Subgroups — C₆₁×#33736 #29537169

Order 15616.

558 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×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₁×#3169	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#2217	C₂	1	4	yes	maximal, complemented
8	C₄×SD₃₂×C₆₁	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#533	—	2	1	no	—
11	C₆₁×#532	—	2	1	no	—
12	C₂×SD₃₂×C₆₁	C₂²	1	4	yes	—
13	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
14	C₆₁×#500	C₂²	1	1	yes	characteristic
15	C₆₁×#500	C₂²	1	1	yes	characteristic
16	C₆₁×#494	C₂²	1	1	yes	characteristic
17	C₆₁×#493	C₂²	1	1	yes	characteristic
18	C₆₁×#487	—	2	1	no	—
19	C₆₁×#486	—	2	1	no	—
20	C₆₁×#467	—	2	1	no	—
21	C₆₁×#467	C₂²	1	2	yes	—
22	C₆₁×#466	—	2	1	no	—
23	C₆₁×#466	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₆₁	—	2	1	no	—
25	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
26	C₄×D₈×C₆₁	—	2	1	no	—
27	C₄×D₈×C₆₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#367	C₂²	1	4	yes	—
30	C₆₁×#358	C₂²	1	4	yes	—
31	C₆₁×#357	C₂²	1	4	yes	—
32	C₄×C₁₆×C₆₁	C₂²	1	2	yes	—
33	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
34	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×#41×C₆₁	—	4	1	no	—
36	C₂×Q₁₆×C₆₁	—	2	1	no	—
37	C₂×Q₁₆×C₆₁	C₂³	1	2	yes	—
38	C₂×D₈×C₆₁	—	2	1	no	—
39	C₂×D₈×C₆₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₆₁	—	2	1	no	—
43	C₄⋊Q₈×C₆₁	—	2	1	no	—
44	C₄⋊Q₈×C₆₁	—	2	1	no	—
45	#127×C₆₁	—	2	1	no	—
46	#125×C₆₁	—	4	1	no	—
47	#124×C₆₁	—	4	1	no	—
48	C₂²⋊Q₈×C₆₁	—	2	2	no	—
49	C₂²⋊Q₈×C₆₁	—	4	1	no	—
50	#121×C₆₁	—	2	2	no	—
51	#121×C₆₁	—	4	1	no	—
52	C₄×Q₈×C₆₁	—	2	2	no	—
53	C₄×Q₈×C₆₁	—	4	1	no	—
54	C₄×D₄×C₆₁	—	2	2	no	—
55	C₄×D₄×C₆₁	—	4	1	no	—
56	#117×C₆₁	—	4	1	no	—
57	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
58	SD₃₂×C₆₁	—	2	8	no	—
59	C₂×C₁₆×C₆₁	—	2	4	no	—
60	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
61	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
62	#107×C₆₁	—	2	1	no	—
63	#107×C₆₁	—	2	1	no	—
64	#107×C₆₁	C₂³	1	2	yes	—
65	#107×C₆₁	C₂³	1	2	yes	—
66	#103×C₆₁	—	2	2	no	—
67	#103×C₆₁	—	2	2	no	—
68	#102×C₆₁	—	2	2	no	—
69	#102×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	#41×C₆₁	—	8	4	no	—
75	C₂×Q₈×C₆₁	—	2	2	no	—
76	C₂×Q₈×C₆₁	—	4	1	no	—
77	C₂×Q₈×C₆₁	—	4	1	no	—
78	C₂×D₄×C₆₁	—	2	2	no	—
79	C₂×D₄×C₆₁	—	4	2	no	—
80	C₂×D₄×C₆₁	—	4	1	no	—
81	C₂×D₄×C₆₁	—	4	1	no	—
82	C₂²×C₄×C₆₁	—	4	2	no	—
83	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
86	Q₁₆×C₆₁	—	2	4	no	—
87	Q₁₆×C₆₁	—	4	2	no	—
88	D₈×C₆₁	—	2	4	no	—
89	D₈×C₆₁	—	4	2	no	—
90	C₂×C₈×C₆₁	—	2	2	no	—
91	C₂×C₈×C₆₁	—	2	2	no	—
92	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₄⋊C₄×C₆₁	—	2	2	no	—
98	C₄⋊C₄×C₆₁	—	4	1	no	—
99	C₄⋊C₄×C₆₁	—	4	1	no	—
100	C₄⋊C₄×C₆₁	—	4	1	no	—
101	C₄⋊C₄×C₆₁	—	4	2	no	—
102	C₄⋊C₄×C₆₁	—	4	2	no	—
103	C₄⋊C₄×C₆₁	—	4	1	no	—
104	C₂²⋊C₄×C₆₁	—	4	2	no	—
105	C₂²⋊C₄×C₆₁	—	4	2	no	—
106	C₂²⋊C₄×C₆₁	—	4	2	no	—
107	C₄²×C₆₁	—	4	2	no	—
108	C₄²×C₆₁	D₈	1	1	yes	characteristic
109	C₄²×C₆₁	D₈	1	1	yes	characteristic
110	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₆₁	—	2	2	no	—
113	C₁₆×C₆₁	#41	1	4	yes	complemented
