Subgroups — C₆₁×#23829 #29527262

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2331	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#2327	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2101	C₂	1	2	yes	maximal
4	C₆₁×#2074	C₂	1	2	yes	maximal
5	C₆₁×#2043	C₂	1	2	yes	maximal
6	C₆₁×#2024	C₂	1	2	yes	maximal
7	C₆₁×#1763	C₂	1	2	yes	maximal
8	C₄×#103×C₆₁	C₂	1	2	yes	maximal
9	C₆₁×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₆₁	C₂²	1	2	yes	—
13	C₂×#103×C₆₁	C₂²	1	2	yes	—
14	C₂×#103×C₆₁	C₂²	1	2	yes	—
15	C₂×#103×C₆₁	C₂²	1	2	yes	—
16	C₂×#103×C₆₁	C₂²	1	2	yes	—
17	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#391	—	2	2	no	—
20	C₆₁×#391	C₂²	1	2	yes	—
21	C₆₁×#389	C₂²	1	2	yes	—
22	C₆₁×#387	—	2	2	no	—
23	C₆₁×#387	—	2	2	no	—
24	C₆₁×#387	—	2	2	no	—
25	C₆₁×#382	C₂²	1	1	yes	characteristic
26	C₆₁×#382	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
28	C₆₁×#376	C₂²	1	1	yes	characteristic
29	C₆₁×#340	C₂²	1	2	yes	—
30	C₆₁×#340	C₂²	1	2	yes	—
31	C₆₁×#340	C₂²	1	2	yes	—
32	C₆₁×#340	C₂²	1	2	yes	—
33	C₆₁×#336	C₂²	1	2	yes	—
34	C₆₁×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₆₁	—	2	1	no	—
36	C₂²×Q₈×C₆₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₆₁	—	2	4	no	—
40	C₄⋊Q₈×C₆₁	—	2	4	no	—
41	C₄×Q₈×C₆₁	—	2	4	no	—
42	C₄×Q₈×C₆₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₆₁	—	2	2	no	—
51	C₂×C₄²×C₆₁	—	2	2	no	—
52	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
55	#103×C₆₁	—	2	4	no	—
56	#103×C₆₁	—	2	4	no	—
57	#103×C₆₁	—	2	4	no	—
58	#103×C₆₁	—	2	4	no	—
59	#97×C₆₁	—	2	2	no	—
60	C₄×C₈×C₆₁	—	2	2	no	—
61	#95×C₆₁	—	2	2	no	—
62	#95×C₆₁	—	2	1	no	—
63	#95×C₆₁	—	2	2	no	—
64	#95×C₆₁	—	2	1	no	—
65	#95×C₆₁	—	2	2	no	—
66	#95×C₆₁	—	2	2	no	—
67	#95×C₆₁	—	2	2	no	—
68	#95×C₆₁	D₄	1	2	yes	—
69	#95×C₆₁	D₄	1	2	yes	—
70	C₂×Q₈×C₆₁	—	2	4	no	—
71	C₂×Q₈×C₆₁	—	2	2	no	—
72	C₂×Q₈×C₆₁	—	2	2	no	—
73	C₂×Q₈×C₆₁	—	2	2	no	—
74	C₂×Q₈×C₆₁	—	2	2	no	—
75	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂×Q₈×C₆₁	—	8	2	no	—
77	C₂×Q₈×C₆₁	#41	1	4	yes	—
78	C₂²×C₄×C₆₁	—	2	2	no	—
79	C₂²×C₄×C₆₁	—	2	2	no	—
80	C₂²×C₄×C₆₁	—	2	2	no	—
81	C₂²×C₄×C₆₁	—	2	2	no	—
82	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₆₁	—	2	2	no	—
88	C₂×C₈×C₆₁	—	2	2	no	—
89	C₂×C₈×C₆₁	—	2	2	no	—
90	C₂×C₈×C₆₁	—	2	2	no	—
91	C₂×C₈×C₆₁	—	2	2	no	—
92	C₂×C₈×C₆₁	—	2	2	no	—
93	C₄⋊C₄×C₆₁	—	2	8	no	—
94	C₄⋊C₄×C₆₁	—	2	4	no	—
95	C₄⋊C₄×C₆₁	—	2	2	no	—
96	C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₄⋊C₄×C₆₁	—	2	2	no	—
100	C₄⋊C₄×C₆₁	—	4	4	no	—
101	C₄⋊C₄×C₆₁	—	8	2	no	—
102	C₄⋊C₄×C₆₁	#41	1	4	yes	—
103	C₄²×C₆₁	—	2	2	no	—
104	C₄²×C₆₁	—	2	2	no	—
105	C₄²×C₆₁	—	4	4	no	—
