Subgroups — C₆₁×#29788 #29533221

Order 15616.

752 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3094	C₂	1	2	yes	maximal, complemented
2	C₂×C₆₁×#466	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2089	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2063	C₂	1	4	yes	maximal, complemented
6	C₆₁×#2045	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1900	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1810	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×#121×C₆₁	C₂²	1	2	yes	—
13	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#482	—	2	4	no	—
15	C₆₁×#481	—	2	2	no	—
16	C₆₁×#480	—	2	2	no	—
17	C₆₁×#466	—	2	2	no	—
18	C₆₁×#466	—	2	2	no	—
19	C₂×#108×C₆₁	C₂²	1	2	yes	—
20	C₆₁×#428	C₂²	1	4	yes	—
21	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
22	C₆₁×#424	—	2	2	no	—
23	C₆₁×#424	C₂²	1	2	yes	—
24	C₆₁×#424	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₁	—	2	2	no	—
26	C₆₁×#418	C₂²	1	4	yes	—
27	C₂×#102×C₆₁	C₂²	1	2	yes	—
28	C₂×#100×C₆₁	C₂²	1	4	yes	—
29	C₆₁×#407	C₂²	1	2	yes	—
30	C₆₁×#343	C₂²	1	4	yes	—
31	C₆₁×#341	C₂²	1	2	yes	—
32	C₂²×D₄×C₆₁	—	2	2	no	—
33	C₂²×D₄×C₆₁	C₂³	1	2	yes	—
34	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
35	C₂×D₈×C₆₁	—	2	2	no	—
36	C₂×D₈×C₆₁	—	4	2	no	—
37	C₂×M₁₆×C₆₁	—	2	1	no	—
38	C₂×M₁₆×C₆₁	—	4	2	no	—
39	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
40	C₂×M₁₆×C₆₁	C₂³	1	4	yes	—
41	C₂²×C₈×C₆₁	—	2	1	no	—
42	C₂²×C₈×C₆₁	—	4	2	no	—
43	C₂²×C₈×C₆₁	D₄	1	2	yes	—
44	C₂²×C₈×C₆₁	D₄	1	2	yes	—
45	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
47	#121×C₆₁	—	2	2	no	—
48	#121×C₆₁	—	2	2	no	—
49	#121×C₆₁	—	2	2	no	—
50	#121×C₆₁	—	2	2	no	—
51	#117×C₆₁	—	2	1	no	—
52	#117×C₆₁	—	2	1	no	—
53	#117×C₆₁	D₄	1	2	yes	—
54	#117×C₆₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
59	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
60	C₂×C₄²×C₆₁	—	2	1	no	—
61	#108×C₆₁	—	4	2	no	—
62	#107×C₆₁	—	2	2	no	—
63	#107×C₆₁	—	2	4	no	—
64	#106×C₆₁	—	2	4	no	—
65	C₄⋊C₈×C₆₁	—	2	4	no	—
66	C₄⋊C₈×C₆₁	—	2	4	no	—
67	#102×C₆₁	—	2	4	no	—
68	#102×C₆₁	—	2	4	no	—
69	#102×C₆₁	—	2	4	no	—
70	#100×C₆₁	—	4	4	no	—
71	C₂²⋊C₈×C₆₁	—	2	2	no	—
72	C₂²⋊C₈×C₆₁	—	2	2	no	—
73	C₂⁴×C₆₁	—	2	2	no	—
74	C₂⁴×C₆₁	#41	1	1	yes	characteristic
75	C₂×D₄×C₆₁	—	2	2	no	—
76	C₂×D₄×C₆₁	—	2	2	no	—
77	C₂×D₄×C₆₁	—	2	4	no	—
78	C₂×D₄×C₆₁	—	2	4	no	—
79	C₂×D₄×C₆₁	—	4	2	no	—
80	C₂×D₄×C₆₁	—	4	2	no	—
81	C₂×D₄×C₆₁	—	8	4	no	—
82	C₂²×C₄×C₆₁	—	2	2	no	—
83	C₂²×C₄×C₆₁	—	2	2	no	—
84	C₂²×C₄×C₆₁	—	2	1	no	—
85	C₂²×C₄×C₆₁	—	2	1	no	—
86	C₂²×C₄×C₆₁	—	2	2	no	—
87	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
