Subgroups — C₆₁×#61288 #29564721

Order 15616.

1846 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3511	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3356	C₂	1	4	yes	maximal, complemented
3	C₆₁×#3354	C₂	1	4	yes	maximal, complemented
4	C₆₁×#3340	C₂	1	4	yes	maximal, complemented
5	C₆₁×#3335	C₂	1	4	yes	maximal, complemented
6	C₂×C₆₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#3099	C₂	1	4	yes	maximal, complemented
8	C₆₁×#3079	C₂	1	2	yes	maximal, complemented
9	C₆₁×#3060	C₂	1	2	yes	maximal, complemented
10	C₄×#135×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#2912	C₂	1	2	yes	maximal, complemented
13	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#131×C₆₁	C₂²	1	2	yes	—
15	C₆₁×#556	C₂²	1	4	yes	—
16	C₆₁×#555	C₂²	1	4	yes	—
17	C₆₁×#554	C₂²	1	4	yes	—
18	C₆₁×#553	—	2	4	no	—
19	C₆₁×#548	—	2	4	no	—
20	C₆₁×#547	C₂²	1	4	yes	—
21	C₆₁×#542	—	2	4	no	—
22	C₆₁×#537	C₂²	1	4	yes	—
23	C₆₁×#533	—	2	2	no	—
24	C₆₁×#532	—	2	2	no	—
25	C₆₁×#532	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
27	C₆₁×#520	C₂²	1	4	yes	—
28	C₄×#41×C₆₁	C₂²	1	2	yes	—
29	C₆₁×#498	C₂²	1	4	yes	—
30	C₆₁×#498	C₂²	1	4	yes	—
31	C₆₁×#485	C₂²	1	8	yes	—
32	C₆₁×#483	C₂²	1	8	yes	—
33	C₆₁×#482	C₂²	1	8	yes	—
34	C₆₁×#481	C₂²	1	8	yes	—
35	C₆₁×#477	C₂²	1	4	yes	—
36	C₆₁×#476	C₂²	1	4	yes	—
37	C₆₁×#475	C₂²	1	4	yes	—
38	C₆₁×#474	C₂²	1	4	yes	—
39	C₆₁×#470	C₂²	1	4	yes	—
40	C₆₁×#469	C₂²	1	4	yes	—
41	C₆₁×#465	C₂²	1	8	yes	—
42	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
43	C₄×SD₁₆×C₆₁	C₂²	1	2	yes	—
44	C₄×SD₁₆×C₆₁	C₂²	1	2	yes	—
45	C₄×D₈×C₆₁	C₂²	1	2	yes	—
46	C₆₁×#436	C₂²	1	4	yes	—
47	C₈×D₄×C₆₁	C₂²	1	4	yes	—
48	C₆₁×#428	C₂²	1	4	yes	—
49	C₆₁×#428	C₂²	1	8	yes	—
50	C₆₁×#427	C₂²	1	4	yes	—
51	C₆₁×#427	C₂²	1	1	yes	characteristic
52	C₂×#106×C₆₁	C₂²	1	2	yes	—
53	C₂×#106×C₆₁	C₂²	1	4	yes	—
54	C₂×#106×C₆₁	C₂²	1	2	yes	—
55	C₆₁×#423	C₂²	1	1	yes	characteristic
56	C₆₁×#417	C₂²	1	4	yes	—
57	C₆₁×#416	C₂²	1	2	yes	—
58	C₆₁×#408	C₂²	1	2	yes	—
59	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
60	C₂×#41×C₆₁	—	2	4	no	—
61	C₂×#41×C₆₁	C₂³	1	2	yes	—
62	C₂×#41×C₆₁	C₂³	1	2	yes	—
63	C₂²×Q₈×C₆₁	—	2	2	no	—
64	#135×C₆₁	—	2	4	no	—
65	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
66	C₂×SD₁₆×C₆₁	C₂³	1	2	yes	—
67	C₂×D₈×C₆₁	C₂³	1	1	yes	characteristic
68	#131×C₆₁	—	2	8	no	—
69	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
