Subgroups — C₆₁×#13129 #29516562

Order 15616.

306 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#2117	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2115	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2110	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#1764	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#1378	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×#104×C₆₁	C₄	1	1	yes	characteristic
14	C₆₁×#416	C₄	1	1	yes	characteristic
15	C₆₁×#405	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×C₄⋊C₁₆	—	2	2	no	—
18	C₆₁×#350	—	2	2	no	—
19	C₆₁×#349	—	2	2	no	—
20	C₆₁×#347	—	2	2	no	—
21	C₆₁×#341	C₄	1	1	yes	characteristic
22	C₆₁×#339	C₄	1	1	yes	characteristic
23	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
24	#131×C₆₁	—	2	1	no	—
25	#131×C₆₁	—	2	1	no	—
26	#131×C₆₁	—	2	1	no	—
27	#131×C₆₁	—	2	1	no	—
28	C₂×M₁₆×C₆₁	—	2	1	no	—
29	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	—	2	1	no	—
32	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₆₁	—	2	1	no	—
37	M₃₂×C₆₁	—	2	1	no	—
38	M₃₂×C₆₁	—	2	1	no	—
39	M₃₂×C₆₁	—	2	1	no	—
40	M₃₂×C₆₁	—	2	1	no	—
41	M₃₂×C₆₁	—	2	1	no	—
42	M₃₂×C₆₁	—	2	1	no	—
43	M₃₂×C₆₁	D₄	1	2	yes	—
44	C₂×C₁₆×C₆₁	—	2	1	no	—
45	C₂×C₁₆×C₆₁	—	2	1	no	—
46	C₂×C₁₆×C₆₁	—	2	1	no	—
47	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
48	#104×C₆₁	—	2	1	no	—
49	#104×C₆₁	—	2	1	no	—
50	#103×C₆₁	—	2	1	no	—
51	#102×C₆₁	—	2	1	no	—
52	#97×C₆₁	—	2	1	no	—
53	C₄×C₈×C₆₁	—	2	1	no	—
54	C₄×C₈×C₆₁	—	2	1	no	—
55	C₄×C₈×C₆₁	—	2	1	no	—
56	#41×C₆₁	—	2	1	no	—
57	#41×C₆₁	—	2	1	no	—
58	#41×C₆₁	—	2	1	no	—
59	#41×C₆₁	—	2	1	no	—
60	C₂×Q₈×C₆₁	M₁₆	1	1	yes	characteristic
61	C₂×D₄×C₆₁	—	2	1	no	—
62	C₂×D₄×C₆₁	M₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₆₁	—	2	1	no	—
64	C₂²×C₄×C₆₁	—	2	1	no	—
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	M₁₆×C₆₁	—	2	1	no	—
67	M₁₆×C₆₁	—	2	1	no	—
68	M₁₆×C₆₁	—	2	1	no	—
69	M₁₆×C₆₁	—	2	1	no	—
70	M₁₆×C₆₁	—	4	1	no	—
71	M₁₆×C₆₁	—	4	1	no	—
72	C₂×C₈×C₆₁	—	2	1	no	—
73	C₂×C₈×C₆₁	—	2	1	no	—
74	C₂×C₈×C₆₁	—	2	1	no	—
75	C₂×C₈×C₆₁	—	2	1	no	—
76	C₂×C₈×C₆₁	—	2	1	no	—
77	C₂×C₈×C₆₁	—	4	1	no	—
78	C₂×C₈×C₆₁	—	4	1	no	—
79	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
82	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
84	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
85	C₄⋊C₄×C₆₁	—	2	1	no	—
86	C₂²⋊C₄×C₆₁	—	2	1	no	—
87	C₄²×C₆₁	—	2	1	no	—
88	C₄²×C₆₁	—	2	1	no	—
89	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
90	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
91	C₁₆×C₆₁	—	2	2	no	—
92	C₁₆×C₆₁	—	2	2	no	—
93	C₁₆×C₆₁	—	4	1	no	—
94	C₁₆×C₆₁	—	4	1	no	—
95	C₂³×C₆₁	—	2	1	no	—
96	C₂³×C₆₁	#131	1	1	yes	characteristic
97	Q₈×C₆₁	—	2	1	no	—
98	Q₈×C₆₁	—	2	1	no	—
99	D₄×C₆₁	—	2	1	no	—
100	D₄×C₆₁	—	2	1	no	—
101	D₄×C₆₁	—	4	1	no	—
102	D₄×C₆₁	—	4	1	no	—
