Subgroups — C₆₁×#17519 #29520952

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3482	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#131×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1494	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1491	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1489	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1485	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1483	C₂	1	2	yes	maximal, complemented
10	C₆₁×#1476	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
12	C₂×#131×C₆₁	—	2	1	no	—
13	C₆₁×#557	C₄	1	1	yes	characteristic
14	C₆₁×#556	—	2	1	no	—
15	C₆₁×#556	C₄	1	1	yes	characteristic
16	C₆₁×#552	—	2	2	no	—
17	C₆₁×#552	C₄	1	2	yes	—
18	D₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
19	C₆₁×#548	C₄	1	2	yes	—
20	C₆₁×#547	—	2	1	no	—
21	C₆₁×#547	C₄	1	1	yes	characteristic
22	C₆₁×#533	—	2	1	no	—
23	C₆₁×#533	C₂²	1	1	yes	characteristic
24	C₆₁×#529	—	2	2	no	—
25	C₆₁×#529	C₂²	1	2	yes	—
26	C₄×#41×C₆₁	—	2	1	no	—
27	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
28	C₄×#41×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
30	C₆₁×#446	—	2	1	no	—
31	C₆₁×#435	—	2	2	no	—
32	C₈×D₄×C₆₁	—	2	1	no	—
33	C₆₁×#432	C₂²	1	2	yes	—
34	C₆₁×#431	C₂²	1	1	yes	characteristic
35	C₆₁×#423	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
37	C₆₁×#405	—	2	2	no	—
38	C₆₁×#405	—	2	1	no	—
39	C₄×M₁₆×C₆₁	C₂²	1	2	yes	—
40	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
43	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
45	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
46	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
47	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
48	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
52	C₆₁×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₆₁	—	2	1	no	—
54	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂×#41×C₆₁	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
57	#131×C₆₁	—	2	8	no	—
58	C₂×M₁₆×C₆₁	—	2	2	no	—
59	C₂×M₁₆×C₆₁	—	2	1	no	—
60	C₂×M₁₆×C₆₁	—	2	1	no	—
61	C₂²×C₈×C₆₁	—	2	2	no	—
62	C₂²×C₈×C₆₁	—	2	1	no	—
63	C₂²×C₈×C₆₁	—	2	2	no	—
64	C₄⋊Q₈×C₆₁	—	2	1	no	—
65	C₄⋊Q₈×C₆₁	—	2	1	no	—
66	C₄⋊Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₄⋊Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
68	#126×C₆₁	—	2	2	no	—
69	#126×C₆₁	—	2	2	no	—
70	#126×C₆₁	—	2	2	no	—
71	#125×C₆₁	—	2	2	no	—
72	#125×C₆₁	—	2	2	no	—
73	#125×C₆₁	—	2	1	no	—
74	#125×C₆₁	C₂×C₄	1	2	yes	—
75	#124×C₆₁	—	2	2	no	—
76	#124×C₆₁	C₂×C₄	1	2	yes	—
77	#123×C₆₁	—	2	2	no	—
78	#123×C₆₁	—	2	2	no	—
79	#123×C₆₁	—	2	2	no	—
80	C₂²⋊Q₈×C₆₁	—	2	1	no	—
81	C₂²⋊Q₈×C₆₁	—	2	1	no	—
82	C₂²⋊Q₈×C₆₁	—	2	1	no	—
83	C₂²⋊Q₈×C₆₁	—	2	2	no	—
84	C₂²⋊Q₈×C₆₁	—	2	2	no	—
85	C₂²⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
86	C₂²⋊Q₈×C₆₁	C₂×C₄	1	2	yes	—
87	#121×C₆₁	—	2	2	no	—
88	#121×C₆₁	C₂×C₄	1	2	yes	—
89	C₄×Q₈×C₆₁	—	2	1	no	—
90	C₄×Q₈×C₆₁	—	2	1	no	—
91	C₄×Q₈×C₆₁	—	2	2	no	—
92	C₄×Q₈×C₆₁	—	2	2	no	—
93	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
94	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
95	C₄×Q₈×C₆₁	C₂×C₄	1	2	yes	—
96	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
97	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
98	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
99	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
