Subgroups — C₆₁×#25830 #29529263

Order 15616.

1054 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3503	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3079	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2868	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2060	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1859	C₂	1	4	yes	maximal
9	C₄×#104×C₆₁	C₂	1	1	yes	maximal, characteristic
10	Q₈²×C₆₁	—	2	1	no	—
11	C₆₁×#557	—	2	1	no	—
12	C₆₁×#554	—	2	1	no	—
13	C₆₁×#554	—	2	2	no	—
14	D₄×Q₈×C₆₁	—	2	2	no	—
15	D₄×Q₈×C₆₁	—	2	1	no	—
16	C₆₁×#544	—	2	2	no	—
17	C₆₁×#544	C₂²	1	2	yes	—
18	C₆₁×#533	—	2	1	no	—
19	C₆₁×#533	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
21	C₂×C₄⋊Q₈×C₆₁	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₆₁×#520	C₂²	1	2	yes	—
25	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#501	—	2	2	no	—
27	C₆₁×#500	—	2	1	no	—
28	C₆₁×#498	—	2	1	no	—
29	C₆₁×#477	—	2	1	no	—
30	C₆₁×#476	—	2	1	no	—
31	C₆₁×#475	—	2	1	no	—
32	C₆₁×#474	—	2	1	no	—
33	C₆₁×#428	C₂²	1	2	yes	—
34	C₆₁×#428	C₂²	1	4	yes	—
35	C₆₁×#423	C₂²	1	1	yes	characteristic
36	C₂×#104×C₆₁	C₂²	1	2	yes	—
37	C₆₁×#417	C₂²	1	2	yes	—
38	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
39	C₆₁×#384	—	2	2	no	—
40	C₆₁×#384	C₂²	1	4	yes	—
41	C₄³×C₆₁	C₂²	1	1	yes	characteristic
42	C₆₁×#343	C₂²	1	4	yes	—
43	C₆₁×#339	C₂²	1	2	yes	—
44	C₆₁×#339	C₂²	1	4	yes	—
45	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₆₁	—	2	2	no	—
47	C₂²×Q₈×C₆₁	—	2	1	no	—
48	C₂²×Q₈×C₆₁	D₄	1	2	yes	—
49	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
50	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
51	C₄⋊Q₈×C₆₁	—	2	2	no	—
52	C₄⋊Q₈×C₆₁	—	2	2	no	—
53	C₄⋊Q₈×C₆₁	—	2	4	no	—
54	C₄⋊Q₈×C₆₁	—	2	2	no	—
55	C₄⋊Q₈×C₆₁	—	2	2	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	—	2	1	no	—
62	C₄⋊Q₈×C₆₁	—	2	1	no	—
63	C₄⋊Q₈×C₆₁	—	2	1	no	—
64	C₄⋊Q₈×C₆₁	—	4	8	no	—
65	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
66	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
67	#125×C₆₁	—	2	4	no	—
68	#125×C₆₁	—	2	2	no	—
69	#125×C₆₁	—	2	2	no	—
70	#125×C₆₁	D₄	1	2	yes	—
71	C₂²⋊Q₈×C₆₁	—	2	2	no	—
72	C₂²⋊Q₈×C₆₁	—	2	2	no	—
73	C₂²⋊Q₈×C₆₁	—	2	4	no	—
74	C₂²⋊Q₈×C₆₁	—	2	4	no	—
75	C₂²⋊Q₈×C₆₁	—	2	1	no	—
76	C₂²⋊Q₈×C₆₁	—	2	1	no	—
77	C₂²⋊Q₈×C₆₁	—	2	4	no	—
78	C₄×Q₈×C₆₁	—	2	2	no	—
79	C₄×Q₈×C₆₁	—	2	2	no	—
80	C₄×Q₈×C₆₁	—	2	2	no	—
81	C₄×Q₈×C₆₁	—	2	1	no	—
82	C₄×Q₈×C₆₁	—	2	1	no	—
83	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
84	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
