Subgroups — C₆₁×#19958 #29523391

Order 15616.

570 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3360	C₂	1	2	yes	maximal
2	C₆₁×#3356	C₂	1	2	yes	maximal, complemented
3	C₆₁×#1707	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1691	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1684	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1650	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1632	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1589	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#556	C₂²	1	2	yes	—
11	C₆₁×#554	C₂²	1	2	yes	—
12	C₆₁×#500	C₂²	1	1	yes	characteristic
13	C₆₁×#498	C₂²	1	2	yes	—
14	C₆₁×#494	C₂²	1	1	yes	characteristic
15	C₆₁×#488	—	2	2	no	—
16	C₆₁×#487	C₂²	1	2	yes	—
17	C₆₁×#485	—	2	2	no	—
18	C₆₁×#483	—	2	2	no	—
19	C₆₁×#479	—	2	2	no	—
20	C₆₁×#477	C₂²	1	2	yes	—
21	C₆₁×#476	C₂²	1	2	yes	—
22	C₆₁×#475	C₂²	1	2	yes	—
23	C₆₁×#470	—	2	2	no	—
24	C₆₁×#463	—	2	2	no	—
25	C₆₁×#461	C₂²	1	2	yes	—
26	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
28	C₄×SD₁₆×C₆₁	—	2	2	no	—
29	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
30	C₆₁×#436	C₂²	1	2	yes	—
31	C₆₁×#428	—	2	2	no	—
32	C₆₁×#335	C₂²	1	1	yes	characteristic
33	C₆₁×#335	C₂²	1	1	yes	characteristic
34	C₆₁×#333	C₂²	1	2	yes	—
35	C₆₁×#332	C₂²	1	2	yes	—
36	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₆₁×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₆₁	—	4	1	no	—
40	C₂×Q₁₆×C₆₁	D₄	1	1	yes	characteristic
41	C₂×SD₁₆×C₆₁	—	2	2	no	—
42	C₂×D₈×C₆₁	D₄	1	1	yes	characteristic
43	C₂×M₁₆×C₆₁	—	2	2	no	—
44	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
46	#126×C₆₁	—	4	2	no	—
47	#125×C₆₁	—	2	2	no	—
48	#124×C₆₁	—	2	2	no	—
49	C₂²⋊Q₈×C₆₁	—	2	2	no	—
50	C₂²⋊Q₈×C₆₁	—	2	2	no	—
51	C₂²⋊Q₈×C₆₁	—	4	2	no	—
52	C₄×Q₈×C₆₁	—	2	2	no	—
53	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
54	C₄×D₄×C₆₁	—	2	2	no	—
55	C₄×D₄×C₆₁	C₂³	1	2	yes	—
56	#117×C₆₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
58	#107×C₆₁	—	2	2	no	—
59	#107×C₆₁	—	4	1	no	—
60	#107×C₆₁	D₄	1	1	yes	characteristic
61	#107×C₆₁	D₄	1	1	yes	characteristic
62	#106×C₆₁	—	2	2	no	—
63	#106×C₆₁	—	2	2	no	—
64	#106×C₆₁	—	2	2	no	—
65	#106×C₆₁	—	2	2	no	—
66	C₄⋊C₈×C₆₁	—	2	2	no	—
67	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
68	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
69	#103×C₆₁	—	2	2	no	—
70	#103×C₆₁	—	2	2	no	—
71	#103×C₆₁	—	2	2	no	—
72	#103×C₆₁	—	2	2	no	—
73	#103×C₆₁	—	2	2	no	—
74	#103×C₆₁	D₄	1	2	yes	—
75	#102×C₆₁	—	2	2	no	—
76	#102×C₆₁	D₄	1	2	yes	—
77	C₂²⋊C₈×C₆₁	—	2	2	no	—
78	C₄×C₈×C₆₁	—	2	2	no	—
79	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₆₁	—	2	2	no	—
83	C₂×Q₈×C₆₁	—	2	2	no	—
84	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
85	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₁	—	2	2	no	—
87	C₂²×C₄×C₆₁	—	4	2	no	—
88	Q₁₆×C₆₁	—	2	2	no	—
89	Q₁₆×C₆₁	—	8	2	no	—
90	SD₁₆×C₆₁	—	4	4	no	—
91	D₈×C₆₁	—	4	1	no	—
92	M₁₆×C₆₁	—	4	4	no	—
93	C₂×C₈×C₆₁	—	2	1	no	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	2	2	no	—