100	C₄×D₄×C₆₁	—	2	1	no	—
101	C₄×D₄×C₆₁	—	2	1	no	—
102	C₄×D₄×C₆₁	—	2	2	no	—
103	C₄×D₄×C₆₁	—	2	1	no	—
104	C₄×D₄×C₆₁	—	2	2	no	—
105	C₄×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
106	C₄×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
107	C₄×D₄×C₆₁	C₂×C₄	1	2	yes	—
108	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
109	C₄×D₄×C₆₁	D₄	1	2	yes	—
110	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
111	#117×C₆₁	—	2	1	no	—
112	#117×C₆₁	—	2	1	no	—
113	#117×C₆₁	—	2	2	no	—
114	#117×C₆₁	—	2	2	no	—
115	#117×C₆₁	—	2	1	no	—
116	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
117	#117×C₆₁	D₄	1	2	yes	—
118	#117×C₆₁	D₄	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
120	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
121	C₂×C₄²×C₆₁	—	2	2	no	—
122	C₂×C₄²×C₆₁	—	2	1	no	—
123	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
125	C₄⋊C₈×C₆₁	—	2	2	no	—
126	C₄⋊C₈×C₆₁	—	2	1	no	—
127	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
128	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
129	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
130	C₂²⋊C₈×C₆₁	—	2	1	no	—
131	C₂²⋊C₈×C₆₁	—	2	2	no	—
132	#97×C₆₁	—	2	1	no	—
133	#97×C₆₁	—	2	2	no	—
134	#97×C₆₁	—	2	2	no	—
135	C₄×C₈×C₆₁	—	2	1	no	—
136	C₄×C₈×C₆₁	—	2	2	no	—
137	C₄×C₈×C₆₁	—	2	1	no	—
138	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₆₁	C₂³	1	2	yes	—
141	#41×C₆₁	—	2	4	no	—
142	#41×C₆₁	—	2	2	no	—
143	#41×C₆₁	—	4	1	no	—
144	#41×C₆₁	—	4	4	no	—
145	C₂×Q₈×C₆₁	—	2	2	no	—
146	C₂×Q₈×C₆₁	—	2	1	no	—
147	C₂×Q₈×C₆₁	—	2	4	no	—
148	C₂×Q₈×C₆₁	—	4	2	no	—
149	C₂×Q₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
150	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
151	C₂×Q₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
152	C₂×D₄×C₆₁	—	2	1	no	—
153	C₂×D₄×C₆₁	—	2	2	no	—
154	C₂×D₄×C₆₁	C₂²⋊C₄	1	2	yes	—
155	C₂×D₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂×D₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₁	—	2	2	no	—
158	C₂²×C₄×C₆₁	—	2	2	no	—
159	C₂²×C₄×C₆₁	—	2	1	no	—
160	C₂²×C₄×C₆₁	—	2	1	no	—
161	C₂²×C₄×C₆₁	—	2	1	no	—
162	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
163	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
168	M₁₆×C₆₁	—	2	8	no	—
169	M₁₆×C₆₁	—	2	4	no	—
170	M₁₆×C₆₁	—	8	1	no	—
171	C₂×C₈×C₆₁	—	2	8	no	—
172	C₂×C₈×C₆₁	—	2	4	no	—
173	C₂×C₈×C₆₁	—	2	1	no	—
174	C₂×C₈×C₆₁	—	2	1	no	—
175	C₂×C₈×C₆₁	—	2	2	no	—
176	C₂×C₈×C₆₁	—	2	1	no	—
177	C₂×C₈×C₆₁	—	2	1	no	—
178	C₂×C₈×C₆₁	—	2	2	no	—
179	C₂×C₈×C₆₁	—	8	2	no	—
180	C₄⋊C₄×C₆₁	—	2	2	no	—
181	C₄⋊C₄×C₆₁	—	2	2	no	—
182	C₄⋊C₄×C₆₁	—	2	2	no	—
183	C₄⋊C₄×C₆₁	—	2	1	no	—
184	C₄⋊C₄×C₆₁	—	2	2	no	—
185	C₄⋊C₄×C₆₁	—	2	2	no	—
186	C₄⋊C₄×C₆₁	—	2	1	no	—
187	C₄⋊C₄×C₆₁	—	2	1	no	—
188	C₄⋊C₄×C₆₁	—	2	2	no	—
189	C₄⋊C₄×C₆₁	—	2	1	no	—
190	C₄⋊C₄×C₆₁	—	2	2	no	—
191	C₄⋊C₄×C₆₁	—	2	1	no	—
192	C₄⋊C₄×C₆₁	—	2	2	no	—
193	C₄⋊C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
194	C₄⋊C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
195	C₄⋊C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
197	C₄⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
199	C₄⋊C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
200	C₂²⋊C₄×C₆₁	—	2	2	no	—
201	C₂²⋊C₄×C₆₁	—	2	2	no	—
202	C₂²⋊C₄×C₆₁	—	2	1	no	—
203	C₂²⋊C₄×C₆₁	—	2	2	no	—
204	C₂²⋊C₄×C₆₁	—	2	1	no	—
205	C₂²⋊C₄×C₆₁	—	2	2	no	—
206	C₂²⋊C₄×C₆₁	—	2	2	no	—
