Subgroups — C₂⁹×C₃×C₁₆ #72201890

Order 24576.

1788344092 conjugacy classes of subgroups.

The subgroup lattice has 136 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁹×C₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂⁸×C₃×C₁₆	C₂	1	1022	yes	maximal, direct product factor
3	C₂⁹×C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂⁹×C₃×C₄	C₄	1	1	yes	characteristic
5	C₂⁸×C₃×C₈	C₄	1	511	yes	—
6	C₂⁸×C₃×C₈	C₂²	1	511	yes	—
7	C₂⁷×C₃×C₁₆	C₂²	1	173740	yes	direct product factor
8	C₂⁹×C₈	C₂×C₃	1	1	yes	characteristic
9	C₂⁸×C₁₆	C₂×C₃	1	1022	yes	direct product factor
10	C₂⁸×C₃×C₄	C₈	1	511	yes	—
11	C₂⁸×C₃×C₄	C₂×C₄	1	511	yes	—
12	C₂¹⁰×C₃	C₈	1	1	yes	characteristic
13	C₂⁷×C₃×C₈	C₂×C₄	1	130305	yes	—
14	C₂⁷×C₃×C₈	C₂³	1	43435	yes	—
15	C₂⁶×C₃×C₁₆	C₂³	1	6304280	yes	direct product factor
16	C₂⁹×C₄	C₃×C₄	1	1	yes	characteristic
17	C₂⁸×C₈	C₃×C₄	1	511	yes	—
18	C₂⁸×C₈	C₂²×C₃	1	511	yes	—
19	C₂⁷×C₁₆	C₂²×C₃	1	173740	yes	direct product factor
20	C₂⁷×C₃×C₄	C₂×C₈	1	130305	yes	—
21	C₂⁷×C₃×C₄	C₂²×C₄	1	43435	yes	—
22	C₂⁹×C₃	C₁₆	1	512	yes	direct product factor
23	C₂⁹×C₃	C₂×C₈	1	511	yes	—
24	C₂⁶×C₃×C₈	C₂²×C₄	1	5516245	yes	—
25	C₂⁶×C₃×C₈	C₂⁴	1	788035	yes	—
26	C₂⁵×C₃×C₁₆	C₂⁴	1	52955952	yes	direct product factor
27	C₂¹⁰	C₃×C₈	1	1	yes	characteristic
28	C₂⁸×C₄	C₃×C₈	1	511	yes	—
29	C₂⁸×C₄	C₂×C₃×C₄	1	511	yes	—
30	C₂⁷×C₈	C₂×C₃×C₄	1	130305	yes	—
31	C₂⁷×C₈	C₂³×C₃	1	43435	yes	—
32	C₂⁶×C₁₆	C₂³×C₃	1	6304280	yes	direct product factor
33	C₂⁸×C₃	C₂×C₁₆	1	130816	yes	direct product factor
34	C₂⁸×C₃	C₂²×C₈	1	43435	yes	—
35	C₂⁶×C₃×C₄	C₂²×C₈	1	5516245	yes	—
36	C₂⁶×C₃×C₄	C₂³×C₄	1	788035	yes	—
37	C₂⁵×C₃×C₈	C₂³×C₄	1	49646205	yes	—
38	C₂⁵×C₃×C₈	C₂⁵	1	3309747	yes	—
39	C₂⁴×C₃×C₁₆	C₂⁵	1	105911904	yes	direct product factor
40	C₂⁹	C₃×C₁₆	1	512	yes	direct product factor
41	C₂⁹	C₂×C₃×C₈	1	511	yes	—
42	C₂⁷×C₄	C₂×C₃×C₈	1	130305	yes	—
43	C₂⁷×C₄	C₂²×C₃×C₄	1	43435	yes	—
44	C₂⁶×C₈	C₂²×C₃×C₄	1	5516245	yes	—
45	C₂⁶×C₈	C₂⁴×C₃	1	788035	yes	—
46	C₂⁵×C₁₆	C₂⁴×C₃	1	52955952	yes	direct product factor
47	C₂⁷×C₃	C₂²×C₁₆	1	5559680	yes	direct product factor
48	C₂⁷×C₃	C₂³×C₈	1	788035	yes	—
49	C₂⁵×C₃×C₄	C₂³×C₈	1	49646205	yes	—
50	C₂⁵×C₃×C₄	C₂⁴×C₄	1	3309747	yes	—
51	C₂⁴×C₃×C₈	C₂⁴×C₄	1	102602157	yes	—
52	C₂⁴×C₃×C₈	C₂⁶	1	3309747	yes	—
53	C₂³×C₃×C₁₆	C₂⁶	1	50434240	yes	direct product factor
54	C₂⁸	C₂×C₃×C₁₆	1	130816	yes	direct product factor
55	C₂⁸	C₂²×C₃×C₈	1	43435	yes	—
