Subgroups — C₃×#66015 #11755692

Order 960.

472 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4093	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4086	C₂	1	1	yes	maximal, characteristic
5	C₃×D₄×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4032	C₂	1	2	yes	maximal, complemented
7	C₃×#4031	C₂	1	2	yes	maximal, complemented
8	C₃×#4029	C₂	1	2	yes	maximal, complemented
9	C₃×#4027	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	#66015	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
18	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
20	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
25	C₃×#685	C₂²	1	2	yes	—
26	C₃×#685	C₂²	1	2	yes	—
27	C₃×#685	C₂²	1	2	yes	—
28	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
30	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
31	C₃×#683	C₂²	1	1	yes	characteristic
32	C₃×#683	C₂²	1	1	yes	characteristic
33	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂²	1	2	yes	—
36	C₃×#559	—	5	1	no	maximal, Hall
37	C₅×#124	C₂×C₃	1	1	yes	characteristic, complemented
38	#4093	C₂×C₃	1	1	yes	characteristic, complemented
39	Q₈×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
40	#4086	C₂×C₃	1	1	yes	characteristic
41	D₄×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
42	#4032	C₂×C₃	1	2	yes	complemented
43	#4031	C₂×C₃	1	2	yes	complemented
44	#4029	C₂×C₃	1	2	yes	complemented
45	#4027	C₂×C₃	1	2	yes	complemented
46	C₄×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
47	C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂³×C₃×C₅	C₂³	1	2	yes	—
49	C₂²×C₃×D₅	C₂³	1	2	yes	—
50	C₃×C₅×Q₈	—	2	2	no	—
51	C₃×C₅×D₄	—	2	2	no	—
52	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₃×C₂²⋊D₅	—	2	4	no	—
57	C₃×C₂²⋊D₅	—	4	2	no	—
58	C₂×C₃×Dic₅	—	4	2	no	—
59	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₂³	1	2	yes	—
63	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
64	C₃×D₂₀	—	2	2	no	—
65	C₃×C₄×D₅	—	4	2	no	—
66	C₃×Dic₁₀	—	2	2	no	—
67	C₃×#127	—	5	1	no	—
68	C₃×#124	—	5	1	no	—
69	C₃×#124	—	5	2	no	—
70	C₃×#124	D₅	1	1	yes	characteristic, complemented
71	C₃×#123	—	5	2	no	—
72	C₃×#123	—	5	2	no	—
73	C₃×C₄×Q₈	—	5	1	no	—
74	C₃×C₄×Q₈	—	5	1	no	—
75	C₃×C₄×D₄	—	5	1	no	—
76	C₃×C₄×D₄	—	5	2	no	—
77	C₃×C₄×D₄	—	5	1	no	—
78	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
81	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
82	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
83	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
85	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
86	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
87	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
88	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
89	C₂²⋊Dic₅	C₂²×C₃	1	2	yes	—
90	#685	C₂²×C₃	1	2	yes	—
91	#685	C₂²×C₃	1	2	yes	—
92	#685	C₂²×C₃	1	2	yes	—
93	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
94	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
95	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
96	#683	C₂²×C₃	1	1	yes	characteristic
97	#683	C₂²×C₃	1	1	yes	characteristic
98	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
99	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
100	C₄×Dic₅	C₂²×C₃	1	2	yes	—
101	#559	—	5	1	no	Sylow
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	—	2	2	no	—
