Subgroups — C₃×C₄×C₅×C₁₃×C₁₆ #72190821

Order 12480.

232 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₁₃×C₁₆	C₂	1	2	yes	maximal
3	C₄×C₅×C₁₃×C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₅×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂×C₃×C₅×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₃×C₄²×C₅×C₁₃	C₄	1	1	yes	characteristic
7	C₃×C₅×C₁₃×C₁₆	C₄	1	4	yes	direct product factor
8	C₃×C₄×C₁₃×C₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₄×C₅×C₈×C₁₃	C₂×C₃	1	1	yes	characteristic
10	C₂×C₅×C₁₃×C₁₆	C₂×C₃	1	2	yes	—
11	C₂×C₃×C₄×C₅×C₁₃	C₈	1	2	yes	—
12	C₂×C₃×C₄×C₅×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₃×C₅×C₈×C₁₃	C₈	1	2	yes	—
14	C₃×C₅×C₈×C₁₃	C₂×C₄	1	2	yes	—
15	C₂×C₃×C₁₃×C₁₆	C₂×C₅	1	2	yes	—
16	C₃×C₄×C₈×C₁₃	C₂×C₅	1	1	yes	characteristic
17	C₂×C₅×C₈×C₁₃	C₃×C₄	1	1	yes	characteristic
18	C₂×C₅×C₈×C₁₃	C₂²×C₃	1	1	yes	characteristic
19	C₄²×C₅×C₁₃	C₃×C₄	1	1	yes	characteristic
20	C₅×C₁₃×C₁₆	C₃×C₄	1	4	yes	direct product factor
21	C₃×C₄×C₅×C₁₆	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₄×C₁₃×C₁₆	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
23	C₂²×C₃×C₅×C₁₃	C₂×C₈	1	1	yes	characteristic
24	C₃×C₄×C₅×C₁₃	C₁₆	1	4	yes	direct product factor
25	C₃×C₄×C₅×C₁₃	C₄²	1	1	yes	characteristic
26	C₃×C₄×C₅×C₁₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₃×C₈×C₁₃	C₄×C₅	1	1	yes	characteristic
28	C₂×C₃×C₈×C₁₃	C₂²×C₅	1	1	yes	characteristic
29	C₃×C₄²×C₁₃	C₄×C₅	1	1	yes	characteristic
30	C₃×C₁₃×C₁₆	C₄×C₅	1	4	yes	direct product factor
31	C₂×C₄×C₅×C₁₃	C₃×C₈	1	2	yes	—
32	C₂×C₄×C₅×C₁₃	C₂×C₃×C₄	1	1	yes	characteristic
33	C₅×C₈×C₁₃	C₃×C₈	1	2	yes	—
34	C₅×C₈×C₁₃	C₂×C₃×C₄	1	2	yes	—
35	C₂×C₃×C₅×C₁₆	C₂×C₁₃	1	2	yes	—
36	C₃×C₄×C₅×C₈	C₂×C₁₃	1	1	yes	characteristic
37	C₂×C₁₃×C₁₆	C₂×C₃×C₅	1	2	yes	—
38	C₄×C₈×C₁₃	C₂×C₃×C₅	1	1	yes	characteristic
39	C₂×C₃×C₅×C₁₃	C₄×C₈	1	1	yes	characteristic
40	C₂×C₃×C₅×C₁₃	C₂×C₁₆	1	2	yes	—
41	C₄×C₅×C₁₆	C₃×C₁₃	1	1	yes	characteristic, direct product factor, Hall
42	C₂×C₃×C₄×C₁₃	C₅×C₈	1	2	yes	—
43	C₂×C₃×C₄×C₁₃	C₂×C₄×C₅	1	1	yes	characteristic
