Subgroups — C₆₁×#9581 #29513014

Order 15616.

364 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#1962	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1540	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1480	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#1301	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#532	C₂²	1	1	yes	characteristic
9	C₆₁×#431	C₂²	1	1	yes	characteristic
10	C₆₁×#431	C₂²	1	1	yes	characteristic
11	C₆₁×#423	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×#104×C₆₁	—	2	1	no	—
15	C₂×#104×C₆₁	—	2	1	no	—
16	C₂×#97×C₆₁	—	2	1	no	—
17	C₄³×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#336	—	2	1	no	—
19	C₆₁×#335	—	2	2	no	—
20	C₆₁×#332	C₄	1	2	yes	complemented
21	C₆₁×#331	C₄	1	2	yes	complemented
22	C₆₁×D₄⋊C₈	—	2	2	no	—
23	C₂×#41×C₆₁	—	2	1	no	—
24	C₂×M₁₆×C₆₁	—	2	1	no	—
25	C₂²×C₈×C₆₁	—	2	1	no	—
26	C₂²×C₈×C₆₁	—	2	1	no	—
27	C₄⋊Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	#127×C₆₁	C₂×C₄	1	1	yes	characteristic
29	#124×C₆₁	—	2	1	no	—
30	#121×C₆₁	—	2	1	no	—
31	#121×C₆₁	—	2	1	no	—
32	C₄×D₄×C₆₁	—	2	1	no	—
33	C₄×D₄×C₆₁	—	2	1	no	—
34	C₂×C₄²×C₆₁	—	2	1	no	—
35	C₂×C₄²×C₆₁	—	2	1	no	—
36	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₆₁	—	2	1	no	—
40	C₄⋊C₈×C₆₁	—	2	1	no	—
41	C₄⋊C₈×C₆₁	—	2	1	no	—
42	C₄⋊C₈×C₆₁	—	2	1	no	—
43	C₄⋊C₈×C₆₁	—	2	1	no	—
44	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
45	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
46	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
47	#104×C₆₁	—	4	2	no	—
48	#104×C₆₁	—	4	2	no	—
49	C₂²⋊C₈×C₆₁	—	2	1	no	—
50	C₂²⋊C₈×C₆₁	—	2	1	no	—
51	#97×C₆₁	—	4	2	no	—
52	C₄×C₈×C₆₁	—	2	1	no	—
53	C₄×C₈×C₆₁	—	2	1	no	—
54	#41×C₆₁	—	4	1	no	—
55	#41×C₆₁	—	4	1	no	—
56	#41×C₆₁	—	4	1	no	—
57	#41×C₆₁	—	4	1	no	—
58	C₂×Q₈×C₆₁	—	2	1	no	—
59	C₂×D₄×C₆₁	—	2	1	no	—
60	C₂×D₄×C₆₁	—	2	1	no	—
61	C₂×D₄×C₆₁	—	2	1	no	—
62	C₂×D₄×C₆₁	—	4	1	no	—
63	C₂²×C₄×C₆₁	—	2	1	no	—
64	C₂²×C₄×C₆₁	—	2	1	no	—
65	C₂²×C₄×C₆₁	—	4	1	no	—
66	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
68	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
69	M₁₆×C₆₁	—	4	1	no	—
70	M₁₆×C₆₁	—	4	1	no	—
71	C₂×C₈×C₆₁	—	2	2	no	—
72	C₂×C₈×C₆₁	—	2	2	no	—
73	C₂×C₈×C₆₁	—	2	2	no	—
74	C₂×C₈×C₆₁	—	4	1	no	—
75	C₂×C₈×C₆₁	—	4	1	no	—
76	C₂×C₈×C₆₁	—	4	1	no	—
77	C₂×C₈×C₆₁	—	4	1	no	—
78	C₂×C₈×C₆₁	—	4	1	no	—
79	C₂×C₈×C₆₁	—	4	1	no	—
80	C₄⋊C₄×C₆₁	—	2	1	no	—
81	C₄⋊C₄×C₆₁	—	2	1	no	—
82	C₂²⋊C₄×C₆₁	—	4	1	no	—
83	C₂²⋊C₄×C₆₁	—	4	1	no	—
84	C₄²×C₆₁	—	2	1	no	—
85	C₄²×C₆₁	—	2	1	no	—
86	C₄²×C₆₁	—	2	1	no	—
87	C₄²×C₆₁	—	4	2	no	—
88	C₄²×C₆₁	—	4	2	no	—
89	C₄²×C₆₁	D₈	1	2	yes	—
90	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₆₁	#41	1	1	yes	characteristic
93	C₄²×C₆₁	#41	1	1	yes	characteristic
94	C₂³×C₆₁	—	4	1	no	—
95	C₂³×C₆₁	#123	1	1	yes	characteristic
96	Q₈×C₆₁	—	4	1	no	—
97	Q₈×C₆₁	—	4	1	no	—
98	D₄×C₆₁	—	4	1	no	—
99	D₄×C₆₁	—	4	1	no	—
100	D₄×C₆₁	—	4	2	no	—
101	D₄×C₆₁	—	8	1	no	—
102	D₄×C₆₁	—	8	1	no	—
103	D₄×C₆₁	—	8	1	no	—
104	C₂×C₄×C₆₁	—	2	2	no	—
105	C₂×C₄×C₆₁	—	2	2	no	—