105	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
106	C₂×C₃×D₅	—	2	2	no	—
107	C₂×C₃×D₅	—	2	2	no	—
108	C₂×C₃×D₅	—	2	2	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	#41	1	2	yes	—
113	C₃×Dic₅	—	2	1	no	—
114	C₃×Dic₅	—	2	2	no	—
115	C₃×Dic₅	—	2	1	no	—
116	C₃×Dic₅	—	2	1	no	—
117	C₃×Dic₅	#41	1	2	yes	—
118	C₂×C₃×Q₈	—	5	1	no	—
119	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
120	C₂×C₃×D₄	—	5	2	no	—
121	C₂×C₃×D₄	—	5	2	no	—
122	C₂×C₃×D₄	—	5	1	no	—
123	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
124	C₂²×C₃×C₄	—	5	2	no	—
125	C₂²×C₃×C₄	—	5	2	no	—
126	C₃×C₄⋊C₄	—	5	1	no	—
127	C₃×C₄⋊C₄	—	5	1	no	—
128	C₃×C₄⋊C₄	—	5	1	no	—
129	C₃×C₄⋊C₄	—	5	2	no	—
130	C₃×C₄⋊C₄	—	5	1	no	—
131	C₃×C₂²⋊C₄	—	5	2	no	—
132	C₃×C₂²⋊C₄	—	5	2	no	—
133	C₃×C₂²⋊C₄	—	5	2	no	—
134	C₃×C₂²⋊C₄	—	5	2	no	—
135	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
136	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
137	C₃×C₄²	—	5	2	no	—
138	C₃×C₄²	—	5	1	no	—
139	C₃×C₄²	—	5	1	no	—
140	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
141	C₂³×C₅	C₂³×C₃	1	2	yes	—
142	C₂²×D₅	C₂³×C₃	1	2	yes	—
143	C₅×Q₈	—	2	2	no	—
144	C₅×D₄	—	2	2	no	—
145	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
146	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
149	C₂²⋊D₅	—	2	4	no	—
150	C₂²⋊D₅	—	4	2	no	—
151	C₂×Dic₅	—	4	2	no	—
152	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
153	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
154	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
155	C₂×Dic₅	C₂³×C₃	1	2	yes	—
156	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
157	D₂₀	—	2	2	no	—
158	C₄×D₅	—	4	2	no	—
159	Dic₁₀	—	2	2	no	—
160	#127	—	5	1	no	—
161	#124	—	5	1	no	—
162	#124	—	5	2	no	—
163	#124	C₃×D₅	1	1	yes	characteristic, complemented
164	#123	—	5	2	no	—
165	#123	—	5	2	no	—
166	C₄×Q₈	—	5	1	no	—
167	C₄×Q₈	—	5	1	no	—
168	C₄×D₄	—	5	1	no	—
169	C₄×D₄	—	5	2	no	—
170	C₄×D₄	—	5	1	no	—
171	C₂×C₃×C₅	—	4	2	no	—
172	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
173	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
174	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
175	C₃×D₅	—	4	2	no	—
176	C₂³×C₃	—	5	2	no	—
177	C₂³×C₃	C₂²×D₅	1	2	yes	—
178	C₃×Q₈	—	2	2	no	—
179	C₃×Q₈	—	10	2	no	—
180	C₃×D₄	—	2	2	no	—
181	C₃×D₄	—	10	4	no	—
182	C₃×D₄	—	10	2	no	—
183	C₃×D₄	—	20	2	no	—
184	C₂×C₃×C₄	—	5	1	no	—
185	C₂×C₃×C₄	—	5	1	no	—
186	C₂×C₃×C₄	—	5	1	no	—
187	C₂×C₃×C₄	—	5	2	no	—
188	C₂×C₃×C₄	—	5	1	no	—
189	C₂×C₃×C₄	—	20	2	no	—
190	C₂×C₃×C₄	—	20	2	no	—
191	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
192	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
199	C₂×D₅	—	2	2	no	—
200	C₂×D₅	—	2	2	no	—
201	C₂×D₅	—	2	2	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	—	2	1	no	—
205	C₄×C₅	C₃×#41	1	2	yes	—
206	Dic₅	—	2	1	no	—
207	Dic₅	—	2	2	no	—
208	Dic₅	—	2	1	no	—
209	Dic₅	—	2	1	no	—
210	Dic₅	C₃×#41	1	2	yes	—
211	C₂×Q₈	—	5	1	no	—
212	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
213	C₂×D₄	—	5	2	no	—
214	C₂×D₄	—	5	2	no	—
215	C₂×D₄	—	5	1	no	—
216	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
217	C₂²×C₄	—	5	2	no	—
218	C₂²×C₄	—	5	2	no	—
219	C₄⋊C₄	—	5	1	no	—
220	C₄⋊C₄	—	5	1	no	—
221	C₄⋊C₄	—	5	1	no	—
222	C₄⋊C₄	—	5	2	no	—
223	C₄⋊C₄	—	5	1	no	—
224	C₂²⋊C₄	—	5	2	no	—
