Subgroups — C₂²×C₃×C₅×C₁₁×C₁₆ #72189152

Order 10560.

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×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₁₁×C₁₆	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₅×C₁₁×C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₄×C₅×C₁₁	C₄	1	1	yes	characteristic
5	C₂×C₃×C₅×C₈×C₁₁	C₄	1	3	yes	—
6	C₂×C₃×C₅×C₈×C₁₁	C₂²	1	3	yes	—
7	C₃×C₅×C₁₁×C₁₆	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₁₁×C₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₅×C₈×C₁₁	C₂×C₃	1	1	yes	characteristic
10	C₂×C₅×C₁₁×C₁₆	C₂×C₃	1	6	yes	direct product factor
11	C₂³×C₃×C₅×C₁₁	C₈	1	1	yes	characteristic
12	C₂×C₃×C₄×C₅×C₁₁	C₈	1	3	yes	—
13	C₂×C₃×C₄×C₅×C₁₁	C₂×C₄	1	3	yes	—
14	C₃×C₅×C₈×C₁₁	C₂×C₄	1	3	yes	—
15	C₃×C₅×C₈×C₁₁	C₂³	1	1	yes	characteristic
16	C₂²×C₃×C₈×C₁₁	C₂×C₅	1	1	yes	characteristic
17	C₂×C₃×C₁₁×C₁₆	C₂×C₅	1	6	yes	direct product factor
18	C₂²×C₃×C₅×C₁₆	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₄×C₅×C₁₁	C₃×C₄	1	1	yes	characteristic
20	C₂×C₅×C₈×C₁₁	C₃×C₄	1	3	yes	—
21	C₂×C₅×C₈×C₁₁	C₂²×C₃	1	3	yes	—
22	C₅×C₁₁×C₁₆	C₂²×C₃	1	4	yes	direct product factor
23	C₂²×C₁₁×C₁₆	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
24	C₂²×C₃×C₅×C₁₁	C₁₆	1	4	yes	direct product factor
25	C₂²×C₃×C₅×C₁₁	C₂×C₈	1	3	yes	—
26	C₃×C₄×C₅×C₁₁	C₂×C₈	1	3	yes	—
27	C₃×C₄×C₅×C₁₁	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₁₁	C₄×C₅	1	1	yes	characteristic
29	C₂×C₃×C₈×C₁₁	C₄×C₅	1	3	yes	—
30	C₂×C₃×C₈×C₁₁	C₂²×C₅	1	3	yes	—
31	C₃×C₁₁×C₁₆	C₂²×C₅	1	4	yes	direct product factor
32	C₂²×C₃×C₅×C₈	C₂×C₁₁	1	1	yes	characteristic
33	C₂×C₃×C₅×C₁₆	C₂×C₁₁	1	6	yes	direct product factor
34	C₂³×C₅×C₁₁	C₃×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅×C₁₁	C₃×C₈	1	3	yes	—
36	C₂×C₄×C₅×C₁₁	C₂×C₃×C₄	1	3	yes	—
37	C₅×C₈×C₁₁	C₂×C₃×C₄	1	3	yes	—
38	C₅×C₈×C₁₁	C₂³×C₃	1	1	yes	characteristic
39	C₂²×C₈×C₁₁	C₂×C₃×C₅	1	1	yes	characteristic
40	C₂×C₁₁×C₁₆	C₂×C₃×C₅	1	6	yes	direct product factor
41	C₂×C₃×C₅×C₁₁	C₂×C₁₆	1	6	yes	direct product factor
42	C₂×C₃×C₅×C₁₁	C₂²×C₈	1	1	yes	characteristic
43	C₂²×C₅×C₁₆	C₃×C₁₁	1	1	yes	characteristic, direct product factor, Hall
44	C₂³×C₃×C₁₁	C₅×C₈	1	1	yes	characteristic
45	C₂×C₃×C₄×C₁₁	C₅×C₈	1	3	yes	—
46	C₂×C₃×C₄×C₁₁	C₂×C₄×C₅	1	3	yes	—
47	C₃×C₈×C₁₁	C₂×C₄×C₅	1	3	yes	—
48	C₃×C₈×C₁₁	C₂³×C₅	1	1	yes	characteristic
