Subgroups — C₆₁×#19683 #29523116

Order 15616.

724 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3209	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3206	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#3167	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#3141	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1723	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1722	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1718	C₂	1	2	yes	maximal, complemented
10	C₆₁×#1714	C₂	1	2	yes	maximal, complemented
11	C₆₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#533	C₂²	1	1	yes	characteristic
13	C₆₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#501	—	2	2	no	—
17	C₆₁×#500	—	2	2	no	—
18	C₆₁×#500	C₂²	1	2	yes	—
19	C₆₁×#500	C₂²	1	2	yes	—
20	C₆₁×#500	C₂²	1	2	yes	—
21	C₆₁×#498	C₂²	1	2	yes	—
22	C₆₁×#498	C₂²	1	1	yes	characteristic
23	C₆₁×#494	C₂²	1	2	yes	—
24	C₆₁×#492	C₂²	1	1	yes	characteristic
25	C₆₁×#492	C₂²	1	2	yes	—
26	C₆₁×#489	—	2	2	no	—
27	C₆₁×#487	C₂²	1	2	yes	—
28	C₆₁×#487	C₂²	1	1	yes	characteristic
29	C₆₁×#487	C₂²	1	2	yes	—
30	C₆₁×#487	C₂²	1	2	yes	—
31	C₆₁×#486	C₂²	1	2	yes	—
32	C₆₁×#486	C₂²	1	1	yes	characteristic
33	C₆₁×#465	—	2	2	no	—
34	C₆₁×#462	—	2	2	no	—
35	C₆₁×#461	—	2	2	no	—
36	C₆₁×#451	—	2	2	no	—
37	C₆₁×#448	—	2	2	no	—
38	C₄×SD₁₆×C₆₁	—	2	2	no	—
39	C₆₁×#431	C₂²	1	1	yes	characteristic
40	C₆₁×#428	—	2	2	no	—
41	C₂×#107×C₆₁	—	2	2	no	—
42	C₆₁×#417	—	2	2	no	—
43	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
45	C₆₁×#322	C₂²	1	2	yes	—
46	C₆₁×#322	C₂²	1	2	yes	—
47	C₂×#41×C₆₁	—	2	1	no	—
48	C₂²×Q₈×C₆₁	—	2	1	no	—
49	C₂²×Q₈×C₆₁	—	2	1	no	—
50	C₂×Q₁₆×C₆₁	—	2	4	no	—
51	C₂×SD₁₆×C₆₁	—	2	2	no	—
52	C₂×SD₁₆×C₆₁	—	2	4	no	—
53	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
54	C₂²×C₈×C₆₁	D₄	1	2	yes	—
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	—	2	1	no	—
59	C₄⋊Q₈×C₆₁	—	2	1	no	—
60	C₄⋊Q₈×C₆₁	—	2	1	no	—
61	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
63	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
66	#127×C₆₁	C₂³	1	1	yes	characteristic
67	#125×C₆₁	—	2	1	no	—
68	#124×C₆₁	—	2	1	no	—
69	C₂²⋊Q₈×C₆₁	—	2	1	no	—
70	C₂²⋊Q₈×C₆₁	—	2	1	no	—
71	#121×C₆₁	—	2	1	no	—
72	#121×C₆₁	—	2	1	no	—
73	C₄×Q₈×C₆₁	—	2	1	no	—
74	C₄×Q₈×C₆₁	—	2	1	no	—
75	C₄×D₄×C₆₁	—	2	1	no	—
76	C₄×D₄×C₆₁	—	2	1	no	—
77	#117×C₆₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
82	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
83	#107×C₆₁	—	2	4	no	—
84	#107×C₆₁	—	2	4	no	—
85	#107×C₆₁	—	2	4	no	—
