Subgroups — C₆₁×#25596 #29529029

Order 15616.

1064 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3510	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3139	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3039	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2658	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2005	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2004	C₂	1	2	yes	maximal
7	C₆₁×#1995	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1967	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1886	C₂	1	2	yes	maximal, complemented
10	C₆₁×#1885	C₂	1	2	yes	maximal, complemented
11	C₆₁×#1759	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#559	—	2	2	no	—
14	C₆₁×#556	—	2	2	no	—
15	C₆₁×#551	—	2	4	no	—
16	C₆₁×#542	—	2	2	no	—
17	C₆₁×#536	C₂²	1	1	yes	characteristic
18	C₆₁×#535	C₂²	1	1	yes	characteristic
19	C₆₁×#533	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
21	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#529	—	2	2	no	—
23	C₂×#124×C₆₁	C₂²	1	2	yes	—
24	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic
25	C₆₁×#519	C₂²	1	1	yes	characteristic
26	C₆₁×#518	C₂²	1	1	yes	characteristic
27	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#491	—	2	2	no	—
29	C₆₁×#487	—	2	2	no	—
30	C₆₁×#462	—	2	2	no	—
31	C₆₁×#461	—	2	2	no	—
32	C₆₁×#423	C₂²	1	1	yes	characteristic
33	C₂×#104×C₆₁	C₂²	1	2	yes	—
34	C₆₁×#419	C₂²	1	2	yes	—
35	C₆₁×#419	C₂²	1	2	yes	—
36	C₆₁×#419	C₂²	1	2	yes	—
37	C₂×#101×C₆₁	C₂²	1	2	yes	—
38	C₂×#100×C₆₁	C₂²	1	2	yes	—
39	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
40	C₆₁×#399	—	2	4	no	—
41	C₆₁×#376	C₂²	1	1	yes	characteristic
42	C₆₁×#339	C₂²	1	2	yes	—
43	C₆₁×#339	C₂²	1	2	yes	—
44	C₆₁×#339	C₂²	1	2	yes	—
45	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
47	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
49	#137×C₆₁	—	4	2	no	—
50	#137×C₆₁	—	4	2	no	—
51	C₂×Q₁₆×C₆₁	—	2	2	no	—
52	C₂×SD₁₆×C₆₁	—	2	2	no	—
53	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
54	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
55	C₄⋊Q₈×C₆₁	—	2	1	no	—
56	C₄⋊Q₈×C₆₁	—	2	2	no	—
57	C₄⋊Q₈×C₆₁	—	2	1	no	—
58	C₄⋊Q₈×C₆₁	—	2	1	no	—
59	C₄⋊Q₈×C₆₁	—	2	4	no	—
60	C₄⋊Q₈×C₆₁	—	4	2	no	—
61	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
62	#127×C₆₁	—	2	1	no	—
63	#126×C₆₁	—	2	4	no	—
64	#126×C₆₁	—	2	4	no	—
65	#125×C₆₁	D₄	1	2	yes	—
66	#124×C₆₁	—	2	4	no	—
67	#124×C₆₁	—	2	1	no	—
68	#124×C₆₁	—	2	1	no	—
69	#124×C₆₁	—	2	1	no	—
70	#124×C₆₁	—	2	2	no	—
71	#124×C₆₁	—	2	1	no	—
72	#124×C₆₁	—	2	4	no	—
73	#124×C₆₁	—	4	2	no	—
74	#123×C₆₁	—	2	4	no	—
75	#123×C₆₁	—	2	4	no	—
76	#123×C₆₁	—	2	2	no	—
77	C₂²⋊Q₈×C₆₁	—	2	4	no	—
78	C₂²⋊Q₈×C₆₁	—	2	2	no	—
79	C₂²⋊Q₈×C₆₁	—	2	2	no	—
80	#121×C₆₁	—	2	2	no	—
81	#120×C₆₁	—	2	2	no	—
82	C₄×Q₈×C₆₁	—	2	2	no	—
83	C₄×Q₈×C₆₁	—	2	2	no	—
84	C₄×Q₈×C₆₁	—	2	2	no	—
85	C₄×Q₈×C₆₁	D₄	1	2	yes	—
86	C₄×D₄×C₆₁	—	2	4	no	—
87	C₄×D₄×C₆₁	—	2	2	no	—
88	C₄×D₄×C₆₁	—	2	2	no	—
89	C₄×D₄×C₆₁	—	2	2	no	—
