Subgroups — #34209289

Order 20224.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765675	C₂	1	2	yes	maximal
2	#23761600	C₂	1	4	yes	maximal
3	C₄×C₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
4	#19098968	—	2	2	no	—
5	#19098389	—	2	2	no	—
6	#19098389	C₂²	1	4	yes	—
7	C₂×C₃₂×C₇₉	C₂²	1	2	yes	—
8	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
9	#14579161	—	4	2	no	—
10	#14579095	—	2	4	no	—
11	#14579095	—	4	2	no	—
12	C₂×C₁₆×C₇₉	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
14	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
15	C₃₂×C₇₉	Q₈	1	4	yes	complemented
16	C₂×Dic₁₅₈	—	8	2	no	—
17	C₄⋊Dic₇₉	—	4	4	no	—
18	C₄⋊Dic₇₉	—	8	2	no	—
19	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
20	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
21	C₄²×C₇₉	D₈	1	1	yes	characteristic
22	C₁₆×C₇₉	Q₁₆	1	2	yes	—
23	C₁₆×C₇₉	C₂×Q₈	1	2	yes	—
24	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
25	C₂×C₄×C₇₉	D₁₆	1	1	yes	characteristic
26	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
27	C₂×Dic₇₉	—	8	4	no	—
28	Dic₁₅₈	—	16	4	no	—
29	C₈×C₇₉	Q₃₂	1	2	yes	—
30	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
31	C₈×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
32	C₂²×C₇₉	C₂×D₁₆	1	1	yes	characteristic
33	C₄×C₇₉	D₃₂	1	2	yes	—
34	C₄×C₇₉	Q₆₄	1	2	yes	—
35	C₄×C₇₉	#500	1	1	yes	characteristic
36	C₄×C₇₉	C₂×Q₃₂	1	1	yes	characteristic
37	Dic₇₉	—	16	4	no	—
38	#13731	—	79	1	no	maximal, Sylow
39	C₂×C₇₉	#2252	1	1	yes	characteristic
40	C₂×C₇₉	C₂×D₃₂	1	1	yes	characteristic
41	C₂×C₇₉	C₂×Q₆₄	1	1	yes	characteristic
42	#2252	—	79	2	no	—
43	#1423	—	79	4	no	—
44	C₄×C₃₂	D₇₉	1	1	yes	characteristic
45	C₇₉	#13731	1	1	yes	characteristic, complemented, Sylow
46	#500	—	158	2	no	—
47	C₂×C₃₂	C₂×D₇₉	1	2	yes	—
48	#366	—	79	4	no	—
49	#366	—	158	2	no	—
50	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
51	C₄⋊Q₈	—	316	2	no	—
52	C₂×C₁₆	D₃₁₆	1	1	yes	characteristic
53	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
54	#107	—	316	2	no	—
55	#107	—	158	4	no	—
56	C₄×C₈	D₃₁₆	1	1	yes	characteristic
57	C₃₂	Dic₁₅₈	1	4	yes	complemented
58	C₂×Q₈	—	632	2	no	—
59	C₂×C₈	D₆₃₂	1	1	yes	characteristic
60	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
61	C₄⋊C₄	—	316	4	no	—
62	C₄⋊C₄	—	632	2	no	—
63	C₄²	D₆₃₂	1	1	yes	characteristic
64	C₁₆	Dic₃₁₆	1	2	yes	—
65	C₁₆	C₂×Dic₁₅₈	1	2	yes	—
66	Q₈	—	1264	4	no	—
67	C₂×C₄	—	632	4	no	—
68	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
69	C₂×C₄	D₁₂₆₄	1	1	yes	characteristic
70	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
71	C₈	Dic₆₃₂	1	2	yes	—
72	C₈	#14579161	1	1	yes	characteristic
73	C₈	C₂×Dic₃₁₆	1	1	yes	characteristic
74	C₂²	C₂×D₁₂₆₄	1	1	yes	characteristic
75	C₄	—	1264	4	no	—
76	C₄	D₂₅₂₈	1	2	yes	—
77	C₄	Dic₁₂₆₄	1	2	yes	—
78	C₄	#19098968	1	1	yes	characteristic
79	C₄	C₂×Dic₆₃₂	1	1	yes	characteristic
80	C₂	#23765675	1	1	yes	characteristic
81	C₂	C₂×D₂₅₂₈	1	1	yes	characteristic
82	C₂	C₂×Dic₁₂₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂₆₄
commutator, cosocle	C₂×C₁₆×C₇₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₉
Fitting	C₄×C₃₂×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	C₂×D₁₆
hypercenter	C₄×C₃₂	D₇₉
hypocenter	C₇₉	#13731
LCS	G ▷ C₂×C₁₆×C₇₉ ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₇₉	Q₈	semidirect
45. C₇₉	#13731	semidirect
57. C₃₂	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23765675	2
#23761600	2
C₄×C₃₂×C₇₉	2
#13731	79