Subgroups — C₆₁×#33931 #29537364

Order 15616.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3393	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#3383	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2242	C₂	1	2	yes	maximal
4	C₆₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2229	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2224	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#2211	C₂	1	2	yes	maximal
8	C₆₁×#2209	C₂	1	1	yes	maximal, characteristic, complemented
9	Q₈×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×Q₃₂×C₆₁	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#557	—	2	1	no	—
13	C₆₁×#556	—	2	1	no	—
14	C₂×Q₃₂×C₆₁	C₂²	1	2	yes	—
15	C₂×SD₃₂×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#500	C₂²	1	1	yes	characteristic
17	C₆₁×#499	C₂²	1	2	yes	—
18	C₆₁×#495	C₂²	1	2	yes	—
19	C₆₁×#494	C₂²	1	1	yes	characteristic
20	C₆₁×#479	—	2	2	no	—
21	C₆₁×#475	—	2	1	no	—
22	C₆₁×#464	—	2	2	no	—
23	C₆₁×#461	—	2	1	no	—
24	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
26	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
28	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#367	C₂²	1	1	yes	characteristic
30	C₆₁×#366	C₂²	1	2	yes	—
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	2	yes	—
35	C₆₁×#358	C₂²	1	2	yes	—
36	C₆₁×#358	C₂²	1	2	yes	—
37	C₆₁×#357	C₂²	1	1	yes	characteristic
38	C₆₁×#357	C₂²	1	2	yes	—
39	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
41	C₂×Q₁₆×C₆₁	—	2	1	no	—
42	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
43	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×SD₁₆×C₆₁	—	2	2	no	—
45	C₂×D₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₆₁	—	2	1	no	—
47	C₄⋊Q₈×C₆₁	—	2	1	no	—
48	C₄⋊Q₈×C₆₁	—	4	1	no	—
49	#126×C₆₁	—	4	2	no	—
50	#125×C₆₁	—	2	2	no	—
51	#125×C₆₁	—	4	2	no	—
52	#124×C₆₁	—	2	2	no	—
53	C₂²⋊Q₈×C₆₁	—	4	1	no	—
54	C₄×Q₈×C₆₁	—	2	2	no	—
55	C₄×Q₈×C₆₁	—	2	1	no	—
56	C₄×Q₈×C₆₁	—	2	2	no	—
57	C₄×Q₈×C₆₁	—	4	1	no	—
58	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
59	C₄×D₄×C₆₁	—	2	1	no	—
60	#117×C₆₁	—	4	1	no	—
61	Q₃₂×C₆₁	—	2	4	no	—
62	SD₃₂×C₆₁	—	2	2	no	—
63	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
65	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
66	#107×C₆₁	—	2	1	no	—
67	#107×C₆₁	C₂³	1	2	yes	—
68	#107×C₆₁	C₂³	1	1	yes	characteristic
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	2	no	—
74	#103×C₆₁	—	2	2	no	—
75	#103×C₆₁	—	2	1	no	—
76	#102×C₆₁	—	2	1	no	—
77	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂³	1	2	yes	—
79	C₂×Q₈×C₆₁	—	2	1	no	—
80	C₂×Q₈×C₆₁	—	2	2	no	—
81	C₂×Q₈×C₆₁	—	4	1	no	—
82	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
83	C₂×D₄×C₆₁	—	2	1	no	—
84	C₂²×C₄×C₆₁	—	4	1	no	—
85	Q₁₆×C₆₁	—	2	2	no	—
86	Q₁₆×C₆₁	—	2	1	no	—
87	Q₁₆×C₆₁	—	2	2	no	—
88	Q₁₆×C₆₁	—	8	1	no	—
89	Q₁₆×C₆₁	#41	1	2	yes	—
90	SD₁₆×C₆₁	—	8	2	no	—
91	D₈×C₆₁	—	2	1	no	—
92	D₈×C₆₁	—	2	1	no	—
93	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₆₁	—	2	1	no	—
97	C₄⋊C₄×C₆₁	—	2	1	no	—
98	C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₄⋊C₄×C₆₁	—	4	1	no	—
100	C₄⋊C₄×C₆₁	—	4	2	no	—
101	C₄⋊C₄×C₆₁	—	4	1	no	—
102	C₄⋊C₄×C₆₁	—	4	2	no	—