49	C₂²×C₃×C₄×C₅	C₄×C₁₁	1	1	yes	characteristic
50	C₂×C₃×C₅×C₈	C₄×C₁₁	1	3	yes	—
51	C₂×C₃×C₅×C₈	C₂²×C₁₁	1	3	yes	—
52	C₃×C₅×C₁₆	C₂²×C₁₁	1	4	yes	direct product factor
53	C₂²×C₅×C₁₁	C₃×C₁₆	1	4	yes	direct product factor
54	C₂²×C₅×C₁₁	C₂×C₃×C₈	1	3	yes	—
55	C₄×C₅×C₁₁	C₂×C₃×C₈	1	3	yes	—
56	C₄×C₅×C₁₁	C₂²×C₃×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₁₆	C₅×C₁₁	1	1	yes	characteristic, direct product factor, Hall
58	C₂²×C₄×C₁₁	C₃×C₄×C₅	1	1	yes	characteristic
59	C₂×C₈×C₁₁	C₃×C₄×C₅	1	3	yes	—
60	C₂×C₈×C₁₁	C₂²×C₃×C₅	1	3	yes	—
61	C₁₁×C₁₆	C₂²×C₃×C₅	1	4	yes	direct product factor
62	C₃×C₅×C₁₁	C₂²×C₁₆	1	1	yes	characteristic, direct product factor, Hall
63	C₂²×C₅×C₈	C₂×C₃×C₁₁	1	1	yes	characteristic
64	C₂×C₅×C₁₆	C₂×C₃×C₁₁	1	6	yes	direct product factor
65	C₂²×C₃×C₁₁	C₅×C₁₆	1	4	yes	direct product factor
66	C₂²×C₃×C₁₁	C₂×C₅×C₈	1	3	yes	—
67	C₃×C₄×C₁₁	C₂×C₅×C₈	1	3	yes	—
68	C₃×C₄×C₁₁	C₂²×C₄×C₅	1	1	yes	characteristic
69	C₂³×C₃×C₅	C₈×C₁₁	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₈×C₁₁	1	3	yes	—
71	C₂×C₃×C₄×C₅	C₂×C₄×C₁₁	1	3	yes	—
72	C₃×C₅×C₈	C₂×C₄×C₁₁	1	3	yes	—
73	C₃×C₅×C₈	C₂³×C₁₁	1	1	yes	characteristic
74	C₂×C₅×C₁₁	C₂×C₃×C₁₆	1	6	yes	direct product factor
75	C₂×C₅×C₁₁	C₂²×C₃×C₈	1	1	yes	characteristic
76	C₂²×C₃×C₈	C₂×C₅×C₁₁	1	1	yes	characteristic
77	C₂×C₃×C₁₆	C₂×C₅×C₁₁	1	6	yes	direct product factor
78	C₂³×C₁₁	C₃×C₅×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₃×C₅×C₈	1	3	yes	—
80	C₂×C₄×C₁₁	C₂×C₃×C₄×C₅	1	3	yes	—
81	C₈×C₁₁	C₂×C₃×C₄×C₅	1	3	yes	—
82	C₈×C₁₁	C₂³×C₃×C₅	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₃×C₄×C₁₁	1	1	yes	characteristic
84	C₂×C₅×C₈	C₃×C₄×C₁₁	1	3	yes	—
85	C₂×C₅×C₈	C₂²×C₃×C₁₁	1	3	yes	—
86	C₅×C₁₆	C₂²×C₃×C₁₁	1	4	yes	direct product factor
87	C₂×C₃×C₁₁	C₂×C₅×C₁₆	1	6	yes	direct product factor
88	C₂×C₃×C₁₁	C₂²×C₅×C₈	1	1	yes	characteristic
89	C₂²×C₁₆	C₃×C₅×C₁₁	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×C₃×C₅	C₁₁×C₁₆	1	4	yes	direct product factor
91	C₂²×C₃×C₅	C₂×C₈×C₁₁	1	3	yes	—
92	C₃×C₄×C₅	C₂×C₈×C₁₁	1	3	yes	—
93	C₃×C₄×C₅	C₂²×C₄×C₁₁	1	1	yes	characteristic
94	C₅×C₁₁	C₂²×C₃×C₁₆	1	1	yes	characteristic, direct product factor, Hall
95	C₂²×C₃×C₄	C₄×C₅×C₁₁	1	1	yes	characteristic
96	C₂×C₃×C₈	C₄×C₅×C₁₁	1	3	yes	—
97	C₂×C₃×C₈	C₂²×C₅×C₁₁	1	3	yes	—
98	C₃×C₁₆	C₂²×C₅×C₁₁	1	4	yes	direct product factor
