Subgroups — C₃×#91394 #13086525

Order 1440.

338 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#685	C₂	1	1	yes	maximal, characteristic
5	C₃×Dic₃⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6665	C₂	1	1	yes	maximal, characteristic
7	C₃×Dic₃×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4038	—	3	1	no	maximal
9	#91394	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂²×C₃²×D₅	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×Dic₁₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×Dic₁₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
17	C₃×D₅×Dic₃	C₄	1	4	yes	complemented
18	C₃×#877	—	5	1	no	maximal, Hall
19	C₂×C₃×C₄×D₅	—	3	1	no	—
20	C₂×D₅×Dic₃	C₂×C₃	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	—	3	1	no	—
22	C₄×Dic₁₅	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
24	C₃×#685	—	2	1	no	—
25	C₃×#685	—	3	1	no	—
26	C₃×#685	D₃	1	1	yes	characteristic
27	C₃×#685	C₂×C₃	1	1	yes	characteristic
28	C₃×C₄⋊Dic₅	—	3	1	no	—
29	C₃×C₄×Dic₅	—	3	1	no	—
30	C₃×C₄×Dic₅	—	3	1	no	—
31	Dic₃⋊Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
32	#6665	C₂×C₃	1	1	yes	characteristic
33	Dic₃×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₃²×D₅	—	2	1	no	—
36	C₂×C₃²×D₅	—	2	1	no	—
37	C₂×C₃²×D₅	C₂×C₄	1	2	yes	—
38	C₃²×C₄×C₅	—	2	1	no	—
39	C₃×Dic₁₅	—	2	1	no	—
40	C₃×Dic₁₅	C₂×C₄	1	2	yes	—
41	C₃×C₅×Dic₃	—	2	1	no	—
42	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	—
43	C₃²×Dic₅	—	2	1	no	—
44	#4038	—	3	1	no	Hall
45	C₂²×C₃×Dic₃	—	5	1	no	—
46	C₃²×C₂²⋊C₄	—	5	1	no	—
47	C₃×#217	—	5	1	no	—
48	C₃×Dic₃⋊C₄	—	5	1	no	—
49	C₃×Dic₃⋊C₄	D₅	1	1	yes	characteristic, complemented
50	C₃×C₄×Dic₃	—	5	1	no	—
51	C₃×C₄×Dic₃	—	5	1	no	—
52	C₂²×C₃×D₅	—	2	1	no	—
53	C₂²×C₃×D₅	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃×D₅	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	3	1	no	—
57	C₂×C₃×C₄×C₅	—	3	1	no	—
58	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₂×Dic₁₅	C₂²×C₃	1	1	yes	characteristic
61	C₂×Dic₁₅	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃×Dic₅	—	2	1	no	—
65	C₂×C₃×Dic₅	—	3	1	no	—
66	C₂×C₃×Dic₅	—	3	1	no	—
67	C₂×C₃×Dic₅	C₂×D₃	1	1	yes	characteristic
68	C₂×C₃×Dic₅	C₂²×C₃	1	1	yes	characteristic
69	C₃×C₄×D₅	—	3	4	no	—
70	D₅×Dic₃	C₃×C₄	1	4	yes	complemented
71	C₃×#117	—	15	1	no	—
72	#877	—	5	1	no	—
73	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃²×C₅	#41	1	1	yes	characteristic
75	C₂×C₃²×C₅	#41	1	1	yes	characteristic
76	C₃²×D₅	—	2	2	no	—
77	C₂×C₄×D₅	—	3	1	no	—
78	C₅×C₄⋊C₄	—	3	1	no	—
79	#685	—	3	1	no	—
80	#685	C₃×D₃	1	1	yes	characteristic
81	C₄⋊Dic₅	—	3	1	no	—
82	C₄×Dic₅	—	3	1	no	—
83	C₄×Dic₅	—	3	1	no	—
84	C₂³×C₃²	—	5	1	no	—
85	C₂×C₃²×C₄	—	5	1	no	—
86	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
87	C₂×C₃×Dic₃	—	5	1	no	—
88	C₂×C₃×Dic₃	—	5	4	no	—
89	C₂×C₃×Dic₃	—	5	1	no	—
90	C₂×C₃×Dic₃	C₂×D₅	1	1	yes	characteristic
91	C₂×C₃×Dic₃	C₂×D₅	1	1	yes	characteristic
92	C₂²×C₃×C₅	—	2	1	no	—
93	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
94	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
95	C₂×C₃×D₅	—	2	1	no	—
96	C₂×C₃×D₅	—	2	1	no	—
97	C₂×C₃×D₅	—	2	1	no	—
98	C₂×C₃×D₅	—	2	1	no	—
99	C₂×C₃×D₅	—	2	2	no	—
100	C₂×C₃×D₅	—	4	1	no	—
