Subgroups — C₂²×C₁₃×M₆₄ #15720311

Order 3328.

502 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×M₆₄	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₃×C₃₂	C₂	1	2	yes	maximal, complemented
4	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₁₃×C₁₆	C₄	1	6	yes	—
7	C₂²×C₁₃×C₁₆	C₂²	1	6	yes	—
8	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₁₃×M₆₄	C₂²	1	16	yes	direct product factor
10	C₂×C₁₃×C₃₂	C₂²	1	12	yes	complemented
11	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₈	1	6	yes	—
14	C₂²×C₈×C₁₃	C₂×C₄	1	6	yes	—
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₃×C₁₆	C₂×C₄	1	6	yes	—
17	C₂×C₁₃×C₁₆	C₂×C₄	1	12	yes	—
18	C₂×C₁₃×C₁₆	C₂×C₄	1	3	yes	—
19	C₂×C₁₃×C₁₆	C₂³	1	3	yes	—
20	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
21	C₁₃×C₃₂	C₂³	1	8	yes	complemented
22	C₂²×M₆₄	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂⁴×C₁₃	C₁₆	1	1	yes	characteristic
24	C₂²×C₄×C₁₃	C₁₆	1	6	yes	—
25	C₂²×C₄×C₁₃	C₁₆	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	C₂×C₈	1	6	yes	—
27	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₁₃	C₂×C₈	1	12	yes	—
29	C₂×C₈×C₁₃	C₂×C₈	1	6	yes	—
30	C₂×C₈×C₁₃	C₂×C₈	1	3	yes	—
31	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
32	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
33	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
34	C₁₃×C₁₆	C₂²×C₄	1	3	yes	—
35	C₁₃×C₁₆	C₂⁴	1	1	yes	characteristic
36	C₂³×C₁₆	C₂×C₁₃	1	1	yes	characteristic
37	C₂×M₆₄	C₂×C₁₃	1	12	yes	direct product factor
38	C₂²×C₃₂	C₂×C₁₃	1	2	yes	complemented
39	C₂³×C₁₃	—	2	4	no	—
40	C₂³×C₁₃	C₂×C₁₆	1	6	yes	—
41	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₁₃	C₂×C₁₆	1	12	yes	—
43	C₂×C₄×C₁₃	C₂×C₁₆	1	3	yes	—
44	C₂×C₄×C₁₃	C₂×C₁₆	1	6	yes	—
45	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
46	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
47	C₈×C₁₃	C₂²×C₈	1	4	yes	—
48	C₈×C₁₃	C₂²×C₈	1	3	yes	—
49	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
50	C₂³×C₈	C₄×C₁₃	1	1	yes	characteristic
51	C₂²×C₁₆	C₄×C₁₃	1	1	yes	characteristic
52	C₂²×C₁₆	C₄×C₁₃	1	6	yes	—
53	C₂²×C₁₆	C₂²×C₁₃	1	6	yes	—
54	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
55	M₆₄	C₂²×C₁₃	1	16	yes	direct product factor
56	C₂×C₃₂	C₂²×C₁₃	1	12	yes	complemented
57	C₂²×C₁₃	—	2	12	no	—
58	C₂²×C₁₃	M₆₄	1	4	yes	direct product factor
59	C₂²×C₁₃	C₂²×C₁₆	1	4	yes	—
60	C₂²×C₁₃	C₂²×C₁₆	1	3	yes	—
61	C₄×C₁₃	C₂²×C₁₆	1	3	yes	—
62	C₄×C₁₃	C₂²×C₁₆	1	4	yes	—
63	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
64	C₂³×C₄	C₈×C₁₃	1	1	yes	characteristic
65	C₂²×C₈	C₈×C₁₃	1	6	yes	—
66	C₂²×C₈	C₈×C₁₃	1	1	yes	characteristic
