Subgroups — C₆₁×#18750 #29522183

Order 15616.

1040 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3205	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#3103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3037	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1640	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1620	C₂	1	4	yes	maximal, complemented
7	C₆₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#137×C₆₁	—	2	1	no	—
11	C₆₁×#559	—	2	2	no	—
12	C₆₁×#559	—	2	2	no	—
13	C₆₁×#559	—	2	2	no	—
14	C₆₁×#559	C₂²	1	4	yes	—
15	C₆₁×#532	C₂²	1	1	yes	characteristic
16	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×#124×C₆₁	—	2	1	no	—
18	C₂×#124×C₆₁	—	2	1	no	—
19	C₂×#123×C₆₁	—	2	2	no	—
20	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₆₁	—	2	1	no	—
23	C₂×C₄×D₄×C₆₁	—	2	1	no	—
24	C₆₁×#493	C₂²	1	2	yes	—
25	C₆₁×#486	C₂²	1	2	yes	—
26	C₆₁×#481	—	2	1	no	—
27	C₆₁×#475	C₂²	1	2	yes	complemented
28	C₆₁×#464	—	2	2	no	—
29	C₆₁×#462	C₂²	1	2	yes	complemented
30	C₆₁×#460	C₂²	1	2	yes	complemented
31	C₆₁×#460	C₂²	1	2	yes	complemented
32	C₆₁×#459	—	2	2	no	—
33	C₆₁×#448	—	2	1	no	—
34	C₆₁×#447	—	2	2	no	—
35	C₄×SD₁₆×C₆₁	—	2	2	no	—
36	C₆₁×#431	C₂²	1	1	yes	characteristic
37	C₆₁×#431	C₂²	1	1	yes	characteristic
38	C₆₁×#423	C₂²	1	1	yes	characteristic
39	C₆₁×#417	—	2	1	no	—
40	C₆₁×#331	C₂²	1	4	yes	—
41	C₆₁×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₆₁×Q₈⋊C₈	C₂²	1	4	yes	complemented
43	C₂×#41×C₆₁	—	2	1	no	—
44	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
45	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₆₁	—	2	1	no	—
47	C₂²×D₄×C₆₁	—	2	1	no	—
48	C₂²×D₄×C₆₁	D₄	1	2	yes	—
49	C₂³×C₄×C₆₁	—	2	2	no	—
50	#137×C₆₁	—	4	4	no	—
51	C₂×Q₁₆×C₆₁	—	2	2	no	—
52	C₂×SD₁₆×C₆₁	—	2	2	no	—
53	C₂×SD₁₆×C₆₁	—	2	2	no	—
54	C₂×D₈×C₆₁	—	2	4	no	—
55	C₂×M₁₆×C₆₁	—	2	1	no	—
56	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
57	#127×C₆₁	—	2	1	no	—
58	#127×C₆₁	—	2	1	no	—
59	#127×C₆₁	D₄	1	2	yes	—
60	#127×C₆₁	C₂³	1	1	yes	characteristic
61	#127×C₆₁	C₂³	1	2	yes	—
62	#124×C₆₁	—	2	4	no	—
63	#124×C₆₁	—	2	4	no	—
64	#124×C₆₁	—	2	4	no	—
65	#124×C₆₁	—	2	4	no	—
66	#124×C₆₁	—	2	1	no	—
67	#123×C₆₁	—	2	8	no	—
68	#123×C₆₁	—	2	8	no	—
69	#121×C₆₁	—	2	1	no	—
70	#121×C₆₁	—	2	1	no	—
71	C₄×Q₈×C₆₁	—	2	1	no	—
72	C₄×Q₈×C₆₁	—	2	1	no	—
73	C₄×Q₈×C₆₁	—	2	1	no	—
74	C₄×Q₈×C₆₁	—	2	1	no	—
75	C₄×Q₈×C₆₁	D₄	1	2	yes	—
76	C₄×Q₈×C₆₁	D₄	1	2	yes	—
