Subgroups — C₆₁×#11239 #29514672

Order 15616.

682 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#424	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#343	C₂	1	2	yes	maximal
4	C₂×C₆₁×#341	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1736	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
8	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
11	C₂×#107×C₆₁	—	2	2	no	—
12	C₂×#107×C₆₁	—	2	4	no	—
13	C₆₁×#424	—	2	2	no	—
14	C₆₁×#424	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
16	C₆₁×#343	—	2	4	no	—
17	C₆₁×#341	—	2	2	no	—
18	C₆₁×#340	—	2	4	no	—
19	C₂³×C₄×C₆₁	—	2	2	no	—
20	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₆₁	—	2	1	no	—
22	C₂×M₁₆×C₆₁	—	2	2	no	—
23	C₂×M₁₆×C₆₁	—	2	2	no	—
24	C₂×M₁₆×C₆₁	—	2	2	no	—
25	C₂×M₁₆×C₆₁	—	2	2	no	—
26	C₂×M₁₆×C₆₁	—	2	2	no	—
27	C₂×M₁₆×C₆₁	—	2	2	no	—
28	C₂×M₁₆×C₆₁	—	2	2	no	—
29	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₆₁	Q₈	1	4	yes	—
31	C₂²×C₈×C₆₁	—	2	2	no	—
32	C₂²×C₈×C₆₁	—	2	2	no	—
33	C₂²×C₈×C₆₁	—	2	2	no	—
34	C₂²×C₈×C₆₁	—	2	1	no	—
35	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₆₁	D₄	1	2	yes	—
37	C₂²×C₈×C₆₁	D₄	1	2	yes	—
38	#117×C₆₁	—	2	1	no	—
39	#117×C₆₁	—	2	1	no	—
40	#117×C₆₁	C₂×C₄	1	2	yes	—
41	#117×C₆₁	Q₈	1	2	yes	—
42	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₁	D₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
50	C₂×C₄²×C₆₁	—	2	1	no	—
51	#107×C₆₁	—	2	8	no	—
52	C₄⋊C₈×C₆₁	—	2	8	no	—
53	C₂⁴×C₆₁	#41	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	—	2	2	no	—
55	C₂²×C₄×C₆₁	—	2	4	no	—
56	C₂²×C₄×C₆₁	—	4	2	no	—
57	C₂²×C₄×C₆₁	—	4	2	no	—
58	C₂²×C₄×C₆₁	—	8	4	no	—
59	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₆₁	#41	1	2	yes	—
67	C₂²×C₄×C₆₁	#41	1	2	yes	—
68	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
69	M₁₆×C₆₁	—	2	4	no	—
70	M₁₆×C₆₁	—	2	4	no	—
71	M₁₆×C₆₁	—	2	4	no	—
72	M₁₆×C₆₁	—	4	2	no	—
73	M₁₆×C₆₁	—	4	2	no	—
74	M₁₆×C₆₁	—	4	2	no	—
75	C₂×C₈×C₆₁	—	2	4	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	—	2	4	no	—
78	C₂×C₈×C₆₁	—	2	2	no	—
79	C₂×C₈×C₆₁	—	2	2	no	—
80	C₂×C₈×C₆₁	—	2	4	no	—
81	C₂×C₈×C₆₁	—	4	1	no	—
82	C₂×C₈×C₆₁	—	4	2	no	—
83	C₂×C₈×C₆₁	—	4	2	no	—
84	C₂×C₈×C₆₁	—	4	1	no	—
85	C₂×C₈×C₆₁	C₂²⋊C₄	1	4	yes	—
86	C₄⋊C₄×C₆₁	—	2	2	no	—
87	C₄⋊C₄×C₆₁	—	2	4	no	—
88	C₄⋊C₄×C₆₁	—	2	4	no	—
89	C₄⋊C₄×C₆₁	—	4	2	no	—
90	C₄⋊C₄×C₆₁	—	4	2	no	—
91	C₄⋊C₄×C₆₁	C₄⋊C₄	1	4	yes	—
92	C₂²⋊C₄×C₆₁	—	2	2	no	—
93	C₂²⋊C₄×C₆₁	—	2	2	no	—
94	C₄²×C₆₁	—	2	2	no	—
95	C₄²×C₆₁	—	2	2	no	—
96	C₂³×C₆₁	—	2	2	no	—
97	C₂³×C₆₁	—	2	2	no	—
98	C₂³×C₆₁	#117	1	1	yes	characteristic
99	C₂³×C₆₁	#121	1	1	yes	characteristic
100	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂³×C₆₁	#123	1	2	yes	—
102	C₂³×C₆₁	#125	1	2	yes	—
103	C₂×C₄×C₆₁	—	2	4	no	—
104	C₂×C₄×C₆₁	—	4	2	no	—
105	C₂×C₄×C₆₁	—	4	2	no	—
106	C₂×C₄×C₆₁	—	4	4	no	—
107	C₂×C₄×C₆₁	—	8	4	no	—
108	C₂×C₄×C₆₁	—	8	4	no	—
