Subgroups — D₆₁×#1707 #29667658

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 350 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	#498×D₆₁	C₂	1	2	yes	maximal
3	#494×D₆₁	C₂	1	1	yes	maximal, characteristic
4	#487×D₆₁	C₂	1	2	yes	maximal
5	#21445417	C₂	1	1	yes	maximal, characteristic, complemented
6	#21445416	C₂	1	1	yes	maximal, characteristic, complemented
7	#21445415	C₂	1	2	yes	maximal, complemented
8	#21445414	C₂	1	2	yes	maximal, complemented
9	#21445413	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₁	—	2	2	no	—
13	C₄⋊Q₈×D₆₁	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₆₁	C₂²	1	2	yes	—
15	#16840167	C₂²	1	1	yes	characteristic
16	#16840166	—	2	2	no	—
17	#16840165	C₂²	1	2	yes	—
18	#16840160	C₂²	1	2	yes	—
19	#16840159	—	2	4	no	—
20	#16840158	C₂²	1	2	yes	—
21	#16840158	C₂²	1	2	yes	—
22	#16840142	C₂²	1	1	yes	characteristic
23	#16840141	C₂²	1	2	yes	—
24	#16840140	—	2	2	no	—
25	#16840114	C₂²	1	2	yes	—
26	#16840113	C₂²	1	2	yes	—
27	#16840112	C₂²	1	2	yes	—
28	#16840111	—	2	4	no	—
29	#107×D₆₁	—	2	2	no	—
30	#106×D₆₁	—	2	4	no	—
31	#103×D₆₁	—	2	4	no	—
32	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
35	C₈×#92418	C₂²	1	1	yes	characteristic
36	C₈×#92418	C₂²	1	1	yes	characteristic
37	C₈×#92418	C₂²	1	1	yes	characteristic
38	C₆₁×#500	C₂²	1	1	yes	characteristic
39	C₆₁×#498	C₂²	1	2	yes	—
40	C₆₁×#494	C₂²	1	1	yes	characteristic
41	C₆₁×#487	C₂²	1	2	yes	—
42	C₈²×C₆₁	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₆₁	—	2	2	no	—
44	C₂×Q₈×D₆₁	—	2	2	no	—
45	C₂×Dic₂₄₄	—	2	2	no	—
46	C₂×#11758202	—	2	4	no	—
47	Q₁₆×D₆₁	—	4	8	no	—
48	C₂×C₈×D₆₁	D₄	1	2	yes	—
49	C₂×C₈×D₆₁	D₄	1	2	yes	—
50	C₂×C₈×D₆₁	D₄	1	2	yes	—
51	#14519777	C₂³	1	2	yes	—
52	#14519777	C₂³	1	2	yes	—
53	#14519776	—	2	2	no	—
54	#14519776	—	2	2	no	—
55	#14519776	—	2	2	no	—
56	#14519776	—	2	2	no	—
57	#14519775	—	2	2	no	—
58	#14519775	—	2	2	no	—
59	C₄⋊C₄×D₆₁	—	2	2	no	—
60	C₄⋊C₄×D₆₁	—	2	2	no	—
61	C₄⋊C₄×D₆₁	—	2	2	no	—
62	C₄⋊C₄×D₆₁	—	2	2	no	—
63	C₄²×D₆₁	C₂³	1	1	yes	characteristic
64	#14519711	—	2	2	no	—
65	#14519710	—	2	4	no	—
66	#14519709	—	2	4	no	—
67	#14519708	—	2	4	no	—
68	#14519708	—	2	4	no	—
69	#14519702	—	2	4	no	—
70	#14519701	—	2	4	no	—
71	#14519700	—	2	4	no	—
72	C₄×#92418	C₂³	1	1	yes	characteristic
73	C₄×#92418	C₂³	1	1	yes	characteristic
74	C₄×#92418	C₂³	1	1	yes	characteristic
75	C₈×Dic₆₁	D₄	1	2	yes	—
76	C₈×Dic₆₁	D₄	1	2	yes	—
77	C₈×Dic₆₁	D₄	1	2	yes	—
78	C₂×Q₁₆×C₆₁	—	2	2	no	—
79	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₆₁	C₂³	1	2	yes	—
81	#107×C₆₁	—	2	2	no	—
82	#106×C₆₁	—	2	4	no	—
83	#103×C₆₁	—	2	4	no	—
84	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₆₁	—	2	2	no	—
