Subgroups — C₆₁×#24798 #29528231

Order 15616.

846 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3059	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3046	C₂	1	2	yes	maximal
4	C₂×C₆₁×#443	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2913	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2064	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2038	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1882	C₂	1	2	yes	maximal, complemented
11	C₆₁×#1816	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₆₁	C₂²	1	2	yes	—
13	C₂×#135×C₆₁	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#536	C₂²	1	2	yes	—
18	C₆₁×#520	C₂²	1	2	yes	—
19	C₆₁×#473	—	2	4	no	—
20	C₆₁×#470	—	2	2	no	—
21	C₆₁×#467	—	2	1	no	—
22	C₆₁×#467	—	2	1	no	—
23	C₆₁×#464	—	2	2	no	—
24	C₆₁×#463	—	2	2	no	—
25	C₆₁×#462	—	2	2	no	—
26	C₆₁×#461	—	2	2	no	—
27	C₆₁×#443	—	2	4	no	—
28	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
29	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#428	—	2	2	no	—
31	C₆₁×#428	C₂²	1	2	yes	—
32	C₆₁×#427	—	2	1	no	—
33	C₆₁×#427	C₂²	1	1	yes	characteristic
34	C₂×#107×C₆₁	—	2	1	no	—
35	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
36	C₆₁×#424	C₂²	1	2	yes	—
37	C₂×#104×C₆₁	C₂²	1	1	yes	characteristic
38	C₂×#104×C₆₁	C₂²	1	1	yes	characteristic
39	C₆₁×#419	C₂²	1	2	yes	—
40	C₆₁×#417	C₂²	1	2	yes	—
41	C₂×#103×C₆₁	C₂²	1	2	yes	—
42	C₂×#101×C₆₁	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
44	C₆₁×#391	C₂²	1	2	yes	—
45	C₆₁×#343	C₂²	1	2	yes	—
46	C₆₁×#339	C₂²	1	2	yes	—
47	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
48	C₂×#41×C₆₁	C₂³	1	1	yes	characteristic
49	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
50	#137×C₆₁	—	2	4	no	—
51	#137×C₆₁	—	2	4	no	—
52	#135×C₆₁	—	2	2	no	—
53	#135×C₆₁	—	2	2	no	—
54	C₂×Q₁₆×C₆₁	—	2	2	no	—
55	C₂×Q₁₆×C₆₁	—	2	1	no	—
56	C₂×Q₁₆×C₆₁	—	2	1	no	—
57	C₂×Q₁₆×C₆₁	—	2	2	no	—
58	C₂×Q₁₆×C₆₁	—	2	2	no	—
59	C₂×Q₁₆×C₆₁	D₄	1	1	yes	characteristic
60	C₂×SD₁₆×C₆₁	—	2	2	no	—
61	C₂×SD₁₆×C₆₁	D₄	1	1	yes	characteristic
62	C₂×SD₁₆×C₆₁	D₄	1	1	yes	characteristic
63	C₂×D₈×C₆₁	D₄	1	1	yes	characteristic
64	#131×C₆₁	—	2	2	no	—
65	#131×C₆₁	—	2	2	no	—
66	#131×C₆₁	—	2	2	no	—
67	#131×C₆₁	D₄	1	4	yes	—
68	C₂×M₁₆×C₆₁	—	2	1	no	—
69	C₂×M₁₆×C₆₁	—	2	1	no	—
70	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
71	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
72	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₆₁	—	2	1	no	—
74	C₂²×C₈×C₆₁	—	2	1	no	—
75	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
77	C₄⋊Q₈×C₆₁	—	2	2	no	—
78	#124×C₆₁	—	2	2	no	—
