Subgroups — C₆₁×#18680 #29522113

Order 15616.

1000 conjugacy classes of subgroups.

The subgroup lattice has 590 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, complemented
2	C₆₁×#3245	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3036	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1654	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1646	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1625	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1624	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1550	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1492	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₆₁	—	2	1	no	—
13	C₆₁×#559	—	2	2	no	—
14	C₆₁×#556	—	2	1	no	—
15	C₆₁×#555	C₂²	1	2	yes	—
16	C₆₁×#553	—	2	2	no	—
17	C₆₁×#550	—	2	1	no	—
18	C₆₁×#547	C₂²	1	2	yes	—
19	C₆₁×#538	—	2	2	no	—
20	C₆₁×#535	—	2	1	no	—
21	C₆₁×#533	C₂²	1	1	yes	characteristic
22	C₆₁×#532	C₂²	1	1	yes	characteristic
23	C₆₁×#532	C₂²	1	1	yes	characteristic
24	C₆₁×#525	—	2	2	no	—
25	C₆₁×#520	—	2	1	no	—
26	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#498	C₂²	1	1	yes	characteristic, complemented
28	C₆₁×#492	C₂²	1	1	yes	characteristic, complemented
29	C₆₁×#487	C₂²	1	1	yes	characteristic, complemented
30	C₆₁×#486	C₂²	1	1	yes	characteristic, complemented
31	C₆₁×#474	—	2	1	no	—
32	C₆₁×#474	C₂²	1	2	yes	complemented
33	C₆₁×#466	—	2	2	no	—
34	C₆₁×#465	—	2	2	no	—
35	C₆₁×#463	—	2	2	no	—
36	C₆₁×#461	C₂²	1	2	yes	—
37	C₆₁×#459	—	2	1	no	—
38	C₆₁×#459	C₂²	1	2	yes	—
39	C₆₁×#459	C₂²	1	2	yes	complemented
40	C₄×SD₁₆×C₆₁	—	2	2	no	—
41	C₆₁×#423	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
43	C₆₁×#417	—	2	1	no	—
44	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
45	C₆₁×#333	C₂²	1	2	yes	complemented
46	C₆₁×#332	C₂²	1	2	yes	complemented
47	C₆₁×D₄⋊C₈	C₂²	1	2	yes	—
48	C₆₁×D₄⋊C₈	C₂²	1	2	yes	—
49	C₆₁×D₄⋊C₈	C₂²	1	2	yes	complemented
50	C₆₁×D₄⋊C₈	C₂²	1	2	yes	complemented
51	C₂×#41×C₆₁	—	2	2	no	—
52	C₂×#41×C₆₁	—	2	1	no	—
53	C₂×#41×C₆₁	—	2	1	no	—
54	C₂²×D₄×C₆₁	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₆₁	—	2	1	no	—
56	AGL(1,ℤ/8)×C₆₁	—	4	4	no	—
57	C₂×SD₁₆×C₆₁	—	2	2	no	—
58	C₂×SD₁₆×C₆₁	—	2	2	no	—
59	C₂×D₈×C₆₁	—	2	2	no	—
60	C₂×D₈×C₆₁	—	2	2	no	—
61	C₂×M₁₆×C₆₁	—	2	1	no	—
62	C₂²×C₈×C₆₁	—	2	1	no	—
63	C₂²×C₈×C₆₁	—	2	1	no	—
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	—	2	2	no	—
69	#127×C₆₁	D₄	1	1	yes	characteristic
70	#127×C₆₁	C₂³	1	1	yes	characteristic
71	#126×C₆₁	—	2	4	no	—
72	#125×C₆₁	—	2	1	no	—
73	#124×C₆₁	—	2	2	no	—
