Subgroups — C₆₁×#11210 #29514643

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2316	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#343	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1736	C₂	1	2	yes	maximal
5	C₂×C₆₁×#339	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₆₁	C₂²	1	2	yes	—
7	C₂×#117×C₆₁	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₆₁	—	2	2	no	—
9	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#389	—	2	4	no	—
13	C₆₁×#384	—	2	2	no	—
14	C₆₁×#384	C₄	1	4	yes	complemented
15	C₆₁×#380	—	2	2	no	—
16	C₆₁×#343	—	2	2	no	—
17	C₆₁×#340	—	2	4	no	—
18	C₆₁×#339	—	2	4	no	—
19	C₂³×C₄×C₆₁	—	2	1	no	—
20	C₂³×C₄×C₆₁	—	2	1	no	—
21	C₂³×C₄×C₆₁	D₄	1	2	yes	—
22	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
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	Q₈	1	4	yes	—
29	C₂²×C₈×C₆₁	—	2	2	no	—
30	#117×C₆₁	—	2	2	no	—
31	#117×C₆₁	—	2	2	no	—
32	#117×C₆₁	—	2	2	no	—
33	#117×C₆₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
39	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₁	—	4	8	no	—
42	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₆₁	D₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
47	C₂×C₄²×C₆₁	—	2	1	no	—
48	C₂×C₄²×C₆₁	—	2	1	no	—
49	C₂×C₄²×C₆₁	—	2	1	no	—
50	C₂×C₄²×C₆₁	—	2	2	no	—
51	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₁	D₄	1	2	yes	—
53	C₂×C₄²×C₆₁	Q₈	1	2	yes	—
54	#95×C₆₁	—	2	4	no	—
55	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	—	2	4	no	—
57	C₂²×C₄×C₆₁	—	2	4	no	—
58	C₂²×C₄×C₆₁	—	2	2	no	—
59	C₂²×C₄×C₆₁	—	2	2	no	—
60	C₂²×C₄×C₆₁	—	2	2	no	—
61	C₂²×C₄×C₆₁	—	2	2	no	—
62	C₂²×C₄×C₆₁	—	2	2	no	—
63	C₂²×C₄×C₆₁	—	2	2	no	—
64	C₂²×C₄×C₆₁	—	2	4	no	—
65	C₂²×C₄×C₆₁	—	4	4	no	—
66	C₂²×C₄×C₆₁	—	4	4	no	—
67	C₂²×C₄×C₆₁	—	4	1	no	—
68	C₂²×C₄×C₆₁	—	4	1	no	—
69	C₂²×C₄×C₆₁	C₂²⋊C₄	1	4	yes	—
70	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
77	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
78	C₂²×C₄×C₆₁	#41	1	2	yes	—
79	C₂²×C₄×C₆₁	#41	1	2	yes	—
80	M₁₆×C₆₁	—	2	4	no	—
81	M₁₆×C₆₁	—	2	4	no	—
82	M₁₆×C₆₁	—	4	2	no	—
83	M₁₆×C₆₁	—	4	2	no	—
84	C₂×C₈×C₆₁	—	2	4	no	—
85	C₂×C₈×C₆₁	—	4	2	no	—
86	C₂×C₈×C₆₁	—	4	2	no	—
87	C₄⋊C₄×C₆₁	—	2	4	no	—
88	C₄⋊C₄×C₆₁	—	2	4	no	—
89	C₄⋊C₄×C₆₁	—	2	4	no	—
90	C₄⋊C₄×C₆₁	—	2	4	no	—
91	C₄⋊C₄×C₆₁	—	2	4	no	—
92	C₄⋊C₄×C₆₁	—	4	16	no	—
93	C₄⋊C₄×C₆₁	—	4	2	no	—
94	C₄⋊C₄×C₆₁	—	8	4	no	—
95	C₂²⋊C₄×C₆₁	—	4	2	no	—
96	C₂²⋊C₄×C₆₁	—	4	2	no	—
97	C₄²×C₆₁	—	2	2	no	—
98	C₄²×C₆₁	—	2	2	no	—
99	C₄²×C₆₁	—	2	2	no	—
100	C₄²×C₆₁	—	4	2	no	—
101	C₄²×C₆₁	—	4	2	no	—
102	C₄²×C₆₁	C₄⋊C₄	1	4	yes	complemented
103	C₂³×C₆₁	—	2	2	no	—
104	C₂³×C₆₁	—	2	2	no	—
105	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₆₁	#120	1	1	yes	characteristic
107	C₂³×C₆₁	#121	1	1	yes	characteristic
108	C₂³×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂³×C₆₁	#123	1	1	yes	characteristic
110	C₂³×C₆₁	#124	1	1	yes	characteristic
111	C₂³×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₆₁	—	2	4	no	—
113	C₂×C₄×C₆₁	—	2	8	no	—
114	C₂×C₄×C₆₁	—	2	4	no	—
115	C₂×C₄×C₆₁	—	2	8	no	—
116	C₂×C₄×C₆₁	—	2	4	no	—
117	C₂×C₄×C₆₁	—	2	4	no	—
118	C₂×C₄×C₆₁	—	4	4	no	—
119	C₂×C₄×C₆₁	—	4	4	no	—
120	C₂×C₄×C₆₁	—	4	4	no	—
