Subgroups — C₆₁×#18753 #29522186

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#554	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3101	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#3078	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1643	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1632	C₂	1	4	yes	maximal, complemented
7	C₆₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1486	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#554	—	2	2	no	—
11	C₆₁×#554	—	2	2	no	—
12	C₆₁×#554	—	2	2	no	—
13	C₆₁×#554	C₂²	1	4	yes	—
14	C₆₁×#533	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#125×C₆₁	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₆₁	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₆₁	—	2	2	no	—
19	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₆₁	—	2	1	no	—
21	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₁	—	2	1	no	—
23	C₆₁×#494	C₂²	1	2	yes	complemented
24	C₆₁×#487	C₂²	1	2	yes	complemented
25	C₆₁×#483	—	2	1	no	—
26	C₆₁×#479	—	2	2	no	—
27	C₆₁×#477	C₂²	1	2	yes	complemented
28	C₆₁×#476	C₂²	1	2	yes	—
29	C₆₁×#475	C₂²	1	2	yes	complemented
30	C₆₁×#462	—	2	2	no	—
31	C₆₁×#461	C₂²	1	2	yes	—
32	C₆₁×#451	—	2	2	no	—
33	C₆₁×#450	—	2	1	no	—
34	C₄×SD₁₆×C₆₁	—	2	2	no	—
35	C₆₁×#431	C₂²	1	1	yes	characteristic
36	C₆₁×#431	C₂²	1	1	yes	characteristic
37	C₆₁×#428	—	2	1	no	—
38	C₆₁×#423	C₂²	1	1	yes	characteristic
39	C₆₁×#419	—	2	1	no	—
40	C₆₁×#333	C₂²	1	4	yes	complemented
41	C₆₁×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₆₁×D₄⋊C₈	C₂²	1	4	yes	—
43	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
44	C₂²×Q₈×C₆₁	D₄	1	2	yes	—
45	C₂²×D₄×C₆₁	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₁	—	2	2	no	—
47	C₂³×C₄×C₆₁	—	2	1	no	—
48	C₂×Q₁₆×C₆₁	—	2	4	no	—
49	C₂×SD₁₆×C₆₁	—	2	2	no	—
50	C₂×M₁₆×C₆₁	—	2	1	no	—
51	C₄⋊Q₈×C₆₁	—	2	1	no	—
52	C₄⋊Q₈×C₆₁	—	2	1	no	—
53	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
54	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
57	#125×C₆₁	—	2	4	no	—
58	#125×C₆₁	—	2	4	no	—
59	#125×C₆₁	—	2	1	no	—
60	C₂²⋊Q₈×C₆₁	—	2	4	no	—
61	C₂²⋊Q₈×C₆₁	—	2	8	no	—
62	C₂²⋊Q₈×C₆₁	—	2	4	no	—
63	C₂²⋊Q₈×C₆₁	—	2	8	no	—
64	C₂²⋊Q₈×C₆₁	—	2	1	no	—
65	C₂²⋊Q₈×C₆₁	—	2	1	no	—
66	C₄×Q₈×C₆₁	—	2	1	no	—
67	C₄×Q₈×C₆₁	—	2	1	no	—
68	C₄×Q₈×C₆₁	—	2	1	no	—
69	C₄×Q₈×C₆₁	—	2	1	no	—
70	C₄×Q₈×C₆₁	D₄	1	2	yes	—
71	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
72	C₄×D₄×C₆₁	—	2	4	no	—
73	C₄×D₄×C₆₁	—	2	4	no	—
74	C₄×D₄×C₆₁	—	2	1	no	—
75	C₄×D₄×C₆₁	—	2	1	no	—
76	C₄×D₄×C₆₁	D₄	1	2	yes	—
77	C₄×D₄×C₆₁	C₂³	1	2	yes	—
78	#117×C₆₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
80	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
81	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
82	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
86	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
90	C₂×C₄²×C₆₁	—	2	1	no	—
