Subgroups — C₆₁×#20013 #29523446

Order 15616.

572 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3365	C₂	1	2	yes	maximal
2	C₆₁×#3363	C₂	1	2	yes	maximal, complemented
3	C₆₁×#1708	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1684	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1682	C₂	1	2	yes	maximal
6	C₆₁×#1653	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1630	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#1623	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₁₆×C₆₁	C₂	1	2	yes	maximal, complemented
10	C₆₁×#559	C₂²	1	1	yes	characteristic
11	Q₈²×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#557	C₂²	1	2	yes	—
13	C₆₁×#498	C₂²	1	1	yes	characteristic
14	C₆₁×#492	C₂²	1	1	yes	characteristic
15	C₆₁×#488	—	2	2	no	—
16	C₆₁×#487	—	2	2	no	—
17	C₆₁×#487	C₂²	1	2	yes	—
18	C₆₁×#486	C₂²	1	2	yes	—
19	C₆₁×#478	—	2	2	no	—
20	C₆₁×#477	—	2	2	no	—
21	C₆₁×#477	C₂²	1	2	yes	—
22	C₆₁×#475	C₂²	1	2	yes	—
23	C₆₁×#464	—	2	2	no	—
24	C₆₁×#462	—	2	2	no	—
25	C₆₁×#462	C₂²	1	2	yes	—
26	C₆₁×#460	C₂²	1	2	yes	—
27	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
28	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
29	C₄×Q₁₆×C₆₁	—	2	2	no	—
30	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
31	C₄×SD₁₆×C₆₁	—	2	2	no	—
32	C₆₁×#335	C₂²	1	2	yes	—
33	C₆₁×#333	C₂²	1	1	yes	characteristic
34	C₆₁×#332	C₂²	1	2	yes	—
35	C₆₁×#331	C₂²	1	1	yes	characteristic
36	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₆₁	—	2	2	no	—
40	C₂×Q₁₆×C₆₁	D₄	1	2	yes	—
41	C₂×SD₁₆×C₆₁	—	2	2	no	—
42	C₂×SD₁₆×C₆₁	—	4	1	no	—
43	C₄⋊Q₈×C₆₁	—	2	2	no	—
44	C₄⋊Q₈×C₆₁	—	2	2	no	—
45	C₄⋊Q₈×C₆₁	—	4	1	no	—
46	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₆₁	C₂³	1	1	yes	characteristic
48	#127×C₆₁	C₂³	1	1	yes	characteristic
49	#125×C₆₁	—	2	2	no	—
50	#125×C₆₁	—	4	2	no	—
51	#124×C₆₁	—	2	2	no	—
52	#123×C₆₁	—	4	1	no	—
53	C₄×Q₈×C₆₁	—	2	2	no	—
54	C₄×Q₈×C₆₁	—	2	2	no	—
55	C₄×Q₈×C₆₁	—	2	2	no	—
56	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
57	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
58	C₄×D₄×C₆₁	—	2	2	no	—
59	#107×C₆₁	—	2	2	no	—
60	#107×C₆₁	D₄	1	2	yes	—
61	#106×C₆₁	—	2	2	no	—
62	#106×C₆₁	—	4	1	no	—
63	C₄⋊C₈×C₆₁	—	2	2	no	—
64	C₄⋊C₈×C₆₁	—	2	2	no	—
65	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
66	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
67	#103×C₆₁	—	2	2	no	—
68	#103×C₆₁	—	2	2	no	—
69	#103×C₆₁	—	2	2	no	—
70	#103×C₆₁	—	2	2	no	—
71	#103×C₆₁	—	2	2	no	—
72	#103×C₆₁	D₄	1	2	yes	—
73	#103×C₆₁	D₄	1	2	yes	—
74	#102×C₆₁	—	2	2	no	—
75	#102×C₆₁	—	2	2	no	—
76	#102×C₆₁	—	2	2	no	—
77	C₄×C₈×C₆₁	—	2	2	no	—
78	C₄×C₈×C₆₁	—	2	2	no	—
79	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₆₁	C₂³	1	2	yes	—
81	C₂×Q₈×C₆₁	—	2	2	no	—
82	C₂×Q₈×C₆₁	—	2	1	no	—
83	C₂×Q₈×C₆₁	—	4	1	no	—
84	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
85	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₆₁	—	2	2	no	—
87	C₂×D₄×C₆₁	—	2	1	no	—
88	C₂²×C₄×C₆₁	—	4	1	no	—