70	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
71	C₂²×C₈×C₆₁	C₂³	1	4	yes	—
72	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₆₁	—	2	2	no	—
75	C₄⋊Q₈×C₆₁	—	2	2	no	—
76	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
77	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
78	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
79	#127×C₆₁	D₄	1	1	yes	characteristic
80	#126×C₆₁	—	2	8	no	—
81	#125×C₆₁	—	2	4	no	—
82	#124×C₆₁	—	2	4	no	—
83	#124×C₆₁	—	2	8	no	—
84	#124×C₆₁	D₄	1	2	yes	—
85	#123×C₆₁	—	2	8	no	—
86	C₂²⋊Q₈×C₆₁	—	2	4	no	—
87	C₂²⋊Q₈×C₆₁	—	2	4	no	—
88	C₂²⋊Q₈×C₆₁	—	2	8	no	—
89	C₂²⋊Q₈×C₆₁	C₂³	1	8	yes	—
90	#121×C₆₁	—	2	4	no	—
91	#121×C₆₁	—	2	4	no	—
92	#121×C₆₁	—	2	8	no	—
93	#121×C₆₁	D₄	1	4	yes	—
94	#121×C₆₁	C₂³	1	8	yes	—
95	C₄×Q₈×C₆₁	—	2	4	no	—
96	C₄×Q₈×C₆₁	—	2	4	no	—
97	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
98	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
99	C₄×D₄×C₆₁	—	2	4	no	—
100	C₄×D₄×C₆₁	—	2	4	no	—
101	C₄×D₄×C₆₁	—	2	8	no	—
102	C₄×D₄×C₆₁	—	2	2	no	—
103	C₄×D₄×C₆₁	—	2	2	no	—
104	C₄×D₄×C₆₁	C₂³	1	2	yes	—
105	C₄×D₄×C₆₁	C₂³	1	2	yes	—
106	C₄×D₄×C₆₁	C₂³	1	4	yes	—
107	#117×C₆₁	—	2	4	no	—
108	#117×C₆₁	—	2	2	no	—
109	#117×C₆₁	C₂³	1	8	yes	—
110	#117×C₆₁	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
112	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₆₁	C₂³	1	8	yes	—
114	C₂×C₄²×C₆₁	—	2	2	no	—
115	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
116	#107×C₆₁	C₂³	1	2	yes	—
117	#107×C₆₁	C₂³	1	8	yes	—
118	#106×C₆₁	C₂³	1	1	yes	characteristic
119	#106×C₆₁	C₂³	1	4	yes	—
120	#106×C₆₁	C₂³	1	4	yes	—
121	#106×C₆₁	C₂³	1	8	yes	—
122	#106×C₆₁	C₂³	1	8	yes	—
123	#106×C₆₁	C₂³	1	1	yes	characteristic
124	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
125	#103×C₆₁	C₂³	1	2	yes	—
126	#103×C₆₁	C₂³	1	2	yes	—
127	#103×C₆₁	C₂³	1	8	yes	—
128	#102×C₆₁	C₂³	1	2	yes	—
129	#102×C₆₁	C₂³	1	2	yes	—
130	#102×C₆₁	C₂³	1	8	yes	—
131	C₂²⋊C₈×C₆₁	C₂³	1	8	yes	—
132	C₄×C₈×C₆₁	C₂³	1	2	yes	—
133	C₄×C₈×C₆₁	C₂³	1	2	yes	—
134	#41×C₆₁	—	2	4	no	—
135	#41×C₆₁	—	2	4	no	—
136	#41×C₆₁	—	2	4	no	—
137	#41×C₆₁	—	2	4	no	—
138	#41×C₆₁	—	8	8	no	—
139	C₂×Q₈×C₆₁	—	2	4	no	—
140	C₂×Q₈×C₆₁	—	2	4	no	—
141	C₂×Q₈×C₆₁	—	8	4	no	—
142	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
143	C₂×Q₈×C₆₁	C₂⁴	1	2	yes	—
144	C₂×D₄×C₆₁	—	2	2	no	—
