Subgroups — D₅×Dic₃×C₁₆ #14312520

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₈×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×D₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745872	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×#672	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₅×C₁₆	—	3	1	no	maximal, Hall
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₄×D₅×Dic₃	C₄	1	1	yes	characteristic
11	C₂×D₅×C₃⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₅	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
15	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₃×#672	C₂²	1	1	yes	characteristic
18	C₃×D₅×C₁₆	C₄	1	4	yes	complemented
19	#90951	C₄	1	1	yes	characteristic
20	Dic₃×#168	C₄	1	1	yes	characteristic
21	D₅×C₃⋊C₁₆	C₄	1	4	yes	complemented
22	C₂×Dic₃×C₁₆	—	5	1	no	maximal, Hall
23	C₂×D₅×C₁₆	—	3	1	no	—
24	C₂×D₅×C₁₆	D₃	1	1	yes	characteristic
25	C₄×C₈×D₅	—	3	1	no	—
26	C₄×C₅×C₁₆	—	3	1	no	—
27	C₁₆×Dic₅	—	3	1	no	—
28	C₁₆×Dic₅	—	3	1	no	—
29	C₄×#672	—	3	1	no	—
30	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×D₅×Dic₃	C₈	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
35	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
38	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
39	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
40	Dic₃×Dic₅	C₈	1	2	yes	—
41	D₅×C₃⋊C₈	C₈	1	4	yes	—
42	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
43	C₃⋊#672	C₂×C₄	1	2	yes	complemented
44	C₃×#672	C₂×C₄	1	2	yes	complemented
45	C₅×C₃⋊C₁₆	C₂×C₄	1	2	yes	complemented
46	C₂²×C₃×C₁₆	—	5	1	no	—
47	C₂×C₈×Dic₃	—	5	1	no	—
48	C₂²×C₃⋊C₁₆	—	5	1	no	—
49	Dic₃×C₁₆	—	5	1	no	—
50	Dic₃×C₁₆	—	5	1	no	—
51	Dic₃×C₁₆	—	5	1	no	—
52	Dic₃×C₁₆	D₅	1	1	yes	characteristic, direct product factor
53	C₂×C₈×D₅	—	3	1	no	—
54	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
55	C₄²×D₅	—	3	1	no	—
56	C₂×C₅×C₁₆	—	3	1	no	—
57	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
58	C₄×C₅×C₈	—	3	1	no	—
59	C₈×Dic₅	—	3	1	no	—
60	C₈×Dic₅	—	3	1	no	—
61	C₂×#672	—	3	1	no	—
62	C₂×#672	C₂×D₃	1	1	yes	characteristic
63	C₄×#168	—	3	1	no	—
64	D₅×C₁₆	—	3	4	no	—
65	D₅×C₁₆	Dic₃	1	4	yes	direct product factor
66	C₂×C₄×C₁₆	—	15	1	no	Sylow
67	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×Dic₁₅	C₂×C₈	1	2	yes	—
70	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
71	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
72	C₃×C₄×D₅	C₄²	1	2	yes	—
73	C₃×C₄×D₅	C₂×C₈	1	2	yes	—
74	D₅×Dic₃	C₁₆	1	8	yes	direct product factor
75	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₃⋊#168	C₂×C₈	1	2	yes	—
78	C₃×#168	C₄²	1	2	yes	—
79	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	—
80	C₂²×C₃×C₈	—	5	1	no	—
81	C₂×C₄×Dic₃	—	5	1	no	—
82	C₂²×C₃⋊C₈	—	5	1	no	—
83	C₂×C₃×C₁₆	—	5	1	no	—
84	C₂×C₃×C₁₆	—	5	4	no	—
85	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
86	C₈×Dic₃	—	5	1	no	—
87	C₈×Dic₃	—	5	1	no	—
88	C₈×Dic₃	—	5	1	no	—
89	C₈×Dic₃	C₂×D₅	1	1	yes	characteristic
90	C₂×C₃⋊C₁₆	—	5	1	no	—
91	C₂×C₃⋊C₁₆	—	5	4	no	—
92	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
