Subgroups — #14580378

Order 2624.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050123	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₄₁	C₂	1	1	yes	maximal, characteristic
3	#13050098	C₂	1	2	yes	maximal, complemented
4	#13050096	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₄₁	C₂²	1	1	yes	characteristic
7	C₂²×Dic₄₁	—	2	1	no	—
8	M₁₆×C₄₁	C₄	1	4	yes	complemented
9	C₂×C₈×C₄₁	C₂²	1	2	yes	—
10	C₂²⋊Dic₄₁	—	2	1	no	—
11	C₄⋊Dic₄₁	—	2	1	no	—
12	C₄⋊Dic₄₁	C₂²	1	2	yes	—
13	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
15	C₄×Dic₄₁	—	2	1	no	—
16	C₂³×C₄₁	D₄	1	1	yes	characteristic
17	C₂×C₄×C₄₁	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₄₁	Q₈	1	2	yes	—
20	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₄₁	—	2	1	no	—
22	C₂×Dic₄₁	—	2	1	no	—
23	C₂×Dic₄₁	—	2	2	no	—
24	C₂×Dic₄₁	—	4	2	no	—
25	C₈×C₄₁	C₂×C₄	1	4	yes	complemented
26	C₂²×C₄₁	—	2	1	no	—
27	C₂²×C₄₁	—	2	1	no	—
28	C₂²×C₄₁	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₄₁	C₄⋊C₄	1	2	yes	—
31	C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
33	Dic₄₁	—	4	1	no	—
34	Dic₄₁	—	4	2	no	—
35	Dic₄₁	—	4	1	no	—
36	C₂×C₄₁	—	2	2	no	—
37	C₂×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂×C₄₁	#137	1	1	yes	characteristic
40	#428	—	41	1	no	maximal, Sylow
41	C₄₁	#428	1	1	yes	characteristic, complemented, Sylow
42	C₂×M₁₆	D₄₁	1	1	yes	characteristic
43	#117	—	41	1	no	—
44	C₂×C₄⋊C₄	—	41	1	no	—
45	#107	—	41	2	no	—
46	#106	—	41	2	no	—
47	C₂²×C₄	—	82	1	no	—
48	C₂²×C₄	C₂×D₄₁	1	1	yes	characteristic
49	M₁₆	Dic₄₁	1	4	yes	complemented
50	C₂×C₈	C₂×D₄₁	1	2	yes	—
51	C₄⋊C₄	—	41	1	no	—
52	C₄⋊C₄	—	41	2	no	—
53	C₄⋊C₄	—	41	1	no	—
54	C₄⋊C₄	—	82	1	no	—
55	C₂²⋊C₄	—	82	1	no	—
56	C₄²	—	82	1	no	—
57	C₂³	D₁₆₄	1	1	yes	characteristic
58	C₂×C₄	—	82	1	no	—
59	C₂×C₄	—	82	1	no	—
60	C₂×C₄	—	82	2	no	—
61	C₂×C₄	—	164	2	no	—
62	C₂×C₄	Dic₈₂	1	2	yes	—
63	C₂×C₄	D₁₆₄	1	1	yes	characteristic
64	C₂×C₄	C₂×Dic₄₁	1	2	yes	—
65	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
66	C₈	C₂×Dic₄₁	1	4	yes	complemented
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	C₄⋊Dic₄₁	1	2	yes	—
70	C₂²	C₂×D₁₆₄	1	1	yes	characteristic
71	C₄	—	164	2	no	—
72	C₄	—	164	1	no	—
73	C₄	—	164	1	no	—
74	C₄	C₄⋊Dic₄₁	1	2	yes	—
75	C₄	C₂×Dic₈₂	1	1	yes	characteristic
76	C₄	C₂²×Dic₄₁	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×C₄⋊Dic₄₁	1	1	yes	characteristic
79	C₂	#13050192	1	1	yes	characteristic
80	C₂	#13050197	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₆₄
commutator	C₄×C₄₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	C₂×M₁₆×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
prime radical	C₂³×C₄₁	D₄
cosocle	C₂×C₄×C₄₁	C₂³
hypercenter	C₂×M₁₆	D₄₁
hypocenter	C₄₁	#428
LCS	G ▷ C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #13050098	C₂	semidirect
4. #13050096	C₂	semidirect
8. M₁₆×C₄₁	C₄	semidirect
25. C₈×C₄₁	C₂×C₄	semidirect
41. C₄₁	#428	semidirect
49. M₁₆	Dic₄₁	semidirect
66. C₈	C₂×Dic₄₁	semidirect

Maximal subgroups

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

subgroup	index
#13050123	2
C₂×C₄⋊Dic₄₁	2
#13050098	2
#13050096	2
C₂×M₁₆×C₄₁	2
#428	41