207	C₂²⋊C₄×C₆₁	—	2	1	no	—
208	C₂²⋊C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
209	C₂²⋊C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
210	C₂²⋊C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
211	C₄²×C₆₁	—	2	1	no	—
212	C₄²×C₆₁	—	2	2	no	—
213	C₄²×C₆₁	—	2	1	no	—
214	C₄²×C₆₁	—	2	2	no	—
215	C₄²×C₆₁	—	2	1	no	—
216	C₄²×C₆₁	—	2	2	no	—
217	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
218	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
219	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₆₁	C₂×D₄	1	2	yes	—
221	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
222	C₄²×C₆₁	C₂×D₄	1	2	yes	—
223	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
224	C₂³×C₆₁	—	2	1	no	—
225	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂³×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
227	Q₈×C₆₁	—	2	1	no	—
228	Q₈×C₆₁	—	2	2	no	—
229	Q₈×C₆₁	—	2	2	no	—
230	Q₈×C₆₁	—	2	2	no	—
231	Q₈×C₆₁	—	4	4	no	—
232	Q₈×C₆₁	—	4	2	no	—
233	Q₈×C₆₁	#131	1	2	yes	complemented
234	D₄×C₆₁	—	2	4	no	—
235	D₄×C₆₁	—	2	1	no	—
236	D₄×C₆₁	—	2	1	no	—
237	D₄×C₆₁	—	4	2	no	—
238	D₄×C₆₁	—	4	4	no	—
239	D₄×C₆₁	#104	1	4	yes	—
240	C₂×C₄×C₆₁	—	2	1	no	—
241	C₂×C₄×C₆₁	—	2	4	no	—
242	C₂×C₄×C₆₁	—	2	2	no	—
243	C₂×C₄×C₆₁	—	2	1	no	—
244	C₂×C₄×C₆₁	—	2	2	no	—
245	C₂×C₄×C₆₁	—	2	1	no	—
246	C₂×C₄×C₆₁	—	2	2	no	—
247	C₂×C₄×C₆₁	—	2	1	no	—
248	C₂×C₄×C₆₁	—	2	2	no	—
249	C₂×C₄×C₆₁	—	2	1	no	—
250	C₂×C₄×C₆₁	—	2	2	no	—
251	C₂×C₄×C₆₁	—	2	1	no	—
252	C₂×C₄×C₆₁	—	2	1	no	—
253	C₂×C₄×C₆₁	—	2	2	no	—
254	C₂×C₄×C₆₁	—	2	2	no	—
255	C₂×C₄×C₆₁	—	4	2	no	—
256	C₂×C₄×C₆₁	—	4	4	no	—
257	C₂×C₄×C₆₁	—	4	4	no	—
258	C₂×C₄×C₆₁	—	4	1	no	—
259	C₂×C₄×C₆₁	—	4	1	no	—
260	C₂×C₄×C₆₁	—	4	2	no	—
261	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
262	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
265	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
266	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
268	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
270	C₈×C₆₁	—	2	2	no	—
271	C₈×C₆₁	—	2	4	no	—
272	C₈×C₆₁	—	2	2	no	—
273	C₈×C₆₁	—	4	1	no	—
274	C₈×C₆₁	—	4	2	no	—
275	C₈×C₆₁	—	4	1	no	—
276	#17519	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
277	C₂²×C₆₁	—	2	2	no	—
278	C₂²×C₆₁	—	2	2	no	—
279	C₂²×C₆₁	—	2	1	no	—
280	C₂²×C₆₁	—	2	1	no	—
281	C₂²×C₆₁	—	2	1	no	—
282	C₂²×C₆₁	—	4	1	no	—
283	C₂²×C₆₁	—	4	1	no	—
284	C₂²×C₆₁	—	4	1	no	—
285	C₂²×C₆₁	C₂×#104	1	4	yes	—
286	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
287	C₄×C₆₁	—	2	1	no	—
288	C₄×C₆₁	—	2	2	no	—
289	C₄×C₆₁	—	2	2	no	—
290	C₄×C₆₁	—	2	2	no	—
291	C₄×C₆₁	—	2	1	no	—
292	C₄×C₆₁	—	2	1	no	—
293	C₄×C₆₁	—	2	2	no	—
294	C₄×C₆₁	—	4	2	no	—
295	C₄×C₆₁	—	4	1	no	—
296	C₄×C₆₁	—	4	1	no	—
297	C₄×C₆₁	—	4	2	no	—
298	C₄×C₆₁	C₂×#104	1	2	yes	—
299	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
300	C₄×C₆₁	#568	1	1	yes	characteristic
301	C₄×C₆₁	#577	1	1	yes	characteristic
302	C₄×C₆₁	#578	1	1	yes	characteristic
303	#3482	C₂×C₆₁	1	1	yes	characteristic
304	#2990	C₂×C₆₁	1	1	yes	characteristic, complemented
305	C₄×#131	C₂×C₆₁	1	1	yes	characteristic, complemented
306	#1494	C₂×C₆₁	1	2	yes	complemented
307	#1493	C₂×C₆₁	1	1	yes	characteristic, complemented
308	#1491	C₂×C₆₁	1	2	yes	complemented
309	#1489	C₂×C₆₁	1	1	yes	characteristic, complemented
310	#1485	C₂×C₆₁	1	1	yes	characteristic, complemented
311	#1483	C₂×C₆₁	1	2	yes	complemented