92	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
93	D₈×C₆₁	—	4	4	no	—
94	M₁₆×C₆₁	—	2	4	no	—
95	M₁₆×C₆₁	—	2	4	no	—
96	M₁₆×C₆₁	—	2	4	no	—
97	M₁₆×C₆₁	—	4	2	no	—
98	C₂×C₈×C₆₁	—	2	2	no	—
99	C₂×C₈×C₆₁	—	2	2	no	—
100	C₂×C₈×C₆₁	—	2	4	no	—
101	C₂×C₈×C₆₁	—	2	2	no	—
102	C₂×C₈×C₆₁	—	4	2	no	—
103	C₂×C₈×C₆₁	—	4	2	no	—
104	C₂×C₈×C₆₁	—	4	2	no	—
105	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
107	C₄⋊C₄×C₆₁	—	2	1	no	—
108	C₄⋊C₄×C₆₁	—	2	1	no	—
109	C₄⋊C₄×C₆₁	—	2	2	no	—
110	C₄⋊C₄×C₆₁	—	2	2	no	—
111	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
113	C₂²⋊C₄×C₆₁	—	2	2	no	—
114	C₂²⋊C₄×C₆₁	—	2	2	no	—
115	C₂²⋊C₄×C₆₁	—	4	2	no	—
116	C₂²⋊C₄×C₆₁	—	4	2	no	—
117	C₄²×C₆₁	—	2	2	no	—
118	C₄²×C₆₁	—	2	2	no	—
119	C₂³×C₆₁	—	2	2	no	—
120	C₂³×C₆₁	—	2	2	no	—
121	C₂³×C₆₁	—	4	1	no	—
122	C₂³×C₆₁	—	4	2	no	—
123	C₂³×C₆₁	—	4	4	no	—
124	C₂³×C₆₁	—	8	2	no	—
125	C₂³×C₆₁	—	8	2	no	—
126	C₂³×C₆₁	#121	1	1	yes	characteristic
127	C₂³×C₆₁	#121	1	1	yes	characteristic
128	C₂³×C₆₁	C₂×#41	1	1	yes	characteristic
129	D₄×C₆₁	—	4	2	no	—
130	D₄×C₆₁	—	4	2	no	—
131	D₄×C₆₁	—	4	4	no	—
132	D₄×C₆₁	—	8	4	no	—
133	D₄×C₆₁	—	8	4	no	—
134	C₂×C₄×C₆₁	—	2	2	no	—
135	C₂×C₄×C₆₁	—	2	1	no	—
136	C₂×C₄×C₆₁	—	2	1	no	—
137	C₂×C₄×C₆₁	—	2	1	no	—
138	C₂×C₄×C₆₁	—	2	1	no	—
139	C₂×C₄×C₆₁	—	2	2	no	—
140	C₂×C₄×C₆₁	—	2	2	no	—
141	C₂×C₄×C₆₁	—	2	2	no	—
142	C₂×C₄×C₆₁	—	2	2	no	—
143	C₂×C₄×C₆₁	—	4	2	no	—
144	C₂×C₄×C₆₁	—	4	2	no	—
145	C₂×C₄×C₆₁	—	4	2	no	—
146	C₂×C₄×C₆₁	—	8	2	no	—
147	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
148	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
149	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
152	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
153	C₈×C₆₁	—	2	2	no	—
154	C₈×C₆₁	—	2	2	no	—
155	C₈×C₆₁	—	4	2	no	—
156	C₈×C₆₁	—	4	4	no	—
157	#29788	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂²×C₆₁	—	2	1	no	—
159	C₂²×C₆₁	—	2	1	no	—
160	C₂²×C₆₁	—	2	1	no	—
161	C₂²×C₆₁	—	2	1	no	—
162	C₂²×C₆₁	—	2	1	no	—
163	C₂²×C₆₁	—	2	1	no	—
164	C₂²×C₆₁	—	2	1	no	—
165	C₂²×C₆₁	—	4	4	no	—
166	C₂²×C₆₁	—	8	2	no	—
167	C₂²×C₆₁	—	8	2	no	—
168	C₂²×C₆₁	—	8	2	no	—
169	C₂²×C₆₁	—	16	4	no	—
170	C₂²×C₆₁	#472	1	2	yes	—
171	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
172	C₂²×C₆₁	#539	1	2	yes	—
173	C₄×C₆₁	—	2	1	no	—
174	C₄×C₆₁	—	2	1	no	—
175	C₄×C₆₁	—	4	2	no	—
176	C₄×C₆₁	—	4	2	no	—
177	C₄×C₆₁	—	8	2	no	—
178	C₄×C₆₁	#534	1	1	yes	characteristic
179	C₄×C₆₁	#536	1	1	yes	characteristic
180	C₄×C₆₁	#537	1	2	yes	—
181	#3094	C₂×C₆₁	1	2	yes	complemented
182	C₂×#466	C₂×C₆₁	1	1	yes	characteristic
183	C₂×#424	C₂×C₆₁	1	1	yes	characteristic, complemented
184	#2089	C₂×C₆₁	1	2	yes	complemented
