Subgroups — C₆₁×#7584 #29511017

Order 15616.

510 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#124×C₆₁	C₂	1	1	yes	maximal, characteristic
2	C₆₁×#1839	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₄⋊C₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₁×#1272	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1270	C₂	1	2	yes	maximal
6	C₂×#124×C₆₁	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₆₁	C₄	1	1	yes	characteristic
8	C₂×C₄×D₄×C₆₁	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₆₁	—	2	1	no	—
10	C₂×C₄×C₈×C₆₁	—	2	1	no	—
11	C₆₁×#391	C₂²	1	1	yes	characteristic
12	C₆₁×#390	C₂²	1	1	yes	characteristic
13	C₆₁×#388	—	2	1	no	—
14	C₆₁×#388	—	2	1	no	—
15	C₆₁×#382	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊C₄×C₆₁	—	2	1	no	—
17	C₄×C₂²⋊C₄×C₆₁	—	2	1	no	—
18	C₄³×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#336	—	2	1	no	—
20	C₆₁×#336	—	2	1	no	—
21	C₆₁×#336	C₂²	1	2	yes	—
22	C₆₁×#336	C₂²	1	2	yes	—
23	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₁	—	2	1	no	—
26	C₂²×C₈×C₆₁	—	2	2	no	—
27	C₂²×C₈×C₆₁	—	2	2	no	—
28	#124×C₆₁	—	2	2	no	—
29	#124×C₆₁	C₈	1	4	yes	complemented
30	C₄×Q₈×C₆₁	—	4	2	no	—
31	C₄×D₄×C₆₁	—	4	2	no	—
32	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
38	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
40	C₂×C₄²×C₆₁	—	2	1	no	—
41	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₆₁	—	2	4	no	—
49	C₄×C₈×C₆₁	—	4	2	no	—
50	#95×C₆₁	—	2	1	no	—
51	#95×C₆₁	—	2	1	no	—
52	#95×C₆₁	—	2	1	no	—
53	#95×C₆₁	—	2	1	no	—
54	C₂⁴×C₆₁	—	2	1	no	—
55	C₂×Q₈×C₆₁	—	2	1	no	—
56	C₂×Q₈×C₆₁	—	4	1	no	—
57	C₂×Q₈×C₆₁	—	4	1	no	—
58	C₂×Q₈×C₆₁	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₆₁	—	2	1	no	—
60	C₂×D₄×C₆₁	—	4	1	no	—
61	C₂×D₄×C₆₁	—	4	1	no	—
62	C₂×D₄×C₆₁	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₆₁	—	2	1	no	—
64	C₂²×C₄×C₆₁	—	2	1	no	—
65	C₂²×C₄×C₆₁	—	2	1	no	—
66	C₂²×C₄×C₆₁	—	4	2	no	—
67	C₂²×C₄×C₆₁	—	4	2	no	—
68	C₂²×C₄×C₆₁	—	4	2	no	—
69	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₁	—	2	4	no	—
77	C₂×C₈×C₆₁	—	4	2	no	—
78	C₂×C₈×C₆₁	—	4	2	no	—
79	C₂×C₈×C₆₁	—	4	2	no	—
80	C₂×C₈×C₆₁	—	4	2	no	—
81	C₂×C₈×C₆₁	—	4	2	no	—
82	C₄⋊C₄×C₆₁	—	4	1	no	—
83	C₄⋊C₄×C₆₁	—	4	1	no	—
84	C₄⋊C₄×C₆₁	—	4	2	no	—
85	C₂²⋊C₄×C₆₁	—	2	4	no	—
86	C₂²⋊C₄×C₆₁	—	2	4	no	—
87	C₂²⋊C₄×C₆₁	—	4	2	no	—
88	C₂²⋊C₄×C₆₁	—	4	2	no	—
89	C₂²⋊C₄×C₆₁	—	4	1	no	—
90	C₂²⋊C₄×C₆₁	—	4	1	no	—
91	C₄²×C₆₁	—	2	2	no	—
92	C₄²×C₆₁	—	2	2	no	—
93	C₄²×C₆₁	—	2	2	no	—
94	C₄²×C₆₁	—	2	1	no	—
95	C₄²×C₆₁	—	2	1	no	—
96	C₄²×C₆₁	—	4	1	no	—
97	C₄²×C₆₁	—	4	1	no	—
98	C₄²×C₆₁	—	4	2	no	—
99	C₄²×C₆₁	C₂×C₈	1	2	yes	—
100	C₄²×C₆₁	M₁₆	1	2	yes	—
101	C₂³×C₆₁	—	2	2	no	—
102	C₂³×C₆₁	—	4	1	no	—
103	C₂³×C₆₁	—	4	1	no	—
104	C₂³×C₆₁	—	4	1	no	—
105	C₂³×C₆₁	—	4	1	no	—
106	C₂³×C₆₁	—	4	1	no	—
107	C₂³×C₆₁	—	4	1	no	—
108	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	Q₈×C₆₁	—	4	2	no	—
110	Q₈×C₆₁	—	4	2	no	—
111	D₄×C₆₁	—	4	2	no	—
112	D₄×C₆₁	—	4	2	no	—
113	C₂×C₄×C₆₁	—	2	1	no	—
114	C₂×C₄×C₆₁	—	2	1	no	—
115	C₂×C₄×C₆₁	—	2	1	no	—
116	C₂×C₄×C₆₁	—	2	1	no	—
117	C₂×C₄×C₆₁	—	2	1	no	—
118	C₂×C₄×C₆₁	—	2	1	no	—
119	C₂×C₄×C₆₁	—	2	1	no	—
120	C₂×C₄×C₆₁	—	2	1	no	—