93	C₂×C₄×D₅	—	3	2	no	—
94	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
95	C₂×C₅×C₈	—	3	1	no	—
96	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
97	C₄²×C₅	—	3	1	no	—
98	C₄×Dic₅	—	3	1	no	—
99	C₄×Dic₅	—	3	2	no	—
100	C₂×#168	—	3	1	no	—
101	C₂×#168	C₄×D₃	1	1	yes	characteristic
102	C₈×D₅	—	3	4	no	—
103	C₈×D₅	C₄×D₃	1	2	yes	—
104	C₈×D₅	C₂×Dic₃	1	2	yes	—
105	C₅×C₁₆	—	3	2	no	—
106	C₅×C₁₆	C₂×Dic₃	1	2	yes	complemented
107	#672	—	3	2	no	—
108	#672	C₂×Dic₃	1	2	yes	complemented
109	C₂²×C₁₆	—	5	1	no	—
110	C₂²×C₁₆	—	15	1	no	—
111	C₂×C₄×C₈	—	15	1	no	—
112	C₄×C₁₆	—	3	1	no	—
113	C₄×C₁₆	—	15	1	no	—
114	C₄×C₁₆	—	15	1	no	—
115	C₄×C₁₆	—	15	1	no	—
116	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₃×D₅	C₄×C₈	1	2	yes	—
118	C₂×C₃×D₅	C₂×C₁₆	1	4	yes	—
119	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
121	Dic₁₅	C₂×C₁₆	1	4	yes	complemented
122	C₃×Dic₅	C₄×C₈	1	2	yes	—
123	C₅×Dic₃	C₂×C₁₆	1	4	yes	complemented
124	C₂²×C₃×C₄	—	5	1	no	—
125	C₂²×Dic₃	—	5	2	no	—
126	C₂×C₃×C₈	—	5	1	no	—
127	C₂×C₃×C₈	—	5	2	no	—
128	C₂×C₃×C₈	—	5	2	no	—
129	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
130	C₄×Dic₃	—	5	1	no	—
131	C₄×Dic₃	—	5	2	no	—
132	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
133	C₂×C₃⋊C₈	—	5	1	no	—
134	C₂×C₃⋊C₈	—	5	4	no	—
135	C₂×C₃⋊C₈	C₄×D₅	1	1	yes	characteristic
136	C₃×C₁₆	—	5	2	no	—
137	C₃×C₁₆	C₄×D₅	1	2	yes	complemented
138	C₃⋊C₁₆	—	5	2	no	—
139	C₃⋊C₁₆	C₄×D₅	1	2	yes	complemented
140	C₂²×D₅	D₃×C₈	1	1	yes	characteristic
141	C₂×C₄×C₅	—	3	2	no	—
142	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
143	C₂×Dic₅	—	3	2	no	—
144	C₂×Dic₅	D₃×C₈	1	1	yes	characteristic
145	C₄×D₅	—	3	8	no	—
146	C₄×D₅	D₃×C₈	1	2	yes	—
147	C₄×D₅	C₄×Dic₃	1	2	yes	—
148	C₅×C₈	—	3	2	no	—
149	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
150	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
151	#168	—	3	2	no	—
152	#168	C₄×Dic₃	1	2	yes	—
153	C₂²×C₈	—	5	1	no	—
154	C₂²×C₈	—	15	1	no	—
155	C₂×C₄²	—	15	1	no	—
156	C₂×C₁₆	—	3	1	no	—
157	C₂×C₁₆	—	5	1	no	—
158	C₂×C₁₆	—	5	4	no	—
159	C₂×C₁₆	—	15	4	no	—
160	C₂×C₁₆	—	15	1	no	—
161	C₂×C₁₆	D₃×D₅	1	1	yes	characteristic
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	—	15	1	no	—
164	C₄×C₈	—	15	1	no	—
165	C₄×C₈	—	15	1	no	—
166	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂²×C₁₆	1	2	yes	—
168	C₃×D₅	C₄×C₁₆	1	4	yes	complemented
169	C₂³×C₃	—	5	1	no	—
170	C₂×C₃×C₄	—	5	1	no	—
171	C₂×C₃×C₄	—	5	2	no	—
172	C₂×C₃×C₄	—	5	2	no	—
173	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
174	C₂×Dic₃	—	5	2	no	—
175	C₂×Dic₃	—	5	8	no	—
176	C₂×Dic₃	C₈×D₅	1	2	yes	—
177	C₃×C₈	—	5	2	no	—
178	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
179	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
180	C₃⋊C₈	—	5	2	no	—
181	C₃⋊C₈	C₈×D₅	1	2	yes	—
182	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
183	C₂×D₅	D₃×C₁₆	1	4	yes	—
184	C₂×D₅	C₈×Dic₃	1	2	yes	—
185	C₄×C₅	—	3	4	no	—
186	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
187	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
188	Dic₅	—	3	4	no	—
