Subgroups — D₃×C₁₆×Dic₅ #14312522

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×C₅×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	D₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×D₃×#672	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745872	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₆×Dic₅	—	3	1	no	maximal, Hall
9	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
10	C₄×D₃×Dic₅	C₄	1	1	yes	characteristic
11	C₂×D₃×#168	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	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₅×D₃×C₁₆	C₄	1	4	yes	complemented
17	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×C₃×#672	C₂²	1	1	yes	characteristic
19	#90951	C₄	1	1	yes	characteristic
20	Dic₅×C₃⋊C₈	C₄	1	1	yes	characteristic
21	D₃×#672	C₄	1	4	yes	complemented
22	C₄×D₃×C₁₆	—	5	1	no	maximal, Hall
23	C₂²×C₅×C₁₆	—	3	1	no	—
24	C₂×C₈×Dic₅	—	3	1	no	—
25	C₂²×#672	—	3	1	no	—
26	C₁₆×Dic₅	—	3	1	no	—
27	C₁₆×Dic₅	—	3	1	no	—
28	C₁₆×Dic₅	—	3	1	no	—
29	C₁₆×Dic₅	D₃	1	1	yes	characteristic, direct product factor
30	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×D₃×Dic₅	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	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₅×D₃×C₈	C₂×C₄	1	2	yes	—
37	C₅×D₃×C₈	C₂×C₄	1	2	yes	—
38	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
40	Dic₃×Dic₅	C₈	1	2	yes	—
41	D₃×#168	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	C₂×D₃×C₁₆	—	5	1	no	—
47	C₂×D₃×C₁₆	D₅	1	1	yes	characteristic
48	C₄×D₃×C₈	—	5	1	no	—
49	C₃×C₄×C₁₆	—	5	1	no	—
50	Dic₃×C₁₆	—	5	1	no	—
51	Dic₃×C₁₆	—	5	1	no	—
52	C₄×C₃⋊C₁₆	—	5	1	no	—
53	C₂²×C₅×C₈	—	3	1	no	—
54	C₂×C₄×Dic₅	—	3	1	no	—
55	C₂²×#168	—	3	1	no	—
56	C₂×C₅×C₁₆	—	3	1	no	—
57	C₂×C₅×C₁₆	—	3	4	no	—
58	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
59	C₈×Dic₅	—	3	1	no	—
60	C₈×Dic₅	—	3	1	no	—
61	C₈×Dic₅	—	3	1	no	—
62	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
63	C₂×#672	—	3	1	no	—
64	C₂×#672	—	3	4	no	—
65	C₂×#672	C₂×D₃	1	1	yes	characteristic
66	C₂×C₄×C₁₆	—	15	1	no	Sylow
67	C₂²×C₅×D₃	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	1	yes	characteristic
71	C₄×C₅×D₃	C₄²	1	2	yes	—
72	C₄×C₅×D₃	C₂×C₈	1	2	yes	—
73	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
74	D₃×Dic₅	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	C₂×C₈	1	2	yes	—
79	C₅×C₃⋊C₈	C₄²	1	2	yes	—
80	C₂×D₃×C₈	—	5	1	no	—
81	C₂×D₃×C₈	C₂×D₅	1	1	yes	characteristic
82	C₄²×D₃	—	5	1	no	—
83	C₂×C₃×C₁₆	—	5	1	no	—
84	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
85	C₃×C₄×C₈	—	5	1	no	—
86	C₈×Dic₃	—	5	1	no	—
87	C₈×Dic₃	—	5	1	no	—
88	C₂×C₃⋊C₁₆	—	5	1	no	—
89	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
