Subgroups — C₆₁×#24521 #29527954

Order 15616.

976 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3095	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#450	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2924	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₆₁×#418	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2063	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2046	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1977	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1931	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1901	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#1855	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1783	C₂	1	1	yes	maximal, characteristic
12	C₂³×D₄×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₆₁	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#485	—	2	1	no	—
18	C₆₁×#485	—	2	1	no	—
19	C₆₁×#484	—	2	1	no	—
20	C₆₁×#483	—	2	1	no	—
21	C₆₁×#450	—	2	1	no	—
22	C₆₁×#450	—	2	1	no	—
23	C₆₁×#450	—	2	2	no	—
24	C₆₁×#433	—	2	2	no	—
25	C₆₁×#432	—	2	1	no	—
26	C₆₁×#431	—	2	1	no	—
27	C₂×#108×C₆₁	C₂²	1	2	yes	complemented
28	C₆₁×#428	C₂²	1	1	yes	characteristic, complemented
29	C₆₁×#428	C₂²	1	1	yes	characteristic, complemented
30	C₆₁×#424	C₂²	1	2	yes	complemented
31	C₆₁×#419	C₂²	1	1	yes	characteristic, complemented
32	C₆₁×#419	C₂²	1	1	yes	characteristic, complemented
33	C₆₁×#418	—	2	1	no	—
34	C₆₁×#418	—	2	1	no	—
35	C₆₁×#418	C₂²	1	2	yes	—
36	C₆₁×#418	C₂²	1	2	yes	—
37	C₂×#102×C₆₁	—	2	1	no	—
38	C₂×#102×C₆₁	—	2	1	no	—
39	C₂×#102×C₆₁	C₂²	1	2	yes	—
40	C₂×#101×C₆₁	C₂²	1	2	yes	complemented
41	C₂×#100×C₆₁	C₂²	1	2	yes	—
42	C₂×#100×C₆₁	C₂²	1	2	yes	—
43	C₆₁×#407	C₂²	1	1	yes	characteristic
44	C₆₁×#407	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
46	C₆₁×#405	C₂²	1	2	yes	complemented
47	C₆₁×#343	C₂²	1	2	yes	complemented
48	C₆₁×#341	C₂²	1	2	yes	complemented
49	C₂⁵×C₆₁	—	2	1	no	—
50	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₆₁	—	2	4	no	—
52	C₂²×D₄×C₆₁	—	2	1	no	—
53	C₂²×D₄×C₆₁	—	2	1	no	—
54	C₂²×D₄×C₆₁	—	2	1	no	—
55	C₂²×D₄×C₆₁	—	2	1	no	—
56	C₂²×D₄×C₆₁	—	2	1	no	—
57	C₂²×D₄×C₆₁	—	4	2	no	—
58	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
60	C₂×SD₁₆×C₆₁	—	2	4	no	—
61	C₂×SD₁₆×C₆₁	—	2	2	no	—
62	C₂×SD₁₆×C₆₁	—	2	2	no	—
63	C₂×SD₁₆×C₆₁	—	4	2	no	—
64	C₂×M₁₆×C₆₁	—	2	1	no	—
65	C₂×M₁₆×C₆₁	—	4	2	no	—
66	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
