Subgroups — C₆₁×#18697 #29522130

Order 15616.

890 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3253	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1638	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1637	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1628	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1627	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1561	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1492	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#563	—	2	2	no	—
13	C₆₁×#559	—	2	1	no	—
14	Q₈²×C₆₁	—	2	1	no	—
15	C₆₁×#556	—	2	1	no	—
16	C₆₁×#556	C₂²	1	2	yes	—
17	D₄×Q₈×C₆₁	—	2	1	no	—
18	D₄×Q₈×C₆₁	C₂²	1	2	yes	—
19	C₆₁×#542	—	2	2	no	—
20	C₆₁×#542	—	2	2	no	—
21	C₆₁×#536	—	2	2	no	—
22	C₆₁×#533	C₂²	1	1	yes	characteristic
23	C₆₁×#533	C₂²	1	1	yes	characteristic
24	C₆₁×#532	C₂²	1	1	yes	characteristic
25	C₆₁×#532	C₂²	1	1	yes	characteristic
26	C₆₁×#488	—	2	2	no	—
27	C₆₁×#487	C₂²	1	1	yes	characteristic
28	C₆₁×#487	C₂²	1	1	yes	characteristic
29	C₆₁×#486	C₂²	1	1	yes	characteristic
30	C₆₁×#486	C₂²	1	1	yes	characteristic
31	C₆₁×#477	C₂²	1	2	yes	—
32	C₆₁×#476	—	2	1	no	—
33	C₆₁×#476	C₂²	1	2	yes	—
34	C₆₁×#474	—	2	1	no	—
35	C₆₁×#469	—	2	2	no	—
36	C₆₁×#468	—	2	2	no	—
37	C₆₁×#461	—	2	1	no	—
38	C₆₁×#460	C₂²	1	2	yes	—
39	C₆₁×#459	—	2	1	no	—
40	C₆₁×#459	C₂²	1	2	yes	—
41	C₆₁×#423	C₂²	1	1	yes	characteristic
42	C₆₁×#423	C₂²	1	1	yes	characteristic
43	C₆₁×#416	—	2	2	no	—
44	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
45	C₆₁×#332	C₂²	1	2	yes	—
46	C₆₁×#332	C₂²	1	2	yes	—
47	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₆₁×D₄⋊C₈	C₂²	1	2	yes	—
50	C₆₁×D₄⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₆₁	—	2	1	no	—
52	C₂×#41×C₆₁	—	2	1	no	—
53	C₂²×Q₈×C₆₁	—	2	2	no	—
54	C₂×SD₁₆×C₆₁	—	2	2	no	—
55	C₂×D₈×C₆₁	—	2	2	no	—
56	C₂×M₁₆×C₆₁	—	2	1	no	—
57	C₂×M₁₆×C₆₁	—	2	1	no	—
58	C₂²×C₈×C₆₁	—	2	1	no	—
59	C₄⋊Q₈×C₆₁	—	2	4	no	—
60	C₄⋊Q₈×C₆₁	—	2	2	no	—
61	C₄⋊Q₈×C₆₁	—	2	2	no	—
62	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
64	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
65	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
68	#127×C₆₁	D₄	1	1	yes	characteristic
69	#127×C₆₁	C₂³	1	1	yes	characteristic
70	#126×C₆₁	—	2	4	no	—
71	#126×C₆₁	—	2	4	no	—
72	#125×C₆₁	—	2	1	no	—
73	#125×C₆₁	D₄	1	2	yes	—
74	#124×C₆₁	—	2	4	no	—
75	#124×C₆₁	—	2	2	no	—
76	#124×C₆₁	—	2	2	no	—
77	#124×C₆₁	—	2	1	no	—
78	#124×C₆₁	—	2	1	no	—
79	#123×C₆₁	—	2	4	no	—
80	#123×C₆₁	—	2	4	no	—
81	C₂²⋊Q₈×C₆₁	—	2	4	no	—
82	C₂²⋊Q₈×C₆₁	—	2	2	no	—
83	C₂²⋊Q₈×C₆₁	—	2	1	no	—
84	C₂²⋊Q₈×C₆₁	—	2	1	no	—
85	C₂²⋊Q₈×C₆₁	—	2	1	no	—
86	C₂²⋊Q₈×C₆₁	—	2	2	no	—