56	C₂⁶×C₄	C₂²×C₃×C₈	1	5516245	yes	—
57	C₂⁶×C₄	C₂³×C₃×C₄	1	788035	yes	—
58	C₂⁵×C₈	C₂³×C₃×C₄	1	49646205	yes	—
59	C₂⁵×C₈	C₂⁵×C₃	1	3309747	yes	—
60	C₂⁴×C₁₆	C₂⁵×C₃	1	105911904	yes	direct product factor
61	C₂⁶×C₃	C₂³×C₁₆	1	50434240	yes	direct product factor
62	C₂⁶×C₃	C₂⁴×C₈	1	3309747	yes	—
63	C₂⁴×C₃×C₄	C₂⁴×C₈	1	102602157	yes	—
64	C₂⁴×C₃×C₄	C₂⁵×C₄	1	3309747	yes	—
65	C₂³×C₃×C₈	C₂⁵×C₄	1	49646205	yes	—
66	C₂³×C₃×C₈	C₂⁷	1	788035	yes	—
67	C₂²×C₃×C₁₆	C₂⁷	1	5559680	yes	direct product factor
68	C₂⁷	C₂²×C₃×C₁₆	1	5559680	yes	direct product factor
69	C₂⁷	C₂³×C₃×C₈	1	788035	yes	—
70	C₂⁵×C₄	C₂³×C₃×C₈	1	49646205	yes	—
71	C₂⁵×C₄	C₂⁴×C₃×C₄	1	3309747	yes	—
72	C₂⁴×C₈	C₂⁴×C₃×C₄	1	102602157	yes	—
73	C₂⁴×C₈	C₂⁶×C₃	1	3309747	yes	—
74	C₂³×C₁₆	C₂⁶×C₃	1	50434240	yes	direct product factor
75	C₂⁵×C₃	C₂⁴×C₁₆	1	105911904	yes	direct product factor
76	C₂⁵×C₃	C₂⁵×C₈	1	3309747	yes	—
77	C₂³×C₃×C₄	C₂⁵×C₈	1	49646205	yes	—
78	C₂³×C₃×C₄	C₂⁶×C₄	1	788035	yes	—
79	C₂²×C₃×C₈	C₂⁶×C₄	1	5516245	yes	—
80	C₂²×C₃×C₈	C₂⁸	1	43435	yes	—
81	C₂×C₃×C₁₆	C₂⁸	1	130816	yes	direct product factor
82	C₂⁶	C₂³×C₃×C₁₆	1	50434240	yes	direct product factor
83	C₂⁶	C₂⁴×C₃×C₈	1	3309747	yes	—
84	C₂⁴×C₄	C₂⁴×C₃×C₈	1	102602157	yes	—
85	C₂⁴×C₄	C₂⁵×C₃×C₄	1	3309747	yes	—
86	C₂³×C₈	C₂⁵×C₃×C₄	1	49646205	yes	—
87	C₂³×C₈	C₂⁷×C₃	1	788035	yes	—
88	C₂²×C₁₆	C₂⁷×C₃	1	5559680	yes	direct product factor
89	C₂⁴×C₃	C₂⁵×C₁₆	1	52955952	yes	direct product factor
90	C₂⁴×C₃	C₂⁶×C₈	1	788035	yes	—
91	C₂²×C₃×C₄	C₂⁶×C₈	1	5516245	yes	—
92	C₂²×C₃×C₄	C₂⁷×C₄	1	43435	yes	—
93	C₂×C₃×C₈	C₂⁷×C₄	1	130305	yes	—
94	C₂×C₃×C₈	C₂⁹	1	511	yes	—
95	C₃×C₁₆	C₂⁹	1	512	yes	direct product factor
96	C₂⁵	C₂⁴×C₃×C₁₆	1	105911904	yes	direct product factor
97	C₂⁵	C₂⁵×C₃×C₈	1	3309747	yes	—
98	C₂³×C₄	C₂⁵×C₃×C₈	1	49646205	yes	—
99	C₂³×C₄	C₂⁶×C₃×C₄	1	788035	yes	—
100	C₂²×C₈	C₂⁶×C₃×C₄	1	5516245	yes	—
101	C₂²×C₈	C₂⁸×C₃	1	43435	yes	—
102	C₂×C₁₆	C₂⁸×C₃	1	130816	yes	direct product factor
103	C₂³×C₃	C₂⁶×C₁₆	1	6304280	yes	direct product factor
104	C₂³×C₃	C₂⁷×C₈	1	43435	yes	—
105	C₂×C₃×C₄	C₂⁷×C₈	1	130305	yes	—
106	C₂×C₃×C₄	C₂⁸×C₄	1	511	yes	—
107	C₃×C₈	C₂⁸×C₄	1	511	yes	—
108	C₃×C₈	C₂¹⁰	1	1	yes	characteristic
109	C₂⁴	C₂⁵×C₃×C₁₆	1	52955952	yes	direct product factor
110	C₂⁴	C₂⁶×C₃×C₈	1	788035	yes	—
111	C₂²×C₄	C₂⁶×C₃×C₈	1	5516245	yes	—
112	C₂²×C₄	C₂⁷×C₃×C₄	1	43435	yes	—
113	C₂×C₈	C₂⁹×C₃	1	511	yes	—
114	C₂×C₈	C₂⁷×C₃×C₄	1	130305	yes	—
115	C₁₆	C₂⁹×C₃	1	512	yes	direct product factor
