Subgroups — C₃×#70710 #11791359

Order 1152.

605 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#824	C₂	1	2	yes	maximal
4	C₃×#5120	C₂	1	2	yes	maximal
5	C₃×#5101	C₂	1	1	yes	maximal, characteristic
6	C₃×#1979	—	3	1	no	maximal
7	#70710	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
9	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₃×#926	C₂²	1	1	yes	characteristic
12	C₂×C₃×#208	C₂²	1	1	yes	characteristic
13	C₂×C₃×Dic₁₂	—	2	1	no	—
14	C₂×C₃×Dic₁₂	—	2	1	no	—
15	C₂×C₃×Dic₁₂	—	2	2	no	—
16	C₂×C₃×Dic₁₂	C₄	1	4	yes	complemented
17	C₃²×#108	—	2	2	no	—
18	C₃×#824	—	2	4	no	—
19	C₃×#816	—	2	4	no	—
20	C₃×#815	—	2	1	no	—
21	C₃×C₈×Dic₃	—	2	1	no	—
22	C₂²×C₃×Q₁₆	—	3	1	no	—
23	C₂²×Dic₁₂	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#108	—	2	1	no	—
25	C₂×C₃×#108	D₃	1	1	yes	characteristic
26	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic
27	C₃×#419	—	3	2	no	—
28	C₂×C₃×#101	—	3	2	no	—
29	C₃×#405	—	3	1	no	—
30	C₂×#824	C₂×C₃	1	2	yes	—
31	#5120	C₂×C₃	1	2	yes	—
32	#5101	C₂×C₃	1	1	yes	characteristic
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×Dic₃	—	2	2	no	—
35	C₂×C₃×Dic₆	—	2	2	no	—
36	C₂×C₃×Dic₆	—	2	2	no	—
37	C₂×C₃×Dic₆	—	2	2	no	—
38	C₂×C₃×Dic₆	—	2	2	no	—
39	C₂×C₃×Dic₆	—	2	2	no	—
40	C₂×C₃×Dic₆	C₂×C₄	1	4	yes	—
41	C₃²×M₁₆	—	2	2	no	—
42	C₃²×M₁₆	—	2	2	no	—
43	C₂×C₃²×C₈	—	2	2	no	—
44	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
46	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
47	C₂×C₃²×C₈	D₄	1	2	yes	—
48	C₃×#217	—	2	1	no	—
49	C₃×C₄⋊Dic₃	D₄	1	2	yes	—
50	C₃×C₄×Dic₃	—	2	1	no	—
51	C₃×#208	—	2	1	no	—
52	C₃×#208	D₄	1	2	yes	—
53	C₂×C₃×C₃⋊C₈	—	2	1	no	—
54	C₃×Dic₁₂	—	2	4	no	—
55	C₃×Dic₁₂	—	2	4	no	—
56	#1979	—	3	1	no	Sylow
57	C₂²×C₃×Q₈	—	3	2	no	—
58	C₂²×Dic₆	C₂²×C₃	1	2	yes	—
59	C₂×C₃×Q₁₆	—	3	4	no	—
60	C₂×C₃×Q₁₆	—	6	1	no	—
61	C₂×C₃×Q₁₆	—	6	1	no	—
62	C₂×C₃×Q₁₆	—	6	2	no	—
63	C₂×C₃×M₁₆	—	2	2	no	—
64	C₂×C₃×M₁₆	—	3	1	no	—
65	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
66	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
67	C₂²×C₃×C₈	—	2	1	no	—
68	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
70	C₃×#117	—	3	1	no	—
71	#926	C₂²×C₃	1	1	yes	characteristic
72	C₂×#208	C₂²×C₃	1	1	yes	characteristic
73	C₂×Dic₁₂	—	2	1	no	—
74	C₂×Dic₁₂	—	2	1	no	—
75	C₂×Dic₁₂	—	2	2	no	—
76	C₂×Dic₁₂	C₃×C₄	1	4	yes	complemented
77	C₃×#108	—	2	4	no	—
78	C₃×#108	—	2	2	no	—
79	C₃×#108	—	2	2	no	—
80	C₃×#103	—	6	4	no	—
81	C₃×#101	—	6	4	no	—
82	C₃×#97	—	6	1	no	—
83	C₃×C₄×C₈	—	6	1	no	—
84	#824	—	2	4	no	—
85	#816	—	2	4	no	—
86	#815	—	2	1	no	—
87	C₈×Dic₃	—	2	1	no	—
88	C₂³×C₃²	#41	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	#41	1	1	yes	characteristic
95	C₂×C₃×Dic₃	—	2	4	no	—
96	C₂×C₃×Dic₃	—	2	2	no	—
97	C₂×C₃×Dic₃	—	4	2	no	—
98	C₂×C₃×Dic₃	—	4	2	no	—
99	C₃×Dic₆	—	2	4	no	—
100	C₃×Dic₆	—	2	4	no	—
101	C₃×Dic₆	—	4	2	no	—
102	C₃×Dic₆	—	4	2	no	—
103	C₃²×C₈	—	4	2	no	—
104	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
105	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
106	C₃×C₃⋊C₈	—	2	2	no	—
107	C₂²×Q₁₆	—	3	1	no	—
108	C₂×#108	C₃×D₃	1	1	yes	characteristic
109	#419	—	3	2	no	—
110	C₂×#101	—	3	2	no	—
111	#405	—	3	1	no	—
112	C₂×C₃×Q₈	—	3	4	no	—
113	C₂×C₃×Q₈	—	6	2	no	—
114	C₂×C₃×Q₈	—	6	2	no	—
115	C₂×C₃×Q₈	—	6	2	no	—
116	C₂×C₃×Q₈	—	6	2	no	—
117	C₂×C₃×Q₈	—	6	2	no	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	—	6	2	no	—
120	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₂²×Dic₃	—	2	2	no	—
123	C₂×Dic₆	—	2	2	no	—
124	C₂×Dic₆	—	2	2	no	—
125	C₂×Dic₆	—	2	2	no	—
