Subgroups — C₆₁×#25623 #29529056

Order 15616.

1058 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3507	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3152	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#3139	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2005	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1986	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1980	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#1882	C₂	1	4	yes	maximal, complemented
9	C₄×#101×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₆₁	C₂²	1	4	yes	—
11	C₆₁×#559	—	2	1	no	—
12	C₆₁×#557	—	2	1	no	—
13	C₆₁×#556	—	2	1	no	—
14	C₆₁×#556	—	2	2	no	—
15	C₆₁×#554	—	2	1	no	—
16	C₆₁×#552	—	2	2	no	—
17	C₆₁×#539	—	2	2	no	—
18	C₆₁×#536	C₂²	1	2	yes	—
19	C₆₁×#533	C₂²	1	1	yes	characteristic
20	C₆₁×#532	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
22	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#529	—	2	2	no	—
24	C₆₁×#520	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#497	—	2	2	no	—
27	C₆₁×#494	—	2	1	no	—
28	C₆₁×#492	—	2	1	no	—
29	C₆₁×#491	—	2	2	no	—
30	C₆₁×#487	—	2	1	no	—
31	C₆₁×#487	—	2	1	no	—
32	C₂×#104×C₆₁	C₂²	1	4	yes	—
33	C₆₁×#419	C₂²	1	4	yes	—
34	C₂×#101×C₆₁	C₂²	1	2	yes	—
35	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
36	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
37	C₆₁×#398	—	2	2	no	—
38	C₆₁×#391	C₂²	1	2	yes	—
39	C₆₁×#389	—	2	2	no	—
40	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
41	C₆₁×#343	C₂²	1	2	yes	—
42	C₆₁×#339	C₂²	1	4	yes	—
43	C₂×#41×C₆₁	C₂³	1	2	yes	—
44	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
45	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
46	#137×C₆₁	—	2	8	no	—
47	#137×C₆₁	—	4	4	no	—
48	C₂×Q₁₆×C₆₁	—	2	4	no	—
49	C₂×SD₁₆×C₆₁	—	2	4	no	—
50	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
51	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
52	C₄⋊Q₈×C₆₁	—	2	2	no	—
53	C₄⋊Q₈×C₆₁	—	2	1	no	—
54	C₄⋊Q₈×C₆₁	—	2	1	no	—
55	C₄⋊Q₈×C₆₁	—	2	1	no	—
56	C₄⋊Q₈×C₆₁	—	2	1	no	—
57	C₄⋊Q₈×C₆₁	—	4	4	no	—
58	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
59	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
60	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
61	#127×C₆₁	D₄	1	1	yes	characteristic
62	#126×C₆₁	—	2	4	no	—
63	#126×C₆₁	—	2	4	no	—
64	#125×C₆₁	—	2	4	no	—
65	#125×C₆₁	—	2	2	no	—
66	#125×C₆₁	D₄	1	2	yes	—
67	#124×C₆₁	—	2	2	no	—
68	#124×C₆₁	—	2	2	no	—
69	#124×C₆₁	D₄	1	2	yes	—
70	#123×C₆₁	—	2	4	no	—
71	#123×C₆₁	—	2	4	no	—
72	C₂²⋊Q₈×C₆₁	—	2	4	no	—
73	C₂²⋊Q₈×C₆₁	—	2	1	no	—
74	C₂²⋊Q₈×C₆₁	—	2	1	no	—
75	C₂²⋊Q₈×C₆₁	—	2	2	no	—
76	C₂²⋊Q₈×C₆₁	—	2	2	no	—
77	#121×C₆₁	—	2	1	no	—
78	#121×C₆₁	—	2	1	no	—
79	C₄×Q₈×C₆₁	—	2	2	no	—
80	C₄×Q₈×C₆₁	—	2	2	no	—
81	C₄×Q₈×C₆₁	—	2	1	no	—
82	C₄×Q₈×C₆₁	—	2	1	no	—
83	C₄×Q₈×C₆₁	—	2	1	no	—
84	C₄×Q₈×C₆₁	—	2	1	no	—
85	C₄×Q₈×C₆₁	—	2	1	no	—
86	C₄×Q₈×C₆₁	—	2	1	no	—
87	C₄×D₄×C₆₁	—	2	2	no	—
