Subgroups — C₆₁×#25621 #29529054

Order 15616.

1122 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3506	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3152	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3139	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2547	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2004	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, complemented
8	C₆₁×#1882	C₂	1	4	yes	maximal, complemented
9	C₄×#100×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₆₁	C₂²	1	4	yes	—
11	C₆₁×#559	—	2	2	no	—
12	C₆₁×#556	—	2	1	no	—
13	C₆₁×#555	—	2	1	no	—
14	C₆₁×#553	—	2	2	no	—
15	C₆₁×#550	—	2	1	no	—
16	C₆₁×#547	—	2	1	no	—
17	C₆₁×#538	—	2	2	no	—
18	C₆₁×#535	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	1	yes	characteristic
22	C₂×#124×C₆₁	C₂²	1	2	yes	—
23	C₆₁×#525	—	2	2	no	—
24	C₆₁×#520	C₂²	1	2	yes	—
25	C₂×C₄×D₄×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×#100×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×#397	—	2	2	no	—
38	C₆₁×#391	C₂²	1	2	yes	—
39	C₆₁×#380	—	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	—	2	2	no	—
44	C₂×#41×C₆₁	C₂³	1	2	yes	—
45	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
46	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₆₁	D₄	1	2	yes	—
48	#137×C₆₁	—	2	8	no	—
49	#137×C₆₁	—	4	4	no	—
50	C₂×Q₁₆×C₆₁	—	2	4	no	—
51	C₂×SD₁₆×C₆₁	—	2	4	no	—
52	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
53	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
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	—	2	1	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	—	2	2	no	—
62	#127×C₆₁	D₄	1	1	yes	characteristic
63	#126×C₆₁	—	2	4	no	—
64	#125×C₆₁	D₄	1	2	yes	—
65	#124×C₆₁	—	2	4	no	—
66	#124×C₆₁	—	2	2	no	—
67	#124×C₆₁	—	4	4	no	—
68	#124×C₆₁	D₄	1	2	yes	—
69	#123×C₆₁	—	2	4	no	—
70	#123×C₆₁	—	2	4	no	—
71	C₂²⋊Q₈×C₆₁	—	2	2	no	—
72	C₂²⋊Q₈×C₆₁	—	2	1	no	—
73	C₂²⋊Q₈×C₆₁	—	2	1	no	—
74	#121×C₆₁	—	2	4	no	—
75	#121×C₆₁	—	2	2	no	—
76	#121×C₆₁	—	2	1	no	—
77	#121×C₆₁	—	2	1	no	—
78	#121×C₆₁	—	2	2	no	—
79	#121×C₆₁	—	2	2	no	—
80	#120×C₆₁	—	2	4	no	—
81	C₄×Q₈×C₆₁	—	2	2	no	—
82	C₄×Q₈×C₆₁	—	2	1	no	—
83	C₄×Q₈×C₆₁	—	2	1	no	—
84	C₄×D₄×C₆₁	—	2	4	no	—
85	C₄×D₄×C₆₁	—	2	2	no	—
86	C₄×D₄×C₆₁	—	2	4	no	—
87	C₄×D₄×C₆₁	—	2	1	no	—
88	C₄×D₄×C₆₁	—	2	1	no	—
89	C₄×D₄×C₆₁	—	2	1	no	—
90	C₄×D₄×C₆₁	—	2	1	no	—
91	C₄×D₄×C₆₁	—	2	1	no	—
92	C₄×D₄×C₆₁	—	2	1	no	—
93	#117×C₆₁	—	2	2	no	—
94	#117×C₆₁	—	2	2	no	—
95	#117×C₆₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
100	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₆₁	—	2	4	no	—
102	C₂×C₂²⋊C₄×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	#100×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	C₂⁴×C₆₁	C₂×D₄	1	2	yes	—
116	#41×C₆₁	—	2	4	no	—
117	#41×C₆₁	—	2	4	no	—
118	#41×C₆₁	—	8	4	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	4	no	—
124	C₂×Q₈×C₆₁	—	2	4	no	—