90	C₄×D₄×C₆₁	D₄	1	2	yes	—
91	#117×C₆₁	—	2	4	no	—
92	#117×C₆₁	—	2	2	no	—
93	#117×C₆₁	—	2	1	no	—
94	#117×C₆₁	C₂³	1	1	yes	characteristic
95	#117×C₆₁	C₂³	1	1	yes	characteristic
96	#117×C₆₁	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
98	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
100	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
101	C₂×C₂²⋊C₄×C₆₁	D₄	1	4	yes	—
102	C₂×C₄²×C₆₁	—	2	1	no	—
103	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
104	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₈×C₆₁	—	2	2	no	—
106	#104×C₆₁	—	2	4	no	—
107	#103×C₆₁	—	2	4	no	—
108	#103×C₆₁	—	2	4	no	—
109	#103×C₆₁	—	2	4	no	—
110	#101×C₆₁	—	4	2	no	—
111	#100×C₆₁	—	4	2	no	—
112	#97×C₆₁	—	2	1	no	—
113	C₄×C₈×C₆₁	—	2	1	no	—
114	#95×C₆₁	—	2	2	no	—
115	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
116	#41×C₆₁	—	2	2	no	—
117	#41×C₆₁	—	2	2	no	—
118	C₂×Q₈×C₆₁	—	2	2	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	2	no	—
126	C₂×Q₈×C₆₁	—	2	2	no	—
127	C₂×Q₈×C₆₁	—	2	2	no	—
128	C₂×Q₈×C₆₁	—	2	2	no	—
129	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
130	C₂×D₄×C₆₁	—	2	1	no	—
131	C₂×D₄×C₆₁	—	2	4	no	—
132	C₂×D₄×C₆₁	—	2	1	no	—
133	C₂×D₄×C₆₁	—	2	1	no	—
134	C₂×D₄×C₆₁	—	2	2	no	—
135	C₂×D₄×C₆₁	—	2	2	no	—
136	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₁	—	2	2	no	—
138	C₂²×C₄×C₆₁	—	2	4	no	—
139	C₂²×C₄×C₆₁	—	2	1	no	—
140	C₂²×C₄×C₆₁	—	2	2	no	—
141	C₂²×C₄×C₆₁	—	2	2	no	—
142	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
145	Q₁₆×C₆₁	—	4	4	no	—
146	SD₁₆×C₆₁	—	4	4	no	—
147	M₁₆×C₆₁	—	2	2	no	—
148	M₁₆×C₆₁	—	2	2	no	—
149	M₁₆×C₆₁	—	2	2	no	—
150	M₁₆×C₆₁	—	2	2	no	—
151	C₂×C₈×C₆₁	—	2	2	no	—
152	C₂×C₈×C₆₁	—	2	2	no	—
153	C₄⋊C₄×C₆₁	—	2	4	no	—
154	C₄⋊C₄×C₆₁	—	2	4	no	—
155	C₄⋊C₄×C₆₁	—	2	2	no	—
156	C₄⋊C₄×C₆₁	—	2	2	no	—
157	C₄⋊C₄×C₆₁	—	2	2	no	—
158	C₄⋊C₄×C₆₁	—	2	2	no	—
159	C₄⋊C₄×C₆₁	—	4	8	no	—
160	C₄⋊C₄×C₆₁	—	4	2	no	—
161	C₄⋊C₄×C₆₁	—	4	2	no	—
162	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
164	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
165	C₂²⋊C₄×C₆₁	—	2	4	no	—
166	C₂²⋊C₄×C₆₁	—	2	4	no	—
167	C₂²⋊C₄×C₆₁	—	2	4	no	—
168	C₂²⋊C₄×C₆₁	—	2	1	no	—
169	C₂²⋊C₄×C₆₁	—	2	1	no	—
170	C₂²⋊C₄×C₆₁	—	2	1	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	—	4	8	no	—
175	C₄²×C₆₁	—	2	4	no	—
176	C₄²×C₆₁	—	2	2	no	—
177	C₄²×C₆₁	—	2	1	no	—
178	C₄²×C₆₁	—	2	1	no	—
179	C₄²×C₆₁	—	2	1	no	—
180	C₄²×C₆₁	—	2	2	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	2	yes	—
184	C₄²×C₆₁	#41	1	4	yes	—
185	C₂³×C₆₁	—	2	2	no	—
186	C₂³×C₆₁	—	2	1	no	—
187	C₂³×C₆₁	—	2	4	no	—
188	C₂³×C₆₁	—	4	4	no	—
189	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
190	Q₈×C₆₁	—	2	1	no	—
191	Q₈×C₆₁	—	2	1	no	—
192	Q₈×C₆₁	—	4	1	no	—
193	Q₈×C₆₁	—	4	1	no	—
194	Q₈×C₆₁	—	4	2	no	—
195	Q₈×C₆₁	—	4	2	no	—
196	Q₈×C₆₁	—	4	2	no	—
197	Q₈×C₆₁	—	4	2	no	—
198	Q₈×C₆₁	—	4	2	no	—
199	Q₈×C₆₁	—	8	2	no	—
200	D₄×C₆₁	—	2	1	no	—
201	D₄×C₆₁	—	2	1	no	—
202	D₄×C₆₁	—	4	2	no	—
203	D₄×C₆₁	—	4	1	no	—
204	D₄×C₆₁	—	4	1	no	—