90	C₄×C₃⋊C₈	—	5	1	no	—
91	D₃×C₁₆	—	5	4	no	—
92	D₃×C₁₆	Dic₅	1	4	yes	direct product factor
93	C₂²×C₄×C₅	—	3	1	no	—
94	C₂²×Dic₅	—	3	2	no	—
95	C₂×C₅×C₈	—	3	1	no	—
96	C₂×C₅×C₈	—	3	2	no	—
97	C₂×C₅×C₈	—	3	2	no	—
98	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
99	C₄×Dic₅	—	3	1	no	—
100	C₄×Dic₅	—	3	2	no	—
101	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
102	C₂×#168	—	3	1	no	—
103	C₂×#168	—	3	4	no	—
104	C₂×#168	C₄×D₃	1	1	yes	characteristic
105	C₅×C₁₆	—	3	2	no	—
106	C₅×C₁₆	C₄×D₃	1	2	yes	complemented
107	#672	—	3	2	no	—
108	#672	C₄×D₃	1	2	yes	complemented
109	C₂²×C₁₆	—	3	1	no	—
110	C₂²×C₁₆	—	15	1	no	—
111	C₂×C₄×C₈	—	15	1	no	—
112	C₄×C₁₆	—	5	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×C₅×D₃	C₄×C₈	1	2	yes	—
118	C₂×C₅×D₃	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	C₂×C₁₆	1	4	yes	complemented
123	C₅×Dic₃	C₄×C₈	1	2	yes	—
124	C₂×C₄×D₃	—	5	2	no	—
125	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
126	C₂×C₃×C₈	—	5	1	no	—
127	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
128	C₃×C₄²	—	5	1	no	—
129	C₄×Dic₃	—	5	1	no	—
130	C₄×Dic₃	—	5	2	no	—
131	C₂×C₃⋊C₈	—	5	1	no	—
132	C₂×C₃⋊C₈	C₄×D₅	1	1	yes	characteristic
133	D₃×C₈	—	5	4	no	—
134	D₃×C₈	C₄×D₅	1	2	yes	—
135	D₃×C₈	C₂×Dic₅	1	2	yes	—
136	C₃×C₁₆	—	5	2	no	—
137	C₃×C₁₆	C₂×Dic₅	1	2	yes	complemented
138	C₃⋊C₁₆	—	5	2	no	—
139	C₃⋊C₁₆	C₂×Dic₅	1	2	yes	complemented
140	C₂³×C₅	—	3	1	no	—
141	C₂×C₄×C₅	—	3	1	no	—
142	C₂×C₄×C₅	—	3	2	no	—
143	C₂×C₄×C₅	—	3	2	no	—
144	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×Dic₅	—	3	2	no	—
146	C₂×Dic₅	—	3	8	no	—
147	C₂×Dic₅	D₃×C₈	1	2	yes	—
148	C₅×C₈	—	3	2	no	—
149	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
150	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
151	#168	—	3	2	no	—
152	#168	D₃×C₈	1	2	yes	—
153	C₂²×C₈	—	3	1	no	—
154	C₂²×C₈	—	15	1	no	—
155	C₂×C₄²	—	15	1	no	—
156	C₂×C₁₆	—	3	1	no	—
157	C₂×C₁₆	—	3	4	no	—
158	C₂×C₁₆	—	5	1	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	—	5	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	C₅×D₃	C₄×C₁₆	1	4	yes	complemented
169	C₂²×D₃	C₈×D₅	1	1	yes	characteristic
170	C₂×C₃×C₄	—	5	2	no	—
171	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
172	C₂×Dic₃	—	5	2	no	—
173	C₂×Dic₃	C₈×D₅	1	1	yes	characteristic
174	C₄×D₃	—	5	8	no	—
175	C₄×D₃	C₈×D₅	1	2	yes	—
176	C₄×D₃	C₄×Dic₅	1	2	yes	—
177	C₃×C₈	—	5	2	no	—
178	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
179	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
180	C₃⋊C₈	—	5	2	no	—
181	C₃⋊C₈	C₄×Dic₅	1	2	yes	—
182	C₂²×C₅	—	3	4	no	—
183	C₂²×C₅	—	3	2	no	—
184	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
185	C₄×C₅	—	3	2	no	—