114	C₂³×C₆₁	—	4	2	no	—
115	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₆₁	—	4	2	no	—
117	Q₈×C₆₁	—	4	2	no	—
118	Q₈×C₆₁	—	8	2	no	—
119	Q₈×C₆₁	—	8	2	no	—
120	D₄×C₆₁	—	4	2	no	—
121	D₄×C₆₁	—	4	2	no	—
122	D₄×C₆₁	—	8	4	no	—
123	D₄×C₆₁	—	8	2	no	—
124	D₄×C₆₁	—	8	2	no	—
125	C₂×C₄×C₆₁	—	2	1	no	—
126	C₂×C₄×C₆₁	—	2	1	no	—
127	C₂×C₄×C₆₁	—	2	2	no	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	4	2	no	—
130	C₂×C₄×C₆₁	—	4	2	no	—
131	C₂×C₄×C₆₁	—	8	4	no	—
132	C₂×C₄×C₆₁	SD₃₂	1	4	yes	complemented
133	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₆₁	—	2	1	no	—
140	C₈×C₆₁	—	2	1	no	—
141	C₈×C₆₁	#135	1	2	yes	—
142	C₈×C₆₁	C₂×#41	1	2	yes	—
143	#33736	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₆₁	—	2	1	no	—
145	C₂²×C₆₁	—	2	1	no	—
146	C₂²×C₆₁	—	8	2	no	—
147	C₂²×C₆₁	—	8	2	no	—
148	C₂²×C₆₁	—	8	2	no	—
149	C₂²×C₆₁	C₂×SD₃₂	1	2	yes	—
150	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₆₁	—	2	1	no	—
152	C₄×C₆₁	—	2	1	no	—
153	C₄×C₆₁	—	8	2	no	—
154	C₄×C₆₁	—	8	2	no	—
155	C₄×C₆₁	—	8	2	no	—
156	C₄×C₆₁	C₂×SD₃₂	1	2	yes	—
157	C₄×C₆₁	C₂×SD₃₂	1	2	yes	—
158	C₄×C₆₁	#508	1	2	yes	—
159	C₄×C₆₁	#532	1	1	yes	characteristic
160	C₄×C₆₁	C₂×#135	1	1	yes	characteristic
161	#3170	C₂×C₆₁	1	1	yes	characteristic, complemented
162	#3169	C₂×C₆₁	1	1	yes	characteristic
163	#2254	C₂×C₆₁	1	1	yes	characteristic, complemented
164	#2248	C₂×C₆₁	1	1	yes	characteristic, complemented
165	#2241	C₂×C₆₁	1	1	yes	characteristic, complemented
166	#2240	C₂×C₆₁	1	1	yes	characteristic, complemented
167	#2217	C₂×C₆₁	1	4	yes	complemented
168	C₄×SD₃₂	C₂×C₆₁	1	4	yes	complemented
169	C₂×C₄×C₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
170	C₂×C₆₁	—	2	2	no	—
171	C₂×C₆₁	—	16	2	no	—
172	C₂×C₆₁	#3169	1	1	yes	characteristic
173	C₂×C₆₁	C₂²×SD₃₂	1	1	yes	characteristic
174	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
175	#533	—	2	1	no	—
176	#532	—	2	1	no	—
177	C₂×SD₃₂	C₂²×C₆₁	1	4	yes	—
178	C₂²×C₁₆	C₂²×C₆₁	1	2	yes	—
179	#500	C₂²×C₆₁	1	1	yes	characteristic
180	#500	C₂²×C₆₁	1	1	yes	characteristic
181	#494	C₂²×C₆₁	1	1	yes	characteristic
182	#493	C₂²×C₆₁	1	1	yes	characteristic
183	#487	—	2	1	no	—
184	#486	—	2	1	no	—
185	#467	—	2	1	no	—
186	#467	C₂²×C₆₁	1	2	yes	—
187	#466	—	2	1	no	—
188	#466	C₂²×C₆₁	1	2	yes	—
189	C₄×Q₁₆	—	2	1	no	—
190	C₄×Q₁₆	C₂²×C₆₁	1	2	yes	—
191	C₄×D₈	—	2	1	no	—
192	C₄×D₈	C₂²×C₆₁	1	2	yes	—
193	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
194	#367	C₂²×C₆₁	1	4	yes	—
195	#358	C₂²×C₆₁	1	4	yes	—
196	#357	C₂²×C₆₁	1	4	yes	—
197	C₄×C₁₆	C₂²×C₆₁	1	2	yes	—
198	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
199	C₄×C₁₆	C₂²×C₆₁	1	1	yes	characteristic
200	C₆₁	#33736	1	1	yes	characteristic, direct product factor, Sylow
201	C₂×#41	—	4	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	C₂³×C₆₁	1	2	yes	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	C₂³×C₆₁	1	2	yes	—
206	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	#127	—	2	1	no	—
212	#125	—	4	1	no	—
213	#124	—	4	1	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	4	1	no	—
216	#121	—	2	2	no	—
