Subgroups — C₆₁×#33932 #29537365

Order 15616.

484 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3396	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3383	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2242	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2240	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2228	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2211	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2207	C₂	1	1	yes	maximal, characteristic
9	Q₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×Q₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×SD₃₂×C₆₁	C₂	1	2	yes	maximal, complemented
12	C₆₁×#559	—	2	1	no	—
13	C₆₁×#557	—	2	1	no	—
14	C₂×Q₃₂×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×SD₃₂×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#500	C₂²	1	1	yes	characteristic
17	C₆₁×#499	C₂²	1	2	yes	—
18	C₆₁×#495	C₂²	1	2	yes	—
19	C₆₁×#493	C₂²	1	1	yes	characteristic
20	C₆₁×#479	—	2	2	no	—
21	C₆₁×#475	—	2	1	no	—
22	C₆₁×#463	—	2	2	no	—
23	C₆₁×#460	—	2	1	no	—
24	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₆₁	C₂²	1	2	yes	—
29	C₆₁×#367	C₂²	1	2	yes	—
30	C₆₁×#366	C₂²	1	1	yes	characteristic
31	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₆₁×#358	C₂²	1	1	yes	characteristic
34	C₆₁×#358	C₂²	1	1	yes	characteristic
35	C₆₁×#358	C₂²	1	2	yes	—
36	C₆₁×#357	C₂²	1	2	yes	—
37	C₆₁×#357	C₂²	1	2	yes	—
38	C₆₁×#357	C₂²	1	2	yes	—
39	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
43	C₂×SD₁₆×C₆₁	—	2	2	no	—
44	C₂×D₈×C₆₁	—	2	1	no	—
45	C₂×D₈×C₆₁	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₁	—	2	1	no	—
47	C₄⋊Q₈×C₆₁	—	4	1	no	—
48	#127×C₆₁	—	2	1	no	—
49	#125×C₆₁	—	2	2	no	—
50	#125×C₆₁	—	4	2	no	—
51	#124×C₆₁	—	2	2	no	—
52	#124×C₆₁	—	4	1	no	—
53	#123×C₆₁	—	4	2	no	—
54	C₄×Q₈×C₆₁	—	2	1	no	—
55	C₄×Q₈×C₆₁	—	2	2	no	—
56	C₄×Q₈×C₆₁	—	2	1	no	—
57	C₄×Q₈×C₆₁	—	4	1	no	—
58	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
59	C₄×D₄×C₆₁	—	2	2	no	—
60	C₄×D₄×C₆₁	—	4	1	no	—
61	Q₃₂×C₆₁	—	2	2	no	—
62	SD₃₂×C₆₁	—	2	4	no	—
63	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
66	#107×C₆₁	—	2	1	no	—
67	#107×C₆₁	C₂³	1	1	yes	characteristic
68	#107×C₆₁	C₂³	1	2	yes	—
69	#107×C₆₁	C₂³	1	1	yes	characteristic
70	#106×C₆₁	—	2	2	no	—
71	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
72	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
73	#103×C₆₁	—	2	1	no	—
74	#103×C₆₁	—	2	2	no	—
75	#103×C₆₁	—	2	1	no	—
76	#102×C₆₁	—	2	2	no	—
77	C₄×C₈×C₆₁	C₂³	1	2	yes	—
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₆₁	—	2	1	no	—
80	C₂×Q₈×C₆₁	—	2	1	no	—
81	C₂×Q₈×C₆₁	—	4	1	no	—
82	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
83	C₂×D₄×C₆₁	—	2	2	no	—
84	C₂×D₄×C₆₁	—	4	1	no	—
85	C₂×D₄×C₆₁	—	4	1	no	—
86	C₂²×C₄×C₆₁	—	4	2	no	—
87	Q₁₆×C₆₁	—	2	1	no	—
88	Q₁₆×C₆₁	—	2	1	no	—
89	Q₁₆×C₆₁	—	2	1	no	—
90	Q₁₆×C₆₁	#41	1	2	yes	—
91	SD₁₆×C₆₁	—	8	2	no	—
92	D₈×C₆₁	—	2	2	no	—
93	D₈×C₆₁	—	2	2	no	—
94	D₈×C₆₁	—	8	1	no	—
95	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₆₁	—	2	1	no	—
99	C₄⋊C₄×C₆₁	—	2	1	no	—
100	C₄⋊C₄×C₆₁	—	2	2	no	—
101	C₄⋊C₄×C₆₁	—	4	1	no	—
102	C₄⋊C₄×C₆₁	—	4	1	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	1	no	—
107	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
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	C₄²×C₆₁	—	4	1	no	—