189	Dic₅	C₈×Dic₃	1	2	yes	—
190	C₂²×C₄	—	5	1	no	—
191	C₂²×C₄	—	15	2	no	—
192	C₂×C₈	—	3	1	no	—
193	C₂×C₈	—	5	1	no	—
194	C₂×C₈	—	5	2	no	—
195	C₂×C₈	—	5	2	no	—
196	C₂×C₈	—	15	4	no	—
197	C₂×C₈	—	15	1	no	—
198	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
199	C₄²	—	3	1	no	—
200	C₄²	—	15	2	no	—
201	C₄²	—	15	1	no	—
202	C₁₆	—	3	2	no	—
203	C₁₆	—	5	2	no	—
204	C₁₆	—	15	2	no	—
205	C₁₆	D₅×Dic₃	1	2	yes	direct product factor
206	C₃×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Hall
207	C₂²×C₃	—	5	4	no	—
208	C₂²×C₃	—	5	2	no	—
209	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
210	C₃×C₄	—	5	2	no	—
211	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
212	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
213	Dic₃	—	5	4	no	—
214	Dic₃	D₅×C₁₆	1	4	yes	direct product factor
215	C₂×C₅	C₂×D₃×C₁₆	1	2	yes	—
216	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
217	D₅	Dic₃×C₁₆	1	4	yes	direct product factor
218	C₂³	—	5	1	no	—
219	C₂×C₄	—	3	2	no	—
220	C₂×C₄	—	5	1	no	—
221	C₂×C₄	—	5	2	no	—
222	C₂×C₄	—	5	2	no	—
223	C₂×C₄	—	15	8	no	—
224	C₂×C₄	—	15	2	no	—
225	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
226	C₈	—	3	2	no	—
227	C₈	—	5	2	no	—
228	C₈	—	15	2	no	—
229	C₈	C₄×D₃×D₅	1	1	yes	characteristic
230	C₈	C₂×D₅×Dic₃	1	1	yes	characteristic
231	C₂×C₃	—	5	4	no	—
232	C₂×C₃	C₄×C₈×D₅	1	1	yes	characteristic
233	C₂×C₃	C₂×D₅×C₁₆	1	2	yes	—
234	C₅	C₂×Dic₃×C₁₆	1	1	yes	characteristic, complemented, Sylow
235	C₂²	—	5	4	no	—
236	C₂²	—	5	2	no	—
237	C₂²	D₃×C₈×D₅	1	1	yes	characteristic
238	C₄	—	3	4	no	—
239	C₄	—	5	2	no	—
240	C₄	—	15	4	no	—
241	C₄	D₃×C₈×D₅	1	1	yes	characteristic
242	C₄	C₄×D₅×Dic₃	1	1	yes	characteristic
243	C₃	C₄×D₅×C₁₆	1	1	yes	characteristic, complemented, Sylow
244	C₂	—	5	4	no	—
245	C₂	D₃×D₅×C₁₆	1	2	yes	—
246	C₂	C₈×D₅×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₁₆	D₃×D₅
commutator, hypocenter	C₃×C₅	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₂×C₃×C₅×C₁₆	C₂²
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
4. C₁₆×Dic₁₅	C₂	semidirect
5. C₅×Dic₃×C₁₆	C₂	semidirect
6. #11745872	C₂	semidirect
7. Dic₃×#672	C₂	semidirect
18. C₃×D₅×C₁₆	C₄	semidirect
21. D₅×C₃⋊C₁₆	C₄	semidirect
42. C₃×C₅×C₁₆	C₂×C₄	semidirect
43. C₃⋊#672	C₂×C₄	semidirect
44. C₃×#672	C₂×C₄	semidirect
45. C₅×C₃⋊C₁₆	C₂×C₄	semidirect
52. Dic₃×C₁₆	D₅	direct (N × H)
65. D₅×C₁₆	Dic₃	direct (N × H)
74. D₅×Dic₃	C₁₆	direct (N × H)
106. C₅×C₁₆	C₂×Dic₃	semidirect
108. #672	C₂×Dic₃	semidirect
121. Dic₁₅	C₂×C₁₆	semidirect
123. C₅×Dic₃	C₂×C₁₆	semidirect
137. C₃×C₁₆	C₄×D₅	semidirect
139. C₃⋊C₁₆	C₄×D₅	semidirect
168. C₃×D₅	C₄×C₁₆	semidirect
205. C₁₆	D₅×Dic₃	direct (N × H)
206. C₃×C₅	C₂×C₄×C₁₆	semidirect
214. Dic₃	D₅×C₁₆	direct (N × H)
217. D₅	Dic₃×C₁₆	direct (N × H)
234. C₅	C₂×Dic₃×C₁₆	semidirect
243. C₃	C₄×D₅×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₅×C₁₆	2
C₈×D₅×Dic₃	2
C₂×D₅×C₃⋊C₁₆	2
C₁₆×Dic₁₅	2
C₅×Dic₃×C₁₆	2
#11745872	2
Dic₃×#672	2
C₄×D₅×C₁₆	3
C₂×Dic₃×C₁₆	5

Subgroups — D5×Dic3×C16 #14312520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅×Dic₃×C₁₆ #14312520