116	C₂²×C₃	C₂⁷×C₁₆	1	173740	yes	direct product factor
117	C₂²×C₃	C₂⁸×C₈	1	511	yes	—
118	C₃×C₄	C₂⁸×C₈	1	511	yes	—
119	C₃×C₄	C₂⁹×C₄	1	1	yes	characteristic
120	C₂³	C₂⁶×C₃×C₁₆	1	6304280	yes	direct product factor
121	C₂³	C₂⁷×C₃×C₈	1	43435	yes	—
122	C₂×C₄	C₂⁷×C₃×C₈	1	130305	yes	—
123	C₂×C₄	C₂⁸×C₃×C₄	1	511	yes	—
124	C₈	C₂¹⁰×C₃	1	1	yes	characteristic
125	C₈	C₂⁸×C₃×C₄	1	511	yes	—
126	C₂×C₃	C₂⁸×C₁₆	1	1022	yes	direct product factor
127	C₂×C₃	C₂⁹×C₈	1	1	yes	characteristic
128	C₂²	C₂⁷×C₃×C₁₆	1	173740	yes	direct product factor
129	C₂²	C₂⁸×C₃×C₈	1	511	yes	—
130	C₄	C₂⁸×C₃×C₈	1	511	yes	—
131	C₄	C₂⁹×C₃×C₄	1	1	yes	characteristic
132	C₃	C₂⁹×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
133	C₂	C₂⁸×C₃×C₁₆	1	1022	yes	direct product factor
134	C₂	C₂⁹×C₃×C₈	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂¹⁰×C₃
socle, prime radical	C₂¹⁰×C₃	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁸×C₃×C₁₆	C₂	direct (N × H)
3. C₂⁹×C₁₆	C₃	direct (N × H)
7. C₂⁷×C₃×C₁₆	C₂²	direct (N × H)
9. C₂⁸×C₁₆	C₂×C₃	direct (N × H)
15. C₂⁶×C₃×C₁₆	C₂³	direct (N × H)
19. C₂⁷×C₁₆	C₂²×C₃	direct (N × H)
22. C₂⁹×C₃	C₁₆	direct (N × H)
26. C₂⁵×C₃×C₁₆	C₂⁴	direct (N × H)
32. C₂⁶×C₁₆	C₂³×C₃	direct (N × H)
33. C₂⁸×C₃	C₂×C₁₆	direct (N × H)
39. C₂⁴×C₃×C₁₆	C₂⁵	direct (N × H)
40. C₂⁹	C₃×C₁₆	direct (N × H)
46. C₂⁵×C₁₆	C₂⁴×C₃	direct (N × H)
47. C₂⁷×C₃	C₂²×C₁₆	direct (N × H)
53. C₂³×C₃×C₁₆	C₂⁶	direct (N × H)
54. C₂⁸	C₂×C₃×C₁₆	direct (N × H)
60. C₂⁴×C₁₆	C₂⁵×C₃	direct (N × H)
61. C₂⁶×C₃	C₂³×C₁₆	direct (N × H)
67. C₂²×C₃×C₁₆	C₂⁷	direct (N × H)
68. C₂⁷	C₂²×C₃×C₁₆	direct (N × H)
74. C₂³×C₁₆	C₂⁶×C₃	direct (N × H)
75. C₂⁵×C₃	C₂⁴×C₁₆	direct (N × H)
81. C₂×C₃×C₁₆	C₂⁸	direct (N × H)
82. C₂⁶	C₂³×C₃×C₁₆	direct (N × H)
88. C₂²×C₁₆	C₂⁷×C₃	direct (N × H)
89. C₂⁴×C₃	C₂⁵×C₁₆	direct (N × H)
95. C₃×C₁₆	C₂⁹	direct (N × H)
96. C₂⁵	C₂⁴×C₃×C₁₆	direct (N × H)
102. C₂×C₁₆	C₂⁸×C₃	direct (N × H)
103. C₂³×C₃	C₂⁶×C₁₆	direct (N × H)
109. C₂⁴	C₂⁵×C₃×C₁₆	direct (N × H)
115. C₁₆	C₂⁹×C₃	direct (N × H)
116. C₂²×C₃	C₂⁷×C₁₆	direct (N × H)
120. C₂³	C₂⁶×C₃×C₁₆	direct (N × H)
126. C₂×C₃	C₂⁸×C₁₆	direct (N × H)
128. C₂²	C₂⁷×C₃×C₁₆	direct (N × H)
132. C₃	C₂⁹×C₁₆	direct (N × H)
133. C₂	C₂⁸×C₃×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
C₂⁹×C₃×C₈	2
C₂⁸×C₃×C₁₆	2
C₂⁹×C₁₆	3

Subgroups — C29×C3×C16 #72201890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁹×C₃×C₁₆ #72201890