44	C₃×C₈×C₁₃	C₅×C₈	1	2	yes	—
45	C₃×C₈×C₁₃	C₂×C₄×C₅	1	2	yes	—
46	C₂²×C₅×C₁₃	C₂×C₃×C₈	1	1	yes	characteristic
47	C₄×C₅×C₁₃	C₃×C₁₆	1	4	yes	direct product factor
48	C₄×C₅×C₁₃	C₃×C₄²	1	1	yes	characteristic
49	C₄×C₅×C₁₃	C₂×C₃×C₈	1	1	yes	characteristic
50	C₂×C₃×C₅×C₈	C₄×C₁₃	1	1	yes	characteristic
51	C₂×C₃×C₅×C₈	C₂²×C₁₃	1	1	yes	characteristic
52	C₃×C₄²×C₅	C₄×C₁₃	1	1	yes	characteristic
53	C₃×C₅×C₁₆	C₄×C₁₃	1	4	yes	direct product factor
54	C₂×C₈×C₁₃	C₃×C₄×C₅	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂²×C₃×C₅	1	1	yes	characteristic
56	C₄²×C₁₃	C₃×C₄×C₅	1	1	yes	characteristic
57	C₁₃×C₁₆	C₃×C₄×C₅	1	4	yes	direct product factor
58	C₃×C₅×C₁₃	C₄×C₁₆	1	1	yes	characteristic, direct product factor, Hall
59	C₃×C₄×C₁₆	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
60	C₂×C₅×C₁₆	C₂×C₃×C₁₃	1	2	yes	—
61	C₄×C₅×C₈	C₂×C₃×C₁₃	1	1	yes	characteristic
62	C₂²×C₃×C₁₃	C₂×C₅×C₈	1	1	yes	characteristic
63	C₃×C₄×C₁₃	C₅×C₁₆	1	4	yes	direct product factor
64	C₃×C₄×C₁₃	C₄²×C₅	1	1	yes	characteristic
65	C₃×C₄×C₁₃	C₂×C₅×C₈	1	1	yes	characteristic
66	C₂×C₅×C₁₃	C₃×C₄×C₈	1	1	yes	characteristic
67	C₂×C₅×C₁₃	C₂×C₃×C₁₆	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₈×C₁₃	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂×C₄×C₁₃	1	1	yes	characteristic
70	C₃×C₅×C₈	C₈×C₁₃	1	2	yes	—
71	C₃×C₅×C₈	C₂×C₄×C₁₃	1	2	yes	—
72	C₂×C₄×C₁₃	C₃×C₅×C₈	1	2	yes	—
73	C₂×C₄×C₁₃	C₂×C₃×C₄×C₅	1	1	yes	characteristic
74	C₈×C₁₃	C₃×C₅×C₈	1	2	yes	—
75	C₈×C₁₃	C₂×C₃×C₄×C₅	1	2	yes	—
76	C₂×C₃×C₁₆	C₂×C₅×C₁₃	1	2	yes	—
77	C₃×C₄×C₈	C₂×C₅×C₁₃	1	1	yes	characteristic
78	C₂×C₅×C₈	C₃×C₄×C₁₃	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²×C₃×C₁₃	1	1	yes	characteristic
80	C₄²×C₅	C₃×C₄×C₁₃	1	1	yes	characteristic
81	C₅×C₁₆	C₃×C₄×C₁₃	1	4	yes	direct product factor
82	C₂×C₃×C₁₃	C₄×C₅×C₈	1	1	yes	characteristic
83	C₂×C₃×C₁₃	C₂×C₅×C₁₆	1	2	yes	—
84	C₅×C₁₃	C₃×C₄×C₁₆	1	1	yes	characteristic, direct product factor, Hall
85	C₄×C₁₆	C₃×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
86	C₂²×C₃×C₅	C₂×C₈×C₁₃	1	1	yes	characteristic