77	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
78	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
79	C₄×D₄×C₆₁	—	2	4	no	—
80	C₄×D₄×C₆₁	—	2	4	no	—
81	C₄×D₄×C₆₁	—	2	4	no	—
82	C₄×D₄×C₆₁	—	2	4	no	—
83	C₄×D₄×C₆₁	—	2	1	no	—
84	C₄×D₄×C₆₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
92	C₂×C₄²×C₆₁	—	2	1	no	—
93	C₂×C₄²×C₆₁	—	2	1	no	—
94	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
95	#106×C₆₁	—	2	2	no	—
96	C₄⋊C₈×C₆₁	—	2	1	no	—
97	C₄⋊C₈×C₆₁	—	2	1	no	—
98	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
99	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
100	#103×C₆₁	—	2	2	no	—
101	#103×C₆₁	—	2	2	no	—
102	#102×C₆₁	—	2	4	no	—
103	#102×C₆₁	—	2	2	no	—
104	#102×C₆₁	—	2	2	no	—
105	C₂²⋊C₈×C₆₁	—	2	1	no	—
106	C₂²⋊C₈×C₆₁	—	2	1	no	—
107	C₄×C₈×C₆₁	—	2	1	no	—
108	C₄×C₈×C₆₁	C₂³	1	2	yes	—
109	C₂⁴×C₆₁	—	2	2	no	—
110	#41×C₆₁	—	4	2	no	—
111	#41×C₆₁	—	4	2	no	—
112	C₂×Q₈×C₆₁	—	2	1	no	—
113	C₂×Q₈×C₆₁	—	2	1	no	—
114	C₂×Q₈×C₆₁	—	2	2	no	—
115	C₂×Q₈×C₆₁	—	2	2	no	—
116	C₂×Q₈×C₆₁	—	2	1	no	—
117	C₂×Q₈×C₆₁	—	4	2	no	—
118	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
119	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
120	C₂×D₄×C₆₁	—	2	2	no	—
121	C₂×D₄×C₆₁	—	2	2	no	—
122	C₂×D₄×C₆₁	—	2	2	no	—
123	C₂×D₄×C₆₁	—	2	2	no	—
124	C₂×D₄×C₆₁	—	2	2	no	—
125	C₂×D₄×C₆₁	—	2	1	no	—
126	C₂×D₄×C₆₁	—	2	1	no	—
127	C₂×D₄×C₆₁	—	2	1	no	—
128	C₂×D₄×C₆₁	—	4	4	no	—
129	C₂×D₄×C₆₁	—	4	1	no	—
130	C₂×D₄×C₆₁	—	4	2	no	—
131	C₂×D₄×C₆₁	—	4	2	no	—
132	C₂×D₄×C₆₁	—	8	2	no	—
133	C₂×D₄×C₆₁	C₂×D₄	1	4	yes	—
134	C₂²×C₄×C₆₁	—	2	1	no	—
135	C₂²×C₄×C₆₁	—	2	1	no	—
136	C₂²×C₄×C₆₁	—	2	1	no	—
137	C₂²×C₄×C₆₁	—	2	1	no	—
138	C₂²×C₄×C₆₁	—	2	8	no	—
139	C₂²×C₄×C₆₁	—	4	1	no	—
140	C₂²×C₄×C₆₁	—	8	4	no	—
141	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
144	Q₁₆×C₆₁	—	4	4	no	—
145	SD₁₆×C₆₁	—	4	4	no	—
146	SD₁₆×C₆₁	—	4	4	no	—
147	D₈×C₆₁	—	8	4	no	—
148	M₁₆×C₆₁	—	4	2	no	—
149	C₂×C₈×C₆₁	—	2	2	no	—
150	C₂×C₈×C₆₁	—	2	2	no	—
151	C₂×C₈×C₆₁	—	2	2	no	—
152	C₄⋊C₄×C₆₁	—	2	2	no	—
153	C₄⋊C₄×C₆₁	—	2	2	no	—
154	C₄⋊C₄×C₆₁	—	2	1	no	—
155	C₄⋊C₄×C₆₁	—	2	1	no	—
156	C₄⋊C₄×C₆₁	—	2	2	no	—
157	C₄⋊C₄×C₆₁	—	2	2	no	—
158	C₄⋊C₄×C₆₁	—	2	1	no	—
159	C₄⋊C₄×C₆₁	—	2	1	no	—
160	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
162	C₂²⋊C₄×C₆₁	—	2	8	no	—
163	C₂²⋊C₄×C₆₁	—	2	8	no	—
164	C₂²⋊C₄×C₆₁	—	2	8	no	—
165	C₂²⋊C₄×C₆₁	—	4	1	no	—
166	C₂²⋊C₄×C₆₁	—	4	1	no	—
167	C₄²×C₆₁	—	2	2	no	—
