Subgroups — C₆₁×#33902 #29537335

Order 15616.

516 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3338	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3334	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2235	C₂	1	2	yes	maximal, complemented
5	C₆₁×#2233	C₂	1	2	yes	maximal, complemented
6	C₆₁×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2217	C₂	1	2	yes	maximal, complemented
9	C₄×SD₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₆₁×#367	C₂	1	2	yes	maximal, complemented
12	C₆₁×#554	—	2	1	no	—
13	C₆₁×#547	—	2	1	no	—
14	C₂×SD₃₂×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
16	C₆₁×#500	C₂²	1	1	yes	characteristic
17	C₆₁×#500	C₂²	1	1	yes	characteristic
18	C₆₁×#484	—	2	2	no	—
19	C₆₁×#480	—	2	2	no	—
20	C₆₁×#477	—	2	1	no	—
21	C₆₁×#474	—	2	1	no	—
22	C₆₁×#467	C₂²	1	2	yes	—
23	C₆₁×#466	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₆₁	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×#107×C₆₁	C₂²	1	2	yes	—
28	C₂×#107×C₆₁	C₂²	1	2	yes	—
29	C₆₁×#367	C₂²	1	1	yes	characteristic
30	C₆₁×#367	C₂²	1	2	yes	—
31	C₆₁×#367	C₂²	1	2	yes	—
32	C₆₁×#367	C₂²	1	2	yes	—
33	C₆₁×#367	C₂²	1	2	yes	—
34	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₆₁×#358	C₂²	1	1	yes	characteristic
36	C₆₁×#358	C₂²	1	2	yes	—
37	C₆₁×#357	C₂²	1	1	yes	characteristic
38	C₆₁×#357	C₂²	1	2	yes	—
39	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×#41×C₆₁	—	4	1	no	—
42	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
43	C₂×D₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₆₁	—	2	1	no	—
46	C₄⋊Q₈×C₆₁	—	2	1	no	—
47	#125×C₆₁	—	4	1	no	—
48	#123×C₆₁	—	4	2	no	—
49	C₂²⋊Q₈×C₆₁	—	2	2	no	—
50	C₂²⋊Q₈×C₆₁	—	4	1	no	—
51	C₂²⋊Q₈×C₆₁	—	4	2	no	—
52	#121×C₆₁	—	2	2	no	—
53	C₄×Q₈×C₆₁	—	2	1	no	—
54	C₄×D₄×C₆₁	—	2	1	no	—
55	C₄×D₄×C₆₁	—	4	1	no	—
56	C₄×D₄×C₆₁	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
59	SD₃₂×C₆₁	—	2	2	no	—
60	C₂×C₁₆×C₆₁	—	2	4	no	—
61	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
63	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
64	#107×C₆₁	—	2	2	no	—
65	#107×C₆₁	—	2	2	no	—
66	#107×C₆₁	—	2	1	no	—
67	#107×C₆₁	—	2	1	no	—
68	#107×C₆₁	C₂³	1	1	yes	characteristic
69	#107×C₆₁	C₂³	1	2	yes	—
70	#107×C₆₁	C₂³	1	1	yes	characteristic
71	#107×C₆₁	C₂³	1	2	yes	—
72	C₄⋊C₈×C₆₁	D₄	1	1	yes	characteristic
73	#103×C₆₁	—	2	1	no	—
74	#103×C₆₁	—	2	2	no	—
75	#102×C₆₁	—	2	1	no	—
76	#102×C₆₁	—	2	2	no	—
77	C₂²⋊C₈×C₆₁	D₄	1	2	yes	—
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	#41×C₆₁	—	8	4	no	—
80	C₂×Q₈×C₆₁	—	2	1	no	—
81	C₂×Q₈×C₆₁	—	4	1	no	—
82	C₂×Q₈×C₆₁	—	4	1	no	—
83	C₂×D₄×C₆₁	—	2	1	no	—
84	C₂×D₄×C₆₁	—	4	2	no	—
