Subgroups — C₆₁×#8863 #29512296

Order 15616.

374 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×#2921	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1798	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1458	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×#433	—	2	2	no	—
10	C₆₁×#433	—	2	2	no	—
11	C₆₁×#431	C₂²	1	1	yes	characteristic
12	C₆₁×#431	C₂²	1	1	yes	characteristic
13	C₆₁×#423	C₂²	1	1	yes	characteristic
14	C₆₁×#423	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#407	—	2	1	no	—
18	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
19	C₄×M₁₆×C₆₁	—	2	1	no	—
20	C₆₁×#336	—	2	1	no	—
21	C₆₁×#335	—	2	2	no	—
22	C₆₁×#334	—	2	2	no	—
23	C₆₁×#324	C₄	1	2	yes	—
24	C₆₁×#323	C₄	1	2	yes	—
25	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₆₁	—	2	1	no	—
27	C₂×M₁₆×C₆₁	—	2	1	no	—
28	C₂²×C₈×C₆₁	—	2	1	no	—
29	C₂²×C₈×C₆₁	—	2	1	no	—
30	#117×C₆₁	—	2	1	no	—
31	#117×C₆₁	—	2	1	no	—
32	#117×C₆₁	—	2	1	no	—
33	#117×C₆₁	—	2	1	no	—
34	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₆₁	Q₈	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₆₁	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₆₁	—	2	1	no	—
41	C₄⋊C₈×C₆₁	—	2	1	no	—
42	C₄⋊C₈×C₆₁	—	2	1	no	—
43	C₄⋊C₈×C₆₁	—	2	1	no	—
44	C₄⋊C₈×C₆₁	—	2	1	no	—
45	C₄⋊C₈×C₆₁	—	2	1	no	—
46	C₄⋊C₈×C₆₁	—	2	1	no	—
47	C₄⋊C₈×C₆₁	—	2	1	no	—
48	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
49	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
50	C₂²⋊C₈×C₆₁	—	2	4	no	—
51	C₂²⋊C₈×C₆₁	—	4	2	no	—
52	C₂²⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
53	C₂²⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
54	#97×C₆₁	—	2	2	no	—
55	C₄×C₈×C₆₁	—	2	2	no	—
56	C₄×C₈×C₆₁	—	2	1	no	—
57	C₄×C₈×C₆₁	—	2	1	no	—
58	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₁	—	2	1	no	—
60	C₂²×C₄×C₆₁	—	2	1	no	—
61	C₂²×C₄×C₆₁	—	2	1	no	—
62	C₂²×C₄×C₆₁	—	2	1	no	—
63	C₂²×C₄×C₆₁	—	2	1	no	—
64	C₂²×C₄×C₆₁	—	2	1	no	—
65	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
71	M₁₆×C₆₁	—	4	1	no	—
72	M₁₆×C₆₁	—	4	1	no	—
73	M₁₆×C₆₁	—	4	1	no	—
74	M₁₆×C₆₁	—	4	1	no	—
75	C₂×C₈×C₆₁	—	2	2	no	—
76	C₂×C₈×C₆₁	—	2	2	no	—
77	C₂×C₈×C₆₁	—	2	2	no	—
78	C₂×C₈×C₆₁	—	2	2	no	—
79	C₂×C₈×C₆₁	—	4	1	no	—
80	C₂×C₈×C₆₁	—	4	1	no	—
81	C₂×C₈×C₆₁	—	4	1	no	—
82	C₂×C₈×C₆₁	—	4	1	no	—
83	C₂×C₈×C₆₁	—	4	1	no	—
84	C₂×C₈×C₆₁	—	4	1	no	—
85	C₄⋊C₄×C₆₁	—	2	1	no	—
86	C₄⋊C₄×C₆₁	—	2	1	no	—
87	C₄⋊C₄×C₆₁	—	2	1	no	—
88	C₄⋊C₄×C₆₁	—	2	1	no	—
89	C₂²⋊C₄×C₆₁	—	2	1	no	—
90	C₂²⋊C₄×C₆₁	—	2	1	no	—
91	C₂²⋊C₄×C₆₁	—	2	1	no	—
92	C₂²⋊C₄×C₆₁	—	2	1	no	—
93	C₄²×C₆₁	—	2	1	no	—
94	C₄²×C₆₁	—	2	1	no	—
95	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₆₁	#41	1	1	yes	characteristic
98	C₄²×C₆₁	#41	1	1	yes	characteristic
99	C₂³×C₆₁	—	2	1	no	—
100	C₂³×C₆₁	—	2	1	no	—
101	C₂³×C₆₁	—	2	1	no	—
102	C₂³×C₆₁	—	4	1	no	—
103	C₂³×C₆₁	—	4	1	no	—
104	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	—	2	1	no	—
