Subgroups — C₆₁×#19338 #29522771

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₁×#3254	C₂	1	2	yes	maximal
2	C₆₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×#2931	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1693	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1689	C₂	1	2	yes	maximal, complemented
6	C₆₁×#1679	C₂	1	4	yes	maximal, complemented
7	C₆₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#1561	C₂	1	2	yes	maximal, complemented
9	C₂×#131×C₆₁	—	2	1	no	—
10	C₆₁×#533	C₂²	1	2	yes	—
11	C₆₁×#532	C₂²	1	1	yes	characteristic
12	C₆₁×#500	C₂²	1	1	yes	characteristic
13	C₆₁×#494	C₂²	1	1	yes	characteristic
14	C₆₁×#487	C₂²	1	2	yes	—
15	C₆₁×#477	—	2	2	no	—
16	C₆₁×#474	C₂²	1	4	yes	—
17	C₆₁×#468	—	2	4	no	—
18	C₆₁×#467	—	2	1	no	—
19	C₆₁×#467	—	2	1	no	—
20	C₆₁×#461	C₂²	1	4	yes	—
21	C₆₁×#460	—	2	2	no	—
22	C₄×Q₁₆×C₆₁	—	2	1	no	—
23	C₄×Q₁₆×C₆₁	—	2	1	no	—
24	C₆₁×#436	—	2	1	no	—
25	C₆₁×#436	C₂²	1	2	yes	—
26	C₈×D₄×C₆₁	—	2	1	no	—
27	C₈×D₄×C₆₁	C₂²	1	2	yes	—
28	C₆₁×#423	C₂²	1	1	yes	characteristic
29	C₆₁×#423	C₂²	1	2	yes	—
30	C₆₁×#408	—	2	1	no	—
31	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
32	C₆₁×#334	C₂²	1	2	yes	—
33	C₆₁×#334	C₂²	1	2	yes	—
34	C₆₁×#332	C₂²	1	4	yes	—
35	C₆₁×#332	C₂²	1	4	yes	—
36	C₂×#41×C₆₁	—	2	1	no	—
37	C₂×Q₁₆×C₆₁	—	2	1	no	—
38	C₂×Q₁₆×C₆₁	—	2	1	no	—
39	C₂×SD₁₆×C₆₁	—	2	4	no	—
40	#131×C₆₁	—	4	4	no	—
41	C₂×M₁₆×C₆₁	—	2	1	no	—
42	C₂×M₁₆×C₆₁	—	2	2	no	—
43	C₂×M₁₆×C₆₁	—	2	2	no	—
44	C₂²×C₈×C₆₁	—	2	2	no	—
45	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₁	D₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
50	#127×C₆₁	D₄	1	1	yes	characteristic
51	#125×C₆₁	—	2	2	no	—
52	#124×C₆₁	—	2	1	no	—
53	C₂²⋊Q₈×C₆₁	—	2	2	no	—
54	C₂²⋊Q₈×C₆₁	—	2	2	no	—
55	#121×C₆₁	—	2	1	no	—
56	#121×C₆₁	D₄	1	2	yes	—
57	C₄×Q₈×C₆₁	—	2	2	no	—
58	C₄×Q₈×C₆₁	—	2	2	no	—
59	C₄×D₄×C₆₁	—	2	1	no	—
60	C₄×D₄×C₆₁	C₂³	1	2	yes	—
61	#117×C₆₁	—	2	2	no	—
62	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
63	#107×C₆₁	—	2	1	no	—
64	#107×C₆₁	—	2	1	no	—
65	#107×C₆₁	—	2	4	no	—
66	C₄⋊C₈×C₆₁	—	2	1	no	—
67	C₄⋊C₈×C₆₁	—	2	2	no	—
68	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
69	C₄⋊C₈×C₆₁	C₂³	1	4	yes	—
70	#103×C₆₁	—	2	2	no	—
71	#103×C₆₁	—	2	2	no	—
72	#103×C₆₁	—	2	4	no	—
73	#102×C₆₁	—	2	4	no	—
74	C₂²⋊C₈×C₆₁	—	2	2	no	—
75	C₂²⋊C₈×C₆₁	—	2	2	no	—
76	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
77	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
80	#41×C₆₁	—	4	2	no	—
81	#41×C₆₁	—	4	2	no	—
82	C₂×Q₈×C₆₁	—	2	1	no	—
83	C₂×Q₈×C₆₁	—	2	2	no	—
84	C₂×Q₈×C₆₁	—	2	2	no	—
