Subgroups — C₂⁵×C₃³×C₁₆ #72192038

Order 13824.

284984 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₃³×C₁₆	C₂	1	62	yes	maximal, direct product factor
3	C₂⁵×C₃²×C₁₆	C₃	1	13	yes	maximal, direct product factor
4	C₂⁵×C₃³×C₄	C₄	1	1	yes	characteristic
5	C₂⁴×C₃³×C₈	C₄	1	31	yes	—
6	C₂⁴×C₃³×C₈	C₂²	1	31	yes	—
7	C₂³×C₃³×C₁₆	C₂²	1	620	yes	direct product factor
8	C₂⁵×C₃²×C₈	C₂×C₃	1	13	yes	—
9	C₂⁴×C₃²×C₁₆	C₂×C₃	1	806	yes	direct product factor
10	C₂⁵×C₃×C₁₆	C₃²	1	13	yes	direct product factor
11	C₂⁶×C₃³	C₈	1	1	yes	characteristic
12	C₂⁴×C₃³×C₄	C₈	1	31	yes	—
13	C₂⁴×C₃³×C₄	C₂×C₄	1	31	yes	—
14	C₂³×C₃³×C₈	C₂×C₄	1	465	yes	—
15	C₂³×C₃³×C₈	C₂³	1	155	yes	—
16	C₂²×C₃³×C₁₆	C₂³	1	1240	yes	direct product factor
17	C₂⁵×C₃²×C₄	C₃×C₄	1	13	yes	—
18	C₂⁴×C₃²×C₈	C₃×C₄	1	403	yes	—
19	C₂⁴×C₃²×C₈	C₂²×C₃	1	403	yes	—
20	C₂³×C₃²×C₁₆	C₂²×C₃	1	8060	yes	direct product factor
21	C₂⁵×C₃³	C₁₆	1	32	yes	direct product factor
22	C₂⁵×C₃³	C₂×C₈	1	31	yes	—
23	C₂³×C₃³×C₄	C₂×C₈	1	465	yes	—
24	C₂³×C₃³×C₄	C₂²×C₄	1	155	yes	—
25	C₂²×C₃³×C₈	C₂²×C₄	1	1085	yes	—
26	C₂²×C₃³×C₈	C₂⁴	1	155	yes	—
27	C₂×C₃³×C₁₆	C₂⁴	1	496	yes	direct product factor
28	C₂⁵×C₃×C₈	C₂×C₃²	1	13	yes	—
29	C₂⁴×C₃×C₁₆	C₂×C₃²	1	806	yes	direct product factor
30	C₂⁶×C₃²	C₃×C₈	1	13	yes	—
31	C₂⁴×C₃²×C₄	C₃×C₈	1	403	yes	—
32	C₂⁴×C₃²×C₄	C₂×C₃×C₄	1	403	yes	—
33	C₂³×C₃²×C₈	C₂×C₃×C₄	1	6045	yes	—
34	C₂³×C₃²×C₈	C₂³×C₃	1	2015	yes	—
35	C₂²×C₃²×C₁₆	C₂³×C₃	1	16120	yes	direct product factor
36	C₂⁵×C₁₆	C₃³	1	1	yes	characteristic, direct product factor, Sylow
37	C₂⁴×C₃³	C₂×C₁₆	1	496	yes	direct product factor
38	C₂⁴×C₃³	C₂²×C₈	1	155	yes	—
39	C₂²×C₃³×C₄	C₂²×C₈	1	1085	yes	—
40	C₂²×C₃³×C₄	C₂³×C₄	1	155	yes	—
41	C₂×C₃³×C₈	C₂³×C₄	1	465	yes	—
42	C₂×C₃³×C₈	C₂⁵	1	31	yes	—
43	C₃³×C₁₆	C₂⁵	1	32	yes	direct product factor
44	C₂⁵×C₃×C₄	C₃²×C₄	1	13	yes	—
45	C₂⁴×C₃×C₈	C₃²×C₄	1	403	yes	—
46	C₂⁴×C₃×C₈	C₂²×C₃²	1	403	yes	—
47	C₂³×C₃×C₁₆	C₂²×C₃²	1	8060	yes	direct product factor
48	C₂⁵×C₃²	C₃×C₁₆	1	416	yes	direct product factor
49	C₂⁵×C₃²	C₂×C₃×C₈	1	403	yes	—
50	C₂³×C₃²×C₄	C₂×C₃×C₈	1	6045	yes	—
51	C₂³×C₃²×C₄	C₂²×C₃×C₄	1	2015	yes	—