96	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₆₁	—	2	2	no	—
101	C₄⋊C₄×C₆₁	—	2	2	no	—
102	C₄⋊C₄×C₆₁	—	2	2	no	—
103	C₄⋊C₄×C₆₁	—	2	2	no	—
104	C₄⋊C₄×C₆₁	—	2	2	no	—
105	C₄⋊C₄×C₆₁	—	4	2	no	—
106	C₄⋊C₄×C₆₁	—	4	2	no	—
107	C₄⋊C₄×C₆₁	—	4	2	no	—
108	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
109	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
110	C₂²⋊C₄×C₆₁	—	2	2	no	—
111	C₂²⋊C₄×C₆₁	—	4	2	no	—
112	C₂²⋊C₄×C₆₁	—	4	2	no	—
113	C₂²⋊C₄×C₆₁	—	4	2	no	—
114	C₄²×C₆₁	—	2	2	no	—
115	C₄²×C₆₁	—	4	2	no	—
116	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
117	C₂³×C₆₁	—	2	2	no	—
118	Q₈×C₆₁	—	2	2	no	—
119	Q₈×C₆₁	—	2	2	no	—
120	Q₈×C₆₁	—	4	2	no	—
121	Q₈×C₆₁	—	4	2	no	—
122	Q₈×C₆₁	—	8	2	no	—
123	D₄×C₆₁	—	2	2	no	—
124	D₄×C₆₁	—	2	2	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	2	no	—
127	C₂×C₄×C₆₁	—	2	2	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	4	2	no	—
130	C₂×C₄×C₆₁	—	4	2	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	—	16	2	no	—
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
136	C₈×C₆₁	—	2	2	no	—
137	C₈×C₆₁	—	2	1	no	—
138	C₈×C₆₁	—	2	1	no	—
139	C₈×C₆₁	—	2	1	no	—
140	C₈×C₆₁	—	4	2	no	—
141	C₈×C₆₁	—	4	2	no	—
142	C₈×C₆₁	#135	1	2	yes	—
143	#19958	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₆₁	—	4	2	no	—
145	C₂²×C₆₁	—	4	2	no	—
146	C₂²×C₆₁	—	4	2	no	—
147	C₂²×C₆₁	D₄²	1	1	yes	characteristic
148	C₄×C₆₁	—	2	1	no	—
149	C₄×C₆₁	—	4	2	no	—
150	C₄×C₆₁	—	4	2	no	—
151	C₄×C₆₁	—	4	2	no	—
152	C₄×C₆₁	—	8	2	no	—
153	C₄×C₆₁	—	8	2	no	—
154	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
155	C₄×C₆₁	#575	1	1	yes	characteristic
156	C₄×C₆₁	#578	1	1	yes	characteristic
157	C₄×C₆₁	#578	1	1	yes	characteristic
158	#3360	C₂×C₆₁	1	2	yes	—
159	#3356	C₂×C₆₁	1	2	yes	complemented
160	#1707	C₂×C₆₁	1	1	yes	characteristic, complemented
161	#1691	C₂×C₆₁	1	2	yes	complemented
162	#1684	C₂×C₆₁	1	2	yes	complemented
163	#1650	C₂×C₆₁	1	2	yes	complemented
164	#1632	C₂×C₆₁	1	2	yes	complemented
165	#1589	C₂×C₆₁	1	1	yes	characteristic
166	C₈×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
167	C₂×C₆₁	—	8	2	no	—
168	C₂×C₆₁	#3277	1	1	yes	characteristic
169	C₂×C₆₁	#3283	1	1	yes	characteristic
170	C₂×C₆₁	#3291	1	1	yes	characteristic
171	#556	C₂²×C₆₁	1	2	yes	—
172	#554	C₂²×C₆₁	1	2	yes	—
173	#500	C₂²×C₆₁	1	1	yes	characteristic
174	#498	C₂²×C₆₁	1	2	yes	—
175	#494	C₂²×C₆₁	1	1	yes	characteristic
176	#488	—	2	2	no	—
177	#487	C₂²×C₆₁	1	2	yes	—
178	#485	—	2	2	no	—
179	#483	—	2	2	no	—
180	#479	—	2	2	no	—
181	#477	C₂²×C₆₁	1	2	yes	—
182	#476	C₂²×C₆₁	1	2	yes	—
183	#475	C₂²×C₆₁	1	2	yes	—
184	#470	—	2	2	no	—
185	#463	—	2	2	no	—
186	#461	C₂²×C₆₁	1	2	yes	—
187	C₈×Q₈	C₂²×C₆₁	1	2	yes	—
188	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
189	C₄×SD₁₆	—	2	2	no	—
190	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
191	#436	C₂²×C₆₁	1	2	yes	—
192	#428	—	2	2	no	—
193	#335	C₂²×C₆₁	1	1	yes	characteristic
194	#335	C₂²×C₆₁	1	1	yes	characteristic
195	#333	C₂²×C₆₁	1	2	yes	—
196	#332	C₂²×C₆₁	1	2	yes	—
197	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