205	D₄×C₆₁	—	4	2	no	—
206	D₄×C₆₁	—	8	4	no	—
207	C₂×C₄×C₆₁	—	2	2	no	—
208	C₂×C₄×C₆₁	—	2	2	no	—
209	C₂×C₄×C₆₁	—	2	2	no	—
210	C₂×C₄×C₆₁	—	2	1	no	—
211	C₂×C₄×C₆₁	—	2	2	no	—
212	C₂×C₄×C₆₁	—	2	4	no	—
213	C₂×C₄×C₆₁	—	2	4	no	—
214	C₂×C₄×C₆₁	—	4	2	no	—
215	C₂×C₄×C₆₁	—	4	4	no	—
216	C₂×C₄×C₆₁	—	4	2	no	—
217	C₂×C₄×C₆₁	—	4	2	no	—
218	C₂×C₄×C₆₁	—	4	2	no	—
219	C₂×C₄×C₆₁	—	4	2	no	—
220	C₂×C₄×C₆₁	—	8	2	no	—
221	C₂×C₄×C₆₁	—	8	4	no	—
222	C₂×C₄×C₆₁	#120	1	2	yes	—
223	C₂×C₄×C₆₁	#120	1	2	yes	—
224	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
227	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
228	C₈×C₆₁	—	4	2	no	—
229	C₈×C₆₁	—	4	2	no	—
230	#25596	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂²×C₆₁	—	2	2	no	—
232	C₂²×C₆₁	—	4	1	no	—
233	C₂²×C₆₁	—	4	2	no	—
234	C₂²×C₆₁	—	4	2	no	—
235	C₂²×C₆₁	—	4	4	no	—
236	C₂²×C₆₁	—	8	4	no	—
237	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
238	C₂²×C₆₁	#546	1	2	yes	—
239	C₄×C₆₁	—	2	2	no	—
240	C₄×C₆₁	—	2	2	no	—
241	C₄×C₆₁	—	4	4	no	—
242	C₄×C₆₁	—	4	2	no	—
243	C₄×C₆₁	—	4	1	no	—
244	C₄×C₆₁	—	4	2	no	—
245	C₄×C₆₁	—	4	2	no	—
246	C₄×C₆₁	—	4	2	no	—
247	C₄×C₆₁	—	8	2	no	—
248	C₄×C₆₁	#532	1	1	yes	characteristic
249	C₄×C₆₁	#535	1	1	yes	characteristic
250	C₄×C₆₁	#550	1	2	yes	—
251	#3510	C₂×C₆₁	1	1	yes	characteristic
252	#3139	C₂×C₆₁	1	1	yes	characteristic, complemented
253	#3039	C₂×C₆₁	1	1	yes	characteristic, complemented
254	#2658	C₂×C₆₁	1	1	yes	characteristic, complemented
255	#2005	C₂×C₆₁	1	2	yes	complemented
256	#2004	C₂×C₆₁	1	2	yes	—
257	#1995	C₂×C₆₁	1	1	yes	characteristic, complemented
258	#1967	C₂×C₆₁	1	1	yes	characteristic, complemented
259	#1886	C₂×C₆₁	1	2	yes	complemented
260	#1885	C₂×C₆₁	1	2	yes	complemented
261	#1759	C₂×C₆₁	1	1	yes	characteristic, complemented
262	C₂×C₆₁	—	2	2	no	—
263	C₂×C₆₁	—	8	1	no	—
264	C₂×C₆₁	—	8	2	no	—
265	C₂×C₆₁	#2431	1	1	yes	characteristic
266	C₂×C₆₁	#3022	1	2	yes	—
267	C₂×#137	C₂²×C₆₁	1	2	yes	—
268	#559	—	2	2	no	—
269	#556	—	2	2	no	—
270	#551	—	2	4	no	—
271	#542	—	2	2	no	—
272	#536	C₂²×C₆₁	1	1	yes	characteristic
273	#535	C₂²×C₆₁	1	1	yes	characteristic
274	#533	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	C₂×#124	C₂²×C₆₁	1	2	yes	—
279	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic
280	#519	C₂²×C₆₁	1	1	yes	characteristic
281	#518	C₂²×C₆₁	1	1	yes	characteristic
282	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
283	#491	—	2	2	no	—
284	#487	—	2	2	no	—
285	#462	—	2	2	no	—
286	#461	—	2	2	no	—
287	#423	C₂²×C₆₁	1	1	yes	characteristic
288	C₂×#104	C₂²×C₆₁	1	2	yes	—
289	#419	C₂²×C₆₁	1	2	yes	—
290	#419	C₂²×C₆₁	1	2	yes	—
291	#419	C₂²×C₆₁	1	2	yes	—
292	C₂×#101	C₂²×C₆₁	1	2	yes	—
293	C₂×#100	C₂²×C₆₁	1	2	yes	—
294	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
295	#399	—	2	4	no	—
296	#376	C₂²×C₆₁	1	1	yes	characteristic
297	#339	C₂²×C₆₁	1	2	yes	—
298	#339	C₂²×C₆₁	1	2	yes	—
299	#339	C₂²×C₆₁	1	2	yes	—
300	C₆₁	#25596	1	1	yes	characteristic, direct product factor, Sylow
