
#97×C₄₁	2
#442	41