Subgroups — #34175950

Order 20224.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761476	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
3	#19098384	—	2	2	no	—
4	C₂×C₃₂×C₇₉	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₇₉	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
7	#14579101	—	4	2	no	—
8	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₇₉	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
11	C₃₂×C₇₉	—	2	1	no	—
12	C₃₂×C₇₉	C₈	1	2	yes	—
13	C₇₉⋊C₁₆	—	8	2	no	—
14	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
16	C₄²×C₇₉	Q₁₆	1	1	yes	characteristic
17	C₁₆×C₇₉	—	2	1	no	—
18	C₁₆×C₇₉	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₇₉	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₇₉	—	2	1	no	—
21	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
22	C₈×C₇₉	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₇₉	#108	1	1	yes	characteristic
24	C₈×C₇₉	D₁₆	1	1	yes	characteristic
25	C₈×C₇₉	Q₃₂	1	1	yes	characteristic
26	C₂²×C₇₉	#368	1	1	yes	characteristic
27	C₄×C₇₉	—	2	2	no	—
28	C₄×C₇₉	#335	1	1	yes	characteristic
29	C₄×C₇₉	#366	1	1	yes	characteristic
30	#7462	—	79	1	no	maximal, Sylow
31	C₂×C₇₉	—	2	1	no	—
32	C₂×C₇₉	#1371	1	1	yes	characteristic
33	C₄×C₃₂	D₇₉	1	1	yes	characteristic
34	#1373	—	79	2	no	—
35	C₇₉	#7462	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃₂	Dic₇₉	1	1	yes	characteristic
37	C₂×C₃₂	Dic₇₉	1	1	yes	characteristic
38	#364	—	158	2	no	—
39	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
40	M₃₂	—	316	2	no	—
41	C₂×C₁₆	Dic₁₅₈	1	1	yes	characteristic
42	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
43	C₄×C₈	D₃₁₆	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	C₇₉⋊C₈	1	2	yes	—
46	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
47	C₂×C₈	D₆₃₂	1	1	yes	characteristic
48	C₄²	Dic₃₁₆	1	1	yes	characteristic
49	C₁₆	—	2	1	no	—
50	C₁₆	—	632	2	no	—
51	C₁₆	C₂×C₇₉⋊C₈	1	1	yes	characteristic
52	C₁₆	#11929608	1	1	yes	characteristic
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	#14579095	1	1	yes	characteristic
55	C₈	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
56	C₈	#14579097	1	1	yes	characteristic
57	C₈	D₁₂₆₄	1	1	yes	characteristic
58	C₈	Dic₆₃₂	1	1	yes	characteristic
59	C₂²	#19098393	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#19098316	1	1	yes	characteristic
62	C₄	#19098389	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	#23761472	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₂₆₄
commutator	C₁₆×C₇₉	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₇₉
Fitting	C₄×C₃₂×C₇₉	C₂
socle	C₂×C₇₉	#1371
prime radical	C₂²×C₇₉	#368
cosocle	C₄×C₁₆×C₇₉	C₂²
hypercenter	C₄×C₃₂	D₇₉
hypocenter	C₇₉	#7462
LCS	G ▷ C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
35. C₇₉	#7462	semidirect

Maximal subgroups

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

subgroup	index
#23761476	2
C₄×C₃₂×C₇₉	2
#7462	79