113	C₄²×C₆₁	—	4	2	no	—
114	C₄²×C₆₁	—	4	1	no	—
115	C₄²×C₆₁	C₂×D₄	1	2	yes	—
116	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₆₁	—	2	2	no	—
118	C₁₆×C₆₁	—	2	1	no	—
119	C₁₆×C₆₁	—	2	1	no	—
120	C₂³×C₆₁	—	4	2	no	—
121	Q₈×C₆₁	—	2	2	no	—
122	Q₈×C₆₁	—	2	2	no	—
123	Q₈×C₆₁	—	4	1	no	—
124	Q₈×C₆₁	—	4	1	no	—
125	Q₈×C₆₁	—	4	1	no	—
126	Q₈×C₆₁	—	8	2	no	—
127	D₄×C₆₁	—	4	2	no	—
128	D₄×C₆₁	—	4	2	no	—
129	D₄×C₆₁	—	8	2	no	—
130	D₄×C₆₁	—	16	1	no	—
131	C₂×C₄×C₆₁	—	4	1	no	—
132	C₂×C₄×C₆₁	—	4	1	no	—
133	C₂×C₄×C₆₁	—	4	1	no	—
134	C₂×C₄×C₆₁	—	4	1	no	—
135	C₂×C₄×C₆₁	—	4	2	no	—
136	C₂×C₄×C₆₁	—	16	2	no	—
137	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₆₁	—	2	1	no	—
143	C₈×C₆₁	—	2	2	no	—
144	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
145	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
146	#33932	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂²×C₆₁	—	8	2	no	—
148	C₂²×C₆₁	—	8	2	no	—
149	C₂²×C₆₁	—	8	2	no	—
150	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₆₁	—	2	2	no	—
152	C₄×C₆₁	—	2	2	no	—
153	C₄×C₆₁	—	2	1	no	—
154	C₄×C₆₁	—	4	2	no	—
155	C₄×C₆₁	—	8	1	no	—
156	C₄×C₆₁	—	8	2	no	—
157	C₄×C₆₁	—	8	1	no	—
158	C₄×C₆₁	—	8	1	no	—
159	C₄×C₆₁	#508	1	2	yes	—
160	C₄×C₆₁	#548	1	1	yes	characteristic
161	C₄×C₆₁	#577	1	1	yes	characteristic
162	#3396	C₂×C₆₁	1	1	yes	characteristic
163	#3383	C₂×C₆₁	1	1	yes	characteristic, complemented
164	#2242	C₂×C₆₁	1	2	yes	complemented
165	#2240	C₂×C₆₁	1	1	yes	characteristic
166	#2228	C₂×C₆₁	1	2	yes	complemented
167	#2225	C₂×C₆₁	1	1	yes	characteristic, complemented
168	#2211	C₂×C₆₁	1	2	yes	complemented
169	#2207	C₂×C₆₁	1	1	yes	characteristic
170	Q₈×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
171	C₄×Q₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
172	C₄×SD₃₂	C₂×C₆₁	1	2	yes	complemented
173	C₂×C₆₁	—	16	2	no	—
174	C₂×C₆₁	#3358	1	1	yes	characteristic
175	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
176	C₂×C₆₁	#3416	1	1	yes	characteristic
177	#559	—	2	1	no	—
178	#557	—	2	1	no	—
179	C₂×Q₃₂	C₂²×C₆₁	1	1	yes	characteristic
180	C₂×SD₃₂	C₂²×C₆₁	1	2	yes	—
181	#500	C₂²×C₆₁	1	1	yes	characteristic
182	#499	C₂²×C₆₁	1	2	yes	—
183	#495	C₂²×C₆₁	1	2	yes	—
184	#493	C₂²×C₆₁	1	1	yes	characteristic
185	#479	—	2	2	no	—
186	#475	—	2	1	no	—
187	#463	—	2	2	no	—
188	#460	—	2	1	no	—
189	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
190	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
191	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
192	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
193	C₄×D₈	C₂²×C₆₁	1	2	yes	—
194	#367	C₂²×C₆₁	1	2	yes	—
195	#366	C₂²×C₆₁	1	1	yes	characteristic
196	C₄⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
197	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
198	#358	C₂²×C₆₁	1	1	yes	characteristic
199	#358	C₂²×C₆₁	1	1	yes	characteristic
200	#358	C₂²×C₆₁	1	2	yes	—
201	#357	C₂²×C₆₁	1	2	yes	—
202	#357	C₂²×C₆₁	1	2	yes	—
203	#357	C₂²×C₆₁	1	2	yes	—
204	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
205	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
206	C₆₁	#33932	1	1	yes	characteristic, direct product factor, Sylow
207	C₂×Q₁₆	C₂³×C₆₁	1	1	yes	characteristic
208	C₂×Q₁₆	C₂³×C₆₁	1	1	yes	characteristic
209	C₂×SD₁₆	—	2	2	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	C₂³×C₆₁	1	2	yes	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	4	1	no	—
