Subgroups — C₃×#65967 #11755644

Order 960.

580 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#709	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4093	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4042	C₂	1	2	yes	maximal, complemented
7	C₃×#4041	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4039	C₂	1	2	yes	maximal, complemented
9	C₃×#4033	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#4031	C₂	1	2	yes	maximal, complemented
11	C₃×#4030	C₂	1	2	yes	maximal, complemented
12	#65967	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊D₅	C₂²	1	1	yes	characteristic
17	C₃×Q₈×D₅	—	2	4	no	—
18	C₃×#709	—	2	4	no	—
19	C₂×C₃×D₂₀	C₂²	1	1	yes	characteristic
20	C₂×C₃×D₂₀	C₂²	1	2	yes	—
21	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
22	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
23	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
24	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
25	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
26	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
28	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
30	C₃×#685	C₂²	1	2	yes	—
31	C₃×#685	C₂²	1	2	yes	—
32	C₃×#685	C₂²	1	2	yes	—
33	C₃×#685	C₂²	1	1	yes	characteristic
34	C₃×#683	C₂²	1	2	yes	—
35	C₃×#683	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
37	C₃×C₄×Dic₅	C₂²	1	2	yes	—
38	C₃×#548	—	5	1	no	maximal, Hall
39	C₂×Q₈×D₅	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×#709	C₂×C₃	1	1	yes	characteristic, complemented
41	C₅×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
42	#4093	C₂×C₃	1	1	yes	characteristic, complemented
43	C₄×C₂²⋊D₅	C₂×C₃	1	1	yes	characteristic, complemented
44	#4042	C₂×C₃	1	2	yes	complemented
45	#4041	C₂×C₃	1	1	yes	characteristic, complemented
46	#4039	C₂×C₃	1	2	yes	complemented
47	#4033	C₂×C₃	1	1	yes	characteristic, complemented
48	#4031	C₂×C₃	1	2	yes	complemented
49	#4030	C₂×C₃	1	2	yes	complemented
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂²×C₃×D₅	C₂³	1	2	yes	—
52	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
53	C₃×C₅×Q₈	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₃×C₂²⋊D₅	—	2	2	no	—
60	C₃×C₂²⋊D₅	—	2	4	no	—
61	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
63	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
64	C₂×C₃×Dic₅	C₂³	1	2	yes	—
65	C₃×D₂₀	—	2	4	no	—
66	C₃×D₂₀	—	2	2	no	—
67	C₃×C₄×D₅	—	2	2	no	—
68	C₃×C₄×D₅	—	2	4	no	—
69	C₃×C₄×D₅	—	2	4	no	—
70	C₃×Dic₁₀	—	2	4	no	—
71	C₃×Dic₁₀	D₄	1	4	yes	complemented
72	C₂×C₃×#41	—	5	1	no	—
73	C₂²×C₃×Q₈	—	5	1	no	—
74	C₃×#124	—	5	1	no	—
75	C₃×#124	—	5	2	no	—
76	C₃×C₂²⋊Q₈	—	5	2	no	—
77	C₃×C₂²⋊Q₈	D₅	1	1	yes	characteristic, complemented
78	C₃×#121	—	5	1	no	—
79	C₃×#121	—	5	2	no	—
80	C₃×C₄×Q₈	—	5	1	no	—
81	C₃×C₄×D₄	—	5	1	no	—
82	C₃×C₄×D₄	—	5	2	no	—
83	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₂²⋊D₅	C₂²×C₃	1	2	yes	—
