Subgroups — C₃×#65914 #11755591

Order 960.

468 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×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4093	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4092	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4044	C₂	1	2	yes	maximal, complemented
5	C₃×#4041	C₂	1	1	yes	maximal, characteristic
6	C₃×#4040	C₂	1	2	yes	maximal, complemented
7	C₃×#4036	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄×D₂₀	C₂	1	2	yes	maximal, complemented
10	C₃×#4018	C₂	1	2	yes	maximal, complemented
11	C₃×C₄²×D₅	C₂	1	1	yes	maximal, characteristic, complemented
12	#65914	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₂₀	C₂²	1	2	yes	—
16	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
20	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₃×C₄²×C₅	C₂²	1	2	yes	—
24	C₃×#685	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×#685	C₂²	1	2	yes	—
29	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
30	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
31	C₃×#683	C₂²	1	2	yes	—
32	C₃×#683	C₂²	1	2	yes	—
33	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
36	C₃×#529	—	5	1	no	maximal, Hall
37	C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
38	#4093	C₂×C₃	1	1	yes	characteristic, complemented
39	#4092	C₂×C₃	1	1	yes	characteristic, complemented
40	#4044	C₂×C₃	1	2	yes	complemented
41	#4041	C₂×C₃	1	1	yes	characteristic
42	#4040	C₂×C₃	1	2	yes	complemented
43	#4036	C₂×C₃	1	1	yes	characteristic, complemented
44	C₄×D₂₀	C₂×C₃	1	2	yes	complemented
45	C₄×D₂₀	C₂×C₃	1	1	yes	characteristic, complemented
46	#4018	C₂×C₃	1	2	yes	complemented
47	C₄²×D₅	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂²×C₃×D₅	C₂³	1	2	yes	—
49	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
50	C₃×C₅×Q₈	—	2	2	no	—
51	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
57	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
58	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
59	C₂×C₃×Dic₅	C₂³	1	2	yes	—
60	C₃×D₂₀	—	2	4	no	—
61	C₃×D₂₀	—	2	2	no	—
62	C₃×C₄×D₅	—	2	2	no	—
63	C₃×C₄×D₅	—	2	2	no	—
64	C₃×C₄×D₅	—	2	2	no	—
65	C₃×C₄×D₅	—	2	4	no	—
66	C₃×#126	—	5	2	no	—
67	C₃×#125	—	5	1	no	—
68	C₃×#124	—	5	1	no	—
69	C₃×#123	—	5	2	no	—
70	C₃×C₂²⋊Q₈	—	5	1	no	—
71	C₃×#121	—	5	1	no	—
72	C₃×C₄×Q₈	D₅	1	1	yes	characteristic, complemented
73	C₃×C₄×D₄	—	5	1	no	—
74	C₃×C₄×D₄	—	5	2	no	—
75	C₃×#117	—	5	2	no	—
76	C₂×C₃×C₄²	—	5	1	no	—
77	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
78	C₂×D₂₀	C₂²×C₃	1	2	yes	—
79	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
84	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
85	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
86	C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
87	C₄²×C₅	C₂²×C₃	1	2	yes	—
88	#685	C₂²×C₃	1	2	yes	—
89	#685	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	2	yes	—
94	C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
95	#683	C₂²×C₃	1	2	yes	—
96	#683	C₂²×C₃	1	2	yes	—
97	C₄×Dic₅	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	#529	—	5	1	no	Sylow
101	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
102	C₂×C₃×D₅	—	2	1	no	—
103	C₂×C₃×D₅	—	2	1	no	—
104	C₂×C₃×D₅	—	2	2	no	—
105	C₂×C₃×D₅	—	2	2	no	—
