Subgroups — C₆₁×#19940 #29523373

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3369	C₂	1	2	yes	maximal
3	C₆₁×#3342	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1704	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1675	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#1674	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1645	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1638	C₂	1	2	yes	maximal
10	C₆₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
12	Q₈²×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#556	C₂²	1	2	yes	—
14	C₆₁×#500	—	2	1	no	—
15	C₆₁×#500	C₂²	1	1	yes	characteristic
16	C₆₁×#500	C₂²	1	1	yes	characteristic
17	C₆₁×#489	—	2	2	no	—
18	C₆₁×#487	C₂²	1	2	yes	—
19	C₆₁×#487	C₂²	1	2	yes	—
20	C₆₁×#485	—	2	2	no	—
21	C₆₁×#477	—	2	2	no	—
22	C₆₁×#477	C₂²	1	2	yes	—
23	C₆₁×#476	C₂²	1	2	yes	—
24	C₆₁×#475	—	2	2	no	—
25	C₆₁×#475	C₂²	1	2	yes	—
26	C₆₁×#467	—	2	1	no	—
27	C₆₁×#464	—	2	2	no	—
28	C₆₁×#462	C₂²	1	2	yes	—
29	C₆₁×#446	C₂²	1	2	yes	—
30	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₆₁×#441	—	2	2	no	—
32	C₄×Q₁₆×C₆₁	—	2	1	no	—
33	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₆₁	C₂²	1	1	yes	characteristic
36	C₆₁×#427	—	2	1	no	—
37	C₆₁×#335	C₂²	1	1	yes	characteristic
38	C₆₁×#334	C₂²	1	1	yes	characteristic
39	C₆₁×#333	C₂²	1	2	yes	—
40	C₆₁×#332	C₂²	1	2	yes	—
41	C₆₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₆₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₆₁	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆×C₆₁	—	2	2	no	—
46	C₂×Q₁₆×C₆₁	D₄	1	1	yes	characteristic
47	C₂×SD₁₆×C₆₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₁	—	2	1	no	—
49	C₄⋊Q₈×C₆₁	—	2	2	no	—
50	C₄⋊Q₈×C₆₁	—	2	2	no	—
51	C₄⋊Q₈×C₆₁	—	4	2	no	—
52	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
53	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
54	#126×C₆₁	—	4	2	no	—
55	#124×C₆₁	—	2	2	no	—
56	C₂²⋊Q₈×C₆₁	—	2	2	no	—
57	C₄×Q₈×C₆₁	—	2	2	no	—
58	C₄×Q₈×C₆₁	—	2	2	no	—
59	C₄×Q₈×C₆₁	—	2	2	no	—
60	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
61	C₄×Q₈×C₆₁	C₂³	1	1	yes	characteristic
62	C₄×D₄×C₆₁	C₂³	1	1	yes	characteristic
63	#117×C₆₁	—	2	2	no	—
64	#107×C₆₁	—	2	1	no	—
65	#107×C₆₁	—	2	1	no	—
66	#107×C₆₁	—	2	1	no	—
67	#107×C₆₁	—	2	1	no	—
68	#107×C₆₁	—	2	2	no	—
69	#107×C₆₁	D₄	1	1	yes	characteristic
70	#106×C₆₁	—	2	2	no	—
71	#106×C₆₁	—	2	2	no	—
72	#106×C₆₁	D₄	1	1	yes	characteristic
73	C₄⋊C₈×C₆₁	—	2	1	no	—
74	C₄⋊C₈×C₆₁	—	2	2	no	—
75	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
77	C₄⋊C₈×C₆₁	C₂³	1	1	yes	characteristic
78	#103×C₆₁	—	2	2	no	—
79	#103×C₆₁	—	2	2	no	—
80	#103×C₆₁	—	2	2	no	—
81	#103×C₆₁	—	2	2	no	—
82	#103×C₆₁	—	2	2	no	—
83	#103×C₆₁	—	4	2	no	—
84	#103×C₆₁	D₄	1	1	yes	characteristic
85	#103×C₆₁	D₄	1	2	yes	—
86	#102×C₆₁	—	2	2	no	—