109	C₂×C₄×C₆₁	—	8	4	no	—
110	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
114	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
115	C₂×C₄×C₆₁	#121	1	2	yes	—
116	C₂×C₄×C₆₁	#121	1	2	yes	—
117	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
118	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
123	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
124	C₂×C₄×C₆₁	#125	1	2	yes	—
125	C₂×C₄×C₆₁	#126	1	2	yes	—
126	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
128	C₈×C₆₁	—	2	4	no	—
129	C₈×C₆₁	—	4	2	no	—
130	C₈×C₆₁	—	4	4	no	—
131	#11239	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
132	C₂²×C₆₁	—	2	2	no	—
133	C₂²×C₆₁	—	2	2	no	—
134	C₂²×C₆₁	—	2	4	no	—
135	C₂²×C₆₁	—	2	2	no	—
136	C₂²×C₆₁	—	2	2	no	—
137	C₂²×C₆₁	#380	1	1	yes	characteristic
138	C₂²×C₆₁	#387	1	2	yes	—
139	C₂²×C₆₁	#394	1	1	yes	characteristic
140	C₂²×C₆₁	#398	1	2	yes	—
141	C₂²×C₆₁	#398	1	1	yes	characteristic
142	C₂²×C₆₁	#472	1	2	yes	—
143	C₂²×C₆₁	#473	1	2	yes	—
144	C₄×C₆₁	—	4	4	no	—
145	C₄×C₆₁	—	8	4	no	—
146	C₄×C₆₁	#386	1	1	yes	characteristic
147	C₄×C₆₁	#389	1	2	yes	—
148	C₄×C₆₁	#391	1	1	yes	characteristic
149	C₄×C₆₁	#395	1	1	yes	characteristic
150	C₄×C₆₁	#396	1	2	yes	—
151	C₄×C₆₁	#400	1	1	yes	characteristic
152	C₂²×#107	C₂×C₆₁	1	1	yes	characteristic
153	C₂×#424	C₂×C₆₁	1	1	yes	characteristic
154	C₂×#343	C₂×C₆₁	1	2	yes	—
155	C₂×#341	C₂×C₆₁	1	1	yes	characteristic
156	#1736	C₂×C₆₁	1	2	yes	—
157	C₂×C₆₁	—	2	2	no	—
158	C₂×C₆₁	—	2	2	no	—
159	C₂×C₆₁	#1441	1	1	yes	characteristic
160	C₂×C₆₁	#1816	1	2	yes	—
161	C₂×C₆₁	#2045	1	1	yes	characteristic
162	C₂×C₆₁	#2047	1	1	yes	characteristic
163	C₂×C₆₁	#2089	1	2	yes	—
164	C₂²×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
165	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
166	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
167	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
168	C₂²×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
169	C₂×#107	—	2	2	no	—
170	C₂×#107	—	2	4	no	—
171	#424	—	2	2	no	—
172	#424	C₄×C₆₁	1	4	yes	complemented
173	C₂×C₄⋊C₈	—	2	2	no	—
174	#343	—	2	4	no	—
175	#341	—	2	2	no	—
176	#340	—	2	4	no	—
177	C₆₁	#11239	1	1	yes	characteristic, direct product factor, Sylow
178	C₂³×C₄	—	2	2	no	—
179	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	C₂×C₄×C₆₁	1	2	yes	—
189	C₂×M₁₆	Q₈×C₆₁	1	4	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	C₂×C₄×C₆₁	1	2	yes	—
195	C₂²×C₈	D₄×C₆₁	1	2	yes	—
196	C₂²×C₈	D₄×C₆₁	1	2	yes	—
197	#117	—	2	1	no	—
198	#117	—	2	1	no	—
199	#117	C₂×C₄×C₆₁	1	2	yes	—
200	#117	Q₈×C₆₁	1	2	yes	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	D₄×C₆₁	1	4	yes	—
207	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
208	C₂×C₂²⋊C₄	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	#107	—	2	8	no	—
211	C₄⋊C₈	—	2	8	no	—
212	C₂⁴	#41×C₆₁	1	1	yes	characteristic
