Subgroups — C₆₁×#18688 #29522121

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3254	C₂	1	2	yes	maximal, complemented
3	C₆₁×#3147	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1633	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1630	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1623	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1620	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1560	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#1493	C₂	1	2	yes	maximal, complemented
10	C₂×#135×C₆₁	—	2	2	no	—
11	C₆₁×#563	—	2	2	no	—
12	C₆₁×#559	C₂²	1	2	yes	—
13	Q₈²×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#556	—	2	2	no	—
15	D₄×Q₈×C₆₁	—	2	2	no	—
16	C₆₁×#542	—	2	4	no	—
17	C₆₁×#536	—	2	2	no	—
18	C₆₁×#533	C₂²	1	2	yes	—
19	C₆₁×#532	C₂²	1	1	yes	characteristic
20	C₆₁×#532	C₂²	1	1	yes	characteristic
21	C₆₁×#495	—	2	2	no	—
22	C₆₁×#494	C₂²	1	1	yes	characteristic
23	C₆₁×#493	C₂²	1	1	yes	characteristic
24	C₆₁×#492	C₂²	1	1	yes	characteristic
25	C₆₁×#492	C₂²	1	1	yes	characteristic
26	C₆₁×#477	C₂²	1	4	yes	—
27	C₆₁×#474	—	2	2	no	—
28	C₆₁×#468	—	2	4	no	—
29	C₆₁×#461	—	2	2	no	—
30	C₆₁×#460	C₂²	1	4	yes	—
31	C₆₁×#423	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
33	C₆₁×#333	C₂²	1	2	yes	—
34	C₆₁×#331	C₂²	1	2	yes	—
35	C₆₁×Q₈⋊C₈	C₂²	1	4	yes	—
36	C₆₁×Q₈⋊C₈	C₂²	1	4	yes	—
37	C₂×#41×C₆₁	—	2	1	no	—
38	C₂×#41×C₆₁	D₄	1	2	yes	—
39	C₂²×Q₈×C₆₁	—	2	2	no	—
40	#135×C₆₁	—	4	8	no	—
41	C₂×Q₁₆×C₆₁	—	2	2	no	—
42	C₂×SD₁₆×C₆₁	—	2	2	no	—
43	C₂×SD₁₆×C₆₁	—	2	2	no	—
44	C₂×SD₁₆×C₆₁	—	2	4	no	—
45	C₂×D₈×C₆₁	—	2	2	no	—
46	C₂×M₁₆×C₆₁	—	2	2	no	—
47	C₂²×C₈×C₆₁	—	2	1	no	—
48	C₄⋊Q₈×C₆₁	—	2	4	no	—
49	C₄⋊Q₈×C₆₁	—	2	4	no	—
50	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
51	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
52	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
54	#127×C₆₁	C₂³	1	1	yes	characteristic
55	#127×C₆₁	C₂³	1	1	yes	characteristic
56	#126×C₆₁	—	2	8	no	—
57	#125×C₆₁	—	2	2	no	—
58	#124×C₆₁	—	2	4	no	—
59	#124×C₆₁	—	2	4	no	—
60	#124×C₆₁	—	2	1	no	—
61	#124×C₆₁	D₄	1	2	yes	—
62	#123×C₆₁	—	2	4	no	—
63	#123×C₆₁	—	2	4	no	—
64	C₂²⋊Q₈×C₆₁	—	2	4	no	—
65	C₂²⋊Q₈×C₆₁	—	2	8	no	—
66	C₂²⋊Q₈×C₆₁	—	2	2	no	—
67	C₂²⋊Q₈×C₆₁	D₄	1	4	yes	—
68	#121×C₆₁	—	2	2	no	—
69	#121×C₆₁	—	2	2	no	—
70	C₄×Q₈×C₆₁	—	2	4	no	—
71	C₄×Q₈×C₆₁	—	2	2	no	—
72	C₄×Q₈×C₆₁	C₂³	1	4	yes	—
73	C₄×D₄×C₆₁	—	2	4	no	—
74	C₄×D₄×C₆₁	—	2	2	no	—
75	C₄×D₄×C₆₁	—	2	2	no	—
76	#117×C₆₁	—	2	2	no	—
77	#117×C₆₁	—	2	2	no	—
78	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
79	#107×C₆₁	—	2	4	no	—
80	C₄⋊C₈×C₆₁	—	2	2	no	—
81	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
82	#103×C₆₁	—	2	4	no	—
83	#102×C₆₁	—	2	4	no	—
84	C₄×C₈×C₆₁	—	2	1	no	—
85	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
87	#41×C₆₁	—	4	4	no	—
