Subgroups — C₃×#66113 #11755790

Order 960.

496 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4150	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#4147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4096	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#689	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4064	C₂	1	2	yes	maximal, complemented
7	C₃×D₅×Q₁₆	C₂	1	2	yes	maximal, complemented
8	C₃×#4062	C₂	1	2	yes	maximal, complemented
9	C₃×#4061	C₂	1	2	yes	maximal, complemented
10	C₃×#4055	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#4053	C₂	1	1	yes	maximal, characteristic, complemented
12	#66113	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₃×C₅×#41	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₃×#713	—	2	1	no	—
16	C₃×#713	C₂²	1	2	yes	—
17	C₃×Q₈×D₅	—	2	1	no	—
18	C₃×Q₈×D₅	C₂²	1	2	yes	—
19	C₃×Q₈×D₅	C₂²	1	2	yes	—
20	C₃×#711	—	2	2	no	—
21	C₃×#711	C₂²	1	2	yes	—
22	C₃×#709	—	2	2	no	—
23	C₃×#709	—	2	1	no	—
24	C₃×#709	C₂²	1	1	yes	characteristic
25	C₃×#709	C₂²	1	1	yes	characteristic
26	C₂×C₃×Dic₁₀	—	2	1	no	—
27	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
28	C₃×C₅×Q₁₆	C₂²	1	2	yes	—
29	C₃×C₅×SD₁₆	C₂²	1	2	yes	—
30	C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
31	C₃×#689	C₂²	1	2	yes	—
32	C₃×#689	C₂²	1	2	yes	—
33	C₃×#689	C₂²	1	1	yes	characteristic
34	C₃×#688	C₂²	1	1	yes	characteristic
35	C₃×#687	C₂²	1	1	yes	characteristic
36	C₃×#686	C₂²	1	1	yes	characteristic
37	C₂×C₃×#168	C₂²	1	1	yes	characteristic
38	C₃×Dic₂₀	C₂²	1	2	yes	—
39	C₃×#677	C₂²	1	2	yes	—
40	C₃×#676	C₂²	1	2	yes	—
41	C₃×C₈×D₅	C₂²	1	2	yes	—
42	C₃×#578	—	5	1	no	maximal, Hall
43	#4150	C₂×C₃	1	1	yes	characteristic, complemented
44	#4147	C₂×C₃	1	1	yes	characteristic, complemented
45	C₅×#137	C₂×C₃	1	1	yes	characteristic, complemented
46	#4096	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×#689	C₂×C₃	1	1	yes	characteristic, complemented
48	#4064	C₂×C₃	1	2	yes	complemented
49	D₅×Q₁₆	C₂×C₃	1	2	yes	complemented
50	#4062	C₂×C₃	1	2	yes	complemented
51	#4061	C₂×C₃	1	2	yes	complemented
52	#4055	C₂×C₃	1	1	yes	characteristic, complemented
53	#4053	C₂×C₃	1	1	yes	characteristic, complemented
54	C₃×C₅×Q₈	—	2	1	no	—
55	C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
56	C₃×C₅×Q₈	C₂³	1	2	yes	—
57	C₃×C₅×D₄	—	2	1	no	—
