Subgroups — C₆₁×#33743 #29537176

Order 15616.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3176	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2253	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2245	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2244	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2220	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2217	C₂	1	2	yes	maximal, complemented
9	C₆₁×#2175	C₂	1	2	yes	maximal, complemented
10	C₄×SD₃₂×C₆₁	C₂	1	2	yes	maximal, complemented
11	C₆₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#541	—	2	1	no	—
13	C₆₁×#540	—	2	1	no	—
14	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	—
15	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	—
16	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#499	C₂²	1	1	yes	characteristic
19	C₆₁×#499	C₂²	1	1	yes	characteristic
20	C₆₁×#494	C₂²	1	1	yes	characteristic
21	C₆₁×#493	C₂²	1	1	yes	characteristic
22	C₆₁×#491	—	2	1	no	—
23	C₆₁×#490	—	2	1	no	—
24	C₆₁×#470	—	2	1	no	—
25	C₆₁×#469	—	2	1	no	—
26	C₆₁×#467	C₂²	1	2	yes	—
27	C₆₁×#466	C₂²	1	2	yes	—
28	C₆₁×#442	—	2	1	no	—
29	C₆₁×#441	—	2	1	no	—
30	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
31	C₄×D₈×C₆₁	C₂²	1	2	yes	—
32	C₆₁×#405	C₂²	1	1	yes	characteristic
33	C₆₁×#367	C₂²	1	2	yes	—
34	C₆₁×#366	C₂²	1	2	yes	—
35	C₆₁×#358	C₂²	1	2	yes	—
36	C₆₁×#358	C₂²	1	2	yes	—
37	C₆₁×#357	C₂²	1	2	yes	—
38	C₆₁×#357	C₂²	1	2	yes	—
39	C₆₁×#346	C₂²	1	1	yes	characteristic
40	C₆₁×#346	C₂²	1	1	yes	characteristic
41	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
42	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
43	C₂×Q₁₆×C₆₁	—	2	1	no	—
44	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
45	C₂×D₈×C₆₁	—	2	1	no	—
46	C₂×D₈×C₆₁	C₂³	1	2	yes	—
47	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₁	—	2	1	no	—
50	#127×C₆₁	—	2	1	no	—
51	#126×C₆₁	—	4	2	no	—
52	#125×C₆₁	—	2	1	no	—
53	#125×C₆₁	—	2	1	no	—
54	#125×C₆₁	—	4	2	no	—
55	#123×C₆₁	—	4	2	no	—
56	C₂²⋊Q₈×C₆₁	—	2	2	no	—
57	#121×C₆₁	—	2	2	no	—
58	#121×C₆₁	—	4	2	no	—
59	C₄×Q₈×C₆₁	—	2	2	no	—
60	C₄×D₄×C₆₁	—	2	2	no	—
61	#117×C₆₁	D₄	1	1	yes	characteristic
62	SD₃₂×C₆₁	—	2	4	no	—
63	SD₃₂×C₆₁	—	2	4	no	—
64	M₃₂×C₆₁	—	2	2	no	—
65	C₂×C₁₆×C₆₁	—	2	2	no	—
66	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
70	#107×C₆₁	C₂³	1	2	yes	—
71	#107×C₆₁	C₂³	1	2	yes	—
72	#106×C₆₁	—	2	1	no	—
73	#106×C₆₁	—	2	1	no	—
74	#103×C₆₁	—	2	2	no	—
75	#103×C₆₁	—	2	2	no	—
76	#102×C₆₁	—	2	2	no	—
77	#102×C₆₁	—	2	2	no	—
78	#97×C₆₁	D₄	1	1	yes	characteristic
79	#97×C₆₁	D₄	1	1	yes	characteristic
80	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₆₁	—	2	2	no	—
83	C₂×D₄×C₆₁	—	2	2	no	—
84	C₂×D₄×C₆₁	—	4	2	no	—
85	C₂×D₄×C₆₁	—	4	2	no	—
86	C₂²×C₄×C₆₁	—	4	2	no	—
87	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₆₁	—	2	2	no	—
89	Q₁₆×C₆₁	—	2	2	no	—
90	Q₁₆×C₆₁	—	4	2	no	—
91	D₈×C₆₁	—	2	2	no	—
92	D₈×C₆₁	—	2	2	no	—
93	D₈×C₆₁	—	4	2	no	—
94	M₁₆×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	2	1	no	—
96	C₂×C₈×C₆₁	—	2	1	no	—
97	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
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	—	4	2	no	—
108	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