74	#124×C₆₁	—	2	4	no	—
75	#124×C₆₁	—	2	1	no	—
76	#124×C₆₁	—	2	1	no	—
77	#123×C₆₁	—	2	4	no	—
78	#123×C₆₁	—	2	4	no	—
79	C₂²⋊Q₈×C₆₁	—	2	1	no	—
80	C₂²⋊Q₈×C₆₁	—	2	2	no	—
81	C₂²⋊Q₈×C₆₁	D₄	1	2	yes	—
82	#121×C₆₁	—	2	4	no	—
83	#121×C₆₁	—	2	2	no	—
84	#121×C₆₁	—	2	2	no	—
85	#121×C₆₁	—	2	2	no	—
86	#121×C₆₁	—	2	1	no	—
87	#121×C₆₁	—	2	1	no	—
88	#121×C₆₁	—	2	1	no	—
89	#121×C₆₁	D₄	1	2	yes	—
90	#120×C₆₁	—	2	4	no	—
91	C₄×Q₈×C₆₁	—	2	2	no	—
92	C₄×Q₈×C₆₁	—	2	1	no	—
93	C₄×Q₈×C₆₁	D₄	1	2	yes	—
94	C₄×D₄×C₆₁	—	2	4	no	—
95	C₄×D₄×C₆₁	—	2	2	no	—
96	C₄×D₄×C₆₁	—	2	4	no	—
97	C₄×D₄×C₆₁	—	2	1	no	—
98	C₄×D₄×C₆₁	—	2	1	no	—
99	C₄×D₄×C₆₁	—	2	1	no	—
100	C₄×D₄×C₆₁	—	2	1	no	—
101	C₄×D₄×C₆₁	—	2	1	no	—
102	C₄×D₄×C₆₁	D₄	1	2	yes	—
103	C₄×D₄×C₆₁	C₂³	1	2	yes	—
104	C₄×D₄×C₆₁	C₂³	1	2	yes	—
105	#117×C₆₁	—	2	1	no	—
106	#117×C₆₁	—	2	2	no	—
107	#117×C₆₁	—	2	2	no	—
108	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
109	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
111	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
112	#107×C₆₁	—	2	2	no	—
113	#106×C₆₁	—	2	2	no	—
114	C₄⋊C₈×C₆₁	—	2	1	no	—
115	C₄⋊C₈×C₆₁	—	2	1	no	—
116	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
117	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
118	#103×C₆₁	—	2	2	no	—
119	#103×C₆₁	—	2	2	no	—
120	#102×C₆₁	—	2	2	no	—
121	#102×C₆₁	—	2	2	no	—
122	#102×C₆₁	—	2	2	no	—
123	#102×C₆₁	—	2	2	no	—
124	C₄×C₈×C₆₁	—	2	1	no	—
125	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
126	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
127	C₂⁴×C₆₁	—	2	1	no	—
128	#41×C₆₁	—	4	2	no	—
129	#41×C₆₁	—	4	2	no	—
130	#41×C₆₁	—	4	2	no	—
131	#41×C₆₁	—	4	2	no	—
132	#41×C₆₁	—	8	2	no	—
133	#41×C₆₁	—	8	2	no	—
134	C₂×Q₈×C₆₁	—	2	1	no	—
135	C₂×Q₈×C₆₁	—	2	1	no	—
136	C₂×Q₈×C₆₁	—	2	1	no	—
137	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
138	C₂×D₄×C₆₁	—	2	2	no	—
139	C₂×D₄×C₆₁	—	2	4	no	—
140	C₂×D₄×C₆₁	—	2	4	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	—	2	2	no	—
145	C₂×D₄×C₆₁	—	2	2	no	—
146	C₂×D₄×C₆₁	—	2	1	no	—
147	C₂×D₄×C₆₁	—	2	1	no	—
148	C₂×D₄×C₆₁	—	2	1	no	—
149	C₂×D₄×C₆₁	—	4	1	no	—
150	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
151	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₆₁	—	2	4	no	—
153	C₂²×C₄×C₆₁	—	2	1	no	—
154	C₂²×C₄×C₆₁	—	2	2	no	—
155	C₂²×C₄×C₆₁	—	2	2	no	—
156	C₂²×C₄×C₆₁	—	2	2	no	—
157	C₂²×C₄×C₆₁	—	4	1	no	—
158	C₂²×C₄×C₆₁	—	8	1	no	—
159	C₂²×C₄×C₆₁	—	8	1	no	—
160	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
163	SD₁₆×C₆₁	—	4	4	no	—
164	SD₁₆×C₆₁	—	4	4	no	—
165	D₈×C₆₁	—	4	4	no	—
