Subgroups — #16840802

Order 3904.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#6923	C₂	1	2	yes	maximal
2	#14519845	C₂	1	1	yes	maximal, characteristic
3	#14519840	C₂	1	2	yes	maximal
4	#14519831	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×D₆₁	C₂²	1	2	yes	—
7	C₄⋊#6923	—	2	2	no	—
8	C₄⋊#6923	C₂²	1	2	yes	—
9	C₄×#6923	—	2	2	no	—
10	C₄×#6923	C₂²	1	1	yes	characteristic
11	D₆₁⋊C₈	C₂²	1	1	yes	characteristic
12	Q₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
13	#11758202	C₄	1	2	yes	complemented
14	Dic₂₄₄	C₄	1	1	yes	characteristic, complemented
15	C₈×D₆₁	C₂²	1	1	yes	characteristic
16	C₂×#6923	—	2	2	no	—
17	C₂×#6923	D₄	1	2	yes	—
18	Q₈×C₆₁	C₂×C₄	1	2	yes	—
19	C₄×D₆₁	—	2	2	no	—
20	C₄×D₆₁	C₂³	1	1	yes	characteristic
21	Dic₁₂₂	—	2	2	no	—
22	Dic₁₂₂	C₂×C₄	1	2	yes	—
23	#92420	—	2	1	no	—
24	C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	#92418	C₂×C₄	1	1	yes	characteristic
26	C₂×D₆₁	C₂×D₄	1	1	yes	characteristic
27	#6923	—	2	2	no	—
28	#6923	—	4	2	no	—
29	C₄×C₆₁	—	2	2	no	—
30	C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	Dic₆₁	—	2	2	no	—
32	Dic₆₁	#41	1	1	yes	characteristic
33	C₂×C₆₁	C₄×D₄	1	1	yes	characteristic
34	D₆₁	#137	1	2	yes	—
35	#441	—	61	1	no	maximal, Sylow
36	C₆₁	#441	1	1	yes	characteristic, complemented, Sylow
37	C₂×Q₁₆	—	61	1	no	—
38	C₄×Q₈	—	61	2	no	—
39	#106	—	61	1	no	—
40	#103	—	61	2	no	—
41	#97	—	61	1	no	—
42	C₂×Q₈	—	61	2	no	—
43	Q₁₆	—	61	2	no	—
44	Q₁₆	—	61	1	no	—
45	Q₁₆	#6923	1	1	yes	characteristic, complemented
46	C₂×C₈	—	61	1	no	—
47	C₂×C₈	—	61	1	no	—
48	C₄⋊C₄	—	61	2	no	—
49	C₄⋊C₄	—	122	2	no	—
50	C₄²	—	61	1	no	—
51	C₄²	—	122	2	no	—
52	Q₈	—	61	2	no	—
53	Q₈	—	122	2	no	—
54	Q₈	C₂×#6923	1	2	yes	—
55	C₂×C₄	—	61	1	no	—
56	C₂×C₄	—	61	2	no	—
57	C₂×C₄	—	122	2	no	—
58	C₂×C₄	—	122	2	no	—
59	C₈	—	61	1	no	—
60	C₈	—	122	1	no	—
61	C₈	C₂×#6923	1	1	yes	characteristic
62	C₂²	—	61	1	no	—
63	C₄	—	2	2	no	—
64	C₄	—	61	1	no	—
65	C₄	—	122	2	no	—
66	C₄	—	122	2	no	—
67	C₄	—	244	2	no	—
68	C₄	C₂²×#6923	1	1	yes	characteristic
69	C₂	—	61	2	no	—
70	C₂	D₄×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#6923
commutator	C₄×C₆₁	C₂²×C₄
Frattini	C₄	C₂²×#6923
Fitting	Q₁₆×C₆₁	C₄
socle	C₂×C₆₁	C₄×D₄
prime radical	C₂×D₆₁	C₂×D₄
cosocle	C₄×D₆₁	C₂³
hypercenter	Q₁₆	#6923
hypocenter	C₆₁	#441
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
12. Q₁₆×C₆₁	C₄	semidirect
13. #11758202	C₄	semidirect
14. Dic₂₄₄	C₄	semidirect
36. C₆₁	#441	semidirect
45. Q₁₆	#6923	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6923	2
#14519845	2
#14519840	2
#14519831	2
Q₁₆×D₆₁	2
#441	61