58	C₃×C₅×D₄	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₃×C₂²⋊D₅	—	2	1	no	—
63	C₃×C₂²⋊D₅	—	2	2	no	—
64	C₃×C₂²⋊D₅	D₄	1	2	yes	—
65	C₂×C₃×Dic₅	—	2	1	no	—
66	C₂×C₃×Dic₅	—	2	2	no	—
67	C₃×D₂₀	—	2	2	no	—
68	C₃×D₂₀	D₄	1	1	yes	characteristic
69	C₃×D₂₀	C₂³	1	1	yes	characteristic
70	C₃×C₄×D₅	—	2	2	no	—
71	C₃×C₄×D₅	—	2	2	no	—
72	C₃×C₄×D₅	—	2	1	no	—
73	C₃×C₄×D₅	—	2	2	no	—
74	C₃×C₄×D₅	C₂³	1	2	yes	—
75	C₃×Dic₁₀	—	2	1	no	—
76	C₃×Dic₁₀	—	2	2	no	—
77	C₃×Dic₁₀	—	2	2	no	—
78	C₃×Dic₁₀	D₄	1	1	yes	characteristic
79	C₃×Dic₁₀	C₂³	1	2	yes	—
80	C₃×Dic₁₀	C₂³	1	1	yes	characteristic
81	C₃×C₅×C₈	C₂³	1	2	yes	—
82	C₃×#168	C₂³	1	1	yes	characteristic
83	C₃×#168	C₂³	1	1	yes	characteristic
84	C₃×2₋¹⁺⁴	—	5	1	no	—
85	C₃×2₋¹⁺⁴	—	5	1	no	—
86	C₃×#137	—	5	2	no	—
87	C₃×#137	—	5	2	no	—
88	C₃×#137	—	5	1	no	—
89	C₃×#137	D₅	1	1	yes	characteristic, complemented
90	C₃×#135	—	5	1	no	—
91	C₃×#135	—	5	2	no	—
92	C₂×C₃×Q₁₆	—	5	1	no	—
93	C₂×C₃×Q₁₆	—	5	2	no	—
94	C₃×#131	—	5	1	no	—
95	C₅×#41	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
97	#713	—	2	1	no	—
98	#713	C₂²×C₃	1	2	yes	—
99	Q₈×D₅	—	2	1	no	—
100	Q₈×D₅	C₂²×C₃	1	2	yes	—
101	Q₈×D₅	C₂²×C₃	1	2	yes	—
102	#711	—	2	2	no	—
103	#711	C₂²×C₃	1	2	yes	—
104	#709	—	2	2	no	—
105	#709	—	2	1	no	—
106	#709	C₂²×C₃	1	1	yes	characteristic
107	#709	C₂²×C₃	1	1	yes	characteristic
108	C₂×Dic₁₀	—	2	1	no	—
109	C₂×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
110	C₅×Q₁₆	C₂²×C₃	1	2	yes	—
111	C₅×SD₁₆	C₂²×C₃	1	2	yes	—
112	C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
113	#689	C₂²×C₃	1	1	yes	characteristic
114	#689	C₂²×C₃	1	2	yes	—
115	#689	C₂²×C₃	1	2	yes	—
116	#688	C₂²×C₃	1	1	yes	characteristic
117	#687	C₂²×C₃	1	1	yes	characteristic
118	#686	C₂²×C₃	1	1	yes	characteristic
119	C₂×#168	C₂²×C₃	1	1	yes	characteristic
120	Dic₂₀	C₂²×C₃	1	2	yes	—
121	#677	C₂²×C₃	1	2	yes	—
122	#676	C₂²×C₃	1	2	yes	—
123	C₈×D₅	C₂²×C₃	1	2	yes	—
124	#578	—	5	1	no	Sylow
125	C₂²×C₃×C₅	—	2	1	no	—
126	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
127	C₂×C₃×D₅	—	2	1	no	—
128	C₂×C₃×D₅	C₂×D₄	1	2	yes	—
129	C₃×C₄×C₅	—	2	1	no	—
130	C₃×C₄×C₅	—	2	2	no	—
131	C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
132	C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
133	C₃×Dic₅	—	2	1	no	—
134	C₃×Dic₅	—	2	2	no	—
135	C₃×Dic₅	C₂×D₄	1	2	yes	—
136	C₃×#41	—	5	1	no	—
137	C₃×#41	—	5	2	no	—
138	C₃×#41	—	5	2	no	—
139	C₃×#41	—	5	1	no	—