79	#123×C₆₁	—	4	2	no	—
80	C₂²⋊Q₈×C₆₁	—	2	2	no	—
81	C₂²⋊Q₈×C₆₁	—	2	2	no	—
82	C₄×Q₈×C₆₁	—	2	2	no	—
83	C₄×D₄×C₆₁	—	2	2	no	—
84	C₄×D₄×C₆₁	—	4	2	no	—
85	#117×C₆₁	—	2	2	no	—
86	#117×C₆₁	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
88	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
89	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
90	#108×C₆₁	—	2	2	no	—
91	#108×C₆₁	—	2	2	no	—
92	#107×C₆₁	—	2	1	no	—
93	#107×C₆₁	—	2	1	no	—
94	#107×C₆₁	—	2	2	no	—
95	#107×C₆₁	D₄	1	1	yes	characteristic
96	#107×C₆₁	D₄	1	1	yes	characteristic
97	#106×C₆₁	—	2	2	no	—
98	#106×C₆₁	D₄	1	2	yes	—
99	C₄⋊C₈×C₆₁	—	2	2	no	—
100	C₄⋊C₈×C₆₁	—	2	2	no	—
101	#104×C₆₁	—	2	2	no	—
102	#104×C₆₁	—	2	2	no	—
103	#103×C₆₁	—	2	2	no	—
104	#103×C₆₁	—	2	2	no	—
105	#103×C₆₁	—	2	2	no	—
106	#103×C₆₁	—	2	2	no	—
107	#103×C₆₁	—	2	2	no	—
108	#102×C₆₁	—	2	2	no	—
109	#101×C₆₁	—	2	4	no	—
110	C₄×C₈×C₆₁	—	2	2	no	—
111	#95×C₆₁	—	4	2	no	—
112	#41×C₆₁	—	2	1	no	—
113	#41×C₆₁	—	2	1	no	—
114	#41×C₆₁	—	2	1	no	—
115	#41×C₆₁	—	2	1	no	—
116	#41×C₆₁	—	2	1	no	—
117	#41×C₆₁	—	2	1	no	—
118	#41×C₆₁	C₂×D₄	1	2	yes	—
119	#41×C₆₁	#41	1	2	yes	—
120	C₂×Q₈×C₆₁	—	2	2	no	—
121	C₂×Q₈×C₆₁	—	2	2	no	—
122	C₂×Q₈×C₆₁	—	2	2	no	—
123	C₂×Q₈×C₆₁	—	2	2	no	—
124	C₂×Q₈×C₆₁	—	2	2	no	—
125	C₂×Q₈×C₆₁	—	2	2	no	—
126	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
127	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
128	C₂×D₄×C₆₁	—	2	1	no	—
129	C₂×D₄×C₆₁	—	2	1	no	—
130	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
131	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₁	—	2	1	no	—
133	C₂²×C₄×C₆₁	—	2	1	no	—
134	C₂²×C₄×C₆₁	—	2	1	no	—
135	C₂²×C₄×C₆₁	—	2	2	no	—
136	C₂²×C₄×C₆₁	—	2	2	no	—
137	C₂²×C₄×C₆₁	—	4	2	no	—
138	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
139	Q₁₆×C₆₁	—	2	2	no	—
140	Q₁₆×C₆₁	—	4	4	no	—
141	Q₁₆×C₆₁	—	4	2	no	—
142	Q₁₆×C₆₁	—	4	2	no	—
143	SD₁₆×C₆₁	—	2	2	no	—
144	SD₁₆×C₆₁	—	2	2	no	—
145	SD₁₆×C₆₁	—	4	4	no	—
146	D₈×C₆₁	—	2	2	no	—
147	M₁₆×C₆₁	—	2	1	no	—
148	M₁₆×C₆₁	—	2	4	no	—
149	M₁₆×C₆₁	—	2	2	no	—
150	M₁₆×C₆₁	—	2	1	no	—
151	M₁₆×C₆₁	—	2	1	no	—
152	M₁₆×C₆₁	—	2	2	no	—
153	M₁₆×C₆₁	—	4	2	no	—
154	M₁₆×C₆₁	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₆₁	—	2	1	no	—
156	C₂×C₈×C₆₁	—	2	1	no	—
157	C₂×C₈×C₆₁	—	2	1	no	—
158	C₂×C₈×C₆₁	—	2	4	no	—
159	C₂×C₈×C₆₁	—	2	2	no	—
160	C₂×C₈×C₆₁	—	2	1	no	—
161	C₂×C₈×C₆₁	—	4	2	no	—
162	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
163	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
166	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₆₁	#41	1	2	yes	—
168	C₄⋊C₄×C₆₁	—	2	2	no	—