109	C₂²⋊C₄×C₆₁	—	4	2	no	—
110	C₂²⋊C₄×C₆₁	—	4	2	no	—
111	C₂²⋊C₄×C₆₁	—	4	2	no	—
112	C₂²⋊C₄×C₆₁	D₈	1	2	yes	—
113	C₄²×C₆₁	—	4	2	no	—
114	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₆₁	—	2	1	no	—
117	C₁₆×C₆₁	—	2	1	no	—
118	C₁₆×C₆₁	#41	1	2	yes	—
119	C₁₆×C₆₁	#41	1	2	yes	—
120	C₂³×C₆₁	—	4	2	no	—
121	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
122	Q₈×C₆₁	—	4	2	no	—
123	Q₈×C₆₁	—	4	2	no	—
124	D₄×C₆₁	—	4	2	no	—
125	D₄×C₆₁	—	4	2	no	—
126	D₄×C₆₁	—	16	2	no	—
127	D₄×C₆₁	—	16	2	no	—
128	C₂×C₄×C₆₁	—	2	1	no	—
129	C₂×C₄×C₆₁	—	2	1	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	—	16	2	no	—
134	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₆₁	—	2	1	no	—
139	C₈×C₆₁	—	2	1	no	—
140	C₈×C₆₁	#135	1	2	yes	—
141	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
142	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
143	#33743	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₆₁	—	2	2	no	—
145	C₂²×C₆₁	—	2	1	no	—
146	C₂²×C₆₁	—	8	2	no	—
147	C₂²×C₆₁	—	8	2	no	—
148	C₂²×C₆₁	—	8	2	no	—
149	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₆₁	—	2	2	no	—
151	C₄×C₆₁	—	2	2	no	—
152	C₄×C₆₁	—	2	1	no	—
153	C₄×C₆₁	—	8	2	no	—
154	C₄×C₆₁	—	8	2	no	—
155	C₄×C₆₁	—	8	2	no	—
156	C₄×C₆₁	#532	1	1	yes	characteristic
157	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
158	#3177	C₂×C₆₁	1	1	yes	characteristic, complemented
159	#3176	C₂×C₆₁	1	1	yes	characteristic
160	#2253	C₂×C₆₁	1	1	yes	characteristic, complemented
161	#2248	C₂×C₆₁	1	1	yes	characteristic, complemented
162	#2245	C₂×C₆₁	1	1	yes	characteristic, complemented
163	#2244	C₂×C₆₁	1	1	yes	characteristic, complemented
164	#2220	C₂×C₆₁	1	2	yes	complemented
165	#2217	C₂×C₆₁	1	2	yes	complemented
166	#2175	C₂×C₆₁	1	2	yes	complemented
167	C₄×SD₃₂	C₂×C₆₁	1	2	yes	complemented
168	#2108	C₂×C₆₁	1	1	yes	characteristic, complemented
169	C₂×C₆₁	—	4	1	no	—
170	C₂×C₆₁	—	16	2	no	—
171	C₂×C₆₁	#3169	1	1	yes	characteristic
172	C₂×C₆₁	#3416	1	2	yes	—
173	#541	—	2	1	no	—
174	#540	—	2	1	no	—
175	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	—
176	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	—
177	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
178	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
179	#499	C₂²×C₆₁	1	1	yes	characteristic
180	#499	C₂²×C₆₁	1	1	yes	characteristic
181	#494	C₂²×C₆₁	1	1	yes	characteristic
182	#493	C₂²×C₆₁	1	1	yes	characteristic
183	#491	—	2	1	no	—
184	#490	—	2	1	no	—
185	#470	—	2	1	no	—
186	#469	—	2	1	no	—
187	#467	C₂²×C₆₁	1	2	yes	—
188	#466	C₂²×C₆₁	1	2	yes	—
189	#442	—	2	1	no	—
190	#441	—	2	1	no	—
191	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
192	C₄×D₈	C₂²×C₆₁	1	2	yes	—
193	#405	C₂²×C₆₁	1	1	yes	characteristic
194	#367	C₂²×C₆₁	1	2	yes	—
195	#366	C₂²×C₆₁	1	2	yes	—
196	#358	C₂²×C₆₁	1	2	yes	—
197	#358	C₂²×C₆₁	1	2	yes	—
198	#357	C₂²×C₆₁	1	2	yes	—
199	#357	C₂²×C₆₁	1	2	yes	—
200	#346	C₂²×C₆₁	1	1	yes	characteristic
201	#346	C₂²×C₆₁	1	1	yes	characteristic
202	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
203	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
204	C₆₁	#33743	1	1	yes	characteristic, direct product factor, Sylow