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	8	4	no	—
218	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
223	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
225	C₂²×C₄	#41×C₆₁	1	2	yes	—
226	C₂²×C₄	#41×C₆₁	1	2	yes	—
227	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	C₂²⋊C₄×C₆₁	1	4	yes	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	C₄⋊C₄×C₆₁	1	4	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	#117×C₆₁	1	1	yes	characteristic
258	C₂³	#121×C₆₁	1	1	yes	characteristic
259	C₂³	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
260	C₂³	#123×C₆₁	1	2	yes	—
261	C₂³	#125×C₆₁	1	2	yes	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
271	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
272	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
273	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
274	C₂×C₄	#121×C₆₁	1	2	yes	—
275	C₂×C₄	#121×C₆₁	1	2	yes	—
276	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
277	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
278	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
279	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
280	C₂×C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
281	C₂×C₄	#123×C₆₁	1	1	yes	characteristic
282	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
283	C₂×C₄	#125×C₆₁	1	2	yes	—
284	C₂×C₄	#126×C₆₁	1	2	yes	—
285	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
286	C₂×C₄	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
287	C₈	—	2	4	no	—
288	C₈	—	4	2	no	—
289	C₈	—	4	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	4	no	—
295	C₂²	C₆₁×#380	1	1	yes	characteristic
296	C₂²	C₆₁×#387	1	2	yes	—
297	C₂²	C₆₁×#394	1	1	yes	characteristic
298	C₂²	C₆₁×#398	1	2	yes	—
299	C₂²	C₆₁×#398	1	1	yes	characteristic
300	C₂²	C₆₁×#472	1	2	yes	—
301	C₂²	C₆₁×#473	1	2	yes	—
302	C₄	—	4	4	no	—
303	C₄	—	8	4	no	—
304	C₄	C₆₁×#386	1	1	yes	characteristic
305	C₄	C₆₁×#389	1	2	yes	—
306	C₄	C₆₁×#391	1	1	yes	characteristic
307	C₄	C₆₁×#395	1	1	yes	characteristic
308	C₄	C₆₁×#396	1	2	yes	—
309	C₄	C₆₁×#400	1	1	yes	characteristic
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	C₆₁×#1441	1	1	yes	characteristic
313	C₂	C₆₁×#1816	1	2	yes	—
314	C₂	C₆₁×#2045	1	1	yes	characteristic
315	C₂	C₆₁×#2047	1	1	yes	characteristic
316	C₂	C₆₁×#2089	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	#121
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
prime radical	C₂⁴×C₆₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₆₁×#424	C₄	semidirect
131. #11239	C₆₁	direct (N × H)
172. #424	C₄×C₆₁	semidirect
177. C₆₁	#11239	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#107×C₆₁	2
C₂×C₆₁×#424	2
C₂×C₆₁×#343	2
C₂×C₆₁×#341	2
C₆₁×#1736	2
#11239	61

Subgroups — C61×#11239 #29514672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#11239 #29514672