Subgroups — D₆₁×#1579 #29649402

Order 15616.

718 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₆₁	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#21444639	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444638	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444637	C₂	1	1	yes	maximal, characteristic, complemented
6	#21444636	C₂	1	2	yes	maximal, complemented
7	#21444635	C₂	1	2	yes	maximal, complemented
8	#334×D₆₁	C₂	1	1	yes	maximal, characteristic
9	Q₈⋊C₈×D₆₁	C₂	1	2	yes	maximal
10	#322×D₆₁	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₁	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₆₁	C₂²	1	2	yes	—
14	Q₈×#92418	C₂²	1	2	yes	—
15	C₈×Dic₁₂₂	—	2	2	no	—
16	C₈×Dic₁₂₂	C₂²	1	2	yes	—
17	#16839886	—	2	2	no	—
18	#16839886	C₂²	1	2	yes	—
19	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₆₁	C₄	1	1	yes	characteristic
21	#16839856	C₄	1	1	yes	characteristic
22	C₄×#11758202	C₂²	1	2	yes	—
23	#16839854	C₄	1	2	yes	—
24	#107×D₆₁	C₄	1	1	yes	characteristic
25	C₄⋊C₈×D₆₁	—	2	2	no	—
26	C₄⋊C₈×D₆₁	C₂²	1	2	yes	—
27	#103×D₆₁	C₄	1	2	yes	—
28	C₄×C₈×D₆₁	—	2	2	no	—
29	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
32	#16839513	C₂²	1	1	yes	characteristic
33	#16839512	C₂²	1	2	yes	—
34	#16839494	C₂²	1	2	yes	—
35	Q₈⋊#92418	C₂²	1	2	yes	—
36	#16839492	C₂²	1	2	yes	—
37	#16839484	C₂²	1	1	yes	characteristic
38	#16839484	C₂²	1	1	yes	characteristic
39	#16839483	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₆₁	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
42	C₆₁×#334	C₂²	1	1	yes	characteristic
43	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₆₁×#322	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₆₁	C₂×C₄	1	2	yes	—
46	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
47	C₂×#11758202	C₂×C₄	1	2	yes	—
48	Q₁₆×D₆₁	C₈	1	8	yes	complemented
49	C₂×C₈×D₆₁	—	2	2	no	—
50	C₂×C₈×D₆₁	—	2	1	no	—
51	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₆₁	D₄	1	2	yes	—
54	Q₈×Dic₆₁	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₂₂	—	2	2	no	—
56	C₄×Dic₁₂₂	C₂³	1	2	yes	—
57	#14519742	—	2	2	no	—
58	#14519742	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×D₆₁	—	2	2	no	—
60	C₄⋊C₄×D₆₁	C₂×C₄	1	2	yes	—
61	C₄²×D₆₁	—	2	2	no	—
62	C₄²×D₆₁	C₂³	1	1	yes	characteristic
63	#14519711	C₂×C₄	1	1	yes	characteristic
64	#14519710	C₂×C₄	1	2	yes	—
65	C₄⋊#92418	—	2	2	no	—
66	C₄⋊#92418	C₂³	1	2	yes	—
67	#14519706	—	2	2	no	—
68	#14519706	—	2	2	no	—
69	#14519706	—	2	2	no	—
70	#14519702	C₂×C₄	1	2	yes	—
71	#14519701	C₂×C₄	1	2	yes	—
72	#14519700	C₂×C₄	1	2	yes	—
73	C₄×#92418	—	2	2	no	—
74	C₄×#92418	C₂³	1	1	yes	characteristic
75	C₄×#92418	C₂³	1	1	yes	characteristic
76	C₄×#92418	C₂³	1	1	yes	characteristic
77	C₈×Dic₆₁	—	2	2	no	—
78	C₈×Dic₆₁	—	2	1	no	—
79	C₈×Dic₆₁	—	2	2	no	—
80	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₆₁	D₄	1	2	yes	—
83	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
85	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₆₁	—	2	2	no	—
