Subgroups — #34176156

Order 20224.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761611	C₂	1	1	yes	maximal, characteristic, complemented
2	#23761602	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₉×#2262	C₂	1	1	yes	maximal, characteristic
4	#19098389	—	2	1	no	—
5	C₇₉⋊C₆₄	—	2	1	no	—
6	#508×C₇₉	—	2	1	no	—
7	Q₆₄×C₇₉	C₄	1	1	yes	characteristic, complemented
8	SD₆₄×C₇₉	—	2	1	no	—
9	D₃₂×C₇₉	C₄	1	1	yes	characteristic, complemented
10	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
11	#14579095	—	4	1	no	—
12	#135×C₇₉	—	4	1	no	—
13	Q₃₂×C₇₉	—	2	1	no	—
14	SD₃₂×C₇₉	—	4	1	no	—
15	D₁₆×C₇₉	—	2	1	no	—
16	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
17	C₃₂×C₇₉	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₇₉	D₄	1	1	yes	characteristic
19	C₄⋊Dic₇₉	—	8	1	no	—
20	#41×C₇₉	—	8	1	no	—
21	Q₁₆×C₇₉	—	4	1	no	—
22	SD₁₆×C₇₉	—	8	1	no	—
23	D₈×C₇₉	—	4	1	no	—
24	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
25	C₁₆×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₇₉	SD₁₆	1	1	yes	characteristic
27	Q₈×C₇₉	—	8	1	no	—
28	D₄×C₇₉	—	8	1	no	—
29	D₄×C₇₉	—	16	1	no	—
30	C₂×C₄×C₇₉	—	16	1	no	—
31	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
32	C₂×Dic₇₉	—	16	1	no	—
33	C₈×C₇₉	#102	1	1	yes	characteristic
34	C₈×C₇₉	SD₃₂	1	1	yes	characteristic
35	C₂²×C₇₉	—	16	1	no	—
36	C₂²×C₇₉	D₃₂	1	1	yes	characteristic
37	C₄×C₇₉	—	16	1	no	—
38	C₄×C₇₉	#357	1	1	yes	characteristic
39	C₄×C₇₉	SD₆₄	1	1	yes	characteristic
40	Dic₇₉	—	32	1	no	—
41	#7540	—	79	1	no	maximal, Sylow
42	C₂×C₇₉	—	2	1	no	—
43	C₂×C₇₉	—	32	1	no	—
44	C₂×C₇₉	#1415	1	1	yes	characteristic
45	#2262	D₇₉	1	1	yes	characteristic
46	M₁₂₈	—	79	1	no	—
47	#1424	—	79	1	no	—
48	C₇₉	#7540	1	1	yes	characteristic, complemented, Sylow
49	#508	—	2	1	no	—
50	Q₆₄	Dic₇₉	1	1	yes	characteristic, complemented
51	SD₆₄	—	2	1	no	—
52	D₃₂	Dic₇₉	1	1	yes	characteristic, complemented
53	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
54	#366	—	158	1	no	—
55	C₆₄	—	158	1	no	—
56	#135	—	4	1	no	—
57	Q₃₂	—	2	1	no	—
58	SD₃₂	—	4	1	no	—
59	D₁₆	—	2	1	no	—
60	C₂×C₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
61	#107	—	316	1	no	—
62	C₃₂	C₂×Dic₇₉	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₇₉	1	1	yes	characteristic
64	#41	—	8	1	no	—
65	Q₁₆	—	4	1	no	—
66	SD₁₆	—	8	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	#11929609	1	1	yes	characteristic
69	C₄⋊C₄	—	632	1	no	—
70	C₁₆	C₂²⋊Dic₇₉	1	1	yes	characteristic
71	C₁₆	#11929612	1	1	yes	characteristic
72	Q₈	—	8	1	no	—
73	D₄	—	8	1	no	—
74	D₄	—	16	1	no	—
75	C₂×C₄	—	16	1	no	—
76	C₂×C₄	—	1264	1	no	—
77	C₂×C₄	#14579102	1	1	yes	characteristic
78	C₈	#14579082	1	1	yes	characteristic
79	C₈	#14579105	1	1	yes	characteristic
80	C₂²	—	16	1	no	—
81	C₂²	#19098398	1	1	yes	characteristic
82	C₄	—	16	1	no	—
83	C₄	—	2528	1	no	—
84	C₄	#19098364	1	1	yes	characteristic
85	C₄	#19098401	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	32	1	no	—
88	C₂	#23761578	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23761578
commutator	C₃₂×C₇₉	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₉
Fitting, prime radical	C₇₉×#2262	C₂
socle	C₂×C₇₉	#1415
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	#2262	D₇₉
hypocenter	C₇₉	#7540
LCS	G ▷ C₃₂×C₇₉ ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #2262

Complements

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

N	H ≅ G/N	type
1. #23761611	C₂	semidirect
2. #23761602	C₂	semidirect
7. Q₆₄×C₇₉	C₄	semidirect
9. D₃₂×C₇₉	C₄	semidirect
48. C₇₉	#7540	semidirect
50. Q₆₄	Dic₇₉	semidirect
52. D₃₂	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23761611	2
#23761602	2
C₇₉×#2262	2
#7540	79