87	C₃×C₄×C₅	C₁₃×C₁₆	1	4	yes	direct product factor
88	C₃×C₄×C₅	C₄²×C₁₃	1	1	yes	characteristic
89	C₃×C₄×C₅	C₂×C₈×C₁₃	1	1	yes	characteristic
90	C₂²×C₁₃	C₂×C₃×C₅×C₈	1	1	yes	characteristic
91	C₄×C₁₃	C₃×C₅×C₁₆	1	4	yes	direct product factor
92	C₄×C₁₃	C₃×C₄²×C₅	1	1	yes	characteristic
93	C₄×C₁₃	C₂×C₃×C₅×C₈	1	1	yes	characteristic
94	C₂×C₃×C₈	C₄×C₅×C₁₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂²×C₅×C₁₃	1	1	yes	characteristic
96	C₃×C₄²	C₄×C₅×C₁₃	1	1	yes	characteristic
97	C₃×C₁₆	C₄×C₅×C₁₃	1	4	yes	direct product factor
98	C₂×C₄×C₅	C₃×C₈×C₁₃	1	2	yes	—
99	C₂×C₄×C₅	C₂×C₃×C₄×C₁₃	1	1	yes	characteristic
100	C₅×C₈	C₃×C₈×C₁₃	1	2	yes	—
101	C₅×C₈	C₂×C₃×C₄×C₁₃	1	2	yes	—
102	C₃×C₁₃	C₄×C₅×C₁₆	1	1	yes	characteristic, direct product factor, Hall
103	C₂×C₁₆	C₂×C₃×C₅×C₁₃	1	2	yes	—
104	C₄×C₈	C₂×C₃×C₅×C₁₃	1	1	yes	characteristic
105	C₂×C₃×C₅	C₄×C₈×C₁₃	1	1	yes	characteristic
106	C₂×C₃×C₅	C₂×C₁₃×C₁₆	1	2	yes	—
107	C₂×C₁₃	C₃×C₄×C₅×C₈	1	1	yes	characteristic
108	C₂×C₁₃	C₂×C₃×C₅×C₁₆	1	2	yes	—
109	C₂×C₃×C₄	C₅×C₈×C₁₃	1	2	yes	—
110	C₂×C₃×C₄	C₂×C₄×C₅×C₁₃	1	1	yes	characteristic
111	C₃×C₈	C₅×C₈×C₁₃	1	2	yes	—
112	C₃×C₈	C₂×C₄×C₅×C₁₃	1	2	yes	—
113	C₂²×C₅	C₂×C₃×C₈×C₁₃	1	1	yes	characteristic
114	C₄×C₅	C₃×C₁₃×C₁₆	1	4	yes	direct product factor
115	C₄×C₅	C₃×C₄²×C₁₃	1	1	yes	characteristic
116	C₄×C₅	C₂×C₃×C₈×C₁₃	1	1	yes	characteristic
117	C₂×C₈	C₃×C₄×C₅×C₁₃	1	1	yes	characteristic
118	C₂×C₈	C₂²×C₃×C₅×C₁₃	1	1	yes	characteristic
119	C₄²	C₃×C₄×C₅×C₁₃	1	1	yes	characteristic
120	C₁₆	C₃×C₄×C₅×C₁₃	1	4	yes	direct product factor
121	C₃×C₅	C₄×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Hall
122	C₁₃	C₃×C₄×C₅×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
123	C₂²×C₃	C₂×C₅×C₈×C₁₃	1	1	yes	characteristic
124	C₃×C₄	C₅×C₁₃×C₁₆	1	4	yes	direct product factor
125	C₃×C₄	C₄²×C₅×C₁₃	1	1	yes	characteristic
126	C₃×C₄	C₂×C₅×C₈×C₁₃	1	1	yes	characteristic
127	C₂×C₅	C₃×C₄×C₈×C₁₃	1	1	yes	characteristic
128	C₂×C₅	C₂×C₃×C₁₃×C₁₆	1	2	yes	—
129	C₂×C₄	C₃×C₅×C₈×C₁₃	1	2	yes	—
130	C₂×C₄	C₂×C₃×C₄×C₅×C₁₃	1	1	yes	characteristic