166	D₈×C₆₁	—	8	2	no	—
167	M₁₆×C₆₁	—	4	2	no	—
168	C₂×C₈×C₆₁	—	2	1	no	—
169	C₂×C₈×C₆₁	—	2	1	no	—
170	C₂×C₈×C₆₁	—	2	2	no	—
171	C₂×C₈×C₆₁	—	2	2	no	—
172	C₂×C₈×C₆₁	—	4	2	no	—
173	C₂×C₈×C₆₁	—	8	1	no	—
174	C₄⋊C₄×C₆₁	—	2	4	no	—
175	C₄⋊C₄×C₆₁	—	2	1	no	—
176	C₄⋊C₄×C₆₁	—	2	2	no	—
177	C₄⋊C₄×C₆₁	—	2	1	no	—
178	C₄⋊C₄×C₆₁	—	2	2	no	—
179	C₄⋊C₄×C₆₁	—	2	1	no	—
180	C₄⋊C₄×C₆₁	—	2	1	no	—
181	C₄⋊C₄×C₆₁	—	2	1	no	—
182	C₄⋊C₄×C₆₁	—	2	1	no	—
183	C₄⋊C₄×C₆₁	—	2	1	no	—
184	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
186	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
187	C₂²⋊C₄×C₆₁	—	2	4	no	—
188	C₂²⋊C₄×C₆₁	—	2	4	no	—
189	C₂²⋊C₄×C₆₁	—	2	4	no	—
190	C₂²⋊C₄×C₆₁	—	2	2	no	—
191	C₂²⋊C₄×C₆₁	—	2	2	no	—
192	C₂²⋊C₄×C₆₁	—	2	2	no	—
193	C₂²⋊C₄×C₆₁	—	2	2	no	—
194	C₂²⋊C₄×C₆₁	—	2	2	no	—
195	C₂²⋊C₄×C₆₁	—	4	1	no	—
196	C₂²⋊C₄×C₆₁	—	4	1	no	—
197	C₄²×C₆₁	—	2	4	no	—
198	C₄²×C₆₁	—	2	2	no	—
199	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
202	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
203	C₂³×C₆₁	—	2	2	no	—
204	C₂³×C₆₁	—	2	2	no	—
205	C₂³×C₆₁	—	4	2	no	—
206	C₂³×C₆₁	—	4	1	no	—
207	C₂³×C₆₁	—	8	1	no	—
208	C₂³×C₆₁	—	8	1	no	—
209	C₂³×C₆₁	#120	1	1	yes	characteristic
210	Q₈×C₆₁	—	4	2	no	—
211	Q₈×C₆₁	—	4	1	no	—
212	Q₈×C₆₁	—	4	1	no	—
213	Q₈×C₆₁	—	4	2	no	—
214	Q₈×C₆₁	—	4	1	no	—
215	Q₈×C₆₁	—	4	1	no	—
216	D₄×C₆₁	—	2	2	no	—
217	D₄×C₆₁	—	2	2	no	—
218	D₄×C₆₁	—	2	2	no	—
219	D₄×C₆₁	—	2	2	no	—
220	D₄×C₆₁	—	4	2	no	—
221	D₄×C₆₁	—	4	4	no	—
222	D₄×C₆₁	—	4	1	no	—
223	D₄×C₆₁	—	4	1	no	—
224	D₄×C₆₁	—	4	1	no	—
225	D₄×C₆₁	—	4	1	no	—
226	D₄×C₆₁	—	4	2	no	—
227	D₄×C₆₁	—	4	2	no	—
228	D₄×C₆₁	—	8	1	no	—
229	D₄×C₆₁	—	8	4	no	—
230	D₄×C₆₁	—	8	4	no	—
231	D₄×C₆₁	—	8	2	no	—
232	C₂×C₄×C₆₁	—	2	1	no	—
233	C₂×C₄×C₆₁	—	2	1	no	—
234	C₂×C₄×C₆₁	—	2	2	no	—
235	C₂×C₄×C₆₁	—	2	2	no	—
236	C₂×C₄×C₆₁	—	2	2	no	—
237	C₂×C₄×C₆₁	—	2	2	no	—
238	C₂×C₄×C₆₁	—	4	1	no	—
239	C₂×C₄×C₆₁	—	4	2	no	—
240	C₂×C₄×C₆₁	—	4	2	no	—
241	C₂×C₄×C₆₁	—	4	2	no	—
242	C₂×C₄×C₆₁	—	4	2	no	—
243	C₂×C₄×C₆₁	—	4	1	no	—
244	C₂×C₄×C₆₁	—	8	2	no	—
245	C₂×C₄×C₆₁	—	8	1	no	—
246	C₂×C₄×C₆₁	—	8	4	no	—
247	C₂×C₄×C₆₁	—	8	2	no	—
248	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
249	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
250	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
251	C₂×C₄×C₆₁	#135	1	2	yes	—
252	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
253	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
256	C₈×C₆₁	—	2	2	no	—
257	C₈×C₆₁	—	4	1	no	—
258	C₈×C₆₁	—	4	2	no	—
259	C₈×C₆₁	—	4	2	no	—
