Subgroups — C₃×#65966 #11755643

Order 960.

560 conjugacy classes of subgroups.

The subgroup lattice has 320 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	C₂²×C₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
6	C₃×#4043	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4042	C₂	1	2	yes	maximal, complemented
8	C₃×#4034	C₂	1	2	yes	maximal, complemented
9	C₃×#4033	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#4027	C₂	1	2	yes	maximal
11	C₃×#4024	C₂	1	2	yes	maximal, complemented
12	#65966	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	1	yes	characteristic
16	C₂²×C₃×Dic₅	C₂²	1	2	yes	—
17	C₃×Q₈×D₅	—	2	4	no	—
18	C₂×C₃×C₄×D₅	C₂²	1	2	yes	—
19	C₂×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
20	C₂×C₃×Dic₁₀	—	2	4	no	—
21	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
22	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
23	C₂×C₃×Dic₁₀	C₂²	1	2	yes	—
24	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
25	C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
27	C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₃×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
29	C₃×#685	C₂²	1	2	yes	—
30	C₃×#685	C₂²	1	1	yes	characteristic
31	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
32	C₃×#683	C₂²	1	2	yes	—
33	C₃×#683	C₂²	1	2	yes	—
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×D₄×Q₈	—	5	1	no	maximal, Hall
39	C₂×Q₈×D₅	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂²×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
41	C₅×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
42	#4090	C₂×C₃	1	1	yes	characteristic, complemented
43	C₄×C₂²⋊D₅	C₂×C₃	1	1	yes	characteristic
44	#4043	C₂×C₃	1	1	yes	characteristic, complemented
45	#4042	C₂×C₃	1	2	yes	complemented
46	#4034	C₂×C₃	1	2	yes	complemented
47	#4033	C₂×C₃	1	1	yes	characteristic, complemented
48	#4027	C₂×C₃	1	2	yes	—
49	#4024	C₂×C₃	1	2	yes	complemented
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂²×C₃×D₅	C₂³	1	1	yes	characteristic
52	C₃×C₅×Q₈	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	C₂³	1	2	yes	—
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	1	yes	characteristic
58	C₃×C₂²⋊D₅	Q₈	1	4	yes	complemented
59	C₂×C₃×Dic₅	—	2	4	no	—
60	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₂³	1	2	yes	—
62	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
63	C₂×C₃×Dic₅	C₂³	1	2	yes	—
64	C₂×C₃×Dic₅	C₂³	1	1	yes	characteristic
65	C₃×C₄×D₅	—	2	2	no	—
66	C₃×C₄×D₅	—	2	4	no	—
67	C₃×Dic₁₀	—	2	2	no	—
68	C₃×Dic₁₀	—	2	4	no	—
69	C₃×Dic₁₀	—	2	4	no	—
70	C₃×Dic₁₀	D₄	1	4	yes	complemented
71	C₂²×C₃×Q₈	—	5	1	no	—
72	C₂²×C₃×Q₈	—	5	1	no	—
73	C₃×C₄⋊Q₈	—	5	1	no	—
74	C₃×C₄⋊Q₈	—	5	2	no	—
75	C₃×C₂²⋊Q₈	—	5	2	no	—
76	C₃×C₂²⋊Q₈	—	5	1	no	—
77	C₃×C₂²⋊Q₈	—	5	2	no	—
78	C₃×C₂²⋊Q₈	D₅	1	1	yes	characteristic, complemented
79	C₃×C₄×Q₈	—	5	1	no	—
80	C₃×C₄×D₄	—	5	1	no	—