198	D₄⋊C₈	C₂²×C₆₁	1	2	yes	—
199	C₈²	C₂²×C₆₁	1	1	yes	characteristic
200	C₆₁	#19958	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×Q₁₆	—	4	1	no	—
202	C₂×Q₁₆	D₄×C₆₁	1	1	yes	characteristic
203	C₂×SD₁₆	—	2	2	no	—
204	C₂×D₈	D₄×C₆₁	1	1	yes	characteristic
205	C₂×M₁₆	—	2	2	no	—
206	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
207	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
208	#126	—	4	2	no	—
209	#125	—	2	2	no	—
210	#124	—	2	2	no	—
211	C₂²⋊Q₈	—	2	2	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	C₂²⋊Q₈	—	4	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	C₂³×C₆₁	1	2	yes	—
218	#117	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	#107	—	2	2	no	—
221	#107	—	4	1	no	—
222	#107	D₄×C₆₁	1	1	yes	characteristic
223	#107	D₄×C₆₁	1	1	yes	characteristic
224	#106	—	2	2	no	—
225	#106	—	2	2	no	—
226	#106	—	2	2	no	—
227	#106	—	2	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
230	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
231	#103	—	2	2	no	—
232	#103	—	2	2	no	—
233	#103	—	2	2	no	—
234	#103	—	2	2	no	—
235	#103	—	2	2	no	—
236	#103	D₄×C₆₁	1	2	yes	—
237	#102	—	2	2	no	—
238	#102	D₄×C₆₁	1	2	yes	—
239	C₂²⋊C₈	—	2	2	no	—
240	C₄×C₈	—	2	2	no	—
241	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
247	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	8	2	no	—
252	SD₁₆	—	4	4	no	—
253	D₈	—	4	1	no	—
254	M₁₆	—	4	4	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
271	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
279	C₂³	—	2	2	no	—
280	Q₈	—	2	2	no	—
281	Q₈	—	2	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	16	2	no	—
295	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
297	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	4	2	no	—
303	C₈	—	4	2	no	—
304	C₈	#135×C₆₁	1	2	yes	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	D₄²×C₆₁	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	2	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
316	C₄	C₆₁×#575	1	1	yes	characteristic
317	C₄	C₆₁×#578	1	1	yes	characteristic
318	C₄	C₆₁×#578	1	1	yes	characteristic
319	C₂	—	8	2	no	—
320	C₂	C₆₁×#3277	1	1	yes	characteristic
321	C₂	C₆₁×#3283	1	1	yes	characteristic
322	C₂	C₆₁×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₂×D₈×C₆₁	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
2. C₆₁×#3356	C₂	semidirect
3. C₆₁×#1707	C₂	semidirect
4. C₆₁×#1691	C₂	semidirect
5. C₆₁×#1684	C₂	semidirect
6. C₆₁×#1650	C₂	semidirect
7. C₆₁×#1632	C₂	semidirect
9. C₈×Q₁₆×C₆₁	C₂	semidirect
143. #19958	C₆₁	direct (N × H)
159. #3356	C₂×C₆₁	semidirect
160. #1707	C₂×C₆₁	semidirect
161. #1691	C₂×C₆₁	semidirect
162. #1684	C₂×C₆₁	semidirect
163. #1650	C₂×C₆₁	semidirect
164. #1632	C₂×C₆₁	semidirect
166. C₈×Q₁₆	C₂×C₆₁	semidirect
200. C₆₁	#19958	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×#3360	2
C₆₁×#3356	2
C₆₁×#1707	2
C₆₁×#1691	2
C₆₁×#1684	2
C₆₁×#1650	2
C₆₁×#1632	2
C₆₁×#1589	2
C₈×Q₁₆×C₆₁	2
#19958	61

Subgroups — C61×#19958 #29523391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19958 #29523391