85	C₄×D₄×C₆₁	—	2	2	no	—
86	C₄×D₄×C₆₁	—	2	1	no	—
87	C₄×D₄×C₆₁	—	2	4	no	—
88	C₄×D₄×C₆₁	—	2	1	no	—
89	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
90	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
91	#117×C₆₁	—	2	2	no	—
92	#117×C₆₁	—	2	1	no	—
93	#117×C₆₁	C₂³	1	2	yes	—
94	#117×C₆₁	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
96	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₂×C₄⋊C₄×C₆₁	—	2	8	no	—
98	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
100	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
102	C₂×C₄²×C₆₁	—	2	1	no	—
103	C₂×C₄²×C₆₁	—	2	1	no	—
104	C₂×C₄²×C₆₁	—	2	1	no	—
105	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
106	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
107	#107×C₆₁	—	2	2	no	—
108	#107×C₆₁	—	2	4	no	—
109	#106×C₆₁	—	2	2	no	—
110	#106×C₆₁	—	2	4	no	—
111	C₄⋊C₈×C₆₁	—	2	1	no	—
112	C₄⋊C₈×C₆₁	—	2	1	no	—
113	#104×C₆₁	Q₈	1	8	yes	complemented
114	#103×C₆₁	—	2	2	no	—
115	#102×C₆₁	—	2	2	no	—
116	#97×C₆₁	—	2	1	no	—
117	C₄×C₈×C₆₁	—	2	1	no	—
118	#41×C₆₁	—	2	2	no	—
119	#41×C₆₁	C₂×Q₈	1	4	yes	—
120	C₂×Q₈×C₆₁	—	2	4	no	—
121	C₂×Q₈×C₆₁	—	2	1	no	—
122	C₂×Q₈×C₆₁	—	2	2	no	—
123	C₂×Q₈×C₆₁	—	2	1	no	—
124	C₂×Q₈×C₆₁	—	2	2	no	—
125	C₂×Q₈×C₆₁	—	4	4	no	—
126	C₂×Q₈×C₆₁	—	4	4	no	—
127	C₂×Q₈×C₆₁	—	4	1	no	—
128	C₂×Q₈×C₆₁	—	4	1	no	—
129	C₂×Q₈×C₆₁	—	8	4	no	—
130	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
131	C₂×D₄×C₆₁	—	2	1	no	—
132	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₁	—	2	2	no	—
134	C₂²×C₄×C₆₁	—	2	1	no	—
135	C₂²×C₄×C₆₁	—	2	4	no	—
136	C₂²×C₄×C₆₁	—	2	1	no	—
137	C₂²×C₄×C₆₁	—	2	1	no	—
138	C₂²×C₄×C₆₁	—	2	2	no	—
139	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₁	C₂×D₄	1	4	yes	—
142	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
143	M₁₆×C₆₁	—	2	2	no	—
144	M₁₆×C₆₁	—	2	4	no	—
145	M₁₆×C₆₁	C₂×Q₈	1	4	yes	complemented
146	C₂×C₈×C₆₁	—	2	2	no	—
147	C₂×C₈×C₆₁	—	2	2	no	—
148	C₄⋊C₄×C₆₁	—	2	4	no	—
149	C₄⋊C₄×C₆₁	—	2	4	no	—
150	C₄⋊C₄×C₆₁	—	2	4	no	—
151	C₄⋊C₄×C₆₁	—	2	4	no	—
152	C₄⋊C₄×C₆₁	—	2	1	no	—
153	C₄⋊C₄×C₆₁	—	2	2	no	—
154	C₄⋊C₄×C₆₁	—	2	2	no	—
155	C₄⋊C₄×C₆₁	—	2	1	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	2	no	—
160	C₄⋊C₄×C₆₁	—	2	2	no	—
161	C₄⋊C₄×C₆₁	—	2	2	no	—
162	C₄⋊C₄×C₆₁	—	2	2	no	—
163	C₄⋊C₄×C₆₁	—	2	2	no	—
164	C₄⋊C₄×C₆₁	—	4	16	no	—
165	C₄⋊C₄×C₆₁	—	4	1	no	—
166	C₄⋊C₄×C₆₁	—	4	1	no	—
167	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
168	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
169	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