86	#106×C₆₁	—	2	4	no	—
87	#106×C₆₁	—	2	2	no	—
88	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
89	#103×C₆₁	—	2	4	no	—
90	#103×C₆₁	—	2	2	no	—
91	#103×C₆₁	—	2	4	no	—
92	#103×C₆₁	—	2	4	no	—
93	#102×C₆₁	—	2	4	no	—
94	#102×C₆₁	—	2	2	no	—
95	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
96	#97×C₆₁	D₄	1	2	yes	—
97	C₄×C₈×C₆₁	D₄	1	2	yes	—
98	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₆₁	C₂³	1	2	yes	—
101	C₄×C₈×C₆₁	C₂³	1	2	yes	—
102	#41×C₆₁	—	4	4	no	—
103	C₂×Q₈×C₆₁	—	2	1	no	—
104	C₂×Q₈×C₆₁	—	2	1	no	—
105	C₂×Q₈×C₆₁	—	2	2	no	—
106	C₂×Q₈×C₆₁	—	2	2	no	—
107	C₂×Q₈×C₆₁	—	2	1	no	—
108	C₂×Q₈×C₆₁	—	4	1	no	—
109	C₂×Q₈×C₆₁	—	4	1	no	—
110	C₂×Q₈×C₆₁	—	8	2	no	—
111	C₂×Q₈×C₆₁	—	8	2	no	—
112	C₂×D₄×C₆₁	—	2	1	no	—
113	C₂×D₄×C₆₁	—	2	1	no	—
114	C₂×D₄×C₆₁	—	2	1	no	—
115	C₂×D₄×C₆₁	—	4	1	no	—
116	C₂²×C₄×C₆₁	—	4	1	no	—
117	C₂²×C₄×C₆₁	—	4	1	no	—
118	C₂²×C₄×C₆₁	—	4	1	no	—
119	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
122	Q₁₆×C₆₁	—	8	4	no	—
123	SD₁₆×C₆₁	—	4	4	no	—
124	SD₁₆×C₆₁	—	8	4	no	—
125	M₁₆×C₆₁	—	2	4	no	—
126	C₂×C₈×C₆₁	—	2	4	no	—
127	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
129	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
130	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₆₁	—	2	2	no	—
132	C₄⋊C₄×C₆₁	—	2	1	no	—
133	C₄⋊C₄×C₆₁	—	2	1	no	—
134	C₄⋊C₄×C₆₁	—	2	1	no	—
135	C₄⋊C₄×C₆₁	—	2	1	no	—
136	C₄⋊C₄×C₆₁	—	2	1	no	—
137	C₄⋊C₄×C₆₁	—	2	2	no	—
138	C₄⋊C₄×C₆₁	—	2	2	no	—
139	C₄⋊C₄×C₆₁	—	2	2	no	—
140	C₄⋊C₄×C₆₁	—	2	1	no	—
141	C₄⋊C₄×C₆₁	—	4	1	no	—
142	C₄⋊C₄×C₆₁	—	4	1	no	—
143	C₄⋊C₄×C₆₁	—	4	1	no	—
144	C₄⋊C₄×C₆₁	—	4	1	no	—
145	C₄⋊C₄×C₆₁	—	4	1	no	—
146	C₄⋊C₄×C₆₁	—	4	1	no	—
147	C₂²⋊C₄×C₆₁	—	4	1	no	—
148	C₂²⋊C₄×C₆₁	—	4	1	no	—
149	C₂²⋊C₄×C₆₁	—	4	1	no	—
150	C₄²×C₆₁	—	4	1	no	—
151	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
155	C₂³×C₆₁	—	4	1	no	—
156	C₂³×C₆₁	#127	1	1	yes	characteristic
157	Q₈×C₆₁	—	4	2	no	—
158	Q₈×C₆₁	—	4	2	no	—
159	Q₈×C₆₁	—	4	4	no	—
160	Q₈×C₆₁	—	4	2	no	—
161	Q₈×C₆₁	—	4	4	no	—
162	Q₈×C₆₁	—	8	2	no	—
163	Q₈×C₆₁	—	8	2	no	—
164	D₄×C₆₁	—	4	2	no	—
165	D₄×C₆₁	—	4	2	no	—
166	D₄×C₆₁	—	4	2	no	—
167	D₄×C₆₁	—	8	2	no	—
168	C₂×C₄×C₆₁	—	2	2	no	—
169	C₂×C₄×C₆₁	—	2	2	no	—
170	C₂×C₄×C₆₁	—	4	1	no	—
171	C₂×C₄×C₆₁	—	4	1	no	—
172	C₂×C₄×C₆₁	—	4	1	no	—
173	C₂×C₄×C₆₁	—	4	2	no	—
174	C₂×C₄×C₆₁	—	4	2	no	—
175	C₂×C₄×C₆₁	—	8	2	no	—
176	C₂×C₄×C₆₁	—	8	2	no	—
177	C₂×C₄×C₆₁	—	8	2	no	—
178	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
179	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
180	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