88	C₄×D₄×C₆₁	—	2	2	no	—
89	C₄×D₄×C₆₁	—	2	4	no	—
90	C₄×D₄×C₆₁	—	2	1	no	—
91	C₄×D₄×C₆₁	—	2	1	no	—
92	#117×C₆₁	—	2	4	no	—
93	#117×C₆₁	—	2	2	no	—
94	#117×C₆₁	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
96	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
99	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
100	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₆₁	D₄	1	2	yes	—
103	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
104	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
105	#104×C₆₁	—	4	4	no	—
106	#103×C₆₁	—	2	4	no	—
107	#103×C₆₁	—	2	4	no	—
108	#101×C₆₁	—	2	4	no	—
109	#97×C₆₁	—	2	1	no	—
110	#97×C₆₁	—	2	1	no	—
111	C₄×C₈×C₆₁	—	2	1	no	—
112	C₄×C₈×C₆₁	—	2	1	no	—
113	#95×C₆₁	—	2	1	no	—
114	#95×C₆₁	—	2	4	no	—
115	#41×C₆₁	—	2	4	no	—
116	#41×C₆₁	—	2	4	no	—
117	C₂×Q₈×C₆₁	—	2	1	no	—
118	C₂×Q₈×C₆₁	—	2	1	no	—
119	C₂×Q₈×C₆₁	—	2	1	no	—
120	C₂×Q₈×C₆₁	—	2	1	no	—
121	C₂×Q₈×C₆₁	—	2	2	no	—
122	C₂×Q₈×C₆₁	—	2	2	no	—
123	C₂×Q₈×C₆₁	—	2	2	no	—
124	C₂×Q₈×C₆₁	—	2	2	no	—
125	C₂×Q₈×C₆₁	—	2	4	no	—
126	C₂×Q₈×C₆₁	—	2	4	no	—
127	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
128	C₂×D₄×C₆₁	—	2	2	no	—
129	C₂×D₄×C₆₁	—	2	1	no	—
130	C₂×D₄×C₆₁	—	2	1	no	—
131	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₆₁	—	2	2	no	—
133	C₂²×C₄×C₆₁	—	2	4	no	—
134	C₂²×C₄×C₆₁	—	2	2	no	—
135	C₂²×C₄×C₆₁	—	2	2	no	—
136	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
141	Q₁₆×C₆₁	—	4	8	no	—
142	SD₁₆×C₆₁	—	4	8	no	—
143	M₁₆×C₆₁	—	2	2	no	—
144	M₁₆×C₆₁	—	2	2	no	—
145	M₁₆×C₆₁	—	2	2	no	—
146	M₁₆×C₆₁	#41	1	4	yes	—
147	C₂×C₈×C₆₁	—	2	2	no	—
148	C₂×C₈×C₆₁	—	2	2	no	—
149	C₄⋊C₄×C₆₁	—	2	4	no	—
150	C₄⋊C₄×C₆₁	—	2	4	no	—
151	C₄⋊C₄×C₆₁	—	2	4	no	—
152	C₄⋊C₄×C₆₁	—	2	4	no	—
153	C₄⋊C₄×C₆₁	—	2	2	no	—
154	C₄⋊C₄×C₆₁	—	2	2	no	—
155	C₄⋊C₄×C₆₁	—	2	2	no	—
156	C₄⋊C₄×C₆₁	—	2	1	no	—
157	C₄⋊C₄×C₆₁	—	2	1	no	—
158	C₄⋊C₄×C₆₁	—	2	1	no	—
159	C₄⋊C₄×C₆₁	—	2	2	no	—
160	C₄⋊C₄×C₆₁	—	2	1	no	—
161	C₄⋊C₄×C₆₁	—	2	1	no	—
162	C₄⋊C₄×C₆₁	—	2	1	no	—
163	C₄⋊C₄×C₆₁	—	4	8	no	—
164	C₄⋊C₄×C₆₁	—	4	1	no	—
165	C₄⋊C₄×C₆₁	—	4	1	no	—
166	C₄⋊C₄×C₆₁	—	4	1	no	—
167	C₄⋊C₄×C₆₁	—	4	1	no	—
168	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
169	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
170	C₂²⋊C₄×C₆₁	—	2	4	no	—
171	C₂²⋊C₄×C₆₁	—	2	4	no	—
172	C₂²⋊C₄×C₆₁	—	2	4	no	—
173	C₂²⋊C₄×C₆₁	—	2	2	no	—
174	C₂²⋊C₄×C₆₁	—	2	2	no	—
175	C₂²⋊C₄×C₆₁	—	2	2	no	—
176	C₄²×C₆₁	—	2	4	no	—
177	C₄²×C₆₁	—	2	2	no	—
178	C₄²×C₆₁	—	2	2	no	—
179	C₄²×C₆₁	—	2	2	no	—
180	C₄²×C₆₁	—	2	4	no	—
181	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
185	C₂³×C₆₁	—	2	2	no	—
186	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
187	Q₈×C₆₁	—	2	4	no	—
188	Q₈×C₆₁	—	4	2	no	—
189	Q₈×C₆₁	—	4	2	no	—
190	Q₈×C₆₁	—	4	2	no	—
191	Q₈×C₆₁	—	4	4	no	—
192	Q₈×C₆₁	—	4	4	no	—
193	Q₈×C₆₁	—	8	1	no	—
194	D₄×C₆₁	—	2	4	no	—
195	D₄×C₆₁	—	4	2	no	—
196	D₄×C₆₁	—	4	2	no	—
197	D₄×C₆₁	—	4	2	no	—
198	D₄×C₆₁	—	8	1	no	—