85	C₂×D₄×C₆₁	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₆₁	—	4	2	no	—
87	C₂²×C₄×C₆₁	—	4	2	no	—
88	C₂²×C₄×C₆₁	—	4	1	no	—
89	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
90	Q₁₆×C₆₁	—	2	1	no	—
91	Q₁₆×C₆₁	—	2	1	no	—
92	D₈×C₆₁	—	2	1	no	—
93	D₈×C₆₁	—	2	1	no	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	2	2	no	—
96	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₆₁	—	2	1	no	—
100	C₄⋊C₄×C₆₁	—	2	2	no	—
101	C₄⋊C₄×C₆₁	—	2	1	no	—
102	C₄⋊C₄×C₆₁	—	2	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	—	4	2	no	—
108	C₄⋊C₄×C₆₁	—	4	1	no	—
109	C₄⋊C₄×C₆₁	D₈	1	1	yes	characteristic
110	C₂²⋊C₄×C₆₁	—	4	2	no	—
111	C₂²⋊C₄×C₆₁	—	4	1	no	—
112	C₂²⋊C₄×C₆₁	—	4	2	no	—
113	C₂²⋊C₄×C₆₁	D₈	1	2	yes	—
114	C₄²×C₆₁	—	4	1	no	—
115	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₆₁	—	2	2	no	—
117	C₁₆×C₆₁	—	2	1	no	—
118	C₁₆×C₆₁	#41	1	2	yes	complemented
119	C₂³×C₆₁	—	4	1	no	—
120	C₂³×C₆₁	C₂×D₈	1	2	yes	—
121	Q₈×C₆₁	—	4	1	no	—
122	Q₈×C₆₁	—	4	1	no	—
123	Q₈×C₆₁	—	8	2	no	—
124	Q₈×C₆₁	—	16	1	no	—
125	D₄×C₆₁	—	4	1	no	—
126	D₄×C₆₁	—	4	1	no	—
127	D₄×C₆₁	—	8	4	no	—
128	D₄×C₆₁	SD₃₂	1	4	yes	complemented
129	C₂×C₄×C₆₁	—	2	2	no	—
130	C₂×C₄×C₆₁	—	2	2	no	—
131	C₂×C₄×C₆₁	—	4	1	no	—
132	C₂×C₄×C₆₁	—	4	1	no	—
133	C₂×C₄×C₆₁	—	4	2	no	—
134	C₂×C₄×C₆₁	—	4	1	no	—
135	C₂×C₄×C₆₁	—	4	2	no	—
136	C₂×C₄×C₆₁	—	8	4	no	—
137	C₂×C₄×C₆₁	—	8	2	no	—
138	C₂×C₄×C₆₁	—	16	2	no	—
139	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
143	C₈×C₆₁	—	2	2	no	—
144	C₈×C₆₁	—	2	1	no	—
145	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
146	C₈×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
147	#33902	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₆₁	—	2	2	no	—
149	C₂²×C₆₁	—	2	2	no	—
150	C₂²×C₆₁	—	8	1	no	—
151	C₂²×C₆₁	—	8	1	no	—
152	C₂²×C₆₁	—	8	1	no	—
153	C₂²×C₆₁	C₂×SD₃₂	1	4	yes	—
154	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
155	C₄×C₆₁	—	2	2	no	—
156	C₄×C₆₁	—	2	1	no	—
157	C₄×C₆₁	—	8	1	no	—
158	C₄×C₆₁	—	8	1	no	—
159	C₄×C₆₁	—	8	2	no	—
160	C₄×C₆₁	—	8	1	no	—
161	C₄×C₆₁	—	8	2	no	—
162	C₄×C₆₁	C₂×SD₃₂	1	2	yes	—
163	C₄×C₆₁	#547	1	1	yes	characteristic
164	C₄×C₆₁	#578	1	1	yes	characteristic
165	#3338	C₂×C₆₁	1	1	yes	characteristic, complemented
166	#3334	C₂×C₆₁	1	1	yes	characteristic
167	#2254	C₂×C₆₁	1	1	yes	characteristic, complemented
168	#2235	C₂×C₆₁	1	2	yes	complemented
169	#2233	C₂×C₆₁	1	2	yes	complemented
170	#2226	C₂×C₆₁	1	1	yes	characteristic, complemented
171	#2225	C₂×C₆₁	1	1	yes	characteristic, complemented
172	#2217	C₂×C₆₁	1	2	yes	complemented
173	C₄×SD₃₂	C₂×C₆₁	1	1	yes	characteristic, complemented