67	C₂²×C₈	C₂×C₄×C₁₃	1	6	yes	—
68	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
69	C₂×C₁₆	C₂×C₄×C₁₃	1	12	yes	—
70	C₂×C₁₆	C₂×C₄×C₁₃	1	6	yes	—
71	C₂×C₁₆	C₂×C₄×C₁₃	1	3	yes	—
72	C₂×C₁₆	C₂³×C₁₃	1	3	yes	—
73	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
74	C₃₂	C₂³×C₁₃	1	8	yes	complemented
75	C₂×C₁₃	—	2	4	no	—
76	C₂×C₁₃	C₂×M₆₄	1	6	yes	direct product factor
77	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
78	C₂⁴	C₁₃×C₁₆	1	1	yes	characteristic
79	C₂²×C₄	C₁₃×C₁₆	1	6	yes	—
80	C₂²×C₄	C₁₃×C₁₆	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₈×C₁₃	1	6	yes	—
82	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
83	C₂×C₈	C₂×C₈×C₁₃	1	12	yes	—
84	C₂×C₈	C₂×C₈×C₁₃	1	6	yes	—
85	C₂×C₈	C₂×C₈×C₁₃	1	3	yes	—
86	C₂×C₈	C₂²×C₄×C₁₃	1	3	yes	—
87	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
88	C₁₆	C₂²×C₄×C₁₃	1	3	yes	—
89	C₁₆	C₂²×C₄×C₁₃	1	4	yes	—
90	C₁₆	C₂⁴×C₁₃	1	1	yes	characteristic
91	C₁₃	C₂²×M₆₄	1	1	yes	characteristic, direct product factor, Sylow
92	C₂³	—	2	4	no	—
93	C₂³	C₂×C₁₃×C₁₆	1	6	yes	—
94	C₂³	C₂×C₁₃×C₁₆	1	1	yes	characteristic
95	C₂×C₄	C₂×C₁₃×C₁₆	1	12	yes	—
96	C₂×C₄	C₂×C₁₃×C₁₆	1	3	yes	—
97	C₂×C₄	C₂×C₁₃×C₁₆	1	6	yes	—
98	C₂×C₄	C₂²×C₈×C₁₃	1	4	yes	—
99	C₂×C₄	C₂²×C₈×C₁₃	1	3	yes	—
100	C₈	C₂²×C₈×C₁₃	1	3	yes	—
101	C₈	C₂²×C₈×C₁₃	1	4	yes	—
102	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
103	C₂²	—	2	12	no	—
104	C₂²	C₁₃×M₆₄	1	4	yes	direct product factor
105	C₂²	C₂²×C₁₃×C₁₆	1	4	yes	—
106	C₂²	C₂²×C₁₃×C₁₆	1	3	yes	—
107	C₄	C₂²×C₁₃×C₁₆	1	3	yes	—
108	C₄	C₂²×C₁₃×C₁₆	1	4	yes	—
109	C₄	C₂³×C₈×C₁₃	1	1	yes	characteristic
110	C₂	—	2	4	no	—
111	C₂	C₂×C₁₃×M₆₄	1	6	yes	direct product factor
112	C₂	C₂³×C₁₃×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₃×C₁₆	C₂²
commutator	C₂	C₂³×C₁₃×C₁₆
Frattini, cosocle	C₁₆	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂×C₁₆
prime radical	C₂⁴×C₁₃	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×M₆₄	C₂	direct (N × H)
3. C₂²×C₁₃×C₃₂	C₂	semidirect
9. C₁₃×M₆₄	C₂²	direct (N × H)
10. C₂×C₁₃×C₃₂	C₂²	semidirect
21. C₁₃×C₃₂	C₂³	semidirect
22. C₂²×M₆₄	C₁₃	direct (N × H)
37. C₂×M₆₄	C₂×C₁₃	direct (N × H)
38. C₂²×C₃₂	C₂×C₁₃	semidirect
55. M₆₄	C₂²×C₁₃	direct (N × H)
56. C₂×C₃₂	C₂²×C₁₃	semidirect
58. C₂²×C₁₃	M₆₄	direct (N × H)
74. C₃₂	C₂³×C₁₃	semidirect
76. C₂×C₁₃	C₂×M₆₄	direct (N × H)
91. C₁₃	C₂²×M₆₄	direct (N × H)
104. C₂²	C₁₃×M₆₄	direct (N × H)
111. C₂	C₂×C₁₃×M₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×C₁₆	2
C₂×C₁₃×M₆₄	2
C₂²×C₁₃×C₃₂	2
C₂²×M₆₄	13

Subgroups — C22×C13×M64 #15720311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×M₆₄ #15720311