Subgroups — C₆₁×#33951 #29537384

Order 15616.

498 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×#3380	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2255	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2228	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2226	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2226	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2224	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#2177	C₂	1	2	yes	maximal
11	C₄×D₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
12	C₆₁×#554	—	2	1	no	—
13	D₄×Q₈×C₆₁	—	2	1	no	—
14	C₂×D₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×#500	C₂²	1	1	yes	characteristic
16	C₆₁×#500	C₂²	1	2	yes	—
17	C₆₁×#500	C₂²	1	1	yes	characteristic
18	C₆₁×#499	C₂²	1	2	yes	—
19	C₆₁×#478	—	2	2	no	—
20	C₆₁×#476	—	2	1	no	—
21	C₆₁×#474	—	2	1	no	—
22	C₆₁×#474	—	2	2	no	—
23	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₆₁	C₂²	1	2	yes	—
26	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₆₁	C₂²	1	2	yes	—
28	C₆₁×#367	C₂²	1	2	yes	—
29	C₆₁×#367	C₂²	1	2	yes	—
30	C₆₁×#367	C₂²	1	2	yes	—
31	C₆₁×#366	C₂²	1	1	yes	characteristic
32	C₆₁×#366	C₂²	1	2	yes	—
33	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₆₁×#357	C₂²	1	1	yes	characteristic
36	C₆₁×#357	C₂²	1	1	yes	characteristic
37	C₆₁×#357	C₂²	1	2	yes	—
38	C₆₁×#357	C₂²	1	2	yes	—
39	C₆₁×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₆₁	—	4	1	no	—
42	C₂×D₈×C₆₁	C₂³	1	1	yes	characteristic
43	C₂×D₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₄⋊Q₈×C₆₁	—	2	2	no	—
45	C₄⋊Q₈×C₆₁	—	2	1	no	—
46	C₄⋊Q₈×C₆₁	—	2	1	no	—
47	#125×C₆₁	—	2	2	no	—
48	C₂²⋊Q₈×C₆₁	—	4	1	no	—
49	C₂²⋊Q₈×C₆₁	—	4	1	no	—
50	C₂²⋊Q₈×C₆₁	—	4	2	no	—
51	C₂²⋊Q₈×C₆₁	—	4	2	no	—
52	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
53	C₄×D₄×C₆₁	—	2	2	no	—
54	C₄×D₄×C₆₁	—	2	1	no	—
55	C₄×D₄×C₆₁	—	2	1	no	—
56	C₄×D₄×C₆₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
58	D₁₆×C₆₁	Q₈	1	4	yes	complemented
59	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
61	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
62	#107×C₆₁	—	2	2	no	—
63	#107×C₆₁	—	2	1	no	—
64	#107×C₆₁	—	2	1	no	—
65	#107×C₆₁	C₂³	1	1	yes	characteristic
66	#107×C₆₁	C₂³	1	1	yes	characteristic
67	#107×C₆₁	C₂³	1	2	yes	—
68	#107×C₆₁	C₂³	1	2	yes	—
69	#106×C₆₁	—	2	2	no	—
70	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
71	C₄⋊C₈×C₆₁	D₄	1	2	yes	—
72	#102×C₆₁	—	2	2	no	—
73	#102×C₆₁	—	2	1	no	—
74	#102×C₆₁	—	2	2	no	—
75	#102×C₆₁	—	2	1	no	—
76	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₆₁	C₂³	1	2	yes	—
78	C₂×Q₈×C₆₁	—	4	2	no	—
79	C₂×Q₈×C₆₁	—	4	1	no	—
80	C₂×Q₈×C₆₁	—	4	1	no	—
81	C₂×Q₈×C₆₁	—	8	4	no	—
82	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
83	C₂×D₄×C₆₁	—	2	1	no	—
84	C₂×D₄×C₆₁	—	2	1	no	—
85	C₂²×C₄×C₆₁	—	4	1	no	—
86	C₂²×C₄×C₆₁	—	4	2	no	—
87	C₂²×C₄×C₆₁	—	4	1	no	—
88	C₂²×C₄×C₆₁	—	4	2	no	—
89	D₈×C₆₁	—	2	1	no	—
90	D₈×C₆₁	—	2	1	no	—
91	D₈×C₆₁	C₂×Q₈	1	2	yes	—
92	D₈×C₆₁	C₂×Q₈	1	2	yes	complemented
93	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₆₁	—	2	1	no	—
97	C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₄⋊C₄×C₆₁	—	2	2	no	—
99	C₄⋊C₄×C₆₁	—	2	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	1	no	—
104	C₄⋊C₄×C₆₁	—	4	2	no	—
105	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
106	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
107	C₂²⋊C₄×C₆₁	—	4	1	no	—
108	C₂²⋊C₄×C₆₁	—	4	2	no	—