169	C₄⋊C₄×C₆₁	—	2	2	no	—
170	C₄⋊C₄×C₆₁	—	4	2	no	—
171	C₄⋊C₄×C₆₁	—	4	2	no	—
172	C₄⋊C₄×C₆₁	—	4	2	no	—
173	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
175	C₂²⋊C₄×C₆₁	—	2	2	no	—
176	C₂²⋊C₄×C₆₁	—	4	2	no	—
177	C₂²⋊C₄×C₆₁	—	4	2	no	—
178	C₂²⋊C₄×C₆₁	—	4	2	no	—
179	C₄²×C₆₁	—	2	2	no	—
180	C₄²×C₆₁	—	2	1	no	—
181	C₄²×C₆₁	—	2	1	no	—
182	C₄²×C₆₁	—	4	2	no	—
183	C₂³×C₆₁	—	2	1	no	—
184	C₂³×C₆₁	—	2	1	no	—
185	C₂³×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
186	Q₈×C₆₁	—	2	1	no	—
187	Q₈×C₆₁	—	2	1	no	—
188	Q₈×C₆₁	—	2	1	no	—
189	Q₈×C₆₁	—	4	2	no	—
190	Q₈×C₆₁	—	4	2	no	—
191	Q₈×C₆₁	—	4	2	no	—
192	Q₈×C₆₁	—	4	2	no	—
193	Q₈×C₆₁	#121	1	2	yes	—
194	D₄×C₆₁	—	2	2	no	—
195	D₄×C₆₁	—	2	2	no	—
196	D₄×C₆₁	—	2	1	no	—
197	D₄×C₆₁	—	2	1	no	—
198	D₄×C₆₁	—	2	1	no	—
199	D₄×C₆₁	—	4	1	no	—
200	D₄×C₆₁	—	4	1	no	—
201	D₄×C₆₁	#121	1	2	yes	—
202	C₂×C₄×C₆₁	—	2	1	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	—	2	2	no	—
207	C₂×C₄×C₆₁	—	2	2	no	—
208	C₂×C₄×C₆₁	—	2	1	no	—
209	C₂×C₄×C₆₁	—	4	1	no	—
210	C₂×C₄×C₆₁	—	4	1	no	—
211	C₂×C₄×C₆₁	—	4	1	no	—
212	C₂×C₄×C₆₁	—	4	1	no	—
213	C₂×C₄×C₆₁	—	4	2	no	—
214	C₂×C₄×C₆₁	—	4	2	no	—
215	C₂×C₄×C₆₁	—	4	2	no	—
216	C₂×C₄×C₆₁	—	4	2	no	—
217	C₂×C₄×C₆₁	—	8	2	no	—
218	C₂×C₄×C₆₁	—	16	2	no	—
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	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
225	C₈×C₆₁	—	2	2	no	—
226	C₈×C₆₁	—	2	1	no	—
227	C₈×C₆₁	—	2	1	no	—
228	C₈×C₆₁	—	4	1	no	—
229	C₈×C₆₁	—	4	2	no	—
230	C₈×C₆₁	#121	1	2	yes	—
231	C₈×C₆₁	#121	1	2	yes	—
232	#24798	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂²×C₆₁	—	2	1	no	—
234	C₂²×C₆₁	—	2	1	no	—
235	C₂²×C₆₁	—	2	2	no	—
236	C₂²×C₆₁	—	4	1	no	—
237	C₂²×C₆₁	—	4	1	no	—
238	C₂²×C₆₁	—	4	1	no	—
239	C₂²×C₆₁	—	4	1	no	—
240	C₂²×C₆₁	—	4	1	no	—
241	C₂²×C₆₁	#537	1	1	yes	characteristic
242	C₂²×C₆₁	D₄²	1	1	yes	characteristic
243	C₂²×C₆₁	#553	1	1	yes	characteristic
244	C₄×C₆₁	—	2	2	no	—
245	C₄×C₆₁	—	4	2	no	—
246	C₄×C₆₁	—	4	1	no	—
247	C₄×C₆₁	—	4	2	no	—
248	C₄×C₆₁	—	4	2	no	—
249	C₄×C₆₁	—	8	2	no	—
250	C₄×C₆₁	—	8	2	no	—
251	C₄×C₆₁	C₂×#121	1	1	yes	characteristic
252	C₄×C₆₁	C₂×#121	1	1	yes	characteristic
253	C₄×C₆₁	#547	1	1	yes	characteristic
254	C₄×C₆₁	#548	1	1	yes	characteristic
255	C₂×#473	C₂×C₆₁	1	1	yes	characteristic, complemented
256	#3059	C₂×C₆₁	1	1	yes	characteristic, complemented
257	#3046	C₂×C₆₁	1	2	yes	—
258	C₂×#443	C₂×C₆₁	1	1	yes	characteristic
259	#2913	C₂×C₆₁	1	1	yes	characteristic, complemented
260	#2064	C₂×C₆₁	1	2	yes	complemented
261	#2038	C₂×C₆₁	1	2	yes	complemented
262	#1969	C₂×C₆₁	1	1	yes	characteristic, complemented