91	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
92	#107×C₆₁	—	2	2	no	—
93	#106×C₆₁	—	2	2	no	—
94	#106×C₆₁	—	2	2	no	—
95	C₄⋊C₈×C₆₁	—	2	1	no	—
96	C₄⋊C₈×C₆₁	—	2	1	no	—
97	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
98	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
99	#103×C₆₁	—	2	2	no	—
100	#103×C₆₁	—	2	2	no	—
101	#103×C₆₁	—	2	4	no	—
102	#103×C₆₁	—	2	2	no	—
103	#102×C₆₁	—	2	2	no	—
104	C₂²⋊C₈×C₆₁	—	2	1	no	—
105	C₂²⋊C₈×C₆₁	—	2	1	no	—
106	C₄×C₈×C₆₁	—	2	1	no	—
107	C₄×C₈×C₆₁	C₂³	1	2	yes	—
108	C₂⁴×C₆₁	—	2	1	no	—
109	C₂×Q₈×C₆₁	—	2	1	no	—
110	C₂×Q₈×C₆₁	—	2	2	no	—
111	C₂×Q₈×C₆₁	—	2	2	no	—
112	C₂×Q₈×C₆₁	—	2	2	no	—
113	C₂×Q₈×C₆₁	—	2	1	no	—
114	C₂×Q₈×C₆₁	—	2	1	no	—
115	C₂×Q₈×C₆₁	—	8	2	no	—
116	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
117	C₂×Q₈×C₆₁	C₂×D₄	1	4	yes	—
118	C₂×D₄×C₆₁	—	2	4	no	—
119	C₂×D₄×C₆₁	—	2	1	no	—
120	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₆₁	—	2	1	no	—
122	C₂²×C₄×C₆₁	—	2	1	no	—
123	C₂²×C₄×C₆₁	—	2	1	no	—
124	C₂²×C₄×C₆₁	—	2	2	no	—
125	C₂²×C₄×C₆₁	—	2	8	no	—
126	C₂²×C₄×C₆₁	—	2	2	no	—
127	C₂²×C₄×C₆₁	—	2	2	no	—
128	C₂²×C₄×C₆₁	—	4	2	no	—
129	C₂²×C₄×C₆₁	—	4	2	no	—
130	C₂²×C₄×C₆₁	—	8	4	no	—
131	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
134	Q₁₆×C₆₁	—	8	4	no	—
135	SD₁₆×C₆₁	—	4	4	no	—
136	M₁₆×C₆₁	—	4	2	no	—
137	C₂×C₈×C₆₁	—	2	2	no	—
138	C₂×C₈×C₆₁	—	2	2	no	—
139	C₂×C₈×C₆₁	—	2	2	no	—
140	C₄⋊C₄×C₆₁	—	2	8	no	—
141	C₄⋊C₄×C₆₁	—	2	8	no	—
142	C₄⋊C₄×C₆₁	—	2	1	no	—
143	C₄⋊C₄×C₆₁	—	2	2	no	—
144	C₄⋊C₄×C₆₁	—	2	2	no	—
145	C₄⋊C₄×C₆₁	—	2	2	no	—
146	C₄⋊C₄×C₆₁	—	2	1	no	—
147	C₄⋊C₄×C₆₁	—	2	2	no	—
148	C₄⋊C₄×C₆₁	—	2	2	no	—
149	C₄⋊C₄×C₆₁	—	2	2	no	—
150	C₄⋊C₄×C₆₁	—	2	1	no	—
151	C₄⋊C₄×C₆₁	—	2	1	no	—
152	C₄⋊C₄×C₆₁	—	2	1	no	—
153	C₄⋊C₄×C₆₁	—	2	1	no	—
154	C₄⋊C₄×C₆₁	—	4	1	no	—
155	C₄⋊C₄×C₆₁	—	4	1	no	—
156	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
158	C₂²⋊C₄×C₆₁	—	2	8	no	—
159	C₂²⋊C₄×C₆₁	—	2	2	no	—
160	C₂²⋊C₄×C₆₁	—	2	2	no	—
161	C₂²⋊C₄×C₆₁	—	4	1	no	—
162	C₄²×C₆₁	—	2	2	no	—
163	C₄²×C₆₁	—	2	2	no	—
164	C₄²×C₆₁	—	4	1	no	—
165	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
166	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
167	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
169	C₂³×C₆₁	—	2	4	no	—
170	C₂³×C₆₁	—	2	2	no	—
171	C₂³×C₆₁	—	4	2	no	—
172	C₂³×C₆₁	—	4	2	no	—
173	C₂³×C₆₁	#120	1	1	yes	characteristic
174	Q₈×C₆₁	—	2	2	no	—
175	Q₈×C₆₁	—	2	2	no	—
176	Q₈×C₆₁	—	4	4	no	—
177	Q₈×C₆₁	—	4	2	no	—
178	Q₈×C₆₁	—	4	2	no	—
179	Q₈×C₆₁	—	4	2	no	—
180	Q₈×C₆₁	—	8	2	no	—
181	D₄×C₆₁	—	2	4	no	—
182	D₄×C₆₁	—	2	2	no	—
183	D₄×C₆₁	#137	1	4	yes	—
184	C₂×C₄×C₆₁	—	2	2	no	—
185	C₂×C₄×C₆₁	—	2	2	no	—
186	C₂×C₄×C₆₁	—	2	2	no	—
187	C₂×C₄×C₆₁	—	2	2	no	—
188	C₂×C₄×C₆₁	—	2	4	no	—
189	C₂×C₄×C₆₁	—	4	1	no	—
190	C₂×C₄×C₆₁	—	4	2	no	—
191	C₂×C₄×C₆₁	—	4	4	no	—
192	C₂×C₄×C₆₁	—	4	2	no	—
193	C₂×C₄×C₆₁	—	4	2	no	—
194	C₂×C₄×C₆₁	—	4	2	no	—
195	C₂×C₄×C₆₁	—	4	2	no	—