89	Q₁₆×C₆₁	—	2	4	no	—
90	Q₁₆×C₆₁	—	4	4	no	—
91	SD₁₆×C₆₁	—	4	4	no	—
92	SD₁₆×C₆₁	—	8	2	no	—
93	C₂×C₈×C₆₁	—	2	1	no	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	2	2	no	—
96	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
98	C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₄⋊C₄×C₆₁	—	2	2	no	—
100	C₄⋊C₄×C₆₁	—	2	2	no	—
101	C₄⋊C₄×C₆₁	—	2	2	no	—
102	C₄⋊C₄×C₆₁	—	2	2	no	—
103	C₄⋊C₄×C₆₁	—	4	2	no	—
104	C₄⋊C₄×C₆₁	—	4	2	no	—
105	C₄⋊C₄×C₆₁	—	4	2	no	—
106	C₄⋊C₄×C₆₁	—	4	2	no	—
107	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
108	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
109	C₂²⋊C₄×C₆₁	—	4	2	no	—
110	C₂²⋊C₄×C₆₁	—	4	1	no	—
111	C₄²×C₆₁	—	2	2	no	—
112	C₄²×C₆₁	—	2	2	no	—
113	C₄²×C₆₁	—	4	2	no	—
114	C₄²×C₆₁	—	4	1	no	—
115	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
116	C₂³×C₆₁	—	4	1	no	—
117	Q₈×C₆₁	—	2	2	no	—
118	Q₈×C₆₁	—	2	2	no	—
119	Q₈×C₆₁	—	2	2	no	—
120	Q₈×C₆₁	—	4	2	no	—
121	Q₈×C₆₁	—	4	2	no	—
122	Q₈×C₆₁	—	8	1	no	—
123	Q₈×C₆₁	—	16	1	no	—
124	Q₈×C₆₁	#135	1	4	yes	—
125	D₄×C₆₁	—	4	2	no	—
126	D₄×C₆₁	—	4	2	no	—
127	D₄×C₆₁	—	8	1	no	—
128	C₂×C₄×C₆₁	—	2	2	no	—
129	C₂×C₄×C₆₁	—	2	2	no	—
130	C₂×C₄×C₆₁	—	2	2	no	—
131	C₂×C₄×C₆₁	—	2	2	no	—
132	C₂×C₄×C₆₁	—	4	2	no	—
133	C₂×C₄×C₆₁	—	4	1	no	—
134	C₂×C₄×C₆₁	—	16	1	no	—
135	C₂×C₄×C₆₁	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₆₁	—	2	2	no	—
138	C₈×C₆₁	—	2	2	no	—
139	C₈×C₆₁	—	2	2	no	—
140	C₈×C₆₁	—	4	2	no	—
141	C₈×C₆₁	—	4	2	no	—
142	#20013	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₆₁	—	8	2	no	—
144	C₂²×C₆₁	—	8	1	no	—
145	C₂²×C₆₁	D₄²	1	1	yes	characteristic
146	C₄×C₆₁	—	2	4	no	—
147	C₄×C₆₁	—	2	1	no	—
148	C₄×C₆₁	—	4	2	no	—
149	C₄×C₆₁	—	4	2	no	—
150	C₄×C₆₁	—	4	2	no	—
151	C₄×C₆₁	—	8	2	no	—
152	C₄×C₆₁	—	8	1	no	—
153	C₄×C₆₁	C₂×#135	1	2	yes	—
154	C₄×C₆₁	#578	1	2	yes	—
155	#3365	C₂×C₆₁	1	2	yes	—
156	#3363	C₂×C₆₁	1	2	yes	complemented
157	#1708	C₂×C₆₁	1	1	yes	characteristic
158	#1684	C₂×C₆₁	1	2	yes	complemented
159	#1682	C₂×C₆₁	1	2	yes	—
160	#1653	C₂×C₆₁	1	2	yes	complemented
161	#1630	C₂×C₆₁	1	1	yes	characteristic
162	#1623	C₂×C₆₁	1	1	yes	characteristic
163	C₈×Q₁₆	C₂×C₆₁	1	2	yes	complemented
164	C₂×C₆₁	—	16	1	no	—
165	C₂×C₆₁	#3285	1	2	yes	—
166	C₂×C₆₁	#3291	1	1	yes	characteristic
167	#559	C₂²×C₆₁	1	1	yes	characteristic
168	Q₈²	C₂²×C₆₁	1	1	yes	characteristic
169	#557	C₂²×C₆₁	1	2	yes	—
170	#498	C₂²×C₆₁	1	1	yes	characteristic
171	#492	C₂²×C₆₁	1	1	yes	characteristic
172	#488	—	2	2	no	—
173	#487	—	2	2	no	—
174	#487	C₂²×C₆₁	1	2	yes	—
175	#486	C₂²×C₆₁	1	2	yes	—
176	#478	—	2	2	no	—
177	#477	—	2	2	no	—
178	#477	C₂²×C₆₁	1	2	yes	—
179	#475	C₂²×C₆₁	1	2	yes	—
180	#464	—	2	2	no	—
181	#462	—	2	2	no	—
182	#462	C₂²×C₆₁	1	2	yes	—
183	#460	C₂²×C₆₁	1	2	yes	—
184	C₈×Q₈	C₂²×C₆₁	1	2	yes	—
185	C₈×Q₈	C₂²×C₆₁	1	2	yes	—
186	C₄×Q₁₆	—	2	2	no	—
187	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
188	C₄×SD₁₆	—	2	2	no	—
189	#335	C₂²×C₆₁	1	2	yes	—
190	#333	C₂²×C₆₁	1	1	yes	characteristic
191	#332	C₂²×C₆₁	1	2	yes	—