87	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
88	#103×C₆₁	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₆₁	—	2	2	no	—
90	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
94	Q₈×D₆₁	—	2	4	no	—
95	Q₈×D₆₁	C₂×C₈	1	8	yes	—
96	C₂×C₄×D₆₁	—	2	2	no	—
97	C₂×C₄×D₆₁	—	2	2	no	—
98	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
101	C₂×Dic₁₂₂	—	2	2	no	—
102	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
103	Q₁₆×C₆₁	C₂×C₈	1	4	yes	complemented
104	C₂×C₈×C₆₁	—	2	2	no	—
105	C₂×C₈×C₆₁	—	2	1	no	—
106	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
109	C₄⋊C₄×C₆₁	—	2	2	no	—
110	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
111	C₄²×C₆₁	—	2	2	no	—
112	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
113	#11758202	C₂×C₈	1	8	yes	complemented
114	C₄⋊Dic₆₁	—	2	2	no	—
115	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
116	#11758196	—	2	2	no	—
117	#11758196	—	2	2	no	—
118	#11758196	—	2	2	no	—
119	C₄×Dic₆₁	—	2	2	no	—
120	C₄×Dic₆₁	—	2	2	no	—
121	C₄×Dic₆₁	—	2	2	no	—
122	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₆₁	#41	1	1	yes	characteristic
125	C₂×#92418	—	2	2	no	—
126	C₂×#92418	—	2	1	no	—
127	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
128	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
129	C₂×#92418	C₂×D₄	1	2	yes	—
130	Dic₂₄₄	C₂×C₈	1	4	yes	complemented
131	C₈×D₆₁	—	2	2	no	—
132	C₈×D₆₁	—	2	4	no	—
133	C₈×D₆₁	—	4	2	no	—
134	C₈×D₆₁	—	4	4	no	—
135	C₈×D₆₁	C₂×C₈	1	4	yes	—
136	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
137	Q₈×C₆₁	—	2	2	no	—
138	Q₈×C₆₁	C₂²×C₈	1	4	yes	—
139	C₂×C₄×C₆₁	—	2	2	no	—
140	C₂×C₄×C₆₁	—	2	2	no	—
141	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
144	C₂×Dic₆₁	—	2	2	no	—
145	C₂×Dic₆₁	—	2	2	no	—
146	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
147	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
149	C₄×D₆₁	—	2	8	no	—
150	C₄×D₆₁	—	2	2	no	—
151	C₄×D₆₁	—	4	4	no	—
152	C₄×D₆₁	C₂²×C₈	1	2	yes	—
153	C₄×D₆₁	#131	1	2	yes	—
154	C₄×D₆₁	#137	1	4	yes	—
155	Dic₁₂₂	—	2	2	no	—
156	Dic₁₂₂	—	2	8	no	—
157	Dic₁₂₂	C₂²×C₈	1	4	yes	—
158	C₈×C₆₁	—	2	1	no	—
159	C₈×C₆₁	—	2	2	no	—
160	C₈×C₆₁	—	4	1	no	—
161	C₈×C₆₁	—	4	2	no	—
162	C₈×C₆₁	C₂²×C₈	1	2	yes	—
163	#92418	—	2	1	no	—
164	#92418	—	2	2	no	—
165	#92418	—	4	1	no	—
166	#92418	—	4	2	no	—
167	#92418	C₂²×C₈	1	2	yes	—
168	C₂×#1579	—	61	1	no	maximal, Sylow
169	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×D₆₁	C₈×D₄	1	2	yes	—
171	C₂×D₆₁	#441	1	2	yes	—
172	C₂×D₆₁	#444	1	2	yes	—
173	C₄×C₆₁	—	2	4	no	—
174	C₄×C₆₁	—	2	1	no	—
175	C₄×C₆₁	—	4	2	no	—
176	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
177	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
178	C₄×C₆₁	C₂×#137	1	2	yes	—
179	Dic₆₁	—	2	4	no	—
180	Dic₆₁	—	2	1	no	—
181	Dic₆₁	—	4	2	no	—
182	Dic₆₁	C₈×D₄	1	2	yes	—
183	Dic₆₁	#441	1	2	yes	—
184	C₂×C₈×Q₈	—	61	2	no	—
185	C₂×C₄×Q₁₆	—	61	1	no	—
186	#1579	—	61	2	no	—
187	#1579	—	61	1	no	—