81	C₃×C₄×D₄	—	5	2	no	—
82	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₂²⋊D₅	C₂²×C₃	1	1	yes	characteristic
85	C₂²×Dic₅	C₂²×C₃	1	2	yes	—
86	Q₈×D₅	—	2	4	no	—
87	C₂×C₄×D₅	C₂²×C₃	1	2	yes	—
88	C₂×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
89	C₂×Dic₁₀	—	2	4	no	—
90	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
91	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
92	C₂×Dic₁₀	C₂²×C₃	1	2	yes	—
93	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
94	C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
95	C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
96	C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
97	C₂²⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
98	#685	C₂²×C₃	1	2	yes	—
99	#685	C₂²×C₃	1	1	yes	characteristic
100	C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
101	#683	C₂²×C₃	1	2	yes	—
102	#683	C₂²×C₃	1	2	yes	—
103	#683	C₂²×C₃	1	2	yes	—
104	#683	C₂²×C₃	1	1	yes	characteristic
105	C₄×Dic₅	C₂²×C₃	1	2	yes	—
106	C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
107	D₄×Q₈	—	5	1	no	Sylow
108	C₂²×C₃×C₅	—	2	1	no	—
109	C₂²×C₃×C₅	—	2	1	no	—
110	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	complemented
111	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
112	C₂×C₃×D₅	—	2	1	no	—
113	C₂×C₃×D₅	—	2	1	no	—
114	C₂×C₃×D₅	C₂×Q₈	1	2	yes	complemented
115	C₃×C₄×C₅	—	2	1	no	—
116	C₃×C₄×C₅	—	2	2	no	—
117	C₃×C₄×C₅	—	2	2	no	—
118	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
119	C₃×Dic₅	—	2	1	no	—
120	C₃×Dic₅	—	2	2	no	—
121	C₃×Dic₅	—	2	1	no	—
122	C₃×Dic₅	C₂×D₄	1	4	yes	—
123	C₃×Dic₅	C₂×Q₈	1	2	yes	complemented
124	C₂×C₃×Q₈	—	5	1	no	—
125	C₂×C₃×Q₈	—	5	2	no	—
126	C₂×C₃×Q₈	—	5	2	no	—
127	C₂×C₃×Q₈	—	5	1	no	—
128	C₂×C₃×Q₈	—	10	4	no	—
129	C₂×C₃×Q₈	—	10	4	no	—
130	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
131	C₂×C₃×D₄	—	5	1	no	—
132	C₂²×C₃×C₄	—	5	2	no	—
133	C₂²×C₃×C₄	—	5	2	no	—
134	C₂²×C₃×C₄	—	5	1	no	—
135	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
136	C₃×C₄⋊C₄	—	5	1	no	—
137	C₃×C₄⋊C₄	—	5	2	no	—
138	C₃×C₄⋊C₄	—	5	2	no	—
139	C₃×C₄⋊C₄	—	5	2	no	—
140	C₃×C₄⋊C₄	—	5	2	no	—
141	C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
142	C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
143	C₃×C₂²⋊C₄	—	5	1	no	—
144	C₃×C₂²⋊C₄	—	5	1	no	—
145	C₃×C₂²⋊C₄	—	5	2	no	—
146	C₃×C₂²⋊C₄	C₂×D₅	1	2	yes	—
147	C₃×C₄²	—	5	2	no	—
148	C₃×C₄²	—	5	1	no	—
149	C₂³×C₅	C₂³×C₃	1	1	yes	characteristic
150	C₂²×D₅	C₂³×C₃	1	1	yes	characteristic
151	C₅×Q₈	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
154	C₂×C₄×C₅	C₂³×C₃	1	2	yes	—
155	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
157	C₂²⋊D₅	C₃×Q₈	1	4	yes	complemented
158	C₂×Dic₅	—	2	4	no	—
159	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
160	C₂×Dic₅	C₂³×C₃	1	2	yes	—
161	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
162	C₂×Dic₅	C₂³×C₃	1	2	yes	—
