Subgroups — C₂²×C₃×C₅×C₁₃×C₁₆ #72190823

Order 12480.

392 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₁₃×C₁₆	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅×C₁₃×C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₄×C₅×C₁₃	C₄	1	1	yes	characteristic
5	C₂×C₃×C₅×C₈×C₁₃	C₄	1	3	yes	—
6	C₂×C₃×C₅×C₈×C₁₃	C₂²	1	3	yes	—
7	C₃×C₅×C₁₃×C₁₆	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₁₃×C₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₅×C₈×C₁₃	C₂×C₃	1	1	yes	characteristic
10	C₂×C₅×C₁₃×C₁₆	C₂×C₃	1	6	yes	direct product factor
11	C₂³×C₃×C₅×C₁₃	C₈	1	1	yes	characteristic
12	C₂×C₃×C₄×C₅×C₁₃	C₈	1	3	yes	—
13	C₂×C₃×C₄×C₅×C₁₃	C₂×C₄	1	3	yes	—
14	C₃×C₅×C₈×C₁₃	C₂×C₄	1	3	yes	—
15	C₃×C₅×C₈×C₁₃	C₂³	1	1	yes	characteristic
16	C₂²×C₃×C₈×C₁₃	C₂×C₅	1	1	yes	characteristic
17	C₂×C₃×C₁₃×C₁₆	C₂×C₅	1	6	yes	direct product factor
18	C₂²×C₄×C₅×C₁₃	C₃×C₄	1	1	yes	characteristic
19	C₂×C₅×C₈×C₁₃	C₃×C₄	1	3	yes	—
20	C₂×C₅×C₈×C₁₃	C₂²×C₃	1	3	yes	—
21	C₅×C₁₃×C₁₆	C₂²×C₃	1	4	yes	direct product factor
22	C₂²×C₃×C₅×C₁₆	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂²×C₁₃×C₁₆	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
24	C₂²×C₃×C₅×C₁₃	C₁₆	1	4	yes	direct product factor
25	C₂²×C₃×C₅×C₁₃	C₂×C₈	1	3	yes	—
26	C₃×C₄×C₅×C₁₃	C₂×C₈	1	3	yes	—
27	C₃×C₄×C₅×C₁₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₁₃	C₄×C₅	1	1	yes	characteristic
29	C₂×C₃×C₈×C₁₃	C₄×C₅	1	3	yes	—
30	C₂×C₃×C₈×C₁₃	C₂²×C₅	1	3	yes	—
31	C₃×C₁₃×C₁₆	C₂²×C₅	1	4	yes	direct product factor
32	C₂³×C₅×C₁₃	C₃×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅×C₁₃	C₃×C₈	1	3	yes	—
34	C₂×C₄×C₅×C₁₃	C₂×C₃×C₄	1	3	yes	—
35	C₅×C₈×C₁₃	C₂×C₃×C₄	1	3	yes	—
36	C₅×C₈×C₁₃	C₂³×C₃	1	1	yes	characteristic
37	C₂²×C₃×C₅×C₈	C₂×C₁₃	1	1	yes	characteristic
38	C₂×C₃×C₅×C₁₆	C₂×C₁₃	1	6	yes	direct product factor
39	C₂²×C₈×C₁₃	C₂×C₃×C₅	1	1	yes	characteristic
40	C₂×C₁₃×C₁₆	C₂×C₃×C₅	1	6	yes	direct product factor
41	C₂×C₃×C₅×C₁₃	C₂×C₁₆	1	6	yes	direct product factor
42	C₂×C₃×C₅×C₁₃	C₂²×C₈	1	1	yes	characteristic
43	C₂²×C₅×C₁₆	C₃×C₁₃	1	1	yes	characteristic, direct product factor, Hall
44	C₂³×C₃×C₁₃	C₅×C₈	1	1	yes	characteristic
45	C₂×C₃×C₄×C₁₃	C₅×C₈	1	3	yes	—
46	C₂×C₃×C₄×C₁₃	C₂×C₄×C₅	1	3	yes	—
47	C₃×C₈×C₁₃	C₂×C₄×C₅	1	3	yes	—
48	C₃×C₈×C₁₃	C₂³×C₅	1	1	yes	characteristic