86	C₂×C₂²⋊D₅	C₂²×C₃	1	1	yes	characteristic
87	Q₈×D₅	—	2	4	no	—
88	#709	—	2	4	no	—
89	C₂×D₂₀	C₂²×C₃	1	1	yes	characteristic
90	C₂×D₂₀	C₂²×C₃	1	2	yes	—
91	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
92	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
93	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
94	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
95	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
96	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
97	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
98	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
99	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
100	#685	C₂²×C₃	1	2	yes	—
101	#685	C₂²×C₃	1	2	yes	—
102	#685	C₂²×C₃	1	2	yes	—
103	#685	C₂²×C₃	1	1	yes	characteristic
104	#683	C₂²×C₃	1	2	yes	—
105	#683	C₂²×C₃	1	1	yes	characteristic
106	C₄×Dic₅	C₂²×C₃	1	2	yes	—
107	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
108	#548	—	5	1	no	Sylow
109	C₂²×C₃×C₅	—	2	1	no	—
110	C₂²×C₃×C₅	—	2	1	no	—
111	C₂²×C₃×C₅	—	2	1	no	—
112	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
113	C₂×C₃×D₅	—	2	1	no	—
114	C₂×C₃×D₅	—	2	1	no	—
115	C₂×C₃×D₅	—	2	2	no	—
116	C₂×C₃×D₅	—	2	2	no	—
117	C₂×C₃×D₅	—	2	2	no	—
118	C₂×C₃×D₅	#41	1	2	yes	complemented
119	C₃×C₄×C₅	—	2	2	no	—
120	C₃×C₄×C₅	—	2	1	no	—
121	C₃×C₄×C₅	—	2	2	no	—
122	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
123	C₃×Dic₅	—	2	1	no	—
124	C₃×Dic₅	—	2	1	no	—
125	C₃×Dic₅	—	2	1	no	—
126	C₃×Dic₅	C₂×D₄	1	4	yes	—
127	C₃×#41	—	10	4	no	—
128	C₂×C₃×Q₈	—	5	1	no	—
129	C₂×C₃×Q₈	—	5	2	no	—
130	C₂×C₃×Q₈	—	10	4	no	—
131	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
132	C₂×C₃×D₄	—	5	1	no	—
133	C₂×C₃×D₄	—	5	2	no	—
134	C₂×C₃×D₄	—	5	2	no	—
135	C₂×C₃×D₄	—	5	1	no	—
136	C₂²×C₃×C₄	—	5	2	no	—
137	C₂²×C₃×C₄	—	5	1	no	—
138	C₂²×C₃×C₄	—	5	2	no	—
139	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
140	C₃×C₄⋊C₄	—	5	1	no	—
141	C₃×C₄⋊C₄	—	5	2	no	—
142	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
143	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
144	C₃×C₂²⋊C₄	—	5	1	no	—
145	C₃×C₂²⋊C₄	—	5	1	no	—
146	C₃×C₂²⋊C₄	—	5	2	no	—
147	C₃×C₂²⋊C₄	—	5	2	no	—
148	C₃×C₂²⋊C₄	—	5	2	no	—
149	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
150	C₃×C₄²	—	5	2	no	—
151	C₃×C₄²	—	5	1	no	—
152	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
153	C₂²×D₅	C₂³×C₃	1	2	yes	—
154	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
155	C₅×Q₈	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
158	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
160	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
161	C₂²⋊D₅	—	2	2	no	—
162	C₂²⋊D₅	—	2	4	no	—
163	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
164	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
165	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
166	C₂×Dic₅	C₂³×C₃	1	2	yes	—
167	D₂₀	—	2	4	no	—
168	D₂₀	—	2	2	no	—