170	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
171	C₂²⋊C₄×C₆₁	—	2	4	no	—
172	C₂²⋊C₄×C₆₁	—	2	2	no	—
173	C₂²⋊C₄×C₆₁	—	2	1	no	—
174	C₂²⋊C₄×C₆₁	—	2	1	no	—
175	C₂²⋊C₄×C₆₁	—	2	1	no	—
176	C₄²×C₆₁	—	2	4	no	—
177	C₄²×C₆₁	—	2	4	no	—
178	C₄²×C₆₁	—	2	4	no	—
179	C₄²×C₆₁	—	2	1	no	—
180	C₄²×C₆₁	—	2	1	no	—
181	C₄²×C₆₁	—	2	1	no	—
182	C₄²×C₆₁	—	2	2	no	—
183	C₄²×C₆₁	—	2	2	no	—
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	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₆₁	C₂×Q₈	1	4	yes	complemented
189	C₂³×C₆₁	—	2	1	no	—
190	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
191	Q₈×C₆₁	—	2	1	no	—
192	Q₈×C₆₁	—	4	4	no	—
193	Q₈×C₆₁	—	4	2	no	—
194	Q₈×C₆₁	—	4	2	no	—
195	Q₈×C₆₁	—	8	2	no	—
196	Q₈×C₆₁	—	8	2	no	—
197	Q₈×C₆₁	—	8	4	no	—
198	Q₈×C₆₁	—	8	2	no	—
199	Q₈×C₆₁	C₄⋊Q₈	1	2	yes	complemented
200	D₄×C₆₁	—	2	4	no	—
201	D₄×C₆₁	—	2	1	no	—
202	D₄×C₆₁	C₄⋊Q₈	1	2	yes	complemented
203	C₂×C₄×C₆₁	—	2	4	no	—
204	C₂×C₄×C₆₁	—	2	4	no	—
205	C₂×C₄×C₆₁	—	2	1	no	—
206	C₂×C₄×C₆₁	—	2	4	no	—
207	C₂×C₄×C₆₁	—	2	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	1	no	—
212	C₂×C₄×C₆₁	—	2	1	no	—
213	C₂×C₄×C₆₁	—	2	4	no	—
214	C₂×C₄×C₆₁	—	2	2	no	—
215	C₂×C₄×C₆₁	—	4	2	no	—
216	C₂×C₄×C₆₁	—	4	2	no	—
217	C₂×C₄×C₆₁	—	4	4	no	—
218	C₂×C₄×C₆₁	—	4	4	no	—
219	C₂×C₄×C₆₁	—	4	2	no	—
220	C₂×C₄×C₆₁	—	8	4	no	—
221	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
222	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
223	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
224	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
225	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₆₁	C₂²×Q₈	1	2	yes	—
228	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
229	C₈×C₆₁	—	2	4	no	—
230	C₈×C₆₁	—	4	2	no	—
231	#25830	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂²×C₆₁	—	2	1	no	—
233	C₂²×C₆₁	—	2	1	no	—
234	C₂²×C₆₁	—	2	2	no	—
235	C₂²×C₆₁	—	4	1	no	—
236	C₂²×C₆₁	—	4	1	no	—
237	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
238	C₂²×C₆₁	D₄×Q₈	1	2	yes	—
239	C₄×C₆₁	—	2	8	no	—
240	C₄×C₆₁	—	2	2	no	—
241	C₄×C₆₁	—	2	1	no	—
242	C₄×C₆₁	—	2	1	no	—
243	C₄×C₆₁	—	4	1	no	—
244	C₄×C₆₁	—	4	4	no	—
245	C₄×C₆₁	—	4	2	no	—
246	C₄×C₆₁	—	4	1	no	—
247	C₄×C₆₁	—	4	2	no	—
248	C₄×C₆₁	—	8	2	no	—
249	C₄×C₆₁	#453	1	2	yes	—
250	C₄×C₆₁	#456	1	2	yes	—
251	C₄×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
252	C₄×C₆₁	#536	1	1	yes	characteristic
253	C₄×C₆₁	D₄×Q₈	1	2	yes	—
254	#3503	C₂×C₆₁	1	1	yes	characteristic
255	#3165	C₂×C₆₁	1	1	yes	characteristic, complemented
