Subgroups — #34176171

Order 20224.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761600	C₂	1	2	yes	maximal
2	C₂×C₆₄×C₇₉	C₂	1	1	yes	maximal, characteristic
3	#19098389	—	2	2	no	—
4	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
5	C₆₄×C₇₉	C₄	1	2	yes	complemented
6	#14579095	—	4	2	no	—
7	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
8	C₃₂×C₇₉	C₂×C₄	1	1	yes	characteristic
9	C₃₂×C₇₉	Q₈	1	1	yes	characteristic
10	C₄⋊Dic₇₉	—	8	2	no	—
11	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
12	C₁₆×C₇₉	C₄⋊C₄	1	1	yes	characteristic
13	C₁₆×C₇₉	Q₁₆	1	1	yes	characteristic
14	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
15	C₂×Dic₇₉	—	16	2	no	—
16	C₈×C₇₉	#107	1	1	yes	characteristic
17	C₈×C₇₉	Q₃₂	1	1	yes	characteristic
18	C₂²×C₇₉	D₃₂	1	1	yes	characteristic
19	C₄×C₇₉	#366	1	1	yes	characteristic
20	C₄×C₇₉	Q₆₄	1	1	yes	characteristic
21	Dic₇₉	—	32	2	no	—
22	#7546	—	79	1	no	maximal, Sylow
23	C₂×C₇₉	#1423	1	1	yes	characteristic
24	C₂×C₇₉	SD₁₂₈	1	2	yes	—
25	C₂×C₆₄	D₇₉	1	1	yes	characteristic
26	#1423	—	79	2	no	—
27	C₇₉	#7546	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
29	#366	—	158	2	no	—
30	C₆₄	Dic₇₉	1	2	yes	complemented
31	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
32	#107	—	316	2	no	—
33	C₃₂	Dic₁₅₈	1	1	yes	characteristic
34	C₃₂	C₂×Dic₇₉	1	1	yes	characteristic
35	C₂×C₈	D₆₃₂	1	1	yes	characteristic
36	C₄⋊C₄	—	632	2	no	—
37	C₁₆	C₄⋊Dic₇₉	1	1	yes	characteristic
38	C₁₆	Dic₃₁₆	1	1	yes	characteristic
39	C₂×C₄	—	1264	2	no	—
40	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
41	C₈	#14579095	1	1	yes	characteristic
42	C₈	Dic₆₃₂	1	1	yes	characteristic
43	C₂²	D₂₅₂₈	1	1	yes	characteristic
44	C₄	—	2528	2	no	—
45	C₄	#19098389	1	1	yes	characteristic
46	C₄	Dic₁₂₆₄	1	1	yes	characteristic
47	C₂	#23761600	1	1	yes	characteristic
48	C₂	#23761616	1	2	yes	—

	subgroup	G/H
center	C₂²	D₂₅₂₈
commutator	C₃₂×C₇₉	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₉
Fitting	C₂×C₆₄×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	D₃₂
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	C₂×C₆₄	D₇₉
hypocenter	C₇₉	#7546
LCS	G ▷ C₃₂×C₇₉ ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
5. C₆₄×C₇₉	C₄	semidirect
27. C₇₉	#7546	semidirect
30. C₆₄	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23761600	2
C₂×C₆₄×C₇₉	2
#7546	79