Subgroups — D₆₁×#2110 #29832668

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₆₁	C₂	1	2	yes	maximal
2	#405×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#21447424	C₂	1	1	yes	maximal, characteristic, complemented
4	#21447423	C₂	1	2	yes	maximal, complemented
5	#21447422	C₂	1	4	yes	maximal, complemented
6	#347×D₆₁	C₂	1	4	yes	maximal, complemented
7	C₆₁×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₆₁	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
10	#117×D₆₁	C₄	1	1	yes	characteristic
11	C₂×#14519717	C₂²	1	2	yes	—
12	C₂×#14519716	C₄	1	4	yes	complemented
13	M₃₂×D₆₁	C₄	1	4	yes	complemented
14	M₃₂×D₆₁	C₄	1	4	yes	complemented
15	M₃₂×D₆₁	C₂²	1	4	yes	—
16	M₃₂×D₆₁	C₂²	1	4	yes	—
17	C₂×C₁₆×D₆₁	C₄	1	4	yes	complemented
18	#16839720	C₄	1	1	yes	characteristic
19	#16839719	C₂²	1	2	yes	—
20	#16839718	C₄	1	1	yes	characteristic
21	#16839717	C₂²	1	1	yes	characteristic
22	#16839716	C₄	1	2	yes	—
23	#97×D₆₁	C₂²	1	2	yes	—
24	C₄×C₈×D₆₁	C₄	1	2	yes	—
25	#16839540	C₂²	1	4	yes	complemented
26	#16839539	C₂²	1	8	yes	complemented
27	C₂×M₃₂×C₆₁	C₂²	1	2	yes	—
28	C₆₁×#405	C₂²	1	1	yes	characteristic
29	C₆₁×#347	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
32	C₂×#11758189	C₂×C₄	1	2	yes	—
33	M₁₆×D₆₁	—	2	2	no	—
34	M₁₆×D₆₁	—	2	2	no	—
35	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
45	#14519736	C₂×C₄	1	1	yes	characteristic
46	#14519735	—	2	1	no	—
47	#14519734	C₂×C₄	1	2	yes	—
48	#14519733	—	2	1	no	—
49	C₄⋊C₄×D₆₁	—	2	1	no	—
50	C₂²⋊C₄×D₆₁	—	2	1	no	—
51	C₄²×D₆₁	C₂×C₄	1	2	yes	—
52	#14519717	C₂×C₄	1	4	yes	complemented
53	#14519717	C₂³	1	4	yes	—
54	#14519716	C₂×C₄	1	8	yes	—
55	#14519716	C₂×C₄	1	8	yes	complemented
56	C₂×#11758185	C₂×C₄	1	4	yes	complemented
57	C₁₆×D₆₁	C₂×C₄	1	8	yes	—
58	C₁₆×D₆₁	C₂×C₄	1	8	yes	complemented
59	#14519688	C₂×C₄	1	2	yes	—
60	#14519687	C₂×C₄	1	2	yes	—
61	#14519687	C₂³	1	2	yes	—
62	C₄×#92418	C₂×C₄	1	2	yes	—
63	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
64	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
67	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
68	M₃₂×C₆₁	C₂×C₄	1	4	yes	complemented
69	M₃₂×C₆₁	C₂³	1	4	yes	—
70	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
71	#97×C₆₁	C₂³	1	2	yes	—
72	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
73	C₂³×D₆₁	M₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×Dic₆₁	M₁₆	1	1	yes	characteristic
76	C₂×C₄×D₆₁	—	2	2	no	—
77	C₂×C₄×D₆₁	C₄²	1	2	yes	—
78	C₂×C₄×D₆₁	C₄²	1	2	yes	—
79	C₂×C₄×D₆₁	M₁₆	1	1	yes	characteristic
80	C₂×C₄×D₆₁	M₁₆	1	1	yes	characteristic
81	C₂×C₄×D₆₁	M₁₆	1	2	yes	—
82	C₂×C₄×D₆₁	M₁₆	1	2	yes	—
83	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
85	M₁₆×C₆₁	—	2	2	no	—
86	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₆₁	—	2	1	no	—
92	C₂²⋊C₄×C₆₁	—	2	1	no	—
93	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
94	C₂²⋊Dic₆₁	—	2	1	no	—
95	#11758198	—	2	2	no	—