163	C₂×Dic₅	C₂³×C₃	1	1	yes	characteristic
164	C₄×D₅	—	2	2	no	—
165	C₄×D₅	—	2	4	no	—
166	Dic₁₀	—	2	2	no	—
167	Dic₁₀	—	2	4	no	—
168	Dic₁₀	—	2	4	no	—
169	Dic₁₀	C₃×D₄	1	4	yes	complemented
170	C₂²×Q₈	—	5	1	no	—
171	C₂²×Q₈	—	5	1	no	—
172	C₄⋊Q₈	—	5	1	no	—
173	C₄⋊Q₈	—	5	2	no	—
174	C₂²⋊Q₈	—	5	2	no	—
175	C₂²⋊Q₈	—	5	1	no	—
176	C₂²⋊Q₈	—	5	2	no	—
177	C₂²⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
178	C₄×Q₈	—	5	1	no	—
179	C₄×D₄	—	5	1	no	—
180	C₄×D₄	—	5	2	no	—
181	C₂×C₃×C₅	—	2	2	no	—
182	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
184	C₂×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
185	C₃×D₅	—	2	2	no	—
186	C₂³×C₃	—	5	1	no	—
187	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
188	C₃×Q₈	—	2	2	no	—
189	C₃×Q₈	—	5	4	no	—
190	C₃×Q₈	—	10	4	no	—
191	C₃×Q₈	—	10	4	no	—
192	C₃×Q₈	—	10	2	no	—
193	C₃×D₄	—	5	4	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	5	1	no	—
196	C₂×C₃×C₄	—	5	2	no	—
197	C₂×C₃×C₄	—	5	1	no	—
198	C₂×C₃×C₄	—	5	2	no	—
199	C₂×C₃×C₄	—	5	1	no	—
200	C₂×C₃×C₄	—	10	4	no	—
201	C₂×C₃×C₄	—	10	4	no	—
202	C₂×C₃×C₄	—	10	2	no	—
203	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
204	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×D₅	1	2	yes	—
206	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	C₂×C₃×Q₈	1	2	yes	complemented
210	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
211	C₂×D₅	—	2	1	no	—
212	C₂×D₅	—	2	1	no	—
213	C₂×D₅	C₂×C₃×Q₈	1	2	yes	complemented
214	C₄×C₅	—	2	1	no	—
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	C₂×C₃×D₄	1	2	yes	—
218	Dic₅	—	2	1	no	—
219	Dic₅	—	2	2	no	—
220	Dic₅	—	2	1	no	—
221	Dic₅	C₂×C₃×D₄	1	4	yes	—
222	Dic₅	C₂×C₃×Q₈	1	2	yes	complemented
223	C₂×Q₈	—	5	1	no	—
224	C₂×Q₈	—	5	2	no	—
225	C₂×Q₈	—	5	2	no	—
226	C₂×Q₈	—	5	1	no	—
227	C₂×Q₈	—	10	4	no	—
228	C₂×Q₈	—	10	4	no	—
229	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
230	C₂×D₄	—	5	1	no	—
231	C₂²×C₄	—	5	2	no	—
232	C₂²×C₄	—	5	2	no	—
233	C₂²×C₄	—	5	1	no	—
234	C₂²×C₄	C₂×C₃×D₅	1	1	yes	characteristic
235	C₄⋊C₄	—	5	1	no	—
236	C₄⋊C₄	—	5	2	no	—
237	C₄⋊C₄	—	5	2	no	—
238	C₄⋊C₄	—	5	2	no	—
239	C₄⋊C₄	—	5	2	no	—
240	C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
241	C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
242	C₂²⋊C₄	—	5	1	no	—
243	C₂²⋊C₄	—	5	1	no	—
244	C₂²⋊C₄	—	5	2	no	—
245	C₂²⋊C₄	C₂×C₃×D₅	1	2	yes	—
246	C₄²	—	5	2	no	—
247	C₄²	—	5	1	no	—
248	C₃×C₅	D₄×Q₈	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	5	2	no	—
252	C₂²×C₃	—	10	1	no	—
253	C₂²×C₃	—	10	1	no	—
254	C₂²×C₃	Q₈×D₅	1	2	yes	complemented
255	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	2	no	—
259	C₃×C₄	—	5	4	no	—
260	C₃×C₄	—	5	2	no	—
261	C₃×C₄	—	10	1	no	—
262	C₃×C₄	—	10	2	no	—
263	C₃×C₄	—	10	1	no	—
264	C₃×C₄	D₄×D₅	1	2	yes	—
265	C₂×C₅	—	2	2	no	—
266	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
267	C₂×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