263	#1945	C₂×C₆₁	1	1	yes	characteristic, complemented
264	#1882	C₂×C₆₁	1	2	yes	complemented
265	#1816	C₂×C₆₁	1	1	yes	characteristic, complemented
266	C₂×C₆₁	—	2	1	no	—
267	C₂×C₆₁	—	2	1	no	—
268	C₂×C₆₁	—	4	2	no	—
269	C₂×C₆₁	—	8	1	no	—
270	C₂×C₆₁	#2416	1	1	yes	characteristic
271	C₂×C₆₁	#3290	1	1	yes	characteristic
272	C₂×C₆₁	#3293	1	1	yes	characteristic
273	C₂×#137	C₂²×C₆₁	1	2	yes	—
274	C₂×#135	C₂²×C₆₁	1	1	yes	characteristic
275	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
276	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
277	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
278	#536	C₂²×C₆₁	1	2	yes	—
279	#520	C₂²×C₆₁	1	2	yes	—
280	#473	—	2	4	no	—
281	#470	—	2	2	no	—
282	#467	—	2	1	no	—
283	#467	—	2	1	no	—
284	#464	—	2	2	no	—
285	#463	—	2	2	no	—
286	#462	—	2	2	no	—
287	#461	—	2	2	no	—
288	#443	—	2	4	no	—
289	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
290	C₂×#108	C₂²×C₆₁	1	1	yes	characteristic
291	#428	—	2	2	no	—
292	#428	C₂²×C₆₁	1	2	yes	—
293	#427	—	2	1	no	—
294	#427	C₂²×C₆₁	1	1	yes	characteristic
295	C₂×#107	—	2	1	no	—
296	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
297	#424	C₂²×C₆₁	1	2	yes	—
298	C₂×#104	C₂²×C₆₁	1	1	yes	characteristic
299	C₂×#104	C₂²×C₆₁	1	1	yes	characteristic
300	#419	C₂²×C₆₁	1	2	yes	—
301	#417	C₂²×C₆₁	1	2	yes	—
302	C₂×#103	C₂²×C₆₁	1	2	yes	—
303	C₂×#101	C₂²×C₆₁	1	2	yes	—
304	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
305	#391	C₂²×C₆₁	1	2	yes	—
306	#343	C₂²×C₆₁	1	2	yes	—
307	#339	C₂²×C₆₁	1	2	yes	—
308	C₆₁	#24798	1	1	yes	characteristic, direct product factor, Sylow
309	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
310	C₂×#41	C₂³×C₆₁	1	1	yes	characteristic
311	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
312	#137	—	2	4	no	—
313	#137	—	2	4	no	—
314	#135	—	2	2	no	—
315	#135	—	2	2	no	—
316	C₂×Q₁₆	—	2	2	no	—
317	C₂×Q₁₆	—	2	1	no	—
318	C₂×Q₁₆	—	2	1	no	—
319	C₂×Q₁₆	—	2	2	no	—
320	C₂×Q₁₆	—	2	2	no	—
321	C₂×Q₁₆	D₄×C₆₁	1	1	yes	characteristic
322	C₂×SD₁₆	—	2	2	no	—
323	C₂×SD₁₆	D₄×C₆₁	1	1	yes	characteristic
324	C₂×SD₁₆	D₄×C₆₁	1	1	yes	characteristic
325	C₂×D₈	D₄×C₆₁	1	1	yes	characteristic
326	#131	—	2	2	no	—
327	#131	—	2	2	no	—
328	#131	—	2	2	no	—
329	#131	D₄×C₆₁	1	4	yes	—
330	C₂×M₁₆	—	2	1	no	—
331	C₂×M₁₆	—	2	1	no	—
332	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
333	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
334	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
335	C₂²×C₈	—	2	1	no	—
336	C₂²×C₈	—	2	1	no	—
337	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
338	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