52	C₂²×C₃²×C₈	C₂²×C₃×C₄	1	14105	yes	—
53	C₂²×C₃²×C₈	C₂⁴×C₃	1	2015	yes	—
54	C₂×C₃²×C₁₆	C₂⁴×C₃	1	6448	yes	direct product factor
55	C₂⁵×C₈	C₂×C₃³	1	1	yes	characteristic
56	C₂⁴×C₁₆	C₂×C₃³	1	62	yes	direct product factor
57	C₂³×C₃³	C₂²×C₁₆	1	1240	yes	direct product factor
58	C₂³×C₃³	C₂³×C₈	1	155	yes	—
59	C₂×C₃³×C₄	C₂³×C₈	1	465	yes	—
60	C₂×C₃³×C₄	C₂⁴×C₄	1	31	yes	—
61	C₃³×C₈	C₂⁴×C₄	1	31	yes	—
62	C₃³×C₈	C₂⁶	1	1	yes	characteristic
63	C₂⁶×C₃	C₃²×C₈	1	13	yes	—
64	C₂⁴×C₃×C₄	C₃²×C₈	1	403	yes	—
65	C₂⁴×C₃×C₄	C₂×C₃²×C₄	1	403	yes	—
66	C₂³×C₃×C₈	C₂×C₃²×C₄	1	6045	yes	—
67	C₂³×C₃×C₈	C₂³×C₃²	1	2015	yes	—
68	C₂²×C₃×C₁₆	C₂³×C₃²	1	16120	yes	direct product factor
69	C₂⁴×C₃²	C₂×C₃×C₁₆	1	6448	yes	direct product factor
70	C₂⁴×C₃²	C₂²×C₃×C₈	1	2015	yes	—
71	C₂²×C₃²×C₄	C₂²×C₃×C₈	1	14105	yes	—
72	C₂²×C₃²×C₄	C₂³×C₃×C₄	1	2015	yes	—
73	C₂×C₃²×C₈	C₂³×C₃×C₄	1	6045	yes	—
74	C₂×C₃²×C₈	C₂⁵×C₃	1	403	yes	—
75	C₃²×C₁₆	C₂⁵×C₃	1	416	yes	direct product factor
76	C₂⁵×C₄	C₃³×C₄	1	1	yes	characteristic
77	C₂⁴×C₈	C₃³×C₄	1	31	yes	—
78	C₂⁴×C₈	C₂²×C₃³	1	31	yes	—
79	C₂³×C₁₆	C₂²×C₃³	1	620	yes	direct product factor
80	C₂²×C₃³	C₂³×C₁₆	1	620	yes	direct product factor
81	C₂²×C₃³	C₂⁴×C₈	1	31	yes	—
82	C₃³×C₄	C₂⁴×C₈	1	31	yes	—
83	C₃³×C₄	C₂⁵×C₄	1	1	yes	characteristic
84	C₂⁵×C₃	C₃²×C₁₆	1	416	yes	direct product factor
85	C₂⁵×C₃	C₂×C₃²×C₈	1	403	yes	—
86	C₂³×C₃×C₄	C₂×C₃²×C₈	1	6045	yes	—
87	C₂³×C₃×C₄	C₂²×C₃²×C₄	1	2015	yes	—
88	C₂²×C₃×C₈	C₂²×C₃²×C₄	1	14105	yes	—
89	C₂²×C₃×C₈	C₂⁴×C₃²	1	2015	yes	—
90	C₂×C₃×C₁₆	C₂⁴×C₃²	1	6448	yes	direct product factor
91	C₂³×C₃²	C₂²×C₃×C₁₆	1	16120	yes	direct product factor
92	C₂³×C₃²	C₂³×C₃×C₈	1	2015	yes	—
93	C₂×C₃²×C₄	C₂³×C₃×C₈	1	6045	yes	—
94	C₂×C₃²×C₄	C₂⁴×C₃×C₄	1	403	yes	—
95	C₃²×C₈	C₂⁴×C₃×C₄	1	403	yes	—
96	C₃²×C₈	C₂⁶×C₃	1	13	yes	—
97	C₂⁶	C₃³×C₈	1	1	yes	characteristic
98	C₂⁴×C₄	C₃³×C₈	1	31	yes	—
99	C₂⁴×C₄	C₂×C₃³×C₄	1	31	yes	—
100	C₂³×C₈	C₂×C₃³×C₄	1	465	yes	—
101	C₂³×C₈	C₂³×C₃³	1	155	yes	—
102	C₂²×C₁₆	C₂³×C₃³	1	1240	yes	direct product factor
103	C₂×C₃³	C₂⁴×C₁₆	1	62	yes	direct product factor
104	C₂×C₃³	C₂⁵×C₈	1	1	yes	characteristic
