Subgroups — C₆₁×#33781 #29537214

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#495	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2251	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2243	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2211	C₂	1	4	yes	maximal, complemented
6	C₆₁×#2188	C₂	1	2	yes	maximal, complemented
7	C₆₁×#2187	C₂	1	2	yes	maximal
8	C₆₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₆₁	—	2	1	no	—
11	C₂²×D₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#533	—	2	1	no	—
13	C₂×#124×C₆₁	—	2	1	no	—
14	C₂×Q₃₂×C₆₁	C₂²	1	4	yes	complemented
15	C₂×SD₃₂×C₆₁	C₂²	1	4	yes	—
16	C₆₁×#500	C₂²	1	1	yes	characteristic, complemented
17	C₆₁×#495	—	2	1	no	—
18	C₆₁×#495	—	2	1	no	—
19	C₆₁×#495	C₂²	1	2	yes	—
20	C₆₁×#495	C₂²	1	2	yes	—
21	C₆₁×#494	C₂²	1	1	yes	characteristic, complemented
22	C₆₁×#487	—	2	1	no	—
23	C₆₁×#467	—	2	1	no	—
24	C₆₁×#467	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₆₁	—	2	1	no	—
26	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	complemented
27	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×C₄⋊C₁₆	C₂²	1	2	yes	complemented
29	C₆₁×#358	C₂²	1	4	yes	complemented
30	C₆₁×#357	C₂²	1	4	yes	—
31	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₆₁×#346	C₂²	1	2	yes	complemented
33	C₂²×Q₈×C₆₁	—	2	1	no	—
34	C₂²×D₄×C₆₁	—	2	1	no	—
35	C₂×Q₁₆×C₆₁	—	2	1	no	—
36	C₂×Q₁₆×C₆₁	—	2	1	no	—
37	C₂×Q₁₆×C₆₁	—	4	2	no	—
38	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
39	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₆₁	—	2	2	no	—
41	C₂×SD₁₆×C₆₁	—	2	2	no	—
42	C₂×SD₁₆×C₆₁	—	4	4	no	—
43	C₂×D₈×C₆₁	—	2	1	no	—
44	C₂×D₈×C₆₁	—	4	2	no	—
45	C₂×D₈×C₆₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₁	—	2	1	no	—
49	C₄⋊Q₈×C₆₁	—	2	1	no	—
50	#125×C₆₁	—	4	1	no	—
51	#124×C₆₁	—	2	2	no	—
52	#124×C₆₁	—	2	2	no	—
53	#124×C₆₁	—	4	1	no	—
54	#124×C₆₁	—	4	1	no	—
55	C₂²⋊Q₈×C₆₁	—	2	2	no	—
56	C₂²⋊Q₈×C₆₁	—	4	1	no	—
57	C₄×Q₈×C₆₁	—	2	2	no	—
58	C₄×Q₈×C₆₁	—	4	1	no	—
59	#117×C₆₁	—	4	1	no	—
60	C₂×C₂²⋊C₄×C₆₁	—	4	1	no	—
61	C₂×C₂²⋊C₄×C₆₁	—	4	1	no	—
62	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
63	Q₃₂×C₆₁	—	4	4	no	—
64	SD₃₂×C₆₁	—	4	4	no	—
65	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
66	#107×C₆₁	—	2	1	no	—
67	#107×C₆₁	C₂³	1	2	yes	—
68	#103×C₆₁	—	2	2	no	—
69	#103×C₆₁	—	2	2	no	—
70	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
71	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
72	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
74	C₂⁴×C₆₁	—	4	1	no	—
75	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂×Q₈×C₆₁	—	2	2	no	—
77	C₂×Q₈×C₆₁	—	4	2	no	—
78	C₂×Q₈×C₆₁	—	4	2	no	—
79	C₂×Q₈×C₆₁	—	4	1	no	—
80	C₂×Q₈×C₆₁	—	4	1	no	—
81	C₂×D₄×C₆₁	—	2	2	no	—
82	C₂×D₄×C₆₁	—	4	2	no	—
83	C₂×D₄×C₆₁	—	4	2	no	—
84	C₂×D₄×C₆₁	—	4	1	no	—
85	C₂²×C₄×C₆₁	—	4	1	no	—
86	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₆₁	—	2	4	no	—
90	Q₁₆×C₆₁	—	2	4	no	—
91	Q₁₆×C₆₁	—	4	2	no	—
92	Q₁₆×C₆₁	—	4	2	no	—
93	SD₁₆×C₆₁	—	4	4	no	—
94	SD₁₆×C₆₁	—	4	4	no	—
95	D₈×C₆₁	—	2	4	no	—
96	D₈×C₆₁	—	4	2	no	—
97	C₂×C₈×C₆₁	—	2	2	no	—
98	C₂×C₈×C₆₁	—	2	2	no	—
99	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₆₁	—	2	2	no	—
104	C₄⋊C₄×C₆₁	—	4	1	no	—
105	C₄⋊C₄×C₆₁	—	4	1	no	—
106	C₄⋊C₄×C₆₁	—	4	2	no	—
107	C₄⋊C₄×C₆₁	—	4	2	no	—
108	C₄⋊C₄×C₆₁	—	4	1	no	—
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	—	4	2	no	—
