Subgroups — D₆₁×#2225 #29834632

Order 15616.

574 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₆₁	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#21447962	C₂	1	1	yes	maximal, characteristic, complemented
4	#21447961	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447960	C₂	1	2	yes	maximal, complemented
6	#21447959	C₂	1	2	yes	maximal, complemented
7	#21447958	C₂	1	1	yes	maximal, characteristic, complemented
8	#367×D₆₁	C₂	1	2	yes	maximal
9	C₄⋊C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
10	#358×D₆₁	C₂	1	2	yes	maximal
11	C₆₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₁	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×D₆₁	—	2	1	no	—
14	C₄×Q₈×D₆₁	—	2	1	no	—
15	C₂×C₁₆×D₆₁	C₂²	1	2	yes	—
16	#16840167	C₂²	1	1	yes	characteristic
17	#16840166	—	2	1	no	—
18	#16840166	C₂²	1	2	yes	—
19	#16840165	—	2	1	no	—
20	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
21	Q₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
22	#16839856	C₂²	1	1	yes	characteristic
23	C₄×#11758202	—	2	1	no	—
24	#16839854	—	2	1	no	—
25	#107×D₆₁	—	2	1	no	—
26	#107×D₆₁	C₂²	1	2	yes	—
27	#107×D₆₁	C₂²	1	1	yes	characteristic
28	#103×D₆₁	—	2	1	no	—
29	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
30	#16839590	C₂²	1	2	yes	—
31	#16839589	C₂²	1	2	yes	—
32	#16839589	C₂²	1	2	yes	—
33	C₄⋊#11758185	C₂²	1	1	yes	characteristic
34	#16839580	C₂²	1	2	yes	—
35	#16839567	C₂²	1	2	yes	—
36	#16839566	C₂²	1	2	yes	—
37	#16839565	C₂²	1	2	yes	—
38	C₆₁×#500	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
40	C₆₁×#367	C₂²	1	2	yes	—
41	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₆₁×#358	C₂²	1	2	yes	—
43	C₂×Q₈×D₆₁	—	2	1	no	—
44	C₂×Q₈×D₆₁	—	4	1	no	—
45	C₂×Dic₂₄₄	C₂³	1	1	yes	characteristic
46	C₂×#11758202	—	2	1	no	—
47	Q₁₆×D₆₁	—	2	2	no	—
48	Q₁₆×D₆₁	Q₈	1	4	yes	complemented
49	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
50	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
51	#14519777	—	2	1	no	—
52	#14519776	—	2	2	no	—
53	#14519776	—	4	1	no	—
54	#14519775	—	4	1	no	—
55	Q₈×Dic₆₁	—	2	1	no	—
56	C₄×Dic₁₂₂	—	2	1	no	—
57	C₄×Dic₁₂₂	—	4	1	no	—
58	#14519742	—	2	1	no	—
59	#14519742	—	4	1	no	—
60	C₄⋊C₄×D₆₁	—	2	1	no	—
61	C₄⋊C₄×D₆₁	—	2	2	no	—
62	C₄⋊C₄×D₆₁	—	4	1	no	—
63	C₄⋊C₄×D₆₁	—	4	1	no	—
64	C₄²×D₆₁	—	4	1	no	—
65	C₄²×D₆₁	D₄	1	1	yes	characteristic
66	C₂×#11758185	C₂³	1	2	yes	—
67	C₁₆×D₆₁	—	2	4	no	—
68	#14519711	—	2	1	no	—
69	#14519711	C₂³	1	2	yes	—