192	#331	C₂²×C₆₁	1	1	yes	characteristic
193	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
194	Q₈⋊C₈	C₂²×C₆₁	1	2	yes	—
195	C₈²	C₂²×C₆₁	1	1	yes	characteristic
196	C₆₁	#20013	1	1	yes	characteristic, direct product factor, Sylow
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×Q₁₆	D₄×C₆₁	1	2	yes	—
199	C₂×SD₁₆	—	2	2	no	—
200	C₂×SD₁₆	—	4	1	no	—
201	C₄⋊Q₈	—	2	2	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	4	1	no	—
204	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
205	C₄⋊Q₈	C₂³×C₆₁	1	1	yes	characteristic
206	#127	C₂³×C₆₁	1	1	yes	characteristic
207	#125	—	2	2	no	—
208	#125	—	4	2	no	—
209	#124	—	2	2	no	—
210	#123	—	4	1	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
215	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
216	C₄×D₄	—	2	2	no	—
217	#107	—	2	2	no	—
218	#107	D₄×C₆₁	1	2	yes	—
219	#106	—	2	2	no	—
220	#106	—	4	1	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
224	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
225	#103	—	2	2	no	—
226	#103	—	2	2	no	—
227	#103	—	2	2	no	—
228	#103	—	2	2	no	—
229	#103	—	2	2	no	—
230	#103	D₄×C₆₁	1	2	yes	—
231	#103	D₄×C₆₁	1	2	yes	—
232	#102	—	2	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₆₁	1	2	yes	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
243	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	Q₁₆	—	2	4	no	—
248	Q₁₆	—	4	4	no	—
249	SD₁₆	—	4	4	no	—
250	SD₁₆	—	8	2	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
255	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
266	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
274	C₂³	—	4	1	no	—
275	Q₈	—	2	2	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	8	1	no	—
281	Q₈	—	16	1	no	—
282	Q₈	#135×C₆₁	1	4	yes	—
283	D₄	—	4	2	no	—
284	D₄	—	4	2	no	—
285	D₄	—	8	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	16	1	no	—
293	C₂×C₄	C₂²×D₄×C₆₁	1	2	yes	—
294	C₂×C₄	H(5,𝔽₂)×C₆₁	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	2	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	2	no	—
302	C₂²	D₄²×C₆₁	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	4	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	1	no	—
309	C₄	—	8	2	no	—
310	C₄	C₂×#135×C₆₁	1	2	yes	—
311	C₄	C₆₁×#578	1	2	yes	—
312	C₂	—	16	1	no	—
313	C₂	C₆₁×#3285	1	2	yes	—
314	C₂	C₆₁×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	#127×C₆₁	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#3363	C₂	semidirect
4. C₆₁×#1684	C₂	semidirect
6. C₆₁×#1653	C₂	semidirect
9. C₈×Q₁₆×C₆₁	C₂	semidirect
142. #20013	C₆₁	direct (N × H)
156. #3363	C₂×C₆₁	semidirect
158. #1684	C₂×C₆₁	semidirect
160. #1653	C₂×C₆₁	semidirect
163. C₈×Q₁₆	C₂×C₆₁	semidirect
196. C₆₁	#20013	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×#3365	2
C₆₁×#3363	2
C₆₁×#1708	2
C₆₁×#1684	2
C₆₁×#1682	2
C₆₁×#1653	2
C₆₁×#1630	2
C₆₁×#1623	2
C₈×Q₁₆×C₆₁	2
#20013	61

Subgroups — C61×#20013 #29523446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#20013 #29523446