Subgroups — #34174885

Order 20224.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761318	C₂	1	2	yes	maximal
2	C₇₉×#1311	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇₉⋊C₁₆	C₄	1	4	yes	—
4	C₄×C₁₆×C₇₉	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₇₉	C₄	1	1	yes	characteristic
6	C₈²×C₇₉	C₂²	1	1	yes	characteristic
7	C₂×C₇₉⋊C₁₆	—	2	4	no	—
8	C₂×C₁₆×C₇₉	C₈	1	2	yes	—
9	C₂×C₁₆×C₇₉	C₈	1	2	yes	—
10	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
11	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
12	C₇₉⋊C₁₆	—	4	4	no	—
13	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
14	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
15	C₂×C₈×C₇₉	M₁₆	1	4	yes	—
16	C₄²×C₇₉	C₄²	1	1	yes	characteristic
17	C₁₆×C₇₉	—	2	2	no	—
18	C₁₆×C₇₉	C₁₆	1	4	yes	complemented
19	C₂×C₄×C₇₉	C₄×C₈	1	1	yes	characteristic
20	C₂×C₄×C₇₉	#97	1	2	yes	—
21	C₈×C₇₉	—	2	4	no	—
22	C₈×C₇₉	C₂×C₁₆	1	2	yes	—
23	C₈×C₇₉	M₃₂	1	2	yes	—
24	C₂²×C₇₉	#322	1	1	yes	characteristic
25	C₄×C₇₉	C₄×C₁₆	1	1	yes	characteristic
26	C₄×C₇₉	#346	1	1	yes	characteristic
27	C₄×C₇₉	#347	1	4	yes	—
28	#7053	—	79	1	no	maximal, Sylow
29	C₂×C₇₉	#1312	1	1	yes	characteristic
30	C₂×C₇₉	#1313	1	2	yes	—
31	#1312	—	79	2	no	—
32	#1311	D₇₉	1	1	yes	characteristic
33	C₇₉	#7053	1	1	yes	characteristic, complemented, Sylow
34	C₄×C₁₆	—	79	4	no	—
35	C₄×C₁₆	Dic₇₉	1	1	yes	characteristic
36	C₄×C₁₆	Dic₇₉	1	1	yes	characteristic
37	C₈²	C₂×D₇₉	1	1	yes	characteristic
38	C₂×C₁₆	—	158	4	no	—
39	C₂×C₁₆	C₇₉⋊C₈	1	2	yes	—
40	C₂×C₁₆	C₇₉⋊C₈	1	2	yes	—
41	C₄×C₈	C₄×D₇₉	1	2	yes	—
42	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
43	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
44	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
45	C₂×C₈	#11929607	1	4	yes	—
46	C₄²	C₄×Dic₇₉	1	1	yes	characteristic
47	C₁₆	—	2	2	no	—
48	C₁₆	—	316	4	no	—
49	C₁₆	C₇₉⋊C₁₆	1	4	yes	complemented
50	C₂×C₄	C₄×C₇₉⋊C₈	1	1	yes	characteristic
51	C₂×C₄	#14579071	1	2	yes	—
52	C₈	—	2	4	no	—
53	C₈	C₂×C₇₉⋊C₁₆	1	2	yes	—
54	C₈	#14579101	1	2	yes	—
55	C₂²	#19098284	1	1	yes	characteristic
56	C₄	C₄×C₇₉⋊C₁₆	1	1	yes	characteristic
57	C₄	#19098339	1	1	yes	characteristic
58	C₄	#19098340	1	4	yes	—
59	C₂	#23761318	1	1	yes	characteristic
60	C₂	#23761320	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₇₉
commutator	C₄×C₇₉	C₄×C₁₆
Frattini	C₈²	C₂×D₇₉
Fitting	C₇₉×#1311	C₂
socle, prime radical	C₂²×C₇₉	#322
cosocle	C₈²×C₇₉	C₂²
hypercenter	#1311	D₇₉
hypocenter	C₇₉	#7053
LCS	G ▷ C₄×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ #1311

Complements

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

N	H ≅ G/N	type
18. C₁₆×C₇₉	C₁₆	semidirect
33. C₇₉	#7053	semidirect
49. C₁₆	C₇₉⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#23761318	2
C₇₉×#1311	2
#7053	79