Subgroups — C₆₁×#19638 #29523071

Order 15616.

754 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3206	C₂	1	2	yes	maximal, complemented
2	C₆₁×#3169	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#3141	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1719	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1710	C₂	1	2	yes	maximal, complemented
7	C₆₁×#1706	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#532	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#500	—	2	2	no	—
12	C₆₁×#500	C₂²	1	2	yes	—
13	C₆₁×#500	C₂²	1	2	yes	—
14	C₆₁×#500	C₂²	1	1	yes	characteristic
15	C₆₁×#493	C₂²	1	1	yes	characteristic
16	C₆₁×#492	C₂²	1	4	yes	—
17	C₆₁×#489	—	2	4	no	—
18	C₆₁×#487	C₂²	1	4	yes	—
19	C₆₁×#487	C₂²	1	4	yes	—
20	C₆₁×#486	C₂²	1	4	yes	—
21	C₆₁×#486	C₂²	1	2	yes	—
22	C₆₁×#466	—	2	2	no	—
23	C₆₁×#461	—	2	4	no	—
24	C₆₁×#448	—	2	4	no	—
25	C₄×D₈×C₆₁	—	2	2	no	—
26	C₆₁×#431	C₂²	1	1	yes	characteristic
27	C₂×#107×C₆₁	—	2	2	no	—
28	C₆₁×#417	—	2	4	no	—
29	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
31	C₆₁×#322	C₂²	1	2	yes	—
32	C₈²×C₆₁	C₂²	1	2	yes	—
33	C₂×#41×C₆₁	—	2	2	no	—
34	C₂²×Q₈×C₆₁	—	2	2	no	—
35	C₂×SD₁₆×C₆₁	—	2	8	no	—
36	C₂×D₈×C₆₁	—	2	2	no	—
37	C₂×M₁₆×C₆₁	D₄	1	2	yes	—
38	C₂²×C₈×C₆₁	D₄	1	2	yes	—
39	C₄⋊Q₈×C₆₁	—	2	2	no	—
40	C₄⋊Q₈×C₆₁	—	2	2	no	—
41	C₄⋊Q₈×C₆₁	—	2	2	no	—
42	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
43	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
44	#127×C₆₁	C₂³	1	2	yes	—
45	#124×C₆₁	—	2	2	no	—
46	#121×C₆₁	—	2	2	no	—
47	#121×C₆₁	—	2	2	no	—
48	C₄×D₄×C₆₁	—	2	2	no	—
49	C₄×D₄×C₆₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
52	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
53	#107×C₆₁	—	2	4	no	—
54	#107×C₆₁	—	2	4	no	—
55	#107×C₆₁	—	2	2	no	—
56	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
57	#103×C₆₁	—	2	8	no	—
58	#103×C₆₁	—	2	8	no	—
59	#102×C₆₁	—	2	8	no	—
60	#102×C₆₁	—	2	4	no	—
61	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
62	#97×C₆₁	D₄	1	2	yes	—
63	C₄×C₈×C₆₁	D₄	1	2	yes	—
64	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₆₁	C₂³	1	2	yes	—
66	C₄×C₈×C₆₁	C₂³	1	2	yes	—
67	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
68	#41×C₆₁	—	4	8	no	—
69	C₂×Q₈×C₆₁	—	2	2	no	—
70	C₂×Q₈×C₆₁	—	2	4	no	—
71	C₂×Q₈×C₆₁	—	4	2	no	—
72	C₂×Q₈×C₆₁	—	8	4	no	—
73	C₂×D₄×C₆₁	—	2	2	no	—
74	C₂×D₄×C₆₁	—	2	2	no	—
75	C₂×D₄×C₆₁	—	2	2	no	—
76	C₂×D₄×C₆₁	—	4	2	no	—
77	C₂²×C₄×C₆₁	—	4	2	no	—
78	C₂²×C₄×C₆₁	—	4	2	no	—
79	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
82	SD₁₆×C₆₁	—	8	8	no	—
83	D₈×C₆₁	—	4	4	no	—
84	M₁₆×C₆₁	—	2	4	no	—
85	C₂×C₈×C₆₁	—	2	4	no	—
86	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
88	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
90	C₄⋊C₄×C₆₁	—	2	4	no	—
91	C₄⋊C₄×C₆₁	—	2	2	no	—
92	C₄⋊C₄×C₆₁	—	2	2	no	—
93	C₄⋊C₄×C₆₁	—	2	4	no	—
94	C₄⋊C₄×C₆₁	—	4	2	no	—
95	C₄⋊C₄×C₆₁	—	4	2	no	—