140	C₃×#41	—	10	2	no	—
141	C₃×#41	—	10	1	no	—
142	C₃×#41	—	10	2	no	—
143	C₃×#41	—	10	1	no	—
144	C₃×#41	C₂×D₅	1	1	yes	characteristic
145	C₂×C₃×Q₈	—	5	2	no	—
146	C₂×C₃×Q₈	—	5	2	no	—
147	C₂×C₃×Q₈	—	5	1	no	—
148	C₂×C₃×Q₈	—	10	1	no	—
149	C₂×C₃×Q₈	—	10	1	no	—
150	C₂×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
151	C₃×Q₁₆	—	5	1	no	—
152	C₃×Q₁₆	—	5	2	no	—
153	C₃×Q₁₆	—	5	2	no	—
154	C₃×Q₁₆	—	5	2	no	—
155	C₃×Q₁₆	C₂×D₅	1	2	yes	—
156	C₃×SD₁₆	—	5	1	no	—
157	C₃×SD₁₆	—	5	1	no	—
158	C₃×SD₁₆	—	5	2	no	—
159	C₃×SD₁₆	C₂×D₅	1	2	yes	—
160	C₃×D₈	—	5	1	no	—
161	C₃×M₁₆	—	5	2	no	—
162	C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
163	C₂×C₃×C₈	—	5	1	no	—
164	C₂×C₃×C₈	—	5	2	no	—
165	C₅×Q₈	—	2	1	no	—
166	C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
167	C₅×Q₈	C₂³×C₃	1	2	yes	—
168	C₅×D₄	—	2	1	no	—
169	C₅×D₄	C₂³×C₃	1	1	yes	characteristic
170	C₂×C₄×C₅	—	2	1	no	—
171	C₂×C₄×C₅	—	2	1	no	—
172	C₂×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
173	C₂²⋊D₅	—	2	1	no	—
174	C₂²⋊D₅	—	2	2	no	—
175	C₂²⋊D₅	C₃×D₄	1	2	yes	—
176	C₂×Dic₅	—	2	1	no	—
177	C₂×Dic₅	—	2	2	no	—
178	D₂₀	—	2	2	no	—
179	D₂₀	C₃×D₄	1	1	yes	characteristic
180	D₂₀	C₂³×C₃	1	1	yes	characteristic
181	C₄×D₅	—	2	2	no	—
182	C₄×D₅	—	2	2	no	—
183	C₄×D₅	—	2	1	no	—
184	C₄×D₅	—	2	2	no	—
185	C₄×D₅	C₂³×C₃	1	2	yes	—
186	Dic₁₀	—	2	1	no	—
187	Dic₁₀	—	2	2	no	—
188	Dic₁₀	—	2	2	no	—
189	Dic₁₀	C₃×D₄	1	1	yes	characteristic
190	Dic₁₀	C₂³×C₃	1	2	yes	—
191	Dic₁₀	C₂³×C₃	1	1	yes	characteristic
192	C₅×C₈	C₂³×C₃	1	2	yes	—
193	#168	C₂³×C₃	1	1	yes	characteristic
194	#168	C₂³×C₃	1	1	yes	characteristic
195	2₋¹⁺⁴	—	5	1	no	—
196	2₋¹⁺⁴	—	5	1	no	—
197	#137	—	5	2	no	—
198	#137	—	5	2	no	—
199	#137	—	5	1	no	—
200	#137	C₃×D₅	1	1	yes	characteristic, complemented
201	#135	—	5	1	no	—
202	#135	—	5	2	no	—
203	C₂×Q₁₆	—	5	1	no	—
204	C₂×Q₁₆	—	5	2	no	—
205	#131	—	5	1	no	—
206	C₂×C₃×C₅	—	2	1	no	—
207	C₂×C₃×C₅	—	4	1	no	—
208	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
209	C₃×D₅	—	2	2	no	—
210	C₃×D₅	—	4	1	no	—
211	C₃×Q₈	—	2	1	no	—
212	C₃×Q₈	—	5	1	no	—
213	C₃×Q₈	—	5	2	no	—
214	C₃×Q₈	—	5	1	no	—
215	C₃×Q₈	—	10	2	no	—
216	C₃×Q₈	—	10	2	no	—
217	C₃×Q₈	—	10	1	no	—
218	C₃×Q₈	C₂²×D₅	1	1	yes	characteristic
219	C₃×Q₈	C₂²×D₅	1	2	yes	—
220	C₃×D₄	—	2	1	no	—
221	C₃×D₄	—	5	2	no	—
222	C₃×D₄	—	5	1	no	—
223	C₃×D₄	—	5	1	no	—
224	C₃×D₄	—	10	2	no	—
