Subgroups — C₃×#66000 #11755677

Order 960.

504 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4086	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4083	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4076	C₂	1	2	yes	maximal, complemented
5	C₃×#4044	C₂	1	2	yes	maximal, complemented
6	C₃×#4041	C₂	1	2	yes	maximal, complemented
7	C₃×#4030	C₂	1	2	yes	maximal, complemented
8	C₃×#4029	C₂	1	2	yes	maximal, complemented
9	C₃×#4028	C₂	1	2	yes	maximal, complemented
10	#66000	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂³×C₃×D₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	complemented
17	C₂×C₃×D₂₀	C₂²	1	2	yes	—
18	C₂×C₃×D₂₀	C₂²	1	2	yes	—
19	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₃×C₄×D₅	C₂²	1	2	yes	complemented
21	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
22	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
24	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₃×#685	C₂²	1	2	yes	—
26	C₃×#685	C₂²	1	2	yes	—
27	C₃×#685	C₂²	1	2	yes	—
28	C₃×#685	C₂²	1	2	yes	complemented
29	C₃×C₄⋊Dic₅	C₂²	1	2	yes	complemented
30	C₃×#683	C₂²	1	2	yes	complemented
31	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic, complemented
32	C₃×#560	—	5	1	no	maximal, Hall
33	C₅×#123	C₂×C₃	1	1	yes	characteristic, complemented
34	#4086	C₂×C₃	1	1	yes	characteristic, complemented
35	#4083	C₂×C₃	1	1	yes	characteristic, complemented
36	#4076	C₂×C₃	1	2	yes	complemented
37	#4044	C₂×C₃	1	2	yes	complemented
38	#4041	C₂×C₃	1	2	yes	complemented
39	#4030	C₂×C₃	1	2	yes	complemented
40	#4029	C₂×C₃	1	2	yes	complemented
41	#4028	C₂×C₃	1	2	yes	complemented
42	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
43	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
44	C₂²×C₃×D₅	—	4	1	no	—
45	C₂²×C₃×D₅	—	4	2	no	—
46	C₂²×C₃×D₅	C₂³	1	2	yes	—
47	C₂²×C₃×D₅	C₂³	1	2	yes	—
48	C₃×C₅×D₄	—	4	1	no	—
49	C₂×C₃×C₄×C₅	—	4	1	no	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
53	C₃×C₂²⋊D₅	—	4	2	no	—
54	C₃×C₂²⋊D₅	—	4	2	no	—
55	C₃×C₂²⋊D₅	—	4	2	no	—
56	C₂×C₃×Dic₅	C₂³	1	2	yes	—
57	C₂×C₃×Dic₅	C₂³	1	2	yes	—
58	C₃×D₂₀	—	4	2	no	—
59	C₃×D₂₀	—	4	2	no	—
60	C₃×D₂₀	—	4	1	no	—
61	C₃×C₄×D₅	—	4	2	no	—
62	C₃×#127	—	5	1	no	—
63	C₃×#126	—	5	2	no	—
64	C₃×#123	—	5	2	no	—
65	C₃×#123	D₅	1	1	yes	characteristic, complemented
66	C₃×#121	—	5	2	no	—
67	C₃×#121	—	5	2	no	—
68	C₃×#121	—	5	2	no	—
69	C₃×#120	—	5	1	no	—
70	C₃×#120	—	5	2	no	—
71	C₂³×D₅	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
74	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
75	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
76	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	complemented
77	C₂×D₂₀	C₂²×C₃	1	2	yes	—
78	C₂×D₂₀	C₂²×C₃	1	2	yes	—
79	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic, complemented
80	C₂×C₄×D₅	C₂²×C₃	1	2	yes	complemented
81	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	complemented
82	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
83	C₅×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
84	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
85	#685	C₂²×C₃	1	2	yes	—
86	#685	C₂²×C₃	1	2	yes	—
87	#685	C₂²×C₃	1	2	yes	—
88	#685	C₂²×C₃	1	2	yes	complemented
89	C₄⋊Dic₅	C₂²×C₃	1	2	yes	complemented
90	#683	C₂²×C₃	1	2	yes	complemented
91	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
92	#560	—	5	1	no	Sylow
93	C₂²×C₃×C₅	—	2	2	no	—
94	C₂²×C₃×C₅	—	2	2	no	—
95	C₂²×C₃×C₅	—	2	1	no	—
96	C₂²×C₃×C₅	—	2	1	no	—
97	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
98	C₂×C₃×D₅	—	2	2	no	—
99	C₂×C₃×D₅	—	2	2	no	—
100	C₂×C₃×D₅	—	2	2	no	—
101	C₂×C₃×D₅	—	2	2	no	—
102	C₂×C₃×D₅	—	2	2	no	—
103	C₂×C₃×D₅	—	2	2	no	—