312	#1476	C₂×C₆₁	1	1	yes	characteristic, complemented
313	C₂×Q₈⋊C₈	C₂×C₆₁	1	2	yes	complemented
314	C₂×C₆₁	—	2	4	no	—
315	C₂×C₆₁	—	4	1	no	—
316	C₂×C₆₁	—	8	1	no	—
317	C₂×C₆₁	#2880	1	1	yes	characteristic
318	C₂×C₆₁	#2898	1	1	yes	characteristic
319	C₂×C₆₁	C₂²×#104	1	1	yes	characteristic
320	C₂×#131	—	2	1	no	—
321	#557	C₄×C₆₁	1	1	yes	characteristic
322	#556	—	2	1	no	—
323	#556	C₄×C₆₁	1	1	yes	characteristic
324	#552	—	2	2	no	—
325	#552	C₄×C₆₁	1	2	yes	—
326	D₄×Q₈	C₄×C₆₁	1	1	yes	characteristic
327	#548	C₄×C₆₁	1	2	yes	—
328	#547	—	2	1	no	—
329	#547	C₄×C₆₁	1	1	yes	characteristic
330	#533	—	2	1	no	—
331	#533	C₂²×C₆₁	1	1	yes	characteristic
332	#529	—	2	2	no	—
333	#529	C₂²×C₆₁	1	2	yes	—
334	C₄×#41	—	2	1	no	—
335	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
336	C₄×#41	C₂²×C₆₁	1	1	yes	characteristic
337	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
338	#446	—	2	1	no	—
339	#435	—	2	2	no	—
340	C₈×D₄	—	2	1	no	—
341	#432	C₂²×C₆₁	1	2	yes	—
342	#431	C₂²×C₆₁	1	1	yes	characteristic
343	#423	C₂²×C₆₁	1	1	yes	characteristic
344	C₂×C₄⋊C₈	C₂²×C₆₁	1	2	yes	—
345	#405	—	2	2	no	—
346	#405	—	2	1	no	—
347	C₄×M₁₆	C₂²×C₆₁	1	2	yes	—
348	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
349	C₂×C₄×C₈	C₂²×C₆₁	1	2	yes	—
350	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
351	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
352	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
353	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
354	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
355	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
356	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
357	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
358	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
359	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
360	D₄⋊C₈	C₂²×C₆₁	1	2	yes	—
361	C₆₁	#17519	1	1	yes	characteristic, direct product factor, Sylow
362	C₂×#41	—	2	1	no	—
363	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
364	C₂×#41	D₄×C₆₁	1	1	yes	characteristic
365	C₂²×Q₈	C₂×C₄×C₆₁	1	2	yes	—
366	#131	—	2	8	no	—
367	C₂×M₁₆	—	2	2	no	—
368	C₂×M₁₆	—	2	1	no	—
369	C₂×M₁₆	—	2	1	no	—
370	C₂²×C₈	—	2	2	no	—
371	C₂²×C₈	—	2	1	no	—
372	C₂²×C₈	—	2	2	no	—
373	C₄⋊Q₈	—	2	1	no	—
374	C₄⋊Q₈	—	2	1	no	—
375	C₄⋊Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
376	C₄⋊Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
377	#126	—	2	2	no	—
378	#126	—	2	2	no	—
379	#126	—	2	2	no	—
380	#125	—	2	2	no	—
381	#125	—	2	2	no	—
382	#125	—	2	1	no	—
383	#125	C₂×C₄×C₆₁	1	2	yes	—
384	#124	—	2	2	no	—
385	#124	C₂×C₄×C₆₁	1	2	yes	—
386	#123	—	2	2	no	—
387	#123	—	2	2	no	—
388	#123	—	2	2	no	—
389	C₂²⋊Q₈	—	2	1	no	—
390	C₂²⋊Q₈	—	2	1	no	—
391	C₂²⋊Q₈	—	2	1	no	—
392	C₂²⋊Q₈	—	2	2	no	—
393	C₂²⋊Q₈	—	2	2	no	—
394	C₂²⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
395	C₂²⋊Q₈	C₂×C₄×C₆₁	1	2	yes	—
396	#121	—	2	2	no	—
397	#121	C₂×C₄×C₆₁	1	2	yes	—
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	—	2	2	no	—
400	C₄×Q₈	—	2	2	no	—
401	C₄×Q₈	—	2	1	no	—
402	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
403	C₄×Q₈	C₂×C₄×C₆₁	1	2	yes	—
404	C₄×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
405	C₄×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
406	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