256	#3079	C₂×C₆₁	1	1	yes	characteristic
257	#2868	C₂×C₆₁	1	1	yes	characteristic, complemented
258	#2060	C₂×C₆₁	1	4	yes	complemented
259	#1997	C₂×C₆₁	1	1	yes	characteristic, complemented
260	#1994	C₂×C₆₁	1	1	yes	characteristic, complemented
261	#1859	C₂×C₆₁	1	4	yes	—
262	C₄×#104	C₂×C₆₁	1	1	yes	characteristic
263	C₂×C₆₁	—	2	2	no	—
264	C₂×C₆₁	—	4	2	no	—
265	C₂×C₆₁	#2434	1	1	yes	characteristic
266	C₂×C₆₁	C₂×#453	1	1	yes	characteristic
267	C₂×C₆₁	C₂×#456	1	1	yes	characteristic
268	Q₈²	—	2	1	no	—
269	#557	—	2	1	no	—
270	#554	—	2	1	no	—
271	#554	—	2	2	no	—
272	D₄×Q₈	—	2	2	no	—
273	D₄×Q₈	—	2	1	no	—
274	#544	—	2	2	no	—
275	#544	C₂²×C₆₁	1	2	yes	—
276	#533	—	2	1	no	—
277	#533	C₂²×C₆₁	1	1	yes	characteristic
278	C₂×C₄⋊Q₈	—	2	1	no	—
279	C₂×C₄⋊Q₈	—	2	2	no	—
280	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
281	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
282	#520	C₂²×C₆₁	1	2	yes	—
283	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
284	#501	—	2	2	no	—
285	#500	—	2	1	no	—
286	#498	—	2	1	no	—
287	#477	—	2	1	no	—
288	#476	—	2	1	no	—
289	#475	—	2	1	no	—
290	#474	—	2	1	no	—
291	#428	C₂²×C₆₁	1	2	yes	—
292	#428	C₂²×C₆₁	1	4	yes	—
293	#423	C₂²×C₆₁	1	1	yes	characteristic
294	C₂×#104	C₂²×C₆₁	1	2	yes	—
295	#417	C₂²×C₆₁	1	2	yes	—
296	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
297	#384	—	2	2	no	—
298	#384	C₂²×C₆₁	1	4	yes	—
299	C₄³	C₂²×C₆₁	1	1	yes	characteristic
300	#343	C₂²×C₆₁	1	4	yes	—
301	#339	C₂²×C₆₁	1	2	yes	—
302	#339	C₂²×C₆₁	1	4	yes	—
303	C₆₁	#25830	1	1	yes	characteristic, direct product factor, Sylow
304	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
305	C₂²×Q₈	—	2	2	no	—
306	C₂²×Q₈	—	2	1	no	—
307	C₂²×Q₈	D₄×C₆₁	1	2	yes	—
308	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
309	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
310	C₄⋊Q₈	—	2	2	no	—
311	C₄⋊Q₈	—	2	2	no	—
312	C₄⋊Q₈	—	2	4	no	—
313	C₄⋊Q₈	—	2	2	no	—
314	C₄⋊Q₈	—	2	2	no	—
315	C₄⋊Q₈	—	2	1	no	—
316	C₄⋊Q₈	—	2	1	no	—
317	C₄⋊Q₈	—	2	1	no	—
318	C₄⋊Q₈	—	2	1	no	—
319	C₄⋊Q₈	—	2	1	no	—
320	C₄⋊Q₈	—	2	1	no	—
321	C₄⋊Q₈	—	2	1	no	—
322	C₄⋊Q₈	—	2	1	no	—
323	C₄⋊Q₈	—	4	8	no	—
324	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
325	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
326	#125	—	2	4	no	—
327	#125	—	2	2	no	—
328	#125	—	2	2	no	—
329	#125	D₄×C₆₁	1	2	yes	—
330	C₂²⋊Q₈	—	2	2	no	—
331	C₂²⋊Q₈	—	2	2	no	—
332	C₂²⋊Q₈	—	2	4	no	—
333	C₂²⋊Q₈	—	2	4	no	—
334	C₂²⋊Q₈	—	2	1	no	—
335	C₂²⋊Q₈	—	2	1	no	—
336	C₂²⋊Q₈	—	2	4	no	—