225	C₃×D₄	—	10	2	no	—
226	C₃×D₄	—	10	1	no	—
227	C₃×D₄	C₂²×D₅	1	1	yes	characteristic
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	5	2	no	—
231	C₂×C₃×C₄	—	10	2	no	—
232	C₂×C₃×C₄	—	10	2	no	—
233	C₂×C₃×C₄	—	10	2	no	—
234	C₂×C₃×C₄	—	10	2	no	—
235	C₂×C₃×C₄	—	10	1	no	—
236	C₂×C₃×C₄	—	10	1	no	—
237	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
238	C₃×C₈	—	5	1	no	—
239	C₃×C₈	—	5	1	no	—
240	C₃×C₈	C₂²×D₅	1	2	yes	—
241	C₂²×C₅	—	2	1	no	—
242	C₂²×C₅	C₂×C₃×D₄	1	1	yes	characteristic
243	C₂×D₅	—	2	1	no	—
244	C₂×D₅	C₂×C₃×D₄	1	2	yes	—
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	2	2	no	—
247	C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
248	C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
249	Dic₅	—	2	1	no	—
250	Dic₅	—	2	2	no	—
251	Dic₅	C₂×C₃×D₄	1	2	yes	—
252	#41	—	5	1	no	—
253	#41	—	5	2	no	—
254	#41	—	5	2	no	—
255	#41	—	5	1	no	—
256	#41	—	10	2	no	—
257	#41	—	10	1	no	—
258	#41	—	10	2	no	—
259	#41	—	10	1	no	—
260	#41	C₂×C₃×D₅	1	1	yes	characteristic
261	C₂×Q₈	—	5	2	no	—
262	C₂×Q₈	—	5	2	no	—
263	C₂×Q₈	—	5	1	no	—
264	C₂×Q₈	—	10	1	no	—
265	C₂×Q₈	—	10	1	no	—
266	C₂×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
267	Q₁₆	—	5	1	no	—
268	Q₁₆	—	5	2	no	—
269	Q₁₆	—	5	2	no	—
270	Q₁₆	—	5	2	no	—
271	Q₁₆	C₂×C₃×D₅	1	2	yes	—
272	SD₁₆	—	5	1	no	—
273	SD₁₆	—	5	1	no	—
274	SD₁₆	—	5	2	no	—
275	SD₁₆	C₂×C₃×D₅	1	2	yes	—
276	D₈	—	5	1	no	—
277	M₁₆	—	5	2	no	—
278	M₁₆	C₂×C₃×D₅	1	1	yes	characteristic
279	C₂×C₈	—	5	1	no	—
280	C₂×C₈	—	5	2	no	—
281	C₃×C₅	#578	1	1	yes	characteristic, complemented, Hall
282	C₂²×C₃	—	2	1	no	—
283	C₂²×C₃	—	5	2	no	—
284	C₂²×C₃	—	10	1	no	—
285	C₂²×C₃	D₄×D₅	1	1	yes	characteristic
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	2	no	—
288	C₃×C₄	—	5	2	no	—
289	C₃×C₄	—	10	1	no	—
290	C₃×C₄	—	10	2	no	—
291	C₃×C₄	D₄×D₅	1	1	yes	characteristic
292	C₃×C₄	C₂³×D₅	1	1	yes	characteristic
293	C₂×C₅	—	2	1	no	—
294	C₂×C₅	—	4	1	no	—
295	C₂×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
296	D₅	—	2	2	no	—
297	D₅	—	4	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	5	1	no	—
300	Q₈	—	5	1	no	—
301	Q₈	—	5	2	no	—
302	Q₈	—	10	2	no	—
303	Q₈	—	10	2	no	—
304	Q₈	—	10	1	no	—
305	Q₈	C₂²×C₃×D₅	1	1	yes	characteristic
306	Q₈	C₂²×C₃×D₅	1	2	yes	—
307	D₄	—	2	1	no	—
308	D₄	—	5	2	no	—
309	D₄	—	5	1	no	—
310	D₄	—	5	1	no	—
311	D₄	—	10	2	no	—
312	D₄	—	10	2	no	—