181	C₂×C₄×C₆₁	#137	1	4	yes	—
182	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
185	C₈×C₆₁	—	2	2	no	—
186	C₈×C₆₁	—	2	4	no	—
187	C₈×C₆₁	—	2	4	no	—
188	C₈×C₆₁	#137	1	4	yes	—
189	#19683	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂²×C₆₁	—	2	1	no	—
191	C₂²×C₆₁	—	2	1	no	—
192	C₂²×C₆₁	—	8	1	no	—
193	C₂²×C₆₁	—	8	1	no	—
194	C₂²×C₆₁	—	8	1	no	—
195	C₂²×C₆₁	#497	1	2	yes	—
196	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
197	C₄×C₆₁	—	2	1	no	—
198	C₄×C₆₁	—	2	1	no	—
199	C₄×C₆₁	—	8	1	no	—
200	C₄×C₆₁	—	8	1	no	—
201	C₄×C₆₁	—	8	2	no	—
202	C₄×C₆₁	—	8	2	no	—
203	C₄×C₆₁	—	8	1	no	—
204	C₄×C₆₁	#497	1	2	yes	—
205	C₄×C₆₁	C₂×#137	1	2	yes	—
206	C₄×C₆₁	C₂×#137	1	2	yes	—
207	C₄×C₆₁	#576	1	2	yes	—
208	#3209	C₂×C₆₁	1	1	yes	characteristic, complemented
209	#3206	C₂×C₆₁	1	1	yes	characteristic
210	#3167	C₂×C₆₁	1	1	yes	characteristic
211	#3165	C₂×C₆₁	1	1	yes	characteristic, complemented
212	C₂×#500	C₂×C₆₁	1	1	yes	characteristic, complemented
213	#3141	C₂×C₆₁	1	1	yes	characteristic
214	#1723	C₂×C₆₁	1	2	yes	complemented
215	#1722	C₂×C₆₁	1	2	yes	complemented
216	#1718	C₂×C₆₁	1	2	yes	complemented
217	#1714	C₂×C₆₁	1	2	yes	complemented
218	#1451	C₂×C₆₁	1	1	yes	characteristic, complemented
219	C₂×C₆₁	—	2	2	no	—
220	C₂×C₆₁	—	16	1	no	—
221	C₂×C₆₁	C₂×#497	1	1	yes	characteristic
222	C₂×C₆₁	#3152	1	1	yes	characteristic
223	C₂×C₆₁	#3155	1	1	yes	characteristic
224	#533	C₂²×C₆₁	1	1	yes	characteristic
225	#532	C₂²×C₆₁	1	1	yes	characteristic
226	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
227	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
228	#501	—	2	2	no	—
229	#500	—	2	2	no	—
230	#500	C₂²×C₆₁	1	2	yes	—
231	#500	C₂²×C₆₁	1	2	yes	—
232	#500	C₂²×C₆₁	1	2	yes	—
233	#498	C₂²×C₆₁	1	2	yes	—
234	#498	C₂²×C₆₁	1	1	yes	characteristic
235	#494	C₂²×C₆₁	1	2	yes	—
236	#492	C₂²×C₆₁	1	1	yes	characteristic
237	#492	C₂²×C₆₁	1	2	yes	—
238	#489	—	2	2	no	—
239	#487	C₂²×C₆₁	1	2	yes	—
240	#487	C₂²×C₆₁	1	1	yes	characteristic
241	#487	C₂²×C₆₁	1	2	yes	—
242	#487	C₂²×C₆₁	1	2	yes	—
243	#486	C₂²×C₆₁	1	2	yes	—
244	#486	C₂²×C₆₁	1	1	yes	characteristic
245	#465	—	2	2	no	—
246	#462	—	2	2	no	—
247	#461	—	2	2	no	—
248	#451	—	2	2	no	—
249	#448	—	2	2	no	—
250	C₄×SD₁₆	—	2	2	no	—
251	#431	C₂²×C₆₁	1	1	yes	characteristic
252	#428	—	2	2	no	—
253	C₂×#107	—	2	2	no	—
254	#417	—	2	2	no	—
255	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
256	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
257	#322	C₂²×C₆₁	1	2	yes	—
258	#322	C₂²×C₆₁	1	2	yes	—