103	C₄⋊C₄×C₆₁	—	4	2	no	—
104	C₄⋊C₄×C₆₁	—	4	2	no	—
105	C₄⋊C₄×C₆₁	—	4	1	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	1	no	—
111	C₂²⋊C₄×C₆₁	—	4	2	no	—
112	C₄²×C₆₁	—	4	2	no	—
113	C₄²×C₆₁	—	4	2	no	—
114	C₄²×C₆₁	—	4	1	no	—
115	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₆₁	C₂×D₄	1	2	yes	—
117	C₁₆×C₆₁	—	2	1	no	—
118	C₁₆×C₆₁	—	2	2	no	—
119	C₁₆×C₆₁	—	2	1	no	—
120	C₂³×C₆₁	—	4	1	no	—
121	Q₈×C₆₁	—	2	2	no	—
122	Q₈×C₆₁	—	2	2	no	—
123	Q₈×C₆₁	—	4	1	no	—
124	Q₈×C₆₁	—	4	2	no	—
125	Q₈×C₆₁	—	4	2	no	—
126	Q₈×C₆₁	—	8	2	no	—
127	D₄×C₆₁	—	4	1	no	—
128	D₄×C₆₁	—	4	1	no	—
129	C₂×C₄×C₆₁	—	4	1	no	—
130	C₂×C₄×C₆₁	—	4	1	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	—	4	1	no	—
133	C₂×C₄×C₆₁	—	4	2	no	—
134	C₂×C₄×C₆₁	—	8	2	no	—
135	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₆₁	—	2	2	no	—
141	C₈×C₆₁	—	2	1	no	—
142	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
143	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
144	#33931	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₆₁	—	8	1	no	—
146	C₂²×C₆₁	—	8	1	no	—
147	C₂²×C₆₁	—	8	1	no	—
148	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₆₁	—	2	2	no	—
150	C₄×C₆₁	—	2	2	no	—
151	C₄×C₆₁	—	2	1	no	—
152	C₄×C₆₁	—	4	2	no	—
153	C₄×C₆₁	—	8	2	no	—
154	C₄×C₆₁	—	8	1	no	—
155	C₄×C₆₁	—	8	2	no	—
156	C₄×C₆₁	—	8	1	no	—
157	C₄×C₆₁	#508	1	2	yes	—
158	C₄×C₆₁	#548	1	1	yes	characteristic
159	C₄×C₆₁	#577	1	1	yes	characteristic
160	#3393	C₂×C₆₁	1	1	yes	characteristic
161	#3383	C₂×C₆₁	1	1	yes	characteristic, complemented
162	#2242	C₂×C₆₁	1	2	yes	—
163	#2241	C₂×C₆₁	1	1	yes	characteristic, complemented
164	#2229	C₂×C₆₁	1	2	yes	complemented
165	#2224	C₂×C₆₁	1	1	yes	characteristic
166	#2211	C₂×C₆₁	1	2	yes	—
167	#2209	C₂×C₆₁	1	1	yes	characteristic, complemented
168	Q₈×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
169	C₄×Q₃₂	C₂×C₆₁	1	2	yes	complemented
170	C₄×SD₃₂	C₂×C₆₁	1	1	yes	characteristic
171	C₂×C₆₁	—	16	1	no	—
172	C₂×C₆₁	#3358	1	1	yes	characteristic
173	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
174	C₂×C₆₁	#3417	1	1	yes	characteristic
175	#557	—	2	1	no	—
176	#556	—	2	1	no	—
177	C₂×Q₃₂	C₂²×C₆₁	1	2	yes	—
178	C₂×SD₃₂	C₂²×C₆₁	1	1	yes	characteristic
179	#500	C₂²×C₆₁	1	1	yes	characteristic
180	#499	C₂²×C₆₁	1	2	yes	—
181	#495	C₂²×C₆₁	1	2	yes	—
182	#494	C₂²×C₆₁	1	1	yes	characteristic
183	#479	—	2	2	no	—
184	#475	—	2	1	no	—
185	#464	—	2	2	no	—
186	#461	—	2	1	no	—
187	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
188	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
189	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
190	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
191	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
192	#367	C₂²×C₆₁	1	1	yes	characteristic
193	#366	C₂²×C₆₁	1	2	yes	—
194	C₄⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
195	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
196	#358	C₂²×C₆₁	1	1	yes	characteristic
197	#358	C₂²×C₆₁	1	2	yes	—
198	#358	C₂²×C₆₁	1	2	yes	—
199	#358	C₂²×C₆₁	1	2	yes	—
200	#357	C₂²×C₆₁	1	1	yes	characteristic
201	#357	C₂²×C₆₁	1	2	yes	—
202	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