109	C₂²⋊C₄×C₆₁	—	4	1	no	—
110	C₂²⋊C₄×C₆₁	—	4	2	no	—
111	C₄²×C₆₁	C₂×D₄	1	2	yes	—
112	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₆₁	—	2	1	no	—
114	C₁₆×C₆₁	—	2	2	no	—
115	C₁₆×C₆₁	C₂×Q₈	1	2	yes	complemented
116	C₂³×C₆₁	—	4	1	no	—
117	C₂³×C₆₁	—	4	1	no	—
118	Q₈×C₆₁	—	2	2	no	—
119	Q₈×C₆₁	—	8	4	no	—
120	Q₈×C₆₁	—	8	2	no	—
121	Q₈×C₆₁	—	16	1	no	—
122	D₄×C₆₁	—	2	2	no	—
123	D₄×C₆₁	—	2	2	no	—
124	D₄×C₆₁	—	4	1	no	—
125	D₄×C₆₁	—	4	1	no	—
126	C₂×C₄×C₆₁	—	4	1	no	—
127	C₂×C₄×C₆₁	—	4	2	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	4	1	no	—
130	C₂×C₄×C₆₁	—	8	4	no	—
131	C₂×C₄×C₆₁	—	8	2	no	—
132	C₂×C₄×C₆₁	—	8	2	no	—
133	C₂×C₄×C₆₁	—	16	2	no	—
134	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₆₁	—	2	2	no	—
140	C₈×C₆₁	—	2	1	no	—
141	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
142	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
143	#33951	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₆₁	—	4	2	no	—
145	C₂²×C₆₁	—	4	2	no	—
146	C₂²×C₆₁	—	8	1	no	—
147	C₂²×C₆₁	—	8	1	no	—
148	C₂²×C₆₁	—	8	1	no	—
149	C₂²×C₆₁	—	8	1	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	—	8	1	no	—
155	C₄×C₆₁	—	8	2	no	—
156	C₄×C₆₁	—	8	1	no	—
157	C₄×C₆₁	—	8	2	no	—
158	C₄×C₆₁	#509	1	2	yes	—
159	C₄×C₆₁	D₄×Q₈	1	1	yes	characteristic
160	C₄×C₆₁	#578	1	1	yes	characteristic
161	#3380	C₂×C₆₁	1	1	yes	characteristic, complemented
162	Q₈×D₈	C₂×C₆₁	1	1	yes	characteristic, complemented
163	#2255	C₂×C₆₁	1	2	yes	complemented
164	#2254	C₂×C₆₁	1	1	yes	characteristic, complemented
165	#2228	C₂×C₆₁	1	2	yes	complemented
166	#2226	C₂×C₆₁	1	2	yes	complemented
167	#2226	C₂×C₆₁	1	1	yes	characteristic, complemented
168	#2224	C₂×C₆₁	1	1	yes	characteristic, complemented
169	#2183	C₂×C₆₁	1	1	yes	characteristic, complemented
170	#2177	C₂×C₆₁	1	2	yes	—
171	C₄×D₁₆	C₂×C₆₁	1	1	yes	characteristic
172	C₂×C₆₁	—	8	2	no	—
173	C₂×C₆₁	—	8	2	no	—
174	C₂×C₆₁	Q₈×D₈	1	1	yes	characteristic
175	C₂×C₆₁	C₂×#509	1	1	yes	characteristic
176	C₂×C₆₁	#3416	1	1	yes	characteristic
177	#554	—	2	1	no	—
178	D₄×Q₈	—	2	1	no	—
179	C₂×D₁₆	C₂²×C₆₁	1	1	yes	characteristic
180	#500	C₂²×C₆₁	1	1	yes	characteristic
181	#500	C₂²×C₆₁	1	2	yes	—
182	#500	C₂²×C₆₁	1	1	yes	characteristic
183	#499	C₂²×C₆₁	1	2	yes	—
184	#478	—	2	2	no	—
185	#476	—	2	1	no	—
186	#474	—	2	1	no	—
187	#474	—	2	2	no	—
188	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
189	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
190	C₄×D₈	C₂²×C₆₁	1	2	yes	—
191	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
192	C₄×D₈	C₂²×C₆₁	1	2	yes	—
193	#367	C₂²×C₆₁	1	2	yes	—
194	#367	C₂²×C₆₁	1	2	yes	—
195	#367	C₂²×C₆₁	1	2	yes	—
196	#366	C₂²×C₆₁	1	1	yes	characteristic
197	#366	C₂²×C₆₁	1	2	yes	—
198	C₄⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
199	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
200	#357	C₂²×C₆₁	1	1	yes	characteristic
201	#357	C₂²×C₆₁	1	1	yes	characteristic
202	#357	C₂²×C₆₁	1	2	yes	—
203	#357	C₂²×C₆₁	1	2	yes	—
204	#346	C₂²×C₆₁	1	2	yes	—
205	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
206	C₆₁	#33951	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²×Q₈	—	4	1	no	—
208	C₂×D₈	C₂³×C₆₁	1	1	yes	characteristic
209	C₂×D₈	C₂³×C₆₁	1	1	yes	characteristic
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	#125	—	2	2	no	—
214	C₂²⋊Q₈	—	4	1	no	—
215	C₂²⋊Q₈	—	4	1	no	—
216	C₂²⋊Q₈	—	4	2	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	—	2	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	4	1	no	—