96	C₄⋊Dic₆₁	—	2	1	no	—
97	#11758196	—	2	2	no	—
98	C₄×Dic₆₁	—	2	2	no	—
99	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
100	#11758194	—	2	2	no	—
101	C₂×#92418	C₄²	1	2	yes	—
102	C₂×#92418	C₄²	1	2	yes	—
103	C₂×#92418	C₂²×C₄	1	2	yes	—
104	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
105	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
106	#11758189	—	2	4	no	—
107	C₈×D₆₁	—	2	2	no	—
108	C₈×D₆₁	—	2	2	no	—
109	C₈×D₆₁	C₄²	1	1	yes	characteristic
110	C₈×D₆₁	C₄²	1	1	yes	characteristic
111	C₈×D₆₁	C₄²	1	2	yes	—
112	C₈×D₆₁	C₄²	1	1	yes	characteristic
113	C₈×D₆₁	C₄²	1	1	yes	characteristic
114	C₈×D₆₁	C₄²	1	2	yes	—
115	C₈×D₆₁	C₂²×C₄	1	2	yes	—
116	C₈×D₆₁	C₂²×C₄	1	2	yes	—
117	C₈×D₆₁	C₂²×C₄	1	2	yes	—
118	C₈×D₆₁	C₂²×C₄	1	2	yes	—
119	C₁₆×C₆₁	C₂²×C₄	1	8	yes	—
120	#11758185	C₂²×C₄	1	8	yes	—
121	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
122	C₂²×D₆₁	—	4	2	no	—
123	C₂²×D₆₁	#97	1	2	yes	—
124	C₂²×D₆₁	#97	1	2	yes	—
125	C₂²×D₆₁	C₂×M₁₆	1	1	yes	characteristic
126	C₂²×D₆₁	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₆₁	—	2	2	no	—
128	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
132	C₂×Dic₆₁	—	2	2	no	—
133	C₂×Dic₆₁	#97	1	2	yes	—
134	C₂×Dic₆₁	#97	1	2	yes	—
135	C₂×Dic₆₁	C₂×M₁₆	1	1	yes	characteristic
136	C₂×Dic₆₁	C₂×M₁₆	1	1	yes	characteristic
137	C₄×D₆₁	—	4	4	no	—
138	C₄×D₆₁	#97	1	2	yes	—
139	C₄×D₆₁	#97	1	2	yes	—
140	C₄×D₆₁	#97	1	2	yes	—
141	C₄×D₆₁	#97	1	2	yes	—
142	C₄×D₆₁	C₂×C₄²	1	2	yes	—
143	C₄×D₆₁	C₂×C₄²	1	1	yes	characteristic
144	C₄×D₆₁	C₂×C₄²	1	1	yes	characteristic
145	C₄×D₆₁	C₂×M₁₆	1	1	yes	characteristic
146	C₄×D₆₁	C₂×M₁₆	1	1	yes	characteristic
147	C₄×D₆₁	C₂×M₁₆	1	2	yes	—
148	C₈×C₆₁	—	2	2	no	—
149	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
150	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
151	C₈×C₆₁	C₂³×C₄	1	2	yes	—
152	#92418	—	2	2	no	—
153	#92418	C₂×C₄²	1	2	yes	—
154	#92418	C₂×C₄²	1	1	yes	characteristic
155	#92418	C₂×C₄²	1	1	yes	characteristic
156	C₂×#2110	—	61	1	no	maximal, Sylow
157	C₂²×C₆₁	—	4	1	no	—
158	C₂²×C₆₁	C₂×#97	1	2	yes	—
159	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
160	C₂×D₆₁	—	2	2	no	—
161	C₂×D₆₁	—	2	4	no	—
162	C₂×D₆₁	—	4	2	no	—
163	C₂×D₆₁	—	4	1	no	—
164	C₂×D₆₁	C₂×#97	1	1	yes	characteristic
165	C₂×D₆₁	C₂×#97	1	2	yes	—
166	C₂×D₆₁	C₂×#97	1	1	yes	characteristic
167	C₄×C₆₁	—	4	2	no	—
168	C₄×C₆₁	C₂×#97	1	2	yes	—
169	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
170	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
171	Dic₆₁	—	4	2	no	—
172	Dic₆₁	C₂×#97	1	1	yes	characteristic
173	Dic₆₁	C₂×#97	1	1	yes	characteristic
174	Dic₆₁	C₂×#97	1	2	yes	—
175	C₂²×M₃₂	—	61	2	no	—
176	C₂×#405	—	61	1	no	—
177	#2110	—	61	2	no	—
178	#2110	—	61	1	no	—
179	#2110	—	61	4	no	—
180	#2110	D₆₁	1	1	yes	characteristic, direct product factor
181	C₂×#347	—	61	4	no	—
182	C₂×C₆₁	—	2	1	no	—
183	C₂×C₆₁	—	2	2	no	—
184	C₂×C₆₁	C₂²×#97	1	1	yes	characteristic