88	#41×C₆₁	—	4	4	no	—
89	C₂×Q₈×C₆₁	—	2	4	no	—
90	C₂×Q₈×C₆₁	—	2	2	no	—
91	C₂×Q₈×C₆₁	—	2	1	no	—
92	C₂×Q₈×C₆₁	—	8	4	no	—
93	C₂×Q₈×C₆₁	C₂×D₄	1	4	yes	—
94	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
95	C₂×D₄×C₆₁	—	2	2	no	—
96	C₂×D₄×C₆₁	—	2	2	no	—
97	C₂×D₄×C₆₁	—	2	2	no	—
98	C₂×D₄×C₆₁	—	2	1	no	—
99	C₂×D₄×C₆₁	—	4	1	no	—
100	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₁	—	2	4	no	—
102	C₂²×C₄×C₆₁	—	2	2	no	—
103	C₂²×C₄×C₆₁	—	4	1	no	—
104	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
106	Q₁₆×C₆₁	—	4	4	no	—
107	SD₁₆×C₆₁	—	4	4	no	—
108	SD₁₆×C₆₁	—	4	4	no	—
109	SD₁₆×C₆₁	—	8	4	no	—
110	D₈×C₆₁	—	4	4	no	—
111	M₁₆×C₆₁	—	8	2	no	—
112	C₂×C₈×C₆₁	—	2	4	no	—
113	C₂×C₈×C₆₁	—	2	1	no	—
114	C₂×C₈×C₆₁	—	2	1	no	—
115	C₂×C₈×C₆₁	—	2	4	no	—
116	C₄⋊C₄×C₆₁	—	2	8	no	—
117	C₄⋊C₄×C₆₁	—	2	2	no	—
118	C₄⋊C₄×C₆₁	—	2	2	no	—
119	C₄⋊C₄×C₆₁	—	2	2	no	—
120	C₄⋊C₄×C₆₁	—	2	4	no	—
121	C₄⋊C₄×C₆₁	—	2	2	no	—
122	C₄⋊C₄×C₆₁	—	2	4	no	—
123	C₄⋊C₄×C₆₁	—	2	2	no	—
124	C₄⋊C₄×C₆₁	C₂×D₄	1	4	yes	—
125	C₂²⋊C₄×C₆₁	—	2	4	no	—
126	C₂²⋊C₄×C₆₁	—	2	8	no	—
127	C₂²⋊C₄×C₆₁	—	2	4	no	—
128	C₂²⋊C₄×C₆₁	—	2	4	no	—
129	C₂²⋊C₄×C₆₁	—	4	2	no	—
130	C₄²×C₆₁	—	2	4	no	—
131	C₄²×C₆₁	—	2	4	no	—
132	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₆₁	C₂×D₄	1	2	yes	—
134	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
135	C₂³×C₆₁	—	2	2	no	—
136	C₂³×C₆₁	—	4	1	no	—
137	C₂³×C₆₁	#120	1	1	yes	characteristic
138	Q₈×C₆₁	—	2	4	no	—
139	Q₈×C₆₁	—	2	4	no	—
140	Q₈×C₆₁	—	4	2	no	—
141	Q₈×C₆₁	—	4	2	no	—
142	Q₈×C₆₁	—	4	4	no	—
143	Q₈×C₆₁	—	8	4	no	—
144	D₄×C₆₁	—	4	2	no	—
145	D₄×C₆₁	—	4	2	no	—
146	D₄×C₆₁	—	4	2	no	—
147	D₄×C₆₁	—	4	2	no	—
148	D₄×C₆₁	—	4	2	no	—
149	D₄×C₆₁	—	4	2	no	—
150	D₄×C₆₁	—	8	2	no	—
151	D₄×C₆₁	—	8	2	no	—
152	C₂×C₄×C₆₁	—	2	2	no	—
153	C₂×C₄×C₆₁	—	2	2	no	—
154	C₂×C₄×C₆₁	—	2	2	no	—
155	C₂×C₄×C₆₁	—	2	4	no	—
156	C₂×C₄×C₆₁	—	2	2	no	—
157	C₂×C₄×C₆₁	—	2	4	no	—
158	C₂×C₄×C₆₁	—	4	4	no	—
159	C₂×C₄×C₆₁	—	4	1	no	—
160	C₂×C₄×C₆₁	—	8	4	no	—
161	C₂×C₄×C₆₁	—	8	2	no	—
162	C₂×C₄×C₆₁	#120	1	2	yes	—
163	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
164	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
166	C₈×C₆₁	—	2	2	no	—
167	C₈×C₆₁	—	2	2	no	—
168	C₈×C₆₁	—	4	4	no	—
169	#18688	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂²×C₆₁	—	2	2	no	—
171	C₂²×C₆₁	—	2	1	no	—
172	C₂²×C₆₁	—	4	4	no	—
173	C₂²×C₆₁	—	4	2	no	—
174	C₂²×C₆₁	—	8	2	no	—
175	C₂²×C₆₁	—	8	1	no	—
176	C₂²×C₆₁	C₂×#120	1	1	yes	characteristic
177	C₄×C₆₁	—	2	2	no	—
178	C₄×C₆₁	—	2	1	no	—
179	C₄×C₆₁	—	4	4	no	—
180	C₄×C₆₁	—	4	2	no	—
181	C₄×C₆₁	—	4	4	no	—
182	C₄×C₆₁	—	8	1	no	—
183	C₄×C₆₁	#453	1	2	yes	—
184	C₄×C₆₁	#577	1	2	yes	—
185	C₄×C₆₁	#578	1	2	yes	—
186	#3550	C₂×C₆₁	1	1	yes	characteristic, complemented
187	#3254	C₂×C₆₁	1	2	yes	complemented
188	#3147	C₂×C₆₁	1	1	yes	characteristic