99	C₂²×C₁₁	C₃×C₅×C₁₆	1	4	yes	direct product factor
100	C₂²×C₁₁	C₂×C₃×C₅×C₈	1	3	yes	—
101	C₄×C₁₁	C₂×C₃×C₅×C₈	1	3	yes	—
102	C₄×C₁₁	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
103	C₂³×C₅	C₃×C₈×C₁₁	1	1	yes	characteristic
104	C₂×C₄×C₅	C₃×C₈×C₁₁	1	3	yes	—
105	C₂×C₄×C₅	C₂×C₃×C₄×C₁₁	1	3	yes	—
106	C₅×C₈	C₂×C₃×C₄×C₁₁	1	3	yes	—
107	C₅×C₈	C₂³×C₃×C₁₁	1	1	yes	characteristic
108	C₃×C₁₁	C₂²×C₅×C₁₆	1	1	yes	characteristic, direct product factor, Hall
109	C₂²×C₈	C₂×C₃×C₅×C₁₁	1	1	yes	characteristic
110	C₂×C₁₆	C₂×C₃×C₅×C₁₁	1	6	yes	direct product factor
111	C₂×C₃×C₅	C₂×C₁₁×C₁₆	1	6	yes	direct product factor
112	C₂×C₃×C₅	C₂²×C₈×C₁₁	1	1	yes	characteristic
113	C₂³×C₃	C₅×C₈×C₁₁	1	1	yes	characteristic
114	C₂×C₃×C₄	C₅×C₈×C₁₁	1	3	yes	—
115	C₂×C₃×C₄	C₂×C₄×C₅×C₁₁	1	3	yes	—
116	C₃×C₈	C₂×C₄×C₅×C₁₁	1	3	yes	—
117	C₃×C₈	C₂³×C₅×C₁₁	1	1	yes	characteristic
118	C₂×C₁₁	C₂×C₃×C₅×C₁₆	1	6	yes	direct product factor
119	C₂×C₁₁	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
120	C₂²×C₅	C₃×C₁₁×C₁₆	1	4	yes	direct product factor
121	C₂²×C₅	C₂×C₃×C₈×C₁₁	1	3	yes	—
122	C₄×C₅	C₂×C₃×C₈×C₁₁	1	3	yes	—
123	C₄×C₅	C₂²×C₃×C₄×C₁₁	1	1	yes	characteristic
124	C₂²×C₄	C₃×C₄×C₅×C₁₁	1	1	yes	characteristic
125	C₂×C₈	C₃×C₄×C₅×C₁₁	1	3	yes	—
126	C₂×C₈	C₂²×C₃×C₅×C₁₁	1	3	yes	—
127	C₁₆	C₂²×C₃×C₅×C₁₁	1	4	yes	direct product factor
128	C₃×C₅	C₂²×C₁₁×C₁₆	1	1	yes	characteristic, direct product factor, Hall
129	C₂²×C₃	C₅×C₁₁×C₁₆	1	4	yes	direct product factor
130	C₂²×C₃	C₂×C₅×C₈×C₁₁	1	3	yes	—
131	C₃×C₄	C₂×C₅×C₈×C₁₁	1	3	yes	—
132	C₃×C₄	C₂²×C₄×C₅×C₁₁	1	1	yes	characteristic
133	C₁₁	C₂²×C₃×C₅×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×C₅	C₂×C₃×C₁₁×C₁₆	1	6	yes	direct product factor
135	C₂×C₅	C₂²×C₃×C₈×C₁₁	1	1	yes	characteristic
136	C₂³	C₃×C₅×C₈×C₁₁	1	1	yes	characteristic
137	C₂×C₄	C₃×C₅×C₈×C₁₁	1	3	yes	—
138	C₂×C₄	C₂×C₃×C₄×C₅×C₁₁	1	3	yes	—
139	C₈	C₂×C₃×C₄×C₅×C₁₁	1	3	yes	—
140	C₈	C₂³×C₃×C₅×C₁₁	1	1	yes	characteristic
141	C₂×C₃	C₂×C₅×C₁₁×C₁₆	1	6	yes	direct product factor
142	C₂×C₃	C₂²×C₅×C₈×C₁₁	1	1	yes	characteristic
143	C₅	C₂²×C₃×C₁₁×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²	C₃×C₅×C₁₁×C₁₆	1	4	yes	direct product factor
145	C₂²	C₂×C₃×C₅×C₈×C₁₁	1	3	yes	—
146	C₄	C₂×C₃×C₅×C₈×C₁₁	1	3	yes	—
147	C₄	C₂²×C₃×C₄×C₅×C₁₁	1	1	yes	characteristic
