Subgroups — C₆₁×#29539 #29532972

Order 15616.

688 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2295	C₂	1	2	yes	maximal
2	C₆₁×#2087	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2068	C₂	1	2	yes	maximal
4	C₆₁×#2056	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1928	C₂	1	2	yes	maximal
6	C₆₁×#1921	C₂	1	2	yes	maximal
7	C₆₁×#1878	C₂	1	2	yes	maximal
8	C₆₁×#1863	C₂	1	2	yes	maximal
9	C₆₁×#1831	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
12	C₂×#107×C₆₁	C₂²	1	2	yes	—
13	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#103×C₆₁	C₂²	1	2	yes	—
17	C₂×#103×C₆₁	C₂²	1	2	yes	—
18	C₂×#103×C₆₁	C₂²	1	2	yes	—
19	C₆₁×#391	C₂²	1	2	yes	—
20	C₆₁×#391	C₂²	1	2	yes	—
21	C₆₁×#389	C₂²	1	2	yes	—
22	C₆₁×#389	C₂²	1	2	yes	—
23	C₆₁×#387	—	2	2	no	—
24	C₆₁×#387	—	2	2	no	—
25	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
26	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
27	C₆₁×#376	—	2	2	no	—
28	C₄³×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#340	C₂²	1	1	yes	characteristic
30	C₆₁×#340	C₂²	1	2	yes	—
31	C₆₁×#340	C₂²	1	1	yes	characteristic
32	C₆₁×#340	C₂²	1	2	yes	—
33	C₆₁×#336	C₂²	1	2	yes	—
34	C₆₁×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
36	C₂×Q₁₆×C₆₁	—	2	4	no	—
37	C₂×Q₁₆×C₆₁	—	4	1	no	—
38	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
41	C₄×Q₈×C₆₁	—	2	4	no	—
42	C₄×Q₈×C₆₁	—	2	4	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	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
49	C₂×C₄²×C₆₁	—	2	2	no	—
50	C₂×C₄²×C₆₁	—	2	2	no	—
51	C₂×C₄²×C₆₁	D₄	1	2	yes	—
52	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
56	#107×C₆₁	—	2	4	no	—
57	#107×C₆₁	—	4	1	no	—
58	C₄⋊C₈×C₆₁	—	4	1	no	—
59	C₄⋊C₈×C₆₁	—	4	1	no	—
60	#103×C₆₁	—	2	4	no	—
61	#103×C₆₁	—	2	4	no	—
62	#103×C₆₁	—	4	2	no	—
63	#95×C₆₁	—	2	2	no	—
64	#95×C₆₁	—	2	2	no	—
65	#95×C₆₁	—	2	2	no	—
66	#95×C₆₁	—	2	2	no	—
67	#95×C₆₁	—	2	2	no	—
68	#95×C₆₁	—	2	2	no	—
69	#95×C₆₁	—	2	2	no	—
70	C₂×Q₈×C₆₁	—	2	2	no	—
71	C₂×Q₈×C₆₁	—	2	2	no	—
72	C₂×Q₈×C₆₁	—	2	2	no	—
73	C₂×Q₈×C₆₁	—	2	2	no	—
74	C₂×Q₈×C₆₁	—	2	2	no	—
75	C₂×Q₈×C₆₁	#41	1	4	yes	—
76	C₂²×C₄×C₆₁	—	2	2	no	—
77	C₂²×C₄×C₆₁	—	2	2	no	—
78	C₂²×C₄×C₆₁	—	2	2	no	—
79	C₂²×C₄×C₆₁	—	2	2	no	—
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	1	yes	characteristic
83	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
85	Q₁₆×C₆₁	—	4	4	no	—
86	C₂×C₈×C₆₁	—	2	2	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	2	no	—
91	C₂×C₈×C₆₁	—	2	1	no	—
92	C₂×C₈×C₆₁	—	2	1	no	—
93	C₄⋊C₄×C₆₁	—	2	8	no	—
94	C₄⋊C₄×C₆₁	—	2	2	no	—
95	C₄⋊C₄×C₆₁	—	2	2	no	—
96	C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₄⋊C₄×C₆₁	—	4	4	no	—
100	C₄⋊C₄×C₆₁	—	4	4	no	—
101	C₄⋊C₄×C₆₁	#41	1	4	yes	—
102	C₄²×C₆₁	—	2	8	no	—
103	C₄²×C₆₁	—	2	2	no	—
104	C₄²×C₆₁	—	2	2	no	—
105	C₄²×C₆₁	—	2	1	no	—
106	C₄²×C₆₁	—	2	1	no	—
107	C₄²×C₆₁	—	4	2	no	—
108	C₄²×C₆₁	—	4	1	no	—
109	C₄²×C₆₁	—	4	4	no	—
