Subgroups — C₆₁×#19203 #29522636

Order 15616.

774 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3208	C₂	1	2	yes	maximal, complemented
2	C₂×C₆₁×#493	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1688	C₂	1	4	yes	maximal, complemented
5	C₆₁×#1668	C₂	1	4	yes	maximal, complemented
6	C₆₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1545	C₂	1	2	yes	maximal, complemented
8	C₂²×D₈×C₆₁	—	2	1	no	—
9	C₂²×D₈×C₆₁	—	2	1	no	—
10	C₆₁×#533	C₂²	1	1	yes	characteristic
11	C₂×#127×C₆₁	C₂²	1	2	yes	—
12	C₆₁×#493	—	2	1	no	—
13	C₆₁×#493	—	2	1	no	—
14	C₆₁×#493	C₂²	1	2	yes	—
15	C₆₁×#493	C₂²	1	2	yes	—
16	C₆₁×#492	—	2	2	no	—
17	C₆₁×#486	—	2	2	no	—
18	C₆₁×#460	C₂²	1	8	yes	complemented
19	C₆₁×#450	—	2	4	no	—
20	C₆₁×#446	—	2	1	no	—
21	C₆₁×#446	C₂²	1	2	yes	complemented
22	C₈×Q₈×C₆₁	—	2	1	no	—
23	C₈×Q₈×C₆₁	C₂²	1	2	yes	complemented
24	C₆₁×#431	C₂²	1	2	yes	—
25	C₆₁×#424	—	2	1	no	—
26	C₆₁×#423	C₂²	1	1	yes	characteristic
27	C₆₁×#405	—	2	1	no	—
28	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#335	C₂²	1	2	yes	complemented
30	C₆₁×#334	C₂²	1	2	yes	complemented
31	C₆₁×#331	C₂²	1	8	yes	complemented
32	C₂²×D₄×C₆₁	—	2	2	no	—
33	C₂²×D₄×C₆₁	—	2	2	no	—
34	C₂²×D₄×C₆₁	—	2	2	no	—
35	C₂×SD₁₆×C₆₁	—	2	8	no	—
36	C₂×D₈×C₆₁	—	2	2	no	—
37	C₂×D₈×C₆₁	—	2	2	no	—
38	C₂×D₈×C₆₁	—	2	2	no	—
39	C₂×D₈×C₆₁	—	2	2	no	—
40	C₂×D₈×C₆₁	—	4	4	no	—
41	C₂×D₈×C₆₁	—	4	4	no	—
42	C₂×M₁₆×C₆₁	—	2	1	no	—
43	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
45	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
47	#127×C₆₁	—	2	2	no	—
48	#127×C₆₁	—	2	2	no	—
49	#127×C₆₁	—	2	2	no	—
50	#127×C₆₁	C₂³	1	4	yes	—
51	#125×C₆₁	—	2	1	no	—
52	C₂²⋊Q₈×C₆₁	—	2	1	no	—
53	C₂²⋊Q₈×C₆₁	D₄	1	2	yes	—
54	C₄×Q₈×C₆₁	—	2	1	no	—
55	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
56	#117×C₆₁	—	2	1	no	—
57	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₆₁	—	2	2	no	—
59	C₄⋊C₈×C₆₁	—	2	2	no	—
60	C₄⋊C₈×C₆₁	—	2	1	no	—
61	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
62	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
63	#102×C₆₁	—	2	8	no	—
64	C₂²⋊C₈×C₆₁	—	2	2	no	—
65	#97×C₆₁	—	2	2	no	—
66	C₄×C₈×C₆₁	—	2	2	no	—
67	C₄×C₈×C₆₁	—	2	2	no	—
68	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
72	C₂⁴×C₆₁	—	4	2	no	—
73	C₂×Q₈×C₆₁	—	2	1	no	—
74	C₂×Q₈×C₆₁	C₂×D₄	1	2	yes	—
75	C₂×D₄×C₆₁	—	2	4	no	—
76	C₂×D₄×C₆₁	—	2	4	no	—
77	C₂×D₄×C₆₁	—	2	4	no	—
78	C₂×D₄×C₆₁	—	4	2	no	—
79	C₂×D₄×C₆₁	—	4	2	no	—
80	C₂×D₄×C₆₁	—	4	2	no	—
81	C₂×D₄×C₆₁	—	4	4	no	—
82	C₂×D₄×C₆₁	—	4	4	no	—
83	C₂×D₄×C₆₁	—	8	4	no	—
84	C₂×D₄×C₆₁	—	8	4	no	—
85	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
88	SD₁₆×C₆₁	—	8	8	no	—
89	D₈×C₆₁	—	4	4	no	—
90	D₈×C₆₁	—	4	4	no	—
91	D₈×C₆₁	—	4	4	no	—
92	D₈×C₆₁	—	4	4	no	—
93	M₁₆×C₆₁	—	4	2	no	—
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	—	2	2	no	—
96	C₂×C₈×C₆₁	—	2	4	no	—
97	C₂×C₈×C₆₁	SD₁₆	1	4	yes	complemented
98	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₆₁	—	2	1	no	—
103	C₄⋊C₄×C₆₁	—	2	1	no	—
104	C₄⋊C₄×C₆₁	—	2	1	no	—
105	C₄⋊C₄×C₆₁	—	2	2	no	—
106	C₄⋊C₄×C₆₁	—	2	2	no	—
107	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
108	C₂²⋊C₄×C₆₁	—	2	2	no	—
109	C₄²×C₆₁	—	2	2	no	—
110	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₆₁	#41	1	1	yes	characteristic
113	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
114	C₂³×C₆₁	—	4	4	no	—
115	C₂³×C₆₁	—	8	4	no	—