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	C₂³×C₆₁	1	2	yes	—
209	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	1	no	—
212	#127	—	2	1	no	—
213	#126	—	4	2	no	—
214	#125	—	2	1	no	—
215	#125	—	2	1	no	—
216	#125	—	4	2	no	—
217	#123	—	4	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	#121	—	2	2	no	—
220	#121	—	4	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×D₄	—	2	2	no	—
223	#117	D₄×C₆₁	1	1	yes	characteristic
224	SD₃₂	—	2	4	no	—
225	SD₃₂	—	2	4	no	—
226	M₃₂	—	2	2	no	—
227	C₂×C₁₆	—	2	2	no	—
228	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
229	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
231	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
232	#107	C₂³×C₆₁	1	2	yes	—
233	#107	C₂³×C₆₁	1	2	yes	—
234	#106	—	2	1	no	—
235	#106	—	2	1	no	—
236	#103	—	2	2	no	—
237	#103	—	2	2	no	—
238	#102	—	2	2	no	—
239	#102	—	2	2	no	—
240	#97	D₄×C₆₁	1	1	yes	characteristic
241	#97	D₄×C₆₁	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	4	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	2	2	no	—
255	D₈	—	4	2	no	—
256	M₁₆	—	2	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
262	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	D₈×C₆₁	1	2	yes	—
275	C₄²	—	4	2	no	—
276	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	#41×C₆₁	1	2	yes	—
281	C₁₆	#41×C₆₁	1	2	yes	—
282	C₂³	—	4	2	no	—
283	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	16	2	no	—
289	D₄	—	16	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	16	2	no	—
296	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
297	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
298	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	#135×C₆₁	1	2	yes	—
303	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
304	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
305	C₂²	—	2	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	C₆₁×#532	1	1	yes	characteristic
318	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
319	C₂	—	4	1	no	—
320	C₂	—	16	2	no	—
321	C₂	C₆₁×#3169	1	1	yes	characteristic
322	C₂	C₆₁×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₆₁×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ #117×C₆₁ ◁ C₆₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3177	C₂	semidirect
3. C₆₁×#2253	C₂	semidirect
4. C₆₁×#2248	C₂	semidirect
5. C₆₁×#2245	C₂	semidirect
6. C₆₁×#2244	C₂	semidirect
7. C₆₁×#2220	C₂	semidirect
8. C₆₁×#2217	C₂	semidirect
9. C₆₁×#2175	C₂	semidirect
10. C₄×SD₃₂×C₆₁	C₂	semidirect
11. C₆₁×#2108	C₂	semidirect
143. #33743	C₆₁	direct (N × H)
158. #3177	C₂×C₆₁	semidirect
160. #2253	C₂×C₆₁	semidirect
161. #2248	C₂×C₆₁	semidirect
162. #2245	C₂×C₆₁	semidirect
163. #2244	C₂×C₆₁	semidirect
164. #2220	C₂×C₆₁	semidirect
165. #2217	C₂×C₆₁	semidirect
166. #2175	C₂×C₆₁	semidirect
167. C₄×SD₃₂	C₂×C₆₁	semidirect
168. #2108	C₂×C₆₁	semidirect
204. C₆₁	#33743	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×#3177	2
C₆₁×#3176	2
C₆₁×#2253	2
C₆₁×#2248	2
C₆₁×#2245	2
C₆₁×#2244	2
C₆₁×#2220	2
C₆₁×#2217	2
C₆₁×#2175	2
C₄×SD₃₂×C₆₁	2
C₆₁×#2108	2
#33743	61

Subgroups — C61×#33743 #29537176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33743 #29537176