Subgroups — D₃×C₄⋊Dic₁₅ #13087416

Order 1440.

394 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×Dic₁₅	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊Dic₁₅)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃⋊Dic₁₅	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊Dic₁₅	—	3	1	no	maximal
7	D₃×C₄⋊Dic₅	—	3	1	no	maximal
8	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊Dic₁₅	C₂²	1	2	yes	—
11	C₂×C₃×Dic₁₅	C₂²	1	2	yes	—
12	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×D₃	C₄	1	4	yes	complemented
14	D₃×Dic₁₅	—	2	4	no	—
15	D₃×C₄⋊Dic₃	—	5	1	no	maximal, Hall
16	C₂²×C₃×C₄×C₅	—	3	1	no	—
17	C₂²×Dic₁₅	—	3	2	no	—
18	C₂×C₄×C₅×D₃	D₃	1	1	yes	characteristic
19	C₂×D₃×Dic₅	—	3	2	no	—
20	C₄⋊Dic₁₅	—	3	1	no	—
21	C₄⋊Dic₁₅	—	3	1	no	—
22	C₄⋊Dic₁₅	—	3	2	no	—
23	C₄⋊Dic₁₅	—	6	1	no	—
24	C₄⋊Dic₁₅	D₃	1	1	yes	characteristic, direct product factor
25	C₃×C₄⋊Dic₅	—	3	1	no	—
26	Dic₃⋊Dic₅	—	3	2	no	—
27	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₅×D₃	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×D₃	D₄	1	2	yes	—
30	C₂×C₃×C₅×D₃	Q₈	1	2	yes	—
31	C₃²×C₄×C₅	C₂×C₄	1	2	yes	complemented
32	C₃⋊Dic₁₅	—	2	2	no	—
33	C₃×Dic₁₅	—	2	2	no	—
34	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	complemented
35	C₂×C₄⋊Dic₅	—	9	1	no	Hall
36	C₂×C₃×C₄×D₃	D₅	1	1	yes	characteristic
37	C₂×D₃×Dic₃	—	5	2	no	—
38	C₄⋊(C₃⋊Dic₃)	—	5	1	no	—
39	C₃×C₄⋊Dic₃	—	5	1	no	—
40	Dic₃⋊Dic₃	—	5	2	no	—
41	C₂³×C₃×C₅	—	3	1	no	—
42	C₂²×C₅×D₃	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	—	2	1	no	—
44	C₂×C₃×C₄×C₅	—	3	1	no	—
45	C₂×C₃×C₄×C₅	—	3	4	no	—
46	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
48	C₂×Dic₁₅	—	3	2	no	—
49	C₂×Dic₁₅	—	3	2	no	—
50	C₂×Dic₁₅	—	6	4	no	—
51	C₂×Dic₁₅	—	6	2	no	—
52	C₂×Dic₁₅	C₂×D₃	1	2	yes	—
53	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
54	C₄×C₅×D₃	Dic₃	1	4	yes	complemented
55	C₂×C₃×Dic₅	—	3	2	no	—
56	D₃×Dic₅	—	6	4	no	—
57	C₂×C₄⋊Dic₃	—	15	1	no	—
58	D₃×C₄⋊C₄	—	15	1	no	—
59	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₃×C₅×D₃	C₄⋊C₄	1	4	yes	complemented
63	C₂²×C₄×C₅	—	3	1	no	—
64	C₂²×Dic₅	—	9	2	no	—
65	C₄⋊Dic₅	—	3	1	no	—
66	C₄⋊Dic₅	—	9	2	no	—
67	C₄⋊Dic₅	—	9	1	no	—
