Subgroups — C₆₁×#33742 #29537175

Order 15616.

508 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3175	C₂	1	2	yes	maximal, complemented
2	C₆₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2243	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2219	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2218	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#2217	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2177	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#2176	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₃₂×C₆₁	C₂	1	2	yes	maximal, complemented
11	C₆₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#542	—	2	2	no	—
13	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	—
15	C₂×D₁₆×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₆₁×#500	C₂²	1	2	yes	—
19	C₆₁×#495	C₂²	1	2	yes	—
20	C₆₁×#489	—	2	2	no	—
21	C₆₁×#468	—	2	2	no	—
22	C₆₁×#467	C₂²	1	2	yes	—
23	C₆₁×#466	C₂²	1	2	yes	—
24	C₆₁×#440	—	2	2	no	—
25	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
26	C₄×D₈×C₆₁	C₂²	1	2	yes	—
27	C₆₁×#405	C₂²	1	1	yes	characteristic
28	C₆₁×#367	C₂²	1	2	yes	—
29	C₆₁×#367	C₂²	1	1	yes	characteristic
30	C₆₁×#367	C₂²	1	1	yes	characteristic
31	C₆₁×#358	C₂²	1	2	yes	—
32	C₆₁×#358	C₂²	1	2	yes	—
33	C₆₁×#357	C₂²	1	2	yes	—
34	C₆₁×#357	C₂²	1	2	yes	—
35	C₆₁×#346	C₂²	1	1	yes	characteristic
36	C₆₁×#346	C₂²	1	1	yes	characteristic
37	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₆₁	—	2	2	no	—
41	C₂×D₈×C₆₁	C₂³	1	2	yes	—
42	C₂×M₁₆×C₆₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈×C₆₁	—	2	2	no	—
45	#126×C₆₁	—	4	2	no	—
46	#124×C₆₁	—	2	2	no	—
47	#124×C₆₁	—	4	2	no	—
48	#123×C₆₁	—	4	2	no	—
49	C₂²⋊Q₈×C₆₁	—	2	2	no	—
50	C₂²⋊Q₈×C₆₁	—	4	2	no	—
51	#121×C₆₁	—	2	2	no	—
52	C₄×Q₈×C₆₁	—	2	2	no	—
53	C₄×D₄×C₆₁	—	2	2	no	—
54	#117×C₆₁	D₄	1	1	yes	characteristic
55	Q₃₂×C₆₁	—	2	2	no	—
56	SD₃₂×C₆₁	—	2	4	no	—
57	D₁₆×C₆₁	—	2	2	no	—
58	M₃₂×C₆₁	—	2	2	no	—
59	C₂×C₁₆×C₆₁	—	2	2	no	—
60	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
64	#107×C₆₁	—	2	2	no	—
65	#107×C₆₁	C₂³	1	2	yes	—
66	#107×C₆₁	C₂³	1	2	yes	—
67	#103×C₆₁	—	2	2	no	—
68	#103×C₆₁	—	2	2	no	—
69	#102×C₆₁	—	2	2	no	—
70	#102×C₆₁	—	2	2	no	—
71	#97×C₆₁	D₄	1	1	yes	characteristic
72	#97×C₆₁	D₄	1	1	yes	characteristic
73	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂×Q₈×C₆₁	—	4	2	no	—
77	C₂×D₄×C₆₁	—	2	2	no	—
78	C₂×D₄×C₆₁	—	4	2	no	—
79	C₂²×C₄×C₆₁	—	4	2	no	—
80	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₆₁	—	2	2	no	—
82	Q₁₆×C₆₁	—	2	2	no	—
83	SD₁₆×C₆₁	—	4	4	no	—
84	D₈×C₆₁	—	2	2	no	—
85	D₈×C₆₁	—	2	2	no	—
86	M₁₆×C₆₁	—	2	2	no	—
87	C₂×C₈×C₆₁	—	2	1	no	—
88	C₂×C₈×C₆₁	—	2	1	no	—
89	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₄⋊C₄×C₆₁	—	2	2	no	—
95	C₄⋊C₄×C₆₁	—	4	2	no	—
96	C₄⋊C₄×C₆₁	—	4	2	no	—
97	C₄⋊C₄×C₆₁	—	4	2	no	—
98	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
99	C₂²⋊C₄×C₆₁	—	4	2	no	—
100	C₂²⋊C₄×C₆₁	—	4	2	no	—
101	C₂²⋊C₄×C₆₁	—	4	2	no	—
102	C₂²⋊C₄×C₆₁	—	4	2	no	—
103	C₂²⋊C₄×C₆₁	—	4	2	no	—
104	C₂²⋊C₄×C₆₁	D₈	1	2	yes	—
105	C₄²×C₆₁	—	4	2	no	—