125	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
126	C₂×D₄×C₆₁	—	2	2	no	—
127	C₂×D₄×C₆₁	—	2	4	no	—
128	C₂×D₄×C₆₁	—	2	4	no	—
129	C₂×D₄×C₆₁	—	2	1	no	—
130	C₂×D₄×C₆₁	—	2	1	no	—
131	C₂×D₄×C₆₁	—	2	1	no	—
132	C₂×D₄×C₆₁	—	2	1	no	—
133	C₂×D₄×C₆₁	—	2	2	no	—
134	C₂×D₄×C₆₁	—	2	2	no	—
135	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₆₁	—	2	4	no	—
137	C₂²×C₄×C₆₁	—	2	2	no	—
138	C₂²×C₄×C₆₁	—	2	2	no	—
139	C₂²×C₄×C₆₁	—	2	2	no	—
140	C₂²×C₄×C₆₁	—	2	2	no	—
141	C₂²×C₄×C₆₁	—	2	2	no	—
142	C₂²×C₄×C₆₁	—	4	4	no	—
143	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
147	Q₁₆×C₆₁	—	4	8	no	—
148	SD₁₆×C₆₁	—	4	8	no	—
149	M₁₆×C₆₁	—	2	2	no	—
150	M₁₆×C₆₁	—	2	2	no	—
151	M₁₆×C₆₁	—	2	2	no	—
152	M₁₆×C₆₁	#41	1	4	yes	complemented
153	C₂×C₈×C₆₁	—	2	2	no	—
154	C₂×C₈×C₆₁	—	2	2	no	—
155	C₄⋊C₄×C₆₁	—	2	4	no	—
156	C₄⋊C₄×C₆₁	—	2	1	no	—
157	C₄⋊C₄×C₆₁	—	2	1	no	—
158	C₄⋊C₄×C₆₁	—	2	2	no	—
159	C₄⋊C₄×C₆₁	—	2	1	no	—
160	C₄⋊C₄×C₆₁	—	2	2	no	—
161	C₄⋊C₄×C₆₁	—	2	1	no	—
162	C₄⋊C₄×C₆₁	—	2	1	no	—
163	C₄⋊C₄×C₆₁	—	2	1	no	—
164	C₄⋊C₄×C₆₁	—	4	2	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	—	4	1	no	—
169	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
170	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
171	C₂²⋊C₄×C₆₁	—	2	4	no	—
172	C₂²⋊C₄×C₆₁	—	2	4	no	—
173	C₂²⋊C₄×C₆₁	—	2	4	no	—
174	C₂²⋊C₄×C₆₁	—	2	2	no	—
175	C₂²⋊C₄×C₆₁	—	2	2	no	—
176	C₂²⋊C₄×C₆₁	—	2	2	no	—
177	C₂²⋊C₄×C₆₁	—	2	2	no	—
178	C₂²⋊C₄×C₆₁	—	2	2	no	—
179	C₂²⋊C₄×C₆₁	—	4	8	no	—
180	C₄²×C₆₁	—	2	4	no	—
181	C₄²×C₆₁	—	2	2	no	—
182	C₄²×C₆₁	—	2	4	no	—
183	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
187	C₂³×C₆₁	—	2	2	no	—
188	C₂³×C₆₁	—	2	2	no	—
189	C₂³×C₆₁	—	2	2	no	—
190	C₂³×C₆₁	—	2	2	no	—
191	C₂³×C₆₁	—	4	2	no	—
192	C₂³×C₆₁	—	4	2	no	—
193	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
194	Q₈×C₆₁	—	2	4	no	—
195	Q₈×C₆₁	—	4	2	no	—
196	Q₈×C₆₁	—	4	4	no	—
197	Q₈×C₆₁	—	4	4	no	—
198	Q₈×C₆₁	—	8	1	no	—
199	D₄×C₆₁	—	2	4	no	—
200	D₄×C₆₁	—	4	2	no	—
201	D₄×C₆₁	—	4	2	no	—
202	D₄×C₆₁	—	4	2	no	—
203	D₄×C₆₁	—	4	2	no	—
204	D₄×C₆₁	—	4	2	no	—
205	D₄×C₆₁	—	8	1	no	—
206	D₄×C₆₁	—	8	4	no	—
207	D₄×C₆₁	—	8	4	no	—
208	C₂×C₄×C₆₁	—	2	2	no	—
209	C₂×C₄×C₆₁	—	2	2	no	—
210	C₂×C₄×C₆₁	—	2	2	no	—
211	C₂×C₄×C₆₁	—	2	2	no	—
212	C₂×C₄×C₆₁	—	2	2	no	—
213	C₂×C₄×C₆₁	—	4	1	no	—
214	C₂×C₄×C₆₁	—	4	4	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	—	4	2	no	—
221	C₂×C₄×C₆₁	—	4	2	no	—
222	C₂×C₄×C₆₁	—	4	2	no	—
223	C₂×C₄×C₆₁	—	8	2	no	—
224	C₂×C₄×C₆₁	—	8	4	no	—
225	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
226	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
227	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
228	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
229	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
232	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
233	C₈×C₆₁	—	2	4	no	—
234	C₈×C₆₁	—	4	2	no	—