96	C₂²⋊C₄×C₆₁	—	4	2	no	—
97	C₂²⋊C₄×C₆₁	—	4	2	no	—
98	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
102	C₂³×C₆₁	—	4	2	no	—
103	C₂³×C₆₁	#127	1	1	yes	characteristic
104	Q₈×C₆₁	—	4	4	no	—
105	Q₈×C₆₁	—	4	8	no	—
106	Q₈×C₆₁	—	8	4	no	—
107	D₄×C₆₁	—	4	4	no	—
108	D₄×C₆₁	—	4	4	no	—
109	D₄×C₆₁	—	4	4	no	—
110	D₄×C₆₁	—	8	4	no	—
111	C₂×C₄×C₆₁	—	2	2	no	—
112	C₂×C₄×C₆₁	—	2	2	no	—
113	C₂×C₄×C₆₁	—	4	2	no	—
114	C₂×C₄×C₆₁	—	4	4	no	—
115	C₂×C₄×C₆₁	—	8	4	no	—
116	C₂×C₄×C₆₁	—	8	4	no	—
117	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
118	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
119	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#135	1	4	yes	—
121	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₆₁	—	2	4	no	—
125	C₈×C₆₁	—	2	2	no	—
126	C₈×C₆₁	—	2	4	no	—
127	C₈×C₆₁	#137	1	4	yes	—
128	#19638	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₆₁	—	2	1	no	—
130	C₂²×C₆₁	—	2	1	no	—
131	C₂²×C₆₁	—	8	2	no	—
132	C₂²×C₆₁	—	8	2	no	—
133	C₂²×C₆₁	—	8	2	no	—
134	C₂²×C₆₁	#495	1	2	yes	—
135	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
136	C₄×C₆₁	—	2	1	no	—
137	C₄×C₆₁	—	2	1	no	—
138	C₄×C₆₁	—	8	2	no	—
139	C₄×C₆₁	—	8	4	no	—
140	C₄×C₆₁	#495	1	2	yes	—
141	C₄×C₆₁	C₂×#135	1	2	yes	—
142	C₄×C₆₁	C₂×#137	1	2	yes	—
143	C₄×C₆₁	#576	1	2	yes	—
144	#3206	C₂×C₆₁	1	2	yes	complemented
145	#3169	C₂×C₆₁	1	1	yes	characteristic
146	C₂×#500	C₂×C₆₁	1	1	yes	characteristic, complemented
147	#3141	C₂×C₆₁	1	2	yes	complemented
148	#1719	C₂×C₆₁	1	4	yes	complemented
149	#1710	C₂×C₆₁	1	2	yes	complemented
150	#1706	C₂×C₆₁	1	2	yes	complemented
151	C₈×M₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
152	C₂×C₆₁	—	2	2	no	—
153	C₂×C₆₁	—	16	2	no	—
154	C₂×C₆₁	C₂×#495	1	1	yes	characteristic
155	C₂×C₆₁	#3152	1	1	yes	characteristic
156	C₂×C₆₁	#3155	1	1	yes	characteristic
157	#532	C₂²×C₆₁	1	2	yes	—
158	C₂×C₄⋊Q₈	C₂²×C₆₁	1	2	yes	—
159	#500	—	2	2	no	—
160	#500	C₂²×C₆₁	1	2	yes	—
161	#500	C₂²×C₆₁	1	2	yes	—
162	#500	C₂²×C₆₁	1	1	yes	characteristic
163	#493	C₂²×C₆₁	1	1	yes	characteristic
164	#492	C₂²×C₆₁	1	4	yes	—
165	#489	—	2	4	no	—
166	#487	C₂²×C₆₁	1	4	yes	—
167	#487	C₂²×C₆₁	1	4	yes	—
168	#486	C₂²×C₆₁	1	4	yes	—
169	#486	C₂²×C₆₁	1	2	yes	—
170	#466	—	2	2	no	—
171	#461	—	2	4	no	—
172	#448	—	2	4	no	—
173	C₄×D₈	—	2	2	no	—
174	#431	C₂²×C₆₁	1	1	yes	characteristic
175	C₂×#107	—	2	2	no	—
176	#417	—	2	4	no	—
177	C₄×M₁₆	C₂²×C₆₁	1	1	yes	characteristic
178	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
179	#322	C₂²×C₆₁	1	2	yes	—
180	C₈²	C₂²×C₆₁	1	2	yes	—
181	C₆₁	#19638	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×#41	—	2	2	no	—
183	C₂²×Q₈	—	2	2	no	—
184	C₂×SD₁₆	—	2	8	no	—
185	C₂×D₈	—	2	2	no	—
186	C₂×M₁₆	D₄×C₆₁	1	2	yes	—
187	C₂²×C₈	D₄×C₆₁	1	2	yes	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	C₂³×C₆₁	1	4	yes	—
192	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
193	#127	C₂³×C₆₁	1	2	yes	—
194	#124	—	2	2	no	—