105	C₂⁴×C₃	C₂×C₃²×C₁₆	1	6448	yes	direct product factor
106	C₂⁴×C₃	C₂²×C₃²×C₈	1	2015	yes	—
107	C₂²×C₃×C₄	C₂²×C₃²×C₈	1	14105	yes	—
108	C₂²×C₃×C₄	C₂³×C₃²×C₄	1	2015	yes	—
109	C₂×C₃×C₈	C₂³×C₃²×C₄	1	6045	yes	—
110	C₂×C₃×C₈	C₂⁵×C₃²	1	403	yes	—
111	C₃×C₁₆	C₂⁵×C₃²	1	416	yes	direct product factor
112	C₂²×C₃²	C₂³×C₃×C₁₆	1	8060	yes	direct product factor
113	C₂²×C₃²	C₂⁴×C₃×C₈	1	403	yes	—
114	C₃²×C₄	C₂⁴×C₃×C₈	1	403	yes	—
115	C₃²×C₄	C₂⁵×C₃×C₄	1	13	yes	—
116	C₂⁵	C₃³×C₁₆	1	32	yes	direct product factor
117	C₂⁵	C₂×C₃³×C₈	1	31	yes	—
118	C₂³×C₄	C₂×C₃³×C₈	1	465	yes	—
119	C₂³×C₄	C₂²×C₃³×C₄	1	155	yes	—
120	C₂²×C₈	C₂²×C₃³×C₄	1	1085	yes	—
121	C₂²×C₈	C₂⁴×C₃³	1	155	yes	—
122	C₂×C₁₆	C₂⁴×C₃³	1	496	yes	direct product factor
123	C₃³	C₂⁵×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₃	C₂²×C₃²×C₁₆	1	16120	yes	direct product factor
125	C₂³×C₃	C₂³×C₃²×C₈	1	2015	yes	—
126	C₂×C₃×C₄	C₂³×C₃²×C₈	1	6045	yes	—
127	C₂×C₃×C₄	C₂⁴×C₃²×C₄	1	403	yes	—
128	C₃×C₈	C₂⁴×C₃²×C₄	1	403	yes	—
129	C₃×C₈	C₂⁶×C₃²	1	13	yes	—
130	C₂×C₃²	C₂⁴×C₃×C₁₆	1	806	yes	direct product factor
131	C₂×C₃²	C₂⁵×C₃×C₈	1	13	yes	—
132	C₂⁴	C₂×C₃³×C₁₆	1	496	yes	direct product factor
133	C₂⁴	C₂²×C₃³×C₈	1	155	yes	—
134	C₂²×C₄	C₂²×C₃³×C₈	1	1085	yes	—
135	C₂²×C₄	C₂³×C₃³×C₄	1	155	yes	—
136	C₂×C₈	C₂³×C₃³×C₄	1	465	yes	—
137	C₂×C₈	C₂⁵×C₃³	1	31	yes	—
138	C₁₆	C₂⁵×C₃³	1	32	yes	direct product factor
139	C₂²×C₃	C₂³×C₃²×C₁₆	1	8060	yes	direct product factor
140	C₂²×C₃	C₂⁴×C₃²×C₈	1	403	yes	—
141	C₃×C₄	C₂⁴×C₃²×C₈	1	403	yes	—
142	C₃×C₄	C₂⁵×C₃²×C₄	1	13	yes	—
143	C₃²	C₂⁵×C₃×C₁₆	1	13	yes	direct product factor
144	C₂³	C₂²×C₃³×C₁₆	1	1240	yes	direct product factor
145	C₂³	C₂³×C₃³×C₈	1	155	yes	—
146	C₂×C₄	C₂³×C₃³×C₈	1	465	yes	—
147	C₂×C₄	C₂⁴×C₃³×C₄	1	31	yes	—
148	C₈	C₂⁴×C₃³×C₄	1	31	yes	—
149	C₈	C₂⁶×C₃³	1	1	yes	characteristic
150	C₂×C₃	C₂⁴×C₃²×C₁₆	1	806	yes	direct product factor
151	C₂×C₃	C₂⁵×C₃²×C₈	1	13	yes	—
152	C₂²	C₂³×C₃³×C₁₆	1	620	yes	direct product factor
153	C₂²	C₂⁴×C₃³×C₈	1	31	yes	—
154	C₄	C₂⁴×C₃³×C₈	1	31	yes	—
155	C₄	C₂⁵×C₃³×C₄	1	1	yes	characteristic
156	C₃	C₂⁵×C₃²×C₁₆	1	13	yes	direct product factor
