Subgroups — C₆₁×#19264 #29522697

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3112	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1678	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1676	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1668	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1551	C₂	1	2	yes	maximal, complemented
9	C₆₁×#532	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₆₁×#500	C₂²	1	1	yes	characteristic
12	C₆₁×#493	C₂²	1	1	yes	characteristic
13	C₆₁×#486	C₂²	1	2	yes	—
14	C₆₁×#480	—	2	2	no	—
15	C₆₁×#477	C₂²	1	4	yes	—
16	C₆₁×#466	—	2	1	no	—
17	C₆₁×#466	—	2	1	no	—
18	C₆₁×#460	C₂²	1	4	yes	—
19	C₆₁×#448	—	2	2	no	—
20	C₈×Q₈×C₆₁	—	2	1	no	—
21	C₈×Q₈×C₆₁	—	2	1	no	—
22	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
23	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
24	C₆₁×#440	—	2	4	no	—
25	C₄×D₈×C₆₁	—	2	1	no	—
26	C₄×D₈×C₆₁	—	2	1	no	—
27	C₆₁×#432	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
29	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₁	—	2	1	no	—
31	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#335	C₂²	1	2	yes	—
33	C₆₁×#334	C₂²	1	2	yes	—
34	C₆₁×#333	C₂²	1	4	yes	—
35	C₆₁×#331	C₂²	1	4	yes	—
36	C₂×#41×C₆₁	—	2	2	no	—
37	C₂²×Q₈×C₆₁	D₄	1	1	yes	characteristic
38	C₂×SD₁₆×C₆₁	—	2	4	no	—
39	C₂×D₈×C₆₁	—	2	1	no	—
40	C₂×D₈×C₆₁	—	2	1	no	—
41	C₂²×C₈×C₆₁	—	2	1	no	—
42	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
45	#127×C₆₁	C₂³	1	2	yes	—
46	#124×C₆₁	—	2	2	no	—
47	#121×C₆₁	—	2	2	no	—
48	#121×C₆₁	—	2	2	no	—
49	C₄×Q₈×C₆₁	—	2	1	no	—
50	C₄×Q₈×C₆₁	—	2	1	no	—
51	C₄×Q₈×C₆₁	D₄	1	2	yes	—
52	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
53	C₄×D₄×C₆₁	—	2	2	no	—
54	C₄×D₄×C₆₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₁	—	2	1	no	—
58	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
59	#107×C₆₁	—	2	1	no	—
60	#107×C₆₁	—	2	1	no	—
61	#107×C₆₁	—	2	4	no	—
62	C₄⋊C₈×C₆₁	—	2	2	no	—
63	C₄⋊C₈×C₆₁	—	2	2	no	—
64	C₄⋊C₈×C₆₁	—	2	1	no	—
65	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
66	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
67	#103×C₆₁	—	2	4	no	—
68	#102×C₆₁	—	2	2	no	—
69	#102×C₆₁	—	2	2	no	—
70	#102×C₆₁	—	2	4	no	—
71	C₂²⋊C₈×C₆₁	—	2	2	no	—
72	#97×C₆₁	—	2	2	no	—
73	C₄×C₈×C₆₁	—	2	2	no	—
74	C₄×C₈×C₆₁	—	2	2	no	—
75	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
76	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
77	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	#41×C₆₁	—	8	4	no	—
80	C₂×Q₈×C₆₁	—	2	4	no	—
81	C₂×Q₈×C₆₁	—	2	1	no	—
82	C₂×Q₈×C₆₁	—	2	2	no	—
83	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
84	C₂×D₄×C₆₁	—	2	2	no	—
85	C₂×D₄×C₆₁	—	2	2	no	—
86	C₂×D₄×C₆₁	—	2	2	no	—
87	C₂×D₄×C₆₁	—	4	2	no	—
88	C₂²×C₄×C₆₁	—	2	1	no	—
89	C₂²×C₄×C₆₁	—	2	1	no	—
90	C₂²×C₄×C₆₁	—	4	2	no	—
91	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
94	SD₁₆×C₆₁	—	4	8	no	—
95	D₈×C₆₁	—	4	2	no	—
96	D₈×C₆₁	—	4	2	no	—
97	C₂×C₈×C₆₁	—	2	2	no	—
98	C₂×C₈×C₆₁	—	2	2	no	—
99	C₂×C₈×C₆₁	—	2	4	no	—
100	C₂×C₈×C₆₁	—	4	1	no	—
101	C₂×C₈×C₆₁	—	4	2	no	—
102	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₆₁	—	2	2	no	—
107	C₄⋊C₄×C₆₁	—	2	1	no	—
108	C₄⋊C₄×C₆₁	—	2	2	no	—