145	C₂×D₄×C₆₁	—	2	8	no	—
146	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
147	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
148	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
149	C₂×D₄×C₆₁	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₆₁	—	2	4	no	—
151	C₂²×C₄×C₆₁	—	2	8	no	—
152	C₂²×C₄×C₆₁	—	2	2	no	—
153	C₂²×C₄×C₆₁	—	2	2	no	—
154	C₂²×C₄×C₆₁	—	2	2	no	—
155	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₆₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₆₁	—	2	2	no	—
159	SD₁₆×C₆₁	—	2	4	no	—
160	D₈×C₆₁	—	2	2	no	—
161	M₁₆×C₆₁	—	2	4	no	—
162	M₁₆×C₆₁	—	2	8	no	—
163	C₂×C₈×C₆₁	—	2	2	no	—
164	C₂×C₈×C₆₁	—	2	8	no	—
165	C₂×C₈×C₆₁	#41	1	4	yes	complemented
166	C₂×C₈×C₆₁	C₂⁴	1	2	yes	—
167	C₂×C₈×C₆₁	C₂⁴	1	4	yes	—
168	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
169	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
170	C₄⋊C₄×C₆₁	—	2	4	no	—
171	C₄⋊C₄×C₆₁	—	2	4	no	—
172	C₄⋊C₄×C₆₁	—	2	4	no	—
173	C₄⋊C₄×C₆₁	—	2	8	no	—
174	C₄⋊C₄×C₆₁	—	2	2	no	—
175	C₄⋊C₄×C₆₁	—	2	2	no	—
176	C₄⋊C₄×C₆₁	—	2	8	no	—
177	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
179	C₄⋊C₄×C₆₁	C₂⁴	1	8	yes	—
180	C₄⋊C₄×C₆₁	C₂⁴	1	2	yes	—
181	C₂²⋊C₄×C₆₁	—	2	2	no	—
182	C₂²⋊C₄×C₆₁	—	2	8	no	—
183	C₂²⋊C₄×C₆₁	—	2	8	no	—
184	C₂²⋊C₄×C₆₁	—	2	2	no	—
185	C₂²⋊C₄×C₆₁	—	2	2	no	—
186	C₂²⋊C₄×C₆₁	—	2	8	no	—
187	C₂²⋊C₄×C₆₁	C₂×D₄	1	8	yes	—
188	C₄²×C₆₁	—	2	2	no	—
189	C₄²×C₆₁	—	2	2	no	—
190	C₄²×C₆₁	—	2	2	no	—
191	C₄²×C₆₁	—	2	8	no	—
192	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₆₁	C₂⁴	1	2	yes	—
195	C₂³×C₆₁	—	2	2	no	—
196	C₂³×C₆₁	C₂²×D₄	1	4	yes	—
197	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
198	Q₈×C₆₁	—	2	4	no	—
199	Q₈×C₆₁	—	2	2	no	—
200	Q₈×C₆₁	—	2	2	no	—
201	Q₈×C₆₁	—	4	8	no	—
202	Q₈×C₆₁	—	8	4	no	—
203	D₄×C₆₁	—	2	8	no	—
204	D₄×C₆₁	—	2	4	no	—
205	D₄×C₆₁	—	2	8	no	—
206	D₄×C₆₁	—	2	2	no	—
207	D₄×C₆₁	—	2	2	no	—
208	D₄×C₆₁	—	4	4	no	—
209	D₄×C₆₁	—	8	8	no	—
210	C₂×C₄×C₆₁	—	2	4	no	—
211	C₂×C₄×C₆₁	—	2	4	no	—
212	C₂×C₄×C₆₁	—	2	4	no	—
213	C₂×C₄×C₆₁	—	2	2	no	—
214	C₂×C₄×C₆₁	—	2	2	no	—
215	C₂×C₄×C₆₁	—	2	2	no	—
216	C₂×C₄×C₆₁	—	2	8	no	—
217	C₂×C₄×C₆₁	—	4	4	no	—
218	C₂×C₄×C₆₁	—	4	4	no	—
219	C₂×C₄×C₆₁	—	4	4	no	—
220	C₂×C₄×C₆₁	—	8	4	no	—
221	C₂×C₄×C₆₁	—	8	8	no	—
222	C₂×C₄×C₆₁	C₂²×D₄	1	4	yes	—
223	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