85	C₂×D₄×C₆₁	—	2	2	no	—
86	C₂×D₄×C₆₁	—	2	1	no	—
87	C₂×D₄×C₆₁	—	2	1	no	—
88	C₂×D₄×C₆₁	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₆₁	—	2	2	no	—
90	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
93	Q₁₆×C₆₁	—	4	2	no	—
94	Q₁₆×C₆₁	—	4	2	no	—
95	SD₁₆×C₆₁	—	8	4	no	—
96	M₁₆×C₆₁	—	4	2	no	—
97	M₁₆×C₆₁	—	4	4	no	—
98	M₁₆×C₆₁	—	8	2	no	—
99	C₂×C₈×C₆₁	—	2	2	no	—
100	C₂×C₈×C₆₁	—	2	2	no	—
101	C₂×C₈×C₆₁	—	2	4	no	—
102	C₂×C₈×C₆₁	—	4	1	no	—
103	C₂×C₈×C₆₁	—	4	4	no	—
104	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
108	C₄⋊C₄×C₆₁	—	2	1	no	—
109	C₄⋊C₄×C₆₁	—	2	2	no	—
110	C₄⋊C₄×C₆₁	—	2	2	no	—
111	C₄⋊C₄×C₆₁	—	2	2	no	—
112	C₄⋊C₄×C₆₁	—	2	2	no	—
113	C₄⋊C₄×C₆₁	—	4	2	no	—
114	C₄⋊C₄×C₆₁	—	4	2	no	—
115	C₄⋊C₄×C₆₁	C₂×D₄	1	2	yes	—
116	C₂²⋊C₄×C₆₁	—	2	2	no	—
117	C₂²⋊C₄×C₆₁	—	2	2	no	—
118	C₂²⋊C₄×C₆₁	—	4	2	no	—
119	C₄²×C₆₁	—	4	2	no	—
120	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₆₁	#41	1	1	yes	characteristic
123	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
124	C₂³×C₆₁	—	2	2	no	—
125	C₂³×C₆₁	#121	1	1	yes	characteristic
126	Q₈×C₆₁	—	4	2	no	—
127	Q₈×C₆₁	—	4	2	no	—
128	Q₈×C₆₁	—	4	2	no	—
129	Q₈×C₆₁	—	4	2	no	—
130	Q₈×C₆₁	—	4	2	no	—
131	D₄×C₆₁	—	2	4	no	—
132	D₄×C₆₁	—	4	2	no	—
133	D₄×C₆₁	—	4	4	no	—
134	D₄×C₆₁	—	4	2	no	—
135	C₂×C₄×C₆₁	—	2	1	no	—
136	C₂×C₄×C₆₁	—	2	1	no	—
137	C₂×C₄×C₆₁	—	2	2	no	—
138	C₂×C₄×C₆₁	—	2	2	no	—
139	C₂×C₄×C₆₁	—	2	2	no	—
140	C₂×C₄×C₆₁	—	4	2	no	—
141	C₂×C₄×C₆₁	—	4	2	no	—
142	C₂×C₄×C₆₁	—	4	2	no	—
143	C₂×C₄×C₆₁	—	4	2	no	—
144	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
145	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
146	C₂×C₄×C₆₁	#121	1	1	yes	characteristic
147	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
150	C₈×C₆₁	—	2	1	no	—
151	C₈×C₆₁	—	2	1	no	—
152	C₈×C₆₁	—	2	1	no	—
153	C₈×C₆₁	—	4	2	no	—
154	C₈×C₆₁	—	4	4	no	—
155	C₈×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
156	#19338	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂²×C₆₁	—	2	1	no	—
158	C₂²×C₆₁	—	2	1	no	—
159	C₂²×C₆₁	—	2	1	no	—
160	C₂²×C₆₁	—	4	2	no	—
161	C₂²×C₆₁	—	4	2	no	—
162	C₂²×C₆₁	—	4	2	no	—
163	C₂²×C₆₁	C₂×#121	1	1	yes	characteristic
164	C₄×C₆₁	—	2	1	no	—
165	C₄×C₆₁	—	2	1	no	—
166	C₄×C₆₁	—	2	1	no	—
167	C₄×C₆₁	—	4	2	no	—
168	C₄×C₆₁	—	8	2	no	—
169	C₄×C₆₁	—	8	2	no	—
170	C₄×C₆₁	#473	1	2	yes	—
171	C₄×C₆₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₄×C₆₁	#575	1	1	yes	characteristic
173	C₄×C₆₁	#577	1	2	yes	—
174	#3254	C₂×C₆₁	1	2	yes	—
175	#3170	C₂×C₆₁	1	1	yes	characteristic, complemented
176	#2931	C₂×C₆₁	1	1	yes	characteristic
