Subgroups — #34175937

Order 20224.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761598	C₂	1	2	yes	maximal
2	C₇₉×#1311	C₂	1	1	yes	maximal, characteristic
3	#19098397	—	2	2	no	—
4	C₄×C₁₆×C₇₉	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
6	C₈²×C₇₉	C₄	1	1	yes	characteristic
7	C₇₉⋊C₃₂	—	4	2	no	—
8	C₂×C₁₆×C₇₉	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₇₉	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₇₉	Q₈	1	1	yes	characteristic
12	C₄×C₈×C₇₉	—	2	1	no	—
13	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₇₉	—	2	2	no	—
15	C₂×C₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
16	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
18	C₁₆×C₇₉	—	2	1	no	—
19	C₁₆×C₇₉	—	2	1	no	—
20	C₁₆×C₇₉	C₁₆	1	2	yes	—
21	C₁₆×C₇₉	SD₁₆	1	2	yes	—
22	C₂×C₄×C₇₉	—	2	1	no	—
23	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₇₉	—	4	2	no	—
25	C₈×C₇₉	#106	1	1	yes	characteristic
26	C₈×C₇₉	#108	1	1	yes	characteristic
27	C₈×C₇₉	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₇₉	M₃₂	1	1	yes	characteristic
29	C₂²×C₇₉	#364	1	1	yes	characteristic
30	C₄×C₇₉	—	2	2	no	—
31	C₄×C₇₉	#334	1	1	yes	characteristic
32	C₄×C₇₉	C₄⋊C₁₆	1	1	yes	characteristic
33	#7456	—	79	1	no	maximal, Sylow
34	C₂×C₇₉	—	2	1	no	—
35	C₂×C₇₉	#1367	1	1	yes	characteristic
36	#1422	—	79	2	no	—
37	#1311	D₇₉	1	1	yes	characteristic
38	C₇₉	#7456	1	1	yes	characteristic, complemented, Sylow
39	M₆₄	—	158	2	no	—
40	C₄×C₁₆	Dic₇₉	1	1	yes	characteristic
41	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
42	C₈²	Dic₇₉	1	1	yes	characteristic
43	C₂×C₁₆	C₇₉⋊C₈	1	1	yes	characteristic
44	C₂×C₁₆	C₇₉⋊C₈	1	1	yes	characteristic
45	C₂×C₁₆	Dic₁₅₈	1	1	yes	characteristic
46	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
49	C₃₂	—	316	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	C₄⋊Dic₇₉	1	1	yes	characteristic
52	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
53	C₄²	#11929608	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	2	1	no	—
56	C₁₆	C₇₉⋊C₁₆	1	2	yes	—
57	C₁₆	#11929613	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
60	C₈	—	4	2	no	—
61	C₈	#14579093	1	1	yes	characteristic
62	C₈	#14579097	1	1	yes	characteristic
63	C₈	C₂×C₇₉⋊C₁₆	1	1	yes	characteristic
64	C₈	#14579101	1	1	yes	characteristic
65	C₂²	#19098384	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	#19098314	1	1	yes	characteristic
68	C₄	C₄⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#23761464	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14579093
commutator	C₈×C₇₉	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₇₉
Fitting	C₇₉×#1311	C₂
socle	C₂×C₇₉	#1367
prime radical	C₂²×C₇₉	#364
cosocle	C₄×C₁₆×C₇₉	C₂²
hypercenter	#1311	D₇₉
hypocenter	C₇₉	#7456
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ #1311

Complements

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

N	H ≅ G/N	type
38. C₇₉	#7456	semidirect

Maximal subgroups

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

subgroup	index
#23761598	2
C₇₉×#1311	2
#7456	79