185	D₆₁	—	2	1	no	—
186	D₆₁	—	2	2	no	—
187	D₆₁	#2110	1	2	yes	direct product factor
188	C₂²×M₁₆	—	61	1	no	—
189	C₂³×C₈	—	61	1	no	—
190	C₂×#117	—	61	1	no	—
191	C₂×M₃₂	—	61	2	no	—
192	C₂×M₃₂	—	61	4	no	—
193	C₂×M₃₂	—	61	4	no	—
194	C₂×M₃₂	—	61	4	no	—
195	C₂×M₃₂	—	61	4	no	—
196	C₂×M₃₂	—	61	4	no	—
197	C₂×M₃₂	C₂×D₆₁	1	2	yes	—
198	C₂²×C₁₆	—	61	4	no	—
199	#405	—	61	1	no	—
200	#405	—	61	1	no	—
201	#405	—	61	1	no	—
202	#405	—	61	2	no	—
203	#405	—	61	2	no	—
204	#405	C₂×D₆₁	1	1	yes	characteristic
205	C₂×#97	—	61	2	no	—
206	C₂×C₄×C₈	—	61	2	no	—
207	#347	—	61	8	no	—
208	#347	—	61	4	no	—
209	#347	C₂×D₆₁	1	4	yes	complemented
210	C₆₁	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
211	C₂³×C₄	—	61	1	no	—
212	C₂×M₁₆	—	61	1	no	—
213	C₂×M₁₆	—	61	2	no	—
214	C₂×M₁₆	—	122	2	no	—
215	C₂×M₁₆	—	122	2	no	—
216	C₂×M₁₆	C₄×D₆₁	1	1	yes	characteristic
217	C₂²×C₈	—	61	1	no	—
218	C₂²×C₈	—	61	1	no	—
219	C₂²×C₈	—	61	1	no	—
220	C₂²×C₈	—	61	2	no	—
221	C₂²×C₈	—	61	2	no	—
222	C₂²×C₈	—	61	2	no	—
223	C₂²×C₈	—	61	2	no	—
224	C₂²×C₈	—	61	1	no	—
225	C₂²×C₈	—	61	1	no	—
226	C₂²×C₈	—	61	2	no	—
227	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
228	#117	—	61	1	no	—
229	#117	—	61	2	no	—
230	#117	—	122	1	no	—
231	#117	—	122	1	no	—
232	#117	C₄×D₆₁	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	122	1	no	—
234	C₂×C₂²⋊C₄	—	122	1	no	—
235	C₂×C₄²	—	61	2	no	—
236	M₃₂	—	61	4	no	—
237	M₃₂	—	61	4	no	—
238	M₃₂	—	61	8	no	—
239	M₃₂	—	61	8	no	—
240	M₃₂	C₄×D₆₁	1	4	yes	complemented
241	M₃₂	C₂²×D₆₁	1	4	yes	—
242	C₂×C₁₆	—	61	4	no	—
243	C₂×C₁₆	—	61	8	no	—
244	C₂×C₁₆	—	61	8	no	—
245	C₂×C₁₆	C₄×D₆₁	1	4	yes	complemented
246	#97	—	61	2	no	—
247	#97	—	61	2	no	—
248	#97	—	61	2	no	—
249	#97	C₂²×D₆₁	1	2	yes	—
250	C₄×C₈	—	61	2	no	—
251	C₄×C₈	—	61	2	no	—
252	C₄×C₈	—	61	2	no	—
253	C₄×C₈	C₄×D₆₁	1	2	yes	—
254	C₂⁴	—	61	1	no	—
255	C₂²×C₄	—	61	1	no	—
256	C₂²×C₄	—	61	1	no	—
257	C₂²×C₄	—	61	1	no	—
258	C₂²×C₄	—	61	2	no	—
259	C₂²×C₄	—	61	2	no	—
260	C₂²×C₄	—	61	1	no	—
261	C₂²×C₄	—	61	1	no	—
262	C₂²×C₄	—	61	2	no	—
263	C₂²×C₄	—	61	2	no	—
264	C₂²×C₄	—	122	2	no	—
265	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
266	M₁₆	—	2	2	no	—
267	M₁₆	—	122	4	no	—
268	M₁₆	—	122	2	no	—
269	C₂×C₈	—	61	1	no	—
270	C₂×C₈	—	61	1	no	—
271	C₂×C₈	—	61	2	no	—
272	C₂×C₈	—	61	2	no	—
273	C₂×C₈	—	61	2	no	—
274	C₂×C₈	—	61	2	no	—
275	C₂×C₈	—	61	2	no	—
276	C₂×C₈	—	61	2	no	—
277	C₂×C₈	—	61	1	no	—
278	C₂×C₈	—	61	1	no	—
279	C₂×C₈	—	61	2	no	—
280	C₂×C₈	—	61	1	no	—
281	C₂×C₈	—	61	1	no	—
282	C₂×C₈	—	61	2	no	—
283	C₂×C₈	—	61	2	no	—
284	C₂×C₈	—	122	2	no	—
285	C₂×C₈	—	122	2	no	—
286	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
287	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
288	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
290	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	122	2	no	—
293	C₄⋊C₄	—	122	1	no	—
294	C₂²⋊C₄	—	2	1	no	—
295	C₂²⋊C₄	—	122	2	no	—