177	#1693	C₂×C₆₁	1	2	yes	complemented
178	#1689	C₂×C₆₁	1	2	yes	complemented
179	#1679	C₂×C₆₁	1	4	yes	complemented
180	#1566	C₂×C₆₁	1	1	yes	characteristic, complemented
181	#1561	C₂×C₆₁	1	2	yes	complemented
182	C₂×C₆₁	—	4	1	no	—
183	C₂×C₆₁	—	8	2	no	—
184	C₂×C₆₁	#3036	1	1	yes	characteristic
185	C₂×C₆₁	#3060	1	1	yes	characteristic
186	C₂×C₆₁	C₂×#473	1	1	yes	characteristic
187	C₂×#131	—	2	1	no	—
188	#533	C₂²×C₆₁	1	2	yes	—
189	#532	C₂²×C₆₁	1	1	yes	characteristic
190	#500	C₂²×C₆₁	1	1	yes	characteristic
191	#494	C₂²×C₆₁	1	1	yes	characteristic
192	#487	C₂²×C₆₁	1	2	yes	—
193	#477	—	2	2	no	—
194	#474	C₂²×C₆₁	1	4	yes	—
195	#468	—	2	4	no	—
196	#467	—	2	1	no	—
197	#467	—	2	1	no	—
198	#461	C₂²×C₆₁	1	4	yes	—
199	#460	—	2	2	no	—
200	C₄×Q₁₆	—	2	1	no	—
201	C₄×Q₁₆	—	2	1	no	—
202	#436	—	2	1	no	—
203	#436	C₂²×C₆₁	1	2	yes	—
204	C₈×D₄	—	2	1	no	—
205	C₈×D₄	C₂²×C₆₁	1	2	yes	—
206	#423	C₂²×C₆₁	1	1	yes	characteristic
207	#423	C₂²×C₆₁	1	2	yes	—
208	#408	—	2	1	no	—
209	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
210	#334	C₂²×C₆₁	1	2	yes	—
211	#334	C₂²×C₆₁	1	2	yes	—
212	#332	C₂²×C₆₁	1	4	yes	—
213	#332	C₂²×C₆₁	1	4	yes	—
214	C₆₁	#19338	1	1	yes	characteristic, direct product factor, Sylow
215	C₂×#41	—	2	1	no	—
216	C₂×Q₁₆	—	2	1	no	—
217	C₂×Q₁₆	—	2	1	no	—
218	C₂×SD₁₆	—	2	4	no	—
219	#131	—	4	4	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂²×C₈	—	2	2	no	—
224	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
225	C₂²×C₈	D₄×C₆₁	1	1	yes	characteristic
226	C₄⋊Q₈	D₄×C₆₁	1	1	yes	characteristic
227	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
228	C₄⋊Q₈	C₂³×C₆₁	1	2	yes	—
229	#127	D₄×C₆₁	1	1	yes	characteristic
230	#125	—	2	2	no	—
231	#124	—	2	1	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	#121	—	2	1	no	—
235	#121	D₄×C₆₁	1	2	yes	—
236	C₄×Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	C₂³×C₆₁	1	2	yes	—
240	#117	—	2	2	no	—
241	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
242	#107	—	2	1	no	—
243	#107	—	2	1	no	—
244	#107	—	2	4	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	2	no	—
247	C₄⋊C₈	C₂³×C₆₁	1	2	yes	—
248	C₄⋊C₈	C₂³×C₆₁	1	4	yes	—
249	#103	—	2	2	no	—
250	#103	—	2	2	no	—
251	#103	—	2	4	no	—
252	#102	—	2	4	no	—
253	C₂²⋊C₈	—	2	2	no	—
254	C₂²⋊C₈	—	2	2	no	—
255	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
256	C₄×C₈	D₄×C₆₁	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
258	C₄×C₈	C₂³×C₆₁	1	1	yes	characteristic
259	#41	—	4	2	no	—
260	#41	—	4	2	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	C₂×D₄×C₆₁	1	2	yes	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