106	C₄²×C₆₁	—	8	2	no	—
107	C₄²×C₆₁	#41	1	2	yes	—
108	C₄²×C₆₁	#41	1	2	yes	—
109	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₆₁	—	2	4	no	—
111	Q₈×C₆₁	—	2	4	no	—
112	Q₈×C₆₁	—	4	2	no	—
113	Q₈×C₆₁	—	4	2	no	—
114	Q₈×C₆₁	—	8	4	no	—
115	C₂×C₄×C₆₁	—	2	2	no	—
116	C₂×C₄×C₆₁	—	2	2	no	—
117	C₂×C₄×C₆₁	—	2	2	no	—
118	C₂×C₄×C₆₁	—	2	1	no	—
119	C₂×C₄×C₆₁	—	2	1	no	—
120	C₂×C₄×C₆₁	—	2	4	no	—
121	C₂×C₄×C₆₁	—	2	4	no	—
122	C₂×C₄×C₆₁	—	2	2	no	—
123	C₂×C₄×C₆₁	—	2	2	no	—
124	C₂×C₄×C₆₁	—	2	2	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	4	2	no	—
127	C₂×C₄×C₆₁	—	4	2	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	4	2	no	—
130	C₂×C₄×C₆₁	—	4	2	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	—	4	2	no	—
133	C₂×C₄×C₆₁	—	4	2	no	—
134	C₂×C₄×C₆₁	—	4	2	no	—
135	C₂×C₄×C₆₁	—	4	2	no	—
136	C₂×C₄×C₆₁	#124	1	4	yes	—
137	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
138	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
139	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
141	C₈×C₆₁	—	4	2	no	—
142	C₈×C₆₁	—	4	2	no	—
143	#23829	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₆₁	#525	1	1	yes	characteristic
145	C₂²×C₆₁	#525	1	1	yes	characteristic
146	C₂²×C₆₁	C₂×#124	1	1	yes	characteristic
147	C₂²×C₆₁	#577	1	2	yes	—
148	C₂²×C₆₁	#578	1	2	yes	—
149	C₄×C₆₁	—	2	4	no	—
150	C₄×C₆₁	—	4	2	no	—
151	C₄×C₆₁	—	4	1	no	—
152	C₄×C₆₁	—	4	4	no	—
153	C₄×C₆₁	—	4	1	no	—
154	C₄×C₆₁	—	4	4	no	—
155	C₄×C₆₁	—	8	2	no	—
156	C₄×C₆₁	—	8	2	no	—
157	C₄×C₆₁	#529	1	2	yes	—
158	C₄×C₆₁	#529	1	2	yes	—
159	#2331	C₂×C₆₁	1	1	yes	characteristic
160	#2327	C₂×C₆₁	1	1	yes	characteristic
161	#2101	C₂×C₆₁	1	2	yes	—
162	#2074	C₂×C₆₁	1	2	yes	—
163	#2043	C₂×C₆₁	1	2	yes	—
164	#2024	C₂×C₆₁	1	2	yes	—
165	#1763	C₂×C₆₁	1	2	yes	—
166	C₄×#103	C₂×C₆₁	1	2	yes	—
167	#1746	C₂×C₆₁	1	1	yes	characteristic
168	C₂×C₆₁	#2398	1	1	yes	characteristic
169	C₂×C₆₁	#3121	1	2	yes	—
170	C₂×C₆₁	#3123	1	1	yes	characteristic
171	C₂×C₆₁	#3127	1	1	yes	characteristic
172	C₂×C₆₁	#3137	1	2	yes	—
173	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
174	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
175	C₂×C₄×Q₈	C₂²×C₆₁	1	2	yes	—
176	C₂×#103	C₂²×C₆₁	1	2	yes	—
177	C₂×#103	C₂²×C₆₁	1	2	yes	—
178	C₂×#103	C₂²×C₆₁	1	2	yes	—
179	C₂×#103	C₂²×C₆₁	1	2	yes	—
180	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
182	#391	—	2	2	no	—
183	#391	C₂²×C₆₁	1	2	yes	—
184	#389	C₂²×C₆₁	1	2	yes	—
185	#387	—	2	2	no	—
186	#387	—	2	2	no	—
187	#387	—	2	2	no	—
188	#382	C₂²×C₆₁	1	1	yes	characteristic
189	#382	C₂²×C₆₁	1	1	yes	characteristic
190	C₄×C₄⋊C₄	C₂²×C₆₁	1	2	yes	—
191	#376	C₂²×C₆₁	1	1	yes	characteristic
