Subgroups — #34175009

Order 20224.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761470	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₉⋊C₁₆)	—	2	2	no	—
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	—	4	2	no	—
8	C₂×C₁₆×C₇₉	—	2	1	no	—
9	C₂×C₁₆×C₇₉	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₇₉	C₈	1	1	yes	characteristic
11	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
13	C₄×C₈×C₇₉	Q₈	1	1	yes	characteristic
14	C₇₉⋊C₁₆	—	8	2	no	—
15	C₂×C₈×C₇₉	—	2	1	no	—
16	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
17	C₂×C₈×C₇₉	M₁₆	1	1	yes	characteristic
18	C₂×C₈×C₇₉	D₈	1	1	yes	characteristic
19	C₂×C₈×C₇₉	Q₁₆	1	1	yes	characteristic
20	C₄²×C₇₉	C₄⋊C₄	1	1	yes	characteristic
21	C₁₆×C₇₉	—	2	2	no	—
22	C₁₆×C₇₉	—	2	1	no	—
23	C₁₆×C₇₉	C₁₆	1	2	yes	complemented
24	C₂×C₄×C₇₉	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₇₉	#107	1	1	yes	characteristic
26	C₂×C₄×C₇₉	#108	1	1	yes	characteristic
27	C₈×C₇₉	—	2	2	no	—
28	C₈×C₇₉	—	2	1	no	—
29	C₈×C₇₉	—	2	1	no	—
30	C₈×C₇₉	C₂×C₁₆	1	1	yes	characteristic
31	C₈×C₇₉	M₃₂	1	1	yes	characteristic
32	C₈×C₇₉	D₁₆	1	1	yes	characteristic
33	C₈×C₇₉	Q₃₂	1	1	yes	characteristic
34	C₂²×C₇₉	#335	1	1	yes	characteristic
35	C₄×C₇₉	—	2	1	no	—
36	C₄×C₇₉	C₄⋊C₁₆	1	1	yes	characteristic
37	C₄×C₇₉	#364	1	1	yes	characteristic
38	C₄×C₇₉	#366	1	1	yes	characteristic
39	C₄×C₇₉	#368	1	1	yes	characteristic
40	#7100	—	79	1	no	maximal, Sylow
41	C₂×C₇₉	#1370	1	1	yes	characteristic
42	C₂×C₇₉	#1371	1	1	yes	characteristic
43	C₂×C₇₉	#1373	1	1	yes	characteristic
44	#1370	—	79	2	no	—
45	C₈×C₁₆	D₇₉	1	1	yes	characteristic
46	C₇₉	#7100	1	1	yes	characteristic, complemented, Sylow
47	C₄⋊C₁₆	—	158	2	no	—
48	C₄×C₁₆	Dic₇₉	1	1	yes	characteristic
49	C₄×C₁₆	Dic₇₉	1	1	yes	characteristic
50	C₈²	C₂×D₇₉	1	1	yes	characteristic
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	316	2	no	—
53	C₂×C₁₆	C₇₉⋊C₈	1	1	yes	characteristic
54	C₂×C₁₆	C₇₉⋊C₈	1	1	yes	characteristic
55	C₄×C₈	Dic₁₅₈	1	1	yes	characteristic
56	C₄×C₈	D₃₁₆	1	1	yes	characteristic
57	C₄×C₈	C₂×Dic₇₉	1	1	yes	characteristic
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
60	C₂×C₈	#11929608	1	1	yes	characteristic
61	C₂×C₈	D₆₃₂	1	1	yes	characteristic
62	C₂×C₈	Dic₃₁₆	1	1	yes	characteristic
63	C₄²	C₄⋊Dic₇₉	1	1	yes	characteristic
64	C₁₆	—	2	2	no	—
65	C₁₆	—	2	1	no	—
66	C₁₆	—	632	2	no	—
67	C₁₆	C₇₉⋊C₁₆	1	2	yes	complemented
68	C₂×C₄	C₄⋊(C₇₉⋊C₈)	1	1	yes	characteristic
69	C₂×C₄	#14579095	1	1	yes	characteristic
70	C₂×C₄	#14579097	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₈	—	2	1	no	—
73	C₈	—	2	1	no	—
74	C₈	C₂×C₇₉⋊C₁₆	1	1	yes	characteristic
75	C₈	#14579101	1	1	yes	characteristic
76	C₈	D₁₂₆₄	1	1	yes	characteristic
77	C₈	Dic₆₃₂	1	1	yes	characteristic
78	C₂²	#19098316	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	C₄⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
81	C₄	#19098384	1	1	yes	characteristic
82	C₄	#19098389	1	1	yes	characteristic
83	C₄	#19098393	1	1	yes	characteristic
84	C₂	#23761470	1	1	yes	characteristic
85	C₂	#23761472	1	1	yes	characteristic
86	C₂	#23761476	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₆₃₂
commutator	C₈×C₇₉	C₂×C₁₆
Frattini	C₈²	C₂×D₇₉
Fitting	C₈×C₁₆×C₇₉	C₂
socle, prime radical	C₂²×C₇₉	#335
cosocle	C₈²×C₇₉	C₂²
hypercenter	C₈×C₁₆	D₇₉
hypocenter	C₇₉	#7100
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
23. C₁₆×C₇₉	C₁₆	semidirect
46. C₇₉	#7100	semidirect
67. 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
#23761470	2
C₈×C₁₆×C₇₉	2
#7100	79