87	#102×C₆₁	D₄	1	1	yes	characteristic
88	C₂²⋊C₈×C₆₁	—	2	1	no	—
89	#97×C₆₁	—	2	2	no	—
90	C₄×C₈×C₆₁	—	2	1	no	—
91	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₆₁	—	2	2	no	—
95	C₂×Q₈×C₆₁	—	2	2	no	—
96	C₂×Q₈×C₆₁	—	4	2	no	—
97	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
98	C₂×Q₈×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂×D₄×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₁	—	2	1	no	—
101	Q₁₆×C₆₁	—	2	2	no	—
102	Q₁₆×C₆₁	—	4	4	no	—
103	SD₁₆×C₆₁	—	2	2	no	—
104	C₂×C₈×C₆₁	—	2	1	no	—
105	C₂×C₈×C₆₁	—	2	2	no	—
106	C₂×C₈×C₆₁	—	2	1	no	—
107	C₂×C₈×C₆₁	—	2	1	no	—
108	C₂×C₈×C₆₁	—	4	2	no	—
109	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
113	C₄⋊C₄×C₆₁	—	2	2	no	—
114	C₄⋊C₄×C₆₁	—	2	1	no	—
115	C₄⋊C₄×C₆₁	—	2	2	no	—
116	C₄⋊C₄×C₆₁	—	2	2	no	—
117	C₄⋊C₄×C₆₁	—	2	2	no	—
118	C₄⋊C₄×C₆₁	—	2	2	no	—
119	C₄⋊C₄×C₆₁	—	4	2	no	—
120	C₄⋊C₄×C₆₁	—	4	2	no	—
121	C₄⋊C₄×C₆₁	—	4	2	no	—
122	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
123	C₄⋊C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
125	C₂²⋊C₄×C₆₁	—	2	1	no	—
126	C₂²⋊C₄×C₆₁	—	4	2	no	—
127	C₂²⋊C₄×C₆₁	—	4	2	no	—
128	C₄²×C₆₁	—	2	1	no	—
129	C₄²×C₆₁	—	2	2	no	—
130	C₄²×C₆₁	—	4	2	no	—
131	C₄²×C₆₁	—	4	2	no	—
132	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
133	C₂³×C₆₁	—	2	1	no	—
134	Q₈×C₆₁	—	2	1	no	—
135	Q₈×C₆₁	—	2	2	no	—
136	Q₈×C₆₁	—	2	2	no	—
137	Q₈×C₆₁	—	4	2	no	—
138	Q₈×C₆₁	—	4	2	no	—
139	Q₈×C₆₁	—	8	2	no	—
140	Q₈×C₆₁	—	16	2	no	—
141	Q₈×C₆₁	#135	1	2	yes	—
142	D₄×C₆₁	—	2	1	no	—
143	D₄×C₆₁	—	2	1	no	—
144	C₂×C₄×C₆₁	—	2	1	no	—
145	C₂×C₄×C₆₁	—	2	2	no	—
146	C₂×C₄×C₆₁	—	2	2	no	—
147	C₂×C₄×C₆₁	—	2	1	no	—
148	C₂×C₄×C₆₁	—	2	1	no	—
149	C₂×C₄×C₆₁	—	4	1	no	—
150	C₂×C₄×C₆₁	—	4	1	no	—
151	C₂×C₄×C₆₁	—	4	2	no	—
152	C₂×C₄×C₆₁	—	4	2	no	—
153	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
156	C₈×C₆₁	—	2	1	no	—
157	C₈×C₆₁	—	2	1	no	—
158	C₈×C₆₁	—	2	1	no	—
159	C₈×C₆₁	—	4	1	no	—
160	C₈×C₆₁	—	4	2	no	—
161	C₈×C₆₁	—	4	1	no	—
162	C₈×C₆₁	—	4	1	no	—
163	C₈×C₆₁	#137	1	2	yes	—
164	#19940	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂²×C₆₁	—	4	1	no	—
166	C₂²×C₆₁	—	4	1	no	—
167	C₂²×C₆₁	—	4	1	no	—
168	C₂²×C₆₁	D₄²	1	1	yes	characteristic
169	C₄×C₆₁	—	2	2	no	—
170	C₄×C₆₁	—	2	1	no	—
171	C₄×C₆₁	—	4	1	no	—
172	C₄×C₆₁	—	4	2	no	—
173	C₄×C₆₁	—	4	2	no	—
174	C₄×C₆₁	—	4	1	no	—
175	C₄×C₆₁	—	8	2	no	—
176	C₄×C₆₁	—	8	2	no	—
177	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
178	C₄×C₆₁	C₂×#137	1	1	yes	characteristic
179	C₄×C₆₁	#578	1	1	yes	characteristic
180	C₄×C₆₁	#578	1	1	yes	characteristic
181	Q₈×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
182	#3369	C₂×C₆₁	1	2	yes	—
183	#3342	C₂×C₆₁	1	1	yes	characteristic
184	#1710	C₂×C₆₁	1	1	yes	characteristic, complemented
185	#1704	C₂×C₆₁	1	2	yes	complemented
186	#1675	C₂×C₆₁	1	1	yes	characteristic
187	#1674	C₂×C₆₁	1	1	yes	characteristic, complemented