49	C₂²×C₅×C₁₃	C₃×C₁₆	1	4	yes	direct product factor
50	C₂²×C₅×C₁₃	C₂×C₃×C₈	1	3	yes	—
51	C₄×C₅×C₁₃	C₂×C₃×C₈	1	3	yes	—
52	C₄×C₅×C₁₃	C₂²×C₃×C₄	1	1	yes	characteristic
53	C₂²×C₃×C₄×C₅	C₄×C₁₃	1	1	yes	characteristic
54	C₂×C₃×C₅×C₈	C₄×C₁₃	1	3	yes	—
55	C₂×C₃×C₅×C₈	C₂²×C₁₃	1	3	yes	—
56	C₃×C₅×C₁₆	C₂²×C₁₃	1	4	yes	direct product factor
57	C₂²×C₄×C₁₃	C₃×C₄×C₅	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₃×C₄×C₅	1	3	yes	—
59	C₂×C₈×C₁₃	C₂²×C₃×C₅	1	3	yes	—
60	C₁₃×C₁₆	C₂²×C₃×C₅	1	4	yes	direct product factor
61	C₃×C₅×C₁₃	C₂²×C₁₆	1	1	yes	characteristic, direct product factor, Hall
62	C₂²×C₃×C₁₆	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
63	C₂²×C₅×C₈	C₂×C₃×C₁₃	1	1	yes	characteristic
64	C₂×C₅×C₁₆	C₂×C₃×C₁₃	1	6	yes	direct product factor
65	C₂²×C₃×C₁₃	C₅×C₁₆	1	4	yes	direct product factor
66	C₂²×C₃×C₁₃	C₂×C₅×C₈	1	3	yes	—
67	C₃×C₄×C₁₃	C₂×C₅×C₈	1	3	yes	—
68	C₃×C₄×C₁₃	C₂²×C₄×C₅	1	1	yes	characteristic
69	C₂×C₅×C₁₃	C₂×C₃×C₁₆	1	6	yes	direct product factor
70	C₂×C₅×C₁₃	C₂²×C₃×C₈	1	1	yes	characteristic
71	C₂³×C₃×C₅	C₈×C₁₃	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₈×C₁₃	1	3	yes	—
73	C₂×C₃×C₄×C₅	C₂×C₄×C₁₃	1	3	yes	—
74	C₃×C₅×C₈	C₂×C₄×C₁₃	1	3	yes	—
75	C₃×C₅×C₈	C₂³×C₁₃	1	1	yes	characteristic
76	C₂³×C₁₃	C₃×C₅×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₃×C₅×C₈	1	3	yes	—
78	C₂×C₄×C₁₃	C₂×C₃×C₄×C₅	1	3	yes	—
79	C₈×C₁₃	C₂×C₃×C₄×C₅	1	3	yes	—
80	C₈×C₁₃	C₂³×C₃×C₅	1	1	yes	characteristic
81	C₂²×C₃×C₈	C₂×C₅×C₁₃	1	1	yes	characteristic
82	C₂×C₃×C₁₆	C₂×C₅×C₁₃	1	6	yes	direct product factor
83	C₂²×C₄×C₅	C₃×C₄×C₁₃	1	1	yes	characteristic
84	C₂×C₅×C₈	C₃×C₄×C₁₃	1	3	yes	—
85	C₂×C₅×C₈	C₂²×C₃×C₁₃	1	3	yes	—
86	C₅×C₁₆	C₂²×C₃×C₁₃	1	4	yes	direct product factor
87	C₂×C₃×C₁₃	C₂×C₅×C₁₆	1	6	yes	direct product factor
88	C₂×C₃×C₁₃	C₂²×C₅×C₈	1	1	yes	characteristic
89	C₅×C₁₃	C₂²×C₃×C₁₆	1	1	yes	characteristic, direct product factor, Hall
90	C₂²×C₁₆	C₃×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²×C₃×C₅	C₁₃×C₁₆	1	4	yes	direct product factor
92	C₂²×C₃×C₅	C₂×C₈×C₁₃	1	3	yes	—
93	C₃×C₄×C₅	C₂×C₈×C₁₃	1	3	yes	—
94	C₃×C₄×C₅	C₂²×C₄×C₁₃	1	1	yes	characteristic
95	C₂²×C₁₃	C₃×C₅×C₁₆	1	4	yes	direct product factor
96	C₂²×C₁₃	C₂×C₃×C₅×C₈	1	3	yes	—
97	C₄×C₁₃	C₂×C₃×C₅×C₈	1	3	yes	—
98	C₄×C₁₃	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₄×C₅×C₁₃	1	1	yes	characteristic