67	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
69	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₆₁	—	2	1	no	—
71	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
72	C₂²⋊Q₈×C₆₁	—	2	1	no	—
73	C₂²⋊Q₈×C₆₁	—	2	1	no	—
74	C₂²⋊Q₈×C₆₁	—	2	1	no	—
75	C₂²⋊Q₈×C₆₁	—	2	1	no	—
76	#117×C₆₁	—	2	1	no	—
77	#117×C₆₁	—	2	1	no	—
78	#117×C₆₁	D₄	1	1	yes	characteristic
79	#117×C₆₁	D₄	1	1	yes	characteristic
80	#117×C₆₁	C₂³	1	1	yes	characteristic
81	#117×C₆₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
88	C₂×C₄²×C₆₁	—	2	1	no	—
89	#108×C₆₁	—	4	2	no	—
90	#107×C₆₁	—	2	1	no	—
91	#107×C₆₁	—	2	1	no	—
92	#106×C₆₁	—	2	1	no	—
93	#106×C₆₁	—	2	1	no	—
94	C₄⋊C₈×C₆₁	—	2	2	no	—
95	C₄⋊C₈×C₆₁	—	2	2	no	—
96	#103×C₆₁	—	2	1	no	—
97	#103×C₆₁	—	2	1	no	—
98	#103×C₆₁	—	2	1	no	—
99	#103×C₆₁	—	2	1	no	—
100	#102×C₆₁	—	2	2	no	—
101	#102×C₆₁	—	2	2	no	—
102	#102×C₆₁	—	2	2	no	—
103	#102×C₆₁	—	2	2	no	—
104	#102×C₆₁	—	2	2	no	—
105	#102×C₆₁	—	2	2	no	—
106	#101×C₆₁	—	4	2	no	—
107	#100×C₆₁	—	2	4	no	—
108	#100×C₆₁	—	4	2	no	—
109	C₂²⋊C₈×C₆₁	—	2	1	no	—
110	C₂²⋊C₈×C₆₁	—	2	1	no	—
111	C₂²⋊C₈×C₆₁	—	2	1	no	—
112	C₂²⋊C₈×C₆₁	—	2	1	no	—
113	C₂²⋊C₈×C₆₁	D₄	1	1	yes	characteristic
114	C₂²⋊C₈×C₆₁	D₄	1	1	yes	characteristic
115	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
116	#97×C₆₁	—	2	2	no	—
117	C₄×C₈×C₆₁	—	2	2	no	—
118	C₂⁴×C₆₁	—	2	2	no	—
119	C₂⁴×C₆₁	—	4	4	no	—
120	C₂⁴×C₆₁	—	4	1	no	—
121	C₂⁴×C₆₁	—	4	1	no	—
122	C₂⁴×C₆₁	—	8	2	no	—
123	C₂⁴×C₆₁	—	8	2	no	—
124	C₂⁴×C₆₁	#41	1	1	yes	characteristic
125	C₂×Q₈×C₆₁	—	2	1	no	—
126	C₂×Q₈×C₆₁	—	2	1	no	—
127	C₂×Q₈×C₆₁	—	2	2	no	—
128	C₂×Q₈×C₆₁	—	2	2	no	—
129	C₂×D₄×C₆₁	—	2	4	no	—
130	C₂×D₄×C₆₁	—	2	4	no	—
131	C₂×D₄×C₆₁	—	2	2	no	—
132	C₂×D₄×C₆₁	—	2	2	no	—
133	C₂×D₄×C₆₁	—	2	2	no	—
134	C₂×D₄×C₆₁	—	2	2	no	—
135	C₂×D₄×C₆₁	—	2	2	no	—
136	C₂×D₄×C₆₁	—	4	4	no	—
137	C₂×D₄×C₆₁	—	4	4	no	—
138	C₂×D₄×C₆₁	—	4	4	no	—
139	C₂×D₄×C₆₁	—	4	2	no	—
140	C₂×D₄×C₆₁	—	4	2	no	—
141	C₂×D₄×C₆₁	—	4	1	no	—
142	C₂×D₄×C₆₁	—	4	1	no	—
143	C₂×D₄×C₆₁	#41	1	4	yes	—
144	C₂²×C₄×C₆₁	—	2	2	no	—
145	C₂²×C₄×C₆₁	—	2	2	no	—
146	C₂²×C₄×C₆₁	—	2	1	no	—
147	C₂²×C₄×C₆₁	—	2	1	no	—
148	C₂²×C₄×C₆₁	—	2	1	no	—
149	C₂²×C₄×C₆₁	—	2	1	no	—
150	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
155	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
156	SD₁₆×C₆₁	—	4	4	no	—
157	SD₁₆×C₆₁	—	4	4	no	—
158	M₁₆×C₆₁	—	2	2	no	—
159	M₁₆×C₆₁	—	2	2	no	—
160	M₁₆×C₆₁	—	2	2	no	—
161	M₁₆×C₆₁	—	2	2	no	—
162	M₁₆×C₆₁	—	2	2	no	—