214	#127	—	2	1	no	—
215	#125	—	2	2	no	—
216	#125	—	4	2	no	—
217	#124	—	2	2	no	—
218	#124	—	4	1	no	—
219	#123	—	4	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	4	1	no	—
224	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	—	4	1	no	—
227	Q₃₂	—	2	2	no	—
228	SD₃₂	—	2	4	no	—
229	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
231	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
232	#107	—	2	1	no	—
233	#107	C₂³×C₆₁	1	1	yes	characteristic
234	#107	C₂³×C₆₁	1	2	yes	—
235	#107	C₂³×C₆₁	1	1	yes	characteristic
236	#106	—	2	2	no	—
237	C₄⋊C₈	D₄×C₆₁	1	1	yes	characteristic
238	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
239	#103	—	2	1	no	—
240	#103	—	2	2	no	—
241	#103	—	2	1	no	—
242	#102	—	2	2	no	—
243	C₄×C₈	C₂³×C₆₁	1	2	yes	—
244	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	D₈×C₆₁	1	1	yes	characteristic
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂²×C₄	—	4	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	#41×C₆₁	1	2	yes	—
257	SD₁₆	—	8	2	no	—
258	D₈	—	2	2	no	—
259	D₈	—	2	2	no	—
260	D₈	—	8	1	no	—
261	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
262	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
263	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
274	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	C₂×D₄×C₆₁	1	2	yes	—
282	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	1	no	—
286	C₂³	—	4	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	8	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	8	2	no	—
296	D₄	—	16	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	16	2	no	—
303	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
304	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
305	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
306	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	2	no	—
310	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
311	C₈	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	2	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	C₆₁×#508	1	2	yes	—
325	C₄	C₆₁×#548	1	1	yes	characteristic
326	C₄	C₆₁×#577	1	1	yes	characteristic
327	C₂	—	16	2	no	—
328	C₂	C₆₁×#3358	1	1	yes	characteristic
329	C₂	C₂×C₆₁×#508	1	1	yes	characteristic
330	C₂	C₆₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₆₁×#493	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×Q₈×C₆₁ ◁ C₈×Q₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#3383	C₂	semidirect
3. C₆₁×#2242	C₂	semidirect
5. C₆₁×#2228	C₂	semidirect
6. C₆₁×#2225	C₂	semidirect
7. C₆₁×#2211	C₂	semidirect
9. Q₈×C₁₆×C₆₁	C₂	semidirect
10. C₄×Q₃₂×C₆₁	C₂	semidirect
11. C₄×SD₃₂×C₆₁	C₂	semidirect
146. #33932	C₆₁	direct (N × H)
163. #3383	C₂×C₆₁	semidirect
164. #2242	C₂×C₆₁	semidirect
166. #2228	C₂×C₆₁	semidirect
167. #2225	C₂×C₆₁	semidirect
168. #2211	C₂×C₆₁	semidirect
170. Q₈×C₁₆	C₂×C₆₁	semidirect
171. C₄×Q₃₂	C₂×C₆₁	semidirect
172. C₄×SD₃₂	C₂×C₆₁	semidirect
206. C₆₁	#33932	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×#3396	2
C₆₁×#3383	2
C₆₁×#2242	2
C₆₁×#2240	2
C₆₁×#2228	2
C₆₁×#2225	2
C₆₁×#2211	2
C₆₁×#2207	2
Q₈×C₁₆×C₆₁	2
C₄×Q₃₂×C₆₁	2
C₄×SD₃₂×C₆₁	2
#33932	61

Subgroups — C61×#33932 #29537365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33932 #29537365