101	C₂×C₃×D₅	—	4	1	no	—
102	C₂×C₃×D₅	C₄×D₃	1	2	yes	—
103	C₂×C₃×D₅	C₂×C₃×C₄	1	2	yes	—
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	—	2	1	no	—
106	C₃×C₄×C₅	—	3	2	no	—
107	C₃×C₄×C₅	—	4	1	no	—
108	C₃×C₄×C₅	—	6	1	no	—
109	Dic₁₅	—	2	1	no	—
110	Dic₁₅	C₂×C₃×C₄	1	2	yes	—
111	C₃×Dic₅	—	2	1	no	—
112	C₃×Dic₅	—	2	1	no	—
113	C₃×Dic₅	—	3	2	no	—
114	C₃×Dic₅	—	4	1	no	—
115	C₃×Dic₅	—	6	1	no	—
116	C₅×Dic₃	—	2	1	no	—
117	C₅×Dic₃	C₂×C₃×C₄	1	2	yes	—
118	C₂²×C₃×C₄	—	15	1	no	—
119	C₂²×Dic₃	—	5	1	no	—
120	C₃×C₄⋊C₄	—	3	1	no	—
121	C₃×C₄⋊C₄	—	15	1	no	—
122	C₃×C₂²⋊C₄	—	5	1	no	—
123	C₃×C₂²⋊C₄	—	5	1	no	—
124	C₃×C₂²⋊C₄	—	10	1	no	—
125	C₃×C₂²⋊C₄	—	15	1	no	—
126	C₃×C₄²	—	15	1	no	—
127	C₃×C₄²	—	15	1	no	—
128	#217	—	5	1	no	—
129	Dic₃⋊C₄	—	5	1	no	—
130	Dic₃⋊C₄	C₃×D₅	1	1	yes	characteristic, complemented
131	C₄×Dic₃	—	5	1	no	—
132	C₄×Dic₃	—	5	1	no	—
133	C₃²×C₅	#117	1	1	yes	characteristic, complemented, Hall
134	C₂²×D₅	C₂×C₃×D₃	1	1	yes	characteristic
135	C₂×C₄×C₅	—	3	1	no	—
136	C₂×C₄×C₅	—	3	1	no	—
137	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂×Dic₅	—	3	1	no	—
139	C₂×Dic₅	—	3	1	no	—
140	C₂×Dic₅	C₂×C₃×D₃	1	1	yes	characteristic
141	C₄×D₅	—	3	4	no	—
142	C₂²×C₃²	—	5	2	no	—
143	C₂²×C₃²	—	10	1	no	—
144	C₂²×C₃²	—	10	1	no	—
145	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
146	C₃²×C₄	—	2	1	no	—
147	C₃²×C₄	—	10	1	no	—
148	C₃×Dic₃	—	2	1	no	—
149	C₃×Dic₃	—	5	2	no	—
150	C₃×Dic₃	—	10	1	no	—
151	C₃×Dic₃	C₄×D₅	1	2	yes	—
152	#117	—	15	1	no	Sylow
153	C₂×C₃×C₅	—	2	1	no	—
154	C₂×C₃×C₅	—	2	1	no	—
155	C₂×C₃×C₅	—	2	1	no	—
156	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂×C₃×C₅	#237	1	1	yes	characteristic
158	C₂×C₃×C₅	#237	1	1	yes	characteristic
159	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
160	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
161	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
162	C₃×D₅	—	2	2	no	—
163	C₃×D₅	—	2	2	no	—
164	C₃×D₅	—	4	2	no	—
165	C₂³×C₃	—	5	1	no	—
166	C₂³×C₃	—	5	1	no	—
167	C₂³×C₃	—	10	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	3	1	no	—
170	C₂×C₃×C₄	—	3	1	no	—
171	C₂×C₃×C₄	—	5	1	no	—
172	C₂×C₃×C₄	—	5	1	no	—
173	C₂×C₃×C₄	—	10	1	no	—
174	C₂×C₃×C₄	—	15	4	no	—
175	C₂×C₃×C₄	—	15	1	no	—
176	C₂×C₃×C₄	—	15	1	no	—
177	C₂×C₃×C₄	D₃×D₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
179	C₂×Dic₃	—	5	1	no	—
180	C₂×Dic₃	—	5	4	no	—
181	C₂×Dic₃	—	5	1	no	—
182	C₂×Dic₃	C₂×C₃×D₅	1	1	yes	characteristic
183	C₂×Dic₃	C₂×C₃×D₅	1	1	yes	characteristic
184	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
185	C₂×D₅	—	2	1	no	—
186	C₂×D₅	—	2	1	no	—
187	C₂×D₅	C₃×C₄×D₃	1	2	yes	—
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	3	2	no	—
190	C₄×C₅	—	6	1	no	—
191	Dic₅	—	2	1	no	—
192	Dic₅	—	3	2	no	—
193	Dic₅	—	6	1	no	—
194	C₂×C₃²	—	10	2	no	—
195	C₂×C₃²	C₂×C₄×D₅	1	1	yes	characteristic
196	C₂×C₃²	#711	1	1	yes	characteristic
197	C₂×C₃²	#713	1	1	yes	characteristic
198	C₂²×C₄	—	15	1	no	—
199	C₄⋊C₄	—	3	1	no	—
200	C₄⋊C₄	—	15	1	no	—
201	C₂²⋊C₄	—	5	1	no	—
202	C₂²⋊C₄	—	15	1	no	—
203	C₄²	—	15	1	no	—
204	C₄²	—	15	1	no	—
205	C₃×C₅	—	2	1	no	—
206	C₃×C₅	#877	1	1	yes	characteristic, complemented
207	C₃×C₅	C₃×#117	1	1	yes	characteristic, complemented