260	#18680	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
261	C₂²×C₆₁	—	2	1	no	—
262	C₂²×C₆₁	—	2	1	no	—
263	C₂²×C₆₁	—	4	2	no	—
264	C₂²×C₆₁	—	4	2	no	—
265	C₂²×C₆₁	—	4	2	no	—
266	C₂²×C₆₁	—	4	2	no	—
267	C₂²×C₆₁	—	4	2	no	—
268	C₂²×C₆₁	—	4	2	no	—
269	C₂²×C₆₁	—	8	4	no	—
270	C₂²×C₆₁	—	8	1	no	—
271	C₂²×C₆₁	—	8	1	no	—
272	C₂²×C₆₁	—	8	1	no	—
273	C₂²×C₆₁	#449	1	2	yes	—
274	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
275	C₄×C₆₁	—	2	1	no	—
276	C₄×C₆₁	—	2	1	no	—
277	C₄×C₆₁	—	2	1	no	—
278	C₄×C₆₁	—	4	2	no	—
279	C₄×C₆₁	—	4	2	no	—
280	C₄×C₆₁	—	4	2	no	—
281	C₄×C₆₁	—	4	2	no	—
282	C₄×C₆₁	—	8	1	no	—
283	C₄×C₆₁	#449	1	2	yes	—
284	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
285	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
286	C₄×C₆₁	#576	1	1	yes	characteristic
287	C₄×C₆₁	#577	1	1	yes	characteristic
288	#3506	C₂×C₆₁	1	1	yes	characteristic, complemented
289	#3245	C₂×C₆₁	1	1	yes	characteristic, complemented
290	#3167	C₂×C₆₁	1	1	yes	characteristic, complemented
291	#3036	C₂×C₆₁	1	1	yes	characteristic, complemented
292	#1654	C₂×C₆₁	1	2	yes	complemented
293	#1646	C₂×C₆₁	1	2	yes	complemented
294	#1625	C₂×C₆₁	1	2	yes	complemented
295	#1624	C₂×C₆₁	1	2	yes	complemented
296	#1550	C₂×C₆₁	1	1	yes	characteristic, complemented
297	#1492	C₂×C₆₁	1	1	yes	characteristic, complemented
298	C₂×D₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic, complemented
299	C₂×C₆₁	—	2	2	no	—
300	C₂×C₆₁	—	8	2	no	—
301	C₂×C₆₁	—	8	2	no	—
302	C₂×C₆₁	—	16	1	no	—
303	C₂×C₆₁	C₂×#449	1	1	yes	characteristic
304	C₂×C₆₁	#3012	1	1	yes	characteristic
305	C₂×C₆₁	#3018	1	1	yes	characteristic
306	C₂×AGL(1,ℤ/8)	—	2	1	no	—
307	#559	—	2	2	no	—
308	#556	—	2	1	no	—
309	#555	C₂²×C₆₁	1	2	yes	—
310	#553	—	2	2	no	—
311	#550	—	2	1	no	—
312	#547	C₂²×C₆₁	1	2	yes	—
313	#538	—	2	2	no	—
314	#535	—	2	1	no	—
315	#533	C₂²×C₆₁	1	1	yes	characteristic
316	#532	C₂²×C₆₁	1	1	yes	characteristic
317	#532	C₂²×C₆₁	1	1	yes	characteristic
318	#525	—	2	2	no	—
319	#520	—	2	1	no	—
320	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
321	#498	C₂²×C₆₁	1	1	yes	characteristic, complemented
322	#492	C₂²×C₆₁	1	1	yes	characteristic, complemented
323	#487	C₂²×C₆₁	1	1	yes	characteristic, complemented
324	#486	C₂²×C₆₁	1	1	yes	characteristic, complemented
325	#474	—	2	1	no	—
326	#474	C₂²×C₆₁	1	2	yes	complemented
327	#466	—	2	2	no	—
328	#465	—	2	2	no	—
329	#463	—	2	2	no	—
330	#461	C₂²×C₆₁	1	2	yes	—
331	#459	—	2	1	no	—
332	#459	C₂²×C₆₁	1	2	yes	—
333	#459	C₂²×C₆₁	1	2	yes	complemented
334	C₄×SD₁₆	—	2	2	no	—