188	#1645	C₂×C₆₁	1	2	yes	complemented
189	#1638	C₂×C₆₁	1	2	yes	—
190	#1591	C₂×C₆₁	1	1	yes	characteristic, complemented
191	C₈×SD₁₆	C₂×C₆₁	1	1	yes	characteristic
192	C₂×C₆₁	—	8	1	no	—
193	C₂×C₆₁	#3276	1	1	yes	characteristic
194	C₂×C₆₁	#3285	1	1	yes	characteristic
195	C₂×C₆₁	#3290	1	1	yes	characteristic
196	Q₈²	C₂²×C₆₁	1	2	yes	—
197	#556	C₂²×C₆₁	1	2	yes	—
198	#500	—	2	1	no	—
199	#500	C₂²×C₆₁	1	1	yes	characteristic
200	#500	C₂²×C₆₁	1	1	yes	characteristic
201	#489	—	2	2	no	—
202	#487	C₂²×C₆₁	1	2	yes	—
203	#487	C₂²×C₆₁	1	2	yes	—
204	#485	—	2	2	no	—
205	#477	—	2	2	no	—
206	#477	C₂²×C₆₁	1	2	yes	—
207	#476	C₂²×C₆₁	1	2	yes	—
208	#475	—	2	2	no	—
209	#475	C₂²×C₆₁	1	2	yes	—
210	#467	—	2	1	no	—
211	#464	—	2	2	no	—
212	#462	C₂²×C₆₁	1	2	yes	—
213	#446	C₂²×C₆₁	1	2	yes	—
214	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
215	#441	—	2	2	no	—
216	C₄×Q₁₆	—	2	1	no	—
217	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
218	C₄×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
219	C₈×D₄	C₂²×C₆₁	1	1	yes	characteristic
220	#427	—	2	1	no	—
221	#335	C₂²×C₆₁	1	1	yes	characteristic
222	#334	C₂²×C₆₁	1	1	yes	characteristic
223	#333	C₂²×C₆₁	1	2	yes	—
224	#332	C₂²×C₆₁	1	2	yes	—
225	Q₈⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
226	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
227	D₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
228	C₈²	C₂²×C₆₁	1	1	yes	characteristic
229	C₆₁	#19940	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×Q₁₆	—	2	2	no	—
231	C₂×Q₁₆	D₄×C₆₁	1	1	yes	characteristic
232	C₂×SD₁₆	D₄×C₆₁	1	1	yes	characteristic
233	C₂²×C₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	2	no	—
235	C₄⋊Q₈	—	2	2	no	—
236	C₄⋊Q₈	—	4	2	no	—
237	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
238	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
239	#126	—	4	2	no	—
240	#124	—	2	2	no	—
241	C₂²⋊Q₈	—	2	2	no	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×Q₈	—	2	2	no	—
244	C₄×Q₈	—	2	2	no	—
245	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
246	C₄×Q₈	C₂³×C₆₁	1	1	yes	characteristic
247	C₄×D₄	C₂³×C₆₁	1	1	yes	characteristic
248	#117	—	2	2	no	—
249	#107	—	2	1	no	—
250	#107	—	2	1	no	—
251	#107	—	2	1	no	—
252	#107	—	2	1	no	—
253	#107	—	2	2	no	—
254	#107	D₄×C₆₁	1	1	yes	characteristic
255	#106	—	2	2	no	—
256	#106	—	2	2	no	—
257	#106	D₄×C₆₁	1	1	yes	characteristic
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	2	2	no	—
260	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
261	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
262	C₄⋊C₈	C₂³×C₆₁	1	1	yes	characteristic
263	#103	—	2	2	no	—
264	#103	—	2	2	no	—
265	#103	—	2	2	no	—
266	#103	—	2	2	no	—
267	#103	—	2	2	no	—
268	#103	—	4	2	no	—
269	#103	D₄×C₆₁	1	1	yes	characteristic
270	#103	D₄×C₆₁	1	2	yes	—