113	C₂²⋊C₄×C₆₁	—	4	2	no	—
114	C₄²×C₆₁	—	4	2	no	—
115	C₄²×C₆₁	D₈	1	1	yes	characteristic
116	C₄²×C₆₁	D₈	1	1	yes	characteristic
117	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₆₁	—	2	4	no	—
120	C₂³×C₆₁	—	4	2	no	—
121	C₂³×C₆₁	—	8	2	no	—
122	C₂³×C₆₁	—	8	2	no	—
123	C₂³×C₆₁	—	8	2	no	—
124	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
125	Q₈×C₆₁	—	4	2	no	—
126	Q₈×C₆₁	—	4	2	no	—
127	Q₈×C₆₁	—	4	2	no	—
128	Q₈×C₆₁	—	4	2	no	—
129	Q₈×C₆₁	—	8	2	no	—
130	Q₈×C₆₁	—	8	2	no	—
131	D₄×C₆₁	—	4	2	no	—
132	D₄×C₆₁	—	4	2	no	—
133	D₄×C₆₁	—	8	2	no	—
134	C₂×C₄×C₆₁	—	2	2	no	—
135	C₂×C₄×C₆₁	—	2	1	no	—
136	C₂×C₄×C₆₁	—	2	1	no	—
137	C₂×C₄×C₆₁	—	4	1	no	—
138	C₂×C₄×C₆₁	—	4	2	no	—
139	C₂×C₄×C₆₁	—	4	2	no	—
140	C₂×C₄×C₆₁	—	4	2	no	—
141	C₂×C₄×C₆₁	—	8	2	no	—
142	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₆₁	—	2	1	no	—
149	C₈×C₆₁	—	2	1	no	—
150	C₈×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
151	C₈×C₆₁	H(5,𝔽₂)	1	2	yes	—
152	#33781	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂²×C₆₁	—	2	1	no	—
154	C₂²×C₆₁	—	2	1	no	—
155	C₂²×C₆₁	—	8	2	no	—
156	C₂²×C₆₁	—	8	2	no	—
157	C₂²×C₆₁	—	8	2	no	—
158	C₂²×C₆₁	—	16	4	no	—
159	C₂²×C₆₁	#510	1	2	yes	—
160	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₆₁	—	2	1	no	—
162	C₄×C₆₁	—	2	1	no	—
163	C₄×C₆₁	—	2	1	no	—
164	C₄×C₆₁	—	2	1	no	—
165	C₄×C₆₁	—	2	1	no	—
166	C₄×C₆₁	—	8	2	no	—
167	C₄×C₆₁	—	8	2	no	—
168	C₄×C₆₁	—	8	2	no	—
169	C₄×C₆₁	#535	1	1	yes	characteristic
170	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
171	#3170	C₂×C₆₁	1	1	yes	characteristic, complemented
172	C₂×#495	C₂×C₆₁	1	1	yes	characteristic
173	#2251	C₂×C₆₁	1	2	yes	complemented
174	#2243	C₂×C₆₁	1	2	yes	complemented
175	#2211	C₂×C₆₁	1	4	yes	complemented
176	#2188	C₂×C₆₁	1	2	yes	complemented
177	#2187	C₂×C₆₁	1	2	yes	—
178	#2163	C₂×C₆₁	1	1	yes	characteristic, complemented
179	C₂×C₆₁	—	2	2	no	—
180	C₂×C₆₁	—	16	2	no	—
181	C₂×C₆₁	#3205	1	1	yes	characteristic
182	C₂×C₆₁	C₂×#510	1	1	yes	characteristic
183	C₂×C₆₁	#3414	1	1	yes	characteristic
184	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
185	C₂²×SD₁₆	—	2	1	no	—
186	C₂²×D₈	C₂²×C₆₁	1	1	yes	characteristic
187	#533	—	2	1	no	—
188	C₂×#124	—	2	1	no	—
189	C₂×Q₃₂	C₂²×C₆₁	1	4	yes	complemented
190	C₂×SD₃₂	C₂²×C₆₁	1	4	yes	—
191	#500	C₂²×C₆₁	1	1	yes	characteristic, complemented
192	#495	—	2	1	no	—
193	#495	—	2	1	no	—
194	#495	C₂²×C₆₁	1	2	yes	—
195	#495	C₂²×C₆₁	1	2	yes	—
196	#494	C₂²×C₆₁	1	1	yes	characteristic, complemented
197	#487	—	2	1	no	—
198	#467	—	2	1	no	—
199	#467	C₂²×C₆₁	1	2	yes	—
200	C₄×Q₁₆	—	2	1	no	—
201	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	complemented
202	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
203	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	complemented
204	#358	C₂²×C₆₁	1	4	yes	complemented
205	#357	C₂²×C₆₁	1	4	yes	—
206	C₂²⋊C₁₆	C₂²×C₆₁	1	2	yes	—
207	#346	C₂²×C₆₁	1	2	yes	complemented
208	C₆₁	#33781	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²×Q₈	—	2	1	no	—
210	C₂²×D₄	—	2	1	no	—
211	C₂×Q₁₆	—	2	1	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	—	4	2	no	—
214	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
215	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×SD₁₆	—	4	4	no	—
219	C₂×D₈	—	2	1	no	—
220	C₂×D₈	—	4	2	no	—
221	C₂×D₈	C₂³×C₆₁	1	2	yes	—