337	C₄×Q₈	—	2	2	no	—
338	C₄×Q₈	—	2	2	no	—
339	C₄×Q₈	—	2	2	no	—
340	C₄×Q₈	—	2	1	no	—
341	C₄×Q₈	—	2	1	no	—
342	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
343	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
344	C₄×D₄	—	2	2	no	—
345	C₄×D₄	—	2	1	no	—
346	C₄×D₄	—	2	1	no	—
347	C₄×D₄	—	2	4	no	—
348	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
349	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
350	#117	—	2	2	no	—
351	#117	—	2	1	no	—
352	#117	C₂³×C₆₁	1	1	yes	characteristic
353	#117	C₂³×C₆₁	1	2	yes	—
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₄⋊C₄	—	2	2	no	—
356	C₂×C₄⋊C₄	—	2	8	no	—
357	C₂×C₄⋊C₄	—	2	1	no	—
358	C₂×C₄⋊C₄	D₄×C₆₁	1	2	yes	—
359	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
360	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
361	C₂×C₄²	—	2	1	no	—
362	C₂×C₄²	—	2	1	no	—
363	C₂×C₄²	—	2	1	no	—
364	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
365	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
366	#107	—	2	2	no	—
367	#107	—	2	4	no	—
368	#106	—	2	2	no	—
369	#106	—	2	4	no	—
370	C₄⋊C₈	—	2	1	no	—
371	C₄⋊C₈	—	2	1	no	—
372	#104	Q₈×C₆₁	1	8	yes	complemented
373	#103	—	2	2	no	—
374	#102	—	2	2	no	—
375	#97	—	2	1	no	—
376	C₄×C₈	—	2	1	no	—
377	#41	—	2	2	no	—
378	#41	C₂×Q₈×C₆₁	1	4	yes	—
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	1	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	4	4	no	—
385	C₂×Q₈	—	4	4	no	—
386	C₂×Q₈	—	4	1	no	—
387	C₂×Q₈	—	4	1	no	—
388	C₂×Q₈	—	8	4	no	—
389	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₆₁	1	4	yes	—
401	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	4	no	—
404	M₁₆	C₂×Q₈×C₆₁	1	4	yes	complemented
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	4	16	no	—
424	C₄⋊C₄	—	4	1	no	—
425	C₄⋊C₄	—	4	1	no	—
426	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
427	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
428	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
429	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	2	2	no	—
432	C₂²⋊C₄	—	2	1	no	—
433	C₂²⋊C₄	—	2	1	no	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₄²	—	2	4	no	—
436	C₄²	—	2	4	no	—
437	C₄²	—	2	4	no	—
438	C₄²	—	2	1	no	—
439	C₄²	—	2	1	no	—
440	C₄²	—	2	1	no	—
441	C₄²	—	2	2	no	—
442	C₄²	—	2	2	no	—
443	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
444	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
445	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
446	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