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	5	2	no	—
210	C₂²×C₃	—	5	2	no	—
211	C₂²×C₃	—	10	2	no	—
212	C₂²×C₃	—	10	1	no	—
213	C₂²×C₃	—	10	1	no	—
214	C₂²×C₃	—	10	1	no	—
215	C₂²×C₃	—	10	1	no	—
216	C₂²×C₃	—	20	1	no	—
217	C₂²×C₃	—	20	1	no	—
218	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
219	C₂²×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	3	2	no	—
223	C₃×C₄	—	4	1	no	—
224	C₃×C₄	—	6	1	no	—
225	C₃×C₄	—	10	1	no	—
226	C₃×C₄	—	10	1	no	—
227	C₃×C₄	—	15	2	no	—
228	C₃×C₄	—	20	1	no	—
229	C₃×C₄	—	30	1	no	—
230	Dic₃	—	2	1	no	—
231	Dic₃	—	5	2	no	—
232	Dic₃	—	10	1	no	—
233	Dic₃	C₃×C₄×D₅	1	2	yes	—
234	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
235	C₂×C₅	C₃×#237	1	1	yes	characteristic
236	C₂×C₅	C₃×#237	1	1	yes	characteristic
237	D₅	—	2	2	no	—
238	C₃²	#4038	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	5	1	no	—
240	C₂×C₄	—	3	1	no	—
241	C₂×C₄	—	3	1	no	—
242	C₂×C₄	—	5	1	no	—
243	C₂×C₄	—	15	4	no	—
244	C₂×C₄	—	15	1	no	—
245	C₂×C₄	—	15	1	no	—
246	C₂×C₄	C₃×D₃×D₅	1	1	yes	characteristic
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	10	2	no	—
251	C₂×C₃	—	10	2	no	—
252	C₂×C₃	—	20	2	no	—
253	C₂×C₃	#6766	1	1	yes	characteristic
254	C₂×C₃	C₄×D₃×D₅	1	1	yes	characteristic
255	C₂×C₃	#6780	1	1	yes	characteristic
256	C₂×C₃	C₂×C₃×C₄×D₅	1	1	yes	characteristic
257	C₂×C₃	C₃×#711	1	1	yes	characteristic
258	C₂×C₃	C₃×#713	1	1	yes	characteristic
259	C₅	C₃×#877	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	5	2	no	—
261	C₂²	—	10	1	no	—
262	C₂²	—	10	1	no	—
263	C₂²	C₂×C₃×D₃×D₅	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	3	2	no	—
266	C₄	—	6	1	no	—
267	C₄	—	10	1	no	—
268	C₄	—	15	2	no	—
269	C₄	—	30	1	no	—
270	C₃	—	2	1	no	—
271	C₃	#91394	1	1	yes	characteristic, direct product factor
272	C₃	C₃×#4038	1	1	yes	characteristic, complemented
273	C₂	—	10	2	no	—
274	C₂	C₃×#6766	1	1	yes	characteristic
275	C₂	C₃×C₄×D₃×D₅	1	1	yes	characteristic
276	C₂	C₃×#6780	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂²×C₃×C₄
Frattini	C₂²	C₂×C₃×D₃×D₅
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂²×C₃²×D₅	C₂²
cosocle	C₂²×C₃×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄	D₃×D₅
hypocenter	C₃×C₅	C₃×#117
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×Dic₁₅	C₂	semidirect
3. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
5. C₃×Dic₃⋊Dic₅	C₂	semidirect
7. C₃×Dic₃×Dic₅	C₂	semidirect
9. #91394	C₃	direct (N × H)
17. C₃×D₅×Dic₃	C₄	semidirect
22. C₄×Dic₁₅	C₂×C₃	semidirect
23. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
31. Dic₃⋊Dic₅	C₂×C₃	semidirect
33. Dic₃×Dic₅	C₂×C₃	semidirect
49. C₃×Dic₃⋊C₄	D₅	semidirect
70. D₅×Dic₃	C₃×C₄	semidirect
130. Dic₃⋊C₄	C₃×D₅	semidirect
133. C₃²×C₅	#117	semidirect
206. C₃×C₅	#877	semidirect
207. C₃×C₅	C₃×#117	semidirect
238. C₃²	#4038	semidirect
259. C₅	C₃×#877	semidirect
271. C₃	#91394	direct (N × H)
272. C₃	C₃×#4038	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×Dic₃	2
C₃×C₄×Dic₁₅	2
C₃×C₅×Dic₃⋊C₄	2
C₃²×#685	2
C₃×Dic₃⋊Dic₅	2
C₃×#6665	2
C₃×Dic₃×Dic₅	2
C₃×#4038	3
#91394	3
C₃×#877	5

Subgroups — C3×#91394 #13086525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#91394 #13086525