199	C₂×C₄×C₆₁	—	2	2	no	—
200	C₂×C₄×C₆₁	—	2	4	no	—
201	C₂×C₄×C₆₁	—	2	2	no	—
202	C₂×C₄×C₆₁	—	2	2	no	—
203	C₂×C₄×C₆₁	—	2	2	no	—
204	C₂×C₄×C₆₁	—	2	2	no	—
205	C₂×C₄×C₆₁	—	2	2	no	—
206	C₂×C₄×C₆₁	—	2	2	no	—
207	C₂×C₄×C₆₁	—	2	2	no	—
208	C₂×C₄×C₆₁	—	4	1	no	—
209	C₂×C₄×C₆₁	—	4	2	no	—
210	C₂×C₄×C₆₁	—	4	2	no	—
211	C₂×C₄×C₆₁	—	4	4	no	—
212	C₂×C₄×C₆₁	—	4	2	no	—
213	C₂×C₄×C₆₁	—	4	2	no	—
214	C₂×C₄×C₆₁	—	4	2	no	—
215	C₂×C₄×C₆₁	—	4	2	no	—
216	C₂×C₄×C₆₁	—	8	4	no	—
217	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
218	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
219	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
220	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
221	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
224	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
225	C₈×C₆₁	—	2	4	no	—
226	C₈×C₆₁	—	4	2	no	—
227	#25623	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂²×C₆₁	—	2	1	no	—
229	C₂²×C₆₁	—	2	1	no	—
230	C₂²×C₆₁	—	4	2	no	—
231	C₂²×C₆₁	—	4	2	no	—
232	C₂²×C₆₁	—	4	2	no	—
233	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
234	C₂²×C₆₁	#532	1	1	yes	characteristic
235	C₂²×C₆₁	#535	1	1	yes	characteristic
236	C₄×C₆₁	—	2	1	no	—
237	C₄×C₆₁	—	2	1	no	—
238	C₄×C₆₁	—	2	1	no	—
239	C₄×C₆₁	—	4	4	no	—
240	C₄×C₆₁	—	4	2	no	—
241	C₄×C₆₁	—	4	2	no	—
242	C₄×C₆₁	—	4	2	no	—
243	C₄×C₆₁	—	4	2	no	—
244	C₄×C₆₁	—	4	2	no	—
245	C₄×C₆₁	—	8	2	no	—
246	C₄×C₆₁	#456	1	2	yes	—
247	C₄×C₆₁	#546	1	2	yes	—
248	C₄×C₆₁	#550	1	2	yes	—
249	#3507	C₂×C₆₁	1	1	yes	characteristic
250	#3152	C₂×C₆₁	1	1	yes	characteristic
251	#3139	C₂×C₆₁	1	1	yes	characteristic, complemented
252	#2546	C₂×C₆₁	1	1	yes	characteristic, complemented
253	#2005	C₂×C₆₁	1	4	yes	complemented
254	#1986	C₂×C₆₁	1	1	yes	characteristic, complemented
255	#1980	C₂×C₆₁	1	1	yes	characteristic
256	#1882	C₂×C₆₁	1	4	yes	complemented
257	C₄×#101	C₂×C₆₁	1	1	yes	characteristic, complemented
258	C₂×C₆₁	—	2	1	no	—
259	C₂×C₆₁	—	2	1	no	—
260	C₂×C₆₁	—	8	2	no	—
261	C₂×C₆₁	#2431	1	1	yes	characteristic
262	C₂×C₆₁	C₂×#456	1	1	yes	characteristic
263	C₂×C₆₁	#3022	1	1	yes	characteristic
264	C₂×#137	C₂²×C₆₁	1	4	yes	—
265	#559	—	2	1	no	—
266	#557	—	2	1	no	—
267	#556	—	2	1	no	—
268	#556	—	2	2	no	—
269	#554	—	2	1	no	—
270	#552	—	2	2	no	—
271	#539	—	2	2	no	—
272	#536	C₂²×C₆₁	1	2	yes	—
273	#533	C₂²×C₆₁	1	1	yes	characteristic
274	#532	C₂²×C₆₁	1	1	yes	characteristic
275	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
276	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
277	#529	—	2	2	no	—
278	#520	C₂²×C₆₁	1	2	yes	—
279	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
280	#497	—	2	2	no	—
281	#494	—	2	1	no	—
282	#492	—	2	1	no	—
283	#491	—	2	2	no	—
284	#487	—	2	1	no	—
285	#487	—	2	1	no	—
286	C₂×#104	C₂²×C₆₁	1	4	yes	—
287	#419	C₂²×C₆₁	1	4	yes	—
288	C₂×#101	C₂²×C₆₁	1	2	yes	—
289	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