87	C₂²⋊Q₈×C₆₁	—	2	4	no	—
88	C₂²⋊Q₈×C₆₁	D₄	1	2	yes	—
89	#121×C₆₁	—	2	1	no	—
90	#121×C₆₁	—	2	1	no	—
91	#121×C₆₁	—	2	1	no	—
92	#121×C₆₁	D₄	1	2	yes	—
93	C₄×Q₈×C₆₁	—	2	2	no	—
94	C₄×Q₈×C₆₁	—	2	2	no	—
95	C₄×Q₈×C₆₁	—	2	1	no	—
96	C₄×Q₈×C₆₁	—	2	1	no	—
97	C₄×Q₈×C₆₁	—	2	1	no	—
98	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
99	C₄×D₄×C₆₁	—	2	2	no	—
100	C₄×D₄×C₆₁	—	2	2	no	—
101	C₄×D₄×C₆₁	—	2	1	no	—
102	C₄×D₄×C₆₁	—	2	1	no	—
103	C₄×D₄×C₆₁	—	2	1	no	—
104	C₄×D₄×C₆₁	C₂³	1	2	yes	—
105	#117×C₆₁	—	2	1	no	—
106	#117×C₆₁	—	2	2	no	—
107	#117×C₆₁	D₄	1	2	yes	—
108	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
109	#107×C₆₁	—	2	2	no	—
110	#106×C₆₁	—	2	2	no	—
111	C₄⋊C₈×C₆₁	—	2	1	no	—
112	C₄⋊C₈×C₆₁	—	2	1	no	—
113	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
114	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
115	#103×C₆₁	—	2	2	no	—
116	#103×C₆₁	—	2	2	no	—
117	#103×C₆₁	—	2	2	no	—
118	#102×C₆₁	—	2	2	no	—
119	#102×C₆₁	—	2	2	no	—
120	#102×C₆₁	—	2	2	no	—
121	#102×C₆₁	—	2	2	no	—
122	#102×C₆₁	—	2	2	no	—
123	C₄×C₈×C₆₁	—	2	1	no	—
124	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
126	#41×C₆₁	—	4	4	no	—
127	#41×C₆₁	—	8	1	no	—
128	#41×C₆₁	—	8	1	no	—
129	C₂×Q₈×C₆₁	—	2	4	no	—
130	C₂×Q₈×C₆₁	—	2	1	no	—
131	C₂×Q₈×C₆₁	—	2	1	no	—
132	C₂×Q₈×C₆₁	—	2	1	no	—
133	C₂×Q₈×C₆₁	—	2	1	no	—
134	C₂×Q₈×C₆₁	—	2	1	no	—
135	C₂×Q₈×C₆₁	—	8	2	no	—
136	C₂×Q₈×C₆₁	—	8	2	no	—
137	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
138	C₂×D₄×C₆₁	—	2	2	no	—
139	C₂×D₄×C₆₁	—	2	1	no	—
140	C₂×D₄×C₆₁	—	2	1	no	—
141	C₂×D₄×C₆₁	—	2	1	no	—
142	C₂×D₄×C₆₁	—	2	1	no	—
143	C₂×D₄×C₆₁	—	2	1	no	—
144	C₂×D₄×C₆₁	—	4	1	no	—
145	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₆₁	—	2	4	no	—
147	C₂²×C₄×C₆₁	—	2	2	no	—
148	C₂²×C₄×C₆₁	—	4	1	no	—
149	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
152	SD₁₆×C₆₁	—	8	2	no	—
153	D₈×C₆₁	—	8	2	no	—
154	M₁₆×C₆₁	—	8	1	no	—
155	M₁₆×C₆₁	—	8	1	no	—
156	C₂×C₈×C₆₁	—	2	1	no	—
157	C₂×C₈×C₆₁	—	2	1	no	—
158	C₂×C₈×C₆₁	—	2	2	no	—
159	C₂×C₈×C₆₁	—	2	2	no	—
160	C₂×C₈×C₆₁	—	4	2	no	—
161	C₄⋊C₄×C₆₁	—	2	4	no	—
162	C₄⋊C₄×C₆₁	—	2	4	no	—
163	C₄⋊C₄×C₆₁	—	2	1	no	—
164	C₄⋊C₄×C₆₁	—	2	1	no	—
165	C₄⋊C₄×C₆₁	—	2	1	no	—
166	C₄⋊C₄×C₆₁	—	2	1	no	—
167	C₄⋊C₄×C₆₁	—	2	2	no	—
168	C₄⋊C₄×C₆₁	—	2	2	no	—
169	C₄⋊C₄×C₆₁	—	2	2	no	—
170	C₄⋊C₄×C₆₁	—	2	1	no	—
171	C₄⋊C₄×C₆₁	—	2	1	no	—
172	C₄⋊C₄×C₆₁	—	2	2	no	—
173	C₄⋊C₄×C₆₁	—	2	1	no	—
174	C₄⋊C₄×C₆₁	—	2	1	no	—
175	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
176	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
177	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
178	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
179	C₂²⋊C₄×C₆₁	—	2	4	no	—
