Subgroups — C₃×#66049 #11755726

Order 960.

668 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#711	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4086	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4085	C₂	1	4	yes	maximal, complemented
6	C₃×#4083	C₂	1	4	yes	maximal, complemented
7	C₃×#4082	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4021	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#4018	C₂	1	1	yes	maximal, characteristic, complemented
10	#66049	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂³×C₃×D₅	C₂²	1	2	yes	—
12	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₃×C₅×D₄	C₂²	1	4	yes	complemented
15	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂×C₃×C₂²⋊D₅	C₂²	1	4	yes	complemented
18	C₂²×C₃×Dic₅	C₂²	1	2	yes	complemented
19	C₃×#711	—	2	4	no	—
20	C₃×D₄×D₅	—	2	4	no	—
21	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
23	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
24	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
25	C₃×C₂²⋊Dic₅	C₂²	1	4	yes	complemented
26	C₃×#685	C₂²	1	2	yes	—
27	C₃×#685	C₂²	1	4	yes	complemented
28	C₃×#683	C₂²	1	2	yes	complemented
29	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic, complemented
30	C₃×#535	—	5	1	no	maximal, Hall
31	C₂×#711	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×D₄×D₅	C₂×C₃	1	1	yes	characteristic, complemented
33	C₅×#127	C₂×C₃	1	1	yes	characteristic, complemented
34	#4086	C₂×C₃	1	1	yes	characteristic, complemented
35	#4085	C₂×C₃	1	4	yes	complemented
36	#4083	C₂×C₃	1	4	yes	complemented
37	#4082	C₂×C₃	1	1	yes	characteristic, complemented
38	#4021	C₂×C₃	1	1	yes	characteristic, complemented
39	#4018	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂³×C₃×C₅	C₂³	1	2	yes	—
41	C₂³×C₃×C₅	C₂³	1	2	yes	—
42	C₂²×C₃×D₅	—	2	4	no	—
43	C₂²×C₃×D₅	—	4	4	no	—
44	C₂²×C₃×D₅	C₂³	1	2	yes	—
45	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
46	C₃×C₅×D₄	—	2	2	no	—
47	C₃×C₅×D₄	—	2	2	no	—
48	C₃×C₅×D₄	—	4	4	no	—
49	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
51	C₃×C₂²⋊D₅	—	2	4	no	—
52	C₃×C₂²⋊D₅	—	2	4	no	—
53	C₃×C₂²⋊D₅	—	4	4	no	—
54	C₂×C₃×Dic₅	—	2	4	no	—
55	C₂×C₃×Dic₅	C₂³	1	2	yes	—
56	C₂×C₃×Dic₅	C₂³	1	2	yes	—
57	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
58	C₃×D₂₀	—	2	2	no	—
59	C₃×C₄×D₅	D₄	1	4	yes	complemented
60	C₃×Dic₁₀	—	2	2	no	—
61	C₂×C₃×#41	—	5	1	no	—
62	C₂²×C₃×D₄	—	5	1	no	—
63	C₃×#127	—	5	1	no	—
64	C₃×#127	D₅	1	1	yes	characteristic, complemented
65	C₃×#124	—	5	1	no	—
66	C₃×#124	—	5	1	no	—
67	C₃×#121	—	5	4	no	—
68	C₃×#120	—	5	4	no	—
69	C₃×#117	—	5	1	no	—
70	C₂³×D₅	C₂²×C₃	1	2	yes	—
