Subgroups — #34176080

Order 20224.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23761611	C₂	1	2	yes	maximal, complemented
2	C₂×M₆₄×C₇₉	C₂	1	1	yes	maximal, characteristic
3	C₇₉⋊C₆₄	—	2	2	no	—
4	C₂²×C₁₆×C₇₉	C₄	1	1	yes	characteristic
5	M₆₄×C₇₉	—	2	2	no	—
6	C₂×C₃₂×C₇₉	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₇₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₇₉	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₇₉	—	2	1	no	—
10	C₂×C₁₆×C₇₉	—	2	1	no	—
11	C₂×C₁₆×C₇₉	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₇₉	—	2	1	no	—
14	C₃₂×C₇₉	D₄	1	2	yes	—
15	C₂²×C₄×C₇₉	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₇₉	—	2	1	no	—
17	C₂×C₈×C₇₉	—	2	1	no	—
18	C₂×C₈×C₇₉	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₇₉	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₇₉	—	2	1	no	—
21	C₁₆×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₁₆×C₇₉	M₁₆	1	1	yes	characteristic
23	C₂³×C₇₉	C₃₂	1	1	yes	characteristic
24	C₂×C₄×C₇₉	—	2	1	no	—
25	C₂×C₄×C₇₉	—	2	1	no	—
26	C₂×C₄×C₇₉	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₇₉	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₇₉	—	2	1	no	—
29	C₈×C₇₉	C₂²⋊C₈	1	1	yes	characteristic
30	C₈×C₇₉	M₃₂	1	1	yes	characteristic
31	C₂²×C₇₉	—	2	1	no	—
32	C₂²×C₇₉	—	4	1	no	—
33	C₂²×C₇₉	C₂×C₃₂	1	1	yes	characteristic
34	C₄×C₇₉	—	2	1	no	—
35	C₄×C₇₉	C₂²⋊C₁₆	1	1	yes	characteristic
36	C₄×C₇₉	M₆₄	1	1	yes	characteristic
37	#7513	—	79	1	no	maximal, Sylow
38	C₂×C₇₉	—	2	1	no	—
39	C₂×C₇₉	—	4	1	no	—
40	C₂×C₇₉	C₂²⋊C₃₂	1	1	yes	characteristic
41	C₂×M₆₄	D₇₉	1	1	yes	characteristic
42	M₁₂₈	—	79	2	no	—
43	C₇₉	#7513	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₁₆	Dic₇₉	1	1	yes	characteristic
45	M₆₄	—	2	2	no	—
46	C₂×C₃₂	Dic₇₉	1	1	yes	characteristic
47	C₂×C₃₂	C₂×D₇₉	1	1	yes	characteristic
48	C₆₄	—	158	2	no	—
49	C₂²×C₈	C₇₉⋊C₈	1	1	yes	characteristic
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	C₇₉⋊C₈	1	1	yes	characteristic
53	C₂×C₁₆	C₂×Dic₇₉	1	1	yes	characteristic
54	C₃₂	—	2	1	no	—
55	C₃₂	C₂²⋊D₇₉	1	2	yes	—
56	C₂²×C₄	C₇₉⋊C₁₆	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	C₇₉⋊C₁₆	1	1	yes	characteristic
60	C₂×C₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	C₂²⋊Dic₇₉	1	1	yes	characteristic
63	C₁₆	#11929608	1	1	yes	characteristic
64	C₂³	C₇₉⋊C₃₂	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	C₇₉⋊C₃₂	1	1	yes	characteristic
68	C₂×C₄	C₂×C₇₉⋊C₁₆	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
71	C₈	#14579101	1	1	yes	characteristic
72	C₂²	—	2	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₇₉⋊C₃₂	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	C₂²⋊(C₇₉⋊C₁₆)	1	1	yes	characteristic
77	C₄	#19098397	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	4	1	no	—
80	C₂	C₂²⋊(C₇₉⋊C₃₂)	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊Dic₇₉
commutator	C₂²×C₇₉	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₇₉
Fitting	C₂×M₆₄×C₇₉	C₂
socle	C₂×C₇₉	C₂²⋊C₃₂
prime radical	C₂³×C₇₉	C₃₂
cosocle	C₂×C₃₂×C₇₉	C₂²
hypercenter	C₂×M₆₄	D₇₉
hypocenter	C₇₉	#7513
LCS	G ▷ C₂²×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
1. #23761611	C₂	semidirect
43. C₇₉	#7513	semidirect

Maximal subgroups

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

subgroup	index
#23761611	2
C₂×M₆₄×C₇₉	2
#7513	79