Subgroups — #34176166

Order 20224.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761611	C₂	1	1	yes	maximal, characteristic
2	C₃₂×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	C₇₉×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₉⋊C₃₂	C₄	1	1	yes	characteristic
5	C₁₆×Dic₇₉	C₄	1	1	yes	characteristic
6	C₇₉⋊C₆₄	—	2	1	no	—
7	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
8	C₆₄×C₇₉	—	2	1	no	—
9	C₂×C₇₉⋊C₁₆	C₈	1	1	yes	characteristic
10	C₈×Dic₇₉	C₈	1	1	yes	characteristic
11	C₇₉⋊C₃₂	—	2	1	no	—
12	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₇₉	D₄	1	1	yes	characteristic
14	C₃₂×C₇₉	Q₈	1	1	yes	characteristic
15	C₂×C₇₉⋊C₈	C₁₆	1	1	yes	characteristic
16	C₄×Dic₇₉	C₁₆	1	1	yes	characteristic
17	C₇₉⋊C₁₆	—	2	1	no	—
18	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₇₉	C₄⋊C₄	1	1	yes	characteristic
20	C₁₆×C₇₉	M₁₆	1	1	yes	characteristic
21	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
22	C₂×Dic₇₉	—	2	1	no	—
23	C₈×C₇₉	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₇₉	M₃₂	1	1	yes	characteristic
25	C₇₉⋊C₈	C₃₂	1	2	yes	—
26	C₂²×C₇₉	M₆₄	1	1	yes	characteristic
27	C₄×C₇₉	C₄⋊C₁₆	1	1	yes	characteristic
28	C₄×C₇₉	C₂×C₃₂	1	1	yes	characteristic
29	Dic₇₉	—	2	2	no	—
30	#7544	—	79	1	no	maximal, Sylow
31	C₂×C₇₉	—	2	1	no	—
32	C₂×C₇₉	C₄⋊C₃₂	1	1	yes	characteristic
33	M₁₂₈	—	79	1	no	—
34	M₁₂₈	D₇₉	1	1	yes	characteristic
35	C₄×C₃₂	—	79	1	no	—
36	C₇₉	#7544	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃₂	—	79	1	no	—
38	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
39	C₄×C₁₆	—	79	1	no	—
40	C₆₄	—	2	1	no	—
41	C₆₄	—	158	1	no	—
42	C₂×C₁₆	—	79	1	no	—
43	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
44	C₄×C₈	—	79	1	no	—
45	C₃₂	—	158	1	no	—
46	C₃₂	Dic₁₅₈	1	1	yes	characteristic
47	C₃₂	C₂²⋊D₇₉	1	1	yes	characteristic
48	C₂×C₈	—	79	1	no	—
49	C₂×C₈	C₈×D₇₉	1	1	yes	characteristic
50	C₄²	—	79	1	no	—
51	C₁₆	—	158	1	no	—
52	C₁₆	Dic₇₉⋊C₄	1	1	yes	characteristic
53	C₁₆	#11929607	1	1	yes	characteristic
54	C₂×C₄	—	158	1	no	—
55	C₂×C₄	C₁₆×D₇₉	1	1	yes	characteristic
56	C₈	—	79	2	no	—
57	C₈	Dic₇₉⋊C₈	1	1	yes	characteristic
58	C₈	#14579100	1	1	yes	characteristic
59	C₂²	#19098396	1	1	yes	characteristic
60	C₄	—	158	2	no	—
61	C₄	Dic₇₉⋊C₁₆	1	1	yes	characteristic
62	C₄	C₃₂×D₇₉	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	Dic₇₉⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂²⋊D₇₉
commutator	C₄×C₇₉	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₇₉
Fitting	C₇₉×M₁₂₈	C₂
socle	C₂×C₇₉	C₄⋊C₃₂
prime radical	C₂²×C₇₉	M₆₄
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	M₁₂₈	D₇₉
hypocenter	C₇₉	#7544
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
36. C₇₉	#7544	semidirect

Maximal subgroups

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

subgroup	index
#23761611	2
C₃₂×Dic₇₉	2
C₇₉×M₁₂₈	2
#7544	79