Subgroups — C₆₁×#10152 #29513585

Order 15616.

642 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	C₆₁×#3205	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#1492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₆₁	—	2	2	no	—
9	C₆₁×#532	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×#127×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#493	—	2	1	no	—
13	C₆₁×#492	—	2	2	no	—
14	C₆₁×#492	—	2	2	no	—
15	C₆₁×#489	—	2	2	no	—
16	C₆₁×#487	—	2	1	no	—
17	C₆₁×#486	—	2	1	no	—
18	C₆₁×#486	—	2	1	no	—
19	C₆₁×#459	—	2	2	no	—
20	C₆₁×#447	—	2	2	no	—
21	C₆₁×#431	C₂²	1	1	yes	characteristic
22	C₆₁×#423	C₂²	1	1	yes	characteristic
23	C₆₁×#417	—	2	2	no	—
24	C₂×#100×C₆₁	—	2	2	no	—
25	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#332	—	2	2	no	—
28	C₆₁×D₄⋊C₈	—	2	2	no	—
29	C₂×#41×C₆₁	—	2	1	no	—
30	C₂²×Q₈×C₆₁	—	2	1	no	—
31	C₂²×D₄×C₆₁	—	2	1	no	—
32	C₂²×D₄×C₆₁	—	2	1	no	—
33	C₂²×D₄×C₆₁	D₄	1	2	yes	—
34	C₂×SD₁₆×C₆₁	—	2	8	no	—
35	C₂×SD₁₆×C₆₁	—	4	8	no	—
36	C₂×D₈×C₆₁	—	2	4	no	—
37	C₂×M₁₆×C₆₁	—	2	1	no	—
38	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
39	C₂²×C₈×C₆₁	—	2	1	no	—
40	C₄⋊Q₈×C₆₁	—	2	1	no	—
41	C₄⋊Q₈×C₆₁	—	2	1	no	—
42	C₄⋊Q₈×C₆₁	—	2	1	no	—
43	C₄⋊Q₈×C₆₁	D₄	1	2	yes	—
44	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
45	#127×C₆₁	—	2	1	no	—
46	#127×C₆₁	—	2	1	no	—
47	#127×C₆₁	—	2	1	no	—
48	#127×C₆₁	—	2	1	no	—
49	#127×C₆₁	D₄	1	1	yes	characteristic
50	#124×C₆₁	—	2	1	no	—
51	#121×C₆₁	—	2	1	no	—
52	#121×C₆₁	—	2	1	no	—
53	C₄×D₄×C₆₁	—	2	1	no	—
54	C₄×D₄×C₆₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
57	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₆₁	—	2	1	no	—
59	C₄⋊C₈×C₆₁	—	2	1	no	—
60	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
61	#103×C₆₁	—	2	4	no	—
62	#102×C₆₁	—	2	4	no	—
63	#102×C₆₁	—	2	4	no	—
64	#100×C₆₁	—	4	4	no	—
65	C₂²⋊C₈×C₆₁	—	2	1	no	—
66	#97×C₆₁	—	2	1	no	—
67	C₄×C₈×C₆₁	—	2	1	no	—
68	C₄×C₈×C₆₁	—	2	1	no	—
69	C₄×C₈×C₆₁	—	2	1	no	—
70	C₄×C₈×C₆₁	D₄	1	2	yes	—
71	C₂⁴×C₆₁	—	2	2	no	—
72	#41×C₆₁	—	4	2	no	—
73	#41×C₆₁	—	4	2	no	—
74	C₂×Q₈×C₆₁	—	2	1	no	—
75	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂×Q₈×C₆₁	—	4	4	no	—
77	C₂×Q₈×C₆₁	—	4	1	no	—
78	C₂×D₄×C₆₁	—	2	1	no	—
79	C₂×D₄×C₆₁	—	2	2	no	—
80	C₂×D₄×C₆₁	—	2	1	no	—
81	C₂×D₄×C₆₁	—	2	2	no	—
82	C₂×D₄×C₆₁	—	2	1	no	—
83	C₂×D₄×C₆₁	—	4	4	no	—
84	C₂×D₄×C₆₁	—	4	1	no	—
85	C₂×D₄×C₆₁	—	4	1	no	—
86	C₂×D₄×C₆₁	—	4	1	no	—
87	C₂×D₄×C₆₁	—	4	1	no	—
88	C₂×D₄×C₆₁	—	4	1	no	—
89	C₂×D₄×C₆₁	—	4	2	no	—
90	C₂×D₄×C₆₁	—	4	2	no	—
91	C₂×D₄×C₆₁	—	8	2	no	—
92	C₂²×C₄×C₆₁	—	4	1	no	—
93	C₂²×C₄×C₆₁	—	4	1	no	—
94	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
97	SD₁₆×C₆₁	—	4	16	no	—
98	D₈×C₆₁	—	8	4	no	—
99	M₁₆×C₆₁	—	2	2	no	—
100	M₁₆×C₆₁	—	2	2	no	—
101	M₁₆×C₆₁	—	4	2	no	—
102	C₂×C₈×C₆₁	—	2	4	no	—
103	C₂×C₈×C₆₁	—	2	2	no	—
104	C₂×C₈×C₆₁	—	2	2	no	—
105	C₂×C₈×C₆₁	—	2	2	no	—
106	C₂×C₈×C₆₁	—	4	1	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	1	no	—
111	C₄⋊C₄×C₆₁	—	4	1	no	—
112	C₄⋊C₄×C₆₁	—	4	1	no	—
113	C₂²⋊C₄×C₆₁	—	4	1	no	—
114	C₂²⋊C₄×C₆₁	—	4	1	no	—
115	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
119	C₂³×C₆₁	—	4	1	no	—
120	C₂³×C₆₁	—	4	2	no	—
121	C₂³×C₆₁	—	4	4	no	—
122	C₂³×C₆₁	—	8	2	no	—
123	C₂³×C₆₁	—	8	2	no	—
124	C₂³×C₆₁	#121	1	1	yes	characteristic
125	Q₈×C₆₁	—	4	4	no	—
126	Q₈×C₆₁	—	4	2	no	—
127	Q₈×C₆₁	—	8	2	no	—
128	D₄×C₆₁	—	4	4	no	—
129	D₄×C₆₁	—	4	2	no	—
130	D₄×C₆₁	—	4	2	no	—
131	D₄×C₆₁	—	4	4	no	—
132	D₄×C₆₁	—	4	2	no	—