68	C₂²×C₃×D₃	C₂×D₅	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
70	C₂×C₃⋊Dic₃	—	5	2	no	—
71	C₂×C₃×Dic₃	—	5	2	no	—
72	C₂×C₃×Dic₃	C₂×D₅	1	1	yes	characteristic
73	C₃×C₄×D₃	Dic₅	1	4	yes	complemented
74	D₃×Dic₃	—	10	4	no	—
75	C₂²×C₃×C₅	—	2	1	no	—
76	C₂²×C₃×C₅	—	3	2	no	—
77	C₂²×C₃×C₅	—	3	2	no	—
78	C₂²×C₃×C₅	—	3	2	no	—
79	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
80	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
81	C₂×C₅×D₃	Dic₆	1	2	yes	—
82	C₂×C₅×D₃	D₁₂	1	2	yes	—
83	C₂×C₅×D₃	C₂×Dic₃	1	2	yes	—
84	C₃×C₄×C₅	—	2	2	no	—
85	C₃×C₄×C₅	—	3	2	no	—
86	C₃×C₄×C₅	C₄×D₃	1	2	yes	complemented
87	C₃×C₄×C₅	C₂×Dic₃	1	2	yes	complemented
88	Dic₁₅	—	2	2	no	—
89	Dic₁₅	—	6	2	no	—
90	Dic₁₅	—	6	2	no	—
91	Dic₁₅	—	12	2	no	—
92	C₃×Dic₅	—	6	2	no	—
93	C₅×Dic₃	C₂×Dic₃	1	2	yes	complemented
94	C₂²×C₃×C₄	—	3	1	no	—
95	C₂²×Dic₃	—	15	2	no	—
96	C₂×C₄×D₃	—	15	2	no	—
97	C₂×C₄×D₃	D₁₅	1	1	yes	characteristic
98	C₃×C₄⋊C₄	—	15	1	no	—
99	C₄⋊Dic₃	—	5	1	no	—
100	C₄⋊Dic₃	—	15	2	no	—
101	C₄⋊Dic₃	—	15	1	no	—
102	C₄⋊Dic₃	—	15	1	no	—
103	C₄⋊Dic₃	—	30	1	no	—
104	Dic₃⋊C₄	—	15	2	no	—
105	C₃²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
106	C₂³×C₅	—	3	1	no	—
107	C₂×C₄×C₅	—	3	1	no	—
108	C₂×C₄×C₅	—	3	4	no	—
109	C₂×C₄×C₅	D₃²	1	1	yes	characteristic
110	C₂×Dic₅	—	3	2	no	—
111	C₂×Dic₅	—	9	2	no	—
112	C₂×Dic₅	—	18	4	no	—
113	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
114	C₂×C₃×D₃	Dic₁₀	1	2	yes	—
115	C₂×C₃×D₃	D₂₀	1	2	yes	—
116	C₂×C₃×D₃	C₂×Dic₅	1	2	yes	—
117	C₃²×C₄	C₂×Dic₅	1	2	yes	complemented
118	C₃⋊Dic₃	—	10	2	no	—
119	C₃×Dic₃	—	10	2	no	—
120	C₃×Dic₃	C₂×Dic₅	1	2	yes	complemented
121	C₂×C₄⋊C₄	—	45	1	no	Sylow
122	C₂×C₃×C₅	—	2	1	no	—
123	C₂×C₃×C₅	—	2	1	no	—
124	C₂×C₃×C₅	—	2	1	no	—
125	C₂×C₃×C₅	—	3	4	no	—
126	C₂×C₃×C₅	C₂×Dic₆	1	1	yes	characteristic
127	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
128	C₂×C₃×C₅	C₂×D₁₂	1	1	yes	characteristic
129	C₂×C₃×C₅	D₃×D₄	1	1	yes	characteristic
130	C₂×C₃×C₅	D₃×Q₈	1	1	yes	characteristic
131	C₂×C₃×C₅	C₂²×Dic₃	1	1	yes	characteristic
132	C₅×D₃	C₄⋊Dic₃	1	4	yes	complemented
133	C₂³×C₃	—	3	1	no	—
134	C₂²×D₃	C₂×D₁₅	1	1	yes	characteristic
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	3	1	no	—
137	C₂×C₃×C₄	—	3	4	no	—
138	C₂×C₃×C₄	—	15	2	no	—