335	#423	C₂²×C₆₁	1	1	yes	characteristic
336	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
337	#417	—	2	1	no	—
338	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
339	#333	C₂²×C₆₁	1	2	yes	complemented
340	#332	C₂²×C₆₁	1	2	yes	complemented
341	D₄⋊C₈	C₂²×C₆₁	1	2	yes	—
342	D₄⋊C₈	C₂²×C₆₁	1	2	yes	—
343	D₄⋊C₈	C₂²×C₆₁	1	2	yes	complemented
344	D₄⋊C₈	C₂²×C₆₁	1	2	yes	complemented
345	C₆₁	#18680	1	1	yes	characteristic, direct product factor, Sylow
346	C₂×#41	—	2	2	no	—
347	C₂×#41	—	2	1	no	—
348	C₂×#41	—	2	1	no	—
349	C₂²×D₄	D₄×C₆₁	1	1	yes	characteristic
350	C₂³×C₄	—	2	1	no	—
351	AGL(1,ℤ/8)	—	4	4	no	—
352	C₂×SD₁₆	—	2	2	no	—
353	C₂×SD₁₆	—	2	2	no	—
354	C₂×D₈	—	2	2	no	—
355	C₂×D₈	—	2	2	no	—
356	C₂×M₁₆	—	2	1	no	—
357	C₂²×C₈	—	2	1	no	—
358	C₂²×C₈	—	2	1	no	—
359	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
360	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
361	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
362	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
363	#127	—	2	2	no	—
364	#127	D₄×C₆₁	1	1	yes	characteristic
365	#127	C₂³×C₆₁	1	1	yes	characteristic
366	#126	—	2	4	no	—
367	#125	—	2	1	no	—
368	#124	—	2	2	no	—
369	#124	—	2	4	no	—
370	#124	—	2	1	no	—
371	#124	—	2	1	no	—
372	#123	—	2	4	no	—
373	#123	—	2	4	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	C₂²⋊Q₈	—	2	2	no	—
376	C₂²⋊Q₈	D₄×C₆₁	1	2	yes	—
377	#121	—	2	4	no	—
378	#121	—	2	2	no	—
379	#121	—	2	2	no	—
380	#121	—	2	2	no	—
381	#121	—	2	1	no	—
382	#121	—	2	1	no	—
383	#121	—	2	1	no	—
384	#121	D₄×C₆₁	1	2	yes	—
385	#120	—	2	4	no	—
386	C₄×Q₈	—	2	2	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	D₄×C₆₁	1	2	yes	—
389	C₄×D₄	—	2	4	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	2	4	no	—
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	—	2	1	no	—
394	C₄×D₄	—	2	1	no	—
395	C₄×D₄	—	2	1	no	—
396	C₄×D₄	—	2	1	no	—
397	C₄×D₄	D₄×C₆₁	1	2	yes	—
398	C₄×D₄	C₂³×C₆₁	1	2	yes	—
399	C₄×D₄	C₂³×C₆₁	1	2	yes	—
400	#117	—	2	1	no	—
401	#117	—	2	2	no	—
402	#117	—	2	2	no	—
403	C₂×C₄⋊C₄	—	2	2	no	—
404	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
407	#107	—	2	2	no	—
408	#106	—	2	2	no	—
409	C₄⋊C₈	—	2	1	no	—
410	C₄⋊C₈	—	2	1	no	—
411	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
412	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
413	#103	—	2	2	no	—
414	#103	—	2	2	no	—
415	#102	—	2	2	no	—
416	#102	—	2	2	no	—
417	#102	—	2	2	no	—
418	#102	—	2	2	no	—
419	C₄×C₈	—	2	1	no	—
420	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
421	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