227	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
228	C₂×C₄×C₆₁	C₂⁵	1	1	yes	characteristic
229	C₈×C₆₁	—	2	4	no	—
230	C₈×C₆₁	—	2	2	no	—
231	C₈×C₆₁	C₂×#41	1	2	yes	—
232	C₈×C₆₁	C₂×#41	1	2	yes	—
233	#61288	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂²×C₆₁	—	2	2	no	—
235	C₂²×C₆₁	—	2	4	no	—
236	C₂²×C₆₁	—	2	8	no	—
237	C₂²×C₆₁	—	4	2	no	—
238	C₂²×C₆₁	—	4	2	no	—
239	C₂²×C₆₁	—	4	2	no	—
240	C₂²×C₆₁	#547	1	2	yes	—
241	C₂²×C₆₁	C₂³×D₄	1	1	yes	characteristic
242	C₄×C₆₁	—	2	4	no	—
243	C₄×C₆₁	—	2	2	no	—
244	C₄×C₆₁	—	4	2	no	—
245	C₄×C₆₁	—	4	2	no	—
246	C₄×C₆₁	—	4	2	no	—
247	C₄×C₆₁	—	4	8	no	—
248	C₄×C₆₁	#547	1	2	yes	—
249	C₄×C₆₁	#577	1	4	yes	—
250	C₄×C₆₁	C₂²×#41	1	1	yes	characteristic
251	C₄×C₆₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
252	#3511	C₂×C₆₁	1	2	yes	complemented
253	#3356	C₂×C₆₁	1	4	yes	complemented
254	#3354	C₂×C₆₁	1	4	yes	complemented
255	#3340	C₂×C₆₁	1	4	yes	complemented
256	#3335	C₂×C₆₁	1	4	yes	complemented
257	C₂×#498	C₂×C₆₁	1	1	yes	characteristic, complemented
258	#3099	C₂×C₆₁	1	4	yes	complemented
259	#3079	C₂×C₆₁	1	2	yes	complemented
260	#3060	C₂×C₆₁	1	2	yes	complemented
261	C₄×#135	C₂×C₆₁	1	1	yes	characteristic, complemented
262	#2931	C₂×C₆₁	1	1	yes	characteristic, complemented
263	#2912	C₂×C₆₁	1	2	yes	complemented
264	C₂×C₆₁	—	2	2	no	—
265	C₂×C₆₁	—	4	4	no	—
266	C₂×C₆₁	—	8	2	no	—
267	C₂×C₆₁	C₂×#547	1	1	yes	characteristic
268	C₂×C₆₁	C₂×#577	1	2	yes	—
269	C₂×#135	C₂²×C₆₁	1	1	yes	characteristic
270	C₂×#131	C₂²×C₆₁	1	2	yes	—
271	#556	C₂²×C₆₁	1	4	yes	—
272	#555	C₂²×C₆₁	1	4	yes	—
273	#554	C₂²×C₆₁	1	4	yes	—
274	#553	—	2	4	no	—
275	#548	—	2	4	no	—
276	#547	C₂²×C₆₁	1	4	yes	—
277	#542	—	2	4	no	—
278	#537	C₂²×C₆₁	1	4	yes	—
279	#533	—	2	2	no	—
280	#532	—	2	2	no	—
281	#532	C₂²×C₆₁	1	1	yes	characteristic
282	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
283	#520	C₂²×C₆₁	1	4	yes	—
284	C₄×#41	C₂²×C₆₁	1	2	yes	—
285	#498	C₂²×C₆₁	1	4	yes	—
286	#498	C₂²×C₆₁	1	4	yes	—
287	#485	C₂²×C₆₁	1	8	yes	—
288	#483	C₂²×C₆₁	1	8	yes	—
289	#482	C₂²×C₆₁	1	8	yes	—
290	#481	C₂²×C₆₁	1	8	yes	—
291	#477	C₂²×C₆₁	1	4	yes	—
292	#476	C₂²×C₆₁	1	4	yes	—
293	#475	C₂²×C₆₁	1	4	yes	—
294	#474	C₂²×C₆₁	1	4	yes	—
295	#470	C₂²×C₆₁	1	4	yes	—
296	#469	C₂²×C₆₁	1	4	yes	—
297	#465	C₂²×C₆₁	1	8	yes	—
298	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
299	C₄×SD₁₆	C₂²×C₆₁	1	2	yes	—
300	C₄×SD₁₆	C₂²×C₆₁	1	2	yes	—