169	C₄×D₅	—	2	2	no	—
170	C₄×D₅	—	2	4	no	—
171	C₄×D₅	—	2	4	no	—
172	Dic₁₀	—	2	4	no	—
173	Dic₁₀	C₃×D₄	1	4	yes	complemented
174	C₂×#41	—	5	1	no	—
175	C₂²×Q₈	—	5	1	no	—
176	#124	—	5	1	no	—
177	#124	—	5	2	no	—
178	C₂²⋊Q₈	—	5	2	no	—
179	C₂²⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
180	#121	—	5	1	no	—
181	#121	—	5	2	no	—
182	C₄×Q₈	—	5	1	no	—
183	C₄×D₄	—	5	1	no	—
184	C₄×D₄	—	5	2	no	—
185	C₂×C₃×C₅	—	4	1	no	—
186	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
188	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
189	C₃×D₅	—	2	2	no	—
190	C₃×D₅	—	4	2	no	—
191	C₂³×C₃	—	5	1	no	—
192	C₂³×C₃	—	5	2	no	—
193	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
194	C₃×Q₈	—	2	2	no	—
195	C₃×Q₈	—	5	4	no	—
196	C₃×Q₈	—	10	4	no	—
197	C₃×D₄	—	10	4	no	—
198	C₃×D₄	—	10	4	no	—
199	C₃×D₄	—	10	2	no	—
200	C₃×D₄	—	10	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	5	2	no	—
203	C₂×C₃×C₄	—	5	1	no	—
204	C₂×C₃×C₄	—	5	1	no	—
205	C₂×C₃×C₄	—	5	1	no	—
206	C₂×C₃×C₄	—	10	4	no	—
207	C₂×C₃×C₄	—	10	4	no	—
208	C₂×C₃×C₄	—	10	2	no	—
209	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
210	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
211	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	2	1	no	—
215	C₂²×C₅	—	2	1	no	—
216	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
217	C₂×D₅	—	2	1	no	—
218	C₂×D₅	—	2	1	no	—
219	C₂×D₅	—	2	2	no	—
220	C₂×D₅	—	2	2	no	—
221	C₂×D₅	—	2	2	no	—
222	C₂×D₅	C₃×#41	1	2	yes	complemented
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
227	Dic₅	—	2	1	no	—
228	Dic₅	—	2	1	no	—
229	Dic₅	—	2	1	no	—
230	Dic₅	C₂×C₃×D₄	1	4	yes	—
231	#41	—	10	4	no	—
232	C₂×Q₈	—	5	1	no	—
233	C₂×Q₈	—	5	2	no	—
234	C₂×Q₈	—	10	4	no	—
235	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
236	C₂×D₄	—	5	1	no	—
237	C₂×D₄	—	5	2	no	—
238	C₂×D₄	—	5	2	no	—
239	C₂×D₄	—	5	1	no	—
240	C₂²×C₄	—	5	2	no	—
241	C₂²×C₄	—	5	1	no	—
242	C₂²×C₄	—	5	2	no	—
243	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
244	C₄⋊C₄	—	5	1	no	—
245	C₄⋊C₄	—	5	2	no	—
246	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
247	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
248	C₂²⋊C₄	—	5	1	no	—
249	C₂²⋊C₄	—	5	1	no	—
250	C₂²⋊C₄	—	5	2	no	—
251	C₂²⋊C₄	—	5	2	no	—
252	C₂²⋊C₄	—	5	2	no	—
253	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
254	C₄²	—	5	2	no	—
255	C₄²	—	5	1	no	—
256	C₃×C₅	#548	1	1	yes	characteristic, complemented, Hall
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	5	2	no	—
261	C₂²×C₃	—	10	1	no	—
262	C₂²×C₃	—	10	1	no	—
263	C₂²×C₃	—	10	2	no	—
264	C₂²×C₃	—	10	2	no	—
265	C₂²×C₃	—	10	2	no	—
266	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	5	4	no	—
271	C₃×C₄	—	10	1	no	—
272	C₃×C₄	—	10	1	no	—
273	C₃×C₄	—	10	1	no	—
274	C₃×C₄	D₄×D₅	1	2	yes	—
275	C₂×C₅	—	4	1	no	—
276	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
277	C₂×C₅	C₂×C₃×#41	1	1	yes	characteristic
278	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