168	C₄²×C₆₁	—	2	2	no	—
169	C₄²×C₆₁	—	2	2	no	—
170	C₄²×C₆₁	—	2	2	no	—
171	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
175	C₂³×C₆₁	—	2	4	no	—
176	C₂³×C₆₁	—	4	4	no	—
177	C₂³×C₆₁	—	4	4	no	—
178	C₂³×C₆₁	—	4	4	no	—
179	C₂³×C₆₁	—	4	1	no	—
180	C₂³×C₆₁	#120	1	1	yes	characteristic
181	Q₈×C₆₁	—	2	2	no	—
182	Q₈×C₆₁	—	2	2	no	—
183	Q₈×C₆₁	—	2	2	no	—
184	Q₈×C₆₁	—	2	2	no	—
185	Q₈×C₆₁	—	4	2	no	—
186	Q₈×C₆₁	—	4	2	no	—
187	Q₈×C₆₁	—	4	1	no	—
188	Q₈×C₆₁	—	4	1	no	—
189	D₄×C₆₁	—	4	4	no	—
190	D₄×C₆₁	—	4	4	no	—
191	D₄×C₆₁	—	4	2	no	—
192	D₄×C₆₁	—	4	2	no	—
193	D₄×C₆₁	—	4	1	no	—
194	D₄×C₆₁	—	4	2	no	—
195	D₄×C₆₁	—	4	1	no	—
196	D₄×C₆₁	—	4	4	no	—
197	D₄×C₆₁	—	4	2	no	—
198	D₄×C₆₁	—	4	4	no	—
199	D₄×C₆₁	—	8	2	no	—
200	D₄×C₆₁	—	8	2	no	—
201	C₂×C₄×C₆₁	—	2	2	no	—
202	C₂×C₄×C₆₁	—	2	2	no	—
203	C₂×C₄×C₆₁	—	2	2	no	—
204	C₂×C₄×C₆₁	—	2	2	no	—
205	C₂×C₄×C₆₁	—	2	2	no	—
206	C₂×C₄×C₆₁	—	4	1	no	—
207	C₂×C₄×C₆₁	—	4	2	no	—
208	C₂×C₄×C₆₁	—	4	2	no	—
209	C₂×C₄×C₆₁	—	4	2	no	—
210	C₂×C₄×C₆₁	—	4	2	no	—
211	C₂×C₄×C₆₁	—	4	1	no	—
212	C₂×C₄×C₆₁	—	8	8	no	—
213	C₂×C₄×C₆₁	—	8	2	no	—
214	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
215	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
216	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
217	C₂×C₄×C₆₁	#135	1	2	yes	—
218	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
219	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
222	C₈×C₆₁	—	4	2	no	—
223	C₈×C₆₁	—	4	2	no	—
224	C₈×C₆₁	—	4	2	no	—
225	#18750	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂²×C₆₁	—	2	1	no	—
227	C₂²×C₆₁	—	2	1	no	—
228	C₂²×C₆₁	—	4	4	no	—
229	C₂²×C₆₁	—	4	4	no	—
230	C₂²×C₆₁	—	4	4	no	—
231	C₂²×C₆₁	—	8	8	no	—
232	C₂²×C₆₁	—	8	1	no	—
233	C₂²×C₆₁	—	8	1	no	—
234	C₂²×C₆₁	—	8	1	no	—
235	C₂²×C₆₁	#449	1	2	yes	—
236	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
237	C₄×C₆₁	—	2	1	no	—
238	C₄×C₆₁	—	2	1	no	—
239	C₄×C₆₁	—	2	1	no	—
240	C₄×C₆₁	—	4	2	no	—
241	C₄×C₆₁	—	4	2	no	—
242	C₄×C₆₁	—	4	2	no	—
243	C₄×C₆₁	—	4	2	no	—
244	C₄×C₆₁	—	8	1	no	—
245	C₄×C₆₁	#449	1	2	yes	—
246	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
247	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
248	C₄×C₆₁	#575	1	2	yes	—
249	C₂×#559	C₂×C₆₁	1	1	yes	characteristic, complemented
250	#3205	C₂×C₆₁	1	1	yes	characteristic
251	#3103	C₂×C₆₁	1	1	yes	characteristic, complemented
252	#3037	C₂×C₆₁	1	1	yes	characteristic, complemented
253	#1640	C₂×C₆₁	1	4	yes	complemented