106	C₂×C₄×C₆₁	—	2	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	—	4	1	no	—
111	C₂×C₄×C₆₁	—	4	1	no	—
112	C₂×C₄×C₆₁	—	4	1	no	—
113	C₂×C₄×C₆₁	—	8	1	no	—
114	C₂×C₄×C₆₁	—	8	1	no	—
115	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
116	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
119	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
120	C₂×C₄×C₆₁	#126	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
123	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
125	C₈×C₆₁	—	4	2	no	—
126	C₈×C₆₁	—	4	2	no	—
127	C₈×C₆₁	—	4	2	no	—
128	C₈×C₆₁	—	8	1	no	—
129	C₈×C₆₁	—	8	1	no	—
130	#9581	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₆₁	—	2	1	no	—
132	C₂²×C₆₁	—	2	1	no	—
133	C₂²×C₆₁	—	8	1	no	—
134	C₂²×C₆₁	—	8	1	no	—
135	C₂²×C₆₁	—	8	1	no	—
136	C₂²×C₆₁	#388	1	1	yes	characteristic
137	C₂²×C₆₁	#480	1	1	yes	characteristic
138	C₂²×C₆₁	#482	1	1	yes	characteristic
139	C₄×C₆₁	—	2	1	no	—
140	C₄×C₆₁	—	2	1	no	—
141	C₄×C₆₁	—	2	1	no	—
142	C₄×C₆₁	—	4	2	no	—
143	C₄×C₆₁	—	4	2	no	—
144	C₄×C₆₁	—	8	1	no	—
145	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
146	C₄×C₆₁	#440	1	1	yes	characteristic
147	C₄×C₆₁	#444	1	2	yes	—
148	C₄×C₆₁	#459	1	1	yes	characteristic
149	C₄×C₆₁	#464	1	1	yes	characteristic
150	#1962	C₂×C₆₁	1	1	yes	characteristic
151	#1767	C₂×C₆₁	1	1	yes	characteristic, complemented
152	#1571	C₂×C₆₁	1	1	yes	characteristic, complemented
153	#1540	C₂×C₆₁	1	1	yes	characteristic
154	#1480	C₂×C₆₁	1	1	yes	characteristic
155	#1301	C₂×C₆₁	1	1	yes	characteristic, complemented
156	#1276	C₂×C₆₁	1	1	yes	characteristic, complemented
157	C₂×C₆₁	—	2	1	no	—
158	C₂×C₆₁	—	2	1	no	—
159	C₂×C₆₁	—	16	1	no	—
160	C₂×C₆₁	#1925	1	1	yes	characteristic
161	C₂×C₆₁	#1932	1	1	yes	characteristic
162	C₂×C₆₁	#1938	1	1	yes	characteristic
163	#532	C₂²×C₆₁	1	1	yes	characteristic
164	#431	C₂²×C₆₁	1	1	yes	characteristic
165	#431	C₂²×C₆₁	1	1	yes	characteristic
166	#423	C₂²×C₆₁	1	1	yes	characteristic
167	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
168	C₂×C₄⋊C₈	C₂²×C₆₁	1	1	yes	characteristic
169	C₂×#104	—	2	1	no	—
170	C₂×#104	—	2	1	no	—
171	C₂×#97	—	2	1	no	—
172	C₄³	C₂²×C₆₁	1	1	yes	characteristic
173	#336	—	2	1	no	—
174	#335	—	2	2	no	—
175	#332	C₄×C₆₁	1	2	yes	complemented
176	#331	C₄×C₆₁	1	2	yes	complemented
177	D₄⋊C₈	—	2	2	no	—
178	C₆₁	#9581	1	1	yes	characteristic, direct product factor, Sylow
179	C₂×#41	—	2	1	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₄⋊Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
184	#127	C₂×C₄×C₆₁	1	1	yes	characteristic
185	#124	—	2	1	no	—
186	#121	—	2	1	no	—
187	#121	—	2	1	no	—
188	C₄×D₄	—	2	1	no	—
189	C₄×D₄	—	2	1	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
193	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	2	1	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
201	C₄⋊C₈	C₂×C₄×C₆₁	1	2	yes	—
202	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
203	#104	—	4	2	no	—
204	#104	—	4	2	no	—
205	C₂²⋊C₈	—	2	1	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	#97	—	4	2	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	#41	—	4	1	no	—