290	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
291	#398	—	2	2	no	—
292	#391	C₂²×C₆₁	1	2	yes	—
293	#389	—	2	2	no	—
294	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
295	#343	C₂²×C₆₁	1	2	yes	—
296	#339	C₂²×C₆₁	1	4	yes	—
297	C₆₁	#25623	1	1	yes	characteristic, direct product factor, Sylow
298	C₂×#41	C₂³×C₆₁	1	2	yes	—
299	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
300	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
301	#137	—	2	8	no	—
302	#137	—	4	4	no	—
303	C₂×Q₁₆	—	2	4	no	—
304	C₂×SD₁₆	—	2	4	no	—
305	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
306	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
307	C₄⋊Q₈	—	2	2	no	—
308	C₄⋊Q₈	—	2	1	no	—
309	C₄⋊Q₈	—	2	1	no	—
310	C₄⋊Q₈	—	2	1	no	—
311	C₄⋊Q₈	—	2	1	no	—
312	C₄⋊Q₈	—	4	4	no	—
313	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
314	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
315	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
316	#127	D₄×C₆₁	1	1	yes	characteristic
317	#126	—	2	4	no	—
318	#126	—	2	4	no	—
319	#125	—	2	2	no	—
320	#125	—	2	4	no	—
321	#125	D₄×C₆₁	1	2	yes	—
322	#124	—	2	2	no	—
323	#124	—	2	2	no	—
324	#124	D₄×C₆₁	1	2	yes	—
325	#123	—	2	4	no	—
326	#123	—	2	4	no	—
327	C₂²⋊Q₈	—	2	4	no	—
328	C₂²⋊Q₈	—	2	1	no	—
329	C₂²⋊Q₈	—	2	1	no	—
330	C₂²⋊Q₈	—	2	2	no	—
331	C₂²⋊Q₈	—	2	2	no	—
332	#121	—	2	1	no	—
333	#121	—	2	1	no	—
334	C₄×Q₈	—	2	2	no	—
335	C₄×Q₈	—	2	2	no	—
336	C₄×Q₈	—	2	1	no	—
337	C₄×Q₈	—	2	1	no	—
338	C₄×Q₈	—	2	1	no	—
339	C₄×Q₈	—	2	1	no	—
340	C₄×Q₈	—	2	1	no	—
341	C₄×Q₈	—	2	1	no	—
342	C₄×D₄	—	2	2	no	—
343	C₄×D₄	—	2	2	no	—
344	C₄×D₄	—	2	4	no	—
345	C₄×D₄	—	2	1	no	—
346	C₄×D₄	—	2	1	no	—
347	#117	—	2	4	no	—
348	#117	—	2	2	no	—
349	#117	C₂³×C₆₁	1	2	yes	—
350	C₂×C₄⋊C₄	—	2	1	no	—
351	C₂×C₄⋊C₄	—	2	2	no	—
352	C₂×C₄⋊C₄	—	2	1	no	—
353	C₂×C₄⋊C₄	—	2	4	no	—
354	C₂×C₄⋊C₄	—	2	1	no	—
355	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
356	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
357	C₂×C₄²	D₄×C₆₁	1	2	yes	—
358	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
359	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
360	#104	—	4	4	no	—
361	#103	—	2	4	no	—
362	#103	—	2	4	no	—
363	#101	—	2	4	no	—
364	#97	—	2	1	no	—
365	#97	—	2	1	no	—
366	C₄×C₈	—	2	1	no	—
367	C₄×C₈	—	2	1	no	—
368	#95	—	2	1	no	—
369	#95	—	2	4	no	—
370	#41	—	2	4	no	—
371	#41	—	2	4	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	—	2	1	no	—
375	C₂×Q₈	—	2	1	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	2	2	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	2	4	no	—
382	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
394	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
395	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
396	Q₁₆	—	4	8	no	—
397	SD₁₆	—	4	8	no	—
398	M₁₆	—	2	2	no	—
399	M₁₆	—	2	2	no	—
400	M₁₆	—	2	2	no	—