339	C₄⋊Q₈	—	2	2	no	—
340	#124	—	2	2	no	—
341	#123	—	4	2	no	—
342	C₂²⋊Q₈	—	2	2	no	—
343	C₂²⋊Q₈	—	2	2	no	—
344	C₄×Q₈	—	2	2	no	—
345	C₄×D₄	—	2	2	no	—
346	C₄×D₄	—	4	2	no	—
347	#117	—	2	2	no	—
348	#117	C₂³×C₆₁	1	2	yes	—
349	C₂×C₄⋊C₄	—	2	2	no	—
350	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
351	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
352	#108	—	2	2	no	—
353	#108	—	2	2	no	—
354	#107	—	2	1	no	—
355	#107	—	2	1	no	—
356	#107	—	2	2	no	—
357	#107	D₄×C₆₁	1	1	yes	characteristic
358	#107	D₄×C₆₁	1	1	yes	characteristic
359	#106	—	2	2	no	—
360	#106	D₄×C₆₁	1	2	yes	—
361	C₄⋊C₈	—	2	2	no	—
362	C₄⋊C₈	—	2	2	no	—
363	#104	—	2	2	no	—
364	#104	—	2	2	no	—
365	#103	—	2	2	no	—
366	#103	—	2	2	no	—
367	#103	—	2	2	no	—
368	#103	—	2	2	no	—
369	#103	—	2	2	no	—
370	#102	—	2	2	no	—
371	#101	—	2	4	no	—
372	C₄×C₈	—	2	2	no	—
373	#95	—	4	2	no	—
374	#41	—	2	1	no	—
375	#41	—	2	1	no	—
376	#41	—	2	1	no	—
377	#41	—	2	1	no	—
378	#41	—	2	1	no	—
379	#41	—	2	1	no	—
380	#41	C₂×D₄×C₆₁	1	2	yes	—
381	#41	#41×C₆₁	1	2	yes	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
389	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
393	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
401	Q₁₆	—	2	2	no	—
402	Q₁₆	—	4	4	no	—
403	Q₁₆	—	4	2	no	—
404	Q₁₆	—	4	2	no	—
405	SD₁₆	—	2	2	no	—
406	SD₁₆	—	2	2	no	—
407	SD₁₆	—	4	4	no	—
408	D₈	—	2	2	no	—
409	M₁₆	—	2	1	no	—
410	M₁₆	—	2	4	no	—
411	M₁₆	—	2	2	no	—
412	M₁₆	—	2	1	no	—
413	M₁₆	—	2	1	no	—
414	M₁₆	—	2	2	no	—
415	M₁₆	—	4	2	no	—
416	M₁₆	C₂×D₄×C₆₁	1	2	yes	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	4	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	4	2	no	—
424	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
425	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
426	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
427	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
428	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
429	C₂×C₈	#41×C₆₁	1	2	yes	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	—	4	2	no	—
433	C₄⋊C₄	—	4	2	no	—
434	C₄⋊C₄	—	4	2	no	—
435	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
436	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	—	4	2	no	—
439	C₂²⋊C₄	—	4	2	no	—
440	C₂²⋊C₄	—	4	2	no	—
441	C₄²	—	2	2	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	4	2	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	2	1	no	—
447	C₂³	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
448	Q₈	—	2	1	no	—
449	Q₈	—	2	1	no	—
450	Q₈	—	2	1	no	—
451	Q₈	—	4	2	no	—
452	Q₈	—	4	2	no	—
