Subgroups — #14580779

Order 2624.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050186	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₃₂₈	C₂	1	2	yes	maximal, complemented
4	#13050109	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#10620306	—	2	1	no	—
7	C₂×Dic₈₂	—	2	1	no	—
8	C₂×C₈×C₄₁	C₂²	1	1	yes	characteristic
9	Dic₁₆₄	—	2	1	no	—
10	Dic₁₆₄	C₂²	1	2	yes	—
11	Dic₁₆₄	C₂²	1	1	yes	characteristic
12	D₃₂₈	C₂²	1	1	yes	characteristic
13	#10620273	—	2	1	no	—
14	C₁₆×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
16	C₂²⋊D₄₁	—	4	1	no	—
17	C₂×Dic₄₁	—	4	1	no	—
18	D₁₆₄	—	2	1	no	—
19	C₄×D₄₁	—	4	1	no	—
20	Dic₈₂	—	2	1	no	—
21	Dic₈₂	—	2	2	no	—
22	Dic₈₂	—	4	1	no	—
23	C₈×C₄₁	D₄	1	1	yes	characteristic
24	C₈×C₄₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄₁	D₈	1	1	yes	characteristic
26	C₂×D₄₁	—	4	1	no	—
27	C₄×C₄₁	D₈	1	1	yes	characteristic
28	C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
29	Dic₄₁	—	4	2	no	—
30	Dic₄₁	—	4	1	no	—
31	C₂×C₄₁	—	2	1	no	—
32	C₂×C₄₁	C₂×D₈	1	1	yes	characteristic
33	D₄₁	—	8	1	no	—
34	#510	—	41	1	no	maximal, Sylow
35	C₄₁	#510	1	1	yes	characteristic, complemented, Sylow
36	#135	—	41	1	no	—
37	C₂×Q₁₆	—	41	1	no	—
38	Q₃₂	—	41	2	no	—
39	SD₃₂	—	41	2	no	—
40	M₃₂	D₄₁	1	1	yes	characteristic, complemented
41	#41	—	82	1	no	—
42	C₂×Q₈	—	82	1	no	—
43	Q₁₆	—	41	2	no	—
44	Q₁₆	—	41	1	no	—
45	Q₁₆	—	82	1	no	—
46	SD₁₆	—	82	1	no	—
47	D₈	—	41	1	no	—
48	C₂×C₈	C₂×D₄₁	1	1	yes	characteristic
49	C₁₆	C₂×D₄₁	1	2	yes	—
50	Q₈	—	82	1	no	—
51	Q₈	—	82	2	no	—
52	Q₈	—	164	1	no	—
53	D₄	—	82	1	no	—
54	D₄	—	164	1	no	—
55	C₂×C₄	—	164	1	no	—
56	C₂×C₄	—	164	1	no	—
57	C₂×C₄	D₁₆₄	1	1	yes	characteristic
58	C₈	D₁₆₄	1	1	yes	characteristic
59	C₈	C₂²×D₄₁	1	1	yes	characteristic
60	C₂²	—	164	1	no	—
61	C₂²	D₃₂₈	1	1	yes	characteristic
62	C₄	—	164	1	no	—
63	C₄	—	164	2	no	—
64	C₄	D₃₂₈	1	1	yes	characteristic
65	C₄	C₂×D₁₆₄	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	328	1	no	—
68	C₂	C₂×D₃₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃₂₈
commutator, cosocle	C₈×C₄₁	C₂³
Frattini	C₈	C₂²×D₄₁
Fitting	M₃₂×C₄₁	C₂
socle	C₂×C₄₁	C₂×D₈
prime radical	#13050186	C₂
hypercenter	M₃₂	D₄₁
hypocenter	C₄₁	#510
LCS	G ▷ C₈×C₄₁ ▷ C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₆₄	C₂	semidirect
3. Dic₃₂₈	C₂	semidirect
4. #13050109	C₂	semidirect
5. M₃₂×C₄₁	C₂	semidirect
35. C₄₁	#510	semidirect
40. M₃₂	D₄₁	semidirect

Maximal subgroups

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

subgroup	index
#13050186	2
C₂×Dic₁₆₄	2
Dic₃₂₈	2
#13050109	2
M₃₂×C₄₁	2
#510	41