213	#41	—	4	1	no	—
214	C₂×Q₈	—	2	1	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	—	4	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
223	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
224	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
225	M₁₆	—	4	1	no	—
226	M₁₆	—	4	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	4	1	no	—
239	C₂²⋊C₄	—	4	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	4	2	no	—
244	C₄²	—	4	2	no	—
245	C₄²	D₈×C₆₁	1	2	yes	—
246	C₄²	C₂²×C₄×C₆₁	1	1	yes	characteristic
247	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
248	C₄²	#41×C₆₁	1	1	yes	characteristic
249	C₄²	#41×C₆₁	1	1	yes	characteristic
250	C₂³	—	4	1	no	—
251	C₂³	#123×C₆₁	1	1	yes	characteristic
252	Q₈	—	4	1	no	—
253	Q₈	—	4	1	no	—
254	D₄	—	4	1	no	—
255	D₄	—	4	1	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	1	no	—
258	D₄	—	8	1	no	—
259	D₄	—	8	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	#117×C₆₁	1	1	yes	characteristic
272	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
273	C₂×C₄	C₄×D₄×C₆₁	1	1	yes	characteristic
274	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
275	C₂×C₄	#124×C₆₁	1	1	yes	characteristic
276	C₂×C₄	#126×C₆₁	1	1	yes	characteristic
277	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
278	C₂×C₄	#135×C₆₁	1	1	yes	characteristic
279	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
280	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
281	C₈	—	4	2	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₈	—	8	1	no	—
285	C₈	—	8	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	8	1	no	—
289	C₂²	—	8	1	no	—
290	C₂²	—	8	1	no	—
291	C₂²	C₆₁×#388	1	1	yes	characteristic
292	C₂²	C₆₁×#480	1	1	yes	characteristic
293	C₂²	C₆₁×#482	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	8	1	no	—
300	C₄	C₄×D₈×C₆₁	1	1	yes	characteristic
301	C₄	C₆₁×#440	1	1	yes	characteristic
302	C₄	C₆₁×#444	1	2	yes	—
303	C₄	C₆₁×#459	1	1	yes	characteristic
304	C₄	C₆₁×#464	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	16	1	no	—
308	C₂	C₆₁×#1925	1	1	yes	characteristic
309	C₂	C₆₁×#1932	1	1	yes	characteristic
310	C₂	C₆₁×#1938	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	#121
commutator	C₄²	C₂²×C₄×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
socle	C₂²×C₆₁	#388
prime radical	C₆₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#1767	C₂	semidirect
3. C₆₁×#1571	C₂	semidirect
6. C₆₁×#1301	C₂	semidirect
7. C₆₁×#1276	C₂	semidirect
20. C₆₁×#332	C₄	semidirect
21. C₆₁×#331	C₄	semidirect
130. #9581	C₆₁	direct (N × H)
151. #1767	C₂×C₆₁	semidirect
152. #1571	C₂×C₆₁	semidirect
155. #1301	C₂×C₆₁	semidirect
156. #1276	C₂×C₆₁	semidirect
175. #332	C₄×C₆₁	semidirect
176. #331	C₄×C₆₁	semidirect
178. C₆₁	#9581	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×#1962	2
C₆₁×#1767	2
C₆₁×#1571	2
C₆₁×#1540	2
C₆₁×#1480	2
C₆₁×#1301	2
C₆₁×#1276	2
#9581	61

Subgroups — C61×#9581 #29513014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#9581 #29513014