401	M₁₆	#41×C₆₁	1	4	yes	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	4	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	4	8	no	—
419	C₄⋊C₄	—	4	1	no	—
420	C₄⋊C₄	—	4	1	no	—
421	C₄⋊C₄	—	4	1	no	—
422	C₄⋊C₄	—	4	1	no	—
423	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
424	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
425	C₂²⋊C₄	—	2	4	no	—
426	C₂²⋊C₄	—	2	4	no	—
427	C₂²⋊C₄	—	2	4	no	—
428	C₂²⋊C₄	—	2	2	no	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₂²⋊C₄	—	2	2	no	—
431	C₄²	—	2	4	no	—
432	C₄²	—	2	2	no	—
433	C₄²	—	2	2	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	2	4	no	—
436	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
437	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
438	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
439	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
440	C₂³	—	2	2	no	—
441	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
442	Q₈	—	2	4	no	—
443	Q₈	—	4	2	no	—
444	Q₈	—	4	2	no	—
445	Q₈	—	4	2	no	—
446	Q₈	—	4	4	no	—
447	Q₈	—	4	4	no	—
448	Q₈	—	8	1	no	—
449	D₄	—	2	4	no	—
450	D₄	—	4	2	no	—
451	D₄	—	4	2	no	—
452	D₄	—	4	2	no	—
453	D₄	—	8	1	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	4	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	8	4	no	—
472	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
473	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
474	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
475	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
478	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
479	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
480	C₈	—	2	4	no	—
481	C₈	—	4	2	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
488	C₂²	C₆₁×#532	1	1	yes	characteristic
489	C₂²	C₆₁×#535	1	1	yes	characteristic
490	C₄	—	2	1	no	—
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	4	4	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	2	no	—
499	C₄	—	8	2	no	—
500	C₄	C₆₁×#456	1	2	yes	—
501	C₄	C₆₁×#546	1	2	yes	—
502	C₄	C₆₁×#550	1	2	yes	—
503	C₂	—	2	1	no	—
504	C₂	—	2	1	no	—
505	C₂	—	8	2	no	—
506	C₂	C₆₁×#2431	1	1	yes	characteristic
507	C₂	C₂×C₆₁×#456	1	1	yes	characteristic
508	C₂	C₆₁×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₆₁×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#3139	C₂	semidirect
4. C₆₁×#2546	C₂	semidirect
5. C₆₁×#2005	C₂	semidirect
6. C₆₁×#1986	C₂	semidirect
8. C₆₁×#1882	C₂	semidirect
9. C₄×#101×C₆₁	C₂	semidirect
227. #25623	C₆₁	direct (N × H)
251. #3139	C₂×C₆₁	semidirect
252. #2546	C₂×C₆₁	semidirect
253. #2005	C₂×C₆₁	semidirect
254. #1986	C₂×C₆₁	semidirect
256. #1882	C₂×C₆₁	semidirect
257. C₄×#101	C₂×C₆₁	semidirect
297. C₆₁	#25623	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#3507	2
C₆₁×#3152	2
C₆₁×#3139	2
C₆₁×#2546	2
C₆₁×#2005	2
C₆₁×#1986	2
C₆₁×#1980	2
C₆₁×#1882	2
C₄×#101×C₆₁	2
#25623	61

Subgroups — C61×#25623 #29529056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25623 #29529056