Subgroups — C₂×Dic₃⋊Dic₁₅ #13087553

Order 1440.

422 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₁₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊Dic₁₅	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄⋊Dic₁₅	—	3	1	no	maximal
6	C₂×Dic₃⋊Dic₅	—	3	1	no	maximal
7	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊Dic₁₅	—	2	2	no	—
9	C₂×C₃⋊Dic₁₅	C₂²	1	2	yes	—
10	C₂×C₃×Dic₁₅	—	2	2	no	—
11	C₂×C₃×Dic₁₅	C₂²	1	2	yes	—
12	C₂×C₃×C₅×Dic₃	C₄	1	4	yes	complemented
13	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
14	C₂×Dic₃⋊Dic₃	—	5	1	no	maximal, Hall
15	C₂²×C₃×C₄×C₅	—	3	1	no	—
16	C₂²×Dic₁₅	—	3	1	no	—
17	C₂²×Dic₁₅	—	3	1	no	—
18	C₂²×Dic₁₅	—	6	1	no	—
19	C₂²×Dic₁₅	D₃	1	1	yes	characteristic
20	C₂²×C₅×Dic₃	D₃	1	1	yes	characteristic
21	C₂²×C₃×Dic₅	—	3	1	no	—
22	C₄⋊Dic₁₅	—	3	4	no	—
23	Dic₃⋊Dic₅	—	3	4	no	—
24	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃²×C₅	D₄	1	2	yes	—
26	C₂²×C₃²×C₅	Q₈	1	2	yes	—
27	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
28	C₃⋊Dic₁₅	—	2	2	no	—
29	C₃×Dic₁₅	—	2	2	no	—
30	C₃×C₅×Dic₃	C₂×C₄	1	4	yes	complemented
31	C₂×C₄⋊Dic₅	—	9	1	no	Hall
32	C₂²×C₃⋊Dic₃	—	5	1	no	—
33	C₂²×C₃×Dic₃	—	5	1	no	—
34	C₂²×C₃×Dic₃	D₅	1	1	yes	characteristic
35	Dic₃⋊Dic₃	—	5	4	no	—
36	C₂³×C₃×C₅	—	2	1	no	—
37	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
38	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	—	3	4	no	—
40	C₂×C₃×C₄×C₅	—	3	2	no	—
41	C₂×Dic₁₅	—	2	2	no	—
42	C₂×Dic₁₅	—	3	2	no	—
43	C₂×Dic₁₅	—	3	2	no	—
44	C₂×Dic₁₅	—	6	2	no	—
45	C₂×Dic₁₅	—	6	4	no	—
46	C₂×Dic₁₅	—	6	2	no	—
47	C₂×Dic₁₅	—	6	2	no	—
48	C₂×Dic₁₅	C₂×D₃	1	2	yes	—
49	C₂×C₅×Dic₃	Dic₃	1	4	yes	complemented
50	C₂×C₅×Dic₃	C₂×D₃	1	2	yes	—
51	C₂×C₃×Dic₅	—	3	2	no	—
52	C₂×C₃×Dic₅	—	6	2	no	—
53	C₂×C₄⋊Dic₃	—	15	1	no	—
54	C₂×Dic₃⋊C₄	—	15	1	no	—
55	C₂×C₃²×C₅	C₄⋊C₄	1	4	yes	complemented
56	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
58	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	3	1	no	—
60	C₂²×Dic₅	—	3	1	no	—
61	C₂²×Dic₅	—	9	1	no	—
62	C₄⋊Dic₅	—	9	4	no	—
63	C₂³×C₃²	C₂×D₅	1	1	yes	characteristic
64	C₂×C₃⋊Dic₃	—	5	2	no	—
65	C₂×C₃⋊Dic₃	—	10	2	no	—
66	C₂×C₃×Dic₃	—	5	2	no	—
67	C₂×C₃×Dic₃	—	10	2	no	—
68	C₂×C₃×Dic₃	Dic₅	1	4	yes	complemented
69	C₂×C₃×Dic₃	C₂×D₅	1	2	yes	—
70	C₂²×C₃×C₅	—	2	2	no	—
71	C₂²×C₃×C₅	—	2	2	no	—
72	C₂²×C₃×C₅	—	2	2	no	—
73	C₂²×C₃×C₅	—	2	1	no	—
74	C₂²×C₃×C₅	Dic₆	1	2	yes	—
75	C₂²×C₃×C₅	Dic₆	1	2	yes	—
76	C₂²×C₃×C₅	C₄×D₃	1	2	yes	—
77	C₂²×C₃×C₅	D₁₂	1	2	yes	—
78	C₂²×C₃×C₅	C₂×Dic₃	1	2	yes	—
79	C₂²×C₃×C₅	#67	1	2	yes	—
80	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
82	C₃×C₄×C₅	—	3	4	no	—
83	Dic₁₅	—	2	2	no	—
84	Dic₁₅	—	6	4	no	—
85	Dic₁₅	—	6	2	no	—
86	Dic₁₅	—	6	2	no	—
87	C₃×Dic₅	—	6	2	no	—
88	C₅×Dic₃	C₂×Dic₃	1	4	yes	complemented