186	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
187	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
188	Dic₅	—	3	4	no	—
189	Dic₅	D₃×C₁₆	1	4	yes	direct product factor
190	C₂²×C₄	—	3	1	no	—
191	C₂²×C₄	—	15	2	no	—
192	C₂×C₈	—	3	1	no	—
193	C₂×C₈	—	3	2	no	—
194	C₂×C₈	—	3	2	no	—
195	C₂×C₈	—	5	1	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	—	5	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	D₃×Dic₅	1	2	yes	direct product factor
206	C₃×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Hall
207	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
208	C₂×D₃	D₅×C₁₆	1	4	yes	—
209	C₂×D₃	C₈×Dic₅	1	2	yes	—
210	C₃×C₄	—	5	4	no	—
211	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
212	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
213	Dic₃	—	5	4	no	—
214	Dic₃	C₈×Dic₅	1	2	yes	—
215	C₂×C₅	—	3	4	no	—
216	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
217	C₂×C₅	C₂×D₃×C₁₆	1	2	yes	—
218	C₂³	—	3	1	no	—
219	C₂×C₄	—	3	1	no	—
220	C₂×C₄	—	3	2	no	—
221	C₂×C₄	—	3	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×D₃×Dic₅	1	1	yes	characteristic
231	C₂×C₃	C₂×D₅×C₁₆	1	2	yes	—
232	C₂×C₃	C₂×C₈×Dic₅	1	1	yes	characteristic
233	D₃	C₁₆×Dic₅	1	4	yes	direct product factor
234	C₅	C₄×D₃×C₁₆	1	1	yes	characteristic, complemented, Sylow
235	C₂²	—	3	4	no	—
236	C₂²	—	3	2	no	—
237	C₂²	D₃×C₈×D₅	1	1	yes	characteristic
238	C₄	—	3	2	no	—
239	C₄	—	5	4	no	—
240	C₄	—	15	4	no	—
241	C₄	D₃×C₈×D₅	1	1	yes	characteristic
242	C₄	C₄×D₃×Dic₅	1	1	yes	characteristic
243	C₃	C₂×C₁₆×Dic₅	1	1	yes	characteristic, complemented, Sylow
244	C₂	—	3	4	no	—
245	C₂	D₃×D₅×C₁₆	1	2	yes	—
246	C₂	D₃×C₈×Dic₅	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×C₅×D₃	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. C₃×C₁₆×Dic₅	C₂	semidirect
6. #11745872	C₂	semidirect
7. Dic₅×C₃⋊C₁₆	C₂	semidirect
16. C₅×D₃×C₁₆	C₄	semidirect
21. D₃×#672	C₄	semidirect
29. C₁₆×Dic₅	D₃	direct (N × H)
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
74. D₃×Dic₅	C₁₆	direct (N × H)
92. D₃×C₁₆	Dic₅	direct (N × H)
106. C₅×C₁₆	C₄×D₃	semidirect
108. #672	C₄×D₃	semidirect
121. Dic₁₅	C₂×C₁₆	semidirect
122. C₃×Dic₅	C₂×C₁₆	semidirect
137. C₃×C₁₆	C₂×Dic₅	semidirect
139. C₃⋊C₁₆	C₂×Dic₅	semidirect
168. C₅×D₃	C₄×C₁₆	semidirect
189. Dic₅	D₃×C₁₆	direct (N × H)
205. C₁₆	D₃×Dic₅	direct (N × H)
206. C₃×C₅	C₂×C₄×C₁₆	semidirect
233. D₃	C₁₆×Dic₅	direct (N × H)
234. C₅	C₄×D₃×C₁₆	semidirect
243. C₃	C₂×C₁₆×Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₃×C₁₆	2
D₃×C₈×Dic₅	2
C₂×D₃×#672	2
C₁₆×Dic₁₅	2
C₃×C₁₆×Dic₅	2
#11745872	2
Dic₅×C₃⋊C₁₆	2
C₂×C₁₆×Dic₅	3
C₄×D₃×C₁₆	5

Subgroups — D3×C16×Dic5 #14312522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₁₆×Dic₅ #14312522