180	C₂²⋊C₄×C₆₁	—	2	4	no	—
181	C₂²⋊C₄×C₆₁	—	2	4	no	—
182	C₂²⋊C₄×C₆₁	—	2	2	no	—
183	C₂²⋊C₄×C₆₁	—	2	2	no	—
184	C₂²⋊C₄×C₆₁	—	2	2	no	—
185	C₂²⋊C₄×C₆₁	—	2	2	no	—
186	C₂²⋊C₄×C₆₁	—	4	1	no	—
187	C₂²⋊C₄×C₆₁	—	4	1	no	—
188	C₄²×C₆₁	—	2	4	no	—
189	C₄²×C₆₁	—	2	2	no	—
190	C₄²×C₆₁	—	2	2	no	—
191	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
195	C₂³×C₆₁	—	2	2	no	—
196	C₂³×C₆₁	—	4	1	no	—
197	C₂³×C₆₁	#120	1	1	yes	characteristic
198	Q₈×C₆₁	—	2	2	no	—
199	Q₈×C₆₁	—	2	2	no	—
200	Q₈×C₆₁	—	4	2	no	—
201	Q₈×C₆₁	—	4	1	no	—
202	Q₈×C₆₁	—	4	1	no	—
203	Q₈×C₆₁	—	4	1	no	—
204	Q₈×C₆₁	—	4	1	no	—
205	Q₈×C₆₁	—	4	2	no	—
206	Q₈×C₆₁	—	8	1	no	—
207	Q₈×C₆₁	—	8	4	no	—
208	D₄×C₆₁	—	2	2	no	—
209	D₄×C₆₁	—	2	2	no	—
210	D₄×C₆₁	—	4	2	no	—
211	D₄×C₆₁	—	4	1	no	—
212	D₄×C₆₁	—	4	1	no	—
213	D₄×C₆₁	—	4	1	no	—
214	D₄×C₆₁	—	4	1	no	—
215	D₄×C₆₁	—	4	2	no	—
216	D₄×C₆₁	—	8	1	no	—
217	D₄×C₆₁	—	8	2	no	—
218	D₄×C₆₁	—	8	2	no	—
219	C₂×C₄×C₆₁	—	2	1	no	—
220	C₂×C₄×C₆₁	—	2	1	no	—
221	C₂×C₄×C₆₁	—	2	1	no	—
222	C₂×C₄×C₆₁	—	2	1	no	—
223	C₂×C₄×C₆₁	—	2	2	no	—
224	C₂×C₄×C₆₁	—	2	2	no	—
225	C₂×C₄×C₆₁	—	2	2	no	—
226	C₂×C₄×C₆₁	—	2	2	no	—
227	C₂×C₄×C₆₁	—	2	2	no	—
228	C₂×C₄×C₆₁	—	2	2	no	—
229	C₂×C₄×C₆₁	—	4	2	no	—
230	C₂×C₄×C₆₁	—	4	2	no	—
231	C₂×C₄×C₆₁	—	4	1	no	—
232	C₂×C₄×C₆₁	—	8	4	no	—
233	C₂×C₄×C₆₁	—	8	2	no	—
234	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
235	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
236	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
237	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
240	C₈×C₆₁	—	4	1	no	—
241	C₈×C₆₁	—	4	1	no	—
242	C₈×C₆₁	—	4	2	no	—
243	C₈×C₆₁	—	4	2	no	—
244	#18697	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
245	C₂²×C₆₁	—	2	1	no	—
246	C₂²×C₆₁	—	2	1	no	—
247	C₂²×C₆₁	—	2	1	no	—
248	C₂²×C₆₁	—	4	2	no	—
249	C₂²×C₆₁	—	4	2	no	—
250	C₂²×C₆₁	—	4	2	no	—
251	C₂²×C₆₁	—	8	1	no	—
252	C₂²×C₆₁	—	8	1	no	—
253	C₂²×C₆₁	—	8	1	no	—
254	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
255	C₄×C₆₁	—	2	1	no	—
256	C₄×C₆₁	—	2	1	no	—
257	C₄×C₆₁	—	2	1	no	—
258	C₄×C₆₁	—	4	2	no	—
259	C₄×C₆₁	—	4	2	no	—
260	C₄×C₆₁	—	4	2	no	—
261	C₄×C₆₁	—	4	2	no	—
262	C₄×C₆₁	—	4	2	no	—
263	C₄×C₆₁	—	8	1	no	—
264	C₄×C₆₁	#454	1	2	yes	—
265	C₄×C₆₁	#576	1	1	yes	characteristic
266	C₄×C₆₁	#577	1	1	yes	characteristic
267	C₄×C₆₁	#577	1	1	yes	characteristic
268	C₄×C₆₁	#578	1	1	yes	characteristic
269	#3550	C₂×C₆₁	1	1	yes	characteristic, complemented
270	#3254	C₂×C₆₁	1	1	yes	characteristic, complemented
271	#3253	C₂×C₆₁	1	1	yes	characteristic
272	#3131	C₂×C₆₁	1	1	yes	characteristic, complemented
273	#1638	C₂×C₆₁	1	2	yes	complemented
274	#1637	C₂×C₆₁	1	2	yes	complemented