163	M₁₆×C₆₁	—	2	2	no	—
164	M₁₆×C₆₁	—	4	2	no	—
165	C₂×C₈×C₆₁	—	2	2	no	—
166	C₂×C₈×C₆₁	—	2	2	no	—
167	C₂×C₈×C₆₁	—	2	2	no	—
168	C₂×C₈×C₆₁	—	2	1	no	—
169	C₂×C₈×C₆₁	—	2	1	no	—
170	C₂×C₈×C₆₁	—	2	2	no	—
171	C₂×C₈×C₆₁	—	4	2	no	—
172	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
173	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₆₁	—	2	1	no	—
175	C₄⋊C₄×C₆₁	—	2	1	no	—
176	C₄⋊C₄×C₆₁	—	2	1	no	—
177	C₄⋊C₄×C₆₁	—	2	1	no	—
178	C₄⋊C₄×C₆₁	—	4	1	no	—
179	C₄⋊C₄×C₆₁	—	4	1	no	—
180	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
181	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
182	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
183	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
184	C₂²⋊C₄×C₆₁	—	2	1	no	—
185	C₂²⋊C₄×C₆₁	—	2	1	no	—
186	C₂²⋊C₄×C₆₁	—	2	1	no	—
187	C₂²⋊C₄×C₆₁	—	2	1	no	—
188	C₂²⋊C₄×C₆₁	—	4	1	no	—
189	C₂²⋊C₄×C₆₁	—	4	1	no	—
190	C₄²×C₆₁	—	2	1	no	—
191	C₄²×C₆₁	—	2	1	no	—
192	C₄²×C₆₁	—	2	1	no	—
193	C₄²×C₆₁	—	2	1	no	—
194	C₂³×C₆₁	—	2	2	no	—
195	C₂³×C₆₁	—	2	2	no	—
196	C₂³×C₆₁	—	2	4	no	—
197	C₂³×C₆₁	—	4	1	no	—
198	C₂³×C₆₁	—	4	4	no	—
199	C₂³×C₆₁	—	4	4	no	—
200	C₂³×C₆₁	—	4	2	no	—
201	C₂³×C₆₁	—	4	2	no	—
202	C₂³×C₆₁	—	8	2	no	—
203	C₂³×C₆₁	—	8	2	no	—
204	C₂³×C₆₁	—	8	2	no	—
205	C₂³×C₆₁	—	8	2	no	—
206	C₂³×C₆₁	—	8	2	no	—
207	C₂³×C₆₁	—	8	16	no	—
208	C₂³×C₆₁	—	8	2	no	—
209	C₂³×C₆₁	#121	1	1	yes	characteristic
210	C₂³×C₆₁	#121	1	1	yes	characteristic
211	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
212	Q₈×C₆₁	—	4	1	no	—
213	Q₈×C₆₁	—	4	1	no	—
214	Q₈×C₆₁	—	4	2	no	—
215	D₄×C₆₁	—	2	4	no	—
216	D₄×C₆₁	—	2	4	no	—
217	D₄×C₆₁	—	4	4	no	—
218	D₄×C₆₁	—	4	4	no	—
219	D₄×C₆₁	—	4	2	no	—
220	D₄×C₆₁	—	4	2	no	—
221	C₂×C₄×C₆₁	—	2	2	no	—
222	C₂×C₄×C₆₁	—	2	1	no	—
223	C₂×C₄×C₆₁	—	2	1	no	—
224	C₂×C₄×C₆₁	—	2	1	no	—
225	C₂×C₄×C₆₁	—	2	1	no	—
226	C₂×C₄×C₆₁	—	2	2	no	—
227	C₂×C₄×C₆₁	—	2	2	no	—
228	C₂×C₄×C₆₁	—	2	1	no	—
229	C₂×C₄×C₆₁	—	2	1	no	—
230	C₂×C₄×C₆₁	—	2	1	no	—
231	C₂×C₄×C₆₁	—	2	1	no	—
232	C₂×C₄×C₆₁	—	4	2	no	—
233	C₂×C₄×C₆₁	—	4	1	no	—
234	C₂×C₄×C₆₁	—	4	1	no	—
235	C₂×C₄×C₆₁	—	4	2	no	—
236	C₂×C₄×C₆₁	—	8	1	no	—
237	C₂×C₄×C₆₁	—	8	1	no	—
238	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
239	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
240	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
243	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
244	C₈×C₆₁	—	2	2	no	—
245	C₈×C₆₁	—	2	2	no	—
246	C₈×C₆₁	—	4	1	no	—