407	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
408	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	1	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
415	C₄×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
416	C₄×D₄	C₂×C₄×C₆₁	1	2	yes	—
417	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
418	C₄×D₄	D₄×C₆₁	1	2	yes	—
419	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
420	#117	—	2	1	no	—
421	#117	—	2	1	no	—
422	#117	—	2	2	no	—
423	#117	—	2	1	no	—
424	#117	—	2	2	no	—
425	#117	C₂×C₄×C₆₁	1	1	yes	characteristic
426	#117	D₄×C₆₁	1	2	yes	—
427	#117	D₄×C₆₁	1	1	yes	characteristic
428	C₂×C₄⋊C₄	—	2	2	no	—
429	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	2	yes	—
430	C₂×C₄²	—	2	2	no	—
431	C₂×C₄²	—	2	1	no	—
432	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
433	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
434	C₄⋊C₈	—	2	2	no	—
435	C₄⋊C₈	—	2	1	no	—
436	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
437	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
438	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
439	C₂²⋊C₈	—	2	1	no	—
440	C₂²⋊C₈	—	2	2	no	—
441	#97	—	2	1	no	—
442	#97	—	2	2	no	—
443	#97	—	2	2	no	—
444	C₄×C₈	—	2	1	no	—
445	C₄×C₈	—	2	2	no	—
446	C₄×C₈	—	2	1	no	—
447	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
448	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
449	C₄×C₈	C₂³×C₆₁	1	2	yes	—
450	#41	—	2	4	no	—
451	#41	—	2	2	no	—
452	#41	—	4	1	no	—
453	#41	—	4	4	no	—
454	C₂×Q₈	—	2	2	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	—	2	4	no	—
457	C₂×Q₈	—	4	2	no	—
458	C₂×Q₈	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
459	C₂×Q₈	C₂²×C₄×C₆₁	1	2	yes	—
460	C₂×Q₈	C₂²×C₄×C₆₁	1	1	yes	characteristic
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	C₂²⋊C₄×C₆₁	1	2	yes	—
464	C₂×D₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
465	C₂×D₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	C₂²×C₄×C₆₁	1	2	yes	—
472	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
474	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
476	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
477	M₁₆	—	2	8	no	—
478	M₁₆	—	2	4	no	—
479	M₁₆	—	8	1	no	—
480	C₂×C₈	—	2	8	no	—
481	C₂×C₈	—	2	4	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	2	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	2	no	—
488	C₂×C₈	—	8	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
503	C₄⋊C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
504	C₄⋊C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
505	C₄⋊C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
506	C₄⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
507	C₄⋊C₄	C₂²×C₄×C₆₁	1	2	yes	—
508	C₄⋊C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	1	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	1	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
518	C₂²⋊C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
519	C₂²⋊C₄	C₂²⋊C₄×C₆₁	1	2	yes	—
520	C₄²	—	2	1	no	—
521	C₄²	—	2	2	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	2	1	no	—
525	C₄²	—	2	2	no	—
526	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
527	C₄²	C₂²×C₄×C₆₁	1	2	yes	—
528	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
529	C₄²	C₂×D₄×C₆₁	1	2	yes	—
530	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
531	C₄²	C₂×D₄×C₆₁	1	2	yes	—