235	#25621	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂²×C₆₁	—	2	1	no	—
237	C₂²×C₆₁	—	2	1	no	—
238	C₂²×C₆₁	—	4	4	no	—
239	C₂²×C₆₁	—	4	2	no	—
240	C₂²×C₆₁	—	4	2	no	—
241	C₂²×C₆₁	—	4	2	no	—
242	C₂²×C₆₁	—	4	2	no	—
243	C₂²×C₆₁	—	4	2	no	—
244	C₂²×C₆₁	—	4	2	no	—
245	C₂²×C₆₁	—	8	2	no	—
246	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
247	C₂²×C₆₁	#532	1	1	yes	characteristic
248	C₂²×C₆₁	#535	1	1	yes	characteristic
249	C₄×C₆₁	—	2	1	no	—
250	C₄×C₆₁	—	2	1	no	—
251	C₄×C₆₁	—	2	1	no	—
252	C₄×C₆₁	—	4	2	no	—
253	C₄×C₆₁	—	4	2	no	—
254	C₄×C₆₁	—	4	2	no	—
255	C₄×C₆₁	—	4	2	no	—
256	C₄×C₆₁	—	8	2	no	—
257	C₄×C₆₁	—	8	2	no	—
258	C₄×C₆₁	#455	1	2	yes	—
259	C₄×C₆₁	#546	1	2	yes	—
260	C₄×C₆₁	#550	1	2	yes	—
261	#3506	C₂×C₆₁	1	1	yes	characteristic
262	#3152	C₂×C₆₁	1	1	yes	characteristic, complemented
263	#3139	C₂×C₆₁	1	1	yes	characteristic, complemented
264	#2547	C₂×C₆₁	1	1	yes	characteristic, complemented
265	#2004	C₂×C₆₁	1	4	yes	complemented
266	#1986	C₂×C₆₁	1	1	yes	characteristic, complemented
267	#1980	C₂×C₆₁	1	1	yes	characteristic, complemented
268	#1882	C₂×C₆₁	1	4	yes	complemented
269	C₄×#100	C₂×C₆₁	1	1	yes	characteristic, complemented
270	C₂×C₆₁	—	2	1	no	—
271	C₂×C₆₁	—	2	1	no	—
272	C₂×C₆₁	—	8	2	no	—
273	C₂×C₆₁	—	8	2	no	—
274	C₂×C₆₁	#2431	1	1	yes	characteristic
275	C₂×C₆₁	C₂×#455	1	1	yes	characteristic
276	C₂×C₆₁	#3022	1	1	yes	characteristic
277	C₂×#137	C₂²×C₆₁	1	4	yes	—
278	#559	—	2	2	no	—
279	#556	—	2	1	no	—
280	#555	—	2	1	no	—
281	#553	—	2	2	no	—
282	#550	—	2	1	no	—
283	#547	—	2	1	no	—
284	#538	—	2	2	no	—
285	#535	C₂²×C₆₁	1	2	yes	—
286	#533	C₂²×C₆₁	1	1	yes	characteristic
287	#532	C₂²×C₆₁	1	1	yes	characteristic
288	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
289	C₂×#124	C₂²×C₆₁	1	2	yes	—
290	#525	—	2	2	no	—
291	#520	C₂²×C₆₁	1	2	yes	—
292	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
293	#497	—	2	2	no	—
294	#494	—	2	1	no	—
295	#492	—	2	1	no	—
296	#491	—	2	2	no	—
297	#487	—	2	1	no	—
298	#487	—	2	1	no	—
299	C₂×#104	C₂²×C₆₁	1	4	yes	—
300	#419	C₂²×C₆₁	1	4	yes	—
301	C₂×#100	C₂²×C₆₁	1	2	yes	—
302	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
303	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
304	#397	—	2	2	no	—
305	#391	C₂²×C₆₁	1	2	yes	—
306	#380	—	2	2	no	—
307	C₄×C₄⋊C₄	C₂²×C₆₁	1	1	yes	characteristic
308	#343	C₂²×C₆₁	1	2	yes	—
309	#339	C₂²×C₆₁	1	4	yes	—
310	C₆₁	#25621	1	1	yes	characteristic, direct product factor, Sylow
311	C₂×#41	—	2	2	no	—
312	C₂×#41	C₂³×C₆₁	1	2	yes	—
313	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
314	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
315	C₂³×C₄	D₄×C₆₁	1	2	yes	—
316	#137	—	2	8	no	—
317	#137	—	4	4	no	—
318	C₂×Q₁₆	—	2	4	no	—
319	C₂×SD₁₆	—	2	4	no	—
320	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
321	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
322	C₄⋊Q₈	—	2	1	no	—
323	C₄⋊Q₈	—	2	1	no	—
324	C₄⋊Q₈	—	2	1	no	—
325	C₄⋊Q₈	—	2	1	no	—