247	C₈×C₆₁	—	4	2	no	—
248	C₈×C₆₁	—	4	1	no	—
249	C₈×C₆₁	—	4	2	no	—
250	#24521	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂²×C₆₁	—	2	1	no	—
252	C₂²×C₆₁	—	2	1	no	—
253	C₂²×C₆₁	—	2	1	no	—
254	C₂²×C₆₁	—	2	1	no	—
255	C₂²×C₆₁	—	2	1	no	—
256	C₂²×C₆₁	—	2	1	no	—
257	C₂²×C₆₁	—	2	1	no	—
258	C₂²×C₆₁	—	4	4	no	—
259	C₂²×C₆₁	—	4	4	no	—
260	C₂²×C₆₁	—	8	4	no	—
261	C₂²×C₆₁	—	8	4	no	—
262	C₂²×C₆₁	—	8	8	no	—
263	C₂²×C₆₁	—	8	8	no	—
264	C₂²×C₆₁	—	8	2	no	—
265	C₂²×C₆₁	—	8	2	no	—
266	C₂²×C₆₁	#471	1	2	yes	—
267	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
268	C₂²×C₆₁	#551	1	2	yes	—
269	C₄×C₆₁	—	2	1	no	—
270	C₄×C₆₁	—	2	1	no	—
271	C₄×C₆₁	—	4	1	no	—
272	C₄×C₆₁	—	4	1	no	—
273	C₄×C₆₁	—	4	1	no	—
274	C₄×C₆₁	—	4	1	no	—
275	C₄×C₆₁	—	8	1	no	—
276	C₄×C₆₁	—	8	1	no	—
277	C₄×C₆₁	#525	1	1	yes	characteristic
278	C₄×C₆₁	#535	1	1	yes	characteristic
279	C₄×C₆₁	#546	1	2	yes	—
280	#3095	C₂×C₆₁	1	1	yes	characteristic, complemented
281	C₂×#450	C₂×C₆₁	1	1	yes	characteristic
282	#2924	C₂×C₆₁	1	1	yes	characteristic, complemented
283	C₂×#418	C₂×C₆₁	1	1	yes	characteristic
284	#2063	C₂×C₆₁	1	2	yes	complemented
285	#2046	C₂×C₆₁	1	2	yes	complemented
286	#1977	C₂×C₆₁	1	2	yes	complemented
287	#1931	C₂×C₆₁	1	2	yes	complemented
288	#1901	C₂×C₆₁	1	1	yes	characteristic, complemented
289	#1855	C₂×C₆₁	1	1	yes	characteristic, complemented
290	#1783	C₂×C₆₁	1	1	yes	characteristic
291	C₂×C₆₁	—	2	2	no	—
292	C₂×C₆₁	—	2	2	no	—
293	C₂×C₆₁	—	4	1	no	—
294	C₂×C₆₁	—	8	4	no	—
295	C₂×C₆₁	#2411	1	1	yes	characteristic
296	C₂×C₆₁	C₂×#471	1	1	yes	characteristic
297	C₂×C₆₁	#3068	1	1	yes	characteristic
298	C₂³×D₄	C₂²×C₆₁	1	1	yes	characteristic
299	C₂²×SD₁₆	C₂²×C₆₁	1	2	yes	—
300	C₂²×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
301	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
302	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
303	#485	—	2	1	no	—
304	#485	—	2	1	no	—
305	#484	—	2	1	no	—
306	#483	—	2	1	no	—
307	#450	—	2	1	no	—
308	#450	—	2	1	no	—
309	#450	—	2	2	no	—
310	#433	—	2	2	no	—
311	#432	—	2	1	no	—
312	#431	—	2	1	no	—
313	C₂×#108	C₂²×C₆₁	1	2	yes	complemented
314	#428	C₂²×C₆₁	1	1	yes	characteristic, complemented
315	#428	C₂²×C₆₁	1	1	yes	characteristic, complemented
316	#424	C₂²×C₆₁	1	2	yes	complemented
317	#419	C₂²×C₆₁	1	1	yes	characteristic, complemented
318	#419	C₂²×C₆₁	1	1	yes	characteristic, complemented
319	#418	—	2	1	no	—
320	#418	—	2	1	no	—
321	#418	C₂²×C₆₁	1	2	yes	—
