Subgroups — #34209296

Order 20224.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765675	C₂	1	2	yes	maximal
2	#23761602	C₂	1	4	yes	maximal
3	C₄×C₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098968	—	2	2	no	—
5	#19098389	—	2	2	no	—
6	#19098389	C₂²	1	4	yes	—
7	C₂×C₃₂×C₇₉	C₂²	1	2	yes	—
8	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
9	#14579161	—	4	2	no	—
10	#14579095	—	2	4	no	—
11	#14579095	—	4	2	no	—
12	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
14	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
15	C₃₂×C₇₉	Q₈	1	4	yes	complemented
16	C₂×Dic₁₅₈	—	8	2	no	—
17	C₄⋊Dic₇₉	—	4	4	no	—
18	C₄⋊Dic₇₉	—	8	2	no	—
19	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
20	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
21	C₄²×C₇₉	D₈	1	1	yes	characteristic
22	C₁₆×C₇₉	Q₁₆	1	2	yes	—
23	C₁₆×C₇₉	C₂×Q₈	1	2	yes	—
24	C₂×C₄×C₇₉	D₁₆	1	2	yes	—
25	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
26	C₂×Dic₇₉	—	8	4	no	—
27	Dic₁₅₈	—	16	4	no	—
28	C₈×C₇₉	Q₃₂	1	2	yes	—
29	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
30	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
31	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
32	C₄×C₇₉	SD₆₄	1	4	yes	—
33	C₄×C₇₉	#500	1	1	yes	characteristic
34	C₄×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
35	Dic₇₉	—	16	4	no	—
36	#13733	—	79	1	no	maximal, Sylow
37	C₂×C₇₉	#2252	1	1	yes	characteristic
38	C₂×C₇₉	C₂×SD₆₄	1	2	yes	—
39	#2252	—	79	2	no	—
40	#1424	—	79	4	no	—
41	C₄×C₃₂	D₇₉	1	1	yes	characteristic
42	C₇₉	#13733	1	1	yes	characteristic, complemented, Sylow
43	#500	—	158	2	no	—
44	C₂×C₃₂	C₂×D₇₉	1	2	yes	—
45	#366	—	79	4	no	—
46	#366	—	158	2	no	—
47	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
48	C₄⋊Q₈	—	316	2	no	—
49	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
50	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
51	#107	—	316	2	no	—
52	#107	—	158	4	no	—
53	C₄×C₈	D₃₁₆	1	1	yes	characteristic
54	C₃₂	Dic₁₅₈	1	4	yes	complemented
55	C₂×Q₈	—	632	2	no	—
56	C₂×C₈	D₆₃₂	1	1	yes	characteristic
57	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
58	C₄⋊C₄	—	316	4	no	—
59	C₄⋊C₄	—	632	2	no	—
60	C₄²	D₆₃₂	1	1	yes	characteristic
61	C₁₆	Dic₃₁₆	1	2	yes	—
62	C₁₆	C₂×Dic₁₅₈	1	2	yes	—
63	Q₈	—	1264	4	no	—
64	C₂×C₄	—	632	4	no	—
65	C₂×C₄	D₁₂₆₄	1	2	yes	—
66	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
67	C₈	Dic₆₃₂	1	2	yes	—
68	C₈	#14579161	1	1	yes	characteristic
69	C₈	C₂×Dic₃₁₆	1	1	yes	characteristic
70	C₂²	C₂×D₁₂₆₄	1	1	yes	characteristic
71	C₄	—	1264	4	no	—
72	C₄	#19098402	1	4	yes	—
73	C₄	#19098968	1	1	yes	characteristic
74	C₄	C₂×Dic₆₃₂	1	1	yes	characteristic
75	C₂	#23765675	1	1	yes	characteristic
76	C₂	C₂×#19098402	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₂₆₄
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₄×C₃₂×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	C₂×D₁₆
hypercenter	C₄×C₃₂	D₇₉
hypocenter	C₇₉	#13733
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₇₉	Q₈	semidirect
42. C₇₉	#13733	semidirect
54. C₃₂	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23765675	2
#23761602	2
C₄×C₃₂×C₇₉	2
#13733	79