268	C₂×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
269	D₅	—	2	2	no	—
270	C₂³	—	5	1	no	—
271	C₂³	C₂²×C₃×D₅	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	5	4	no	—
274	Q₈	—	10	4	no	—
275	Q₈	—	10	4	no	—
276	Q₈	—	10	2	no	—
277	D₄	—	5	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	5	1	no	—
280	C₂×C₄	—	5	2	no	—
281	C₂×C₄	—	5	1	no	—
282	C₂×C₄	—	5	1	no	—
283	C₂×C₄	—	5	2	no	—
284	C₂×C₄	—	10	4	no	—
285	C₂×C₄	—	10	4	no	—
286	C₂×C₄	—	10	2	no	—
287	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
288	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₃×D₅	1	2	yes	—
290	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	10	2	no	—
293	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
294	C₂×C₃	C₂×Q₈×D₅	1	1	yes	characteristic
295	C₂×C₃	#4150	1	1	yes	characteristic
296	C₅	C₃×D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	5	2	no	—
300	C₂²	—	10	1	no	—
301	C₂²	—	10	1	no	—
302	C₂²	C₃×Q₈×D₅	1	2	yes	complemented
303	C₂²	C₂³×C₃×D₅	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	5	4	no	—
308	C₄	—	5	2	no	—
309	C₄	—	10	2	no	—
310	C₄	—	10	1	no	—
311	C₄	—	10	1	no	—
312	C₄	C₃×D₄×D₅	1	2	yes	—
313	C₃	#65966	1	1	yes	characteristic, direct product factor, Sylow
314	C₂	—	2	2	no	—
315	C₂	—	10	2	no	—
316	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic
317	C₂	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
318	C₂	C₃×#4150	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
prime radical	C₂×C₃×C₂²⋊D₅	C₂²
hypercenter	C₃×C₂²⋊Q₈	D₅
hypocenter	C₅	C₃×D₄×Q₈
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. C₂²×C₃×Dic₁₀	C₂	semidirect
3. C₃×C₅×C₂²⋊Q₈	C₂	semidirect
4. C₃×#4090	C₂	semidirect
6. C₃×#4043	C₂	semidirect
7. C₃×#4042	C₂	semidirect
8. C₃×#4034	C₂	semidirect
9. C₃×#4033	C₂	semidirect
11. C₃×#4024	C₂	semidirect
12. #65966	C₃	direct (N × H)
39. C₂×Q₈×D₅	C₂×C₃	semidirect
40. C₂²×Dic₁₀	C₂×C₃	semidirect
41. C₅×C₂²⋊Q₈	C₂×C₃	semidirect
42. #4090	C₂×C₃	semidirect
44. #4043	C₂×C₃	semidirect
45. #4042	C₂×C₃	semidirect
46. #4034	C₂×C₃	semidirect
47. #4033	C₂×C₃	semidirect
49. #4024	C₂×C₃	semidirect
58. C₃×C₂²⋊D₅	Q₈	semidirect
70. C₃×Dic₁₀	D₄	semidirect
78. C₃×C₂²⋊Q₈	D₅	semidirect
110. C₂²×C₃×C₅	C₂×Q₈	semidirect
114. C₂×C₃×D₅	C₂×Q₈	semidirect
123. C₃×Dic₅	C₂×Q₈	semidirect
157. C₂²⋊D₅	C₃×Q₈	semidirect
169. Dic₁₀	C₃×D₄	semidirect
177. C₂²⋊Q₈	C₃×D₅	semidirect
209. C₂²×C₅	C₂×C₃×Q₈	semidirect
213. C₂×D₅	C₂×C₃×Q₈	semidirect
222. Dic₅	C₂×C₃×Q₈	semidirect
248. C₃×C₅	D₄×Q₈	semidirect
254. C₂²×C₃	Q₈×D₅	semidirect
296. C₅	C₃×D₄×Q₈	semidirect
302. C₂²	C₃×Q₈×D₅	semidirect
313. C₃	#65966	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
C₂²×C₃×Dic₁₀	2
C₃×C₅×C₂²⋊Q₈	2
C₃×#4090	2
C₃×C₄×C₂²⋊D₅	2
C₃×#4043	2
C₃×#4042	2
C₃×#4034	2
C₃×#4033	2
C₃×#4027	2
C₃×#4024	2
#65966	3
C₃×D₄×Q₈	5

Subgroups — C3×#65966 #11755643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#65966 #11755643