301	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
302	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
303	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
304	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
305	#137	—	4	2	no	—
306	#137	—	4	2	no	—
307	C₂×Q₁₆	—	2	2	no	—
308	C₂×SD₁₆	—	2	2	no	—
309	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
310	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
311	C₄⋊Q₈	—	2	1	no	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	—	2	1	no	—
314	C₄⋊Q₈	—	2	1	no	—
315	C₄⋊Q₈	—	2	4	no	—
316	C₄⋊Q₈	—	4	2	no	—
317	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
318	#127	—	2	1	no	—
319	#126	—	2	4	no	—
320	#126	—	2	4	no	—
321	#125	D₄×C₆₁	1	2	yes	—
322	#124	—	2	4	no	—
323	#124	—	2	1	no	—
324	#124	—	2	1	no	—
325	#124	—	2	1	no	—
326	#124	—	2	1	no	—
327	#124	—	2	2	no	—
328	#124	—	2	4	no	—
329	#124	—	4	2	no	—
330	#123	—	2	4	no	—
331	#123	—	2	4	no	—
332	#123	—	2	2	no	—
333	C₂²⋊Q₈	—	2	4	no	—
334	C₂²⋊Q₈	—	2	2	no	—
335	C₂²⋊Q₈	—	2	2	no	—
336	#121	—	2	2	no	—
337	#120	—	2	2	no	—
338	C₄×Q₈	—	2	2	no	—
339	C₄×Q₈	—	2	2	no	—
340	C₄×Q₈	—	2	2	no	—
341	C₄×Q₈	D₄×C₆₁	1	2	yes	—
342	C₄×D₄	—	2	4	no	—
343	C₄×D₄	—	2	2	no	—
344	C₄×D₄	—	2	2	no	—
345	C₄×D₄	—	2	2	no	—
346	C₄×D₄	D₄×C₆₁	1	2	yes	—
347	#117	—	2	4	no	—
348	#117	—	2	2	no	—
349	#117	—	2	1	no	—
350	#117	C₂³×C₆₁	1	1	yes	characteristic
351	#117	C₂³×C₆₁	1	1	yes	characteristic
352	#117	C₂³×C₆₁	1	1	yes	characteristic
353	C₂×C₄⋊C₄	—	2	4	no	—
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₂²⋊C₄	—	2	2	no	—
356	C₂×C₂²⋊C₄	—	2	4	no	—
357	C₂×C₂²⋊C₄	D₄×C₆₁	1	4	yes	—
358	C₂×C₄²	—	2	1	no	—
359	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
360	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
361	C₄⋊C₈	—	2	2	no	—
362	#104	—	2	4	no	—
363	#103	—	2	4	no	—
364	#103	—	2	4	no	—
365	#103	—	2	4	no	—
366	#101	—	4	2	no	—
367	#100	—	4	2	no	—
368	#97	—	2	1	no	—
369	C₄×C₈	—	2	1	no	—
370	#95	—	2	2	no	—
371	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
372	#41	—	2	2	no	—
373	#41	—	2	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	1	no	—
376	C₂×Q₈	—	2	1	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	2	2	no	—
380	C₂×Q₈	—	2	2	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	4	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
399	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
400	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
401	Q₁₆	—	4	4	no	—
402	SD₁₆	—	4	4	no	—
403	M₁₆	—	2	2	no	—
404	M₁₆	—	2	2	no	—
405	M₁₆	—	2	2	no	—
406	M₁₆	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	2	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	4	8	no	—
416	C₄⋊C₄	—	4	2	no	—
417	C₄⋊C₄	—	4	2	no	—
418	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
419	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
420	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
421	C₂²⋊C₄	—	2	4	no	—