139	C₂×C₃×C₄	D₃×D₅	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
141	C₂×Dic₃	—	5	2	no	—
142	C₂×Dic₃	—	15	2	no	—
143	C₂×Dic₃	—	15	2	no	—
144	C₂×Dic₃	—	30	2	no	—
145	C₂×Dic₃	—	30	4	no	—
146	C₂×Dic₃	C₂×D₁₅	1	1	yes	characteristic
147	C₄×D₃	—	30	4	no	—
148	C₄×D₃	Dic₁₅	1	4	yes	complemented
149	C₂²×C₅	—	3	2	no	—
150	C₂²×C₅	—	3	2	no	—
151	C₂²×C₅	—	3	2	no	—
152	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
153	C₄×C₅	—	3	2	no	—
154	C₄×C₅	D₃×Dic₃	1	2	yes	complemented
155	Dic₅	—	6	2	no	—
156	Dic₅	—	18	2	no	—
157	C₂×C₃²	C₂×Dic₁₀	1	1	yes	characteristic
158	C₂×C₃²	C₂×D₂₀	1	1	yes	characteristic
159	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
160	C₃×D₃	C₄⋊Dic₅	1	4	yes	complemented
161	C₂²×C₄	—	3	1	no	—
162	C₂²×C₄	—	45	2	no	—
163	C₄⋊C₄	—	15	1	no	—
164	C₄⋊C₄	—	45	2	no	—
165	C₄⋊C₄	—	45	1	no	—
166	C₃×C₅	—	2	1	no	—
167	C₃×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic, complemented
168	C₃×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic, complemented
169	C₂²×C₃	—	2	1	no	—
170	C₂²×C₃	—	3	2	no	—
171	C₂²×C₃	—	3	2	no	—
172	C₂²×C₃	—	3	2	no	—
173	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
174	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
175	C₂×D₃	Dic₃₀	1	2	yes	—
176	C₂×D₃	D₆₀	1	2	yes	—
177	C₂×D₃	C₂×Dic₁₅	1	2	yes	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	3	2	no	—
180	C₃×C₄	—	30	2	no	—
181	C₃×C₄	D₃×Dic₅	1	2	yes	complemented
182	C₃×C₄	C₂×Dic₁₅	1	2	yes	complemented
183	Dic₃	—	10	2	no	—
184	Dic₃	—	30	2	no	—
185	Dic₃	—	30	2	no	—
186	Dic₃	—	60	2	no	—
187	Dic₃	C₂×Dic₁₅	1	2	yes	complemented
188	C₂×C₅	—	3	4	no	—
189	C₂×C₅	D₃×Dic₆	1	1	yes	characteristic
190	C₂×C₅	D₃×D₁₂	1	1	yes	characteristic
191	C₂×C₅	C₂×D₃×Dic₃	1	1	yes	characteristic
192	C₃²	C₂×C₄⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	3	1	no	—
194	C₂×C₄	—	3	1	no	—
195	C₂×C₄	—	3	4	no	—
196	C₂×C₄	—	15	2	no	—
197	C₂×C₄	—	45	2	no	—
198	C₂×C₄	—	90	4	no	—
199	C₂×C₄	D₃×D₁₅	1	1	yes	characteristic
200	C₂×C₃	—	2	1	no	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	3	4	no	—
204	C₂×C₃	D₃×Dic₁₀	1	1	yes	characteristic
205	C₂×C₃	D₃×D₂₀	1	1	yes	characteristic
206	C₂×C₃	C₂×D₃×Dic₅	1	1	yes	characteristic