110	C₄²×C₆₁	Q₁₆	1	4	yes	complemented
111	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₆₁	—	2	4	no	—
113	Q₈×C₆₁	—	2	4	no	—
114	Q₈×C₆₁	—	4	2	no	—
115	Q₈×C₆₁	—	4	2	no	—
116	C₂×C₄×C₆₁	—	2	2	no	—
117	C₂×C₄×C₆₁	—	2	1	no	—
118	C₂×C₄×C₆₁	—	2	2	no	—
119	C₂×C₄×C₆₁	—	2	1	no	—
120	C₂×C₄×C₆₁	—	2	2	no	—
121	C₂×C₄×C₆₁	—	2	4	no	—
122	C₂×C₄×C₆₁	—	2	4	no	—
123	C₂×C₄×C₆₁	—	2	1	no	—
124	C₂×C₄×C₆₁	—	2	2	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	2	no	—
127	C₂×C₄×C₆₁	—	4	2	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	4	2	no	—
130	C₂×C₄×C₆₁	—	4	2	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	—	4	2	no	—
133	C₂×C₄×C₆₁	—	4	2	no	—
134	C₂×C₄×C₆₁	—	4	2	no	—
135	C₂×C₄×C₆₁	—	4	2	no	—
136	C₂×C₄×C₆₁	—	4	2	no	—
137	C₂×C₄×C₆₁	C₂×Q₁₆	1	4	yes	—
138	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
139	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
140	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
141	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
143	C₈×C₆₁	—	4	1	no	—
144	C₈×C₆₁	—	4	1	no	—
145	C₈×C₆₁	—	4	2	no	—
146	#29539	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂²×C₆₁	#525	1	2	yes	—
148	C₂²×C₆₁	#537	1	1	yes	characteristic
149	C₂²×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
150	C₂²×C₆₁	#575	1	2	yes	—
151	C₂²×C₆₁	#575	1	1	yes	characteristic
152	C₄×C₆₁	—	2	4	no	—
153	C₄×C₆₁	—	2	2	no	—
154	C₄×C₆₁	—	4	2	no	—
155	C₄×C₆₁	—	4	4	no	—
156	C₄×C₆₁	—	4	4	no	—
157	C₄×C₆₁	—	4	1	no	—
158	C₄×C₆₁	—	8	2	no	—
159	C₄×C₆₁	—	8	2	no	—
160	C₄×C₆₁	#551	1	1	yes	characteristic
161	C₄×C₆₁	#552	1	2	yes	—
162	C₄×C₆₁	#555	1	1	yes	characteristic
163	#2295	C₂×C₆₁	1	2	yes	—
164	#2087	C₂×C₆₁	1	1	yes	characteristic
165	#2068	C₂×C₆₁	1	2	yes	—
166	#2056	C₂×C₆₁	1	1	yes	characteristic
167	#1928	C₂×C₆₁	1	2	yes	—
168	#1921	C₂×C₆₁	1	2	yes	—
169	#1878	C₂×C₆₁	1	2	yes	—
170	#1863	C₂×C₆₁	1	2	yes	—
171	#1831	C₂×C₆₁	1	1	yes	characteristic
172	C₂×C₆₁	#2609	1	1	yes	characteristic
173	C₂×C₆₁	#3104	1	2	yes	—
174	C₂×C₆₁	#3111	1	2	yes	—
175	C₂×C₆₁	#3249	1	1	yes	characteristic
176	C₂×C₆₁	#3256	1	1	yes	characteristic
177	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
178	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
179	C₂×#107	C₂²×C₆₁	1	2	yes	—
180	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
181	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
182	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
183	C₂×#103	C₂²×C₆₁	1	2	yes	—
184	C₂×#103	C₂²×C₆₁	1	2	yes	—
185	C₂×#103	C₂²×C₆₁	1	2	yes	—
186	#391	C₂²×C₆₁	1	2	yes	—
187	#391	C₂²×C₆₁	1	2	yes	—
188	#389	C₂²×C₆₁	1	2	yes	—
189	#389	C₂²×C₆₁	1	2	yes	—
190	#387	—	2	2	no	—
191	#387	—	2	2	no	—
192	C₄×C₄⋊C₄	—	2	2	no	—
193	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
194	#376	—	2	2	no	—
195	C₄³	C₂²×C₆₁	1	1	yes	characteristic
196	#340	C₂²×C₆₁	1	1	yes	characteristic
197	#340	C₂²×C₆₁	1	2	yes	—