222	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
223	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	#125	—	4	1	no	—
227	#124	—	2	2	no	—
228	#124	—	2	2	no	—
229	#124	—	4	1	no	—
230	#124	—	4	1	no	—
231	C₂²⋊Q₈	—	2	2	no	—
232	C₂²⋊Q₈	—	4	1	no	—
233	C₄×Q₈	—	2	2	no	—
234	C₄×Q₈	—	4	1	no	—
235	#117	—	4	1	no	—
236	C₂×C₂²⋊C₄	—	4	1	no	—
237	C₂×C₂²⋊C₄	—	4	1	no	—
238	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
239	Q₃₂	—	4	4	no	—
240	SD₃₂	—	4	4	no	—
241	C₂×C₁₆	C₂³×C₆₁	1	4	yes	—
242	#107	—	2	1	no	—
243	#107	C₂³×C₆₁	1	2	yes	—
244	#103	—	2	2	no	—
245	#103	—	2	2	no	—
246	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
247	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
250	C₂⁴	—	4	1	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
263	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
265	Q₁₆	—	2	4	no	—
266	Q₁₆	—	2	4	no	—
267	Q₁₆	—	4	2	no	—
268	Q₁₆	—	4	2	no	—
269	SD₁₆	—	4	4	no	—
270	SD₁₆	—	4	4	no	—
271	D₈	—	2	4	no	—
272	D₈	—	4	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
278	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	D₈×C₆₁	1	1	yes	characteristic
292	C₄²	D₈×C₆₁	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
294	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
295	C₁₆	—	2	4	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	8	2	no	—
298	C₂³	—	8	2	no	—
299	C₂³	—	8	2	no	—
300	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	8	2	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
319	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
320	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
321	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
327	C₈	H(5,𝔽₂)×C₆₁	1	2	yes	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	16	4	no	—
334	C₂²	C₆₁×#510	1	2	yes	—
335	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	2	no	—
344	C₄	C₆₁×#535	1	1	yes	characteristic
345	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	16	2	no	—
348	C₂	C₆₁×#3205	1	1	yes	characteristic
349	C₂	C₂×C₆₁×#510	1	1	yes	characteristic
350	C₂	C₆₁×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
prime radical	C₂²×D₈×C₆₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ C₂×C₄×C₈×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3170	C₂	semidirect
3. C₆₁×#2251	C₂	semidirect
4. C₆₁×#2243	C₂	semidirect
5. C₆₁×#2211	C₂	semidirect
6. C₆₁×#2188	C₂	semidirect
8. C₆₁×#2163	C₂	semidirect
14. C₂×Q₃₂×C₆₁	C₂²	semidirect
16. C₆₁×#500	C₂²	semidirect
21. C₆₁×#494	C₂²	semidirect
26. C₄×Q₁₆×C₆₁	C₂²	semidirect
28. C₆₁×C₄⋊C₁₆	C₂²	semidirect
29. C₆₁×#358	C₂²	semidirect
32. C₆₁×#346	C₂²	semidirect
152. #33781	C₆₁	direct (N × H)
171. #3170	C₂×C₆₁	semidirect
173. #2251	C₂×C₆₁	semidirect
174. #2243	C₂×C₆₁	semidirect
175. #2211	C₂×C₆₁	semidirect
176. #2188	C₂×C₆₁	semidirect
178. #2163	C₂×C₆₁	semidirect
189. C₂×Q₃₂	C₂²×C₆₁	semidirect
191. #500	C₂²×C₆₁	semidirect
196. #494	C₂²×C₆₁	semidirect
201. C₄×Q₁₆	C₂²×C₆₁	semidirect
203. C₄⋊C₁₆	C₂²×C₆₁	semidirect
204. #358	C₂²×C₆₁	semidirect
207. #346	C₂²×C₆₁	semidirect
208. C₆₁	#33781	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×#3170	2
C₂×C₆₁×#495	2
C₆₁×#2251	2
C₆₁×#2243	2
C₆₁×#2211	2
C₆₁×#2188	2
C₆₁×#2187	2
C₆₁×#2163	2
#33781	61

Subgroups — C61×#33781 #29537214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33781 #29537214