225	C₂²⋊C₄	—	5	2	no	—
226	C₂²⋊C₄	—	5	2	no	—
227	C₂²⋊C₄	—	5	2	no	—
228	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
229	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
230	C₄²	—	5	2	no	—
231	C₄²	—	5	1	no	—
232	C₄²	—	5	1	no	—
233	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
234	C₃×C₅	#559	1	1	yes	characteristic, complemented, Hall
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	10	2	no	—
239	C₂²×C₃	—	10	2	no	—
240	C₂²×C₃	—	10	2	no	—
241	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	5	2	no	—
246	C₃×C₄	—	10	1	no	—
247	C₃×C₄	—	10	2	no	—
248	C₃×C₄	—	10	1	no	—
249	C₃×C₄	—	10	1	no	—
250	C₃×C₄	#711	1	2	yes	—
251	C₂×C₅	—	4	2	no	—
252	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
253	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
254	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
255	D₅	—	4	2	no	—
256	C₂³	—	5	2	no	—
257	C₂³	C₂²×C₃×D₅	1	2	yes	—
258	Q₈	—	2	2	no	—
259	Q₈	—	10	2	no	—
260	D₄	—	2	2	no	—
261	D₄	—	10	4	no	—
262	D₄	—	10	2	no	—
263	D₄	—	20	2	no	—
264	C₂×C₄	—	5	1	no	—
265	C₂×C₄	—	5	1	no	—
266	C₂×C₄	—	5	2	no	—
267	C₂×C₄	—	5	1	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	20	2	no	—
270	C₂×C₄	—	20	2	no	—
271	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
272	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
274	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
275	C₂×C₃	—	4	2	no	—
276	C₂×C₃	—	20	2	no	—
277	C₂×C₃	C₂×#711	1	1	yes	characteristic
278	C₂×C₃	D₅×#41	1	1	yes	characteristic
279	C₂×C₃	#4149	1	1	yes	characteristic
280	C₅	C₃×#559	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	10	2	no	—
285	C₂²	—	10	2	no	—
286	C₂²	—	10	2	no	—
287	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	5	2	no	—
292	C₄	—	10	1	no	—
293	C₄	—	10	2	no	—
294	C₄	—	10	1	no	—
295	C₄	—	10	1	no	—
296	C₄	C₃×#711	1	2	yes	—
297	C₃	#66015	1	1	yes	characteristic, direct product factor, Sylow
298	C₂	—	4	2	no	—
299	C₂	—	20	2	no	—
300	C₂	C₂×C₃×#711	1	1	yes	characteristic
301	C₂	C₃×D₅×#41	1	1	yes	characteristic
302	C₂	C₃×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₃×#4086	C₂
hypercenter	C₃×#124	D₅
hypocenter	C₅	C₃×#559
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#124

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#124	C₂	semidirect
2. C₃×#4093	C₂	semidirect
3. C₃×Q₈×Dic₅	C₂	semidirect
5. C₃×D₄×Dic₅	C₂	semidirect
6. C₃×#4032	C₂	semidirect
7. C₃×#4031	C₂	semidirect
8. C₃×#4029	C₂	semidirect
9. C₃×#4027	C₂	semidirect
10. C₃×C₄×D₂₀	C₂	semidirect
11. C₃×C₄×Dic₁₀	C₂	semidirect
12. #66015	C₃	direct (N × H)
37. C₅×#124	C₂×C₃	semidirect
38. #4093	C₂×C₃	semidirect
39. Q₈×Dic₅	C₂×C₃	semidirect
41. D₄×Dic₅	C₂×C₃	semidirect
42. #4032	C₂×C₃	semidirect
43. #4031	C₂×C₃	semidirect
44. #4029	C₂×C₃	semidirect
45. #4027	C₂×C₃	semidirect
46. C₄×D₂₀	C₂×C₃	semidirect
47. C₄×Dic₁₀	C₂×C₃	semidirect
70. C₃×#124	D₅	semidirect
163. #124	C₃×D₅	semidirect
234. C₃×C₅	#559	semidirect
280. C₅	C₃×#559	semidirect
297. C₃	#66015	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#124	2
C₃×#4093	2
C₃×Q₈×Dic₅	2
C₃×#4086	2
C₃×D₄×Dic₅	2
C₃×#4032	2
C₃×#4031	2
C₃×#4029	2
C₃×#4027	2
C₃×C₄×D₂₀	2
C₃×C₄×Dic₁₀	2
#66015	3
C₃×#559	5

Subgroups — C3×#66015 #11755692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66015 #11755692