106	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
108	C₁₆×C₆₁	—	2	1	no	—
109	C₁₆×C₆₁	—	2	1	no	—
110	C₁₆×C₆₁	#41	1	2	yes	—
111	C₁₆×C₆₁	#41	1	2	yes	—
112	C₂³×C₆₁	—	4	2	no	—
113	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₆₁	—	4	2	no	—
115	Q₈×C₆₁	—	4	2	no	—
116	Q₈×C₆₁	—	16	2	no	—
117	D₄×C₆₁	—	4	2	no	—
118	D₄×C₆₁	—	4	2	no	—
119	D₄×C₆₁	—	16	2	no	—
120	C₂×C₄×C₆₁	—	2	1	no	—
121	C₂×C₄×C₆₁	—	2	1	no	—
122	C₂×C₄×C₆₁	—	4	2	no	—
123	C₂×C₄×C₆₁	—	4	2	no	—
124	C₂×C₄×C₆₁	—	4	2	no	—
125	C₂×C₄×C₆₁	—	16	2	no	—
126	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₆₁	—	2	1	no	—
131	C₈×C₆₁	—	2	1	no	—
132	C₈×C₆₁	#135	1	2	yes	—
133	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
134	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
135	#33742	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂²×C₆₁	—	2	2	no	—
137	C₂²×C₆₁	—	2	1	no	—
138	C₂²×C₆₁	—	8	2	no	—
139	C₂²×C₆₁	—	8	2	no	—
140	C₂²×C₆₁	—	8	2	no	—
141	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
142	C₄×C₆₁	—	2	2	no	—
143	C₄×C₆₁	—	2	2	no	—
144	C₄×C₆₁	—	2	1	no	—
145	C₄×C₆₁	—	8	2	no	—
146	C₄×C₆₁	—	8	2	no	—
147	C₄×C₆₁	—	8	2	no	—
148	C₄×C₆₁	#532	1	1	yes	characteristic
149	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
150	#3175	C₂×C₆₁	1	2	yes	complemented
151	#2254	C₂×C₆₁	1	1	yes	characteristic, complemented
152	#2249	C₂×C₆₁	1	1	yes	characteristic, complemented
153	#2243	C₂×C₆₁	1	2	yes	complemented
154	#2219	C₂×C₆₁	1	1	yes	characteristic, complemented
155	#2218	C₂×C₆₁	1	1	yes	characteristic
156	#2217	C₂×C₆₁	1	2	yes	complemented
157	#2177	C₂×C₆₁	1	1	yes	characteristic, complemented
158	#2176	C₂×C₆₁	1	1	yes	characteristic, complemented
159	C₄×SD₃₂	C₂×C₆₁	1	2	yes	complemented
160	#2108	C₂×C₆₁	1	1	yes	characteristic, complemented
161	C₂×C₆₁	—	4	1	no	—
162	C₂×C₆₁	—	16	2	no	—
163	C₂×C₆₁	#3169	1	1	yes	characteristic
164	C₂×C₆₁	#3416	1	2	yes	—
165	#542	—	2	2	no	—
166	C₂×Q₃₂	C₂²×C₆₁	1	1	yes	characteristic
167	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	—
168	C₂×D₁₆	C₂²×C₆₁	1	1	yes	characteristic
169	C₂×M₃₂	C₂²×C₆₁	1	1	yes	characteristic
170	C₂²×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
171	#500	C₂²×C₆₁	1	2	yes	—
172	#495	C₂²×C₆₁	1	2	yes	—
173	#489	—	2	2	no	—
174	#468	—	2	2	no	—
175	#467	C₂²×C₆₁	1	2	yes	—
176	#466	C₂²×C₆₁	1	2	yes	—
177	#440	—	2	2	no	—
178	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
179	C₄×D₈	C₂²×C₆₁	1	2	yes	—
180	#405	C₂²×C₆₁	1	1	yes	characteristic
181	#367	C₂²×C₆₁	1	2	yes	—
182	#367	C₂²×C₆₁	1	1	yes	characteristic
183	#367	C₂²×C₆₁	1	1	yes	characteristic
184	#358	C₂²×C₆₁	1	2	yes	—
185	#358	C₂²×C₆₁	1	2	yes	—
186	#357	C₂²×C₆₁	1	2	yes	—
187	#357	C₂²×C₆₁	1	2	yes	—
188	#346	C₂²×C₆₁	1	1	yes	characteristic
189	#346	C₂²×C₆₁	1	1	yes	characteristic
190	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
191	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
192	C₆₁	#33742	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
194	C₂×SD₁₆	—	2	2	no	—
195	C₂×D₈	C₂³×C₆₁	1	2	yes	—
196	C₂×M₁₆	D₄×C₆₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	2	no	—