116	C₂³×C₆₁	—	8	4	no	—
117	C₂³×C₆₁	—	8	4	no	—
118	C₂³×C₆₁	#121	1	1	yes	characteristic
119	Q₈×C₆₁	—	2	4	no	—
120	Q₈×C₆₁	—	4	2	no	—
121	D₄×C₆₁	—	4	4	no	—
122	D₄×C₆₁	—	4	4	no	—
123	D₄×C₆₁	—	4	8	no	—
124	D₄×C₆₁	—	8	4	no	—
125	D₄×C₆₁	—	8	4	no	—
126	D₄×C₆₁	—	8	4	no	—
127	D₄×C₆₁	—	8	4	no	—
128	C₂×C₄×C₆₁	—	2	2	no	—
129	C₂×C₄×C₆₁	—	2	2	no	—
130	C₂×C₄×C₆₁	—	2	2	no	—
131	C₂×C₄×C₆₁	—	2	2	no	—
132	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
137	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
140	C₈×C₆₁	—	2	1	no	—
141	C₈×C₆₁	—	2	1	no	—
142	C₈×C₆₁	—	4	2	no	—
143	C₈×C₆₁	—	4	4	no	—
144	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
145	C₈×C₆₁	C₂×SD₁₆	1	2	yes	—
146	#19203	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂²×C₆₁	—	2	1	no	—
148	C₂²×C₆₁	—	2	1	no	—
149	C₂²×C₆₁	—	8	4	no	—
150	C₂²×C₆₁	—	8	4	no	—
151	C₂²×C₆₁	—	8	4	no	—
152	C₂²×C₆₁	—	16	8	no	—
153	C₂²×C₆₁	#460	1	2	yes	—
154	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
155	C₄×C₆₁	—	2	1	no	—
156	C₄×C₆₁	—	2	1	no	—
157	C₄×C₆₁	—	4	2	no	—
158	C₄×C₆₁	—	4	2	no	—
159	C₄×C₆₁	#460	1	2	yes	—
160	C₄×C₆₁	#472	1	2	yes	—
161	C₄×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
162	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₄×C₆₁	#576	1	2	yes	—
164	#3208	C₂×C₆₁	1	2	yes	complemented
165	C₂×#493	C₂×C₆₁	1	1	yes	characteristic
166	#2960	C₂×C₆₁	1	1	yes	characteristic, complemented
167	#1688	C₂×C₆₁	1	4	yes	complemented
168	#1668	C₂×C₆₁	1	4	yes	complemented
169	#1564	C₂×C₆₁	1	1	yes	characteristic, complemented
170	#1545	C₂×C₆₁	1	2	yes	complemented
171	C₂×C₆₁	—	2	2	no	—
172	C₂×C₆₁	—	16	4	no	—
173	C₂×C₆₁	C₂×#460	1	1	yes	characteristic
174	C₂×C₆₁	#3058	1	1	yes	characteristic
175	C₂×C₆₁	C₂×#472	1	1	yes	characteristic
176	C₂²×D₈	—	2	1	no	—
177	C₂²×D₈	—	2	1	no	—
178	#533	C₂²×C₆₁	1	1	yes	characteristic
179	C₂×#127	C₂²×C₆₁	1	2	yes	—
180	#493	—	2	1	no	—
181	#493	—	2	1	no	—
182	#493	C₂²×C₆₁	1	2	yes	—
183	#493	C₂²×C₆₁	1	2	yes	—
184	#492	—	2	2	no	—
185	#486	—	2	2	no	—
186	#460	C₂²×C₆₁	1	8	yes	complemented
187	#450	—	2	4	no	—
188	#446	—	2	1	no	—
189	#446	C₂²×C₆₁	1	2	yes	complemented
190	C₈×Q₈	—	2	1	no	—
191	C₈×Q₈	C₂²×C₆₁	1	2	yes	complemented
192	#431	C₂²×C₆₁	1	2	yes	—
193	#424	—	2	1	no	—
194	#423	C₂²×C₆₁	1	1	yes	characteristic
195	#405	—	2	1	no	—
196	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
197	#335	C₂²×C₆₁	1	2	yes	complemented
198	#334	C₂²×C₆₁	1	2	yes	complemented
199	#331	C₂²×C₆₁	1	8	yes	complemented
200	C₆₁	#19203	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	—	2	2	no	—
203	C₂²×D₄	—	2	2	no	—
204	C₂×SD₁₆	—	2	8	no	—
205	C₂×D₈	—	2	2	no	—
206	C₂×D₈	—	2	2	no	—
207	C₂×D₈	—	2	2	no	—
208	C₂×D₈	—	2	2	no	—
209	C₂×D₈	—	4	4	no	—
210	C₂×D₈	—	4	4	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
213	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
214	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
215	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
216	#127	—	2	2	no	—
217	#127	—	2	2	no	—
218	#127	—	2	2	no	—
219	#127	C₂³×C₆₁	1	4	yes	—
220	#125	—	2	1	no	—
221	C₂²⋊Q₈	—	2	1	no	—
222	C₂²⋊Q₈	D₄×C₆₁	1	2	yes	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×Q₈	C₂³×C₆₁	1	2	yes	—
225	#117	—	2	1	no	—
226	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
231	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