254	#1620	C₂×C₆₁	1	4	yes	complemented
255	#1545	C₂×C₆₁	1	1	yes	characteristic, complemented
256	#1487	C₂×C₆₁	1	1	yes	characteristic, complemented
257	#1479	C₂×C₆₁	1	1	yes	characteristic, complemented
258	C₂×C₆₁	—	2	2	no	—
259	C₂×C₆₁	—	8	4	no	—
260	C₂×C₆₁	—	16	1	no	—
261	C₂×C₆₁	C₂×#449	1	1	yes	characteristic
262	C₂×C₆₁	#3005	1	1	yes	characteristic
263	C₂×C₆₁	#3019	1	1	yes	characteristic
264	C₂×#137	—	2	1	no	—
265	#559	—	2	2	no	—
266	#559	—	2	2	no	—
267	#559	—	2	2	no	—
268	#559	C₂²×C₆₁	1	4	yes	—
269	#532	C₂²×C₆₁	1	1	yes	characteristic
270	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
271	C₂×#124	—	2	1	no	—
272	C₂×#124	—	2	1	no	—
273	C₂×#123	—	2	2	no	—
274	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
275	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
276	C₂×C₄×D₄	—	2	1	no	—
277	C₂×C₄×D₄	—	2	1	no	—
278	#493	C₂²×C₆₁	1	2	yes	—
279	#486	C₂²×C₆₁	1	2	yes	—
280	#481	—	2	1	no	—
281	#475	C₂²×C₆₁	1	2	yes	complemented
282	#464	—	2	2	no	—
283	#462	C₂²×C₆₁	1	2	yes	complemented
284	#460	C₂²×C₆₁	1	2	yes	complemented
285	#460	C₂²×C₆₁	1	2	yes	complemented
286	#459	—	2	2	no	—
287	#448	—	2	1	no	—
288	#447	—	2	2	no	—
289	C₄×SD₁₆	—	2	2	no	—
290	#431	C₂²×C₆₁	1	1	yes	characteristic
291	#431	C₂²×C₆₁	1	1	yes	characteristic
292	#423	C₂²×C₆₁	1	1	yes	characteristic
293	#417	—	2	1	no	—
294	#331	C₂²×C₆₁	1	4	yes	—
295	Q₈⋊C₈	C₂²×C₆₁	1	4	yes	complemented
296	Q₈⋊C₈	C₂²×C₆₁	1	4	yes	complemented
297	C₆₁	#18750	1	1	yes	characteristic, direct product factor, Sylow
298	C₂×#41	—	2	1	no	—
299	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
300	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	D₄×C₆₁	1	2	yes	—
304	C₂³×C₄	—	2	2	no	—
305	#137	—	4	4	no	—
306	C₂×Q₁₆	—	2	2	no	—
307	C₂×SD₁₆	—	2	2	no	—
308	C₂×SD₁₆	—	2	2	no	—
309	C₂×D₈	—	2	4	no	—
310	C₂×M₁₆	—	2	1	no	—
311	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
312	#127	—	2	1	no	—
313	#127	—	2	1	no	—
314	#127	D₄×C₆₁	1	2	yes	—
315	#127	C₂³×C₆₁	1	1	yes	characteristic
316	#127	C₂³×C₆₁	1	2	yes	—
317	#124	—	2	4	no	—
318	#124	—	2	4	no	—
319	#124	—	2	4	no	—
320	#124	—	2	4	no	—
321	#124	—	2	1	no	—
322	#123	—	2	8	no	—
323	#123	—	2	8	no	—
324	#121	—	2	1	no	—
325	#121	—	2	1	no	—
326	C₄×Q₈	—	2	1	no	—
327	C₄×Q₈	—	2	1	no	—
328	C₄×Q₈	—	2	1	no	—
329	C₄×Q₈	—	2	1	no	—
330	C₄×Q₈	D₄×C₆₁	1	2	yes	—
331	C₄×Q₈	D₄×C₆₁	1	2	yes	—
332	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