199	#126	—	4	2	no	—
200	#124	—	2	2	no	—
201	#124	—	4	2	no	—
202	#123	—	4	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	4	2	no	—
205	#121	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×D₄	—	2	2	no	—
208	#117	D₄×C₆₁	1	1	yes	characteristic
209	Q₃₂	—	2	2	no	—
210	SD₃₂	—	2	4	no	—
211	D₁₆	—	2	2	no	—
212	M₃₂	—	2	2	no	—
213	C₂×C₁₆	—	2	2	no	—
214	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
215	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
216	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
217	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
218	#107	—	2	2	no	—
219	#107	C₂³×C₆₁	1	2	yes	—
220	#107	C₂³×C₆₁	1	2	yes	—
221	#103	—	2	2	no	—
222	#103	—	2	2	no	—
223	#102	—	2	2	no	—
224	#102	—	2	2	no	—
225	#97	D₄×C₆₁	1	1	yes	characteristic
226	#97	D₄×C₆₁	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	SD₁₆	—	4	4	no	—
238	D₈	—	2	2	no	—
239	D₈	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
246	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	D₈×C₆₁	1	2	yes	—
259	C₄²	—	4	2	no	—
260	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
261	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
262	C₁₆	—	2	1	no	—
263	C₁₆	—	2	1	no	—
264	C₁₆	#41×C₆₁	1	2	yes	—
265	C₁₆	#41×C₆₁	1	2	yes	—
266	C₂³	—	4	2	no	—
267	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	16	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	16	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	16	2	no	—
280	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
281	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
282	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	#135×C₆₁	1	2	yes	—
287	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
288	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
289	C₂²	—	2	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	C₆₁×#532	1	1	yes	characteristic
302	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₆₁×#3169	1	1	yes	characteristic
306	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×#2218	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×#3175	C₂	semidirect
2. C₆₁×#2254	C₂	semidirect
3. C₆₁×#2249	C₂	semidirect
4. C₆₁×#2243	C₂	semidirect
5. C₆₁×#2219	C₂	semidirect
7. C₆₁×#2217	C₂	semidirect
8. C₆₁×#2177	C₂	semidirect
9. C₆₁×#2176	C₂	semidirect
10. C₄×SD₃₂×C₆₁	C₂	semidirect
11. C₆₁×#2108	C₂	semidirect
135. #33742	C₆₁	direct (N × H)
150. #3175	C₂×C₆₁	semidirect
151. #2254	C₂×C₆₁	semidirect
152. #2249	C₂×C₆₁	semidirect
153. #2243	C₂×C₆₁	semidirect
154. #2219	C₂×C₆₁	semidirect
156. #2217	C₂×C₆₁	semidirect
157. #2177	C₂×C₆₁	semidirect
158. #2176	C₂×C₆₁	semidirect
159. C₄×SD₃₂	C₂×C₆₁	semidirect
160. #2108	C₂×C₆₁	semidirect
192. C₆₁	#33742	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×#3175	2
C₆₁×#2254	2
C₆₁×#2249	2
C₆₁×#2243	2
C₆₁×#2219	2
C₆₁×#2218	2
C₆₁×#2217	2
C₆₁×#2177	2
C₆₁×#2176	2
C₄×SD₃₂×C₆₁	2
C₆₁×#2108	2
#33742	61

Subgroups — C61×#33742 #29537175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33742 #29537175