195	#121	—	2	2	no	—
196	#121	—	2	2	no	—
197	C₄×D₄	—	2	2	no	—
198	C₄×D₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
202	#107	—	2	4	no	—
203	#107	—	2	4	no	—
204	#107	—	2	2	no	—
205	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
206	#103	—	2	8	no	—
207	#103	—	2	8	no	—
208	#102	—	2	8	no	—
209	#102	—	2	4	no	—
210	C₂²⋊C₈	D₄×C₆₁	1	2	yes	—
211	#97	D₄×C₆₁	1	2	yes	—
212	C₄×C₈	D₄×C₆₁	1	2	yes	—
213	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₆₁	1	2	yes	—
215	C₄×C₈	C₂³×C₆₁	1	2	yes	—
216	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
217	#41	—	4	8	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	—	8	4	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	—	4	2	no	—
228	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
231	SD₁₆	—	8	8	no	—
232	D₈	—	4	4	no	—
233	M₁₆	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
236	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
237	C₂×C₈	C₂×D₄×C₆₁	1	4	yes	—
238	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
248	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
250	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
251	C₂³	—	4	2	no	—
252	C₂³	#127×C₆₁	1	1	yes	characteristic
253	Q₈	—	4	4	no	—
254	Q₈	—	4	8	no	—
255	Q₈	—	8	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	4	no	—
259	D₄	—	8	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
267	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
268	C₂×C₄	#127×C₆₁	1	1	yes	characteristic
269	C₂×C₄	#135×C₆₁	1	4	yes	—
270	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	2	2	no	—
275	C₈	—	2	4	no	—
276	C₈	#137×C₆₁	1	4	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	C₆₁×#495	1	2	yes	—
283	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	8	2	no	—
287	C₄	—	8	4	no	—
288	C₄	C₆₁×#495	1	2	yes	—
289	C₄	C₂×#135×C₆₁	1	2	yes	—
290	C₄	C₂×#137×C₆₁	1	2	yes	—
291	C₄	C₆₁×#576	1	2	yes	—
292	C₂	—	2	2	no	—
293	C₂	—	16	2	no	—
294	C₂	C₂×C₆₁×#495	1	1	yes	characteristic
295	C₂	C₆₁×#3152	1	1	yes	characteristic
296	C₂	C₆₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#3169	C₂
LCS	G ▷ 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
1. C₆₁×#3206	C₂	semidirect
3. C₂×C₆₁×#500	C₂	semidirect
4. C₆₁×#3141	C₂	semidirect
5. C₆₁×#1719	C₂	semidirect
6. C₆₁×#1710	C₂	semidirect
7. C₆₁×#1706	C₂	semidirect
8. C₈×M₁₆×C₆₁	C₂	semidirect
128. #19638	C₆₁	direct (N × H)
144. #3206	C₂×C₆₁	semidirect
146. C₂×#500	C₂×C₆₁	semidirect
147. #3141	C₂×C₆₁	semidirect
148. #1719	C₂×C₆₁	semidirect
149. #1710	C₂×C₆₁	semidirect
150. #1706	C₂×C₆₁	semidirect
151. C₈×M₁₆	C₂×C₆₁	semidirect
181. C₆₁	#19638	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×#3206	2
C₆₁×#3169	2
C₂×C₆₁×#500	2
C₆₁×#3141	2
C₆₁×#1719	2
C₆₁×#1710	2
C₆₁×#1706	2
C₈×M₁₆×C₆₁	2
#19638	61

Subgroups — C61×#19638 #29523071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19638 #29523071