104	C₂×C₃×D₅	—	4	4	no	—
105	C₃×C₄×C₅	—	2	2	no	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×Dic₅	—	2	2	no	—
109	C₃×Dic₅	—	2	2	no	—
110	C₂⁴×C₃	—	5	1	no	—
111	C₂×C₃×D₄	—	5	2	no	—
112	C₂×C₃×D₄	—	5	2	no	—
113	C₂×C₃×D₄	—	5	2	no	—
114	C₂×C₃×D₄	—	5	2	no	—
115	C₂×C₃×D₄	—	5	2	no	—
116	C₂×C₃×D₄	—	5	1	no	—
117	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
118	C₂²×C₃×C₄	—	5	2	no	—
119	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic, complemented
120	C₃×C₄⋊C₄	—	5	2	no	—
121	C₃×C₄⋊C₄	—	5	2	no	—
122	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	complemented
123	C₃×C₂²⋊C₄	—	5	1	no	—
124	C₃×C₂²⋊C₄	—	5	2	no	—
125	C₃×C₂²⋊C₄	—	5	2	no	—
126	C₃×C₂²⋊C₄	—	5	2	no	—
127	C₃×C₂²⋊C₄	—	5	2	no	—
128	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
129	C₃×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
130	C₃×C₄²	—	5	1	no	—
131	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
132	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
133	C₂²×D₅	—	4	1	no	—
134	C₂²×D₅	—	4	2	no	—
135	C₂²×D₅	C₂³×C₃	1	2	yes	—
136	C₂²×D₅	C₂³×C₃	1	2	yes	—
137	C₅×D₄	—	4	1	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
141	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
142	C₂²⋊D₅	—	4	2	no	—
143	C₂²⋊D₅	—	4	2	no	—
144	C₂²⋊D₅	—	4	2	no	—
145	C₂×Dic₅	C₂³×C₃	1	2	yes	—
146	C₂×Dic₅	C₂³×C₃	1	2	yes	—
147	D₂₀	—	4	2	no	—
148	D₂₀	—	4	2	no	—
149	D₂₀	—	4	1	no	—
150	C₄×D₅	—	4	2	no	—
151	#127	—	5	1	no	—
152	#126	—	5	2	no	—
153	#123	—	5	2	no	—
154	#123	C₃×D₅	1	1	yes	characteristic, complemented
155	#121	—	5	2	no	—
156	#121	—	5	2	no	—
157	#121	—	5	2	no	—
158	#120	—	5	1	no	—
159	#120	—	5	2	no	—
160	C₂×C₃×C₅	—	4	1	no	—
161	C₂×C₃×C₅	—	4	1	no	—
162	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
164	C₃×D₅	—	4	2	no	—
165	C₃×D₅	—	4	2	no	—
166	C₂³×C₃	—	5	2	no	—
167	C₂³×C₃	—	5	2	no	—
168	C₂³×C₃	—	20	1	no	—
169	C₂³×C₃	—	20	2	no	—
170	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
171	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
172	C₃×D₄	—	4	1	no	—
173	C₃×D₄	—	20	2	no	—
174	C₃×D₄	—	20	1	no	—
175	C₃×D₄	—	20	2	no	—
176	C₃×D₄	—	20	2	no	—
177	C₃×D₄	—	20	2	no	—
178	C₃×D₄	—	20	2	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	—	5	2	no	—
181	C₂×C₃×C₄	—	5	2	no	—
182	C₂×C₃×C₄	—	20	2	no	—
183	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
185	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	—	2	1	no	—
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
191	C₂×D₅	—	2	2	no	—
192	C₂×D₅	—	2	2	no	—
193	C₂×D₅	—	2	2	no	—
194	C₂×D₅	—	2	2	no	—
195	C₂×D₅	—	2	2	no	—
196	C₂×D₅	—	2	2	no	—
197	C₂×D₅	—	4	4	no	—
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	—	2	1	no	—
201	Dic₅	—	2	2	no	—
202	Dic₅	—	2	2	no	—
203	C₂⁴	—	5	1	no	—
204	C₂×D₄	—	5	2	no	—
205	C₂×D₄	—	5	2	no	—
206	C₂×D₄	—	5	2	no	—
207	C₂×D₄	—	5	2	no	—
208	C₂×D₄	—	5	2	no	—
209	C₂×D₄	—	5	1	no	—
210	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
211	C₂²×C₄	—	5	2	no	—
212	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic, complemented
213	C₄⋊C₄	—	5	2	no	—
214	C₄⋊C₄	—	5	2	no	—
215	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	complemented
216	C₂²⋊C₄	—	5	1	no	—
217	C₂²⋊C₄	—	5	2	no	—
218	C₂²⋊C₄	—	5	2	no	—
219	C₂²⋊C₄	—	5	2	no	—
220	C₂²⋊C₄	—	5	2	no	—
221	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
222	C₂²⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic, complemented
223	C₄²	—	5	1	no	—
224	C₃×C₅	#560	1	1	yes	characteristic, complemented, Hall
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	10	2	no	—
230	C₂²×C₃	—	10	2	no	—
231	C₂²×C₃	—	10	2	no	—
232	C₂²×C₃	—	10	2	no	—
233	C₂²×C₃	—	10	2	no	—