126	C₂×Dic₆	—	2	2	no	—
127	C₂×Dic₆	—	2	2	no	—
128	C₂×Dic₆	C₂×C₃×C₄	1	4	yes	—
129	C₃×Q₁₆	—	6	4	no	—
130	C₃×Q₁₆	—	6	4	no	—
131	C₃×M₁₆	—	2	4	no	—
132	C₃×M₁₆	—	2	2	no	—
133	C₃×M₁₆	—	2	2	no	—
134	C₃×M₁₆	—	2	2	no	—
135	C₃×M₁₆	—	2	2	no	—
136	C₃×M₁₆	—	3	2	no	—
137	C₃×M₁₆	—	4	2	no	—
138	C₃×M₁₆	—	6	1	no	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	2	2	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	—	2	2	no	—
145	C₂×C₃×C₈	—	4	2	no	—
146	C₂×C₃×C₈	—	6	1	no	—
147	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
148	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
149	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
150	C₂×C₃×C₈	#67	1	2	yes	—
151	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
152	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
153	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
154	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
155	C₃×C₄⋊C₄	—	3	2	no	—
156	C₃×C₂²⋊C₄	—	6	1	no	—
157	C₃×C₄²	—	6	1	no	—
158	#217	—	2	1	no	—
159	C₄⋊Dic₃	C₃×D₄	1	2	yes	—
160	C₄×Dic₃	—	2	1	no	—
161	#208	—	2	1	no	—
162	#208	C₃×D₄	1	2	yes	—
163	C₂×C₃⋊C₈	—	2	1	no	—
164	Dic₁₂	—	2	4	no	—
165	Dic₁₂	—	2	4	no	—
166	C₂²×C₃²	—	2	2	no	—
167	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
168	C₂²×C₃²	#121	1	1	yes	characteristic
169	C₂²×C₃²	#124	1	1	yes	characteristic
170	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
172	C₃²×C₄	#121	1	1	yes	characteristic
173	C₃²×C₄	#127	1	1	yes	characteristic
174	C₃×Dic₃	—	4	4	no	—
175	C₃×Dic₃	—	4	2	no	—
176	C₂²×Q₈	—	3	2	no	—
177	C₂×Q₁₆	—	3	4	no	—
178	C₂×Q₁₆	—	6	1	no	—
179	C₂×Q₁₆	—	6	1	no	—
180	C₂×Q₁₆	—	6	2	no	—
181	C₂×M₁₆	—	3	1	no	—
182	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
183	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
184	#117	—	3	1	no	—
185	#108	—	2	2	no	—
186	#103	—	6	4	no	—
187	#101	—	6	4	no	—
188	#97	—	6	1	no	—
189	C₄×C₈	—	6	1	no	—
190	C₂³×C₃	—	2	1	no	—
191	C₂³×C₃	#239	1	1	yes	characteristic
192	C₂³×C₃	C₃×#41	1	1	yes	characteristic
193	C₃×Q₈	—	6	4	no	—
194	C₃×Q₈	—	6	4	no	—
195	C₃×Q₈	—	12	2	no	—
196	C₃×Q₈	—	12	2	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	6	4	no	—
204	C₂×C₃×C₄	—	6	2	no	—
205	C₂×C₃×C₄	—	12	2	no	—
206	C₂×C₃×C₄	—	12	2	no	—
207	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
209	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
211	C₂×C₃×C₄	#239	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
218	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
219	C₂×Dic₃	—	2	4	no	—
220	C₂×Dic₃	—	2	2	no	—
221	C₂×Dic₃	—	4	2	no	—
222	C₂×Dic₃	—	4	2	no	—
223	Dic₆	—	2	4	no	—
224	Dic₆	—	2	4	no	—
225	Dic₆	—	4	2	no	—
226	Dic₆	—	4	2	no	—
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	2	2	no	—
229	C₃×C₈	—	4	4	no	—
230	C₃×C₈	—	4	2	no	—
231	C₃×C₈	—	4	2	no	—
232	C₃×C₈	—	6	2	no	—
233	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
234	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
235	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
236	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
237	C₃⋊C₈	—	2	2	no	—
238	C₂×C₃²	—	2	1	no	—
239	C₂×C₃²	—	2	1	no	—
240	C₂×C₃²	#390	1	1	yes	characteristic
241	C₂×C₃²	#473	1	2	yes	—
242	C₂×Q₈	—	3	4	no	—
243	C₂×Q₈	—	6	2	no	—
244	C₂×Q₈	—	6	2	no	—
245	C₂×Q₈	—	6	2	no	—
246	C₂×Q₈	—	6	2	no	—
247	C₂×Q₈	—	6	2	no	—
248	C₂²×C₄	—	6	2	no	—
249	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
250	Q₁₆	—	6	4	no	—
251	Q₁₆	—	6	4	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	3	2	no	—