133	D₄×C₆₁	—	8	2	no	—
134	D₄×C₆₁	—	8	2	no	—
135	D₄×C₆₁	—	8	2	no	—
136	D₄×C₆₁	—	8	2	no	—
137	D₄×C₆₁	—	8	1	no	—
138	D₄×C₆₁	—	8	1	no	—
139	C₂×C₄×C₆₁	—	2	2	no	—
140	C₂×C₄×C₆₁	—	2	2	no	—
141	C₂×C₄×C₆₁	—	4	1	no	—
142	C₂×C₄×C₆₁	—	4	2	no	—
143	C₂×C₄×C₆₁	—	8	2	no	—
144	C₂×C₄×C₆₁	—	8	1	no	—
145	C₂×C₄×C₆₁	—	8	1	no	—
146	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
147	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
148	C₂×C₄×C₆₁	#120	1	1	yes	characteristic
149	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
150	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
151	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
152	C₂×C₄×C₆₁	#135	1	2	yes	—
153	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
154	C₈×C₆₁	—	2	4	no	—
155	C₈×C₆₁	—	4	2	no	—
156	C₈×C₆₁	—	4	2	no	—
157	C₈×C₆₁	—	8	1	no	—
158	#10152	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂²×C₆₁	—	2	1	no	—
160	C₂²×C₆₁	—	2	1	no	—
161	C₂²×C₆₁	—	8	2	no	—
162	C₂²×C₆₁	—	8	1	no	—
163	C₂²×C₆₁	—	8	1	no	—
164	C₂²×C₆₁	—	8	2	no	—
165	C₂²×C₆₁	—	8	1	no	—
166	C₂²×C₆₁	—	8	2	no	—
167	C₂²×C₆₁	—	16	4	no	—
168	C₂²×C₆₁	#392	1	1	yes	characteristic
169	C₂²×C₆₁	#463	1	2	yes	—
170	C₄×C₆₁	—	2	1	no	—
171	C₄×C₆₁	—	2	1	no	—
172	C₄×C₆₁	—	8	2	no	—
173	C₄×C₆₁	—	8	1	no	—
174	C₄×C₆₁	#449	1	2	yes	—
175	C₄×C₆₁	#453	1	2	yes	—
176	C₄×C₆₁	#471	1	2	yes	—
177	C₄×C₆₁	#495	1	1	yes	characteristic
178	C₄×C₆₁	#496	1	1	yes	characteristic
179	#3205	C₂×C₆₁	1	1	yes	characteristic
180	C₂×#492	C₂×C₆₁	1	1	yes	characteristic, complemented
181	#3141	C₂×C₆₁	1	1	yes	characteristic, complemented
182	#1965	C₂×C₆₁	1	1	yes	characteristic, complemented
183	#1549	C₂×C₆₁	1	1	yes	characteristic, complemented
184	#1492	C₂×C₆₁	1	1	yes	characteristic, complemented
185	#1278	C₂×C₆₁	1	1	yes	characteristic, complemented
186	C₂×C₆₁	—	2	2	no	—
187	C₂×C₆₁	—	16	1	no	—
188	C₂×C₆₁	—	16	2	no	—
189	C₂×C₆₁	#2002	1	1	yes	characteristic
190	C₂×C₆₁	#2012	1	1	yes	characteristic
191	C₂×C₆₁	#2019	1	1	yes	characteristic
192	C₂²×SD₁₆	—	2	2	no	—
193	#532	C₂²×C₆₁	1	1	yes	characteristic
194	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
195	C₂×#127	C₂²×C₆₁	1	1	yes	characteristic
196	#493	—	2	1	no	—
197	#492	—	2	2	no	—
198	#492	—	2	2	no	—
199	#489	—	2	2	no	—
200	#487	—	2	1	no	—
201	#486	—	2	1	no	—
202	#486	—	2	1	no	—
203	#459	—	2	2	no	—
204	#447	—	2	2	no	—
205	#431	C₂²×C₆₁	1	1	yes	characteristic
206	#423	C₂²×C₆₁	1	1	yes	characteristic
207	#417	—	2	2	no	—
208	C₂×#100	—	2	2	no	—
209	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
211	#332	—	2	2	no	—
212	D₄⋊C₈	—	2	2	no	—
213	C₆₁	#10152	1	1	yes	characteristic, direct product factor, Sylow
214	C₂×#41	—	2	1	no	—
215	C₂²×Q₈	—	2	1	no	—
216	C₂²×D₄	—	2	1	no	—
217	C₂²×D₄	—	2	1	no	—
218	C₂²×D₄	D₄×C₆₁	1	2	yes	—
219	C₂×SD₁₆	—	2	8	no	—
220	C₂×SD₁₆	—	4	8	no	—
221	C₂×D₈	—	2	4	no	—
222	C₂×M₁₆	—	2	1	no	—
223	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
224	C₂²×C₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	D₄×C₆₁	1	2	yes	—
229	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
230	#127	—	2	1	no	—
231	#127	—	2	1	no	—
232	#127	—	2	1	no	—
233	#127	—	2	1	no	—
234	#127	D₄×C₆₁	1	1	yes	characteristic
235	#124	—	2	1	no	—
236	#121	—	2	1	no	—
237	#121	—	2	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
246	#103	—	2	4	no	—
247	#102	—	2	4	no	—
248	#102	—	2	4	no	—
249	#100	—	4	4	no	—
250	C₂²⋊C₈	—	2	1	no	—