196	C₂×C₄×C₆₁	—	4	4	no	—
197	C₂×C₄×C₆₁	—	4	1	no	—
198	C₂×C₄×C₆₁	—	4	1	no	—
199	C₂×C₄×C₆₁	—	8	8	no	—
200	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
201	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
202	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
203	C₂×C₄×C₆₁	#135	1	2	yes	—
204	C₂×C₄×C₆₁	#137	1	2	yes	—
205	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
208	C₈×C₆₁	—	4	2	no	—
209	C₈×C₆₁	—	4	2	no	—
210	C₈×C₆₁	—	4	2	no	—
211	#18753	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂²×C₆₁	—	2	1	no	—
213	C₂²×C₆₁	—	2	1	no	—
214	C₂²×C₆₁	—	2	4	no	—
215	C₂²×C₆₁	—	4	8	no	—
216	C₂²×C₆₁	—	4	2	no	—
217	C₂²×C₆₁	—	4	2	no	—
218	C₂²×C₆₁	#452	1	2	yes	—
219	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
220	C₄×C₆₁	—	2	1	no	—
221	C₄×C₆₁	—	2	1	no	—
222	C₄×C₆₁	—	2	1	no	—
223	C₄×C₆₁	—	4	2	no	—
224	C₄×C₆₁	—	4	2	no	—
225	C₄×C₆₁	—	4	2	no	—
226	C₄×C₆₁	—	4	4	no	—
227	C₄×C₆₁	—	8	1	no	—
228	C₄×C₆₁	—	8	1	no	—
229	C₄×C₆₁	#452	1	2	yes	—
230	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
231	C₄×C₆₁	C₂×#137	1	1	yes	characteristic
232	C₄×C₆₁	C₂×#137	1	2	yes	—
233	C₂×#554	C₂×C₆₁	1	1	yes	characteristic
234	#3209	C₂×C₆₁	1	1	yes	characteristic, complemented
235	#3101	C₂×C₆₁	1	1	yes	characteristic
236	#3078	C₂×C₆₁	1	1	yes	characteristic, complemented
237	#1643	C₂×C₆₁	1	4	yes	complemented
238	#1632	C₂×C₆₁	1	4	yes	complemented
239	#1548	C₂×C₆₁	1	1	yes	characteristic, complemented
240	#1486	C₂×C₆₁	1	1	yes	characteristic
241	#1479	C₂×C₆₁	1	1	yes	characteristic, complemented
242	C₂×C₆₁	—	2	2	no	—
243	C₂×C₆₁	—	4	4	no	—
244	C₂×C₆₁	C₂×#452	1	1	yes	characteristic
245	C₂×C₆₁	#3004	1	1	yes	characteristic
246	C₂×C₆₁	#3019	1	1	yes	characteristic
247	#554	—	2	2	no	—
248	#554	—	2	2	no	—
249	#554	—	2	2	no	—
250	#554	C₂²×C₆₁	1	4	yes	—
251	#533	C₂²×C₆₁	1	1	yes	characteristic
252	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
253	C₂×#125	—	2	1	no	—
254	C₂×C₂²⋊Q₈	—	2	1	no	—
255	C₂×C₂²⋊Q₈	—	2	2	no	—
256	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
257	C₂×C₄×D₄	—	2	1	no	—
258	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
259	C₂²×C₄⋊C₄	—	2	1	no	—
260	#494	C₂²×C₆₁	1	2	yes	complemented
261	#487	C₂²×C₆₁	1	2	yes	complemented
262	#483	—	2	1	no	—
263	#479	—	2	2	no	—
264	#477	C₂²×C₆₁	1	2	yes	complemented
265	#476	C₂²×C₆₁	1	2	yes	—
266	#475	C₂²×C₆₁	1	2	yes	complemented
267	#462	—	2	2	no	—
268	#461	C₂²×C₆₁	1	2	yes	—
269	#451	—	2	2	no	—
270	#450	—	2	1	no	—
271	C₄×SD₁₆	—	2	2	no	—
272	#431	C₂²×C₆₁	1	1	yes	characteristic
273	#431	C₂²×C₆₁	1	1	yes	characteristic
274	#428	—	2	1	no	—
275	#423	C₂²×C₆₁	1	1	yes	characteristic
276	#419	—	2	1	no	—
277	#333	C₂²×C₆₁	1	4	yes	complemented
278	Q₈⋊C₈	C₂²×C₆₁	1	4	yes	complemented
279	D₄⋊C₈	C₂²×C₆₁	1	4	yes	—
280	C₆₁	#18753	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
282	C₂²×Q₈	D₄×C₆₁	1	2	yes	—
283	C₂²×D₄	D₄×C₆₁	1	1	yes	characteristic