203	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
204	C₆₁	#33931	1	1	yes	characteristic, direct product factor, Sylow
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
207	C₂×Q₁₆	C₂³×C₆₁	1	1	yes	characteristic
208	C₂×SD₁₆	—	2	2	no	—
209	C₂×D₈	C₂³×C₆₁	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	4	1	no	—
213	#126	—	4	2	no	—
214	#125	—	2	2	no	—
215	#125	—	4	2	no	—
216	#124	—	2	2	no	—
217	C₂²⋊Q₈	—	4	1	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	2	2	no	—
221	C₄×Q₈	—	4	1	no	—
222	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
223	C₄×D₄	—	2	1	no	—
224	#117	—	4	1	no	—
225	Q₃₂	—	2	4	no	—
226	SD₃₂	—	2	2	no	—
227	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
228	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
229	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
230	#107	—	2	1	no	—
231	#107	C₂³×C₆₁	1	2	yes	—
232	#107	C₂³×C₆₁	1	1	yes	characteristic
233	#107	C₂³×C₆₁	1	1	yes	characteristic
234	#106	—	2	2	no	—
235	C₄⋊C₈	D₄×C₆₁	1	1	yes	characteristic
236	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
237	#103	—	2	2	no	—
238	#103	—	2	2	no	—
239	#103	—	2	1	no	—
240	#102	—	2	1	no	—
241	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₆₁	1	2	yes	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	D₈×C₆₁	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	8	1	no	—
253	Q₁₆	#41×C₆₁	1	2	yes	—
254	SD₁₆	—	8	2	no	—
255	D₈	—	2	1	no	—
256	D₈	—	2	1	no	—
257	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
259	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
272	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₆₁	1	2	yes	—
281	C₁₆	—	2	1	no	—
282	C₁₆	—	2	2	no	—
283	C₁₆	—	2	1	no	—
284	C₂³	—	4	1	no	—
285	Q₈	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	8	2	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
300	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
302	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
307	C₈	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
308	C₂²	—	8	1	no	—
309	C₂²	—	8	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	1	no	—
320	C₄	C₆₁×#508	1	2	yes	—
321	C₄	C₆₁×#548	1	1	yes	characteristic
322	C₄	C₆₁×#577	1	1	yes	characteristic
323	C₂	—	16	1	no	—
324	C₂	C₆₁×#3358	1	1	yes	characteristic
325	C₂	C₂×C₆₁×#508	1	1	yes	characteristic
326	C₂	C₆₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₂×D₈×C₆₁	C₂³
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
4. C₆₁×#2241	C₂	semidirect
5. C₆₁×#2229	C₂	semidirect
8. C₆₁×#2209	C₂	semidirect
9. Q₈×C₁₆×C₆₁	C₂	semidirect
10. C₄×Q₃₂×C₆₁	C₂	semidirect
144. #33931	C₆₁	direct (N × H)
161. #3383	C₂×C₆₁	semidirect
163. #2241	C₂×C₆₁	semidirect
164. #2229	C₂×C₆₁	semidirect
167. #2209	C₂×C₆₁	semidirect
168. Q₈×C₁₆	C₂×C₆₁	semidirect
169. C₄×Q₃₂	C₂×C₆₁	semidirect
204. C₆₁	#33931	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×#3393	2
C₆₁×#3383	2
C₆₁×#2242	2
C₆₁×#2241	2
C₆₁×#2229	2
C₆₁×#2224	2
C₆₁×#2211	2
C₆₁×#2209	2
Q₈×C₁₆×C₆₁	2
C₄×Q₃₂×C₆₁	2
C₄×SD₃₂×C₆₁	2
#33931	61

Subgroups — C61×#33931 #29537364

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33931 #29537364