Subgroups — C₆₁×#33933 #29537366

Order 15616.

532 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2244	C₂	1	2	yes	maximal, complemented
4	C₆₁×#2240	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2228	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2206	C₂	1	2	yes	maximal, complemented
8	C₆₁×#2206	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, complemented
11	C₄×D₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#554	—	2	1	no	—
13	C₆₁×#550	—	2	1	no	—
14	C₂×D₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×D₁₆×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#500	C₂²	1	1	yes	characteristic
17	C₆₁×#499	C₂²	1	2	yes	—
18	C₆₁×#493	C₂²	1	1	yes	characteristic
19	C₆₁×#493	C₂²	1	2	yes	—
20	C₆₁×#478	—	2	2	no	—
21	C₆₁×#476	—	2	1	no	—
22	C₆₁×#459	—	2	1	no	—
23	C₆₁×#459	—	2	2	no	—
24	C₈×Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₄×D₈×C₆₁	C₂²	1	2	yes	—
27	C₄×D₈×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×#357	C₂²	1	1	yes	characteristic
34	C₆₁×#357	C₂²	1	1	yes	characteristic
35	C₆₁×#357	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	C₆₁×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×#41×C₆₁	—	4	1	no	—
42	C₂×D₈×C₆₁	—	2	2	no	—
43	C₂×D₈×C₆₁	—	2	1	no	—
44	C₂×D₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×D₈×C₆₁	C₂³	1	2	yes	—
46	C₂×D₈×C₆₁	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₁	—	2	1	no	—
48	#127×C₆₁	—	2	2	no	—
49	#127×C₆₁	—	2	1	no	—
50	#125×C₆₁	—	2	2	no	—
51	C₂²⋊Q₈×C₆₁	—	4	1	no	—
52	C₂²⋊Q₈×C₆₁	—	4	2	no	—
53	#121×C₆₁	—	4	1	no	—
54	#121×C₆₁	—	4	2	no	—
55	C₄×Q₈×C₆₁	D₄	1	1	yes	characteristic
56	C₄×D₄×C₆₁	—	2	2	no	—
57	C₄×D₄×C₆₁	—	2	1	no	—
58	C₄×D₄×C₆₁	—	2	1	no	—
59	C₄×D₄×C₆₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
61	D₁₆×C₆₁	—	2	2	no	—
62	D₁₆×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	#102×C₆₁	—	2	2	no	—
74	#102×C₆₁	—	2	1	no	—
75	#102×C₆₁	—	2	2	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	#41×C₆₁	—	8	4	no	—
80	C₂×Q₈×C₆₁	—	4	1	no	—
81	C₂×Q₈×C₆₁	D₈	1	1	yes	characteristic
82	C₂×D₄×C₆₁	—	2	1	no	—
83	C₂×D₄×C₆₁	—	2	2	no	—
84	C₂×D₄×C₆₁	—	2	1	no	—
85	C₂×D₄×C₆₁	—	4	2	no	—
86	C₂×D₄×C₆₁	—	4	1	no	—
87	C₂×D₄×C₆₁	—	4	1	no	—
88	C₂×D₄×C₆₁	—	4	2	no	—
89	C₂²×C₄×C₆₁	—	4	1	no	—
90	C₂²×C₄×C₆₁	—	4	2	no	—
91	C₂²×C₄×C₆₁	—	4	1	no	—
92	C₂²×C₄×C₆₁	—	4	2	no	—
93	D₈×C₆₁	—	2	1	no	—
94	D₈×C₆₁	—	2	2	no	—
95	D₈×C₆₁	—	2	1	no	—
96	D₈×C₆₁	—	2	1	no	—
97	D₈×C₆₁	—	2	2	no	—
98	D₈×C₆₁	—	8	2	no	—
99	D₈×C₆₁	—	8	1	no	—
100	D₈×C₆₁	#41	1	2	yes	complemented
101	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₆₁	—	2	1	no	—
105	C₄⋊C₄×C₆₁	—	2	1	no	—
106	C₄⋊C₄×C₆₁	—	2	2	no	—
107	C₄⋊C₄×C₆₁	—	4	2	no	—
108	C₄⋊C₄×C₆₁	—	4	1	no	—
109	C₄⋊C₄×C₆₁	—	4	2	no	—
110	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
111	C₄⋊C₄×C₆₁	D₈	1	2	yes	—
112	C₂²⋊C₄×C₆₁	—	4	1	no	—
113	C₂²⋊C₄×C₆₁	—	4	2	no	—
114	C₂²⋊C₄×C₆₁	—	4	1	no	—
115	C₂²⋊C₄×C₆₁	—	4	2	no	—
116	C₄²×C₆₁	C₂×D₄	1	2	yes	—
117	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₆₁	—	2	1	no	—
119	C₁₆×C₆₁	—	2	2	no	—
120	C₁₆×C₆₁	—	2	1	no	—
121	C₂³×C₆₁	—	4	1	no	—
122	C₂³×C₆₁	—	4	2	no	—
123	C₂³×C₆₁	—	4	1	no	—
124	Q₈×C₆₁	—	2	2	no	—
125	Q₈×C₆₁	—	16	1	no	—
126	D₄×C₆₁	—	2	2	no	—
127	D₄×C₆₁	—	4	1	no	—
128	D₄×C₆₁	—	4	2	no	—
129	D₄×C₆₁	—	4	1	no	—
130	D₄×C₆₁	—	4	1	no	—
131	D₄×C₆₁	—	4	2	no	—
132	D₄×C₆₁	—	8	4	no	—
133	D₄×C₆₁	—	8	2	no	—
134	D₄×C₆₁	—	16	2	no	—
135	D₄×C₆₁	—	16	1	no	—