284	C₂³×C₄	—	2	2	no	—
285	C₂³×C₄	—	2	1	no	—
286	C₂×Q₁₆	—	2	4	no	—
287	C₂×SD₁₆	—	2	2	no	—
288	C₂×M₁₆	—	2	1	no	—
289	C₄⋊Q₈	—	2	1	no	—
290	C₄⋊Q₈	—	2	1	no	—
291	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
292	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
293	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
294	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
295	#125	—	2	4	no	—
296	#125	—	2	4	no	—
297	#125	—	2	1	no	—
298	C₂²⋊Q₈	—	2	4	no	—
299	C₂²⋊Q₈	—	2	8	no	—
300	C₂²⋊Q₈	—	2	4	no	—
301	C₂²⋊Q₈	—	2	8	no	—
302	C₂²⋊Q₈	—	2	1	no	—
303	C₂²⋊Q₈	—	2	1	no	—
304	C₄×Q₈	—	2	1	no	—
305	C₄×Q₈	—	2	1	no	—
306	C₄×Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	D₄×C₆₁	1	2	yes	—
309	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
310	C₄×D₄	—	2	4	no	—
311	C₄×D₄	—	2	4	no	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	—	2	1	no	—
314	C₄×D₄	D₄×C₆₁	1	2	yes	—
315	C₄×D₄	C₂³×C₆₁	1	2	yes	—
316	#117	—	2	1	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	2	4	no	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	4	no	—
324	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
325	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	—	2	1	no	—
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
330	#107	—	2	2	no	—
331	#106	—	2	2	no	—
332	#106	—	2	2	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	—	2	1	no	—
335	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
336	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
337	#103	—	2	2	no	—
338	#103	—	2	2	no	—
339	#103	—	2	4	no	—
340	#103	—	2	2	no	—
341	#102	—	2	2	no	—
342	C₂²⋊C₈	—	2	1	no	—
343	C₂²⋊C₈	—	2	1	no	—
344	C₄×C₈	—	2	1	no	—
345	C₄×C₈	C₂³×C₆₁	1	2	yes	—
346	C₂⁴	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	1	no	—
352	C₂×Q₈	—	2	1	no	—
353	C₂×Q₈	—	8	2	no	—
354	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
355	C₂×Q₈	C₂×D₄×C₆₁	1	4	yes	—
356	C₂×D₄	—	2	4	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	8	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	4	2	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	8	4	no	—
369	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
372	Q₁₆	—	8	4	no	—
373	SD₁₆	—	4	4	no	—
374	M₁₆	—	4	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₄⋊C₄	—	2	8	no	—
379	C₄⋊C₄	—	2	8	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	1	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	4	1	no	—
393	C₄⋊C₄	—	4	1	no	—
394	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
395	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
396	C₂²⋊C₄	—	2	8	no	—
397	C₂²⋊C₄	—	2	2	no	—
398	C₂²⋊C₄	—	2	2	no	—
399	C₂²⋊C₄	—	4	1	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	4	1	no	—
403	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
404	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
405	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