301	C₄×D₈	C₂²×C₆₁	1	2	yes	—
302	#436	C₂²×C₆₁	1	4	yes	—
303	C₈×D₄	C₂²×C₆₁	1	4	yes	—
304	#428	C₂²×C₆₁	1	4	yes	—
305	#428	C₂²×C₆₁	1	8	yes	—
306	#427	C₂²×C₆₁	1	4	yes	—
307	#427	C₂²×C₆₁	1	1	yes	characteristic
308	C₂×#106	C₂²×C₆₁	1	4	yes	—
309	C₂×#106	C₂²×C₆₁	1	2	yes	—
310	C₂×#106	C₂²×C₆₁	1	2	yes	—
311	#423	C₂²×C₆₁	1	1	yes	characteristic
312	#417	C₂²×C₆₁	1	4	yes	—
313	#416	C₂²×C₆₁	1	2	yes	—
314	#408	C₂²×C₆₁	1	2	yes	—
315	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
316	C₆₁	#61288	1	1	yes	characteristic, direct product factor, Sylow
317	C₂×#41	—	2	4	no	—
318	C₂×#41	C₂³×C₆₁	1	2	yes	—
319	C₂×#41	C₂³×C₆₁	1	2	yes	—
320	C₂²×Q₈	—	2	2	no	—
321	#135	—	2	4	no	—
322	C₂×Q₁₆	C₂³×C₆₁	1	1	yes	characteristic
323	C₂×SD₁₆	C₂³×C₆₁	1	2	yes	—
324	C₂×D₈	C₂³×C₆₁	1	1	yes	characteristic
325	#131	—	2	8	no	—
326	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
327	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
328	C₂²×C₈	C₂³×C₆₁	1	4	yes	—
329	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
331	C₄⋊Q₈	—	2	2	no	—
332	C₄⋊Q₈	—	2	2	no	—
333	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
334	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
335	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
336	#127	D₄×C₆₁	1	1	yes	characteristic
337	#126	—	2	8	no	—
338	#125	—	2	4	no	—
339	#124	—	2	4	no	—
340	#124	—	2	8	no	—
341	#124	D₄×C₆₁	1	2	yes	—
342	#123	—	2	8	no	—
343	C₂²⋊Q₈	—	2	4	no	—
344	C₂²⋊Q₈	—	2	4	no	—
345	C₂²⋊Q₈	—	2	8	no	—
346	C₂²⋊Q₈	C₂³×C₆₁	1	8	yes	—
347	#121	—	2	4	no	—
348	#121	—	2	4	no	—
349	#121	—	2	8	no	—
350	#121	D₄×C₆₁	1	4	yes	—
351	#121	C₂³×C₆₁	1	8	yes	—
352	C₄×Q₈	—	2	4	no	—
353	C₄×Q₈	—	2	4	no	—
354	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
355	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
356	C₄×D₄	—	2	4	no	—
357	C₄×D₄	—	2	8	no	—
358	C₄×D₄	—	2	4	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	C₂³×C₆₁	1	2	yes	—
362	C₄×D₄	C₂³×C₆₁	1	2	yes	—
363	C₄×D₄	C₂³×C₆₁	1	4	yes	—
364	#117	—	2	4	no	—
365	#117	—	2	2	no	—
366	#117	C₂³×C₆₁	1	2	yes	—
367	#117	C₂³×C₆₁	1	8	yes	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	C₂³×C₆₁	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₆₁	1	8	yes	—
371	C₂×C₄²	—	2	2	no	—
372	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
373	#107	C₂³×C₆₁	1	2	yes	—
374	#107	C₂³×C₆₁	1	8	yes	—