89	C₂²×C₃×C₄	—	3	1	no	—
90	C₂²×C₃×C₄	—	15	1	no	—
91	C₂²×Dic₃	—	5	1	no	—
92	C₂²×Dic₃	—	15	1	no	—
93	C₂²×Dic₃	—	15	1	no	—
94	C₂²×Dic₃	—	30	1	no	—
95	C₂²×Dic₃	D₁₅	1	1	yes	characteristic
96	C₄⋊Dic₃	—	15	4	no	—
97	Dic₃⋊C₄	—	15	4	no	—
98	C₃²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
99	C₂³×C₅	D₃²	1	1	yes	characteristic
100	C₂×C₄×C₅	—	3	4	no	—
101	C₂×C₄×C₅	—	3	2	no	—
102	C₂×Dic₅	—	3	2	no	—
103	C₂×Dic₅	—	6	2	no	—
104	C₂×Dic₅	—	9	2	no	—
105	C₂×Dic₅	—	18	2	no	—
106	C₂²×C₃²	Dic₁₀	1	2	yes	—
107	C₂²×C₃²	D₂₀	1	2	yes	—
108	C₂²×C₃²	C₂×Dic₅	1	2	yes	—
109	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
110	C₃⋊Dic₃	—	10	2	no	—
111	C₃×Dic₃	—	10	2	no	—
112	C₃×Dic₃	C₂×Dic₅	1	4	yes	complemented
113	C₂×C₄⋊C₄	—	45	1	no	Sylow
114	C₂×C₃×C₅	—	2	4	no	—
115	C₂×C₃×C₅	—	2	1	no	—
116	C₂×C₃×C₅	—	2	1	no	—
117	C₂×C₃×C₅	—	2	1	no	—
118	C₂×C₃×C₅	Dic₃⋊C₄	1	4	yes	complemented
119	C₂×C₃×C₅	C₄⋊Dic₃	1	4	yes	complemented
120	C₂×C₃×C₅	C₂×Dic₆	1	1	yes	characteristic
121	C₂×C₃×C₅	C₂×Dic₆	1	1	yes	characteristic
122	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
123	C₂×C₃×C₅	C₂×D₁₂	1	1	yes	characteristic
124	C₂×C₃×C₅	C₂²×Dic₃	1	1	yes	characteristic
125	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
126	C₂³×C₃	—	2	1	no	—
127	C₂³×C₃	D₃×D₅	1	1	yes	characteristic
128	C₂³×C₃	C₂×D₁₅	1	1	yes	characteristic
129	C₂×C₃×C₄	—	3	4	no	—
130	C₂×C₃×C₄	—	3	2	no	—
131	C₂×C₃×C₄	—	15	2	no	—
132	C₂×C₃×C₄	—	30	2	no	—
133	C₂×Dic₃	—	5	2	no	—
134	C₂×Dic₃	—	10	2	no	—
135	C₂×Dic₃	—	15	2	no	—
136	C₂×Dic₃	—	15	2	no	—
137	C₂×Dic₃	—	30	2	no	—
138	C₂×Dic₃	—	30	4	no	—
139	C₂×Dic₃	—	30	2	no	—
140	C₂×Dic₃	—	30	2	no	—
141	C₂×Dic₃	Dic₁₅	1	4	yes	complemented
142	C₂×Dic₃	C₂×D₁₅	1	2	yes	—
143	C₂²×C₅	D₃×Dic₃	1	2	yes	—
144	C₂²×C₅	#626	1	2	yes	—
145	C₂²×C₅	#627	1	2	yes	—
146	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
147	C₄×C₅	—	3	4	no	—
148	Dic₅	—	6	2	no	—
149	Dic₅	—	18	2	no	—
150	C₂×C₃²	C₄⋊Dic₅	1	4	yes	complemented
151	C₂×C₃²	C₂×Dic₁₀	1	1	yes	characteristic
152	C₂×C₃²	C₂×D₂₀	1	1	yes	characteristic
153	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
154	C₂²×C₄	—	3	1	no	—
155	C₂²×C₄	—	15	1	no	—
156	C₂²×C₄	—	45	1	no	—
157	C₄⋊C₄	—	45	4	no	—
158	C₃×C₅	—	2	1	no	—
159	C₃×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic, complemented
160	C₃×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic, complemented
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	2	2	no	—
163	C₂²×C₃	—	2	2	no	—
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	D₃×Dic₅	1	2	yes	—
166	C₂²×C₃	#1202	1	2	yes	—
167	C₂²×C₃	#1204	1	2	yes	—
168	C₂²×C₃	Dic₃₀	1	2	yes	—
169	C₂²×C₃	D₆₀	1	2	yes	—
170	C₂²×C₃	C₂×Dic₁₅	1	2	yes	—
171	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