296	C₂²⋊C₄	—	122	1	no	—
297	C₄²	—	61	2	no	—
298	C₄²	—	122	2	no	—
299	C₄²	C₂×C₄×D₆₁	1	2	yes	—
300	C₁₆	—	61	8	no	—
301	C₁₆	C₂×C₄×D₆₁	1	8	yes	—
302	C₂³	—	61	2	no	—
303	C₂³	—	61	1	no	—
304	C₂³	—	61	2	no	—
305	C₂³	—	61	1	no	—
306	C₂³	—	244	2	no	—
307	C₂³	M₁₆×D₆₁	1	1	yes	characteristic
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	61	2	no	—
310	C₂×C₄	—	61	2	no	—
311	C₂×C₄	—	61	1	no	—
312	C₂×C₄	—	61	1	no	—
313	C₂×C₄	—	61	1	no	—
314	C₂×C₄	—	61	2	no	—
315	C₂×C₄	—	61	1	no	—
316	C₂×C₄	—	61	2	no	—
317	C₂×C₄	—	61	2	no	—
318	C₂×C₄	—	61	2	no	—
319	C₂×C₄	—	61	2	no	—
320	C₂×C₄	—	61	1	no	—
321	C₂×C₄	—	61	1	no	—
322	C₂×C₄	—	61	2	no	—
323	C₂×C₄	—	122	2	no	—
324	C₂×C₄	—	244	4	no	—
325	C₂×C₄	C₄²×D₆₁	1	2	yes	—
326	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
327	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
328	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	61	1	no	—
331	C₈	—	61	1	no	—
332	C₈	—	61	2	no	—
333	C₈	—	122	2	no	—
334	C₈	C₄²×D₆₁	1	1	yes	characteristic
335	C₈	C₄²×D₆₁	1	1	yes	characteristic
336	C₈	C₂²×C₄×D₆₁	1	2	yes	—
337	C₂²	—	4	1	no	—
338	C₂²	—	61	1	no	—
339	C₂²	—	61	1	no	—
340	C₂²	—	61	2	no	—
341	C₂²	—	122	2	no	—
342	C₂²	—	122	4	no	—
343	C₂²	—	244	1	no	—
344	C₂²	—	244	2	no	—
345	C₂²	#97×D₆₁	1	2	yes	—
346	C₂²	C₂×M₁₆×D₆₁	1	1	yes	characteristic
347	C₄	—	4	2	no	—
348	C₄	—	61	1	no	—
349	C₄	—	61	1	no	—
350	C₄	—	61	2	no	—
351	C₄	—	244	2	no	—
352	C₄	#97×D₆₁	1	2	yes	—
353	C₄	C₂×C₄²×D₆₁	1	1	yes	characteristic
354	C₄	C₂×M₁₆×D₆₁	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	61	2	no	—
358	C₂	—	122	1	no	—
359	C₂	—	122	2	no	—
360	C₂	C₂×#97×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄²×D₆₁
commutator	C₄×C₆₁	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2110	C₂
socle	C₂×C₆₁	C₂²×#97
prime radical	C₂³×D₆₁	M₁₆
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2110	D₆₁
hypocenter	C₆₁	C₂×#2110
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #21447424	C₂	semidirect
4. #21447423	C₂	semidirect
5. #21447422	C₂	semidirect
6. #347×D₆₁	C₂	semidirect
7. C₆₁×#2110	C₂	semidirect
12. C₂×#14519716	C₄	semidirect
13. M₃₂×D₆₁	C₄	semidirect
14. M₃₂×D₆₁	C₄	semidirect
17. C₂×C₁₆×D₆₁	C₄	semidirect
25. #16839540	C₂²	semidirect
26. #16839539	C₂²	semidirect
29. C₆₁×#347	C₂²	semidirect
52. #14519717	C₂×C₄	semidirect
55. #14519716	C₂×C₄	semidirect
56. C₂×#11758185	C₂×C₄	semidirect
58. C₁₆×D₆₁	C₂×C₄	semidirect
68. M₃₂×C₆₁	C₂×C₄	semidirect
70. C₂×C₁₆×C₆₁	C₂×C₄	semidirect
180. #2110	D₆₁	direct (N × H)
187. D₆₁	#2110	direct (N × H)
209. #347	C₂×D₆₁	semidirect
210. C₆₁	C₂×#2110	semidirect
240. M₃₂	C₄×D₆₁	semidirect
245. C₂×C₁₆	C₄×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₆₁	2
#405×D₆₁	2
#21447424	2
#21447423	2
#21447422	2
#347×D₆₁	2
C₆₁×#2110	2
C₂×#2110	61

Subgroups — D61×#2110 #29832668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁×#2110 #29832668