100	C₂×C₃×C₈	C₄×C₅×C₁₃	1	3	yes	—
101	C₂×C₃×C₈	C₂²×C₅×C₁₃	1	3	yes	—
102	C₃×C₁₆	C₂²×C₅×C₁₃	1	4	yes	direct product factor
103	C₂³×C₅	C₃×C₈×C₁₃	1	1	yes	characteristic
104	C₂×C₄×C₅	C₃×C₈×C₁₃	1	3	yes	—
105	C₂×C₄×C₅	C₂×C₃×C₄×C₁₃	1	3	yes	—
106	C₅×C₈	C₂×C₃×C₄×C₁₃	1	3	yes	—
107	C₅×C₈	C₂³×C₃×C₁₃	1	1	yes	characteristic
108	C₃×C₁₃	C₂²×C₅×C₁₆	1	1	yes	characteristic, direct product factor, Hall
109	C₂²×C₈	C₂×C₃×C₅×C₁₃	1	1	yes	characteristic
110	C₂×C₁₆	C₂×C₃×C₅×C₁₃	1	6	yes	direct product factor
111	C₂×C₃×C₅	C₂×C₁₃×C₁₆	1	6	yes	direct product factor
112	C₂×C₃×C₅	C₂²×C₈×C₁₃	1	1	yes	characteristic
113	C₂×C₁₃	C₂×C₃×C₅×C₁₆	1	6	yes	direct product factor
114	C₂×C₁₃	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
115	C₂³×C₃	C₅×C₈×C₁₃	1	1	yes	characteristic
116	C₂×C₃×C₄	C₅×C₈×C₁₃	1	3	yes	—
117	C₂×C₃×C₄	C₂×C₄×C₅×C₁₃	1	3	yes	—
118	C₃×C₈	C₂×C₄×C₅×C₁₃	1	3	yes	—
119	C₃×C₈	C₂³×C₅×C₁₃	1	1	yes	characteristic
120	C₂²×C₅	C₃×C₁₃×C₁₆	1	4	yes	direct product factor
121	C₂²×C₅	C₂×C₃×C₈×C₁₃	1	3	yes	—
122	C₄×C₅	C₂×C₃×C₈×C₁₃	1	3	yes	—
123	C₄×C₅	C₂²×C₃×C₄×C₁₃	1	1	yes	characteristic
124	C₂²×C₄	C₃×C₄×C₅×C₁₃	1	1	yes	characteristic
125	C₂×C₈	C₃×C₄×C₅×C₁₃	1	3	yes	—
126	C₂×C₈	C₂²×C₃×C₅×C₁₃	1	3	yes	—
127	C₁₆	C₂²×C₃×C₅×C₁₃	1	4	yes	direct product factor
128	C₃×C₅	C₂²×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Hall
129	C₁₃	C₂²×C₃×C₅×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
130	C₂²×C₃	C₅×C₁₃×C₁₆	1	4	yes	direct product factor
131	C₂²×C₃	C₂×C₅×C₈×C₁₃	1	3	yes	—
132	C₃×C₄	C₂×C₅×C₈×C₁₃	1	3	yes	—
133	C₃×C₄	C₂²×C₄×C₅×C₁₃	1	1	yes	characteristic
134	C₂×C₅	C₂×C₃×C₁₃×C₁₆	1	6	yes	direct product factor
135	C₂×C₅	C₂²×C₃×C₈×C₁₃	1	1	yes	characteristic
136	C₂³	C₃×C₅×C₈×C₁₃	1	1	yes	characteristic
137	C₂×C₄	C₃×C₅×C₈×C₁₃	1	3	yes	—
138	C₂×C₄	C₂×C₃×C₄×C₅×C₁₃	1	3	yes	—
139	C₈	C₂×C₃×C₄×C₅×C₁₃	1	3	yes	—
140	C₈	C₂³×C₃×C₅×C₁₃	1	1	yes	characteristic
141	C₂×C₃	C₂×C₅×C₁₃×C₁₆	1	6	yes	direct product factor
142	C₂×C₃	C₂²×C₅×C₈×C₁₃	1	1	yes	characteristic
143	C₅	C₂²×C₃×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²	C₃×C₅×C₁₃×C₁₆	1	4	yes	direct product factor
145	C₂²	C₂×C₃×C₅×C₈×C₁₃	1	3	yes	—
146	C₄	C₂×C₃×C₅×C₈×C₁₃	1	3	yes	—
147	C₄	C₂²×C₃×C₄×C₅×C₁₃	1	1	yes	characteristic
148	C₃	C₂²×C₅×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