172	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
173	C₃×C₄	—	3	4	no	—
174	C₃×C₄	—	30	2	no	—
175	Dic₃	—	10	2	no	—
176	Dic₃	—	30	4	no	—
177	Dic₃	—	30	2	no	—
178	Dic₃	—	30	2	no	—
179	Dic₃	C₂×Dic₁₅	1	4	yes	complemented
180	C₂×C₅	Dic₃⋊Dic₃	1	4	yes	complemented
181	C₂×C₅	C₂×D₃×Dic₃	1	1	yes	characteristic
182	C₂×C₅	C₂×#626	1	1	yes	characteristic
183	C₂×C₅	C₂×#627	1	1	yes	characteristic
184	C₃²	C₂×C₄⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
185	C₂³	D₃×D₁₅	1	1	yes	characteristic
186	C₂×C₄	—	3	4	no	—
187	C₂×C₄	—	3	2	no	—
188	C₂×C₄	—	15	2	no	—
189	C₂×C₄	—	30	2	no	—
190	C₂×C₄	—	45	2	no	—
191	C₂×C₄	—	90	2	no	—
192	C₂×C₃	—	2	4	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	—	2	1	no	—
195	C₂×C₃	—	2	1	no	—
196	C₂×C₃	Dic₃⋊Dic₅	1	4	yes	complemented
197	C₂×C₃	C₄⋊Dic₁₅	1	4	yes	complemented
198	C₂×C₃	C₂×D₃×Dic₅	1	1	yes	characteristic
199	C₂×C₃	C₂×#1202	1	1	yes	characteristic
200	C₂×C₃	C₂×#1204	1	1	yes	characteristic
201	C₂×C₃	C₂×Dic₃₀	1	1	yes	characteristic
202	C₂×C₃	C₂×D₆₀	1	1	yes	characteristic
203	C₂×C₃	C₂²×Dic₁₅	1	1	yes	characteristic
204	C₅	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
205	C₂²	D₃×Dic₁₅	1	2	yes	—
206	C₂²	#67151	1	2	yes	—
207	C₂²	#67153	1	2	yes	—
208	C₂²	C₂×D₃×D₁₅	1	1	yes	characteristic
209	C₄	—	3	4	no	—
210	C₄	—	30	2	no	—
211	C₄	—	90	2	no	—
212	C₃	—	2	1	no	—
213	C₃	C₂×Dic₃⋊Dic₅	1	1	yes	characteristic, complemented
214	C₃	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic, complemented
215	C₂	Dic₃⋊Dic₁₅	1	4	yes	direct product factor
216	C₂	C₂×D₃×Dic₁₅	1	1	yes	characteristic
217	C₂	C₂×#67151	1	1	yes	characteristic
218	C₂	C₂×#67153	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. Dic₃⋊Dic₁₅	C₂	direct (N × H)
12. C₂×C₃×C₅×Dic₃	C₄	semidirect
30. C₃×C₅×Dic₃	C₂×C₄	semidirect
49. C₂×C₅×Dic₃	Dic₃	semidirect
55. C₂×C₃²×C₅	C₄⋊C₄	semidirect
68. C₂×C₃×Dic₃	Dic₅	semidirect
88. C₅×Dic₃	C₂×Dic₃	semidirect
98. C₃²×C₅	C₂×C₄⋊C₄	semidirect
112. C₃×Dic₃	C₂×Dic₅	semidirect
118. C₂×C₃×C₅	Dic₃⋊C₄	semidirect
119. C₂×C₃×C₅	C₄⋊Dic₃	semidirect
141. C₂×Dic₃	Dic₁₅	semidirect
150. C₂×C₃²	C₄⋊Dic₅	semidirect
159. C₃×C₅	C₂×Dic₃⋊C₄	semidirect
160. C₃×C₅	C₂×C₄⋊Dic₃	semidirect
179. Dic₃	C₂×Dic₁₅	semidirect
180. C₂×C₅	Dic₃⋊Dic₃	semidirect
184. C₃²	C₂×C₄⋊Dic₅	semidirect
196. C₂×C₃	Dic₃⋊Dic₅	semidirect
197. C₂×C₃	C₄⋊Dic₁₅	semidirect
204. C₅	C₂×Dic₃⋊Dic₃	semidirect
213. C₃	C₂×Dic₃⋊Dic₅	semidirect
214. C₃	C₂×C₄⋊Dic₁₅	semidirect
215. C₂	Dic₃⋊Dic₁₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊Dic₁₅	2
C₂²×C₃×Dic₁₅	2
C₂²×C₃×C₅×Dic₃	2
Dic₃⋊Dic₁₅	2
C₂×C₄⋊Dic₁₅	3
C₂×Dic₃⋊Dic₅	3
C₂×Dic₃⋊Dic₃	5

Subgroups — C2×Dic3⋊Dic15 #13087553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₃⋊Dic₁₅ #13087553