271	C₂²×C₄	#41×C₆₁	1	1	yes	characteristic
272	Q₁₆	—	4	2	no	—
273	Q₁₆	—	4	2	no	—
274	SD₁₆	—	8	4	no	—
275	M₁₆	—	4	2	no	—
276	M₁₆	—	4	4	no	—
277	M₁₆	—	8	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	4	no	—
283	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
284	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
285	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
286	C₂×C₈	C₂×D₄×C₆₁	1	1	yes	characteristic
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	C₂×D₄×C₆₁	1	2	yes	—
295	C₂²⋊C₄	—	2	2	no	—
296	C₂²⋊C₄	—	2	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₄²	—	4	2	no	—
299	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
300	C₄²	C₂×D₄×C₆₁	1	1	yes	characteristic
301	C₄²	#41×C₆₁	1	1	yes	characteristic
302	C₄²	C₂⁴×C₆₁	1	1	yes	characteristic
303	C₂³	—	2	2	no	—
304	C₂³	#121×C₆₁	1	1	yes	characteristic
305	Q₈	—	4	2	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	D₄	—	2	4	no	—
311	D₄	—	4	2	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
324	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
325	C₂×C₄	#121×C₆₁	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
328	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	2	1	no	—
331	C₈	—	2	1	no	—
332	C₈	—	4	2	no	—
333	C₈	—	4	4	no	—
334	C₈	AGL(1,ℤ/8)×C₆₁	1	2	yes	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	C₂×#121×C₆₁	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	C₆₁×#473	1	2	yes	—
349	C₄	C₂×AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
350	C₄	C₆₁×#575	1	1	yes	characteristic
351	C₄	C₆₁×#577	1	2	yes	—
352	C₂	—	4	1	no	—
353	C₂	—	8	2	no	—
354	C₂	C₆₁×#3036	1	1	yes	characteristic
355	C₂	C₆₁×#3060	1	1	yes	characteristic
356	C₂	C₂×C₆₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₁
prime radical	C₆₁×#532	C₂²
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
2. C₆₁×#3170	C₂	semidirect
4. C₆₁×#1693	C₂	semidirect
5. C₆₁×#1689	C₂	semidirect
6. C₆₁×#1679	C₂	semidirect
7. C₆₁×#1566	C₂	semidirect
8. C₆₁×#1561	C₂	semidirect
156. #19338	C₆₁	direct (N × H)
175. #3170	C₂×C₆₁	semidirect
177. #1693	C₂×C₆₁	semidirect
178. #1689	C₂×C₆₁	semidirect
179. #1679	C₂×C₆₁	semidirect
180. #1566	C₂×C₆₁	semidirect
181. #1561	C₂×C₆₁	semidirect
214. C₆₁	#19338	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×#3254	2
C₆₁×#3170	2
C₆₁×#2931	2
C₆₁×#1693	2
C₆₁×#1689	2
C₆₁×#1679	2
C₆₁×#1566	2
C₆₁×#1561	2
#19338	61

Subgroups — C61×#19338 #29522771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#19338 #29522771