422	C₂²⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₂²⋊C₄	—	2	1	no	—
425	C₂²⋊C₄	—	2	1	no	—
426	C₂²⋊C₄	—	2	1	no	—
427	C₂²⋊C₄	—	2	4	no	—
428	C₂²⋊C₄	—	2	4	no	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₂²⋊C₄	—	4	8	no	—
431	C₄²	—	2	4	no	—
432	C₄²	—	2	2	no	—
433	C₄²	—	2	1	no	—
434	C₄²	—	2	1	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	2	no	—
437	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
438	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
439	C₄²	C₂×D₄×C₆₁	1	2	yes	—
440	C₄²	#41×C₆₁	1	4	yes	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	4	4	no	—
445	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
446	Q₈	—	2	1	no	—
447	Q₈	—	2	1	no	—
448	Q₈	—	4	1	no	—
449	Q₈	—	4	1	no	—
450	Q₈	—	4	2	no	—
451	Q₈	—	4	2	no	—
452	Q₈	—	4	2	no	—
453	Q₈	—	4	2	no	—
454	Q₈	—	4	2	no	—
455	Q₈	—	8	2	no	—
456	D₄	—	2	1	no	—
457	D₄	—	2	1	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	1	no	—
461	D₄	—	4	2	no	—
462	D₄	—	8	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	8	2	no	—
477	C₂×C₄	—	8	4	no	—
478	C₂×C₄	#120×C₆₁	1	2	yes	—
479	C₂×C₄	#120×C₆₁	1	2	yes	—
480	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
481	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
482	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
483	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
484	C₈	—	4	2	no	—
485	C₈	—	4	2	no	—
486	C₂²	—	2	2	no	—
487	C₂²	—	4	1	no	—
488	C₂²	—	4	2	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	4	no	—
491	C₂²	—	8	4	no	—
492	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
493	C₂²	C₆₁×#546	1	2	yes	—
494	C₄	—	2	2	no	—
495	C₄	—	2	2	no	—
496	C₄	—	4	4	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	1	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	8	2	no	—
503	C₄	C₆₁×#532	1	1	yes	characteristic
504	C₄	C₆₁×#535	1	1	yes	characteristic
505	C₄	C₆₁×#550	1	2	yes	—
506	C₂	—	2	2	no	—
507	C₂	—	8	2	no	—
508	C₂	—	8	1	no	—
509	C₂	C₆₁×#2431	1	1	yes	characteristic
510	C₂	C₆₁×#3022	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₆₁×#535	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
2. C₆₁×#3139	C₂	semidirect
3. C₆₁×#3039	C₂	semidirect
4. C₆₁×#2658	C₂	semidirect
5. C₆₁×#2005	C₂	semidirect
7. C₆₁×#1995	C₂	semidirect
8. C₆₁×#1967	C₂	semidirect
9. C₆₁×#1886	C₂	semidirect
10. C₆₁×#1885	C₂	semidirect
11. C₆₁×#1759	C₂	semidirect
230. #25596	C₆₁	direct (N × H)
252. #3139	C₂×C₆₁	semidirect
253. #3039	C₂×C₆₁	semidirect
254. #2658	C₂×C₆₁	semidirect
255. #2005	C₂×C₆₁	semidirect
257. #1995	C₂×C₆₁	semidirect
258. #1967	C₂×C₆₁	semidirect
259. #1886	C₂×C₆₁	semidirect
260. #1885	C₂×C₆₁	semidirect
261. #1759	C₂×C₆₁	semidirect
300. C₆₁	#25596	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×#3510	2
C₆₁×#3139	2
C₆₁×#3039	2
C₆₁×#2658	2
C₆₁×#2005	2
C₆₁×#2004	2
C₆₁×#1995	2
C₆₁×#1967	2
C₆₁×#1886	2
C₆₁×#1885	2
C₆₁×#1759	2
#25596	61

Subgroups — C61×#25596 #29529029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25596 #29529029