121	C₂×C₄×C₆₁	—	2	1	no	—
122	C₂×C₄×C₆₁	—	2	1	no	—
123	C₂×C₄×C₆₁	—	2	1	no	—
124	C₂×C₄×C₆₁	—	2	1	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	1	no	—
129	C₂×C₄×C₆₁	—	4	1	no	—
130	C₂×C₄×C₆₁	—	4	1	no	—
131	C₂×C₄×C₆₁	—	4	1	no	—
132	C₂×C₄×C₆₁	—	4	1	no	—
133	C₂×C₄×C₆₁	—	4	1	no	—
134	C₂×C₄×C₆₁	—	4	1	no	—
135	C₂×C₄×C₆₁	—	4	1	no	—
136	C₂×C₄×C₆₁	—	8	4	no	—
137	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
138	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
139	C₂×C₄×C₆₁	#100	1	2	yes	—
140	C₂×C₄×C₆₁	#104	1	2	yes	—
141	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
143	C₂×C₄×C₆₁	#135	1	2	yes	—
144	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₆₁	#137	1	1	yes	characteristic
146	C₈×C₆₁	—	4	4	no	—
147	C₈×C₆₁	—	8	2	no	—
148	#7584	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂²×C₆₁	—	4	2	no	—
150	C₂²×C₆₁	—	4	2	no	—
151	C₂²×C₆₁	—	4	2	no	—
152	C₂²×C₆₁	—	8	1	no	—
153	C₂²×C₆₁	—	8	1	no	—
154	C₂²×C₆₁	—	8	1	no	—
155	C₂²×C₆₁	—	8	1	no	—
156	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
157	C₂²×C₆₁	#410	1	1	yes	characteristic
158	C₂²×C₆₁	C₂×#100	1	1	yes	characteristic
159	C₂²×C₆₁	#416	1	1	yes	characteristic
160	C₂²×C₆₁	#417	1	1	yes	characteristic
161	C₂²×C₆₁	C₂×#104	1	1	yes	characteristic
162	C₂²×C₆₁	#421	1	1	yes	characteristic
163	C₄×C₆₁	—	2	2	no	—
164	C₄×C₆₁	—	2	1	no	—
165	C₄×C₆₁	—	2	1	no	—
166	C₄×C₆₁	—	2	2	no	—
167	C₄×C₆₁	—	2	2	no	—
168	C₄×C₆₁	—	2	1	no	—
169	C₄×C₆₁	—	2	1	no	—
170	C₄×C₆₁	—	2	2	no	—
171	C₄×C₆₁	—	2	2	no	—
172	C₄×C₆₁	—	4	2	no	—
173	C₄×C₆₁	—	8	1	no	—
174	C₄×C₆₁	—	8	1	no	—
175	C₄×#124	C₂×C₆₁	1	1	yes	characteristic
176	#1839	C₂×C₆₁	1	1	yes	characteristic, complemented
177	C₄×C₄⋊C₈	C₂×C₆₁	1	1	yes	characteristic, complemented
178	#1272	C₂×C₆₁	1	2	yes	complemented
179	#1270	C₂×C₆₁	1	2	yes	—
180	C₂×C₆₁	—	8	2	no	—
181	C₂×C₆₁	#1460	1	1	yes	characteristic
182	C₂×C₆₁	#1476	1	1	yes	characteristic
183	C₂×C₆₁	#1477	1	1	yes	characteristic
184	C₂×C₆₁	#1509	1	1	yes	characteristic
185	C₂×C₆₁	#1521	1	1	yes	characteristic
186	C₂×C₆₁	#1527	1	1	yes	characteristic
187	C₂×C₆₁	#1551	1	1	yes	characteristic
188	C₂×#124	C₄×C₆₁	1	1	yes	characteristic
189	C₂×C₄×Q₈	C₄×C₆₁	1	1	yes	characteristic
190	C₂×C₄×D₄	C₄×C₆₁	1	1	yes	characteristic
191	C₂×C₄⋊C₈	—	2	1	no	—
192	C₂×C₄×C₈	—	2	1	no	—
193	#391	C₂²×C₆₁	1	1	yes	characteristic
194	#390	C₂²×C₆₁	1	1	yes	characteristic
195	#388	—	2	1	no	—
196	#388	—	2	1	no	—
197	#382	C₄×C₆₁	1	1	yes	characteristic
198	C₄×C₂²⋊C₄	—	2	1	no	—
199	C₄×C₂²⋊C₄	—	2	1	no	—
200	C₄³	C₂²×C₆₁	1	1	yes	characteristic
201	#336	—	2	1	no	—
202	#336	—	2	1	no	—
203	#336	C₂²×C₆₁	1	2	yes	—
204	#336	C₂²×C₆₁	1	2	yes	—
205	C₆₁	#7584	1	1	yes	characteristic, direct product factor, Sylow
206	C₂²×Q₈	C₂×C₄×C₆₁	1	1	yes	characteristic
207	C₂²×D₄	C₂×C₄×C₆₁	1	1	yes	characteristic
208	C₂³×C₄	—	2	1	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	2	2	no	—
211	#124	—	2	2	no	—
212	#124	C₈×C₆₁	1	4	yes	complemented
213	C₄×Q₈	—	4	2	no	—
214	C₄×D₄	—	4	2	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
217	C₂×C₄⋊C₄	C₂×C₄×C₆₁	1	1	yes	characteristic
218	C₂×C₂²⋊C₄	—	2	1	no	—
