Subgroups — #34175604

Order 20224.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761536	C₂	1	1	yes	maximal, characteristic
2	#23761320	C₂	1	1	yes	maximal, characteristic
3	C₇₉×#1398	C₂	1	1	yes	maximal, characteristic
4	#19098397	C₄	1	2	yes	—
5	#19098339	C₄	1	1	yes	characteristic
6	#19098284	C₄	1	1	yes	characteristic
7	M₆₄×C₇₉	C₄	1	2	yes	—
8	#346×C₇₉	C₂²	1	1	yes	characteristic
9	C₂×C₇₉⋊C₁₆	—	2	1	no	—
10	C₄×C₇₉⋊C₈	—	2	1	no	—
11	C₇₉⋊C₃₂	—	2	2	no	—
12	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
14	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₇₉	C₈	1	4	yes	complemented
16	C₂×C₇₉⋊C₈	—	4	1	no	—
17	C₇₉⋊C₁₆	—	4	1	no	—
18	C₂×C₈×C₇₉	C₄²	1	1	yes	characteristic
19	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
20	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
21	C₁₆×C₇₉	C₂×C₈	1	2	yes	—
22	C₁₆×C₇₉	M₁₆	1	2	yes	—
23	C₂×C₄×C₇₉	—	2	1	no	—
24	C₂×C₄×C₇₉	#97	1	1	yes	characteristic
25	C₈×C₇₉	—	2	1	no	—
26	C₈×C₇₉	C₄×C₈	1	1	yes	characteristic
27	C₈×C₇₉	#97	1	1	yes	characteristic
28	C₇₉⋊C₈	—	8	1	no	—
29	C₂²×C₇₉	#347	1	1	yes	characteristic
30	C₄×C₇₉	—	4	1	no	—
31	C₄×C₇₉	#322	1	1	yes	characteristic
32	C₄×C₇₉	#347	1	1	yes	characteristic
33	#7333	—	79	1	no	maximal, Sylow
34	C₂×C₇₉	—	2	1	no	—
35	C₂×C₇₉	#1313	1	1	yes	characteristic
36	#1398	—	79	1	no	—
37	#1398	D₇₉	1	1	yes	characteristic
38	#1313	—	79	1	no	—
39	C₇₉	#7333	1	1	yes	characteristic, complemented, Sylow
40	M₆₄	—	79	2	no	—
41	M₆₄	Dic₇₉	1	2	yes	—
42	#346	—	79	1	no	—
43	#346	C₂×D₇₉	1	1	yes	characteristic
44	#322	—	79	1	no	—
45	C₂×C₁₆	—	158	1	no	—
46	C₂×C₁₆	C₄×D₇₉	1	1	yes	characteristic
47	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
48	C₄×C₈	—	158	1	no	—
49	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
50	C₃₂	—	158	2	no	—
51	C₃₂	C₇₉⋊C₈	1	4	yes	complemented
52	C₂×C₈	—	316	1	no	—
53	C₂×C₈	C₄×Dic₇₉	1	1	yes	characteristic
54	C₂×C₈	#11929607	1	1	yes	characteristic
55	C₄²	#11929607	1	1	yes	characteristic
56	C₁₆	—	316	1	no	—
57	C₁₆	C₂×C₇₉⋊C₈	1	2	yes	—
58	C₁₆	#11929607	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	#14579071	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	632	1	no	—
63	C₈	C₄×C₇₉⋊C₈	1	1	yes	characteristic
64	C₈	#14579071	1	1	yes	characteristic
65	C₂²	#19098340	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#19098284	1	1	yes	characteristic
68	C₄	#19098340	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#23761320	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19098284
commutator	C₈×C₇₉	C₄×C₈
Frattini	#346	C₂×D₇₉
Fitting	C₇₉×#1398	C₂
socle	C₂×C₇₉	#1313
prime radical	C₂²×C₇₉	#347
cosocle	#346×C₇₉	C₂²
hypercenter	#1398	D₇₉
hypocenter	C₇₉	#7333
LCS	G ▷ C₈×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ #346 ◁ #1398

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₇₉	C₈	semidirect
39. C₇₉	#7333	semidirect
51. C₃₂	C₇₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#23761536	2
#23761320	2
C₇₉×#1398	2
#7333	79