333	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
334	C₄×D₄	—	2	4	no	—
335	C₄×D₄	—	2	4	no	—
336	C₄×D₄	—	2	4	no	—
337	C₄×D₄	—	2	4	no	—
338	C₄×D₄	—	2	1	no	—
339	C₄×D₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
343	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
344	C₂×C₂²⋊C₄	—	2	2	no	—
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
350	#106	—	2	2	no	—
351	C₄⋊C₈	—	2	1	no	—
352	C₄⋊C₈	—	2	1	no	—
353	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
354	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
355	#103	—	2	2	no	—
356	#103	—	2	2	no	—
357	#102	—	2	4	no	—
358	#102	—	2	2	no	—
359	#102	—	2	2	no	—
360	C₂²⋊C₈	—	2	1	no	—
361	C₂²⋊C₈	—	2	1	no	—
362	C₄×C₈	—	2	1	no	—
363	C₄×C₈	C₂³×C₆₁	1	2	yes	—
364	C₂⁴	—	2	2	no	—
365	#41	—	4	2	no	—
366	#41	—	4	2	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	4	2	no	—
373	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
374	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	4	4	no	—
384	C₂×D₄	—	4	1	no	—
385	C₂×D₄	—	4	2	no	—
386	C₂×D₄	—	4	2	no	—
387	C₂×D₄	—	8	2	no	—
388	C₂×D₄	C₂×D₄×C₆₁	1	4	yes	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	8	no	—
394	C₂²×C₄	—	4	1	no	—
395	C₂²×C₄	—	8	4	no	—
396	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
399	Q₁₆	—	4	4	no	—
400	SD₁₆	—	4	4	no	—
401	SD₁₆	—	4	4	no	—
402	D₈	—	8	4	no	—
403	M₁₆	—	4	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
416	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
417	C₂²⋊C₄	—	2	8	no	—
418	C₂²⋊C₄	—	2	8	no	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₂²⋊C₄	—	4	1	no	—
421	C₂²⋊C₄	—	4	1	no	—
422	C₄²	—	2	2	no	—
423	C₄²	—	2	2	no	—
424	C₄²	—	2	2	no	—
425	C₄²	—	2	2	no	—
426	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
427	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
428	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
429	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
430	C₂³	—	2	4	no	—
431	C₂³	—	4	4	no	—
432	C₂³	—	4	4	no	—
433	C₂³	—	4	4	no	—
434	C₂³	—	4	1	no	—
435	C₂³	#120×C₆₁	1	1	yes	characteristic
436	Q₈	—	2	2	no	—
437	Q₈	—	2	2	no	—
438	Q₈	—	2	2	no	—
439	Q₈	—	2	2	no	—
440	Q₈	—	4	2	no	—
441	Q₈	—	4	2	no	—
442	Q₈	—	4	1	no	—
443	Q₈	—	4	1	no	—
444	D₄	—	4	4	no	—
445	D₄	—	4	4	no	—
446	D₄	—	4	2	no	—
447	D₄	—	4	2	no	—
448	D₄	—	4	1	no	—
449	D₄	—	4	2	no	—
450	D₄	—	4	1	no	—
451	D₄	—	4	4	no	—
452	D₄	—	4	2	no	—