71	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₅×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
73	C₂×C₅×D₄	C₂²×C₃	1	4	yes	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	4	yes	complemented
77	C₂²×Dic₅	C₂²×C₃	1	2	yes	complemented
78	#711	—	2	4	no	—
79	D₄×D₅	—	2	4	no	—
80	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
82	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic, complemented
83	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
84	C₂²⋊Dic₅	C₂²×C₃	1	4	yes	complemented
85	#685	C₂²×C₃	1	2	yes	—
86	#685	C₂²×C₃	1	4	yes	complemented
87	#683	C₂²×C₃	1	2	yes	complemented
88	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
89	#535	—	5	1	no	Sylow
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	—	2	2	no	—
92	C₂²×C₃×C₅	—	2	4	no	—
93	C₂²×C₃×C₅	—	2	4	no	—
94	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
95	C₂×C₃×D₅	—	2	2	no	—
96	C₂×C₃×D₅	—	2	2	no	—
97	C₂×C₃×D₅	—	2	4	no	—
98	C₂×C₃×D₅	—	4	8	no	—
99	C₂×C₃×D₅	C₂×D₄	1	2	yes	—
100	C₃×C₄×C₅	—	2	2	no	—
101	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
102	C₃×Dic₅	—	2	2	no	—
103	C₃×Dic₅	—	2	2	no	—
104	C₃×Dic₅	C₂×D₄	1	2	yes	—
105	C₂⁴×C₃	—	5	2	no	—
106	C₃×#41	—	10	4	no	—
107	C₂×C₃×Q₈	—	5	1	no	—
108	C₂×C₃×D₄	—	5	2	no	—
109	C₂×C₃×D₄	—	5	2	no	—
110	C₂×C₃×D₄	—	5	4	no	—
111	C₂×C₃×D₄	—	5	1	no	—
112	C₂×C₃×D₄	—	10	4	no	—
113	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic
114	C₂×C₃×D₄	C₂×D₅	1	1	yes	characteristic, complemented
115	C₂×C₃×D₄	C₂×D₅	1	4	yes	complemented
116	C₂²×C₃×C₄	—	5	1	no	—
117	C₂²×C₃×C₄	—	5	2	no	—
118	C₃×C₄⋊C₄	—	5	2	no	—
119	C₃×C₂²⋊C₄	—	5	4	no	—
120	C₃×C₂²⋊C₄	—	5	2	no	—
121	C₃×C₂²⋊C₄	—	5	4	no	—
122	C₃×C₄²	—	5	1	no	—
123	C₃×C₄²	C₂×D₅	1	1	yes	characteristic, complemented
124	C₂³×C₅	C₂³×C₃	1	2	yes	—
125	C₂³×C₅	C₂³×C₃	1	2	yes	—
126	C₂²×D₅	—	2	4	no	—
127	C₂²×D₅	—	4	4	no	—
128	C₂²×D₅	C₂³×C₃	1	2	yes	—
129	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
130	C₅×D₄	—	2	2	no	—
131	C₅×D₄	—	2	2	no	—
132	C₅×D₄	—	4	4	no	—
133	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
134	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
135	C₂²⋊D₅	—	2	4	no	—
136	C₂²⋊D₅	—	2	4	no	—
137	C₂²⋊D₅	—	4	4	no	—
138	C₂×Dic₅	—	2	4	no	—
139	C₂×Dic₅	C₂³×C₃	1	2	yes	—
140	C₂×Dic₅	C₂³×C₃	1	2	yes	—
141	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
142	D₂₀	—	2	2	no	—
143	C₄×D₅	C₃×D₄	1	4	yes	complemented
144	Dic₁₀	—	2	2	no	—
145	C₂×#41	—	5	1	no	—
146	C₂²×D₄	—	5	1	no	—
147	#127	—	5	1	no	—
148	#127	C₃×D₅	1	1	yes	characteristic, complemented
149	#124	—	5	1	no	—
150	#124	—	5	1	no	—
151	#121	—	5	4	no	—
152	#120	—	5	4	no	—
153	#117	—	5	1	no	—
154	C₂×C₃×C₅	—	4	2	no	—
155	C₂×C₃×C₅	—	4	2	no	—
156	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