232	#102	—	2	8	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	#97	—	2	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
238	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
241	C₂⁴	—	4	2	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	C₂×D₄×C₆₁	1	2	yes	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	4	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂×D₄	—	8	4	no	—
253	C₂×D₄	—	8	4	no	—
254	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
256	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
257	SD₁₆	—	8	8	no	—
258	D₈	—	4	4	no	—
259	D₈	—	4	4	no	—
260	D₈	—	4	4	no	—
261	D₈	—	4	4	no	—
262	M₁₆	—	4	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	SD₁₆×C₆₁	1	4	yes	complemented
267	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
281	C₄²	#41×C₆₁	1	1	yes	characteristic
282	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
283	C₂³	—	4	4	no	—
284	C₂³	—	8	4	no	—
285	C₂³	—	8	4	no	—
286	C₂³	—	8	4	no	—
287	C₂³	#121×C₆₁	1	1	yes	characteristic
288	Q₈	—	2	4	no	—
289	Q₈	—	4	2	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	8	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	D₄	—	8	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
302	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
303	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
304	C₂×C₄	C₂×SD₁₆×C₆₁	1	2	yes	—
305	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
306	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	2	no	—
312	C₈	—	4	4	no	—
313	C₈	C₂×SD₁₆×C₆₁	1	2	yes	—
314	C₈	C₂×SD₁₆×C₆₁	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	8	4	no	—
318	C₂²	—	8	4	no	—
319	C₂²	—	8	4	no	—
320	C₂²	—	16	8	no	—
321	C₂²	C₆₁×#460	1	2	yes	—
322	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	C₆₁×#460	1	2	yes	—
328	C₄	C₆₁×#472	1	2	yes	—
329	C₄	C₂²×SD₁₆×C₆₁	1	1	yes	characteristic
330	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
331	C₄	C₆₁×#576	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	16	4	no	—
334	C₂	C₂×C₆₁×#460	1	1	yes	characteristic
335	C₂	C₆₁×#3058	1	1	yes	characteristic
336	C₂	C₂×C₆₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₂×C₆₁×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₁ ◁ C₂×C₄²×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₁×#3208	C₂	semidirect
3. C₆₁×#2960	C₂	semidirect
4. C₆₁×#1688	C₂	semidirect
5. C₆₁×#1668	C₂	semidirect
6. C₆₁×#1564	C₂	semidirect
7. C₆₁×#1545	C₂	semidirect
18. C₆₁×#460	C₂²	semidirect
21. C₆₁×#446	C₂²	semidirect
23. C₈×Q₈×C₆₁	C₂²	semidirect
29. C₆₁×#335	C₂²	semidirect
30. C₆₁×#334	C₂²	semidirect
31. C₆₁×#331	C₂²	semidirect
97. C₂×C₈×C₆₁	SD₁₆	semidirect
146. #19203	C₆₁	direct (N × H)
164. #3208	C₂×C₆₁	semidirect
166. #2960	C₂×C₆₁	semidirect
167. #1688	C₂×C₆₁	semidirect
168. #1668	C₂×C₆₁	semidirect
169. #1564	C₂×C₆₁	semidirect
170. #1545	C₂×C₆₁	semidirect
186. #460	C₂²×C₆₁	semidirect
189. #446	C₂²×C₆₁	semidirect
191. C₈×Q₈	C₂²×C₆₁	semidirect
197. #335	C₂²×C₆₁	semidirect
198. #334	C₂²×C₆₁	semidirect
199. #331	C₂²×C₆₁	semidirect
200. C₆₁	#19203	direct (N × H)
266. C₂×C₈	SD₁₆×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₆₁×#3208	2
C₂×C₆₁×#493	2
C₆₁×#2960	2
C₆₁×#1688	2
C₆₁×#1668	2
C₆₁×#1564	2
C₆₁×#1545	2
#19203	61

Subgroups — C61×#19203 #29522636

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19203 #29522636