109	C₄⋊C₄×C₆₁	—	2	2	no	—
110	C₄⋊C₄×C₆₁	—	2	2	no	—
111	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
112	C₂²⋊C₄×C₆₁	—	4	2	no	—
113	C₂²⋊C₄×C₆₁	—	4	2	no	—
114	C₄²×C₆₁	—	2	2	no	—
115	C₄²×C₆₁	—	2	2	no	—
116	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₆₁	#41	1	1	yes	characteristic
119	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
120	C₂³×C₆₁	—	4	2	no	—
121	C₂³×C₆₁	#121	1	1	yes	characteristic
122	Q₈×C₆₁	—	2	4	no	—
123	Q₈×C₆₁	—	2	2	no	—
124	Q₈×C₆₁	—	8	2	no	—
125	Q₈×C₆₁	#135	1	4	yes	—
126	D₄×C₆₁	—	4	4	no	—
127	D₄×C₆₁	—	4	2	no	—
128	D₄×C₆₁	—	4	2	no	—
129	D₄×C₆₁	—	8	2	no	—
130	D₄×C₆₁	—	8	4	no	—
131	C₂×C₄×C₆₁	—	2	4	no	—
132	C₂×C₄×C₆₁	—	2	2	no	—
133	C₂×C₄×C₆₁	—	2	2	no	—
134	C₂×C₄×C₆₁	—	2	2	no	—
135	C₂×C₄×C₆₁	—	2	1	no	—
136	C₂×C₄×C₆₁	—	2	1	no	—
137	C₂×C₄×C₆₁	—	2	2	no	—
138	C₂×C₄×C₆₁	—	4	2	no	—
139	C₂×C₄×C₆₁	—	4	2	no	—
140	C₂×C₄×C₆₁	—	8	4	no	—
141	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
142	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
143	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
147	C₈×C₆₁	—	2	1	no	—
148	C₈×C₆₁	—	2	1	no	—
149	C₈×C₆₁	—	2	1	no	—
150	C₈×C₆₁	—	4	4	no	—
151	C₈×C₆₁	—	4	2	no	—
152	C₈×C₆₁	#137	1	2	yes	—
153	#19264	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂²×C₆₁	—	2	1	no	—
155	C₂²×C₆₁	—	2	1	no	—
156	C₂²×C₆₁	—	8	2	no	—
157	C₂²×C₆₁	—	8	2	no	—
158	C₂²×C₆₁	—	8	2	no	—
159	C₂²×C₆₁	#472	1	2	yes	—
160	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
161	C₄×C₆₁	—	2	1	no	—
162	C₄×C₆₁	—	2	4	no	—
163	C₄×C₆₁	—	2	1	no	—
164	C₄×C₆₁	—	2	1	no	—
165	C₄×C₆₁	—	2	1	no	—
166	C₄×C₆₁	—	4	2	no	—
167	C₄×C₆₁	—	8	2	no	—
168	C₄×C₆₁	C₂×#135	1	2	yes	—
169	C₄×C₆₁	C₂×#137	1	1	yes	characteristic
170	C₄×C₆₁	#575	1	1	yes	characteristic
171	#3169	C₂×C₆₁	1	1	yes	characteristic
172	#3112	C₂×C₆₁	1	2	yes	complemented
173	C₂×C₈×Q₈	C₂×C₆₁	1	1	yes	characteristic, complemented
174	#1678	C₂×C₆₁	1	4	yes	complemented
175	#1676	C₂×C₆₁	1	2	yes	complemented
176	#1668	C₂×C₆₁	1	2	yes	complemented
177	#1568	C₂×C₆₁	1	1	yes	characteristic, complemented
178	#1551	C₂×C₆₁	1	2	yes	complemented
179	C₂×C₆₁	—	2	2	no	—
180	C₂×C₆₁	—	16	2	no	—
181	C₂×C₆₁	#3037	1	1	yes	characteristic
182	C₂×C₆₁	#3050	1	1	yes	characteristic
183	C₂×C₆₁	C₂×#472	1	1	yes	characteristic
184	#532	C₂²×C₆₁	1	2	yes	—
185	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
186	#500	C₂²×C₆₁	1	1	yes	characteristic
187	#493	C₂²×C₆₁	1	1	yes	characteristic
188	#486	C₂²×C₆₁	1	2	yes	—
189	#480	—	2	2	no	—
190	#477	C₂²×C₆₁	1	4	yes	—
191	#466	—	2	1	no	—
192	#466	—	2	1	no	—
193	#460	C₂²×C₆₁	1	4	yes	—
194	#448	—	2	2	no	—
195	C₈×Q₈	—	2	1	no	—
196	C₈×Q₈	—	2	1	no	—
197	C₈×Q₈	C₂²×C₆₁	1	2	yes	—
198	C₈×Q₈	C₂²×C₆₁	1	2	yes	—
199	#440	—	2	4	no	—
200	C₄×D₈	—	2	1	no	—
201	C₄×D₈	—	2	1	no	—
202	#432	C₂²×C₆₁	1	2	yes	—
203	C₂×C₄⋊C₈	—	2	1	no	—
204	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
205	C₂×C₄×C₈	—	2	1	no	—
206	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
207	#335	C₂²×C₆₁	1	2	yes	—
208	#334	C₂²×C₆₁	1	2	yes	—
209	#333	C₂²×C₆₁	1	4	yes	—
210	#331	C₂²×C₆₁	1	4	yes	—
211	C₆₁	#19264	1	1	yes	characteristic, direct product factor, Sylow
212	C₂×#41	—	2	2	no	—
213	C₂²×Q₈	D₄×C₆₁	1	1	yes	characteristic