158	C₃×D₅	—	2	2	no	—
159	C₃×D₅	—	4	2	no	—
160	C₂³×C₃	—	5	1	no	—
161	C₂³×C₃	—	5	2	no	—
162	C₂³×C₃	—	10	4	no	—
163	C₂³×C₃	—	20	4	no	—
164	C₂³×C₃	C₂²×D₅	1	2	yes	—
165	C₂³×C₃	C₂²×D₅	1	2	yes	—
166	C₃×Q₈	—	10	2	no	—
167	C₃×D₄	—	2	2	no	—
168	C₃×D₄	—	2	2	no	—
169	C₃×D₄	—	4	4	no	—
170	C₃×D₄	—	10	4	no	—
171	C₃×D₄	—	10	4	no	—
172	C₃×D₄	—	10	2	no	—
173	C₃×D₄	—	20	4	no	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	5	4	no	—
176	C₂×C₃×C₄	—	5	2	no	—
177	C₂×C₃×C₄	—	5	2	no	—
178	C₂×C₃×C₄	—	10	4	no	—
179	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
180	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	4	no	—
184	C₂²×C₅	—	2	4	no	—
185	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
186	C₂×D₅	—	2	2	no	—
187	C₂×D₅	—	2	2	no	—
188	C₂×D₅	—	2	4	no	—
189	C₂×D₅	—	4	8	no	—
190	C₂×D₅	C₂×C₃×D₄	1	2	yes	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
193	Dic₅	—	2	2	no	—
194	Dic₅	—	2	2	no	—
195	Dic₅	C₂×C₃×D₄	1	2	yes	—
196	C₂⁴	—	5	2	no	—
197	#41	—	10	4	no	—
198	C₂×Q₈	—	5	1	no	—
199	C₂×D₄	—	5	2	no	—
200	C₂×D₄	—	5	2	no	—
201	C₂×D₄	—	5	4	no	—
202	C₂×D₄	—	5	1	no	—
203	C₂×D₄	—	10	4	no	—
204	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic
205	C₂×D₄	C₂×C₃×D₅	1	1	yes	characteristic, complemented
206	C₂×D₄	C₂×C₃×D₅	1	4	yes	complemented
207	C₂²×C₄	—	5	1	no	—
208	C₂²×C₄	—	5	2	no	—
209	C₄⋊C₄	—	5	2	no	—
210	C₂²⋊C₄	—	5	4	no	—
211	C₂²⋊C₄	—	5	2	no	—
212	C₂²⋊C₄	—	5	4	no	—
213	C₄²	—	5	1	no	—
214	C₄²	C₂×C₃×D₅	1	1	yes	characteristic, complemented
215	C₃×C₅	#535	1	1	yes	characteristic, complemented, Hall
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	2	4	no	—
219	C₂²×C₃	—	2	4	no	—
220	C₂²×C₃	—	5	2	no	—
221	C₂²×C₃	—	10	2	no	—
222	C₂²×C₃	—	10	2	no	—
223	C₂²×C₃	—	10	4	no	—
224	C₂²×C₃	—	20	8	no	—
225	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	5	2	no	—
228	C₃×C₄	—	10	2	no	—
229	C₃×C₄	—	10	2	no	—
230	C₃×C₄	D₄×D₅	1	2	yes	—
231	C₂×C₅	—	4	2	no	—
232	C₂×C₅	—	4	2	no	—
233	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
234	C₂×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
235	D₅	—	2	2	no	—
236	D₅	—	4	2	no	—
237	C₂³	—	5	1	no	—
238	C₂³	—	5	2	no	—
239	C₂³	—	10	4	no	—
240	C₂³	—	20	4	no	—
241	C₂³	C₂²×C₃×D₅	1	2	yes	—
242	C₂³	C₂²×C₃×D₅	1	2	yes	—
243	Q₈	—	10	2	no	—
244	D₄	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	4	4	no	—
247	D₄	—	10	4	no	—
248	D₄	—	10	4	no	—
249	D₄	—	10	2	no	—
250	D₄	—	20	4	no	—
251	C₂×C₄	—	5	1	no	—
252	C₂×C₄	—	5	4	no	—
253	C₂×C₄	—	5	2	no	—
254	C₂×C₄	—	5	2	no	—
255	C₂×C₄	—	10	4	no	—
256	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
257	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	—	4	2	no	—