103	C₂×C₄×C₆₁	—	2	1	no	—
104	C₂×C₄×C₆₁	—	2	1	no	—
105	C₂×C₄×C₆₁	—	2	1	no	—
106	C₂×C₄×C₆₁	—	4	2	no	—
107	C₂×C₄×C₆₁	—	4	1	no	—
108	C₂×C₄×C₆₁	—	4	1	no	—
109	C₂×C₄×C₆₁	—	4	1	no	—
110	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
111	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
116	C₈×C₆₁	—	4	1	no	—
117	C₈×C₆₁	—	4	1	no	—
118	C₈×C₆₁	—	4	1	no	—
119	C₈×C₆₁	#121	1	1	yes	characteristic
120	C₈×C₆₁	#123	1	1	yes	characteristic
121	C₈×C₆₁	#124	1	1	yes	characteristic
122	C₈×C₆₁	#126	1	1	yes	characteristic
123	#13129	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₆₁	—	4	1	no	—
125	C₂²×C₆₁	—	4	1	no	—
126	C₂²×C₆₁	—	8	1	no	—
127	C₂²×C₆₁	#433	1	1	yes	characteristic
128	C₂²×C₆₁	#435	1	1	yes	characteristic
129	C₂²×C₆₁	#435	1	1	yes	characteristic
130	C₄×C₆₁	—	2	2	no	—
131	C₄×C₆₁	—	4	1	no	—
132	C₄×C₆₁	—	4	1	no	—
133	C₄×C₆₁	—	8	1	no	—
134	C₄×C₆₁	#388	1	1	yes	characteristic
135	C₄×C₆₁	#432	1	1	yes	characteristic
136	C₄×C₆₁	#435	1	1	yes	characteristic
137	C₄×C₆₁	#436	1	1	yes	characteristic
138	#2150	C₂×C₆₁	1	1	yes	characteristic, complemented
139	#2117	C₂×C₆₁	1	1	yes	characteristic
140	#2115	C₂×C₆₁	1	1	yes	characteristic
141	#2110	C₂×C₆₁	1	1	yes	characteristic, complemented
142	#1764	C₂×C₆₁	1	1	yes	characteristic
143	#1378	C₂×C₆₁	1	1	yes	characteristic, complemented
144	#1376	C₂×C₆₁	1	1	yes	characteristic, complemented
145	C₂×C₆₁	—	2	1	no	—
146	C₂×C₆₁	—	2	1	no	—
147	C₂×C₆₁	—	2	1	no	—
148	C₂×C₆₁	—	8	1	no	—
149	C₂×C₆₁	#1924	1	1	yes	characteristic
150	C₂×#131	C₂²×C₆₁	1	1	yes	characteristic
151	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
152	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
153	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
154	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
155	C₂×#104	C₄×C₆₁	1	1	yes	characteristic
156	#416	C₄×C₆₁	1	1	yes	characteristic
157	#405	C₂²×C₆₁	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
159	C₄⋊C₁₆	—	2	2	no	—
160	#350	—	2	2	no	—
161	#349	—	2	2	no	—
162	#347	—	2	2	no	—
163	#341	C₄×C₆₁	1	1	yes	characteristic
164	#339	C₄×C₆₁	1	1	yes	characteristic
165	C₆₁	#13129	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×#41	C₂×C₄×C₆₁	1	1	yes	characteristic
167	#131	—	2	1	no	—
168	#131	—	2	1	no	—
169	#131	—	2	1	no	—
170	#131	—	2	1	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
173	C₂×M₁₆	C₂×C₄×C₆₁	1	1	yes	characteristic
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	C₂×C₄×C₆₁	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
177	#117	C₂×C₄×C₆₁	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
179	M₃₂	—	2	1	no	—
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	2	1	no	—
184	M₃₂	—	2	1	no	—
185	M₃₂	—	2	1	no	—
186	M₃₂	D₄×C₆₁	1	2	yes	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	D₄×C₆₁	1	2	yes	—
191	#104	—	2	1	no	—
