Subgroups — C₆₁×#29771 #29533204

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#467	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2855	C₂	1	2	yes	maximal
4	C₆₁×#2087	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2068	C₂	1	4	yes	maximal, complemented
6	C₆₁×#2056	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1892	C₂	1	2	yes	maximal
8	C₆₁×#1808	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
13	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#467	—	2	4	no	—
15	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#107×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₁	—	2	4	no	—
18	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
20	C₂×#103×C₆₁	C₂²	1	2	yes	—
21	C₂×#103×C₆₁	C₂²	1	4	yes	—
22	C₆₁×#391	C₂²	1	4	yes	—
23	C₆₁×#389	C₂²	1	4	yes	—
24	C₆₁×#387	—	2	4	no	—
25	C₆₁×#385	—	2	2	no	—
26	C₆₁×#381	—	2	2	no	—
27	C₆₁×#380	C₂²	1	2	yes	—
28	C₆₁×#340	C₂²	1	2	yes	—
29	C₆₁×#340	C₂²	1	4	yes	—
30	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
31	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
34	C₂×Q₁₆×C₆₁	—	2	2	no	—
35	C₂×Q₁₆×C₆₁	—	4	2	no	—
36	C₂²×C₈×C₆₁	—	2	2	no	—
37	C₂²×C₈×C₆₁	—	4	2	no	—
38	C₂²×C₈×C₆₁	—	4	2	no	—
39	C₂²×C₈×C₆₁	D₄	1	4	yes	—
40	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
43	C₂²⋊Q₈×C₆₁	—	2	4	no	—
44	C₂²⋊Q₈×C₆₁	—	2	4	no	—
45	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
46	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
51	C₂×C₄²×C₆₁	—	2	2	no	—
52	C₂×C₄²×C₆₁	—	2	2	no	—
53	C₂×C₄²×C₆₁	D₄	1	2	yes	—
54	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
55	#107×C₆₁	—	2	2	no	—
56	#107×C₆₁	—	4	2	no	—
57	C₄⋊C₈×C₆₁	—	2	4	no	—
58	C₄⋊C₈×C₆₁	—	2	4	no	—
59	#103×C₆₁	—	2	4	no	—
60	#103×C₆₁	—	4	4	no	—
61	#95×C₆₁	—	2	2	no	—
62	#95×C₆₁	—	2	4	no	—
63	#95×C₆₁	—	2	4	no	—
64	#95×C₆₁	—	2	4	no	—
65	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
66	C₂×Q₈×C₆₁	—	2	2	no	—
67	C₂×Q₈×C₆₁	—	2	2	no	—
68	C₂×Q₈×C₆₁	—	2	4	no	—
69	C₂×Q₈×C₆₁	—	2	4	no	—
70	C₂²×C₄×C₆₁	—	2	2	no	—
71	C₂²×C₄×C₆₁	—	2	2	no	—
72	C₂²×C₄×C₆₁	—	2	2	no	—
73	C₂²×C₄×C₆₁	—	2	2	no	—
74	C₂²×C₄×C₆₁	—	2	2	no	—
75	C₂²×C₄×C₆₁	—	2	4	no	—
76	C₂²×C₄×C₆₁	—	2	2	no	—
77	C₂²×C₄×C₆₁	—	2	2	no	—
78	C₂²×C₄×C₆₁	—	4	2	no	—
79	C₂²×C₄×C₆₁	Q₁₆	1	4	yes	complemented
80	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₆₁	#41	1	2	yes	—
85	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
86	Q₁₆×C₆₁	—	4	4	no	—
87	C₂×C₈×C₆₁	—	2	2	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₂×C₈×C₆₁	—	2	2	no	—
90	C₂×C₈×C₆₁	—	2	4	no	—
91	C₂×C₈×C₆₁	—	4	4	no	—
92	C₂×C₈×C₆₁	—	4	4	no	—
93	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
95	C₄⋊C₄×C₆₁	—	2	2	no	—
96	C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₄⋊C₄×C₆₁	—	2	4	no	—
98	C₄⋊C₄×C₆₁	—	2	4	no	—
99	C₄⋊C₄×C₆₁	—	4	8	no	—
100	C₄⋊C₄×C₆₁	—	4	4	no	—
101	C₂²⋊C₄×C₆₁	—	4	4	no	—
102	C₂²⋊C₄×C₆₁	—	4	4	no	—
103	C₄²×C₆₁	—	2	4	no	—
104	C₄²×C₆₁	—	2	2	no	—
105	C₄²×C₆₁	—	2	2	no	—
106	C₂³×C₆₁	—	2	2	no	—
107	C₂³×C₆₁	—	2	2	no	—
108	C₂³×C₆₁	#121	1	2	yes	—
109	C₂³×C₆₁	C₂×Q₁₆	1	2	yes	—
110	C₂³×C₆₁	#135	1	2	yes	—
111	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₆₁	—	4	2	no	—
