Subgroups — C₆₁×#29427 #29532860

Order 15616.

724 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×#2323	C₂	1	2	yes	maximal
2	C₆₁×#2075	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2036	C₂	1	2	yes	maximal
4	C₆₁×#2001	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1928	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1918	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1878	C₂	1	2	yes	maximal, complemented
8	C₆₁×#1863	C₂	1	2	yes	maximal, complemented
9	C₆₁×#1818	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Q₈×C₆₁	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
13	C₂×#117×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#107×C₆₁	C₂²	1	2	yes	—
15	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×#103×C₆₁	C₂²	1	2	yes	—
17	C₂×#103×C₆₁	C₂²	1	2	yes	—
18	C₂×#103×C₆₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₆₁×#391	C₂²	1	2	yes	—
22	C₆₁×#389	—	2	2	no	—
23	C₆₁×#389	C₂²	1	2	yes	—
24	C₆₁×#388	—	2	2	no	—
25	C₆₁×#388	—	2	2	no	—
26	C₆₁×#387	—	2	2	no	—
27	C₆₁×#380	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
29	C₆₁×#340	C₂²	1	2	yes	—
30	C₆₁×#340	C₂²	1	2	yes	—
31	C₆₁×#340	C₂²	1	1	yes	characteristic
32	C₆₁×#340	C₂²	1	1	yes	characteristic
33	C₆₁×#336	C₂²	1	2	yes	—
34	C₆₁×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
36	C₂³×C₄×C₆₁	C₂³	1	1	yes	characteristic
37	C₂×Q₁₆×C₆₁	—	2	4	no	—
38	C₂×Q₁₆×C₆₁	—	4	1	no	—
39	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
42	C₂²⋊Q₈×C₆₁	—	2	4	no	—
43	C₂²⋊Q₈×C₆₁	—	2	4	no	—
44	C₄×Q₈×C₆₁	—	4	2	no	—
45	C₄×Q₈×C₆₁	—	4	2	no	—
46	#117×C₆₁	—	4	1	no	—
47	#117×C₆₁	—	4	1	no	—
48	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
59	C₂×C₄²×C₆₁	—	2	2	no	—
60	C₂×C₄²×C₆₁	—	2	2	no	—
61	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
62	#107×C₆₁	—	2	4	no	—
63	#107×C₆₁	—	4	1	no	—
64	#103×C₆₁	—	2	4	no	—
65	#103×C₆₁	—	2	4	no	—
66	#103×C₆₁	—	4	2	no	—
67	C₂²⋊C₈×C₆₁	—	4	1	no	—
68	C₂²⋊C₈×C₆₁	—	4	1	no	—
69	#95×C₆₁	—	2	2	no	—
70	#95×C₆₁	—	2	2	no	—
71	#95×C₆₁	—	2	2	no	—
72	#95×C₆₁	—	2	2	no	—
73	#95×C₆₁	—	2	2	no	—
74	C₂⁴×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂×Q₈×C₆₁	—	2	2	no	—
77	C₂×Q₈×C₆₁	—	2	2	no	—
78	C₂×Q₈×C₆₁	—	2	2	no	—
79	C₂×Q₈×C₆₁	—	2	2	no	—
80	C₂×Q₈×C₆₁	#41	1	4	yes	—
81	C₂²×C₄×C₆₁	—	2	2	no	—
82	C₂²×C₄×C₆₁	—	2	2	no	—
83	C₂²×C₄×C₆₁	—	2	2	no	—
84	C₂²×C₄×C₆₁	—	2	2	no	—
85	C₂²×C₄×C₆₁	—	2	1	no	—
86	C₂²×C₄×C₆₁	—	2	1	no	—
87	C₂²×C₄×C₆₁	—	2	2	no	—
88	C₂²×C₄×C₆₁	—	2	2	no	—
89	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
93	Q₁₆×C₆₁	—	4	4	no	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	2	2	no	—
96	C₂×C₈×C₆₁	—	2	2	no	—
97	C₂×C₈×C₆₁	—	2	2	no	—
98	C₂×C₈×C₆₁	—	2	2	no	—
99	C₂×C₈×C₆₁	—	2	1	no	—
100	C₂×C₈×C₆₁	—	2	1	no	—
101	C₄⋊C₄×C₆₁	—	2	8	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	—	2	2	no	—
106	C₄⋊C₄×C₆₁	—	2	2	no	—
107	C₄⋊C₄×C₆₁	—	4	1	no	—
108	C₄⋊C₄×C₆₁	—	4	1	no	—
109	C₄⋊C₄×C₆₁	—	4	4	no	—
110	C₄⋊C₄×C₆₁	—	8	2	no	—
111	C₄⋊C₄×C₆₁	#41	1	4	yes	—
112	C₂²⋊C₄×C₆₁	—	2	8	no	—
113	C₂²⋊C₄×C₆₁	—	4	2	no	—
114	C₂²⋊C₄×C₆₁	—	8	2	no	—
115	C₄²×C₆₁	—	2	2	no	—
116	C₄²×C₆₁	—	2	2	no	—
117	C₄²×C₆₁	—	4	4	no	—
118	C₄²×C₆₁	—	8	2	no	—
119	C₂³×C₆₁	—	2	2	no	—
120	C₂³×C₆₁	—	2	1	no	—
121	C₂³×C₆₁	—	2	1	no	—
122	C₂³×C₆₁	—	2	2	no	—
123	C₂³×C₆₁	—	2	1	no	—
124	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