106	C₂×C₃×D₅	—	2	2	no	—
107	C₂×C₃×D₅	#41	1	2	yes	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	2	2	no	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	#41	1	2	yes	—
112	C₃×C₄×C₅	#41	1	2	yes	—
113	C₃×Dic₅	—	2	1	no	—
114	C₃×Dic₅	—	2	1	no	—
115	C₃×Dic₅	—	2	2	no	—
116	C₃×Dic₅	—	2	1	no	—
117	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
118	C₂×C₃×D₄	—	5	2	no	—
119	C₂×C₃×D₄	—	5	1	no	—
120	C₂²×C₃×C₄	—	5	1	no	—
121	C₂²×C₃×C₄	—	5	1	no	—
122	C₂²×C₃×C₄	—	5	1	no	—
123	C₂²×C₃×C₄	—	5	2	no	—
124	C₃×C₄⋊C₄	—	5	2	no	—
125	C₃×C₄⋊C₄	—	5	2	no	—
126	C₃×C₄⋊C₄	—	5	1	no	—
127	C₃×C₄⋊C₄	—	5	2	no	—
128	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
129	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
130	C₃×C₂²⋊C₄	—	5	2	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×C₄²	—	5	1	no	—
136	C₃×C₄²	—	5	1	no	—
137	C₃×C₄²	—	5	1	no	—
138	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
139	C₃×C₄²	C₂×D₅	1	2	yes	—
140	C₂²×D₅	C₂³×C₃	1	2	yes	—
141	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
142	C₅×Q₈	—	2	2	no	—
143	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
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×Dic₅	C₂³×C₃	1	1	yes	characteristic
149	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
150	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
151	C₂×Dic₅	C₂³×C₃	1	2	yes	—
152	D₂₀	—	2	4	no	—
153	D₂₀	—	2	2	no	—
154	C₄×D₅	—	2	2	no	—
155	C₄×D₅	—	2	2	no	—
156	C₄×D₅	—	2	2	no	—
157	C₄×D₅	—	2	4	no	—
158	#126	—	5	2	no	—
159	#125	—	5	1	no	—
160	#124	—	5	1	no	—
161	#123	—	5	2	no	—
162	C₂²⋊Q₈	—	5	1	no	—
163	#121	—	5	1	no	—
164	C₄×Q₈	C₃×D₅	1	1	yes	characteristic, complemented
165	C₄×D₄	—	5	1	no	—
166	C₄×D₄	—	5	2	no	—
167	#117	—	5	2	no	—
168	C₂×C₄²	—	5	1	no	—
169	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
170	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
172	C₃×D₅	—	2	2	no	—
173	C₃×D₅	—	4	2	no	—
174	C₂³×C₃	—	5	1	no	—
175	C₂³×C₃	—	5	2	no	—
176	C₃×Q₈	—	2	2	no	—
177	C₃×D₄	—	10	4	no	—
178	C₃×D₄	—	10	2	no	—
179	C₂×C₃×C₄	—	5	1	no	—
180	C₂×C₃×C₄	—	5	1	no	—
181	C₂×C₃×C₄	—	5	1	no	—
182	C₂×C₃×C₄	—	5	2	no	—
183	C₂×C₃×C₄	—	10	2	no	—
184	C₂×C₃×C₄	—	10	2	no	—
185	C₂×C₃×C₄	—	10	2	no	—
186	C₂×C₃×C₄	—	10	4	no	—
187	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
189	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
190	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
192	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
193	C₂×D₅	—	2	1	no	—
194	C₂×D₅	—	2	1	no	—
195	C₂×D₅	—	2	2	no	—
196	C₂×D₅	—	2	2	no	—
197	C₂×D₅	—	2	2	no	—
198	C₂×D₅	C₃×#41	1	2	yes	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	C₃×#41	1	2	yes	—
203	C₄×C₅	C₃×#41	1	2	yes	—
204	Dic₅	—	2	1	no	—
205	Dic₅	—	2	1	no	—
206	Dic₅	—	2	2	no	—
207	Dic₅	—	2	1	no	—
208	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
209	C₂×D₄	—	5	2	no	—
210	C₂×D₄	—	5	1	no	—
211	C₂²×C₄	—	5	1	no	—
212	C₂²×C₄	—	5	1	no	—
213	C₂²×C₄	—	5	1	no	—
214	C₂²×C₄	—	5	2	no	—
215	C₄⋊C₄	—	5	2	no	—
216	C₄⋊C₄	—	5	2	no	—
217	C₄⋊C₄	—	5	1	no	—
218	C₄⋊C₄	—	5	2	no	—
219	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
220	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
221	C₂²⋊C₄	—	5	2	no	—
222	C₂²⋊C₄	—	5	2	no	—
223	C₂²⋊C₄	—	5	2	no	—
224	C₂²⋊C₄	—	5	2	no	—