275	#1628	C₂×C₆₁	1	2	yes	complemented
276	#1627	C₂×C₆₁	1	2	yes	complemented
277	#1561	C₂×C₆₁	1	1	yes	characteristic, complemented
278	#1493	C₂×C₆₁	1	1	yes	characteristic, complemented
279	#1492	C₂×C₆₁	1	1	yes	characteristic, complemented
280	C₂×C₆₁	—	4	1	no	—
281	C₂×C₆₁	—	8	2	no	—
282	C₂×C₆₁	—	16	1	no	—
283	C₂×C₆₁	#3012	1	1	yes	characteristic
284	C₂×C₆₁	#3013	1	1	yes	characteristic
285	C₂×C₆₁	C₂×#454	1	1	yes	characteristic
286	#563	—	2	2	no	—
287	#559	—	2	1	no	—
288	Q₈²	—	2	1	no	—
289	#556	—	2	1	no	—
290	#556	C₂²×C₆₁	1	2	yes	—
291	D₄×Q₈	—	2	1	no	—
292	D₄×Q₈	C₂²×C₆₁	1	2	yes	—
293	#542	—	2	2	no	—
294	#542	—	2	2	no	—
295	#536	—	2	2	no	—
296	#533	C₂²×C₆₁	1	1	yes	characteristic
297	#533	C₂²×C₆₁	1	1	yes	characteristic
298	#532	C₂²×C₆₁	1	1	yes	characteristic
299	#532	C₂²×C₆₁	1	1	yes	characteristic
300	#488	—	2	2	no	—
301	#487	C₂²×C₆₁	1	1	yes	characteristic
302	#487	C₂²×C₆₁	1	1	yes	characteristic
303	#486	C₂²×C₆₁	1	1	yes	characteristic
304	#486	C₂²×C₆₁	1	1	yes	characteristic
305	#477	C₂²×C₆₁	1	2	yes	—
306	#476	—	2	1	no	—
307	#476	C₂²×C₆₁	1	2	yes	—
308	#474	—	2	1	no	—
309	#469	—	2	2	no	—
310	#468	—	2	2	no	—
311	#461	—	2	1	no	—
312	#460	C₂²×C₆₁	1	2	yes	—
313	#459	—	2	1	no	—
314	#459	C₂²×C₆₁	1	2	yes	—
315	#423	C₂²×C₆₁	1	1	yes	characteristic
316	#423	C₂²×C₆₁	1	1	yes	characteristic
317	#416	—	2	2	no	—
318	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
319	#332	C₂²×C₆₁	1	2	yes	—
320	#332	C₂²×C₆₁	1	2	yes	—
321	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
322	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
323	D₄⋊C₈	C₂²×C₆₁	1	2	yes	—
324	D₄⋊C₈	C₂²×C₆₁	1	2	yes	—
325	C₆₁	#18697	1	1	yes	characteristic, direct product factor, Sylow
326	C₂×#41	—	2	1	no	—
327	C₂×#41	—	2	1	no	—
328	C₂²×Q₈	—	2	2	no	—
329	C₂×SD₁₆	—	2	2	no	—
330	C₂×D₈	—	2	2	no	—
331	C₂×M₁₆	—	2	1	no	—
332	C₂×M₁₆	—	2	1	no	—
333	C₂²×C₈	—	2	1	no	—
334	C₄⋊Q₈	—	2	4	no	—
335	C₄⋊Q₈	—	2	2	no	—
336	C₄⋊Q₈	—	2	2	no	—
337	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
338	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
339	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
340	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
341	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
342	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
343	#127	D₄×C₆₁	1	1	yes	characteristic
344	#127	C₂³×C₆₁	1	1	yes	characteristic
345	#126	—	2	4	no	—
346	#126	—	2	4	no	—
347	#125	—	2	1	no	—
348	#125	D₄×C₆₁	1	2	yes	—
349	#124	—	2	4	no	—
350	#124	—	2	2	no	—
351	#124	—	2	2	no	—
352	#124	—	2	1	no	—
353	#124	—	2	1	no	—
354	#123	—	2	4	no	—
355	#123	—	2	4	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	—	2	2	no	—