189	#1633	C₂×C₆₁	1	2	yes	complemented
190	#1630	C₂×C₆₁	1	2	yes	complemented
191	#1623	C₂×C₆₁	1	2	yes	complemented
192	#1620	C₂×C₆₁	1	2	yes	complemented
193	#1560	C₂×C₆₁	1	1	yes	characteristic, complemented
194	#1493	C₂×C₆₁	1	2	yes	complemented
195	C₂×C₆₁	—	4	1	no	—
196	C₂×C₆₁	—	8	2	no	—
197	C₂×C₆₁	—	16	1	no	—
198	C₂×C₆₁	#3013	1	2	yes	—
199	C₂×C₆₁	C₂×#453	1	1	yes	characteristic
200	C₂×#135	—	2	2	no	—
201	#563	—	2	2	no	—
202	#559	C₂²×C₆₁	1	2	yes	—
203	Q₈²	C₂²×C₆₁	1	2	yes	—
204	#556	—	2	2	no	—
205	D₄×Q₈	—	2	2	no	—
206	#542	—	2	4	no	—
207	#536	—	2	2	no	—
208	#533	C₂²×C₆₁	1	2	yes	—
209	#532	C₂²×C₆₁	1	1	yes	characteristic
210	#532	C₂²×C₆₁	1	1	yes	characteristic
211	#495	—	2	2	no	—
212	#494	C₂²×C₆₁	1	1	yes	characteristic
213	#493	C₂²×C₆₁	1	1	yes	characteristic
214	#492	C₂²×C₆₁	1	1	yes	characteristic
215	#492	C₂²×C₆₁	1	1	yes	characteristic
216	#477	C₂²×C₆₁	1	4	yes	—
217	#474	—	2	2	no	—
218	#468	—	2	4	no	—
219	#461	—	2	2	no	—
220	#460	C₂²×C₆₁	1	4	yes	—
221	#423	C₂²×C₆₁	1	2	yes	—
222	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
223	#333	C₂²×C₆₁	1	2	yes	—
224	#331	C₂²×C₆₁	1	2	yes	—
225	Q₈⋊C₈	C₂²×C₆₁	1	4	yes	—
226	Q₈⋊C₈	C₂²×C₆₁	1	4	yes	—
227	C₆₁	#18688	1	1	yes	characteristic, direct product factor, Sylow
228	C₂×#41	—	2	1	no	—
229	C₂×#41	D₄×C₆₁	1	2	yes	—
230	C₂²×Q₈	—	2	2	no	—
231	#135	—	4	8	no	—
232	C₂×Q₁₆	—	2	2	no	—
233	C₂×SD₁₆	—	2	2	no	—
234	C₂×SD₁₆	—	2	2	no	—
235	C₂×SD₁₆	—	2	4	no	—
236	C₂×D₈	—	2	2	no	—
237	C₂×M₁₆	—	2	2	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₄⋊Q₈	—	2	4	no	—
240	C₄⋊Q₈	—	2	4	no	—
241	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
242	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
243	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
244	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
245	#127	C₂³×C₆₁	1	1	yes	characteristic
246	#127	C₂³×C₆₁	1	1	yes	characteristic
247	#126	—	2	8	no	—
248	#125	—	2	2	no	—
249	#124	—	2	4	no	—
250	#124	—	2	4	no	—
251	#124	—	2	1	no	—
252	#124	D₄×C₆₁	1	2	yes	—
253	#123	—	2	4	no	—
254	#123	—	2	4	no	—
255	C₂²⋊Q₈	—	2	4	no	—
256	C₂²⋊Q₈	—	2	8	no	—
257	C₂²⋊Q₈	—	2	2	no	—
258	C₂²⋊Q₈	D₄×C₆₁	1	4	yes	—
259	#121	—	2	2	no	—
260	#121	—	2	2	no	—
261	C₄×Q₈	—	2	4	no	—
262	C₄×Q₈	—	2	2	no	—
263	C₄×Q₈	C₂³×C₆₁	1	4	yes	—
264	C₄×D₄	—	2	4	no	—
265	C₄×D₄	—	2	2	no	—
266	C₄×D₄	—	2	2	no	—
267	#117	—	2	2	no	—
268	#117	—	2	2	no	—
269	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
270	#107	—	2	4	no	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
273	#103	—	2	4	no	—
274	#102	—	2	4	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	#41	—	4	4	no	—
279	#41	—	4	4	no	—
280	C₂×Q₈	—	2	4	no	—
281	C₂×Q₈	—	2	2	no	—
282	C₂×Q₈	—	2	1	no	—
283	C₂×Q₈	—	8	4	no	—
284	C₂×Q₈	C₂×D₄×C₆₁	1	4	yes	—