125	Q₈×C₆₁	—	2	4	no	—
126	Q₈×C₆₁	—	2	4	no	—
127	Q₈×C₆₁	—	4	2	no	—
128	Q₈×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	2	4	no	—
130	C₂×C₄×C₆₁	—	2	4	no	—
131	C₂×C₄×C₆₁	—	2	2	no	—
132	C₂×C₄×C₆₁	—	2	2	no	—
133	C₂×C₄×C₆₁	—	2	2	no	—
134	C₂×C₄×C₆₁	—	2	1	no	—
135	C₂×C₄×C₆₁	—	2	2	no	—
136	C₂×C₄×C₆₁	—	2	4	no	—
137	C₂×C₄×C₆₁	—	2	2	no	—
138	C₂×C₄×C₆₁	—	4	2	no	—
139	C₂×C₄×C₆₁	—	4	2	no	—
140	C₂×C₄×C₆₁	—	4	2	no	—
141	C₂×C₄×C₆₁	—	4	2	no	—
142	C₂×C₄×C₆₁	—	4	1	no	—
143	C₂×C₄×C₆₁	—	4	2	no	—
144	C₂×C₄×C₆₁	—	4	1	no	—
145	C₂×C₄×C₆₁	—	4	2	no	—
146	C₂×C₄×C₆₁	—	4	2	no	—
147	C₂×C₄×C₆₁	—	4	2	no	—
148	C₂×C₄×C₆₁	—	4	2	no	—
149	C₂×C₄×C₆₁	—	4	2	no	—
150	C₂×C₄×C₆₁	—	4	2	no	—
151	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
152	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
153	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
155	C₈×C₆₁	—	4	1	no	—
156	C₈×C₆₁	—	4	2	no	—
157	C₈×C₆₁	—	4	1	no	—
158	#29427	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂²×C₆₁	—	4	1	no	—
160	C₂²×C₆₁	—	4	2	no	—
161	C₂²×C₆₁	—	4	2	no	—
162	C₂²×C₆₁	—	4	1	no	—
163	C₂²×C₆₁	—	4	1	no	—
164	C₂²×C₆₁	#525	1	2	yes	—
165	C₂²×C₆₁	#534	1	1	yes	characteristic
166	C₂²×C₆₁	#576	1	1	yes	characteristic
167	C₂²×C₆₁	#578	1	2	yes	—
168	C₂²×C₆₁	#578	1	1	yes	characteristic
169	C₄×C₆₁	—	4	1	no	—
170	C₄×C₆₁	—	4	4	no	—
171	C₄×C₆₁	—	4	4	no	—
172	C₄×C₆₁	—	4	2	no	—
173	C₄×C₆₁	—	4	2	no	—
174	C₄×C₆₁	—	8	2	no	—
175	C₄×C₆₁	—	8	2	no	—
176	C₄×C₆₁	#551	1	1	yes	characteristic
177	C₄×C₆₁	#553	1	2	yes	—
178	C₄×C₆₁	#559	1	1	yes	characteristic
179	#2323	C₂×C₆₁	1	2	yes	—
180	#2075	C₂×C₆₁	1	1	yes	characteristic
181	#2036	C₂×C₆₁	1	2	yes	—
182	#2001	C₂×C₆₁	1	1	yes	characteristic
183	#1928	C₂×C₆₁	1	2	yes	complemented
184	#1918	C₂×C₆₁	1	2	yes	complemented
185	#1878	C₂×C₆₁	1	2	yes	complemented
186	#1863	C₂×C₆₁	1	2	yes	complemented
187	#1818	C₂×C₆₁	1	1	yes	characteristic
188	C₂×C₆₁	—	8	1	no	—
189	C₂×C₆₁	#2600	1	1	yes	characteristic
190	C₂×C₆₁	#3124	1	2	yes	—
191	C₂×C₆₁	#3127	1	2	yes	—
192	C₂×C₆₁	#3202	1	1	yes	characteristic
193	C₂×C₆₁	#3203	1	1	yes	characteristic
194	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
195	C₂×C₂²⋊Q₈	C₂²×C₆₁	1	2	yes	—
196	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
197	C₂×#117	C₂²×C₆₁	1	1	yes	characteristic
198	C₂×#107	C₂²×C₆₁	1	2	yes	—
199	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
200	C₂×#103	C₂²×C₆₁	1	2	yes	—
201	C₂×#103	C₂²×C₆₁	1	2	yes	—
202	C₂×#103	C₂²×C₆₁	1	2	yes	—
203	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
204	C₂×C₂²⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
205	#391	C₂²×C₆₁	1	2	yes	—
206	#389	—	2	2	no	—
207	#389	C₂²×C₆₁	1	2	yes	—
208	#388	—	2	2	no	—
209	#388	—	2	2	no	—
210	#387	—	2	2	no	—
211	#380	C₂²×C₆₁	1	2	yes	—
212	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
213	#340	C₂²×C₆₁	1	2	yes	—
214	#340	C₂²×C₆₁	1	1	yes	characteristic
215	#340	C₂²×C₆₁	1	2	yes	—
216	#340	C₂²×C₆₁	1	1	yes	characteristic
217	#336	C₂²×C₆₁	1	2	yes	—
218	#336	C₂²×C₆₁	1	2	yes	—
219	C₆₁	#29427	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
221	C₂³×C₄	C₂³×C₆₁	1	1	yes	characteristic
222	C₂×Q₁₆	—	2	4	no	—
223	C₂×Q₁₆	—	4	1	no	—
224	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
227	C₂²⋊Q₈	—	2	4	no	—
228	C₂²⋊Q₈	—	2	4	no	—
229	C₄×Q₈	—	4	2	no	—