234	C₂²×C₃	—	10	2	no	—
235	C₂²×C₃	—	20	4	no	—
236	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	2	1	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	10	2	no	—
241	C₃×C₄	—	10	2	no	—
242	C₂×C₅	—	4	1	no	—
243	C₂×C₅	—	4	1	no	—
244	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
245	C₂×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
246	D₅	—	4	2	no	—
247	D₅	—	4	2	no	—
248	C₂³	—	5	2	no	—
249	C₂³	—	5	2	no	—
250	C₂³	—	20	1	no	—
251	C₂³	—	20	2	no	—
252	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
253	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
254	D₄	—	4	1	no	—
255	D₄	—	20	2	no	—
256	D₄	—	20	1	no	—
257	D₄	—	20	2	no	—
258	D₄	—	20	2	no	—
259	D₄	—	20	2	no	—
260	D₄	—	20	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	5	2	no	—
263	C₂×C₄	—	5	2	no	—
264	C₂×C₄	—	20	2	no	—
265	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
267	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
268	C₂×C₃	—	4	1	no	—
269	C₂×C₃	—	4	1	no	—
270	C₂×C₃	—	20	2	no	—
271	C₂×C₃	—	20	2	no	—
272	C₂×C₃	#4144	1	1	yes	characteristic
273	C₂×C₃	#4149	1	2	yes	—
274	C₅	C₃×#560	1	1	yes	characteristic, complemented, Sylow
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	10	2	no	—
280	C₂²	—	10	2	no	—
281	C₂²	—	10	2	no	—
282	C₂²	—	10	2	no	—
283	C₂²	—	10	2	no	—
284	C₂²	—	10	2	no	—
285	C₂²	—	20	4	no	—
286	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	10	2	no	—
291	C₄	—	10	2	no	—
292	C₃	#66000	1	1	yes	characteristic, direct product factor, Sylow
293	C₂	—	4	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	20	2	no	—
296	C₂	—	20	2	no	—
297	C₂	C₃×#4144	1	1	yes	characteristic
298	C₂	C₃×#4149	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₅×#123	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×#123	D₅
hypocenter	C₅	C₃×#560
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#123

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#123	C₂	semidirect
2. C₃×#4086	C₂	semidirect
3. C₃×#4083	C₂	semidirect
4. C₃×#4076	C₂	semidirect
5. C₃×#4044	C₂	semidirect
6. C₃×#4041	C₂	semidirect
7. C₃×#4030	C₂	semidirect
8. C₃×#4029	C₂	semidirect
9. C₃×#4028	C₂	semidirect
10. #66000	C₃	direct (N × H)
13. C₂²×C₃×C₄×C₅	C₂²	semidirect
16. C₂×C₃×C₂²⋊D₅	C₂²	semidirect
19. C₂×C₃×D₂₀	C₂²	semidirect
20. C₂×C₃×C₄×D₅	C₂²	semidirect
21. C₃×C₅×C₄⋊C₄	C₂²	semidirect
23. C₃×C₅×C₂²⋊C₄	C₂²	semidirect
28. C₃×#685	C₂²	semidirect
29. C₃×C₄⋊Dic₅	C₂²	semidirect
30. C₃×#683	C₂²	semidirect
31. C₃×C₄×Dic₅	C₂²	semidirect
33. C₅×#123	C₂×C₃	semidirect
34. #4086	C₂×C₃	semidirect
35. #4083	C₂×C₃	semidirect
36. #4076	C₂×C₃	semidirect
37. #4044	C₂×C₃	semidirect
38. #4041	C₂×C₃	semidirect
39. #4030	C₂×C₃	semidirect
40. #4029	C₂×C₃	semidirect
41. #4028	C₂×C₃	semidirect
65. C₃×#123	D₅	semidirect
73. C₂²×C₄×C₅	C₂²×C₃	semidirect
76. C₂×C₂²⋊D₅	C₂²×C₃	semidirect
79. C₂×D₂₀	C₂²×C₃	semidirect
80. C₂×C₄×D₅	C₂²×C₃	semidirect
81. C₅×C₄⋊C₄	C₂²×C₃	semidirect
83. C₅×C₂²⋊C₄	C₂²×C₃	semidirect
88. #685	C₂²×C₃	semidirect
89. C₄⋊Dic₅	C₂²×C₃	semidirect
90. #683	C₂²×C₃	semidirect
91. C₄×Dic₅	C₂²×C₃	semidirect
119. C₂²×C₃×C₄	C₂×D₅	semidirect
122. C₃×C₄⋊C₄	C₂×D₅	semidirect
129. C₃×C₂²⋊C₄	C₂×D₅	semidirect
154. #123	C₃×D₅	semidirect
212. C₂²×C₄	C₂×C₃×D₅	semidirect
215. C₄⋊C₄	C₂×C₃×D₅	semidirect
222. C₂²⋊C₄	C₂×C₃×D₅	semidirect
224. C₃×C₅	#560	semidirect
274. C₅	C₃×#560	semidirect
292. C₃	#66000	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×#123	2
C₃×#4086	2
C₃×#4083	2
C₃×#4076	2
C₃×#4044	2
C₃×#4041	2
C₃×#4030	2
C₃×#4029	2
C₃×#4028	2
#66000	3
C₃×#560	5

Subgroups — C3×#66000 #11755677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66000 #11755677