70	#14519711	C₂³	1	1	yes	characteristic
71	#14519710	—	2	2	no	—
72	#14519710	—	2	1	no	—
73	#14519710	—	2	2	no	—
74	#14519702	—	2	1	no	—
75	#14519701	—	2	1	no	—
76	#14519700	—	2	1	no	—
77	C₄×#92418	D₄	1	1	yes	characteristic
78	C₈×Dic₆₁	D₄	1	1	yes	characteristic
79	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
80	C₂×Q₁₆×C₆₁	C₂³	1	1	yes	characteristic
81	C₄⋊Q₈×C₆₁	—	2	1	no	—
82	C₄×Q₈×C₆₁	—	2	1	no	—
83	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
84	#107×C₆₁	—	2	1	no	—
85	#107×C₆₁	C₂³	1	2	yes	—
86	#107×C₆₁	C₂³	1	1	yes	characteristic
87	#103×C₆₁	—	2	1	no	—
88	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
89	C₂×Q₈×C₆₁	—	2	1	no	—
90	C₂×Q₈×C₆₁	—	4	1	no	—
91	Q₈×D₆₁	—	2	4	no	—
92	Q₈×D₆₁	—	4	2	no	—
93	Q₈×D₆₁	—	8	4	no	—
94	C₂×C₄×D₆₁	—	4	1	no	—
95	C₂×C₄×D₆₁	—	4	2	no	—
96	C₂×C₄×D₆₁	—	4	1	no	—
97	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
98	C₂×C₄×D₆₁	D₈	1	1	yes	characteristic
99	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
100	C₂×Dic₁₂₂	—	2	1	no	—
101	C₂×Dic₁₂₂	—	4	1	no	—
102	C₂×Dic₁₂₂	—	4	1	no	—
103	C₂×Dic₁₂₂	—	4	2	no	—
104	Q₁₆×C₆₁	—	2	1	no	—
105	Q₁₆×C₆₁	C₂×Q₈	1	2	yes	complemented
106	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₆₁	—	2	1	no	—
109	C₄⋊C₄×C₆₁	—	2	2	no	—
110	C₄⋊C₄×C₆₁	—	4	1	no	—
111	C₄⋊C₄×C₆₁	—	4	1	no	—
112	C₄²×C₆₁	—	4	1	no	—
113	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
114	#11758202	—	2	4	no	—
115	C₄⋊Dic₆₁	—	2	1	no	—
116	C₄⋊Dic₆₁	—	2	2	no	—
117	C₄⋊Dic₆₁	—	4	1	no	—
118	C₄⋊Dic₆₁	—	4	1	no	—
119	#11758196	—	4	1	no	—
120	#11758196	—	4	1	no	—
121	#11758196	—	4	2	no	—
122	#11758196	—	4	1	no	—
123	#11758196	—	4	2	no	—
124	C₄×Dic₆₁	—	4	1	no	—
125	C₄×Dic₆₁	—	4	1	no	—
126	C₄×Dic₆₁	—	4	1	no	—
127	C₄×Dic₆₁	D₈	1	1	yes	characteristic
128	C₄×Dic₆₁	D₈	1	1	yes	characteristic
129	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
130	C₂×#92418	C₂×D₄	1	1	yes	characteristic
131	C₂×#92418	C₂×D₄	1	1	yes	characteristic
132	Dic₂₄₄	—	2	1	no	—
133	Dic₂₄₄	C₂×Q₈	1	2	yes	complemented
134	C₈×D₆₁	—	2	2	no	—
135	C₈×D₆₁	C₂×Q₈	1	2	yes	—
136	C₈×D₆₁	#41	1	2	yes	—
137	C₁₆×C₆₁	—	2	2	no	—
138	#11758185	—	2	2	no	—
139	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₆₁	—	2	2	no	—
141	Q₈×C₆₁	—	4	1	no	—
142	Q₈×C₆₁	—	8	2	no	—
143	C₂×C₄×C₆₁	—	4	1	no	—
144	C₂×C₄×C₆₁	—	4	1	no	—
145	C₂×C₄×C₆₁	—	4	2	no	—
146	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₆₁	—	4	1	no	—
150	C₂×Dic₆₁	—	4	1	no	—