358	C₂²⋊Q₈	—	2	1	no	—
359	C₂²⋊Q₈	—	2	1	no	—
360	C₂²⋊Q₈	—	2	1	no	—
361	C₂²⋊Q₈	—	2	2	no	—
362	C₂²⋊Q₈	—	2	4	no	—
363	C₂²⋊Q₈	D₄×C₆₁	1	2	yes	—
364	#121	—	2	1	no	—
365	#121	—	2	1	no	—
366	#121	—	2	1	no	—
367	#121	D₄×C₆₁	1	2	yes	—
368	C₄×Q₈	—	2	2	no	—
369	C₄×Q₈	—	2	2	no	—
370	C₄×Q₈	—	2	1	no	—
371	C₄×Q₈	—	2	1	no	—
372	C₄×Q₈	—	2	1	no	—
373	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	C₂³×C₆₁	1	2	yes	—
380	#117	—	2	1	no	—
381	#117	—	2	2	no	—
382	#117	D₄×C₆₁	1	2	yes	—
383	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
384	#107	—	2	2	no	—
385	#106	—	2	2	no	—
386	C₄⋊C₈	—	2	1	no	—
387	C₄⋊C₈	—	2	1	no	—
388	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
389	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
390	#103	—	2	2	no	—
391	#103	—	2	2	no	—
392	#103	—	2	2	no	—
393	#102	—	2	2	no	—
394	#102	—	2	2	no	—
395	#102	—	2	2	no	—
396	#102	—	2	2	no	—
397	#102	—	2	2	no	—
398	C₄×C₈	—	2	1	no	—
399	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
400	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
401	#41	—	4	4	no	—
402	#41	—	8	1	no	—
403	#41	—	8	1	no	—
404	C₂×Q₈	—	2	4	no	—
405	C₂×Q₈	—	2	1	no	—
406	C₂×Q₈	—	2	1	no	—
407	C₂×Q₈	—	2	1	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	1	no	—
410	C₂×Q₈	—	8	2	no	—
411	C₂×Q₈	—	8	2	no	—
412	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	4	1	no	—
420	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	4	1	no	—
424	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
427	SD₁₆	—	8	2	no	—
428	D₈	—	8	2	no	—
429	M₁₆	—	8	1	no	—
430	M₁₆	—	8	1	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	1	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	4	2	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	4	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	1	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
451	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
452	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
453	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	4	no	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	—	2	2	no	—
458	C₂²⋊C₄	—	2	2	no	—
459	C₂²⋊C₄	—	2	2	no	—
460	C₂²⋊C₄	—	2	2	no	—
461	C₂²⋊C₄	—	4	1	no	—
462	C₂²⋊C₄	—	4	1	no	—
463	C₄²	—	2	4	no	—
464	C₄²	—	2	2	no	—
465	C₄²	—	2	2	no	—
466	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
467	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
468	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
469	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
470	C₂³	—	2	2	no	—
471	C₂³	—	4	1	no	—
472	C₂³	#120×C₆₁	1	1	yes	characteristic
473	Q₈	—	2	2	no	—
474	Q₈	—	2	2	no	—
475	Q₈	—	4	2	no	—