113	Q₈×C₆₁	—	4	2	no	—
114	Q₈×C₆₁	—	4	4	no	—
115	C₂×C₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	2	2	no	—
117	C₂×C₄×C₆₁	—	2	2	no	—
118	C₂×C₄×C₆₁	—	2	8	no	—
119	C₂×C₄×C₆₁	—	2	2	no	—
120	C₂×C₄×C₆₁	—	2	2	no	—
121	C₂×C₄×C₆₁	—	2	4	no	—
122	C₂×C₄×C₆₁	—	2	2	no	—
123	C₂×C₄×C₆₁	—	2	4	no	—
124	C₂×C₄×C₆₁	—	4	4	no	—
125	C₂×C₄×C₆₁	—	4	2	no	—
126	C₂×C₄×C₆₁	—	4	2	no	—
127	C₂×C₄×C₆₁	—	4	4	no	—
128	C₂×C₄×C₆₁	—	4	4	no	—
129	C₂×C₄×C₆₁	—	4	4	no	—
130	C₂×C₄×C₆₁	—	4	4	no	—
131	C₂×C₄×C₆₁	—	4	4	no	—
132	C₂×C₄×C₆₁	#121	1	2	yes	—
133	C₂×C₄×C₆₁	C₂×Q₁₆	1	4	yes	—
134	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
137	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
138	C₈×C₆₁	—	2	2	no	—
139	C₈×C₆₁	—	2	2	no	—
140	C₈×C₆₁	—	4	2	no	—
141	#29771	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂²×C₆₁	—	2	2	no	—
143	C₂²×C₆₁	—	2	4	no	—
144	C₂²×C₆₁	—	2	4	no	—
145	C₂²×C₆₁	—	2	2	no	—
146	C₂²×C₆₁	#467	1	4	yes	—
147	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
148	C₂²×C₆₁	#537	1	2	yes	—
149	C₂²×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
150	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
151	C₂²×C₆₁	#575	1	2	yes	—
152	C₄×C₆₁	—	2	2	no	—
153	C₄×C₆₁	—	2	4	no	—
154	C₄×C₆₁	—	4	2	no	—
155	C₄×C₆₁	—	8	2	no	—
156	C₄×C₆₁	—	8	4	no	—
157	C₄×C₆₁	—	8	2	no	—
158	C₄×C₆₁	#534	1	1	yes	characteristic
159	C₄×C₆₁	#536	1	1	yes	characteristic
160	C₄×C₆₁	#539	1	2	yes	—
161	C₂×#467	C₂×C₆₁	1	1	yes	characteristic
162	C₂²×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic
163	#2855	C₂×C₆₁	1	2	yes	—
164	#2087	C₂×C₆₁	1	2	yes	complemented
165	#2068	C₂×C₆₁	1	4	yes	complemented
166	#2056	C₂×C₆₁	1	2	yes	complemented
167	#1892	C₂×C₆₁	1	2	yes	—
168	#1808	C₂×C₆₁	1	1	yes	characteristic
169	C₂×C₆₁	—	2	4	no	—
170	C₂×C₆₁	#2618	1	1	yes	characteristic
171	C₂×C₆₁	C₂×#467	1	1	yes	characteristic
172	C₂×C₆₁	#3054	1	1	yes	characteristic
173	C₂×C₆₁	#3249	1	2	yes	—
174	C₂×C₆₁	#3250	1	2	yes	—
175	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
176	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
177	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
178	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
179	C₂²×C₄²	C₂²×C₆₁	1	1	yes	characteristic
180	#467	—	2	4	no	—
181	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
182	C₂×#107	C₂²×C₆₁	1	2	yes	—
183	C₂×C₄⋊C₈	—	2	4	no	—
184	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
185	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
186	C₂×#103	C₂²×C₆₁	1	2	yes	—
187	C₂×#103	C₂²×C₆₁	1	4	yes	—
188	#391	C₂²×C₆₁	1	4	yes	—
189	#389	C₂²×C₆₁	1	4	yes	—
190	#387	—	2	4	no	—
191	#385	—	2	2	no	—
192	#381	—	2	2	no	—
193	#380	C₂²×C₆₁	1	2	yes	—
194	#340	C₂²×C₆₁	1	2	yes	—
195	#340	C₂²×C₆₁	1	4	yes	—
196	C₆₁	#29771	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
198	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
199	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
200	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
201	C₂×Q₁₆	—	2	2	no	—
202	C₂×Q₁₆	—	4	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	4	2	no	—