121	C₂×C₄×C₆₁	—	4	2	no	—
122	C₂×C₄×C₆₁	—	4	4	no	—
123	C₂×C₄×C₆₁	—	4	4	no	—
124	C₂×C₄×C₆₁	—	4	4	no	—
125	C₂×C₄×C₆₁	—	8	2	no	—
126	C₂×C₄×C₆₁	—	8	2	no	—
127	C₂×C₄×C₆₁	—	8	2	no	—
128	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
130	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₆₁	#121	1	2	yes	—
133	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₆₁	#121	1	2	yes	—
135	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
139	C₂×C₄×C₆₁	#123	1	2	yes	—
140	C₂×C₄×C₆₁	#125	1	2	yes	—
141	C₂×C₄×C₆₁	#125	1	2	yes	—
142	C₂×C₄×C₆₁	#126	1	2	yes	—
143	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
144	C₈×C₆₁	—	4	4	no	—
145	#11210	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₆₁	—	2	2	no	—
147	C₂²×C₆₁	—	2	2	no	—
148	C₂²×C₆₁	—	2	1	no	—
149	C₂²×C₆₁	—	2	1	no	—
150	C₂²×C₆₁	—	2	1	no	—
151	C₂²×C₆₁	—	2	1	no	—
152	C₂²×C₆₁	—	2	2	no	—
153	C₂²×C₆₁	—	2	2	no	—
154	C₂²×C₆₁	#380	1	1	yes	characteristic
155	C₂²×C₆₁	#386	1	1	yes	characteristic
156	C₂²×C₆₁	#391	1	1	yes	characteristic
157	C₂²×C₆₁	#394	1	1	yes	characteristic
158	C₂²×C₆₁	#395	1	1	yes	characteristic
159	C₂²×C₆₁	#398	1	1	yes	characteristic
160	C₂²×C₆₁	#400	1	1	yes	characteristic
161	C₂²×C₆₁	#453	1	2	yes	—
162	C₂²×C₆₁	#456	1	2	yes	—
163	C₄×C₆₁	—	2	8	no	—
164	C₄×C₆₁	—	4	4	no	—
165	C₄×C₆₁	—	8	2	no	—
166	C₄×C₆₁	—	8	4	no	—
167	C₄×C₆₁	#387	1	2	yes	—
168	C₄×C₆₁	#389	1	2	yes	—
169	C₄×C₆₁	#396	1	2	yes	—
170	C₄×C₆₁	#398	1	2	yes	—
171	#2316	C₂×C₆₁	1	1	yes	characteristic
172	C₂×#384	C₂×C₆₁	1	1	yes	characteristic
173	C₂×#343	C₂×C₆₁	1	1	yes	characteristic
174	#1736	C₂×C₆₁	1	2	yes	—
175	C₂×#339	C₂×C₆₁	1	2	yes	—
176	C₂×C₆₁	—	2	1	no	—
177	C₂×C₆₁	—	2	1	no	—
178	C₂×C₆₁	—	2	1	no	—
179	C₂×C₆₁	—	2	1	no	—
180	C₂×C₆₁	#1441	1	1	yes	characteristic
181	C₂×C₆₁	#1886	1	1	yes	characteristic
182	C₂×C₆₁	#1887	1	1	yes	characteristic
183	C₂×C₆₁	#2039	1	2	yes	—
184	C₂×C₆₁	#2060	1	2	yes	—
185	C₂²×M₁₆	C₂²×C₆₁	1	2	yes	—
186	C₂×#117	C₂²×C₆₁	1	2	yes	—
187	C₂²×C₄⋊C₄	—	2	2	no	—
188	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
189	C₂²×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
190	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
191	#389	—	2	4	no	—
192	#384	—	2	2	no	—
193	#384	C₄×C₆₁	1	4	yes	complemented
194	#380	—	2	2	no	—
195	#343	—	2	2	no	—
196	#340	—	2	4	no	—
197	#339	—	2	4	no	—
198	C₆₁	#11210	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³×C₄	—	2	1	no	—
200	C₂³×C₄	—	2	1	no	—
201	C₂³×C₄	D₄×C₆₁	1	2	yes	—
202	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	2	2	no	—
208	C₂×M₁₆	Q₈×C₆₁	1	4	yes	—
209	C₂²×C₈	—	2	2	no	—
210	#117	—	2	2	no	—
211	#117	—	2	2	no	—
212	#117	—	2	2	no	—
213	#117	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	8	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	4	no	—
221	C₂×C₄⋊C₄	—	4	8	no	—
222	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
223	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
224	C₂×C₄⋊C₄	D₄×C₆₁	1	4	yes	—
225	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₄²	—	2	1	no	—
228	C₂×C₄²	—	2	1	no	—
229	C₂×C₄²	—	2	1	no	—
230	C₂×C₄²	—	2	2	no	—
231	C₂×C₄²	C₂×C₄×C₆₁	1	2	yes	—
232	C₂×C₄²	D₄×C₆₁	1	2	yes	—
233	C₂×C₄²	Q₈×C₆₁	1	2	yes	—