174	D₄×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
175	C₂×#367	C₂×C₆₁	1	2	yes	complemented
176	C₂×C₆₁	—	2	4	no	—
177	C₂×C₆₁	—	16	1	no	—
178	C₂×C₆₁	#3334	1	1	yes	characteristic
179	C₂×C₆₁	C₂²×SD₃₂	1	1	yes	characteristic
180	C₂×C₆₁	#3416	1	1	yes	characteristic
181	#554	—	2	1	no	—
182	#547	—	2	1	no	—
183	C₂×SD₃₂	C₂²×C₆₁	1	1	yes	characteristic
184	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
185	#500	C₂²×C₆₁	1	1	yes	characteristic
186	#500	C₂²×C₆₁	1	1	yes	characteristic
187	#484	—	2	2	no	—
188	#480	—	2	2	no	—
189	#477	—	2	1	no	—
190	#474	—	2	1	no	—
191	#467	C₂²×C₆₁	1	2	yes	—
192	#466	C₂²×C₆₁	1	2	yes	—
193	C₄×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
194	C₄×D₈	C₂²×C₆₁	1	1	yes	characteristic
195	C₈×D₄	C₂²×C₆₁	1	1	yes	characteristic
196	C₂×#107	C₂²×C₆₁	1	2	yes	—
197	C₂×#107	C₂²×C₆₁	1	2	yes	—
198	#367	C₂²×C₆₁	1	1	yes	characteristic
199	#367	C₂²×C₆₁	1	2	yes	—
200	#367	C₂²×C₆₁	1	2	yes	—
201	#367	C₂²×C₆₁	1	2	yes	—
202	#367	C₂²×C₆₁	1	2	yes	—
203	C₄⋊C₁₆	C₂²×C₆₁	1	1	yes	characteristic
204	#358	C₂²×C₆₁	1	1	yes	characteristic
205	#358	C₂²×C₆₁	1	2	yes	—
206	#357	C₂²×C₆₁	1	1	yes	characteristic
207	#357	C₂²×C₆₁	1	2	yes	—
208	C₂²⋊C₁₆	C₂²×C₆₁	1	2	yes	—
209	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
210	C₆₁	#33902	1	1	yes	characteristic, direct product factor, Sylow
211	C₂×#41	—	4	1	no	—
212	C₂×Q₁₆	C₂³×C₆₁	1	1	yes	characteristic
213	C₂×D₈	C₂³×C₆₁	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₆₁	1	2	yes	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	#125	—	4	1	no	—
218	#123	—	4	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	—	4	1	no	—
221	C₂²⋊Q₈	—	4	2	no	—
222	#121	—	2	2	no	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	—	4	1	no	—
226	C₄×D₄	D₄×C₆₁	1	1	yes	characteristic
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	SD₃₂	—	2	2	no	—
230	C₂×C₁₆	—	2	4	no	—
231	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
232	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
233	C₂×C₁₆	C₂³×C₆₁	1	1	yes	characteristic
234	#107	—	2	2	no	—
235	#107	—	2	2	no	—
236	#107	—	2	1	no	—
237	#107	—	2	1	no	—
238	#107	C₂³×C₆₁	1	1	yes	characteristic
239	#107	C₂³×C₆₁	1	2	yes	—
240	#107	C₂³×C₆₁	1	1	yes	characteristic
241	#107	C₂³×C₆₁	1	2	yes	—
242	C₄⋊C₈	D₄×C₆₁	1	1	yes	characteristic
243	#103	—	2	1	no	—
244	#103	—	2	2	no	—
245	#102	—	2	1	no	—
246	#102	—	2	2	no	—
247	C₂²⋊C₈	D₄×C₆₁	1	2	yes	—
248	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
249	#41	—	8	4	no	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	D₈×C₆₁	1	1	yes	characteristic
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