106	C₂×C₄×C₆₁	—	2	1	no	—
107	C₂×C₄×C₆₁	—	2	1	no	—
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	2	1	no	—
110	C₂×C₄×C₆₁	—	4	1	no	—
111	C₂×C₄×C₆₁	—	4	1	no	—
112	C₂×C₄×C₆₁	—	4	1	no	—
113	C₂×C₄×C₆₁	—	4	1	no	—
114	C₂×C₄×C₆₁	—	4	1	no	—
115	C₂×C₄×C₆₁	—	4	1	no	—
116	C₂×C₄×C₆₁	—	4	1	no	—
117	C₂×C₄×C₆₁	—	4	1	no	—
118	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₆₁	#123	1	1	yes	characteristic
124	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
125	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
126	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
128	C₈×C₆₁	—	4	2	no	—
129	C₈×C₆₁	—	4	2	no	—
130	C₈×C₆₁	—	4	2	no	—
131	C₈×C₆₁	—	4	2	no	—
132	C₈×C₆₁	—	8	1	no	—
133	C₈×C₆₁	—	8	1	no	—
134	#8863	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₆₁	—	2	1	no	—
136	C₂²×C₆₁	—	2	1	no	—
137	C₂²×C₆₁	—	4	2	no	—
138	C₂²×C₆₁	—	4	1	no	—
139	C₂²×C₆₁	—	4	1	no	—
140	C₂²×C₆₁	—	4	1	no	—
141	C₂²×C₆₁	—	8	1	no	—
142	C₂²×C₆₁	#385	1	1	yes	characteristic
143	C₂²×C₆₁	#427	1	1	yes	characteristic
144	C₂²×C₆₁	#428	1	1	yes	characteristic
145	C₄×C₆₁	—	2	1	no	—
146	C₄×C₆₁	—	2	1	no	—
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	2	1	no	—
149	C₄×C₆₁	—	4	1	no	—
150	C₄×C₆₁	—	4	1	no	—
151	C₄×C₆₁	—	4	1	no	—
152	C₄×C₆₁	#449	1	1	yes	characteristic
153	C₄×C₆₁	#452	1	1	yes	characteristic
154	C₄×C₆₁	#482	1	1	yes	characteristic
155	C₄×C₆₁	#485	1	1	yes	characteristic
156	#2921	C₂×C₆₁	1	1	yes	characteristic
157	#1798	C₂×C₆₁	1	1	yes	characteristic
158	#1571	C₂×C₆₁	1	1	yes	characteristic, complemented
159	#1570	C₂×C₆₁	1	1	yes	characteristic, complemented
160	#1458	C₂×C₆₁	1	1	yes	characteristic
161	#1301	C₂×C₆₁	1	1	yes	characteristic, complemented
162	#1301	C₂×C₆₁	1	1	yes	characteristic, complemented
163	C₂×C₆₁	—	2	1	no	—
164	C₂×C₆₁	—	2	1	no	—
165	C₂×C₆₁	—	8	1	no	—
166	C₂×C₆₁	#1854	1	1	yes	characteristic
167	C₂×C₆₁	#1858	1	1	yes	characteristic
168	C₂×C₆₁	#1868	1	1	yes	characteristic
169	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
170	#433	—	2	2	no	—
171	#433	—	2	2	no	—
172	#431	C₂²×C₆₁	1	1	yes	characteristic
173	#431	C₂²×C₆₁	1	1	yes	characteristic
174	#423	C₂²×C₆₁	1	1	yes	characteristic
175	#423	C₂²×C₆₁	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
178	#407	—	2	1	no	—
179	C₂×C₂²⋊C₈	—	2	1	no	—
180	C₄×M₁₆	—	2	1	no	—
181	#336	—	2	1	no	—
182	#335	—	2	2	no	—
183	#334	—	2	2	no	—
184	#324	C₄×C₆₁	1	2	yes	—
185	#323	C₄×C₆₁	1	2	yes	—
186	C₆₁	#8863	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³×C₄	D₄×C₆₁	1	1	yes	characteristic
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	#117	—	2	1	no	—
193	#117	—	2	1	no	—
194	#117	—	2	1	no	—
195	#117	—	2	1	no	—
196	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
197	C₂×C₄⋊C₄	Q₈×C₆₁	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