476	Q₈	—	4	1	no	—
477	Q₈	—	4	1	no	—
478	Q₈	—	4	1	no	—
479	Q₈	—	4	1	no	—
480	Q₈	—	4	2	no	—
481	Q₈	—	8	1	no	—
482	Q₈	—	8	4	no	—
483	D₄	—	2	2	no	—
484	D₄	—	2	2	no	—
485	D₄	—	4	2	no	—
486	D₄	—	4	1	no	—
487	D₄	—	4	1	no	—
488	D₄	—	4	1	no	—
489	D₄	—	4	1	no	—
490	D₄	—	4	2	no	—
491	D₄	—	8	1	no	—
492	D₄	—	8	2	no	—
493	D₄	—	8	2	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	8	4	no	—
508	C₂×C₄	—	8	2	no	—
509	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
510	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
511	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
513	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
514	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
515	C₈	—	4	1	no	—
516	C₈	—	4	1	no	—
517	C₈	—	4	2	no	—
518	C₈	—	4	2	no	—
519	C₂²	—	2	1	no	—
520	C₂²	—	2	1	no	—
521	C₂²	—	2	1	no	—
522	C₂²	—	4	2	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	4	2	no	—
525	C₂²	—	8	1	no	—
526	C₂²	—	8	1	no	—
527	C₂²	—	8	1	no	—
528	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
529	C₄	—	2	1	no	—
530	C₄	—	2	1	no	—
531	C₄	—	2	1	no	—
532	C₄	—	4	2	no	—
533	C₄	—	4	2	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	2	no	—
536	C₄	—	4	2	no	—
537	C₄	—	8	1	no	—
538	C₄	C₆₁×#454	1	2	yes	—
539	C₄	C₆₁×#576	1	1	yes	characteristic
540	C₄	C₆₁×#577	1	1	yes	characteristic
541	C₄	C₆₁×#577	1	1	yes	characteristic
542	C₄	C₆₁×#578	1	1	yes	characteristic
543	C₂	—	4	1	no	—
544	C₂	—	8	2	no	—
545	C₂	—	16	1	no	—
546	C₂	C₆₁×#3012	1	1	yes	characteristic
547	C₂	C₆₁×#3013	1	1	yes	characteristic
548	C₂	C₂×C₆₁×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#3253	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ 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
1. C₆₁×#3550	C₂	semidirect
2. C₆₁×#3254	C₂	semidirect
4. C₆₁×#3131	C₂	semidirect
5. C₆₁×#1638	C₂	semidirect
6. C₆₁×#1637	C₂	semidirect
7. C₆₁×#1628	C₂	semidirect
8. C₆₁×#1627	C₂	semidirect
9. C₆₁×#1561	C₂	semidirect
10. C₆₁×#1493	C₂	semidirect
11. C₆₁×#1492	C₂	semidirect
244. #18697	C₆₁	direct (N × H)
269. #3550	C₂×C₆₁	semidirect
270. #3254	C₂×C₆₁	semidirect
272. #3131	C₂×C₆₁	semidirect
273. #1638	C₂×C₆₁	semidirect
274. #1637	C₂×C₆₁	semidirect
275. #1628	C₂×C₆₁	semidirect
276. #1627	C₂×C₆₁	semidirect
277. #1561	C₂×C₆₁	semidirect
278. #1493	C₂×C₆₁	semidirect
279. #1492	C₂×C₆₁	semidirect
325. C₆₁	#18697	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×#3550	2
C₆₁×#3254	2
C₆₁×#3253	2
C₆₁×#3131	2
C₆₁×#1638	2
C₆₁×#1637	2
C₆₁×#1628	2
C₆₁×#1627	2
C₆₁×#1561	2
C₆₁×#1493	2
C₆₁×#1492	2
#18697	61

Subgroups — C61×#18697 #29522130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18697 #29522130