322	#418	C₂²×C₆₁	1	2	yes	—
323	C₂×#102	—	2	1	no	—
324	C₂×#102	—	2	1	no	—
325	C₂×#102	C₂²×C₆₁	1	2	yes	—
326	C₂×#101	C₂²×C₆₁	1	2	yes	complemented
327	C₂×#100	C₂²×C₆₁	1	2	yes	—
328	C₂×#100	C₂²×C₆₁	1	2	yes	—
329	#407	C₂²×C₆₁	1	1	yes	characteristic
330	#407	C₂²×C₆₁	1	1	yes	characteristic
331	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
332	#405	C₂²×C₆₁	1	2	yes	complemented
333	#343	C₂²×C₆₁	1	2	yes	complemented
334	#341	C₂²×C₆₁	1	2	yes	complemented
335	C₆₁	#24521	1	1	yes	characteristic, direct product factor, Sylow
336	C₂⁵	—	2	1	no	—
337	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
338	C₂²×D₄	—	2	4	no	—
339	C₂²×D₄	—	2	1	no	—
340	C₂²×D₄	—	2	1	no	—
341	C₂²×D₄	—	2	1	no	—
342	C₂²×D₄	—	2	1	no	—
343	C₂²×D₄	—	2	1	no	—
344	C₂²×D₄	—	4	2	no	—
345	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
346	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
347	C₂×SD₁₆	—	2	4	no	—
348	C₂×SD₁₆	—	2	2	no	—
349	C₂×SD₁₆	—	2	2	no	—
350	C₂×SD₁₆	—	4	2	no	—
351	C₂×M₁₆	—	2	1	no	—
352	C₂×M₁₆	—	4	2	no	—
353	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
354	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
355	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
356	C₂×M₁₆	C₂³×C₆₁	1	1	yes	characteristic
357	C₂²×C₈	—	2	1	no	—
358	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
359	C₂²⋊Q₈	—	2	1	no	—
360	C₂²⋊Q₈	—	2	1	no	—
361	C₂²⋊Q₈	—	2	1	no	—
362	C₂²⋊Q₈	—	2	1	no	—
363	#117	—	2	1	no	—
364	#117	—	2	1	no	—
365	#117	D₄×C₆₁	1	1	yes	characteristic
366	#117	D₄×C₆₁	1	1	yes	characteristic
367	#117	C₂³×C₆₁	1	1	yes	characteristic
368	#117	C₂³×C₆₁	1	1	yes	characteristic
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
371	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
372	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
373	C₂×C₂²⋊C₄	—	2	1	no	—
374	C₂×C₂²⋊C₄	—	2	1	no	—
375	C₂×C₄²	—	2	1	no	—
376	#108	—	4	2	no	—
377	#107	—	2	1	no	—
378	#107	—	2	1	no	—
379	#106	—	2	1	no	—
380	#106	—	2	1	no	—
381	C₄⋊C₈	—	2	2	no	—
382	C₄⋊C₈	—	2	2	no	—
383	#103	—	2	1	no	—
384	#103	—	2	1	no	—
385	#103	—	2	1	no	—
386	#103	—	2	1	no	—
387	#102	—	2	2	no	—
388	#102	—	2	2	no	—
389	#102	—	2	2	no	—
390	#102	—	2	2	no	—
391	#102	—	2	2	no	—
392	#102	—	2	2	no	—
393	#101	—	4	2	no	—
394	#100	—	2	4	no	—
395	#100	—	4	2	no	—
396	C₂²⋊C₈	—	2	1	no	—
397	C₂²⋊C₈	—	2	1	no	—
398	C₂²⋊C₈	—	2	1	no	—
399	C₂²⋊C₈	—	2	1	no	—
400	C₂²⋊C₈	D₄×C₆₁	1	1	yes	characteristic