149	C₂	C₂×C₃×C₅×C₁₃×C₁₆	1	6	yes	direct product factor
150	C₂	C₂²×C₃×C₅×C₈×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂³×C₃×C₅×C₁₃
socle, prime radical	C₂³×C₃×C₅×C₁₃	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×C₁₃×C₁₆	C₂	direct (N × H)
3. C₂²×C₅×C₁₃×C₁₆	C₃	direct (N × H)
7. C₃×C₅×C₁₃×C₁₆	C₂²	direct (N × H)
8. C₂²×C₃×C₁₃×C₁₆	C₅	direct (N × H)
10. C₂×C₅×C₁₃×C₁₆	C₂×C₃	direct (N × H)
17. C₂×C₃×C₁₃×C₁₆	C₂×C₅	direct (N × H)
21. C₅×C₁₃×C₁₆	C₂²×C₃	direct (N × H)
22. C₂²×C₃×C₅×C₁₆	C₁₃	direct (N × H)
23. C₂²×C₁₃×C₁₆	C₃×C₅	direct (N × H)
24. C₂²×C₃×C₅×C₁₃	C₁₆	direct (N × H)
31. C₃×C₁₃×C₁₆	C₂²×C₅	direct (N × H)
38. C₂×C₃×C₅×C₁₆	C₂×C₁₃	direct (N × H)
40. C₂×C₁₃×C₁₆	C₂×C₃×C₅	direct (N × H)
41. C₂×C₃×C₅×C₁₃	C₂×C₁₆	direct (N × H)
43. C₂²×C₅×C₁₆	C₃×C₁₃	direct (N × H)
49. C₂²×C₅×C₁₃	C₃×C₁₆	direct (N × H)
56. C₃×C₅×C₁₆	C₂²×C₁₃	direct (N × H)
60. C₁₃×C₁₆	C₂²×C₃×C₅	direct (N × H)
61. C₃×C₅×C₁₃	C₂²×C₁₆	direct (N × H)
62. C₂²×C₃×C₁₆	C₅×C₁₃	direct (N × H)
64. C₂×C₅×C₁₆	C₂×C₃×C₁₃	direct (N × H)
65. C₂²×C₃×C₁₃	C₅×C₁₆	direct (N × H)
69. C₂×C₅×C₁₃	C₂×C₃×C₁₆	direct (N × H)
82. C₂×C₃×C₁₆	C₂×C₅×C₁₃	direct (N × H)
86. C₅×C₁₆	C₂²×C₃×C₁₃	direct (N × H)
87. C₂×C₃×C₁₃	C₂×C₅×C₁₆	direct (N × H)
89. C₅×C₁₃	C₂²×C₃×C₁₆	direct (N × H)
90. C₂²×C₁₆	C₃×C₅×C₁₃	direct (N × H)
91. C₂²×C₃×C₅	C₁₃×C₁₆	direct (N × H)
95. C₂²×C₁₃	C₃×C₅×C₁₆	direct (N × H)
102. C₃×C₁₆	C₂²×C₅×C₁₃	direct (N × H)
108. C₃×C₁₃	C₂²×C₅×C₁₆	direct (N × H)
110. C₂×C₁₆	C₂×C₃×C₅×C₁₃	direct (N × H)
111. C₂×C₃×C₅	C₂×C₁₃×C₁₆	direct (N × H)
113. C₂×C₁₃	C₂×C₃×C₅×C₁₆	direct (N × H)
120. C₂²×C₅	C₃×C₁₃×C₁₆	direct (N × H)
127. C₁₆	C₂²×C₃×C₅×C₁₃	direct (N × H)
128. C₃×C₅	C₂²×C₁₃×C₁₆	direct (N × H)
129. C₁₃	C₂²×C₃×C₅×C₁₆	direct (N × H)
130. C₂²×C₃	C₅×C₁₃×C₁₆	direct (N × H)
134. C₂×C₅	C₂×C₃×C₁₃×C₁₆	direct (N × H)
141. C₂×C₃	C₂×C₅×C₁₃×C₁₆	direct (N × H)
143. C₅	C₂²×C₃×C₁₃×C₁₆	direct (N × H)
144. C₂²	C₃×C₅×C₁₃×C₁₆	direct (N × H)
148. C₃	C₂²×C₅×C₁₃×C₁₆	direct (N × H)
149. C₂	C₂×C₃×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
C₂²×C₃×C₅×C₈×C₁₃	2
C₂×C₃×C₅×C₁₃×C₁₆	2
C₂²×C₅×C₁₃×C₁₆	3
C₂²×C₃×C₁₃×C₁₆	5
C₂²×C₃×C₅×C₁₆	13

Subgroups — C22×C3×C5×C13×C16 #72190823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅×C₁₃×C₁₆ #72190823