148	C₃	C₂²×C₅×C₁₁×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
149	C₂	C₂×C₃×C₅×C₁₁×C₁₆	1	6	yes	direct product factor
150	C₂	C₂²×C₃×C₅×C₈×C₁₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂³×C₃×C₅×C₁₁
socle, prime radical	C₂³×C₃×C₅×C₁₁	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×C₁₁×C₁₆	C₂	direct (N × H)
3. C₂²×C₅×C₁₁×C₁₆	C₃	direct (N × H)
7. C₃×C₅×C₁₁×C₁₆	C₂²	direct (N × H)
8. C₂²×C₃×C₁₁×C₁₆	C₅	direct (N × H)
10. C₂×C₅×C₁₁×C₁₆	C₂×C₃	direct (N × H)
17. C₂×C₃×C₁₁×C₁₆	C₂×C₅	direct (N × H)
18. C₂²×C₃×C₅×C₁₆	C₁₁	direct (N × H)
22. C₅×C₁₁×C₁₆	C₂²×C₃	direct (N × H)
23. C₂²×C₁₁×C₁₆	C₃×C₅	direct (N × H)
24. C₂²×C₃×C₅×C₁₁	C₁₆	direct (N × H)
31. C₃×C₁₁×C₁₆	C₂²×C₅	direct (N × H)
33. C₂×C₃×C₅×C₁₆	C₂×C₁₁	direct (N × H)
40. C₂×C₁₁×C₁₆	C₂×C₃×C₅	direct (N × H)
41. C₂×C₃×C₅×C₁₁	C₂×C₁₆	direct (N × H)
43. C₂²×C₅×C₁₆	C₃×C₁₁	direct (N × H)
52. C₃×C₅×C₁₆	C₂²×C₁₁	direct (N × H)
53. C₂²×C₅×C₁₁	C₃×C₁₆	direct (N × H)
57. C₂²×C₃×C₁₆	C₅×C₁₁	direct (N × H)
61. C₁₁×C₁₆	C₂²×C₃×C₅	direct (N × H)
62. C₃×C₅×C₁₁	C₂²×C₁₆	direct (N × H)
64. C₂×C₅×C₁₆	C₂×C₃×C₁₁	direct (N × H)
65. C₂²×C₃×C₁₁	C₅×C₁₆	direct (N × H)
74. C₂×C₅×C₁₁	C₂×C₃×C₁₆	direct (N × H)
77. C₂×C₃×C₁₆	C₂×C₅×C₁₁	direct (N × H)
86. C₅×C₁₆	C₂²×C₃×C₁₁	direct (N × H)
87. C₂×C₃×C₁₁	C₂×C₅×C₁₆	direct (N × H)
89. C₂²×C₁₆	C₃×C₅×C₁₁	direct (N × H)
90. C₂²×C₃×C₅	C₁₁×C₁₆	direct (N × H)
94. C₅×C₁₁	C₂²×C₃×C₁₆	direct (N × H)
98. C₃×C₁₆	C₂²×C₅×C₁₁	direct (N × H)
99. C₂²×C₁₁	C₃×C₅×C₁₆	direct (N × H)
108. C₃×C₁₁	C₂²×C₅×C₁₆	direct (N × H)
110. C₂×C₁₆	C₂×C₃×C₅×C₁₁	direct (N × H)
111. C₂×C₃×C₅	C₂×C₁₁×C₁₆	direct (N × H)
118. C₂×C₁₁	C₂×C₃×C₅×C₁₆	direct (N × H)
120. C₂²×C₅	C₃×C₁₁×C₁₆	direct (N × H)
127. C₁₆	C₂²×C₃×C₅×C₁₁	direct (N × H)
128. C₃×C₅	C₂²×C₁₁×C₁₆	direct (N × H)
129. C₂²×C₃	C₅×C₁₁×C₁₆	direct (N × H)
133. C₁₁	C₂²×C₃×C₅×C₁₆	direct (N × H)
134. C₂×C₅	C₂×C₃×C₁₁×C₁₆	direct (N × H)
141. C₂×C₃	C₂×C₅×C₁₁×C₁₆	direct (N × H)
143. C₅	C₂²×C₃×C₁₁×C₁₆	direct (N × H)
144. C₂²	C₃×C₅×C₁₁×C₁₆	direct (N × H)
148. C₃	C₂²×C₅×C₁₁×C₁₆	direct (N × H)
149. C₂	C₂×C₃×C₅×C₁₁×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×C₁₁	2
C₂×C₃×C₅×C₁₁×C₁₆	2
C₂²×C₅×C₁₁×C₁₆	3
C₂²×C₃×C₁₁×C₁₆	5
C₂²×C₃×C₅×C₁₆	11

Subgroups — C22×C3×C5×C11×C16 #72189152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₅×C₁₁×C₁₆ #72189152