234	#95	—	2	4	no	—
235	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	C₂²⋊C₄×C₆₁	1	4	yes	—
250	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
254	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×Q₈×C₆₁	1	2	yes	—
256	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
257	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
258	C₂²×C₄	#41×C₆₁	1	2	yes	—
259	C₂²×C₄	#41×C₆₁	1	2	yes	—
260	M₁₆	—	2	4	no	—
261	M₁₆	—	2	4	no	—
262	M₁₆	—	4	2	no	—
263	M₁₆	—	4	2	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	4	16	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	8	4	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	4	2	no	—
281	C₄²	—	4	2	no	—
282	C₄²	C₄⋊C₄×C₆₁	1	4	yes	complemented
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
286	C₂³	#120×C₆₁	1	1	yes	characteristic
287	C₂³	#121×C₆₁	1	1	yes	characteristic
288	C₂³	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
289	C₂³	#123×C₆₁	1	1	yes	characteristic
290	C₂³	#124×C₆₁	1	1	yes	characteristic
291	C₂³	C₄⋊Q₈×C₆₁	1	1	yes	characteristic
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	2	8	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
309	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
310	C₂×C₄	C₄×D₄×C₆₁	1	2	yes	—
311	C₂×C₄	C₄×Q₈×C₆₁	1	2	yes	—
312	C₂×C₄	#121×C₆₁	1	2	yes	—
313	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
314	C₂×C₄	#121×C₆₁	1	2	yes	—
315	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
316	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
317	C₂×C₄	C₂²⋊Q₈×C₆₁	1	2	yes	—
318	C₂×C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
319	C₂×C₄	#123×C₆₁	1	2	yes	—
320	C₂×C₄	#125×C₆₁	1	2	yes	—
321	C₂×C₄	#125×C₆₁	1	2	yes	—
322	C₂×C₄	#126×C₆₁	1	2	yes	—
323	C₂×C₄	C₄⋊Q₈×C₆₁	1	2	yes	—
324	C₈	—	4	4	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	C₆₁×#380	1	1	yes	characteristic
334	C₂²	C₆₁×#386	1	1	yes	characteristic
335	C₂²	C₆₁×#391	1	1	yes	characteristic
336	C₂²	C₆₁×#394	1	1	yes	characteristic
337	C₂²	C₆₁×#395	1	1	yes	characteristic
338	C₂²	C₆₁×#398	1	1	yes	characteristic
339	C₂²	C₆₁×#400	1	1	yes	characteristic
340	C₂²	C₆₁×#453	1	2	yes	—
341	C₂²	C₆₁×#456	1	2	yes	—
342	C₄	—	2	8	no	—
343	C₄	—	4	4	no	—
344	C₄	—	8	4	no	—
345	C₄	—	8	2	no	—
346	C₄	C₆₁×#387	1	2	yes	—
347	C₄	C₆₁×#389	1	2	yes	—
348	C₄	C₆₁×#396	1	2	yes	—
349	C₄	C₆₁×#398	1	2	yes	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	C₆₁×#1441	1	1	yes	characteristic
355	C₂	C₆₁×#1886	1	1	yes	characteristic
356	C₂	C₆₁×#1887	1	1	yes	characteristic
357	C₂	C₆₁×#2039	1	2	yes	—
358	C₂	C₆₁×#2060	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	#120
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂³×C₄	C₂³×C₆₁
prime radical	C₂⁴×C₆₁	C₂×D₄
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×#384	C₄	semidirect
102. C₄²×C₆₁	C₄⋊C₄	semidirect
145. #11210	C₆₁	direct (N × H)
193. #384	C₄×C₆₁	semidirect
198. C₆₁	#11210	direct (N × H)
282. C₄²	C₄⋊C₄×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#2316	2
C₂×C₆₁×#384	2
C₂×C₆₁×#343	2
C₆₁×#1736	2
C₂×C₆₁×#339	2
#11210	61

Subgroups — C61×#11210 #29514643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#11210 #29514643