198	#340	C₂²×C₆₁	1	1	yes	characteristic
199	#340	C₂²×C₆₁	1	2	yes	—
200	#336	C₂²×C₆₁	1	2	yes	—
201	#336	C₂²×C₆₁	1	2	yes	—
202	C₆₁	#29539	1	1	yes	characteristic, direct product factor, Sylow
203	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
204	C₂×Q₁₆	—	2	4	no	—
205	C₂×Q₁₆	—	4	1	no	—
206	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
209	C₄×Q₈	—	2	4	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	D₄×C₆₁	1	2	yes	—
220	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
221	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
223	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
224	#107	—	2	4	no	—
225	#107	—	4	1	no	—
226	C₄⋊C₈	—	4	1	no	—
227	C₄⋊C₈	—	4	1	no	—
228	#103	—	2	4	no	—
229	#103	—	2	4	no	—
230	#103	—	4	2	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	#41×C₆₁	1	4	yes	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
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	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×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	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₄⋊C₄	—	2	8	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	#41×C₆₁	1	4	yes	—
270	C₄²	—	2	8	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	4	2	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	4	no	—
278	C₄²	Q₁₆×C₆₁	1	4	yes	complemented
279	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
280	Q₈	—	2	4	no	—
281	Q₈	—	2	4	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	C₂×Q₁₆×C₆₁	1	4	yes	—
306	C₂×C₄	C₂×Q₁₆×C₆₁	1	2	yes	—
307	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
308	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
309	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
310	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
311	C₈	—	4	1	no	—
312	C₈	—	4	1	no	—
313	C₈	—	4	2	no	—
314	C₂²	C₆₁×#525	1	2	yes	—
315	C₂²	C₆₁×#537	1	1	yes	characteristic
316	C₂²	C₂²×Q₁₆×C₆₁	1	1	yes	characteristic
317	C₂²	C₆₁×#575	1	2	yes	—
318	C₂²	C₆₁×#575	1	1	yes	characteristic
319	C₄	—	2	4	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	4	4	no	—
323	C₄	—	4	4	no	—
324	C₄	—	4	1	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	C₆₁×#551	1	1	yes	characteristic
328	C₄	C₆₁×#552	1	2	yes	—
329	C₄	C₆₁×#555	1	1	yes	characteristic
330	C₂	C₆₁×#2609	1	1	yes	characteristic
331	C₂	C₆₁×#3104	1	2	yes	—
332	C₂	C₆₁×#3111	1	2	yes	—
333	C₂	C₆₁×#3249	1	1	yes	characteristic
334	C₂	C₆₁×#3256	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
110. C₄²×C₆₁	Q₁₆	semidirect
146. #29539	C₆₁	direct (N × H)
202. C₆₁	#29539	direct (N × H)
278. C₄²	Q₁₆×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#2295	2
C₆₁×#2087	2
C₆₁×#2068	2
C₆₁×#2056	2
C₆₁×#1928	2
C₆₁×#1921	2
C₆₁×#1878	2
C₆₁×#1863	2
C₆₁×#1831	2
#29539	61

Subgroups — C61×#29539 #29532972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29539 #29532972