225	C₂²⋊C₄	—	5	2	no	—
226	C₄²	—	5	1	no	—
227	C₄²	—	5	1	no	—
228	C₄²	—	5	1	no	—
229	C₄²	C₂×C₃×D₅	1	1	yes	characteristic
230	C₄²	C₂×C₃×D₅	1	2	yes	—
231	C₃×C₅	#529	1	1	yes	characteristic, complemented, Hall
232	C₂²×C₃	—	5	2	no	—
233	C₂²×C₃	—	10	1	no	—
234	C₂²×C₃	—	10	1	no	—
235	C₂²×C₃	—	10	2	no	—
236	C₂²×C₃	—	10	2	no	—
237	C₂²×C₃	—	10	2	no	—
238	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	10	1	no	—
243	C₃×C₄	—	10	1	no	—
244	C₃×C₄	—	10	2	no	—
245	C₃×C₄	—	10	1	no	—
246	C₃×C₄	#709	1	2	yes	—
247	C₃×C₄	#713	1	2	yes	—
248	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
249	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
250	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
251	D₅	—	2	2	no	—
252	D₅	—	4	2	no	—
253	C₂³	—	5	1	no	—
254	C₂³	—	5	2	no	—
255	Q₈	—	2	2	no	—
256	D₄	—	10	4	no	—
257	D₄	—	10	2	no	—
258	C₂×C₄	—	5	1	no	—
259	C₂×C₄	—	5	1	no	—
260	C₂×C₄	—	5	1	no	—
261	C₂×C₄	—	5	2	no	—
262	C₂×C₄	—	10	2	no	—
263	C₂×C₄	—	10	2	no	—
264	C₂×C₄	—	10	2	no	—
265	C₂×C₄	—	10	4	no	—
266	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
268	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
269	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
270	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
271	C₂×C₃	—	10	2	no	—
272	C₂×C₃	—	20	2	no	—
273	C₂×C₃	C₂×#709	1	1	yes	characteristic
274	C₂×C₃	C₂×#713	1	1	yes	characteristic
275	C₂×C₃	D₅×#41	1	1	yes	characteristic
276	C₅	C₃×#529	1	1	yes	characteristic, complemented, Sylow
277	C₂²	—	5	2	no	—
278	C₂²	—	10	1	no	—
279	C₂²	—	10	1	no	—
280	C₂²	—	10	2	no	—
281	C₂²	—	10	2	no	—
282	C₂²	—	10	2	no	—
283	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	10	1	no	—
288	C₄	—	10	1	no	—
289	C₄	—	10	2	no	—
290	C₄	—	10	1	no	—
291	C₄	C₃×#709	1	2	yes	—
292	C₄	C₃×#713	1	2	yes	—
293	C₃	#65914	1	1	yes	characteristic, direct product factor, Sylow
294	C₂	—	10	2	no	—
295	C₂	—	20	2	no	—
296	C₂	C₂×C₃×#709	1	1	yes	characteristic
297	C₂	C₂×C₃×#713	1	1	yes	characteristic
298	C₂	C₃×D₅×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₅
Fitting	C₃×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
prime radical	C₃×#4041	C₂
hypercenter	C₃×C₄×Q₈	D₅
hypocenter	C₅	C₃×#529
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₄×C₅×Q₈	C₂	semidirect
2. C₃×#4093	C₂	semidirect
3. C₃×#4092	C₂	semidirect
4. C₃×#4044	C₂	semidirect
6. C₃×#4040	C₂	semidirect
7. C₃×#4036	C₂	semidirect
8. C₃×C₄×D₂₀	C₂	semidirect
9. C₃×C₄×D₂₀	C₂	semidirect
10. C₃×#4018	C₂	semidirect
11. C₃×C₄²×D₅	C₂	semidirect
12. #65914	C₃	direct (N × H)
37. C₄×C₅×Q₈	C₂×C₃	semidirect
38. #4093	C₂×C₃	semidirect
39. #4092	C₂×C₃	semidirect
40. #4044	C₂×C₃	semidirect
42. #4040	C₂×C₃	semidirect
43. #4036	C₂×C₃	semidirect
44. C₄×D₂₀	C₂×C₃	semidirect
45. C₄×D₂₀	C₂×C₃	semidirect
46. #4018	C₂×C₃	semidirect
47. C₄²×D₅	C₂×C₃	semidirect
72. C₃×C₄×Q₈	D₅	semidirect
164. C₄×Q₈	C₃×D₅	semidirect
231. C₃×C₅	#529	semidirect
276. C₅	C₃×#529	semidirect
293. C₃	#65914	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×C₄×C₅×Q₈	2
C₃×#4093	2
C₃×#4092	2
C₃×#4044	2
C₃×#4041	2
C₃×#4040	2
C₃×#4036	2
C₃×C₄×D₂₀	2
C₃×#4018	2
C₃×C₄²×D₅	2
#65914	3
C₃×#529	5

Subgroups — C3×#65914 #11755591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65914 #11755591