205	C₂²×C₈	—	4	2	no	—
206	C₂²×C₈	D₄×C₆₁	1	4	yes	—
207	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
208	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
210	C₂²⋊Q₈	—	2	4	no	—
211	C₂²⋊Q₈	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	D₄×C₆₁	1	2	yes	—
221	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
222	#107	—	2	2	no	—
223	#107	—	4	2	no	—
224	C₄⋊C₈	—	2	4	no	—
225	C₄⋊C₈	—	2	4	no	—
226	#103	—	2	4	no	—
227	#103	—	4	4	no	—
228	#95	—	2	2	no	—
229	#95	—	2	4	no	—
230	#95	—	2	4	no	—
231	#95	—	2	4	no	—
232	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂²×C₄	—	2	2	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	4	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	Q₁₆×C₆₁	1	4	yes	complemented
247	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
251	C₂²×C₄	#41×C₆₁	1	2	yes	—
252	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
253	Q₁₆	—	4	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	4	4	no	—
259	C₂×C₈	—	4	4	no	—
260	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
261	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	8	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₂²⋊C₄	—	4	4	no	—
269	C₂²⋊C₄	—	4	4	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	#121×C₆₁	1	2	yes	—
276	C₂³	C₂×Q₁₆×C₆₁	1	2	yes	—
277	C₂³	#135×C₆₁	1	2	yes	—
278	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	Q₈	—	4	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	#121×C₆₁	1	2	yes	—
300	C₂×C₄	C₂×Q₁₆×C₆₁	1	4	yes	—
301	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
302	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
303	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
304	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
305	C₈	—	2	2	no	—
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	2	4	no	—
312	C₂²	C₆₁×#467	1	4	yes	—
313	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
314	C₂²	C₆₁×#537	1	2	yes	—
315	C₂²	C₂²×Q₁₆×C₆₁	1	1	yes	characteristic
316	C₂²	C₂×#135×C₆₁	1	1	yes	characteristic
317	C₂²	C₆₁×#575	1	2	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	4	no	—
324	C₄	C₆₁×#534	1	1	yes	characteristic
325	C₄	C₆₁×#536	1	1	yes	characteristic
326	C₄	C₆₁×#539	1	2	yes	—
327	C₂	—	2	4	no	—
328	C₂	C₆₁×#2618	1	1	yes	characteristic
329	C₂	C₂×C₆₁×#467	1	1	yes	characteristic
330	C₂	C₆₁×#3054	1	1	yes	characteristic
331	C₂	C₆₁×#3249	1	2	yes	—
332	C₂	C₆₁×#3250	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂⁴×C₆₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂²×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₁×#2087	C₂	semidirect
5. C₆₁×#2068	C₂	semidirect
6. C₆₁×#2056	C₂	semidirect
79. C₂²×C₄×C₆₁	Q₁₆	semidirect
141. #29771	C₆₁	direct (N × H)
164. #2087	C₂×C₆₁	semidirect
165. #2068	C₂×C₆₁	semidirect
166. #2056	C₂×C₆₁	semidirect
196. C₆₁	#29771	direct (N × H)
246. C₂²×C₄	Q₁₆×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#467	2
C₂²×C₄⋊C₈×C₆₁	2
C₆₁×#2855	2
C₆₁×#2087	2
C₆₁×#2068	2
C₆₁×#2056	2
C₆₁×#1892	2
C₆₁×#1808	2
#29771	61

Subgroups — C61×#29771 #29533204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29771 #29533204