406	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
407	C₂³	—	2	4	no	—
408	C₂³	—	2	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	#120×C₆₁	1	1	yes	characteristic
412	Q₈	—	2	2	no	—
413	Q₈	—	2	2	no	—
414	Q₈	—	4	4	no	—
415	Q₈	—	4	2	no	—
416	Q₈	—	4	2	no	—
417	Q₈	—	4	2	no	—
418	Q₈	—	8	2	no	—
419	D₄	—	2	4	no	—
420	D₄	—	2	2	no	—
421	D₄	#137×C₆₁	1	4	yes	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	4	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	4	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	8	8	no	—
438	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
439	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
440	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
441	C₂×C₄	#135×C₆₁	1	2	yes	—
442	C₂×C₄	#137×C₆₁	1	2	yes	—
443	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
444	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
445	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
446	C₈	—	4	2	no	—
447	C₈	—	4	2	no	—
448	C₈	—	4	2	no	—
449	C₂²	—	2	1	no	—
450	C₂²	—	2	1	no	—
451	C₂²	—	2	4	no	—
452	C₂²	—	4	8	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	2	no	—
455	C₂²	C₆₁×#452	1	2	yes	—
456	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
457	C₄	—	2	1	no	—
458	C₄	—	2	1	no	—
459	C₄	—	2	1	no	—
460	C₄	—	4	2	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	2	no	—
463	C₄	—	4	4	no	—
464	C₄	—	8	1	no	—
465	C₄	—	8	1	no	—
466	C₄	C₆₁×#452	1	2	yes	—
467	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
468	C₄	C₂×#137×C₆₁	1	1	yes	characteristic
469	C₄	C₂×#137×C₆₁	1	2	yes	—
470	C₂	—	2	2	no	—
471	C₂	—	4	4	no	—
472	C₂	C₂×C₆₁×#452	1	1	yes	characteristic
473	C₂	C₆₁×#3004	1	1	yes	characteristic
474	C₂	C₆₁×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₂²×D₄×C₆₁	D₄
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
2. C₆₁×#3209	C₂	semidirect
4. C₆₁×#3078	C₂	semidirect
5. C₆₁×#1643	C₂	semidirect
6. C₆₁×#1632	C₂	semidirect
7. C₆₁×#1548	C₂	semidirect
9. C₆₁×#1479	C₂	semidirect
23. C₆₁×#494	C₂²	semidirect
24. C₆₁×#487	C₂²	semidirect
27. C₆₁×#477	C₂²	semidirect
29. C₆₁×#475	C₂²	semidirect
40. C₆₁×#333	C₂²	semidirect
41. C₆₁×Q₈⋊C₈	C₂²	semidirect
211. #18753	C₆₁	direct (N × H)
234. #3209	C₂×C₆₁	semidirect
236. #3078	C₂×C₆₁	semidirect
237. #1643	C₂×C₆₁	semidirect
238. #1632	C₂×C₆₁	semidirect
239. #1548	C₂×C₆₁	semidirect
241. #1479	C₂×C₆₁	semidirect
260. #494	C₂²×C₆₁	semidirect
261. #487	C₂²×C₆₁	semidirect
264. #477	C₂²×C₆₁	semidirect
266. #475	C₂²×C₆₁	semidirect
277. #333	C₂²×C₆₁	semidirect
278. Q₈⋊C₈	C₂²×C₆₁	semidirect
280. C₆₁	#18753	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#554	2
C₆₁×#3209	2
C₆₁×#3101	2
C₆₁×#3078	2
C₆₁×#1643	2
C₆₁×#1632	2
C₆₁×#1548	2
C₆₁×#1486	2
C₆₁×#1479	2
#18753	61

Subgroups — C61×#18753 #29522186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18753 #29522186