326	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
327	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
328	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
329	#127	—	2	2	no	—
330	#127	D₄×C₆₁	1	1	yes	characteristic
331	#126	—	2	4	no	—
332	#125	D₄×C₆₁	1	2	yes	—
333	#124	—	2	4	no	—
334	#124	—	2	2	no	—
335	#124	—	4	4	no	—
336	#124	D₄×C₆₁	1	2	yes	—
337	#123	—	2	4	no	—
338	#123	—	2	4	no	—
339	C₂²⋊Q₈	—	2	2	no	—
340	C₂²⋊Q₈	—	2	1	no	—
341	C₂²⋊Q₈	—	2	1	no	—
342	#121	—	2	2	no	—
343	#121	—	2	4	no	—
344	#121	—	2	1	no	—
345	#121	—	2	1	no	—
346	#121	—	2	2	no	—
347	#121	—	2	2	no	—
348	#120	—	2	4	no	—
349	C₄×Q₈	—	2	2	no	—
350	C₄×Q₈	—	2	1	no	—
351	C₄×Q₈	—	2	1	no	—
352	C₄×D₄	—	2	4	no	—
353	C₄×D₄	—	2	2	no	—
354	C₄×D₄	—	2	4	no	—
355	C₄×D₄	—	2	1	no	—
356	C₄×D₄	—	2	1	no	—
357	C₄×D₄	—	2	1	no	—
358	C₄×D₄	—	2	1	no	—
359	C₄×D₄	—	2	1	no	—
360	C₄×D₄	—	2	1	no	—
361	#117	—	2	2	no	—
362	#117	—	2	2	no	—
363	#117	C₂³×C₆₁	1	2	yes	—
364	C₂×C₄⋊C₄	—	2	2	no	—
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
369	C₂×C₂²⋊C₄	—	2	4	no	—
370	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
371	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
372	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
373	#104	—	4	4	no	—
374	#103	—	2	4	no	—
375	#103	—	2	4	no	—
376	#100	—	2	4	no	—
377	#97	—	2	1	no	—
378	#97	—	2	1	no	—
379	C₄×C₈	—	2	1	no	—
380	C₄×C₈	—	2	1	no	—
381	#95	—	2	1	no	—
382	#95	—	2	4	no	—
383	C₂⁴	C₂×D₄×C₆₁	1	2	yes	—
384	#41	—	2	4	no	—
385	#41	—	2	4	no	—
386	#41	—	8	4	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	1	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	—	2	2	no	—
391	C₂×Q₈	—	2	4	no	—
392	C₂×Q₈	—	2	4	no	—
393	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	4	no	—
396	C₂×D₄	—	2	4	no	—
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	2	1	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	1	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	4	4	no	—
411	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
412	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
413	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
414	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
415	Q₁₆	—	4	8	no	—
416	SD₁₆	—	4	8	no	—
417	M₁₆	—	2	2	no	—
418	M₁₆	—	2	2	no	—
419	M₁₆	—	2	2	no	—
420	M₁₆	#41×C₆₁	1	4	yes	complemented
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	4	2	no	—
433	C₄⋊C₄	—	4	1	no	—
434	C₄⋊C₄	—	4	1	no	—
435	C₄⋊C₄	—	4	1	no	—
436	C₄⋊C₄	—	4	1	no	—
437	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
438	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
439	C₂²⋊C₄	—	2	4	no	—
440	C₂²⋊C₄	—	2	4	no	—
441	C₂²⋊C₄	—	2	4	no	—
442	C₂²⋊C₄	—	2	2	no	—
443	C₂²⋊C₄	—	2	2	no	—
444	C₂²⋊C₄	—	2	2	no	—
445	C₂²⋊C₄	—	2	2	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	4	8	no	—
448	C₄²	—	2	4	no	—
449	C₄²	—	2	2	no	—
450	C₄²	—	2	4	no	—