219	C₂×C₂²⋊C₄	—	2	1	no	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	2	1	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
225	C₂×C₄²	C₂×C₄×C₆₁	1	1	yes	characteristic
226	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
227	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
228	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
229	C₂×C₄²	D₄×C₆₁	1	1	yes	characteristic
230	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
231	C₄⋊C₈	—	2	4	no	—
232	C₄×C₈	—	4	2	no	—
233	#95	—	2	1	no	—
234	#95	—	2	1	no	—
235	#95	—	2	1	no	—
236	#95	—	2	1	no	—
237	C₂⁴	—	2	1	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	C₂×C₈×C₆₁	1	2	yes	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	C₂×C₈×C₆₁	1	2	yes	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
253	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
254	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
255	C₂²×C₄	C₂²⋊C₄×C₆₁	1	1	yes	characteristic
256	C₂²×C₄	C₂²×C₄×C₆₁	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	2	4	no	—
269	C₂²⋊C₄	—	2	4	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	4	2	no	—
282	C₄²	C₂×C₈×C₆₁	1	2	yes	—
283	C₄²	M₁₆×C₆₁	1	2	yes	—
284	C₂³	—	2	2	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	4	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	C₂×C₂²⋊C₄×C₆₁	1	1	yes	characteristic
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
321	C₂×C₄	C₂²⋊C₈×C₆₁	1	2	yes	—
322	C₂×C₄	#100×C₆₁	1	2	yes	—
323	C₂×C₄	#104×C₆₁	1	2	yes	—
324	C₂×C₄	C₂²×C₈×C₆₁	1	1	yes	characteristic
325	C₂×C₄	C₂×M₁₆×C₆₁	1	1	yes	characteristic
326	C₂×C₄	#135×C₆₁	1	2	yes	—
327	C₂×C₄	AGL(1,ℤ/8)×C₆₁	1	1	yes	characteristic
328	C₂×C₄	#137×C₆₁	1	1	yes	characteristic
329	C₈	—	4	4	no	—
330	C₈	—	8	2	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	C₂×C₂²⋊C₈×C₆₁	1	1	yes	characteristic
339	C₂²	C₆₁×#410	1	1	yes	characteristic
340	C₂²	C₂×#100×C₆₁	1	1	yes	characteristic
341	C₂²	C₆₁×#416	1	1	yes	characteristic
342	C₂²	C₆₁×#417	1	1	yes	characteristic
343	C₂²	C₂×#104×C₆₁	1	1	yes	characteristic
344	C₂²	C₆₁×#421	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	8	1	no	—
356	C₄	—	8	1	no	—
357	C₂	—	8	2	no	—
358	C₂	C₆₁×#1460	1	1	yes	characteristic
359	C₂	C₆₁×#1476	1	1	yes	characteristic
360	C₂	C₆₁×#1477	1	1	yes	characteristic
361	C₂	C₆₁×#1509	1	1	yes	characteristic
362	C₂	C₆₁×#1521	1	1	yes	characteristic
363	C₂	C₆₁×#1527	1	1	yes	characteristic
364	C₂	C₆₁×#1551	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₆₁
Frattini, cosocle	C₂×C₄²	C₂³×C₆₁
prime radical	C₂²×D₄×C₆₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₄³×C₆₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₁×#1839	C₂	semidirect
3. C₄×C₄⋊C₈×C₆₁	C₂	semidirect
4. C₆₁×#1272	C₂	semidirect
29. #124×C₆₁	C₈	semidirect
148. #7584	C₆₁	direct (N × H)
176. #1839	C₂×C₆₁	semidirect
177. C₄×C₄⋊C₈	C₂×C₆₁	semidirect
178. #1272	C₂×C₆₁	semidirect
205. C₆₁	#7584	direct (N × H)
212. #124	C₈×C₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#124×C₆₁	2
C₆₁×#1839	2
C₄×C₄⋊C₈×C₆₁	2
C₆₁×#1272	2
C₆₁×#1270	2
#7584	61

Subgroups — C61×#7584 #29511017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#7584 #29511017