Subgroups — C₂×C₃×C₈²×C₇₉ #72207121

Order 30336.

560 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₈×C₇₉	C₂	1	3	yes	maximal
2	C₃×C₈²×C₇₉	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₄²×C₇₉	C₂²	1	1	yes	characteristic
4	C₂²×C₃×C₈×C₇₉	C₄	1	6	yes	—
5	C₃×C₄×C₈×C₇₉	C₄	1	6	yes	—
6	C₃×C₄×C₈×C₇₉	C₂²	1	6	yes	—
7	C₂²×C₃×C₄×C₇₉	C₂×C₄	1	3	yes	—
8	C₃×C₄²×C₇₉	C₂×C₄	1	3	yes	—
9	C₃×C₄²×C₇₉	C₂³	1	1	yes	characteristic
10	C₂×C₃×C₈×C₇₉	C₈	1	24	yes	direct product factor
11	C₂×C₃×C₈×C₇₉	C₂×C₄	1	12	yes	—
12	C₂×C₈²×C₇₉	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₄×C₈×C₇₉	C₂×C₃	1	3	yes	—
14	C₈²×C₇₉	C₂×C₃	1	4	yes	direct product factor
15	C₂²×C₈×C₇₉	C₃×C₄	1	6	yes	—
16	C₂×C₄²×C₇₉	C₂²×C₃	1	1	yes	characteristic
17	C₄×C₈×C₇₉	C₃×C₄	1	6	yes	—
18	C₄×C₈×C₇₉	C₂²×C₃	1	6	yes	—
19	C₂³×C₃×C₇₉	C₄²	1	1	yes	characteristic
20	C₂×C₃×C₄×C₇₉	C₄²	1	3	yes	—
21	C₂×C₃×C₄×C₇₉	C₂×C₈	1	12	yes	—
22	C₂×C₃×C₄×C₇₉	C₂²×C₄	1	3	yes	—
23	C₃×C₈×C₇₉	C₂×C₈	1	24	yes	direct product factor
24	C₂²×C₄×C₇₉	C₂×C₃×C₄	1	3	yes	—
25	C₂×C₈×C₇₉	C₃×C₈	1	24	yes	direct product factor
26	C₂×C₈×C₇₉	C₂×C₃×C₄	1	12	yes	—
27	C₄²×C₇₉	C₂×C₃×C₄	1	3	yes	—
28	C₄²×C₇₉	C₂³×C₃	1	1	yes	characteristic
29	C₂²×C₃×C₇₉	C₄×C₈	1	6	yes	—
30	C₂²×C₃×C₇₉	C₂×C₄²	1	1	yes	characteristic
31	C₃×C₄×C₇₉	C₄×C₈	1	6	yes	—
32	C₃×C₄×C₇₉	C₂²×C₈	1	6	yes	—
33	C₂³×C₇₉	C₃×C₄²	1	1	yes	characteristic
34	C₂×C₄×C₇₉	C₃×C₄²	1	3	yes	—
35	C₂×C₄×C₇₉	C₂×C₃×C₈	1	12	yes	—
36	C₂×C₄×C₇₉	C₂²×C₃×C₄	1	3	yes	—
37	C₈×C₇₉	C₂×C₃×C₈	1	24	yes	direct product factor
38	C₂×C₃×C₇₉	C₈²	1	4	yes	direct product factor
39	C₂×C₃×C₇₉	C₂×C₄×C₈	1	3	yes	—
40	C₂×C₃×C₈²	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Hall
41	C₂²×C₇₉	C₃×C₄×C₈	1	6	yes	—
42	C₂²×C₇₉	C₂×C₃×C₄²	1	1	yes	characteristic
43	C₄×C₇₉	C₃×C₄×C₈	1	6	yes	—
44	C₄×C₇₉	C₂²×C₃×C₈	1	6	yes	—
45	C₃×C₇₉	C₂×C₈²	1	1	yes	characteristic, direct product factor, Hall
46	C₂×C₃×C₄×C₈	C₂×C₇₉	1	3	yes	—
47	C₃×C₈²	C₂×C₇₉	1	4	yes	direct product factor
48	C₂×C₇₉	C₃×C₈²	1	4	yes	direct product factor
49	C₂×C₇₉	C₂×C₃×C₄×C₈	1	3	yes	—
50	C₂×C₈²	C₃×C₇₉	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²×C₃×C₈	C₄×C₇₉	1	6	yes	—
52	C₂×C₃×C₄²	C₂²×C₇₉	1	1	yes	characteristic
53	C₃×C₄×C₈	C₄×C₇₉	1	6	yes	—
54	C₃×C₄×C₈	C₂²×C₇₉	1	6	yes	—
55	C₇₉	C₂×C₃×C₈²	1	1	yes	characteristic, direct product factor, Sylow
56	C₂×C₄×C₈	C₂×C₃×C₇₉	1	3	yes	—
57	C₈²	C₂×C₃×C₇₉	1	4	yes	direct product factor
58	C₂²×C₃×C₄	C₂×C₄×C₇₉	1	3	yes	—