157	C₂	C₂⁴×C₃³×C₁₆	1	62	yes	direct product factor
158	C₂	C₂⁵×C₃³×C₈	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂⁶×C₃³
socle, prime radical	C₂⁶×C₃³	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₃³×C₁₆	C₂	direct (N × H)
3. C₂⁵×C₃²×C₁₆	C₃	direct (N × H)
7. C₂³×C₃³×C₁₆	C₂²	direct (N × H)
9. C₂⁴×C₃²×C₁₆	C₂×C₃	direct (N × H)
10. C₂⁵×C₃×C₁₆	C₃²	direct (N × H)
16. C₂²×C₃³×C₁₆	C₂³	direct (N × H)
20. C₂³×C₃²×C₁₆	C₂²×C₃	direct (N × H)
21. C₂⁵×C₃³	C₁₆	direct (N × H)
27. C₂×C₃³×C₁₆	C₂⁴	direct (N × H)
29. C₂⁴×C₃×C₁₆	C₂×C₃²	direct (N × H)
35. C₂²×C₃²×C₁₆	C₂³×C₃	direct (N × H)
36. C₂⁵×C₁₆	C₃³	direct (N × H)
37. C₂⁴×C₃³	C₂×C₁₆	direct (N × H)
43. C₃³×C₁₆	C₂⁵	direct (N × H)
47. C₂³×C₃×C₁₆	C₂²×C₃²	direct (N × H)
48. C₂⁵×C₃²	C₃×C₁₆	direct (N × H)
54. C₂×C₃²×C₁₆	C₂⁴×C₃	direct (N × H)
56. C₂⁴×C₁₆	C₂×C₃³	direct (N × H)
57. C₂³×C₃³	C₂²×C₁₆	direct (N × H)
68. C₂²×C₃×C₁₆	C₂³×C₃²	direct (N × H)
69. C₂⁴×C₃²	C₂×C₃×C₁₆	direct (N × H)
75. C₃²×C₁₆	C₂⁵×C₃	direct (N × H)
79. C₂³×C₁₆	C₂²×C₃³	direct (N × H)
80. C₂²×C₃³	C₂³×C₁₆	direct (N × H)
84. C₂⁵×C₃	C₃²×C₁₆	direct (N × H)
90. C₂×C₃×C₁₆	C₂⁴×C₃²	direct (N × H)
91. C₂³×C₃²	C₂²×C₃×C₁₆	direct (N × H)
102. C₂²×C₁₆	C₂³×C₃³	direct (N × H)
103. C₂×C₃³	C₂⁴×C₁₆	direct (N × H)
105. C₂⁴×C₃	C₂×C₃²×C₁₆	direct (N × H)
111. C₃×C₁₆	C₂⁵×C₃²	direct (N × H)
112. C₂²×C₃²	C₂³×C₃×C₁₆	direct (N × H)
116. C₂⁵	C₃³×C₁₆	direct (N × H)
122. C₂×C₁₆	C₂⁴×C₃³	direct (N × H)
123. C₃³	C₂⁵×C₁₆	direct (N × H)
124. C₂³×C₃	C₂²×C₃²×C₁₆	direct (N × H)
130. C₂×C₃²	C₂⁴×C₃×C₁₆	direct (N × H)
132. C₂⁴	C₂×C₃³×C₁₆	direct (N × H)
138. C₁₆	C₂⁵×C₃³	direct (N × H)
139. C₂²×C₃	C₂³×C₃²×C₁₆	direct (N × H)
143. C₃²	C₂⁵×C₃×C₁₆	direct (N × H)
144. C₂³	C₂²×C₃³×C₁₆	direct (N × H)
150. C₂×C₃	C₂⁴×C₃²×C₁₆	direct (N × H)
152. C₂²	C₂³×C₃³×C₁₆	direct (N × H)
156. C₃	C₂⁵×C₃²×C₁₆	direct (N × H)
157. C₂	C₂⁴×C₃³×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
C₂⁵×C₃³×C₈	2
C₂⁴×C₃³×C₁₆	2
C₂⁵×C₃²×C₁₆	3

Subgroups — C25×C33×C16 #72192038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×C₃³×C₁₆ #72192038