259	C₆₁	#19683	1	1	yes	characteristic, direct product factor, Sylow
260	C₂×#41	—	2	1	no	—
261	C₂²×Q₈	—	2	1	no	—
262	C₂²×Q₈	—	2	1	no	—
263	C₂×Q₁₆	—	2	4	no	—
264	C₂×SD₁₆	—	2	2	no	—
265	C₂×SD₁₆	—	2	4	no	—
266	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
267	C₂²×C₈	D₄×C₆₁	1	2	yes	—
268	C₄⋊Q₈	—	2	1	no	—
269	C₄⋊Q₈	—	2	1	no	—
270	C₄⋊Q₈	—	2	1	no	—
271	C₄⋊Q₈	—	2	1	no	—
272	C₄⋊Q₈	—	2	1	no	—
273	C₄⋊Q₈	—	2	1	no	—
274	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
275	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
276	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
277	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
278	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
279	#127	C₂³×C₆₁	1	1	yes	characteristic
280	#125	—	2	1	no	—
281	#124	—	2	1	no	—
282	C₂²⋊Q₈	—	2	1	no	—
283	C₂²⋊Q₈	—	2	1	no	—
284	#121	—	2	1	no	—
285	#121	—	2	1	no	—
286	C₄×Q₈	—	2	1	no	—
287	C₄×Q₈	—	2	1	no	—
288	C₄×D₄	—	2	1	no	—
289	C₄×D₄	—	2	1	no	—
290	#117	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
296	#107	—	2	4	no	—
297	#107	—	2	4	no	—
298	#107	—	2	4	no	—
299	#106	—	2	4	no	—
300	#106	—	2	2	no	—
301	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
302	#103	—	2	4	no	—
303	#103	—	2	2	no	—
304	#103	—	2	4	no	—
305	#103	—	2	4	no	—
306	#102	—	2	4	no	—
307	#102	—	2	2	no	—
308	C₂²⋊C₈	D₄×C₆₁	1	2	yes	—
309	#97	D₄×C₆₁	1	2	yes	—
310	C₄×C₈	D₄×C₆₁	1	2	yes	—
311	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
312	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
313	C₄×C₈	C₂³×C₆₁	1	2	yes	—
314	C₄×C₈	C₂³×C₆₁	1	2	yes	—
315	#41	—	4	4	no	—
316	C₂×Q₈	—	2	1	no	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×Q₈	—	4	1	no	—
322	C₂×Q₈	—	4	1	no	—
323	C₂×Q₈	—	8	2	no	—
324	C₂×Q₈	—	8	2	no	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	4	1	no	—
329	C₂²×C₄	—	4	1	no	—
330	C₂²×C₄	—	4	1	no	—
331	C₂²×C₄	—	4	1	no	—
332	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
335	Q₁₆	—	8	4	no	—
336	SD₁₆	—	4	4	no	—
337	SD₁₆	—	8	4	no	—
338	M₁₆	—	2	4	no	—
339	C₂×C₈	—	2	4	no	—
340	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
341	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
342	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
343	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	4	1	no	—
355	C₄⋊C₄	—	4	1	no	—
356	C₄⋊C₄	—	4	1	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	—	4	1	no	—
359	C₄⋊C₄	—	4	1	no	—
360	C₂²⋊C₄	—	4	1	no	—
361	C₂²⋊C₄	—	4	1	no	—
362	C₂²⋊C₄	—	4	1	no	—
363	C₄²	—	4	1	no	—
364	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