199	C₂×C₂²⋊C₄	Q₈×C₆₁	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
202	C₄⋊C₈	—	2	1	no	—
203	C₄⋊C₈	—	2	1	no	—
204	C₄⋊C₈	—	2	1	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
211	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
212	C₂²⋊C₈	—	2	4	no	—
213	C₂²⋊C₈	—	4	2	no	—
214	C₂²⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
215	C₂²⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
216	#97	—	2	2	no	—
217	C₄×C₈	—	2	2	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₂⁴	C₄⋊C₄×C₆₁	1	1	yes	characteristic
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
228	C₂²×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
229	C₂²×C₄	C₄⋊C₄×C₆₁	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×Q₈×C₆₁	1	1	yes	characteristic
233	M₁₆	—	4	1	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	1	no	—
236	M₁₆	—	4	1	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
259	C₄²	#41×C₆₁	1	1	yes	characteristic
260	C₄²	#41×C₆₁	1	1	yes	characteristic
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	—	4	1	no	—
266	C₂³	C₂×C₄⋊C₄×C₆₁	1	1	yes	characteristic
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
281	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
282	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
283	C₂×C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊Q₈×C₆₁	1	1	yes	characteristic
285	C₂×C₄	#123×C₆₁	1	1	yes	characteristic
286	C₂×C₄	#135×C₆₁	1	1	yes	characteristic
287	C₂×C₄	#135×C₆₁	1	1	yes	characteristic
288	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
289	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₈	—	8	1	no	—
295	C₈	—	8	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	8	1	no	—
303	C₂²	C₆₁×#385	1	1	yes	characteristic
304	C₂²	C₆₁×#427	1	1	yes	characteristic
305	C₂²	C₆₁×#428	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	C₆₁×#449	1	1	yes	characteristic
314	C₄	C₆₁×#452	1	1	yes	characteristic
315	C₄	C₆₁×#482	1	1	yes	characteristic
316	C₄	C₆₁×#485	1	1	yes	characteristic
317	C₂	—	2	1	no	—
318	C₂	—	2	1	no	—
319	C₂	—	8	1	no	—
320	C₂	C₆₁×#1854	1	1	yes	characteristic
321	C₂	C₆₁×#1858	1	1	yes	characteristic
322	C₂	C₆₁×#1868	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#120
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#385
prime radical	C₂⁴×C₆₁	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₁×#1571	C₂	semidirect
4. C₆₁×#1570	C₂	semidirect
6. C₆₁×#1301	C₂	semidirect
7. C₆₁×#1301	C₂	semidirect
134. #8863	C₆₁	direct (N × H)
158. #1571	C₂×C₆₁	semidirect
159. #1570	C₂×C₆₁	semidirect
161. #1301	C₂×C₆₁	semidirect
162. #1301	C₂×C₆₁	semidirect
186. C₆₁	#8863	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×#2921	2
C₆₁×#1798	2
C₆₁×#1571	2
C₆₁×#1570	2
C₆₁×#1458	2
C₆₁×#1301	2
#8863	61

Subgroups — C61×#8863 #29512296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#8863 #29512296