185	#2063	C₂×C₆₁	1	4	yes	complemented
186	#2045	C₂×C₆₁	1	2	yes	complemented
187	#1900	C₂×C₆₁	1	2	yes	complemented
188	#1810	C₂×C₆₁	1	1	yes	characteristic, complemented
189	C₂×C₆₁	—	2	2	no	—
190	C₂×C₆₁	—	2	2	no	—
191	C₂×C₆₁	—	4	1	no	—
192	C₂×C₆₁	—	16	2	no	—
193	C₂×C₆₁	#2618	1	1	yes	characteristic
194	C₂×C₆₁	C₂×#472	1	1	yes	characteristic
195	C₂×C₆₁	#3067	1	1	yes	characteristic
196	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
197	C₂²×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
198	C₂³×C₈	C₂²×C₆₁	1	1	yes	characteristic
199	C₂×#121	C₂²×C₆₁	1	2	yes	—
200	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
201	#482	—	2	4	no	—
202	#481	—	2	2	no	—
203	#480	—	2	2	no	—
204	#466	—	2	2	no	—
205	#466	—	2	2	no	—
206	C₂×#108	C₂²×C₆₁	1	2	yes	—
207	#428	C₂²×C₆₁	1	4	yes	—
208	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
209	#424	—	2	2	no	—
210	#424	C₂²×C₆₁	1	2	yes	—
211	#424	C₂²×C₆₁	1	2	yes	—
212	C₂×C₄⋊C₈	—	2	2	no	—
213	#418	C₂²×C₆₁	1	4	yes	—
214	C₂×#102	C₂²×C₆₁	1	2	yes	—
215	C₂×#100	C₂²×C₆₁	1	4	yes	—
216	#407	C₂²×C₆₁	1	2	yes	—
217	#343	C₂²×C₆₁	1	4	yes	—
218	#341	C₂²×C₆₁	1	2	yes	—
219	C₆₁	#29788	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²×D₄	—	2	2	no	—
221	C₂²×D₄	C₂³×C₆₁	1	2	yes	—
222	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
223	C₂×D₈	—	2	2	no	—
224	C₂×D₈	—	4	2	no	—
225	C₂×M₁₆	—	2	1	no	—
226	C₂×M₁₆	—	4	2	no	—
227	C₂×M₁₆	C₂³×C₆₁	1	2	yes	—
228	C₂×M₁₆	C₂³×C₆₁	1	4	yes	—
229	C₂²×C₈	—	2	1	no	—
230	C₂²×C₈	—	4	2	no	—
231	C₂²×C₈	D₄×C₆₁	1	2	yes	—
232	C₂²×C₈	D₄×C₆₁	1	2	yes	—
233	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
234	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
235	#121	—	2	2	no	—
236	#121	—	2	2	no	—
237	#121	—	2	2	no	—
238	#121	—	2	2	no	—
239	#117	—	2	1	no	—
240	#117	—	2	1	no	—
241	#117	D₄×C₆₁	1	2	yes	—
242	#117	C₂³×C₆₁	1	2	yes	—
243	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
244	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
246	C₂×C₂²⋊C₄	—	2	1	no	—
247	C₂×C₂²⋊C₄	—	2	2	no	—
248	C₂×C₄²	—	2	1	no	—
249	#108	—	4	2	no	—
250	#107	—	2	2	no	—
251	#107	—	2	4	no	—
252	#106	—	2	4	no	—
253	C₄⋊C₈	—	2	4	no	—
254	C₄⋊C₈	—	2	4	no	—
255	#102	—	2	4	no	—
256	#102	—	2	4	no	—
257	#102	—	2	4	no	—
258	#100	—	4	4	no	—
259	C₂²⋊C₈	—	2	2	no	—
260	C₂²⋊C₈	—	2	2	no	—
261	C₂⁴	—	2	2	no	—
262	C₂⁴	#41×C₆₁	1	1	yes	characteristic
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	4	no	—
266	C₂×D₄	—	2	4	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	—	8	4	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
279	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