131	C₈	C₃×C₅×C₈×C₁₃	1	2	yes	—
132	C₈	C₂×C₃×C₄×C₅×C₁₃	1	2	yes	—
133	C₂×C₃	C₂×C₅×C₁₃×C₁₆	1	2	yes	—
134	C₂×C₃	C₄×C₅×C₈×C₁₃	1	1	yes	characteristic
135	C₅	C₃×C₄×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²	C₂×C₃×C₅×C₈×C₁₃	1	1	yes	characteristic
137	C₄	C₃×C₅×C₁₃×C₁₆	1	4	yes	direct product factor
138	C₄	C₃×C₄²×C₅×C₁₃	1	1	yes	characteristic
139	C₄	C₂×C₃×C₅×C₈×C₁₃	1	1	yes	characteristic
140	C₃	C₄×C₅×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
141	C₂	C₂×C₃×C₅×C₁₃×C₁₆	1	2	yes	—
142	C₂	C₃×C₄×C₅×C₈×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈	C₂²×C₃×C₅×C₁₃
socle, prime radical	C₂²×C₃×C₅×C₁₃	C₂×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₅×C₁₃×C₁₆	C₃	direct (N × H)
7. C₃×C₅×C₁₃×C₁₆	C₄	direct (N × H)
8. C₃×C₄×C₁₃×C₁₆	C₅	direct (N × H)
20. C₅×C₁₃×C₁₆	C₃×C₄	direct (N × H)
21. C₃×C₄×C₅×C₁₆	C₁₃	direct (N × H)
22. C₄×C₁₃×C₁₆	C₃×C₅	direct (N × H)
24. C₃×C₄×C₅×C₁₃	C₁₆	direct (N × H)
30. C₃×C₁₃×C₁₆	C₄×C₅	direct (N × H)
41. C₄×C₅×C₁₆	C₃×C₁₃	direct (N × H)
47. C₄×C₅×C₁₃	C₃×C₁₆	direct (N × H)
53. C₃×C₅×C₁₆	C₄×C₁₃	direct (N × H)
57. C₁₃×C₁₆	C₃×C₄×C₅	direct (N × H)
58. C₃×C₅×C₁₃	C₄×C₁₆	direct (N × H)
59. C₃×C₄×C₁₆	C₅×C₁₃	direct (N × H)
63. C₃×C₄×C₁₃	C₅×C₁₆	direct (N × H)
81. C₅×C₁₆	C₃×C₄×C₁₃	direct (N × H)
84. C₅×C₁₃	C₃×C₄×C₁₆	direct (N × H)
85. C₄×C₁₆	C₃×C₅×C₁₃	direct (N × H)
87. C₃×C₄×C₅	C₁₃×C₁₆	direct (N × H)
91. C₄×C₁₃	C₃×C₅×C₁₆	direct (N × H)
97. C₃×C₁₆	C₄×C₅×C₁₃	direct (N × H)
102. C₃×C₁₃	C₄×C₅×C₁₆	direct (N × H)
114. C₄×C₅	C₃×C₁₃×C₁₆	direct (N × H)
120. C₁₆	C₃×C₄×C₅×C₁₃	direct (N × H)
121. C₃×C₅	C₄×C₁₃×C₁₆	direct (N × H)
122. C₁₃	C₃×C₄×C₅×C₁₆	direct (N × H)
124. C₃×C₄	C₅×C₁₃×C₁₆	direct (N × H)
135. C₅	C₃×C₄×C₁₃×C₁₆	direct (N × H)
137. C₄	C₃×C₅×C₁₃×C₁₆	direct (N × H)
140. C₃	C₄×C₅×C₁₃×C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×C₈×C₁₃	2
C₂×C₃×C₅×C₁₃×C₁₆	2
C₄×C₅×C₁₃×C₁₆	3
C₃×C₄×C₁₃×C₁₆	5
C₃×C₄×C₅×C₁₆	13

Subgroups — C3×C4×C5×C13×C16 #72190821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₅×C₁₃×C₁₆ #72190821