188	#1579	—	61	1	no	—
189	#1579	—	61	2	no	—
190	#1579	—	61	1	no	—
191	#1579	D₆₁	1	1	yes	characteristic, direct product factor
192	C₂×#334	—	61	1	no	—
193	C₂×Q₈⋊C₈	—	61	2	no	—
194	C₂×#322	—	61	1	no	—
195	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
196	C₂×C₆₁	C₂×#441	1	1	yes	characteristic
197	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
198	D₆₁	#1579	1	4	yes	direct product factor
199	C₂²×Q₁₆	—	61	1	no	—
200	C₂×C₄×Q₈	—	61	2	no	—
201	C₈×Q₈	—	61	2	no	—
202	C₈×Q₈	—	61	2	no	—
203	C₈×Q₈	—	61	2	no	—
204	C₈×Q₈	—	122	2	no	—
205	C₈×Q₈	—	122	2	no	—
206	C₈×Q₈	C₂×D₆₁	1	2	yes	—
207	C₄×Q₁₆	—	61	1	no	—
208	C₄×Q₁₆	—	61	2	no	—
209	C₄×Q₁₆	—	61	1	no	—
210	C₄×Q₁₆	—	61	2	no	—
211	C₄×Q₁₆	—	61	1	no	—
212	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
213	C₂×#107	—	61	1	no	—
214	C₂×C₄⋊C₈	—	61	2	no	—
215	C₂×C₄⋊C₈	—	122	2	no	—
216	C₂×#103	—	61	2	no	—
217	C₂×C₄×C₈	—	61	1	no	—
218	C₂×C₄×C₈	—	61	1	no	—
219	C₂×C₄×C₈	—	61	1	no	—
220	C₂×C₄×C₈	—	122	2	no	—
221	#334	—	61	2	no	—
222	#334	—	61	1	no	—
223	#334	C₂×D₆₁	1	1	yes	characteristic
224	Q₈⋊C₈	—	61	2	no	—
225	Q₈⋊C₈	—	61	2	no	—
226	Q₈⋊C₈	—	61	2	no	—
227	Q₈⋊C₈	C₂×D₆₁	1	2	yes	—
228	#322	—	61	1	no	—
229	#322	—	61	1	no	—
230	#322	—	61	1	no	—
231	#322	C₂×D₆₁	1	1	yes	characteristic
232	C₆₁	C₂×#1579	1	1	yes	characteristic, complemented, Sylow
233	C₂²×Q₈	—	61	2	no	—
234	C₂×Q₁₆	—	61	2	no	—
235	C₂×Q₁₆	—	61	8	no	—
236	C₂×Q₁₆	—	61	1	no	—
237	C₂×Q₁₆	C₄×D₆₁	1	1	yes	characteristic
238	C₂²×C₈	—	61	1	no	—
239	C₂²×C₈	—	61	1	no	—
240	C₂²×C₈	—	61	2	no	—
241	C₂²×C₈	—	122	1	no	—
242	C₂²×C₈	—	122	2	no	—
243	C₄×Q₈	—	61	2	no	—
244	C₄×Q₈	—	61	2	no	—
245	C₄×Q₈	—	61	2	no	—
246	C₄×Q₈	—	122	2	no	—
247	C₄×Q₈	—	122	2	no	—
248	C₄×Q₈	C₂²×D₆₁	1	2	yes	—
249	C₂×C₄⋊C₄	—	61	2	no	—
250	C₂×C₄⋊C₄	—	122	2	no	—
251	C₂×C₄²	—	61	1	no	—
252	C₂×C₄²	—	122	2	no	—
253	#107	—	61	1	no	—
254	#107	—	61	2	no	—
255	#107	C₄×D₆₁	1	1	yes	characteristic
256	C₄⋊C₈	—	2	2	no	—
257	C₄⋊C₈	—	61	2	no	—
258	C₄⋊C₈	—	122	2	no	—
259	C₄⋊C₈	—	122	2	no	—
260	C₄⋊C₈	—	122	2	no	—
261	C₄⋊C₈	—	122	2	no	—
262	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
263	#103	—	61	2	no	—
264	#103	—	61	2	no	—
265	#103	—	61	2	no	—
266	#103	C₄×D₆₁	1	2	yes	—
267	C₄×C₈	—	2	2	no	—
268	C₄×C₈	—	61	1	no	—
269	C₄×C₈	—	61	1	no	—
270	C₄×C₈	—	61	2	no	—
271	C₄×C₈	—	61	1	no	—
272	C₄×C₈	—	61	1	no	—
273	C₄×C₈	—	61	1	no	—
274	C₄×C₈	—	122	2	no	—
275	C₄×C₈	—	122	1	no	—
276	C₄×C₈	—	122	2	no	—
277	C₄×C₈	—	122	2	no	—
278	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
281	C₂×Q₈	—	61	8	no	—
282	C₂×Q₈	—	61	2	no	—
283	C₂×Q₈	—	122	2	no	—
284	C₂×Q₈	—	122	4	no	—
285	C₂×Q₈	C₂×C₄×D₆₁	1	2	yes	—
286	C₂²×C₄	—	61	1	no	—
287	C₂²×C₄	—	61	1	no	—
288	C₂²×C₄	—	61	1	no	—
289	C₂²×C₄	—	122	2	no	—
290	C₂²×C₄	—	122	2	no	—
291	Q₁₆	—	61	8	no	—
292	Q₁₆	—	61	4	no	—
293	Q₁₆	C₈×D₆₁	1	4	yes	complemented
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	61	1	no	—
297	C₂×C₈	—	61	4	no	—
298	C₂×C₈	—	61	1	no	—