136	C₂×C₄×C₆₁	—	4	1	no	—
137	C₂×C₄×C₆₁	—	4	1	no	—
138	C₂×C₄×C₆₁	—	4	2	no	—
139	C₂×C₄×C₆₁	—	8	4	no	—
140	C₂×C₄×C₆₁	—	8	2	no	—
141	C₂×C₄×C₆₁	—	8	2	no	—
142	C₂×C₄×C₆₁	—	16	2	no	—
143	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₆₁	—	2	2	no	—
149	C₈×C₆₁	—	2	1	no	—
150	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
151	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
152	#33933	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂²×C₆₁	—	4	2	no	—
154	C₂²×C₆₁	—	8	1	no	—
155	C₂²×C₆₁	—	8	1	no	—
156	C₂²×C₆₁	—	8	2	no	—
157	C₂²×C₆₁	—	8	1	no	—
158	C₂²×C₆₁	—	8	2	no	—
159	C₂²×C₆₁	—	8	1	no	—
160	C₂²×C₆₁	—	8	1	no	—
161	C₂²×C₆₁	—	8	2	no	—
162	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₆₁	—	2	2	no	—
164	C₄×C₆₁	—	2	2	no	—
165	C₄×C₆₁	—	2	1	no	—
166	C₄×C₆₁	—	8	1	no	—
167	C₄×C₆₁	—	8	1	no	—
168	C₄×C₆₁	—	8	2	no	—
169	C₄×C₆₁	#509	1	2	yes	—
170	C₄×C₆₁	#548	1	1	yes	characteristic
171	C₄×C₆₁	#577	1	1	yes	characteristic
172	#3391	C₂×C₆₁	1	1	yes	characteristic, complemented
173	#3380	C₂×C₆₁	1	1	yes	characteristic, complemented
174	#2244	C₂×C₆₁	1	2	yes	complemented
175	#2240	C₂×C₆₁	1	1	yes	characteristic, complemented
176	#2228	C₂×C₆₁	1	2	yes	complemented
177	#2226	C₂×C₆₁	1	1	yes	characteristic, complemented
178	#2206	C₂×C₆₁	1	2	yes	complemented
179	#2206	C₂×C₆₁	1	1	yes	characteristic, complemented
180	#2183	C₂×C₆₁	1	1	yes	characteristic, complemented
181	#2177	C₂×C₆₁	1	2	yes	complemented
182	C₄×D₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
183	C₂×C₆₁	—	8	2	no	—
184	C₂×C₆₁	—	16	2	no	—
185	C₂×C₆₁	—	16	1	no	—
186	C₂×C₆₁	#3358	1	1	yes	characteristic
187	C₂×C₆₁	C₂×#509	1	1	yes	characteristic
188	C₂×C₆₁	#3415	1	1	yes	characteristic
189	#554	—	2	1	no	—
190	#550	—	2	1	no	—
191	C₂×D₁₆	C₂²×C₆₁	1	1	yes	characteristic
192	C₂×D₁₆	C₂²×C₆₁	1	2	yes	—
193	#500	C₂²×C₆₁	1	1	yes	characteristic
194	#499	C₂²×C₆₁	1	2	yes	—
195	#493	C₂²×C₆₁	1	1	yes	characteristic
196	#493	C₂²×C₆₁	1	2	yes	—
197	#478	—	2	2	no	—
198	#476	—	2	1	no	—
199	#459	—	2	1	no	—
200	#459	—	2	2	no	—
201	C₈×Q₈	C₂²×C₆₁	1	1	yes	characteristic
202	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
203	C₄×D₈	C₂²×C₆₁	1	2	yes	—
204	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
205	C₄×D₈	C₂²×C₆₁	1	2	yes	—
206	#367	C₂²×C₆₁	1	2	yes	—
207	#366	C₂²×C₆₁	1	1	yes	characteristic
208	C₄⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
209	C₄⋊C₁₆	C₂²×C₆₁	1	2	yes	—
210	#357	C₂²×C₆₁	1	1	yes	characteristic
211	#357	C₂²×C₆₁	1	1	yes	characteristic
212	#357	C₂²×C₆₁	1	2	yes	—
213	#357	C₂²×C₆₁	1	2	yes	—
214	#357	C₂²×C₆₁	1	2	yes	—
215	#357	C₂²×C₆₁	1	2	yes	—
216	#346	C₂²×C₆₁	1	2	yes	—
217	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
218	C₆₁	#33933	1	1	yes	characteristic, direct product factor, Sylow
219	C₂×#41	—	4	1	no	—
220	C₂×D₈	—	2	2	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	C₂³×C₆₁	1	1	yes	characteristic
223	C₂×D₈	C₂³×C₆₁	1	2	yes	—
224	C₂×D₈	C₂³×C₆₁	1	1	yes	characteristic
225	C₄⋊Q₈	—	2	1	no	—
226	#127	—	2	2	no	—
227	#127	—	2	1	no	—
228	#125	—	2	2	no	—
229	C₂²⋊Q₈	—	4	1	no	—
230	C₂²⋊Q₈	—	4	2	no	—
231	#121	—	4	1	no	—
232	#121	—	4	2	no	—
233	C₄×Q₈	D₄×C₆₁	1	1	yes	characteristic
234	C₄×D₄	—	2	2	no	—
235	C₄×D₄	—	2	1	no	—
236	C₄×D₄	—	2	1	no	—
237	C₄×D₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	4	1	no	—
239	D₁₆	—	2	2	no	—
240	D₁₆	—	2	4	no	—
241	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