260	C₂×C₃	—	10	2	no	—
261	C₂×C₃	—	20	2	no	—
262	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
263	C₂×C₃	#4144	1	2	yes	—
264	C₅	C₃×#535	1	1	yes	characteristic, complemented, Sylow
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	5	2	no	—
270	C₂²	—	10	2	no	—
271	C₂²	—	10	2	no	—
272	C₂²	—	10	4	no	—
273	C₂²	—	20	8	no	—
274	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	5	2	no	—
277	C₄	—	10	2	no	—
278	C₄	—	10	2	no	—
279	C₄	C₃×D₄×D₅	1	2	yes	—
280	C₃	#66049	1	1	yes	characteristic, direct product factor, Sylow
281	C₂	—	4	2	no	—
282	C₂	—	4	2	no	—
283	C₂	—	10	2	no	—
284	C₂	—	20	2	no	—
285	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic
286	C₂	C₃×#4144	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×#127	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×#127	D₅
hypocenter	C₅	C₃×#535
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#711	C₂	semidirect
2. C₂×C₃×D₄×D₅	C₂	semidirect
3. C₃×C₅×#127	C₂	semidirect
4. C₃×#4086	C₂	semidirect
5. C₃×#4085	C₂	semidirect
6. C₃×#4083	C₂	semidirect
7. C₃×#4082	C₂	semidirect
8. C₃×#4021	C₂	semidirect
9. C₃×#4018	C₂	semidirect
10. #66049	C₃	direct (N × H)
13. C₂×C₃×C₅×D₄	C₂²	semidirect
14. C₂×C₃×C₅×D₄	C₂²	semidirect
17. C₂×C₃×C₂²⋊D₅	C₂²	semidirect
18. C₂²×C₃×Dic₅	C₂²	semidirect
23. C₂×C₃×Dic₁₀	C₂²	semidirect
24. C₃×C₄²×C₅	C₂²	semidirect
25. C₃×C₂²⋊Dic₅	C₂²	semidirect
27. C₃×#685	C₂²	semidirect
28. C₃×#683	C₂²	semidirect
29. C₃×C₄×Dic₅	C₂²	semidirect
31. C₂×#711	C₂×C₃	semidirect
32. C₂×D₄×D₅	C₂×C₃	semidirect
33. C₅×#127	C₂×C₃	semidirect
34. #4086	C₂×C₃	semidirect
35. #4085	C₂×C₃	semidirect
36. #4083	C₂×C₃	semidirect
37. #4082	C₂×C₃	semidirect
38. #4021	C₂×C₃	semidirect
39. #4018	C₂×C₃	semidirect
59. C₃×C₄×D₅	D₄	semidirect
64. C₃×#127	D₅	semidirect
72. C₂×C₅×D₄	C₂²×C₃	semidirect
73. C₂×C₅×D₄	C₂²×C₃	semidirect
76. C₂×C₂²⋊D₅	C₂²×C₃	semidirect
77. C₂²×Dic₅	C₂²×C₃	semidirect
82. C₂×Dic₁₀	C₂²×C₃	semidirect
83. C₄²×C₅	C₂²×C₃	semidirect
84. C₂²⋊Dic₅	C₂²×C₃	semidirect
86. #685	C₂²×C₃	semidirect
87. #683	C₂²×C₃	semidirect
88. C₄×Dic₅	C₂²×C₃	semidirect
114. C₂×C₃×D₄	C₂×D₅	semidirect
115. C₂×C₃×D₄	C₂×D₅	semidirect
123. C₃×C₄²	C₂×D₅	semidirect
143. C₄×D₅	C₃×D₄	semidirect
148. #127	C₃×D₅	semidirect
205. C₂×D₄	C₂×C₃×D₅	semidirect
206. C₂×D₄	C₂×C₃×D₅	semidirect
214. C₄²	C₂×C₃×D₅	semidirect
215. C₃×C₅	#535	semidirect
264. C₅	C₃×#535	semidirect
280. C₃	#66049	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#711	2
C₂×C₃×D₄×D₅	2
C₃×C₅×#127	2
C₃×#4086	2
C₃×#4085	2
C₃×#4083	2
C₃×#4082	2
C₃×#4021	2
C₃×#4018	2
#66049	3
C₃×#535	5

Subgroups — C3×#66049 #11755726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66049 #11755726