532	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
533	C₂³	—	2	1	no	—
534	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
535	C₂³	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
536	Q₈	—	2	1	no	—
537	Q₈	—	2	2	no	—
538	Q₈	—	2	2	no	—
539	Q₈	—	2	2	no	—
540	Q₈	—	4	4	no	—
541	Q₈	—	4	2	no	—
542	Q₈	#131×C₆₁	1	2	yes	complemented
543	D₄	—	2	4	no	—
544	D₄	—	2	1	no	—
545	D₄	—	2	1	no	—
546	D₄	—	4	2	no	—
547	D₄	—	4	4	no	—
548	D₄	#104×C₆₁	1	4	yes	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	—	4	4	no	—
566	C₂×C₄	—	4	4	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	2	no	—
570	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
571	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
572	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
573	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
574	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	2	yes	—
575	C₂×C₄	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
576	C₂×C₄	C₂³×C₄×C₆₁	1	1	yes	characteristic
577	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
578	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
579	C₈	—	2	2	no	—
580	C₈	—	2	4	no	—
581	C₈	—	2	2	no	—
582	C₈	—	4	1	no	—
583	C₈	—	4	2	no	—
584	C₈	—	4	1	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	1	no	—
588	C₂²	—	2	1	no	—
589	C₂²	—	2	1	no	—
590	C₂²	—	4	1	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	4	1	no	—
593	C₂²	C₂×#104×C₆₁	1	4	yes	—
594	C₂²	C₂²×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
595	C₄	—	2	1	no	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	1	no	—
600	C₄	—	2	1	no	—
601	C₄	—	2	2	no	—
602	C₄	—	4	2	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	2	no	—
606	C₄	C₂×#104×C₆₁	1	2	yes	—
607	C₄	C₂×#131×C₆₁	1	1	yes	characteristic
608	C₄	C₆₁×#568	1	1	yes	characteristic
609	C₄	C₆₁×#577	1	1	yes	characteristic
610	C₄	C₆₁×#578	1	1	yes	characteristic
611	C₂	—	2	4	no	—
612	C₂	—	4	1	no	—
613	C₂	—	8	1	no	—
614	C₂	C₆₁×#2880	1	1	yes	characteristic
615	C₂	C₆₁×#2898	1	1	yes	characteristic
616	C₂	C₂²×#104×C₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₁
Frattini, cosocle	C₄²	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂²×C₂²⋊C₄
prime radical	C₆₁×#3482	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₄×#41×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#2990	C₂	semidirect
3. C₄×#131×C₆₁	C₂	semidirect
4. C₆₁×#1494	C₂	semidirect
5. C₆₁×#1493	C₂	semidirect
6. C₆₁×#1491	C₂	semidirect
7. C₆₁×#1489	C₂	semidirect
8. C₆₁×#1485	C₂	semidirect
9. C₆₁×#1483	C₂	semidirect
10. C₆₁×#1476	C₂	semidirect
11. C₂×C₆₁×Q₈⋊C₈	C₂	semidirect
233. Q₈×C₆₁	#131	semidirect
276. #17519	C₆₁	direct (N × H)
304. #2990	C₂×C₆₁	semidirect
305. C₄×#131	C₂×C₆₁	semidirect
306. #1494	C₂×C₆₁	semidirect
307. #1493	C₂×C₆₁	semidirect
308. #1491	C₂×C₆₁	semidirect
309. #1489	C₂×C₆₁	semidirect
310. #1485	C₂×C₆₁	semidirect
311. #1483	C₂×C₆₁	semidirect
312. #1476	C₂×C₆₁	semidirect
313. C₂×Q₈⋊C₈	C₂×C₆₁	semidirect
361. C₆₁	#17519	direct (N × H)
542. Q₈	#131×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3482	2
C₆₁×#2990	2
C₄×#131×C₆₁	2
C₆₁×#1494	2
C₆₁×#1493	2
C₆₁×#1491	2
C₆₁×#1489	2
C₆₁×#1485	2
C₆₁×#1483	2
C₆₁×#1476	2
C₂×C₆₁×Q₈⋊C₈	2
#17519	61

Subgroups — C61×#17519 #29520952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#17519 #29520952