260	Q₁₆	—	2	1	no	—
261	Q₁₆	—	2	1	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
268	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	D₈×C₆₁	1	1	yes	characteristic
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	D₈×C₆₁	1	2	yes	—
284	C₄²	—	4	1	no	—
285	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
286	C₁₆	—	2	2	no	—
287	C₁₆	—	2	1	no	—
288	C₁₆	#41×C₆₁	1	2	yes	complemented
289	C₂³	—	4	1	no	—
290	C₂³	C₂×D₈×C₆₁	1	2	yes	—
291	Q₈	—	4	1	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	8	2	no	—
294	Q₈	—	16	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	8	4	no	—
298	D₄	SD₃₂×C₆₁	1	4	yes	complemented
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	4	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	16	2	no	—
309	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
310	C₂×C₄	C₂×D₈×C₆₁	1	2	yes	—
311	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	2	1	no	—
315	C₈	C₂×#41×C₆₁	1	1	yes	characteristic
316	C₈	2₋¹⁺⁴×C₆₁	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	C₂×SD₃₂×C₆₁	1	4	yes	—
323	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	2	no	—
331	C₄	C₂×SD₃₂×C₆₁	1	2	yes	—
332	C₄	C₆₁×#547	1	1	yes	characteristic
333	C₄	C₆₁×#578	1	1	yes	characteristic
334	C₂	—	2	4	no	—
335	C₂	—	16	1	no	—
336	C₂	C₆₁×#3334	1	1	yes	characteristic
337	C₂	C₂²×SD₃₂×C₆₁	1	1	yes	characteristic
338	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×#3334	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₄×D₄×C₆₁ ◁ C₈×D₄×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3338	C₂	semidirect
3. C₆₁×#2254	C₂	semidirect
4. C₆₁×#2235	C₂	semidirect
5. C₆₁×#2233	C₂	semidirect
6. C₆₁×#2226	C₂	semidirect
7. C₆₁×#2225	C₂	semidirect
8. C₆₁×#2217	C₂	semidirect
9. C₄×SD₃₂×C₆₁	C₂	semidirect
10. D₄×C₁₆×C₆₁	C₂	semidirect
11. C₂×C₆₁×#367	C₂	semidirect
118. C₁₆×C₆₁	#41	semidirect
128. D₄×C₆₁	SD₃₂	semidirect
147. #33902	C₆₁	direct (N × H)
165. #3338	C₂×C₆₁	semidirect
167. #2254	C₂×C₆₁	semidirect
168. #2235	C₂×C₆₁	semidirect
169. #2233	C₂×C₆₁	semidirect
170. #2226	C₂×C₆₁	semidirect
171. #2225	C₂×C₆₁	semidirect
172. #2217	C₂×C₆₁	semidirect
173. C₄×SD₃₂	C₂×C₆₁	semidirect
174. D₄×C₁₆	C₂×C₆₁	semidirect
175. C₂×#367	C₂×C₆₁	semidirect
210. C₆₁	#33902	direct (N × H)
288. C₁₆	#41×C₆₁	semidirect
298. D₄	SD₃₂×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3338	2
C₆₁×#3334	2
C₆₁×#2254	2
C₆₁×#2235	2
C₆₁×#2233	2
C₆₁×#2226	2
C₆₁×#2225	2
C₆₁×#2217	2
C₄×SD₃₂×C₆₁	2
D₄×C₁₆×C₆₁	2
C₂×C₆₁×#367	2
#33902	61

Subgroups — C61×#33902 #29537335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33902 #29537335