365	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
366	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
367	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
368	C₂³	—	4	1	no	—
369	C₂³	#127×C₆₁	1	1	yes	characteristic
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	4	4	no	—
373	Q₈	—	4	2	no	—
374	Q₈	—	4	4	no	—
375	Q₈	—	8	2	no	—
376	Q₈	—	8	2	no	—
377	D₄	—	4	2	no	—
378	D₄	—	4	2	no	—
379	D₄	—	4	2	no	—
380	D₄	—	8	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	8	2	no	—
389	C₂×C₄	—	8	2	no	—
390	C₂×C₄	—	8	2	no	—
391	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
392	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
393	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
394	C₂×C₄	#137×C₆₁	1	4	yes	—
395	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
396	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
397	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
398	C₈	—	2	2	no	—
399	C₈	—	2	4	no	—
400	C₈	—	2	4	no	—
401	C₈	#137×C₆₁	1	4	yes	—
402	C₂²	—	2	1	no	—
403	C₂²	—	2	1	no	—
404	C₂²	—	8	1	no	—
405	C₂²	—	8	1	no	—
406	C₂²	—	8	1	no	—
407	C₂²	C₆₁×#497	1	2	yes	—
408	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
409	C₄	—	2	1	no	—
410	C₄	—	2	1	no	—
411	C₄	—	8	1	no	—
412	C₄	—	8	1	no	—
413	C₄	—	8	2	no	—
414	C₄	—	8	2	no	—
415	C₄	—	8	1	no	—
416	C₄	C₆₁×#497	1	2	yes	—
417	C₄	C₂×#137×C₆₁	1	2	yes	—
418	C₄	C₂×#137×C₆₁	1	2	yes	—
419	C₄	C₆₁×#576	1	2	yes	—
420	C₂	—	2	2	no	—
421	C₂	—	16	1	no	—
422	C₂	C₂×C₆₁×#497	1	1	yes	characteristic
423	C₂	C₆₁×#3152	1	1	yes	characteristic
424	C₂	C₆₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#532	C₂²
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×#3209	C₂	semidirect
4. C₆₁×#3165	C₂	semidirect
5. C₂×C₆₁×#500	C₂	semidirect
7. C₆₁×#1723	C₂	semidirect
8. C₆₁×#1722	C₂	semidirect
9. C₆₁×#1718	C₂	semidirect
10. C₆₁×#1714	C₂	semidirect
11. C₆₁×#1451	C₂	semidirect
189. #19683	C₆₁	direct (N × H)
208. #3209	C₂×C₆₁	semidirect
211. #3165	C₂×C₆₁	semidirect
212. C₂×#500	C₂×C₆₁	semidirect
214. #1723	C₂×C₆₁	semidirect
215. #1722	C₂×C₆₁	semidirect
216. #1718	C₂×C₆₁	semidirect
217. #1714	C₂×C₆₁	semidirect
218. #1451	C₂×C₆₁	semidirect
259. C₆₁	#19683	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×#3209	2
C₆₁×#3206	2
C₆₁×#3167	2
C₆₁×#3165	2
C₂×C₆₁×#500	2
C₆₁×#3141	2
C₆₁×#1723	2
C₆₁×#1722	2
C₆₁×#1718	2
C₆₁×#1714	2
C₆₁×#1451	2
#19683	61

Subgroups — C61×#19683 #29523116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19683 #29523116