242	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
243	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
244	#107	—	2	1	no	—
245	#107	C₂³×C₆₁	1	1	yes	characteristic
246	#107	C₂³×C₆₁	1	2	yes	—
247	#107	C₂³×C₆₁	1	1	yes	characteristic
248	#106	—	2	2	no	—
249	C₄⋊C₈	D₄×C₆₁	1	1	yes	characteristic
250	C₄⋊C₈	D₄×C₆₁	1	2	yes	—
251	#102	—	2	2	no	—
252	#102	—	2	1	no	—
253	#102	—	2	2	no	—
254	#102	—	2	1	no	—
255	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₆₁	1	2	yes	—
257	#41	—	8	4	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	D₈×C₆₁	1	1	yes	characteristic
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	4	2	no	—
271	D₈	—	2	1	no	—
272	D₈	—	2	2	no	—
273	D₈	—	2	1	no	—
274	D₈	—	2	1	no	—
275	D₈	—	2	2	no	—
276	D₈	—	8	2	no	—
277	D₈	—	8	1	no	—
278	D₈	#41×C₆₁	1	2	yes	complemented
279	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
281	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
289	C₄⋊C₄	D₈×C₆₁	1	2	yes	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	1	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₄²	C₂×D₄×C₆₁	1	2	yes	—
295	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	2	no	—
298	C₁₆	—	2	1	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	1	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	16	1	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	1	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	8	4	no	—
311	D₄	—	8	2	no	—
312	D₄	—	16	2	no	—
313	D₄	—	16	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	16	2	no	—
321	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
323	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
324	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	2	1	no	—
328	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
329	C₈	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
330	C₂²	—	4	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	2	no	—
339	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	2	no	—
346	C₄	C₆₁×#509	1	2	yes	—
347	C₄	C₆₁×#548	1	1	yes	characteristic
348	C₄	C₆₁×#577	1	1	yes	characteristic
349	C₂	—	8	2	no	—
350	C₂	—	16	2	no	—
351	C₂	—	16	1	no	—
352	C₂	C₆₁×#3358	1	1	yes	characteristic
353	C₂	C₂×C₆₁×#509	1	1	yes	characteristic
354	C₂	C₆₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
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×#3391	C₂	semidirect
2. C₆₁×#3380	C₂	semidirect
3. C₆₁×#2244	C₂	semidirect
4. C₆₁×#2240	C₂	semidirect
5. C₆₁×#2228	C₂	semidirect
6. C₆₁×#2226	C₂	semidirect
7. C₆₁×#2206	C₂	semidirect
8. C₆₁×#2206	C₂	semidirect
9. C₆₁×#2183	C₂	semidirect
10. C₆₁×#2177	C₂	semidirect
11. C₄×D₁₆×C₆₁	C₂	semidirect
100. D₈×C₆₁	#41	semidirect
152. #33933	C₆₁	direct (N × H)
172. #3391	C₂×C₆₁	semidirect
173. #3380	C₂×C₆₁	semidirect
174. #2244	C₂×C₆₁	semidirect
175. #2240	C₂×C₆₁	semidirect
176. #2228	C₂×C₆₁	semidirect
177. #2226	C₂×C₆₁	semidirect
178. #2206	C₂×C₆₁	semidirect
179. #2206	C₂×C₆₁	semidirect
180. #2183	C₂×C₆₁	semidirect
181. #2177	C₂×C₆₁	semidirect
182. C₄×D₁₆	C₂×C₆₁	semidirect
218. C₆₁	#33933	direct (N × H)
278. D₈	#41×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3391	2
C₆₁×#3380	2
C₆₁×#2244	2
C₆₁×#2240	2
C₆₁×#2228	2
C₆₁×#2226	2
C₆₁×#2206	2
C₆₁×#2183	2
C₆₁×#2177	2
C₄×D₁₆×C₆₁	2
#33933	61

Subgroups — C61×#33933 #29537366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33933 #29537366