255	M₁₆	—	6	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	6	1	no	—
258	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
259	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
260	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
261	C₂×C₈	C₃×#67	1	2	yes	—
262	C₄⋊C₄	—	3	2	no	—
263	C₂²⋊C₄	—	6	1	no	—
264	C₄²	—	6	1	no	—
265	C₂²×C₃	—	2	2	no	—
266	C₂²×C₃	—	2	2	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	4	2	no	—
271	C₂²×C₃	#893	1	1	yes	characteristic
272	C₂²×C₃	#895	1	1	yes	characteristic
273	C₂²×C₃	#947	1	1	yes	characteristic
274	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
275	C₂²×C₃	C₃×#121	1	1	yes	characteristic
276	C₂²×C₃	C₃×#124	1	1	yes	characteristic
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	12	2	no	—
282	C₃×C₄	—	12	4	no	—
283	C₃×C₄	#893	1	1	yes	characteristic
284	C₃×C₄	#895	1	1	yes	characteristic
285	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
286	C₃×C₄	#940	1	1	yes	characteristic
287	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
288	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
289	C₃×C₄	C₃×#121	1	1	yes	characteristic
290	C₃×C₄	C₃×#127	1	1	yes	characteristic
291	Dic₃	—	4	4	no	—
292	Dic₃	—	4	2	no	—
293	C₃²	#1979	1	1	yes	characteristic, complemented, Sylow
294	C₂³	C₃×#239	1	1	yes	characteristic
295	Q₈	—	6	4	no	—
296	Q₈	—	6	4	no	—
297	Q₈	—	12	2	no	—
298	Q₈	—	12	2	no	—
299	C₂×C₄	—	6	4	no	—
300	C₂×C₄	—	6	2	no	—
301	C₂×C₄	—	12	2	no	—
302	C₂×C₄	—	12	2	no	—
303	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
304	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
305	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
306	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
307	C₂×C₄	C₃×#239	1	1	yes	characteristic
308	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	6	2	no	—
311	C₈	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
312	C₈	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	1	no	—
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	2	no	—
319	C₂×C₃	—	4	1	no	—
320	C₂×C₃	—	4	1	no	—
321	C₂×C₃	#4896	1	2	yes	—
322	C₂×C₃	#4988	1	1	yes	characteristic
323	C₂×C₃	C₃×#390	1	1	yes	characteristic
324	C₂×C₃	C₃×#473	1	2	yes	—
325	C₂²	—	2	2	no	—
326	C₂²	C₃×#893	1	1	yes	characteristic
327	C₂²	C₃×#895	1	1	yes	characteristic
328	C₂²	C₃×#947	1	1	yes	characteristic
329	C₄	—	12	4	no	—
330	C₄	—	12	2	no	—
331	C₄	C₃×#893	1	1	yes	characteristic
332	C₄	C₃×#895	1	1	yes	characteristic
333	C₄	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
334	C₄	C₃×#940	1	1	yes	characteristic
335	C₃	—	2	1	no	—
336	C₃	#70710	1	1	yes	characteristic, direct product factor
337	C₃	C₃×#1979	1	1	yes	characteristic, complemented
338	C₂	—	2	1	no	—
339	C₂	—	2	1	no	—
340	C₂	C₃×#4896	1	2	yes	—
341	C₂	C₃×#4988	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#895
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂²×C₃²	#121
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#108	D₃
hypocenter	C₃	C₃×#1979
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#108

Complements

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

N	H ≅ G/N	type
7. #70710	C₃	direct (N × H)
16. C₂×C₃×Dic₁₂	C₄	semidirect
76. C₂×Dic₁₂	C₃×C₄	semidirect
293. C₃²	#1979	semidirect
336. C₃	#70710	direct (N × H)
337. C₃	C₃×#1979	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Dic₁₂	2
C₂×C₃²×#108	2
C₂×C₃×#824	2
C₃×#5120	2
C₃×#5101	2
C₃×#1979	3
#70710	3

Subgroups — C3×#70710 #11791359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70710 #11791359