192	#340	C₂²×C₆₁	1	2	yes	—
193	#340	C₂²×C₆₁	1	2	yes	—
194	#340	C₂²×C₆₁	1	2	yes	—
195	#340	C₂²×C₆₁	1	2	yes	—
196	#336	C₂²×C₆₁	1	2	yes	—
197	#336	C₂²×C₆₁	1	2	yes	—
198	C₆₁	#23829	1	1	yes	characteristic, direct product factor, Sylow
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	C₂³×C₆₁	1	2	yes	—
201	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
202	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
203	C₄⋊Q₈	—	2	4	no	—
204	C₄⋊Q₈	—	2	4	no	—
205	C₄×Q₈	—	2	4	no	—
206	C₄×Q₈	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₆₁	1	2	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
219	#103	—	2	4	no	—
220	#103	—	2	4	no	—
221	#103	—	2	4	no	—
222	#103	—	2	4	no	—
223	#97	—	2	2	no	—
224	C₄×C₈	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	1	no	—
227	#95	—	2	2	no	—
228	#95	—	2	1	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	#95	D₄×C₆₁	1	2	yes	—
233	#95	D₄×C₆₁	1	2	yes	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	8	2	no	—
241	C₂×Q₈	#41×C₆₁	1	4	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
250	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₄⋊C₄	—	2	8	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	—	8	2	no	—
266	C₄⋊C₄	#41×C₆₁	1	4	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	4	4	no	—
270	C₄²	—	8	2	no	—
271	C₄²	#41×C₆₁	1	2	yes	—
272	C₄²	#41×C₆₁	1	2	yes	—
273	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
274	Q₈	—	2	4	no	—
275	Q₈	—	2	4	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	8	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	#124×C₆₁	1	4	yes	—
301	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
302	C₂×C₄	C₂×#41×C₆₁	1	2	yes	—
303	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₂²	C₆₁×#525	1	1	yes	characteristic
308	C₂²	C₆₁×#525	1	1	yes	characteristic
309	C₂²	C₂×#124×C₆₁	1	1	yes	characteristic
310	C₂²	C₆₁×#577	1	2	yes	—
311	C₂²	C₆₁×#578	1	2	yes	—
312	C₄	—	2	4	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₆₁×#529	1	2	yes	—
321	C₄	C₆₁×#529	1	2	yes	—
322	C₂	C₆₁×#2398	1	1	yes	characteristic
323	C₂	C₆₁×#3121	1	2	yes	—
324	C₂	C₆₁×#3123	1	1	yes	characteristic
325	C₂	C₆₁×#3127	1	1	yes	characteristic
326	C₂	C₆₁×#3137	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₆₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
143. #23829	C₆₁	direct (N × H)
198. C₆₁	#23829	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×#2331	2
C₆₁×#2327	2
C₆₁×#2101	2
C₆₁×#2074	2
C₆₁×#2043	2
C₆₁×#2024	2
C₆₁×#1763	2
C₄×#103×C₆₁	2
C₆₁×#1746	2
#23829	61

Subgroups — C61×#23829 #29527262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#23829 #29527262