401	C₂²⋊C₈	D₄×C₆₁	1	1	yes	characteristic
402	C₂²⋊C₈	D₄×C₆₁	1	2	yes	—
403	#97	—	2	2	no	—
404	C₄×C₈	—	2	2	no	—
405	C₂⁴	—	2	2	no	—
406	C₂⁴	—	4	4	no	—
407	C₂⁴	—	4	1	no	—
408	C₂⁴	—	4	1	no	—
409	C₂⁴	—	8	2	no	—
410	C₂⁴	—	8	2	no	—
411	C₂⁴	#41×C₆₁	1	1	yes	characteristic
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	4	4	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	4	no	—
426	C₂×D₄	—	4	2	no	—
427	C₂×D₄	—	4	2	no	—
428	C₂×D₄	—	4	1	no	—
429	C₂×D₄	—	4	1	no	—
430	C₂×D₄	#41×C₆₁	1	4	yes	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
438	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
441	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
443	SD₁₆	—	4	4	no	—
444	SD₁₆	—	4	4	no	—
445	M₁₆	—	2	2	no	—
446	M₁₆	—	2	2	no	—
447	M₁₆	—	2	2	no	—
448	M₁₆	—	2	2	no	—
449	M₁₆	—	2	2	no	—
450	M₁₆	—	2	2	no	—
451	M₁₆	—	4	2	no	—
452	C₂×C₈	—	2	2	no	—
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	2	no	—
458	C₂×C₈	—	4	2	no	—
459	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
460	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	4	1	no	—
466	C₄⋊C₄	—	4	1	no	—
467	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
468	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
469	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
470	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
471	C₂²⋊C₄	—	2	1	no	—
472	C₂²⋊C₄	—	2	1	no	—
473	C₂²⋊C₄	—	2	1	no	—
474	C₂²⋊C₄	—	2	1	no	—
475	C₂²⋊C₄	—	4	1	no	—
476	C₂²⋊C₄	—	4	1	no	—
477	C₄²	—	2	1	no	—
478	C₄²	—	2	1	no	—
479	C₄²	—	2	1	no	—
480	C₄²	—	2	1	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	4	1	no	—
485	C₂³	—	4	4	no	—
486	C₂³	—	4	4	no	—
487	C₂³	—	4	2	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	8	2	no	—
490	C₂³	—	8	2	no	—
491	C₂³	—	8	2	no	—
492	C₂³	—	8	2	no	—
493	C₂³	—	8	2	no	—
494	C₂³	—	8	16	no	—
495	C₂³	—	8	2	no	—
496	C₂³	#121×C₆₁	1	1	yes	characteristic
497	C₂³	#121×C₆₁	1	1	yes	characteristic
498	C₂³	C₂×#41×C₆₁	1	1	yes	characteristic
499	Q₈	—	4	1	no	—
500	Q₈	—	4	2	no	—
501	Q₈	—	4	1	no	—
502	D₄	—	2	4	no	—
503	D₄	—	2	4	no	—
504	D₄	—	4	4	no	—
505	D₄	—	4	4	no	—
506	D₄	—	4	2	no	—
507	D₄	—	4	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	8	1	no	—
524	C₂×C₄	—	8	1	no	—
525	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
526	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
529	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
530	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