375	#106	C₂³×C₆₁	1	1	yes	characteristic
376	#106	C₂³×C₆₁	1	4	yes	—
377	#106	C₂³×C₆₁	1	4	yes	—
378	#106	C₂³×C₆₁	1	8	yes	—
379	#106	C₂³×C₆₁	1	8	yes	—
380	#106	C₂³×C₆₁	1	1	yes	characteristic
381	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
382	#103	C₂³×C₆₁	1	2	yes	—
383	#103	C₂³×C₆₁	1	2	yes	—
384	#103	C₂³×C₆₁	1	8	yes	—
385	#102	C₂³×C₆₁	1	2	yes	—
386	#102	C₂³×C₆₁	1	2	yes	—
387	#102	C₂³×C₆₁	1	8	yes	—
388	C₂²⋊C₈	C₂³×C₆₁	1	8	yes	—
389	C₄×C₈	C₂³×C₆₁	1	2	yes	—
390	C₄×C₈	C₂³×C₆₁	1	2	yes	—
391	#41	—	2	4	no	—
392	#41	—	2	4	no	—
393	#41	—	2	4	no	—
394	#41	—	2	4	no	—
395	#41	—	8	8	no	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	—	2	4	no	—
398	C₂×Q₈	—	8	4	no	—
399	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
400	C₂×Q₈	C₂⁴×C₆₁	1	2	yes	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
404	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
405	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
406	C₂×D₄	C₂⁴×C₆₁	1	2	yes	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
413	C₂²×C₄	C₂⁴×C₆₁	1	4	yes	—
414	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
415	Q₁₆	—	2	2	no	—
416	SD₁₆	—	2	4	no	—
417	D₈	—	2	2	no	—
418	M₁₆	—	2	4	no	—
419	M₁₆	—	2	8	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	8	no	—
422	C₂×C₈	#41×C₆₁	1	4	yes	complemented
423	C₂×C₈	C₂⁴×C₆₁	1	2	yes	—
424	C₂×C₈	C₂⁴×C₆₁	1	4	yes	—
425	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
426	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
435	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
436	C₄⋊C₄	C₂⁴×C₆₁	1	8	yes	—
437	C₄⋊C₄	C₂⁴×C₆₁	1	2	yes	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	2	8	no	—
440	C₂²⋊C₄	—	2	8	no	—
441	C₂²⋊C₄	—	2	2	no	—
442	C₂²⋊C₄	—	2	2	no	—
443	C₂²⋊C₄	—	2	8	no	—
444	C₂²⋊C₄	C₂×D₄×C₆₁	1	8	yes	—
445	C₄²	—	2	2	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	8	no	—
449	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
450	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
451	C₄²	C₂⁴×C₆₁	1	2	yes	—
452	C₂³	—	2	2	no	—
453	C₂³	C₂²×D₄×C₆₁	1	4	yes	—
454	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
455	Q₈	—	2	4	no	—
456	Q₈	—	2	2	no	—
457	Q₈	—	2	2	no	—
458	Q₈	—	4	8	no	—
459	Q₈	—	8	4	no	—
460	D₄	—	2	8	no	—
461	D₄	—	2	4	no	—
462	D₄	—	2	8	no	—
463	D₄	—	2	2	no	—
464	D₄	—	2	2	no	—
465	D₄	—	4	4	no	—