207	C₂×C₃	C₂×Dic₃₀	1	1	yes	characteristic
208	C₂×C₃	C₂×D₆₀	1	1	yes	characteristic
209	C₂×C₃	C₂²×Dic₁₅	1	1	yes	characteristic
210	D₃	C₄⋊Dic₁₅	1	4	yes	direct product factor
211	C₅	D₃×C₄⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
212	C₂²	—	3	2	no	—
213	C₂²	—	3	2	no	—
214	C₂²	—	3	2	no	—
215	C₂²	C₂×D₃×D₁₅	1	1	yes	characteristic
216	C₄	—	3	2	no	—
217	C₄	—	30	2	no	—
218	C₄	—	90	2	no	—
219	C₄	D₃×Dic₁₅	1	2	yes	complemented
220	C₃	—	2	1	no	—
221	C₃	D₃×C₄⋊Dic₅	1	1	yes	characteristic, complemented
222	C₃	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic, complemented
223	C₂	—	3	4	no	—
224	C₂	D₃×Dic₃₀	1	1	yes	characteristic
225	C₂	D₃×D₆₀	1	1	yes	characteristic
226	C₂	C₂×D₃×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×D₁₅
commutator	C₂×C₃²×C₅	C₂²×C₄
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle, cosocle	C₂²×C₃²×C₅	C₂³
prime radical	C₂²×C₃×C₅×D₃	C₂²
hypercenter	C₂×C₄	D₃×D₁₅
hypocenter	C₃²×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊Dic₁₅)	C₂	semidirect
4. C₃×C₄⋊Dic₁₅	C₂	semidirect
5. Dic₃⋊Dic₁₅	C₂	semidirect
13. C₃×C₄×C₅×D₃	C₄	semidirect
24. C₄⋊Dic₁₅	D₃	direct (N × H)
31. C₃²×C₄×C₅	C₂×C₄	semidirect
34. C₃×C₅×Dic₃	C₂×C₄	semidirect
54. C₄×C₅×D₃	Dic₃	semidirect
62. C₃×C₅×D₃	C₄⋊C₄	semidirect
73. C₃×C₄×D₃	Dic₅	semidirect
86. C₃×C₄×C₅	C₄×D₃	semidirect
87. C₃×C₄×C₅	C₂×Dic₃	semidirect
93. C₅×Dic₃	C₂×Dic₃	semidirect
105. C₃²×C₅	C₂×C₄⋊C₄	semidirect
117. C₃²×C₄	C₂×Dic₅	semidirect
120. C₃×Dic₃	C₂×Dic₅	semidirect
132. C₅×D₃	C₄⋊Dic₃	semidirect
148. C₄×D₃	Dic₁₅	semidirect
154. C₄×C₅	D₃×Dic₃	semidirect
160. C₃×D₃	C₄⋊Dic₅	semidirect
167. C₃×C₅	D₃×C₄⋊C₄	semidirect
168. C₃×C₅	C₂×C₄⋊Dic₃	semidirect
181. C₃×C₄	D₃×Dic₅	semidirect
182. C₃×C₄	C₂×Dic₁₅	semidirect
187. Dic₃	C₂×Dic₁₅	semidirect
192. C₃²	C₂×C₄⋊Dic₅	semidirect
210. D₃	C₄⋊Dic₁₅	direct (N × H)
211. C₅	D₃×C₄⋊Dic₃	semidirect
219. C₄	D₃×Dic₁₅	semidirect
221. C₃	D₃×C₄⋊Dic₅	semidirect
222. C₃	C₂×C₄⋊Dic₁₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×D₃	2
C₂×D₃×Dic₁₅	2
C₄⋊(C₃⋊Dic₁₅)	2
C₃×C₄⋊Dic₁₅	2
Dic₃⋊Dic₁₅	2
C₂×C₄⋊Dic₁₅	3
D₃×C₄⋊Dic₅	3
D₃×C₄⋊Dic₃	5

Subgroups — D3×C4⋊Dic15 #13087416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₄⋊Dic₁₅ #13087416