531	C₈	—	2	2	no	—
532	C₈	—	2	2	no	—
533	C₈	—	4	2	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	2	no	—
536	C₈	—	4	1	no	—
537	C₂²	—	2	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	2	1	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	8	4	no	—
547	C₂²	—	8	4	no	—
548	C₂²	—	8	8	no	—
549	C₂²	—	8	8	no	—
550	C₂²	—	8	2	no	—
551	C₂²	—	8	2	no	—
552	C₂²	C₆₁×#471	1	2	yes	—
553	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
554	C₂²	C₆₁×#551	1	2	yes	—
555	C₄	—	2	1	no	—
556	C₄	—	2	1	no	—
557	C₄	—	4	1	no	—
558	C₄	—	4	1	no	—
559	C₄	—	4	1	no	—
560	C₄	—	4	1	no	—
561	C₄	—	8	1	no	—
562	C₄	—	8	1	no	—
563	C₄	C₆₁×#525	1	1	yes	characteristic
564	C₄	C₆₁×#535	1	1	yes	characteristic
565	C₄	C₆₁×#546	1	2	yes	—
566	C₂	—	2	2	no	—
567	C₂	—	2	2	no	—
568	C₂	—	4	1	no	—
569	C₂	—	8	4	no	—
570	C₂	C₆₁×#2411	1	1	yes	characteristic
571	C₂	C₂×C₆₁×#471	1	1	yes	characteristic
572	C₂	C₆₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂³×D₄×C₆₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3095	C₂	semidirect
3. C₆₁×#2924	C₂	semidirect
5. C₆₁×#2063	C₂	semidirect
6. C₆₁×#2046	C₂	semidirect
7. C₆₁×#1977	C₂	semidirect
8. C₆₁×#1931	C₂	semidirect
9. C₆₁×#1901	C₂	semidirect
10. C₆₁×#1855	C₂	semidirect
27. C₂×#108×C₆₁	C₂²	semidirect
28. C₆₁×#428	C₂²	semidirect
29. C₆₁×#428	C₂²	semidirect
30. C₆₁×#424	C₂²	semidirect
31. C₆₁×#419	C₂²	semidirect
32. C₆₁×#419	C₂²	semidirect
40. C₂×#101×C₆₁	C₂²	semidirect
46. C₆₁×#405	C₂²	semidirect
47. C₆₁×#343	C₂²	semidirect
48. C₆₁×#341	C₂²	semidirect
250. #24521	C₆₁	direct (N × H)
280. #3095	C₂×C₆₁	semidirect
282. #2924	C₂×C₆₁	semidirect
284. #2063	C₂×C₆₁	semidirect
285. #2046	C₂×C₆₁	semidirect
286. #1977	C₂×C₆₁	semidirect
287. #1931	C₂×C₆₁	semidirect
288. #1901	C₂×C₆₁	semidirect
289. #1855	C₂×C₆₁	semidirect
313. C₂×#108	C₂²×C₆₁	semidirect
314. #428	C₂²×C₆₁	semidirect
315. #428	C₂²×C₆₁	semidirect
316. #424	C₂²×C₆₁	semidirect
317. #419	C₂²×C₆₁	semidirect
318. #419	C₂²×C₆₁	semidirect
326. C₂×#101	C₂²×C₆₁	semidirect
332. #405	C₂²×C₆₁	semidirect
333. #343	C₂²×C₆₁	semidirect
334. #341	C₂²×C₆₁	semidirect
335. C₆₁	#24521	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₁×#3095	2
C₂×C₆₁×#450	2
C₆₁×#2924	2
C₂×C₆₁×#418	2
C₆₁×#2063	2
C₆₁×#2046	2
C₆₁×#1977	2
C₆₁×#1931	2
C₆₁×#1901	2
C₆₁×#1855	2
C₆₁×#1783	2
#24521	61

Subgroups — C61×#24521 #29527954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#24521 #29527954