151	C₂×Dic₆₁	—	4	2	no	—
152	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
153	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
154	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
155	C₄×D₆₁	—	2	2	no	—
156	C₄×D₆₁	—	4	4	no	—
157	C₄×D₆₁	—	8	4	no	—
158	C₄×D₆₁	—	8	2	no	—
159	C₄×D₆₁	SD₃₂	1	4	yes	—
160	C₄×D₆₁	C₂²⋊Q₈	1	2	yes	—
161	C₄×D₆₁	#137	1	2	yes	—
162	Dic₁₂₂	—	2	2	no	—
163	Dic₁₂₂	—	4	1	no	—
164	Dic₁₂₂	—	4	4	no	—
165	Dic₁₂₂	—	8	2	no	—
166	Dic₁₂₂	—	8	4	no	—
167	C₈×C₆₁	—	2	1	no	—
168	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
169	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
170	#92418	—	2	1	no	—
171	#92418	C₂²⋊Q₈	1	2	yes	—
172	C₂×#2225	—	61	1	no	maximal, Sylow
173	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
174	C₂×D₆₁	#474	1	2	yes	—
175	C₂×D₆₁	C₂×SD₃₂	1	2	yes	—
176	C₂×D₆₁	#510	1	2	yes	—
177	C₄×C₆₁	—	2	1	no	—
178	C₄×C₆₁	—	4	2	no	—
179	C₄×C₆₁	—	8	2	no	—
180	C₄×C₆₁	—	8	1	no	—
181	C₄×C₆₁	C₂×SD₃₂	1	2	yes	—
182	C₄×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
183	C₄×C₆₁	C₂×#137	1	1	yes	characteristic
184	Dic₆₁	—	2	1	no	—
185	Dic₆₁	—	4	2	no	—
186	Dic₆₁	—	8	2	no	—
187	Dic₆₁	—	8	1	no	—
188	Dic₆₁	#474	1	2	yes	—
189	Dic₆₁	C₂×SD₃₂	1	2	yes	—
190	C₂×#500	—	61	1	no	—
191	C₂×C₄×Q₁₆	—	61	1	no	—
192	#2225	—	61	1	no	—
193	#2225	—	61	1	no	—
194	#2225	—	61	2	no	—
195	#2225	—	61	1	no	—
196	#2225	—	61	2	no	—
197	#2225	D₆₁	1	1	yes	characteristic, direct product factor
198	C₂×#367	—	61	2	no	—
199	C₂×C₄⋊C₁₆	—	61	1	no	—
200	C₂×#358	—	61	2	no	—
201	C₂×C₆₁	C₂×#474	1	1	yes	characteristic
202	C₂×C₆₁	C₂²×SD₃₂	1	1	yes	characteristic
203	C₂×C₆₁	C₂×#510	1	1	yes	characteristic
204	D₆₁	#2225	1	4	yes	direct product factor
205	C₂²×Q₁₆	—	61	1	no	—
206	C₂×C₄⋊Q₈	—	122	1	no	—
207	C₂×C₄×Q₈	—	122	1	no	—
208	C₂²×C₁₆	—	61	2	no	—
209	#500	—	61	2	no	—
210	#500	—	61	1	no	—
211	#500	—	122	1	no	—
212	#500	—	122	1	no	—
213	#500	C₂×D₆₁	1	1	yes	characteristic
214	C₄×Q₁₆	—	61	1	no	—
215	C₄×Q₁₆	—	61	1	no	—
216	C₄×Q₁₆	—	61	1	no	—
217	C₄×Q₁₆	—	122	1	no	—
218	C₄×Q₁₆	—	122	1	no	—
219	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
220	C₂×#107	—	61	2	no	—
221	C₂×#107	—	61	1	no	—
222	C₂×#107	—	122	1	no	—
223	C₂×#103	—	122	1	no	—
224	C₂×C₄×C₈	—	61	1	no	—
225	#367	—	61	2	no	—
226	#367	—	61	2	no	—
227	#367	—	61	2	no	—
228	#367	C₂×D₆₁	1	2	yes	—
229	C₄⋊C₁₆	—	61	2	no	—
230	C₄⋊C₁₆	—	61	1	no	—
231	C₄⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