59	C₂×C₃×C₈	C₈×C₇₉	1	24	yes	direct product factor
60	C₂×C₃×C₈	C₂×C₄×C₇₉	1	12	yes	—
61	C₃×C₄²	C₂×C₄×C₇₉	1	3	yes	—
62	C₃×C₄²	C₂³×C₇₉	1	1	yes	characteristic
63	C₂²×C₈	C₃×C₄×C₇₉	1	6	yes	—
64	C₂×C₄²	C₂²×C₃×C₇₉	1	1	yes	characteristic
65	C₄×C₈	C₃×C₄×C₇₉	1	6	yes	—
66	C₄×C₈	C₂²×C₃×C₇₉	1	6	yes	—
67	C₂³×C₃	C₄²×C₇₉	1	1	yes	characteristic
68	C₂×C₃×C₄	C₄²×C₇₉	1	3	yes	—
69	C₂×C₃×C₄	C₂×C₈×C₇₉	1	12	yes	—
70	C₂×C₃×C₄	C₂²×C₄×C₇₉	1	3	yes	—
71	C₃×C₈	C₂×C₈×C₇₉	1	24	yes	direct product factor
72	C₂²×C₄	C₂×C₃×C₄×C₇₉	1	3	yes	—
73	C₂×C₈	C₃×C₈×C₇₉	1	24	yes	direct product factor
74	C₂×C₈	C₂×C₃×C₄×C₇₉	1	12	yes	—
75	C₄²	C₂×C₃×C₄×C₇₉	1	3	yes	—
76	C₄²	C₂³×C₃×C₇₉	1	1	yes	characteristic
77	C₂²×C₃	C₄×C₈×C₇₉	1	6	yes	—
78	C₂²×C₃	C₂×C₄²×C₇₉	1	1	yes	characteristic
79	C₃×C₄	C₄×C₈×C₇₉	1	6	yes	—
80	C₃×C₄	C₂²×C₈×C₇₉	1	6	yes	—
81	C₂³	C₃×C₄²×C₇₉	1	1	yes	characteristic
82	C₂×C₄	C₂×C₃×C₈×C₇₉	1	12	yes	—
83	C₂×C₄	C₃×C₄²×C₇₉	1	3	yes	—
84	C₂×C₄	C₂²×C₃×C₄×C₇₉	1	3	yes	—
85	C₈	C₂×C₃×C₈×C₇₉	1	24	yes	direct product factor
86	C₂×C₃	C₈²×C₇₉	1	4	yes	direct product factor
87	C₂×C₃	C₂×C₄×C₈×C₇₉	1	3	yes	—
88	C₂²	C₃×C₄×C₈×C₇₉	1	6	yes	—
89	C₂²	C₂×C₃×C₄²×C₇₉	1	1	yes	characteristic
90	C₄	C₃×C₄×C₈×C₇₉	1	6	yes	—
91	C₄	C₂²×C₃×C₈×C₇₉	1	6	yes	—
92	C₃	C₂×C₈²×C₇₉	1	1	yes	characteristic, direct product factor, Sylow
93	C₂	C₃×C₈²×C₇₉	1	4	yes	direct product factor
94	C₂	C₂×C₃×C₄×C₈×C₇₉	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄²	C₂³×C₃×C₇₉
socle, prime radical	C₂³×C₃×C₇₉	C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₈²×C₇₉	C₂	direct (N × H)
10. C₂×C₃×C₈×C₇₉	C₈	direct (N × H)
12. C₂×C₈²×C₇₉	C₃	direct (N × H)
14. C₈²×C₇₉	C₂×C₃	direct (N × H)
23. C₃×C₈×C₇₉	C₂×C₈	direct (N × H)
25. C₂×C₈×C₇₉	C₃×C₈	direct (N × H)
37. C₈×C₇₉	C₂×C₃×C₈	direct (N × H)
38. C₂×C₃×C₇₉	C₈²	direct (N × H)
40. C₂×C₃×C₈²	C₇₉	direct (N × H)
45. C₃×C₇₉	C₂×C₈²	direct (N × H)
47. C₃×C₈²	C₂×C₇₉	direct (N × H)
48. C₂×C₇₉	C₃×C₈²	direct (N × H)
50. C₂×C₈²	C₃×C₇₉	direct (N × H)
55. C₇₉	C₂×C₃×C₈²	direct (N × H)
57. C₈²	C₂×C₃×C₇₉	direct (N × H)
59. C₂×C₃×C₈	C₈×C₇₉	direct (N × H)
71. C₃×C₈	C₂×C₈×C₇₉	direct (N × H)
73. C₂×C₈	C₃×C₈×C₇₉	direct (N × H)
85. C₈	C₂×C₃×C₈×C₇₉	direct (N × H)
86. C₂×C₃	C₈²×C₇₉	direct (N × H)
92. C₃	C₂×C₈²×C₇₉	direct (N × H)
93. C₂	C₃×C₈²×C₇₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₈×C₇₉	2
C₃×C₈²×C₇₉	2
C₂×C₈²×C₇₉	3
C₂×C₃×C₈²	79

Subgroups — C2×C3×C82×C79 #72207121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₈²×C₇₉ #72207121