299	C₂×C₈	—	61	2	no	—
300	C₂×C₈	—	122	2	no	—
301	C₂×C₈	—	122	4	no	—
302	C₂×C₈	—	122	2	no	—
303	C₂×C₈	—	122	1	no	—
304	C₂×C₈	—	244	4	no	—
305	C₂×C₈	—	244	2	no	—
306	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
307	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
308	C₂×C₈	D₄×D₆₁	1	2	yes	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	61	2	no	—
311	C₄⋊C₄	—	122	2	no	—
312	C₄⋊C₄	—	122	2	no	—
313	C₄⋊C₄	—	122	2	no	—
314	C₄⋊C₄	—	122	2	no	—
315	C₄⋊C₄	C₂×C₄×D₆₁	1	2	yes	—
316	C₄²	—	2	2	no	—
317	C₄²	—	61	1	no	—
318	C₄²	—	61	1	no	—
319	C₄²	—	61	1	no	—
320	C₄²	—	122	2	no	—
321	C₄²	—	122	2	no	—
322	C₄²	—	122	2	no	—
323	C₄²	C₂³×D₆₁	1	1	yes	characteristic
324	C₂³	—	61	1	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	61	4	no	—
327	Q₈	—	122	8	no	—
328	Q₈	—	122	2	no	—
329	Q₈	C₂×C₈×D₆₁	1	4	yes	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	61	1	no	—
333	C₂×C₄	—	61	1	no	—
334	C₂×C₄	—	61	2	no	—
335	C₂×C₄	—	61	2	no	—
336	C₂×C₄	—	61	4	no	—
337	C₂×C₄	—	61	1	no	—
338	C₂×C₄	—	122	2	no	—
339	C₂×C₄	—	122	8	no	—
340	C₂×C₄	—	122	2	no	—
341	C₂×C₄	—	122	2	no	—
342	C₂×C₄	—	244	4	no	—
343	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
344	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
345	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
346	C₈	—	2	1	no	—
347	C₈	—	2	2	no	—
348	C₈	—	4	1	no	—
349	C₈	—	4	2	no	—
350	C₈	—	61	2	no	—
351	C₈	—	122	1	no	—
352	C₈	—	122	2	no	—
353	C₈	—	244	1	no	—
354	C₈	—	244	2	no	—
355	C₈	C₂×C₈×D₆₁	1	2	yes	—
356	C₂²	—	61	2	no	—
357	C₂²	—	61	2	no	—
358	C₂²	—	61	2	no	—
359	C₂²	C₄×D₄×D₆₁	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	4	no	—
362	C₄	—	4	2	no	—
363	C₄	—	61	2	no	—
364	C₄	—	61	2	no	—
365	C₄	—	122	1	no	—
366	C₄	—	122	4	no	—
367	C₄	—	244	2	no	—
368	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
369	C₄	#131×D₆₁	1	1	yes	characteristic
370	C₄	#137×D₆₁	1	2	yes	—
371	C₂	—	61	4	no	—
372	C₂	C₈×D₄×D₆₁	1	1	yes	characteristic
373	C₂	#441×D₆₁	1	1	yes	characteristic
374	C₂	#444×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1579	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂²×D₆₁	C₄×D₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#1579	D₆₁
hypocenter	C₆₁	C₂×#1579
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #21444639	C₂	semidirect
4. #21444638	C₂	semidirect
5. #21444637	C₂	semidirect
6. #21444636	C₂	semidirect
7. #21444635	C₂	semidirect
11. C₆₁×#1579	C₂	semidirect
48. Q₁₆×D₆₁	C₈	semidirect
103. Q₁₆×C₆₁	C₂×C₈	semidirect
113. #11758202	C₂×C₈	semidirect
130. Dic₂₄₄	C₂×C₈	semidirect
191. #1579	D₆₁	direct (N × H)
198. D₆₁	#1579	direct (N × H)
232. C₆₁	C₂×#1579	semidirect
293. Q₁₆	C₈×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₆₁	2
C₄×Q₁₆×D₆₁	2
#21444639	2
#21444638	2
#21444637	2
#21444636	2
#21444635	2
#334×D₆₁	2
Q₈⋊C₈×D₆₁	2
#322×D₆₁	2
C₆₁×#1579	2
C₂×#1579	61

Subgroups — D61×#1579 #29649402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁×#1579 #29649402