232	#358	—	61	2	no	—
233	#358	—	61	2	no	—
234	#358	—	61	2	no	—
235	#358	C₂×D₆₁	1	2	yes	—
236	C₆₁	C₂×#2225	1	1	yes	characteristic, complemented, Sylow
237	C₂²×Q₈	—	122	1	no	—
238	C₂²×Q₈	—	244	1	no	—
239	C₂×Q₁₆	—	61	4	no	—
240	C₂×Q₁₆	—	61	1	no	—
241	C₂×Q₁₆	—	122	2	no	—
242	C₂×Q₁₆	—	122	1	no	—
243	C₂×Q₁₆	C₂²×D₆₁	1	1	yes	characteristic
244	C₂²×C₈	—	61	1	no	—
245	C₂²×C₈	—	61	1	no	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	122	2	no	—
248	C₄⋊Q₈	—	122	1	no	—
249	C₄⋊Q₈	—	244	1	no	—
250	C₄⋊Q₈	—	244	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	122	1	no	—
253	C₄×Q₈	—	122	1	no	—
254	C₄×Q₈	—	122	1	no	—
255	C₄×Q₈	—	244	1	no	—
256	C₄×Q₈	—	244	1	no	—
257	C₂×C₄⋊C₄	—	122	1	no	—
258	C₂×C₄⋊C₄	—	122	2	no	—
259	C₂×C₄⋊C₄	—	244	1	no	—
260	C₂×C₄⋊C₄	—	244	1	no	—
261	C₂×C₄²	—	61	1	no	—
262	C₂×C₄²	—	244	1	no	—
263	C₂×C₁₆	—	61	2	no	—
264	C₂×C₁₆	—	122	4	no	—
265	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
266	#107	—	2	1	no	—
267	#107	—	61	2	no	—
268	#107	—	61	1	no	—
269	#107	—	122	1	no	—
270	#107	—	122	2	no	—
271	#107	—	122	2	no	—
272	#107	—	122	1	no	—
273	#107	C₂²×D₆₁	1	2	yes	—
274	#107	C₂²×D₆₁	1	1	yes	characteristic
275	#103	—	2	1	no	—
276	#103	—	122	1	no	—
277	#103	—	122	1	no	—
278	#103	—	122	1	no	—
279	C₄×C₈	—	61	1	no	—
280	C₄×C₈	—	61	1	no	—
281	C₄×C₈	—	61	1	no	—
282	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	4	1	no	—
285	C₂×Q₈	—	122	4	no	—
286	C₂×Q₈	—	122	1	no	—
287	C₂×Q₈	—	488	4	no	—
288	C₂×Q₈	—	244	2	no	—
289	C₂×Q₈	—	244	1	no	—
290	C₂×Q₈	—	244	2	no	—
291	C₂×Q₈	—	244	1	no	—
292	C₂²×C₄	—	61	1	no	—
293	C₂²×C₄	—	61	1	no	—
294	C₂²×C₄	—	61	1	no	—
295	C₂²×C₄	—	244	1	no	—
296	C₂²×C₄	—	244	2	no	—
297	C₂²×C₄	—	244	1	no	—
298	Q₁₆	—	2	1	no	—
299	Q₁₆	—	61	2	no	—
300	Q₁₆	—	122	4	no	—
301	Q₁₆	—	122	1	no	—
302	Q₁₆	Q₈×D₆₁	1	2	yes	complemented
303	C₂×C₈	—	61	1	no	—
304	C₂×C₈	—	61	2	no	—
305	C₂×C₈	—	61	2	no	—
306	C₂×C₈	—	61	1	no	—
307	C₂×C₈	—	122	2	no	—
308	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
309	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	122	1	no	—
315	C₄⋊C₄	—	122	2	no	—
316	C₄⋊C₄	—	244	1	no	—
317	C₄⋊C₄	—	244	2	no	—
318	C₄⋊C₄	—	244	1	no	—
319	C₄⋊C₄	—	244	2	no	—
320	C₄⋊C₄	—	244	1	no	—
321	C₄⋊C₄	—	244	1	no	—
322	C₄⋊C₄	—	244	1	no	—
323	C₄²	—	4	1	no	—
324	C₄²	—	61	1	no	—
325	C₄²	—	61	1	no	—
326	C₄²	—	61	1	no	—
327	C₄²	—	244	1	no	—
328	C₄²	—	244	1	no	—
329	C₄²	—	244	1	no	—
330	C₄²	D₄×D₆₁	1	1	yes	characteristic