230	C₄×Q₈	—	4	2	no	—
231	#117	—	4	1	no	—
232	#117	—	4	1	no	—
233	C₂×C₄⋊C₄	—	2	2	no	—
234	C₂×C₄⋊C₄	—	2	2	no	—
235	C₂×C₄⋊C₄	—	2	2	no	—
236	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
237	C₂×C₄⋊C₄	D₄×C₆₁	1	1	yes	characteristic
238	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
239	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
240	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
241	C₂×C₂²⋊C₄	—	2	2	no	—
242	C₂×C₂²⋊C₄	—	2	2	no	—
243	C₂×C₂²⋊C₄	D₄×C₆₁	1	2	yes	—
244	C₂×C₄²	—	2	2	no	—
245	C₂×C₄²	—	2	2	no	—
246	C₂×C₄²	C₂³×C₆₁	1	2	yes	—
247	#107	—	2	4	no	—
248	#107	—	4	1	no	—
249	#103	—	2	4	no	—
250	#103	—	2	4	no	—
251	#103	—	4	2	no	—
252	C₂²⋊C₈	—	4	1	no	—
253	C₂²⋊C₈	—	4	1	no	—
254	#95	—	2	2	no	—
255	#95	—	2	2	no	—
256	#95	—	2	2	no	—
257	#95	—	2	2	no	—
258	#95	—	2	2	no	—
259	C₂⁴	C₂×D₄×C₆₁	1	1	yes	characteristic
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	#41×C₆₁	1	4	yes	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
275	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
277	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
278	Q₁₆	—	4	4	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₄⋊C₄	—	2	8	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	4	no	—
295	C₄⋊C₄	—	8	2	no	—
296	C₄⋊C₄	#41×C₆₁	1	4	yes	—
297	C₂²⋊C₄	—	2	8	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	8	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	4	4	no	—
303	C₄²	—	8	2	no	—
304	C₂³	—	2	1	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	2	2	no	—
307	C₂³	—	2	1	no	—
308	C₂³	—	2	2	no	—
309	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
310	Q₈	—	2	4	no	—
311	Q₈	—	2	4	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
337	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
338	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
339	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
340	C₈	—	4	1	no	—
341	C₈	—	4	2	no	—
342	C₈	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	C₆₁×#525	1	2	yes	—
349	C₂²	C₆₁×#534	1	1	yes	characteristic
350	C₂²	C₆₁×#576	1	1	yes	characteristic
351	C₂²	C₆₁×#578	1	2	yes	—
352	C₂²	C₆₁×#578	1	1	yes	characteristic
353	C₄	—	4	1	no	—
354	C₄	—	4	4	no	—
355	C₄	—	4	4	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	2	no	—
360	C₄	C₆₁×#551	1	1	yes	characteristic
361	C₄	C₆₁×#553	1	2	yes	—
362	C₄	C₆₁×#559	1	1	yes	characteristic
363	C₂	—	8	1	no	—
364	C₂	C₆₁×#2600	1	1	yes	characteristic
365	C₂	C₆₁×#3124	1	2	yes	—
366	C₂	C₆₁×#3127	1	2	yes	—
367	C₂	C₆₁×#3202	1	1	yes	characteristic
368	C₂	C₆₁×#3203	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂⁴×C₆₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₂×#117×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₆₁×#1928	C₂	semidirect
6. C₆₁×#1918	C₂	semidirect
7. C₆₁×#1878	C₂	semidirect
8. C₆₁×#1863	C₂	semidirect
158. #29427	C₆₁	direct (N × H)
183. #1928	C₂×C₆₁	semidirect
184. #1918	C₂×C₆₁	semidirect
185. #1878	C₂×C₆₁	semidirect
186. #1863	C₂×C₆₁	semidirect
219. C₆₁	#29427	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×#2323	2
C₆₁×#2075	2
C₆₁×#2036	2
C₆₁×#2001	2
C₆₁×#1928	2
C₆₁×#1918	2
C₆₁×#1878	2
C₆₁×#1863	2
C₆₁×#1818	2
#29427	61

Subgroups — C61×#29427 #29532860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#29427 #29532860