224	D₁₆	Q₈×C₆₁	1	4	yes	complemented
225	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
226	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
227	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
228	#107	—	2	2	no	—
229	#107	—	2	1	no	—
230	#107	—	2	1	no	—
231	#107	C₂³×C₆₁	1	1	yes	characteristic
232	#107	C₂³×C₆₁	1	1	yes	characteristic
233	#107	C₂³×C₆₁	1	2	yes	—
234	#107	C₂³×C₆₁	1	2	yes	—
235	#106	—	2	2	no	—
236	C₄⋊C₈	D₄×C₆₁	1	1	yes	characteristic
237	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
238	#102	—	2	2	no	—
239	#102	—	2	1	no	—
240	#102	—	2	2	no	—
241	#102	—	2	1	no	—
242	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₆₁	1	2	yes	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	8	4	no	—
248	C₂×Q₈	D₈×C₆₁	1	1	yes	characteristic
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	2	no	—
255	D₈	—	2	1	no	—
256	D₈	—	2	1	no	—
257	D₈	C₂×Q₈×C₆₁	1	2	yes	—
258	D₈	C₂×Q₈×C₆₁	1	2	yes	complemented
259	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
260	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
261	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
272	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	C₂×D₄×C₆₁	1	2	yes	—
278	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
279	C₁₆	—	2	1	no	—
280	C₁₆	—	2	2	no	—
281	C₁₆	C₂×Q₈×C₆₁	1	2	yes	complemented
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	8	4	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	16	1	no	—
288	D₄	—	2	2	no	—
289	D₄	—	2	2	no	—
290	D₄	—	4	1	no	—
291	D₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	8	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	16	2	no	—
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	2	yes	—
303	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	C₂²×Q₈×C₆₁	1	1	yes	characteristic
308	C₈	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	1	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	—	8	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	2	no	—
323	C₄	C₆₁×#509	1	2	yes	—
324	C₄	D₄×Q₈×C₆₁	1	1	yes	characteristic
325	C₄	C₆₁×#578	1	1	yes	characteristic
326	C₂	—	8	2	no	—
327	C₂	—	8	2	no	—
328	C₂	Q₈×D₈×C₆₁	1	1	yes	characteristic
329	C₂	C₂×C₆₁×#509	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	C₂×D₁₆×C₆₁	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
1. C₆₁×#3380	C₂	semidirect
2. Q₈×D₈×C₆₁	C₂	semidirect
3. C₆₁×#2255	C₂	semidirect
4. C₆₁×#2254	C₂	semidirect
5. C₆₁×#2228	C₂	semidirect
6. C₆₁×#2226	C₂	semidirect
7. C₆₁×#2226	C₂	semidirect
8. C₆₁×#2224	C₂	semidirect
9. C₆₁×#2183	C₂	semidirect
58. D₁₆×C₆₁	Q₈	semidirect
92. D₈×C₆₁	C₂×Q₈	semidirect
115. C₁₆×C₆₁	C₂×Q₈	semidirect
143. #33951	C₆₁	direct (N × H)
161. #3380	C₂×C₆₁	semidirect
162. Q₈×D₈	C₂×C₆₁	semidirect
163. #2255	C₂×C₆₁	semidirect
164. #2254	C₂×C₆₁	semidirect
165. #2228	C₂×C₆₁	semidirect
166. #2226	C₂×C₆₁	semidirect
167. #2226	C₂×C₆₁	semidirect
168. #2224	C₂×C₆₁	semidirect
169. #2183	C₂×C₆₁	semidirect
206. C₆₁	#33951	direct (N × H)
224. D₁₆	Q₈×C₆₁	semidirect
258. D₈	C₂×Q₈×C₆₁	semidirect
281. C₁₆	C₂×Q₈×C₆₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₁×#3380	2
Q₈×D₈×C₆₁	2
C₆₁×#2255	2
C₆₁×#2254	2
C₆₁×#2228	2
C₆₁×#2226	2
C₆₁×#2224	2
C₆₁×#2183	2
C₆₁×#2177	2
C₄×D₁₆×C₆₁	2
#33951	61

Subgroups — C61×#33951 #29537384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33951 #29537384