271	#102	—	2	2	no	—
272	#102	D₄×C₆₁	1	1	yes	characteristic
273	C₂²⋊C₈	—	2	1	no	—
274	#97	—	2	2	no	—
275	C₄×C₈	—	2	1	no	—
276	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
277	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
278	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	4	2	no	—
282	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
283	C₂×Q₈	C₂×D₄×C₆₁	1	1	yes	characteristic
284	C₂×D₄	C₂×D₄×C₆₁	1	1	yes	characteristic
285	C₂²×C₄	—	2	1	no	—
286	Q₁₆	—	2	2	no	—
287	Q₁₆	—	4	4	no	—
288	SD₁₆	—	2	2	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	4	2	no	—
294	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
295	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
296	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
297	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
308	C₄⋊C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
309	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
310	C₂²⋊C₄	—	2	1	no	—
311	C₂²⋊C₄	—	4	2	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	4	2	no	—
316	C₄²	—	4	2	no	—
317	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
318	C₂³	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	2	no	—
321	Q₈	—	2	2	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	8	2	no	—
325	Q₈	—	16	2	no	—
326	Q₈	#135×C₆₁	1	2	yes	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
340	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	4	1	no	—
345	C₈	—	4	2	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	1	no	—
348	C₈	#137×C₆₁	1	2	yes	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	D₄²×C₆₁	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	2	no	—
359	C₄	—	8	2	no	—
360	C₄	—	8	2	no	—
361	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
362	C₄	C₂×#137×C₆₁	1	1	yes	characteristic
363	C₄	C₆₁×#578	1	1	yes	characteristic
364	C₄	C₆₁×#578	1	1	yes	characteristic
365	C₂	—	8	1	no	—
366	C₂	C₆₁×#3276	1	1	yes	characteristic
367	C₂	C₆₁×#3285	1	1	yes	characteristic
368	C₂	C₆₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₂×D₄×C₆₁	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×Q₁₆×C₆₁	C₂	semidirect
4. C₆₁×#1710	C₂	semidirect
5. C₆₁×#1704	C₂	semidirect
7. C₆₁×#1674	C₂	semidirect
8. C₆₁×#1645	C₂	semidirect
10. C₆₁×#1591	C₂	semidirect
164. #19940	C₆₁	direct (N × H)
181. Q₈×Q₁₆	C₂×C₆₁	semidirect
184. #1710	C₂×C₆₁	semidirect
185. #1704	C₂×C₆₁	semidirect
187. #1674	C₂×C₆₁	semidirect
188. #1645	C₂×C₆₁	semidirect
190. #1591	C₂×C₆₁	semidirect
229. C₆₁	#19940	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×Q₁₆×C₆₁	2
C₆₁×#3369	2
C₆₁×#3342	2
C₆₁×#1710	2
C₆₁×#1704	2
C₆₁×#1675	2
C₆₁×#1674	2
C₆₁×#1645	2
C₆₁×#1638	2
C₆₁×#1591	2
C₈×SD₁₆×C₆₁	2
#19940	61

Subgroups — C61×#19940 #29523373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19940 #29523373