285	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	4	1	no	—
291	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	4	1	no	—
295	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
296	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
297	Q₁₆	—	4	4	no	—
298	SD₁₆	—	4	4	no	—
299	SD₁₆	—	4	4	no	—
300	SD₁₆	—	8	4	no	—
301	D₈	—	4	4	no	—
302	M₁₆	—	8	2	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	4	no	—
307	C₄⋊C₄	—	2	8	no	—
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	4	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	4	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	C₂×D₄×C₆₁	1	4	yes	—
316	C₂²⋊C₄	—	2	4	no	—
317	C₂²⋊C₄	—	2	8	no	—
318	C₂²⋊C₄	—	2	4	no	—
319	C₂²⋊C₄	—	2	4	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₄²	—	2	4	no	—
322	C₄²	—	2	4	no	—
323	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
324	C₄²	C₂×D₄×C₆₁	1	2	yes	—
325	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
326	C₂³	—	2	2	no	—
327	C₂³	—	4	1	no	—
328	C₂³	#120×C₆₁	1	1	yes	characteristic
329	Q₈	—	2	4	no	—
330	Q₈	—	2	4	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	4	4	no	—
334	Q₈	—	8	4	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	2	no	—
339	D₄	—	4	2	no	—
340	D₄	—	4	2	no	—
341	D₄	—	8	2	no	—
342	D₄	—	8	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	4	4	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	8	4	no	—
352	C₂×C₄	—	8	2	no	—
353	C₂×C₄	#120×C₆₁	1	2	yes	—
354	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	4	4	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	1	no	—
366	C₂²	C₂×#120×C₆₁	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	4	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	4	no	—
372	C₄	—	8	1	no	—
373	C₄	C₆₁×#453	1	2	yes	—
374	C₄	C₆₁×#577	1	2	yes	—
375	C₄	C₆₁×#578	1	2	yes	—
376	C₂	—	4	1	no	—
377	C₂	—	8	2	no	—
378	C₂	—	16	1	no	—
379	C₂	C₆₁×#3013	1	2	yes	—
380	C₂	C₂×C₆₁×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#3147	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
1. C₆₁×#3550	C₂	semidirect
2. C₆₁×#3254	C₂	semidirect
4. C₆₁×#1633	C₂	semidirect
5. C₆₁×#1630	C₂	semidirect
6. C₆₁×#1623	C₂	semidirect
7. C₆₁×#1620	C₂	semidirect
8. C₆₁×#1560	C₂	semidirect
9. C₆₁×#1493	C₂	semidirect
169. #18688	C₆₁	direct (N × H)
186. #3550	C₂×C₆₁	semidirect
187. #3254	C₂×C₆₁	semidirect
189. #1633	C₂×C₆₁	semidirect
190. #1630	C₂×C₆₁	semidirect
191. #1623	C₂×C₆₁	semidirect
192. #1620	C₂×C₆₁	semidirect
193. #1560	C₂×C₆₁	semidirect
194. #1493	C₂×C₆₁	semidirect
227. C₆₁	#18688	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×#3550	2
C₆₁×#3254	2
C₆₁×#3147	2
C₆₁×#1633	2
C₆₁×#1630	2
C₆₁×#1623	2
C₆₁×#1620	2
C₆₁×#1560	2
C₆₁×#1493	2
#18688	61

Subgroups — C61×#18688 #29522121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#18688 #29522121