280	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
281	D₈	—	4	4	no	—
282	M₁₆	—	2	4	no	—
283	M₁₆	—	2	4	no	—
284	M₁₆	—	2	4	no	—
285	M₁₆	—	4	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	—	4	2	no	—
292	C₂×C₈	—	4	2	no	—
293	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
294	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
300	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	2	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	—	4	1	no	—
310	C₂³	—	4	2	no	—
311	C₂³	—	4	4	no	—
312	C₂³	—	8	2	no	—
313	C₂³	—	8	2	no	—
314	C₂³	#121×C₆₁	1	1	yes	characteristic
315	C₂³	#121×C₆₁	1	1	yes	characteristic
316	C₂³	C₂×#41×C₆₁	1	1	yes	characteristic
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	4	no	—
320	D₄	—	8	4	no	—
321	D₄	—	8	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
336	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
337	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
338	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
339	C₂×C₄	H(5,𝔽₂)×C₆₁	1	2	yes	—
340	C₂×C₄	2₋¹⁺⁴×C₆₁	1	2	yes	—
341	C₈	—	2	2	no	—
342	C₈	—	2	2	no	—
343	C₈	—	4	2	no	—
344	C₈	—	4	4	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	4	4	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	16	4	no	—
357	C₂²	C₆₁×#472	1	2	yes	—
358	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
359	C₂²	C₆₁×#539	1	2	yes	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	8	2	no	—
365	C₄	C₆₁×#534	1	1	yes	characteristic
366	C₄	C₆₁×#536	1	1	yes	characteristic
367	C₄	C₆₁×#537	1	2	yes	—
368	C₂	—	2	2	no	—
369	C₂	—	2	2	no	—
370	C₂	—	4	1	no	—
371	C₂	—	16	2	no	—
372	C₂	C₆₁×#2618	1	1	yes	characteristic
373	C₂	C₂×C₆₁×#472	1	1	yes	characteristic
374	C₂	C₆₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂×C₆₁×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂³×C₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3094	C₂	semidirect
3. C₂×C₆₁×#424	C₂	semidirect
4. C₆₁×#2089	C₂	semidirect
5. C₆₁×#2063	C₂	semidirect
6. C₆₁×#2045	C₂	semidirect
7. C₆₁×#1900	C₂	semidirect
8. C₆₁×#1810	C₂	semidirect
157. #29788	C₆₁	direct (N × H)
181. #3094	C₂×C₆₁	semidirect
183. C₂×#424	C₂×C₆₁	semidirect
184. #2089	C₂×C₆₁	semidirect
185. #2063	C₂×C₆₁	semidirect
186. #2045	C₂×C₆₁	semidirect
187. #1900	C₂×C₆₁	semidirect
188. #1810	C₂×C₆₁	semidirect
219. C₆₁	#29788	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×#3094	2
C₂×C₆₁×#466	2
C₂×C₆₁×#424	2
C₆₁×#2089	2
C₆₁×#2063	2
C₆₁×#2045	2
C₆₁×#1900	2
C₆₁×#1810	2
#29788	61

Subgroups — C61×#29788 #29533221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29788 #29533221