279	D₅	—	2	2	no	—
280	D₅	—	4	2	no	—
281	C₂³	—	5	1	no	—
282	C₂³	—	5	2	no	—
283	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	5	4	no	—
286	Q₈	—	10	4	no	—
287	D₄	—	10	4	no	—
288	D₄	—	10	4	no	—
289	D₄	—	10	2	no	—
290	D₄	—	10	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	5	2	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	5	1	no	—
295	C₂×C₄	—	5	1	no	—
296	C₂×C₄	—	10	4	no	—
297	C₂×C₄	—	10	4	no	—
298	C₂×C₄	—	10	2	no	—
299	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
300	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
301	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
303	C₂×C₃	—	4	1	no	—
304	C₂×C₃	—	10	2	no	—
305	C₂×C₃	—	20	2	no	—
306	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
307	C₂×C₃	#4147	1	1	yes	characteristic
308	C₂×C₃	D₅×#41	1	1	yes	characteristic
309	C₅	C₃×#548	1	1	yes	characteristic, complemented, Sylow
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	5	2	no	—
314	C₂²	—	10	1	no	—
315	C₂²	—	10	1	no	—
316	C₂²	—	10	2	no	—
317	C₂²	—	10	2	no	—
318	C₂²	—	10	2	no	—
319	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	5	4	no	—
324	C₄	—	10	1	no	—
325	C₄	—	10	1	no	—
326	C₄	—	10	1	no	—
327	C₄	C₃×D₄×D₅	1	2	yes	—
328	C₃	#65967	1	1	yes	characteristic, direct product factor, Sylow
329	C₂	—	4	1	no	—
330	C₂	—	10	2	no	—
331	C₂	—	20	2	no	—
332	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic
333	C₂	C₃×#4147	1	1	yes	characteristic
334	C₂	C₃×D₅×#41	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×C₂²⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₃×C₂²⋊Q₈	D₅
hypocenter	C₅	C₃×#548
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×Q₈×D₅	C₂	semidirect
2. C₂×C₃×#709	C₂	semidirect
3. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
4. C₃×#4093	C₂	semidirect
5. C₃×C₄×C₂²⋊D₅	C₂	semidirect
6. C₃×#4042	C₂	semidirect
7. C₃×#4041	C₂	semidirect
8. C₃×#4039	C₂	semidirect
9. C₃×#4033	C₂	semidirect
10. C₃×#4031	C₂	semidirect
11. C₃×#4030	C₂	semidirect
12. #65967	C₃	direct (N × H)
39. C₂×Q₈×D₅	C₂×C₃	semidirect
40. C₂×#709	C₂×C₃	semidirect
41. C₅×C₂²⋊Q₈	C₂×C₃	semidirect
42. #4093	C₂×C₃	semidirect
43. C₄×C₂²⋊D₅	C₂×C₃	semidirect
44. #4042	C₂×C₃	semidirect
45. #4041	C₂×C₃	semidirect
46. #4039	C₂×C₃	semidirect
47. #4033	C₂×C₃	semidirect
48. #4031	C₂×C₃	semidirect
49. #4030	C₂×C₃	semidirect
71. C₃×Dic₁₀	D₄	semidirect
77. C₃×C₂²⋊Q₈	D₅	semidirect
118. C₂×C₃×D₅	#41	semidirect
173. Dic₁₀	C₃×D₄	semidirect
179. C₂²⋊Q₈	C₃×D₅	semidirect
222. C₂×D₅	C₃×#41	semidirect
256. C₃×C₅	#548	semidirect
309. C₅	C₃×#548	semidirect
328. C₃	#65967	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×Q₈×D₅	2
C₂×C₃×#709	2
C₃×C₅×C₂²⋊Q₈	2
C₃×#4093	2
C₃×C₄×C₂²⋊D₅	2
C₃×#4042	2
C₃×#4041	2
C₃×#4039	2
C₃×#4033	2
C₃×#4031	2
C₃×#4030	2
#65967	3
C₃×#548	5

Subgroups — C3×#65967 #11755644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65967 #11755644