453	D₄	—	4	4	no	—
454	D₄	—	8	2	no	—
455	D₄	—	8	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	8	8	no	—
468	C₂×C₄	—	8	2	no	—
469	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
470	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
471	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
472	C₂×C₄	#135×C₆₁	1	2	yes	—
473	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
474	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
477	C₈	—	4	2	no	—
478	C₈	—	4	2	no	—
479	C₈	—	4	2	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	4	4	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	4	4	no	—
485	C₂²	—	8	8	no	—
486	C₂²	—	8	1	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	1	no	—
489	C₂²	C₆₁×#449	1	2	yes	—
490	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	8	1	no	—
499	C₄	C₆₁×#449	1	2	yes	—
500	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
501	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
502	C₄	C₆₁×#575	1	2	yes	—
503	C₂	—	2	2	no	—
504	C₂	—	8	4	no	—
505	C₂	—	16	1	no	—
506	C₂	C₂×C₆₁×#449	1	1	yes	characteristic
507	C₂	C₆₁×#3005	1	1	yes	characteristic
508	C₂	C₆₁×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#3205	C₂
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. C₂×C₆₁×#559	C₂	semidirect
3. C₆₁×#3103	C₂	semidirect
4. C₆₁×#3037	C₂	semidirect
5. C₆₁×#1640	C₂	semidirect
6. C₆₁×#1620	C₂	semidirect
7. C₆₁×#1545	C₂	semidirect
8. C₆₁×#1487	C₂	semidirect
9. C₆₁×#1479	C₂	semidirect
27. C₆₁×#475	C₂²	semidirect
29. C₆₁×#462	C₂²	semidirect
30. C₆₁×#460	C₂²	semidirect
31. C₆₁×#460	C₂²	semidirect
41. C₆₁×Q₈⋊C₈	C₂²	semidirect
42. C₆₁×Q₈⋊C₈	C₂²	semidirect
225. #18750	C₆₁	direct (N × H)
249. C₂×#559	C₂×C₆₁	semidirect
251. #3103	C₂×C₆₁	semidirect
252. #3037	C₂×C₆₁	semidirect
253. #1640	C₂×C₆₁	semidirect
254. #1620	C₂×C₆₁	semidirect
255. #1545	C₂×C₆₁	semidirect
256. #1487	C₂×C₆₁	semidirect
257. #1479	C₂×C₆₁	semidirect
281. #475	C₂²×C₆₁	semidirect
283. #462	C₂²×C₆₁	semidirect
284. #460	C₂²×C₆₁	semidirect
285. #460	C₂²×C₆₁	semidirect
295. Q₈⋊C₈	C₂²×C₆₁	semidirect
296. Q₈⋊C₈	C₂²×C₆₁	semidirect
297. C₆₁	#18750	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₆₁×#559	2
C₆₁×#3205	2
C₆₁×#3103	2
C₆₁×#3037	2
C₆₁×#1640	2
C₆₁×#1620	2
C₆₁×#1545	2
C₆₁×#1487	2
C₆₁×#1479	2
#18750	61

Subgroups — C61×#18750 #29522183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18750 #29522183