214	C₂×SD₁₆	—	2	4	no	—
215	C₂×D₈	—	2	1	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
219	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
220	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
221	#127	C₂³×C₆₁	1	2	yes	—
222	#124	—	2	2	no	—
223	#121	—	2	2	no	—
224	#121	—	2	2	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	—	2	1	no	—
227	C₄×Q₈	D₄×C₆₁	1	2	yes	—
228	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	2	2	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
233	C₂×C₄²	—	2	1	no	—
234	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
235	#107	—	2	1	no	—
236	#107	—	2	1	no	—
237	#107	—	2	4	no	—
238	C₄⋊C₈	—	2	2	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
242	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
243	#103	—	2	4	no	—
244	#102	—	2	2	no	—
245	#102	—	2	2	no	—
246	#102	—	2	4	no	—
247	C₂²⋊C₈	—	2	2	no	—
248	#97	—	2	2	no	—
249	C₄×C₈	—	2	2	no	—
250	C₄×C₈	—	2	2	no	—
251	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
252	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
254	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
255	#41	—	8	4	no	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	4	2	no	—
267	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
269	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
270	SD₁₆	—	4	8	no	—
271	D₈	—	4	2	no	—
272	D₈	—	4	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
294	C₄²	#41×C₆₁	1	1	yes	characteristic
295	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
296	C₂³	—	4	2	no	—
297	C₂³	#121×C₆₁	1	1	yes	characteristic
298	Q₈	—	2	4	no	—
299	Q₈	—	2	2	no	—
300	Q₈	—	8	2	no	—
301	Q₈	#135×C₆₁	1	4	yes	—
302	D₄	—	4	4	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
318	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
319	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
322	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	4	4	no	—
327	C₈	—	4	2	no	—
328	C₈	#137×C₆₁	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	C₆₁×#472	1	2	yes	—
335	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₂×#135×C₆₁	1	2	yes	—
344	C₄	C₂×#137×C₆₁	1	1	yes	characteristic
345	C₄	C₆₁×#575	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₆₁×#3037	1	1	yes	characteristic
349	C₂	C₆₁×#3050	1	1	yes	characteristic
350	C₂	C₂×C₆₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#3169	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
2. C₆₁×#3112	C₂	semidirect
3. C₂×C₈×Q₈×C₆₁	C₂	semidirect
4. C₆₁×#1678	C₂	semidirect
5. C₆₁×#1676	C₂	semidirect
6. C₆₁×#1668	C₂	semidirect
7. C₆₁×#1568	C₂	semidirect
8. C₆₁×#1551	C₂	semidirect
153. #19264	C₆₁	direct (N × H)
172. #3112	C₂×C₆₁	semidirect
173. C₂×C₈×Q₈	C₂×C₆₁	semidirect
174. #1678	C₂×C₆₁	semidirect
175. #1676	C₂×C₆₁	semidirect
176. #1668	C₂×C₆₁	semidirect
177. #1568	C₂×C₆₁	semidirect
178. #1551	C₂×C₆₁	semidirect
211. C₆₁	#19264	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×#3169	2
C₆₁×#3112	2
C₂×C₈×Q₈×C₆₁	2
C₆₁×#1678	2
C₆₁×#1676	2
C₆₁×#1668	2
C₆₁×#1568	2
C₆₁×#1551	2
#19264	61

Subgroups — C61×#19264 #29522697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19264 #29522697