451	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
452	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
453	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
454	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
455	C₂³	—	2	2	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	4	2	no	—
460	C₂³	—	4	2	no	—
461	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
462	Q₈	—	2	4	no	—
463	Q₈	—	4	2	no	—
464	Q₈	—	4	4	no	—
465	Q₈	—	4	4	no	—
466	Q₈	—	8	1	no	—
467	D₄	—	2	4	no	—
468	D₄	—	4	2	no	—
469	D₄	—	4	2	no	—
470	D₄	—	4	2	no	—
471	D₄	—	4	2	no	—
472	D₄	—	4	2	no	—
473	D₄	—	8	1	no	—
474	D₄	—	8	4	no	—
475	D₄	—	8	4	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	4	no	—
483	C₂×C₄	—	4	4	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	8	2	no	—
492	C₂×C₄	—	8	4	no	—
493	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
494	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
495	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
496	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
498	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
499	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
500	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
501	C₈	—	2	4	no	—
502	C₈	—	4	2	no	—
503	C₂²	—	2	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	4	4	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	8	2	no	—
513	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
514	C₂²	C₆₁×#532	1	1	yes	characteristic
515	C₂²	C₆₁×#535	1	1	yes	characteristic
516	C₄	—	2	1	no	—
517	C₄	—	2	1	no	—
518	C₄	—	2	1	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	2	no	—
522	C₄	—	4	2	no	—
523	C₄	—	8	2	no	—
524	C₄	—	8	2	no	—
525	C₄	C₆₁×#455	1	2	yes	—
526	C₄	C₆₁×#546	1	2	yes	—
527	C₄	C₆₁×#550	1	2	yes	—
528	C₂	—	2	1	no	—
529	C₂	—	2	1	no	—
530	C₂	—	8	2	no	—
531	C₂	—	8	2	no	—
532	C₂	C₆₁×#2431	1	1	yes	characteristic
533	C₂	C₂×C₆₁×#455	1	1	yes	characteristic
534	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×#3506	C₂
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×#3152	C₂	semidirect
3. C₆₁×#3139	C₂	semidirect
4. C₆₁×#2547	C₂	semidirect
5. C₆₁×#2004	C₂	semidirect
6. C₆₁×#1986	C₂	semidirect
7. C₆₁×#1980	C₂	semidirect
8. C₆₁×#1882	C₂	semidirect
9. C₄×#100×C₆₁	C₂	semidirect
152. M₁₆×C₆₁	#41	semidirect
235. #25621	C₆₁	direct (N × H)
262. #3152	C₂×C₆₁	semidirect
263. #3139	C₂×C₆₁	semidirect
264. #2547	C₂×C₆₁	semidirect
265. #2004	C₂×C₆₁	semidirect
266. #1986	C₂×C₆₁	semidirect
267. #1980	C₂×C₆₁	semidirect
268. #1882	C₂×C₆₁	semidirect
269. C₄×#100	C₂×C₆₁	semidirect
310. C₆₁	#25621	direct (N × H)
420. M₁₆	#41×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3506	2
C₆₁×#3152	2
C₆₁×#3139	2
C₆₁×#2547	2
C₆₁×#2004	2
C₆₁×#1986	2
C₆₁×#1980	2
C₆₁×#1882	2
C₄×#100×C₆₁	2
#25621	61

Subgroups — C61×#25621 #29529054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25621 #29529054