217	#121	—	4	1	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	4	1	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	4	1	no	—
222	#117	—	4	1	no	—
223	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
224	SD₃₂	—	2	8	no	—
225	C₂×C₁₆	—	2	4	no	—
226	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
227	C₂×C₁₆	C₂³×C₆₁	1	2	yes	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	C₂³×C₆₁	1	2	yes	—
231	#107	C₂³×C₆₁	1	2	yes	—
232	#103	—	2	2	no	—
233	#103	—	2	2	no	—
234	#102	—	2	2	no	—
235	#102	—	2	2	no	—
236	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
237	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
240	#41	—	8	4	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
250	C₂²×C₄	D₈×C₆₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
252	Q₁₆	—	2	4	no	—
253	Q₁₆	—	4	2	no	—
254	D₈	—	2	4	no	—
255	D₈	—	4	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
261	C₂×C₈	C₂⁴×C₆₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	D₈×C₆₁	1	1	yes	characteristic
275	C₄²	D₈×C₆₁	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	#41×C₆₁	1	4	yes	complemented
280	C₂³	—	4	2	no	—
281	C₂³	C₂×D₈×C₆₁	1	1	yes	characteristic
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	Q₈	—	8	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	2	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	SD₃₂×C₆₁	1	4	yes	complemented
299	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
300	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₆₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	#135×C₆₁	1	2	yes	—
308	C₈	C₂×#41×C₆₁	1	2	yes	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂×SD₃₂×C₆₁	1	2	yes	—
315	C₂²	C₂²×D₈×C₆₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₂×SD₃₂×C₆₁	1	2	yes	—
322	C₄	C₂×SD₃₂×C₆₁	1	2	yes	—
323	C₄	C₆₁×#508	1	2	yes	—
324	C₄	C₆₁×#532	1	1	yes	characteristic
325	C₄	C₂×#135×C₆₁	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	16	2	no	—
328	C₂	C₆₁×#3169	1	1	yes	characteristic
329	C₂	C₂²×SD₃₂×C₆₁	1	1	yes	characteristic
330	C₂	C₂×C₆₁×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₁
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₆₁×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×#2254	C₂	semidirect
4. C₆₁×#2248	C₂	semidirect
5. C₆₁×#2241	C₂	semidirect
6. C₆₁×#2240	C₂	semidirect
7. C₆₁×#2217	C₂	semidirect
8. C₄×SD₃₂×C₆₁	C₂	semidirect
9. C₂×C₄×C₁₆×C₆₁	C₂	semidirect
113. C₁₆×C₆₁	#41	semidirect
132. C₂×C₄×C₆₁	SD₃₂	semidirect
143. #33736	C₆₁	direct (N × H)
161. #3170	C₂×C₆₁	semidirect
163. #2254	C₂×C₆₁	semidirect
164. #2248	C₂×C₆₁	semidirect
165. #2241	C₂×C₆₁	semidirect
166. #2240	C₂×C₆₁	semidirect
167. #2217	C₂×C₆₁	semidirect
168. C₄×SD₃₂	C₂×C₆₁	semidirect
169. C₂×C₄×C₁₆	C₂×C₆₁	semidirect
200. C₆₁	#33736	direct (N × H)
279. C₁₆	#41×C₆₁	semidirect
298. C₂×C₄	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×#3170	2
C₆₁×#3169	2
C₆₁×#2254	2
C₆₁×#2248	2
C₆₁×#2241	2
C₆₁×#2240	2
C₆₁×#2217	2
C₄×SD₃₂×C₆₁	2
C₂×C₄×C₁₆×C₆₁	2
#33736	61

Subgroups — C61×#33736 #29537169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#33736 #29537169