331	C₁₆	—	2	2	no	—
332	C₁₆	—	122	2	no	—
333	C₂³	—	61	1	no	—
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	8	2	no	—
337	Q₈	—	122	2	no	—
338	Q₈	—	488	4	no	—
339	Q₈	—	488	2	no	—
340	Q₈	—	244	4	no	—
341	Q₈	—	244	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	61	1	no	—
346	C₂×C₄	—	61	1	no	—
347	C₂×C₄	—	61	2	no	—
348	C₂×C₄	—	61	2	no	—
349	C₂×C₄	—	61	4	no	—
350	C₂×C₄	—	61	1	no	—
351	C₂×C₄	—	122	2	no	—
352	C₂×C₄	—	488	2	no	—
353	C₂×C₄	—	488	4	no	—
354	C₂×C₄	—	244	4	no	—
355	C₂×C₄	—	244	1	no	—
356	C₂×C₄	—	244	2	no	—
357	C₂×C₄	—	244	1	no	—
358	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
359	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
360	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
361	C₈	—	2	1	no	—
362	C₈	—	61	2	no	—
363	C₈	—	122	1	no	—
364	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
365	C₈	#41×D₆₁	1	1	yes	characteristic
366	C₂²	—	61	2	no	—
367	C₂²	—	61	2	no	—
368	C₂²	—	61	2	no	—
369	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	8	1	no	—
373	C₄	—	8	2	no	—
374	C₄	—	61	2	no	—
375	C₄	—	61	2	no	—
376	C₄	—	122	1	no	—
377	C₄	—	488	1	no	—
378	C₄	—	488	2	no	—
379	C₄	—	244	2	no	—
380	C₄	SD₃₂×D₆₁	1	2	yes	—
381	C₄	C₂²⋊Q₈×D₆₁	1	1	yes	characteristic
382	C₄	#137×D₆₁	1	1	yes	characteristic
383	C₂	—	61	4	no	—
384	C₂	#474×D₆₁	1	1	yes	characteristic
385	C₂	C₂×SD₃₂×D₆₁	1	1	yes	characteristic
386	C₂	#510×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₁
commutator, cosocle	C₂×C₈×C₆₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2225	C₂
socle	C₂²×C₆₁	C₂²×D₈
prime radical	C₂²×D₆₁	C₂×D₈
hypercenter	#2225	D₆₁
hypocenter	C₆₁	C₂×#2225
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2225

Complements

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

N	H ≅ G/N	type
3. #21447962	C₂	semidirect
4. #21447961	C₂	semidirect
5. #21447960	C₂	semidirect
6. #21447959	C₂	semidirect
7. #21447958	C₂	semidirect
11. C₆₁×#2225	C₂	semidirect
48. Q₁₆×D₆₁	Q₈	semidirect
105. Q₁₆×C₆₁	C₂×Q₈	semidirect
133. Dic₂₄₄	C₂×Q₈	semidirect
197. #2225	D₆₁	direct (N × H)
204. D₆₁	#2225	direct (N × H)
236. C₆₁	C₂×#2225	semidirect
302. Q₁₆	Q₈×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#500×D₆₁	2
C₄×Q₁₆×D₆₁	2
#21447962	2
#21447961	2
#21447960	2
#21447959	2
#21447958	2
#367×D₆₁	2
C₄⋊C₁₆×D₆₁	2
#358×D₆₁	2
C₆₁×#2225	2
C₂×#2225	61

Subgroups — D61×#2225 #29834632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁×#2225 #29834632