422	C₂⁴	—	2	1	no	—
423	#41	—	4	2	no	—
424	#41	—	4	2	no	—
425	#41	—	4	2	no	—
426	#41	—	4	2	no	—
427	#41	—	8	2	no	—
428	#41	—	8	2	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	4	1	no	—
445	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
446	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	4	1	no	—
453	C₂²×C₄	—	8	1	no	—
454	C₂²×C₄	—	8	1	no	—
455	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
458	SD₁₆	—	4	4	no	—
459	SD₁₆	—	4	4	no	—
460	D₈	—	4	4	no	—
461	D₈	—	8	2	no	—
462	M₁₆	—	4	2	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	2	no	—
466	C₂×C₈	—	2	2	no	—
467	C₂×C₈	—	4	2	no	—
468	C₂×C₈	—	8	1	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
480	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
481	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	—	2	2	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	2	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	4	1	no	—
491	C₂²⋊C₄	—	4	1	no	—
492	C₄²	—	2	4	no	—
493	C₄²	—	2	2	no	—
494	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
495	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
496	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
497	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
498	C₂³	—	2	2	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	4	2	no	—
501	C₂³	—	4	1	no	—
502	C₂³	—	8	1	no	—
503	C₂³	—	8	1	no	—
504	C₂³	#120×C₆₁	1	1	yes	characteristic
505	Q₈	—	4	2	no	—
506	Q₈	—	4	1	no	—
507	Q₈	—	4	1	no	—
508	Q₈	—	4	2	no	—
509	Q₈	—	4	1	no	—
510	Q₈	—	4	1	no	—
511	D₄	—	2	2	no	—
512	D₄	—	2	2	no	—
513	D₄	—	2	2	no	—
514	D₄	—	2	2	no	—
515	D₄	—	4	2	no	—
516	D₄	—	4	4	no	—
517	D₄	—	4	1	no	—
518	D₄	—	4	1	no	—
519	D₄	—	4	1	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	2	no	—
522	D₄	—	4	2	no	—
523	D₄	—	8	1	no	—
524	D₄	—	8	4	no	—
525	D₄	—	8	4	no	—
526	D₄	—	8	2	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	8	2	no	—
540	C₂×C₄	—	8	1	no	—
541	C₂×C₄	—	8	4	no	—
542	C₂×C₄	—	8	2	no	—
543	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
544	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
545	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
546	C₂×C₄	#135×C₆₁	1	2	yes	—
547	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
548	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
549	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
550	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
551	C₈	—	2	2	no	—
552	C₈	—	4	1	no	—
553	C₈	—	4	2	no	—
554	C₈	—	4	2	no	—