447	C₄²	C₂×Q₈×C₆₁	1	4	yes	complemented
448	C₂³	—	2	1	no	—
449	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
450	Q₈	—	2	1	no	—
451	Q₈	—	4	4	no	—
452	Q₈	—	4	2	no	—
453	Q₈	—	4	2	no	—
454	Q₈	—	8	2	no	—
455	Q₈	—	8	2	no	—
456	Q₈	—	8	4	no	—
457	Q₈	—	8	2	no	—
458	Q₈	C₄⋊Q₈×C₆₁	1	2	yes	complemented
459	D₄	—	2	4	no	—
460	D₄	—	2	1	no	—
461	D₄	C₄⋊Q₈×C₆₁	1	2	yes	complemented
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	8	4	no	—
480	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
481	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
482	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
483	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
486	C₂×C₄	C₂²×Q₈×C₆₁	1	2	yes	—
487	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
488	C₈	—	2	4	no	—
489	C₈	—	4	2	no	—
490	C₂²	—	2	1	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	1	no	—
495	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
496	C₂²	D₄×Q₈×C₆₁	1	2	yes	—
497	C₄	—	2	8	no	—
498	C₄	—	2	2	no	—
499	C₄	—	2	1	no	—
500	C₄	—	2	1	no	—
501	C₄	—	4	1	no	—
502	C₄	—	4	4	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	1	no	—
505	C₄	—	4	2	no	—
506	C₄	—	8	2	no	—
507	C₄	C₆₁×#453	1	2	yes	—
508	C₄	C₆₁×#456	1	2	yes	—
509	C₄	C₂×C₄⋊Q₈×C₆₁	1	1	yes	characteristic
510	C₄	C₆₁×#536	1	1	yes	characteristic
511	C₄	D₄×Q₈×C₆₁	1	2	yes	—
512	C₂	—	2	2	no	—
513	C₂	—	4	2	no	—
514	C₂	C₆₁×#2434	1	1	yes	characteristic
515	C₂	C₂×C₆₁×#453	1	1	yes	characteristic
516	C₂	C₂×C₆₁×#456	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×#41×C₆₁	C₂³
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×#3165	C₂	semidirect
4. C₆₁×#2868	C₂	semidirect
5. C₆₁×#2060	C₂	semidirect
6. C₆₁×#1997	C₂	semidirect
7. C₆₁×#1994	C₂	semidirect
113. #104×C₆₁	Q₈	semidirect
145. M₁₆×C₆₁	C₂×Q₈	semidirect
188. C₄²×C₆₁	C₂×Q₈	semidirect
199. Q₈×C₆₁	C₄⋊Q₈	semidirect
202. D₄×C₆₁	C₄⋊Q₈	semidirect
231. #25830	C₆₁	direct (N × H)
255. #3165	C₂×C₆₁	semidirect
257. #2868	C₂×C₆₁	semidirect
258. #2060	C₂×C₆₁	semidirect
259. #1997	C₂×C₆₁	semidirect
260. #1994	C₂×C₆₁	semidirect
303. C₆₁	#25830	direct (N × H)
372. #104	Q₈×C₆₁	semidirect
404. M₁₆	C₂×Q₈×C₆₁	semidirect
447. C₄²	C₂×Q₈×C₆₁	semidirect
458. Q₈	C₄⋊Q₈×C₆₁	semidirect
461. D₄	C₄⋊Q₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3503	2
C₆₁×#3165	2
C₆₁×#3079	2
C₆₁×#2868	2
C₆₁×#2060	2
C₆₁×#1997	2
C₆₁×#1994	2
C₆₁×#1859	2
C₄×#104×C₆₁	2
#25830	61

Subgroups — C61×#25830 #29529263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#25830 #29529263