453	Q₈	—	4	2	no	—
454	Q₈	—	4	2	no	—
455	Q₈	#121×C₆₁	1	2	yes	—
456	D₄	—	2	2	no	—
457	D₄	—	2	2	no	—
458	D₄	—	2	1	no	—
459	D₄	—	2	1	no	—
460	D₄	—	2	1	no	—
461	D₄	—	4	1	no	—
462	D₄	—	4	1	no	—
463	D₄	#121×C₆₁	1	2	yes	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	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	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	8	2	no	—
480	C₂×C₄	—	16	2	no	—
481	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
484	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
485	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
486	C₂×C₄	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
487	C₈	—	2	2	no	—
488	C₈	—	2	1	no	—
489	C₈	—	2	1	no	—
490	C₈	—	4	1	no	—
491	C₈	—	4	2	no	—
492	C₈	#121×C₆₁	1	2	yes	—
493	C₈	#121×C₆₁	1	2	yes	—
494	C₂²	—	2	1	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	4	1	no	—
498	C₂²	—	4	1	no	—
499	C₂²	—	4	1	no	—
500	C₂²	—	4	1	no	—
501	C₂²	—	4	1	no	—
502	C₂²	C₆₁×#537	1	1	yes	characteristic
503	C₂²	D₄²×C₆₁	1	1	yes	characteristic
504	C₂²	C₆₁×#553	1	1	yes	characteristic
505	C₄	—	2	2	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	—	4	2	no	—
509	C₄	—	4	1	no	—
510	C₄	—	8	2	no	—
511	C₄	—	8	2	no	—
512	C₄	C₂×#121×C₆₁	1	1	yes	characteristic
513	C₄	C₂×#121×C₆₁	1	1	yes	characteristic
514	C₄	C₆₁×#547	1	1	yes	characteristic
515	C₄	C₆₁×#548	1	1	yes	characteristic
516	C₂	—	2	1	no	—
517	C₂	—	2	1	no	—
518	C₂	—	4	2	no	—
519	C₂	—	8	1	no	—
520	C₂	C₆₁×#2416	1	1	yes	characteristic
521	C₂	C₆₁×#3290	1	1	yes	characteristic
522	C₂	C₆₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×#135×C₆₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂²×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₆₁×#473	C₂	semidirect
2. C₆₁×#3059	C₂	semidirect
5. C₆₁×#2913	C₂	semidirect
6. C₆₁×#2064	C₂	semidirect
7. C₆₁×#2038	C₂	semidirect
8. C₆₁×#1969	C₂	semidirect
9. C₆₁×#1945	C₂	semidirect
10. C₆₁×#1882	C₂	semidirect
11. C₆₁×#1816	C₂	semidirect
232. #24798	C₆₁	direct (N × H)
255. C₂×#473	C₂×C₆₁	semidirect
256. #3059	C₂×C₆₁	semidirect
259. #2913	C₂×C₆₁	semidirect
260. #2064	C₂×C₆₁	semidirect
261. #2038	C₂×C₆₁	semidirect
262. #1969	C₂×C₆₁	semidirect
263. #1945	C₂×C₆₁	semidirect
264. #1882	C₂×C₆₁	semidirect
265. #1816	C₂×C₆₁	semidirect
308. C₆₁	#24798	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×#473	2
C₆₁×#3059	2
C₆₁×#3046	2
C₂×C₆₁×#443	2
C₆₁×#2913	2
C₆₁×#2064	2
C₆₁×#2038	2
C₆₁×#1969	2
C₆₁×#1945	2
C₆₁×#1882	2
C₆₁×#1816	2
#24798	61

Subgroups — C61×#24798 #29528231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24798 #29528231