192	#104	—	2	1	no	—
193	#103	—	2	1	no	—
194	#102	—	2	1	no	—
195	#97	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	2	1	no	—
199	#41	—	2	1	no	—
200	#41	—	2	1	no	—
201	#41	—	2	1	no	—
202	#41	—	2	1	no	—
203	C₂×Q₈	M₁₆×C₆₁	1	1	yes	characteristic
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	M₁₆×C₆₁	1	1	yes	characteristic
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
209	M₁₆	—	2	1	no	—
210	M₁₆	—	2	1	no	—
211	M₁₆	—	2	1	no	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
224	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
225	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
226	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
227	C₂×C₈	#41×C₆₁	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	2	1	no	—
232	C₄²	M₁₆×C₆₁	1	1	yes	characteristic
233	C₄²	M₁₆×C₆₁	1	1	yes	characteristic
234	C₁₆	—	2	2	no	—
235	C₁₆	—	2	2	no	—
236	C₁₆	—	4	1	no	—
237	C₁₆	—	4	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	#131×C₆₁	1	1	yes	characteristic
240	Q₈	—	2	1	no	—
241	Q₈	—	2	1	no	—
242	D₄	—	2	1	no	—
243	D₄	—	2	1	no	—
244	D₄	—	4	1	no	—
245	D₄	—	4	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
256	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
257	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
258	C₂×C₄	#131×C₆₁	1	1	yes	characteristic
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₈	—	4	1	no	—
262	C₈	#121×C₆₁	1	1	yes	characteristic
263	C₈	#123×C₆₁	1	1	yes	characteristic
264	C₈	#124×C₆₁	1	1	yes	characteristic
265	C₈	#126×C₆₁	1	1	yes	characteristic
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	8	1	no	—
269	C₂²	C₆₁×#433	1	1	yes	characteristic
270	C₂²	C₆₁×#435	1	1	yes	characteristic
271	C₂²	C₆₁×#435	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	8	1	no	—
276	C₄	C₆₁×#388	1	1	yes	characteristic
277	C₄	C₆₁×#432	1	1	yes	characteristic
278	C₄	C₆₁×#435	1	1	yes	characteristic
279	C₄	C₆₁×#436	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	1	no	—
284	C₂	C₆₁×#1924	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₆₁	#124
commutator	C₂²×C₄	C₂²×C₄×C₆₁
Frattini, cosocle	C₂²×C₈	C₂³×C₆₁
socle	C₂×C₆₁	#1924
prime radical	C₂×#41×C₆₁	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₆₁ ◁ C₂²×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#2150	C₂	semidirect
4. C₆₁×#2110	C₂	semidirect
6. C₆₁×#1378	C₂	semidirect
7. C₆₁×#1376	C₂	semidirect
123. #13129	C₆₁	direct (N × H)
138. #2150	C₂×C₆₁	semidirect
141. #2110	C₂×C₆₁	semidirect
143. #1378	C₂×C₆₁	semidirect
144. #1376	C₂×C₆₁	semidirect
165. C₆₁	#13129	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×#2150	2
C₆₁×#2117	2
C₆₁×#2115	2
C₆₁×#2110	2
C₆₁×#1764	2
C₆₁×#1378	2
C₆₁×#1376	2
#13129	61

Subgroups — C61×#13129 #29516562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#13129 #29516562