555	C₂²	—	2	1	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	4	2	no	—
558	C₂²	—	4	2	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	4	2	no	—
562	C₂²	—	4	2	no	—
563	C₂²	—	8	4	no	—
564	C₂²	—	8	1	no	—
565	C₂²	—	8	1	no	—
566	C₂²	—	8	1	no	—
567	C₂²	C₆₁×#449	1	2	yes	—
568	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
569	C₄	—	2	1	no	—
570	C₄	—	2	1	no	—
571	C₄	—	2	1	no	—
572	C₄	—	4	2	no	—
573	C₄	—	4	2	no	—
574	C₄	—	4	2	no	—
575	C₄	—	4	2	no	—
576	C₄	—	8	1	no	—
577	C₄	C₆₁×#449	1	2	yes	—
578	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
579	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
580	C₄	C₆₁×#576	1	1	yes	characteristic
581	C₄	C₆₁×#577	1	1	yes	characteristic
582	C₂	—	2	2	no	—
583	C₂	—	8	2	no	—
584	C₂	—	8	2	no	—
585	C₂	—	16	1	no	—
586	C₂	C₂×C₆₁×#449	1	1	yes	characteristic
587	C₂	C₆₁×#3012	1	1	yes	characteristic
588	C₂	C₆₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
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×#3506	C₂	semidirect
2. C₆₁×#3245	C₂	semidirect
3. C₆₁×#3167	C₂	semidirect
4. C₆₁×#3036	C₂	semidirect
5. C₆₁×#1654	C₂	semidirect
6. C₆₁×#1646	C₂	semidirect
7. C₆₁×#1625	C₂	semidirect
8. C₆₁×#1624	C₂	semidirect
9. C₆₁×#1550	C₂	semidirect
10. C₆₁×#1492	C₂	semidirect
11. C₂×C₆₁×D₄⋊C₈	C₂	semidirect
27. C₆₁×#498	C₂²	semidirect
28. C₆₁×#492	C₂²	semidirect
29. C₆₁×#487	C₂²	semidirect
30. C₆₁×#486	C₂²	semidirect
32. C₆₁×#474	C₂²	semidirect
39. C₆₁×#459	C₂²	semidirect
45. C₆₁×#333	C₂²	semidirect
46. C₆₁×#332	C₂²	semidirect
49. C₆₁×D₄⋊C₈	C₂²	semidirect
50. C₆₁×D₄⋊C₈	C₂²	semidirect
260. #18680	C₆₁	direct (N × H)
288. #3506	C₂×C₆₁	semidirect
289. #3245	C₂×C₆₁	semidirect
290. #3167	C₂×C₆₁	semidirect
291. #3036	C₂×C₆₁	semidirect
292. #1654	C₂×C₆₁	semidirect
293. #1646	C₂×C₆₁	semidirect
294. #1625	C₂×C₆₁	semidirect
295. #1624	C₂×C₆₁	semidirect
296. #1550	C₂×C₆₁	semidirect
297. #1492	C₂×C₆₁	semidirect
298. C₂×D₄⋊C₈	C₂×C₆₁	semidirect
321. #498	C₂²×C₆₁	semidirect
322. #492	C₂²×C₆₁	semidirect
323. #487	C₂²×C₆₁	semidirect
324. #486	C₂²×C₆₁	semidirect
326. #474	C₂²×C₆₁	semidirect
333. #459	C₂²×C₆₁	semidirect
339. #333	C₂²×C₆₁	semidirect
340. #332	C₂²×C₆₁	semidirect
343. D₄⋊C₈	C₂²×C₆₁	semidirect
344. D₄⋊C₈	C₂²×C₆₁	semidirect
345. C₆₁	#18680	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×#3506	2
C₆₁×#3245	2
C₆₁×#3167	2
C₆₁×#3036	2
C₆₁×#1654	2
C₆₁×#1646	2
C₆₁×#1625	2
C₆₁×#1624	2
C₆₁×#1550	2
C₆₁×#1492	2
C₂×C₆₁×D₄⋊C₈	2
#18680	61

Subgroups — C61×#18680 #29522113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18680 #29522113