313	D₄	—	10	1	no	—
314	D₄	C₂²×C₃×D₅	1	1	yes	characteristic
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	5	2	no	—
318	C₂×C₄	—	10	2	no	—
319	C₂×C₄	—	10	2	no	—
320	C₂×C₄	—	10	2	no	—
321	C₂×C₄	—	10	2	no	—
322	C₂×C₄	—	10	1	no	—
323	C₂×C₄	—	10	1	no	—
324	C₂×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
325	C₈	—	5	1	no	—
326	C₈	—	5	1	no	—
327	C₈	C₂²×C₃×D₅	1	2	yes	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	4	1	no	—
330	C₂×C₃	—	10	2	no	—
331	C₂×C₃	—	20	1	no	—
332	C₂×C₃	C₂×D₄×D₅	1	1	yes	characteristic
333	C₅	C₃×#578	1	1	yes	characteristic, complemented, Sylow
334	C₂²	—	2	1	no	—
335	C₂²	—	5	2	no	—
336	C₂²	—	10	1	no	—
337	C₂²	C₃×D₄×D₅	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	5	2	no	—
341	C₄	—	10	1	no	—
342	C₄	—	10	2	no	—
343	C₄	C₃×D₄×D₅	1	1	yes	characteristic
344	C₄	C₂³×C₃×D₅	1	1	yes	characteristic
345	C₃	#66113	1	1	yes	characteristic, direct product factor, Sylow
346	C₂	—	2	1	no	—
347	C₂	—	4	1	no	—
348	C₂	—	10	2	no	—
349	C₂	—	20	1	no	—
350	C₂	C₂×C₃×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#137	C₂
socle	C₂×C₃×C₅	C₂²×D₄
hypercenter	C₃×#137	D₅
hypocenter	C₅	C₃×#578
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#137

Complements

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

N	H ≅ G/N	type
1. C₃×#4150	C₂	semidirect
2. C₃×#4147	C₂	semidirect
3. C₃×C₅×#137	C₂	semidirect
4. C₃×#4096	C₂	semidirect
5. C₂×C₃×#689	C₂	semidirect
6. C₃×#4064	C₂	semidirect
7. C₃×D₅×Q₁₆	C₂	semidirect
8. C₃×#4062	C₂	semidirect
9. C₃×#4061	C₂	semidirect
10. C₃×#4055	C₂	semidirect
11. C₃×#4053	C₂	semidirect
12. #66113	C₃	direct (N × H)
43. #4150	C₂×C₃	semidirect
44. #4147	C₂×C₃	semidirect
45. C₅×#137	C₂×C₃	semidirect
46. #4096	C₂×C₃	semidirect
47. C₂×#689	C₂×C₃	semidirect
48. #4064	C₂×C₃	semidirect
49. D₅×Q₁₆	C₂×C₃	semidirect
50. #4062	C₂×C₃	semidirect
51. #4061	C₂×C₃	semidirect
52. #4055	C₂×C₃	semidirect
53. #4053	C₂×C₃	semidirect
89. C₃×#137	D₅	semidirect
200. #137	C₃×D₅	semidirect
281. C₃×C₅	#578	semidirect
333. C₅	C₃×#578	semidirect
345. C₃	#66113	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×#4150	2
C₃×#4147	2
C₃×C₅×#137	2
C₃×#4096	2
C₂×C₃×#689	2
C₃×#4064	2
C₃×D₅×Q₁₆	2
C₃×#4062	2
C₃×#4061	2
C₃×#4055	2
C₃×#4053	2
#66113	3
C₃×#578	5

Subgroups — C3×#66113 #11755790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66113 #11755790