251	#97	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	D₄×C₆₁	1	2	yes	—
256	C₂⁴	—	2	2	no	—
257	#41	—	4	2	no	—
258	#41	—	4	2	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	4	4	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	4	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂×D₄	—	4	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	8	2	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	4	1	no	—
279	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
281	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
282	SD₁₆	—	4	16	no	—
283	D₈	—	8	4	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	2	2	no	—
286	M₁₆	—	4	2	no	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	4	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
301	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
302	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
303	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
304	C₂³	—	4	1	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	4	4	no	—
307	C₂³	—	8	2	no	—
308	C₂³	—	8	2	no	—
309	C₂³	#121×C₆₁	1	1	yes	characteristic
310	Q₈	—	4	4	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	2	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	4	no	—
317	D₄	—	4	2	no	—
318	D₄	—	8	2	no	—
319	D₄	—	8	2	no	—
320	D₄	—	8	2	no	—
321	D₄	—	8	2	no	—
322	D₄	—	8	1	no	—
323	D₄	—	8	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	2	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	—	8	1	no	—
331	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
332	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
333	C₂×C₄	#120×C₆₁	1	1	yes	characteristic
334	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
335	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
336	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
337	C₂×C₄	#135×C₆₁	1	2	yes	—
338	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
339	C₈	—	2	4	no	—
340	C₈	—	4	2	no	—
341	C₈	—	4	2	no	—
342	C₈	—	8	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	8	1	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	1	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	1	no	—
351	C₂²	—	16	4	no	—
352	C₂²	C₆₁×#392	1	1	yes	characteristic
353	C₂²	C₆₁×#463	1	2	yes	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	8	1	no	—
357	C₄	—	8	2	no	—
358	C₄	C₆₁×#449	1	2	yes	—
359	C₄	C₆₁×#453	1	2	yes	—
360	C₄	C₆₁×#471	1	2	yes	—
361	C₄	C₆₁×#495	1	1	yes	characteristic
362	C₄	C₆₁×#496	1	1	yes	characteristic
363	C₂	—	2	2	no	—
364	C₂	—	16	2	no	—
365	C₂	—	16	1	no	—
366	C₂	C₆₁×#2002	1	1	yes	characteristic
367	C₂	C₆₁×#2012	1	1	yes	characteristic
368	C₂	C₆₁×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₆₁×#3205	C₂
LCS	G ▷ C₂×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. C₂×C₆₁×#492	C₂	semidirect
3. C₆₁×#3141	C₂	semidirect
4. C₆₁×#1965	C₂	semidirect
5. C₆₁×#1549	C₂	semidirect
6. C₆₁×#1492	C₂	semidirect
7. C₆₁×#1278	C₂	semidirect
158. #10152	C₆₁	direct (N × H)
180. C₂×#492	C₂×C₆₁	semidirect
181. #3141	C₂×C₆₁	semidirect
182. #1965	C₂×C₆₁	semidirect
183. #1549	C₂×C₆₁	semidirect
184. #1492	C₂×C₆₁	semidirect
185. #1278	C₂×C₆₁	semidirect
213. C₆₁	#10152	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×#3205	2
C₂×C₆₁×#492	2
C₆₁×#3141	2
C₆₁×#1965	2
C₆₁×#1549	2
C₆₁×#1492	2
C₆₁×#1278	2
#10152	61

Subgroups — C61×#10152 #29513585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#10152 #29513585