466	D₄	—	8	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	8	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	8	4	no	—
478	C₂×C₄	—	8	8	no	—
479	C₂×C₄	C₂²×D₄×C₆₁	1	4	yes	—
480	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
481	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
483	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
484	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
485	C₂×C₄	C₂⁵×C₆₁	1	1	yes	characteristic
486	C₈	—	2	4	no	—
487	C₈	—	2	2	no	—
488	C₈	C₂×#41×C₆₁	1	2	yes	—
489	C₈	C₂×#41×C₆₁	1	2	yes	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	8	no	—
492	C₂²	—	2	4	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	C₆₁×#547	1	2	yes	—
497	C₂²	C₂³×D₄×C₆₁	1	1	yes	characteristic
498	C₄	—	2	2	no	—
499	C₄	—	2	4	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	8	no	—
503	C₄	—	4	2	no	—
504	C₄	C₆₁×#547	1	2	yes	—
505	C₄	C₆₁×#577	1	4	yes	—
506	C₄	C₂²×#41×C₆₁	1	1	yes	characteristic
507	C₄	C₂×2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
508	C₂	—	2	2	no	—
509	C₂	—	4	4	no	—
510	C₂	—	8	2	no	—
511	C₂	C₂×C₆₁×#547	1	1	yes	characteristic
512	C₂	C₂×C₆₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₆₁×#532 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₆₁×#3511	C₂	semidirect
2. C₆₁×#3356	C₂	semidirect
3. C₆₁×#3354	C₂	semidirect
4. C₆₁×#3340	C₂	semidirect
5. C₆₁×#3335	C₂	semidirect
6. C₂×C₆₁×#498	C₂	semidirect
7. C₆₁×#3099	C₂	semidirect
8. C₆₁×#3079	C₂	semidirect
9. C₆₁×#3060	C₂	semidirect
10. C₄×#135×C₆₁	C₂	semidirect
11. C₆₁×#2931	C₂	semidirect
12. C₆₁×#2912	C₂	semidirect
165. C₂×C₈×C₆₁	#41	semidirect
233. #61288	C₆₁	direct (N × H)
252. #3511	C₂×C₆₁	semidirect
253. #3356	C₂×C₆₁	semidirect
254. #3354	C₂×C₆₁	semidirect
255. #3340	C₂×C₆₁	semidirect
256. #3335	C₂×C₆₁	semidirect
257. C₂×#498	C₂×C₆₁	semidirect
258. #3099	C₂×C₆₁	semidirect
259. #3079	C₂×C₆₁	semidirect
260. #3060	C₂×C₆₁	semidirect
261. C₄×#135	C₂×C₆₁	semidirect
262. #2931	C₂×C₆₁	semidirect
263. #2912	C₂×C₆₁	semidirect
316. C₆₁	#61288	direct (N × H)
422. C₂×C₈	#41×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3511	2
C₆₁×#3356	2
C₆₁×#3354	2
C₆₁×#3340	2
C₆₁×#3335	2
C₂×C₆₁×#498	2
C₆₁×#3099	2
C₆₁×#3079	2
C₆₁×#3060	2
C₄×#135×C₆₁	2
C₆₁×#2931	2
C₆₁×#2912	2
#61288	61

Subgroups — C61×#61288 #29564721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#61288 #29564721