88	C₂×Q₈×C₆₁	—	2	2	no	—
89	Q₈×D₆₁	—	4	8	no	—
90	Q₈×D₆₁	—	4	8	no	—
91	C₂×C₄×D₆₁	—	4	2	no	—
92	C₂×C₄×D₆₁	—	4	2	no	—
93	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
96	C₂×Dic₁₂₂	—	2	2	no	—
97	C₂×Dic₁₂₂	—	2	2	no	—
98	C₂×Dic₁₂₂	—	4	2	no	—
99	C₂×Dic₁₂₂	—	4	2	no	—
100	Q₁₆×C₆₁	—	4	4	no	—
101	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
104	C₄⋊C₄×C₆₁	—	2	2	no	—
105	C₄⋊C₄×C₆₁	—	2	2	no	—
106	C₄⋊C₄×C₆₁	—	2	2	no	—
107	C₄⋊C₄×C₆₁	—	2	2	no	—
108	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
109	#11758202	—	4	8	no	—
110	C₄⋊Dic₆₁	—	2	2	no	—
111	C₄⋊Dic₆₁	—	2	2	no	—
112	C₄⋊Dic₆₁	—	2	2	no	—
113	C₄⋊Dic₆₁	—	2	2	no	—
114	#11758196	—	4	2	no	—
115	#11758196	—	4	2	no	—
116	#11758196	—	4	2	no	—
117	#11758196	—	4	2	no	—
118	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
119	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
120	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
121	C₂×#92418	C₂×D₄	1	2	yes	—
122	C₂×#92418	C₂×D₄	1	2	yes	—
123	C₂×#92418	C₂×D₄	1	2	yes	—
124	Dic₂₄₄	—	4	4	no	—
125	C₈×D₆₁	—	2	4	no	—
126	C₈×D₆₁	SD₁₆	1	8	yes	—
127	C₈×D₆₁	Q₁₆	1	8	yes	complemented
128	C₂²×D₆₁	#127	1	1	yes	characteristic
129	Q₈×C₆₁	—	4	4	no	—
130	Q₈×C₆₁	—	4	4	no	—
131	C₂×C₄×C₆₁	—	4	2	no	—
132	C₂×C₄×C₆₁	—	4	2	no	—
133	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₆₁	—	4	2	no	—
137	C₂×Dic₆₁	—	4	2	no	—
138	C₂×Dic₆₁	#127	1	1	yes	characteristic
139	C₂×Dic₆₁	#127	1	1	yes	characteristic
140	C₂×Dic₆₁	#127	1	1	yes	characteristic
141	C₄×D₆₁	—	8	4	no	—
142	C₄×D₆₁	—	8	4	no	—
143	C₄×D₆₁	C₂×SD₁₆	1	4	yes	—
144	C₄×D₆₁	C₂×Q₁₆	1	4	yes	—
145	C₄×D₆₁	#135	1	4	yes	—
146	Dic₁₂₂	—	4	4	no	—
147	Dic₁₂₂	—	4	4	no	—
148	Dic₁₂₂	—	8	4	no	—
149	Dic₁₂₂	—	8	4	no	—
150	C₈×C₆₁	—	2	2	no	—
151	C₈×C₆₁	C₂×SD₁₆	1	4	yes	—
152	C₈×C₆₁	C₂×Q₁₆	1	4	yes	complemented
153	#92418	—	2	2	no	—
154	#92418	C₂×SD₁₆	1	4	yes	—
155	#92418	C₂×Q₁₆	1	4	yes	complemented
156	C₂×#1707	—	61	1	no	maximal, Sylow
157	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
158	C₂×D₆₁	#492	1	2	yes	—
159	C₂×D₆₁	#494	1	2	yes	—
160	C₂×D₆₁	#495	1	2	yes	—
161	C₄×C₆₁	—	8	2	no	—
162	C₄×C₆₁	—	8	2	no	—
163	C₄×C₆₁	C₂²×SD₁₆	1	2	yes	—
164	C₄×C₆₁	C₂²×Q₁₆	1	2	yes	—
165	C₄×C₆₁	C₂×#135	1	2	yes	—
166	Dic₆₁	—	8	2	no	—
167	Dic₆₁	—	8	2	no	—
168	Dic₆₁	#492	1	2	yes	—
169	Dic₆₁	#494	1	2	yes	—
170	Dic₆₁	#495	1	2	yes	—
171	C₂×#500	—	61	1	no	—
172	C₂×#498	—	61	2	no	—
173	C₂×#494	—	61	1	no	—
174	C₂×#487	—	61	2	no	—
175	#1707	—	61	1	no	—
176	#1707	—	61	2	no	—
177	#1707	—	61	1	no	—
178	#1707	—	61	2	no	—
179	#1707	—	61	1	no	—
180	#1707	D₆₁	1	1	yes	characteristic, direct product factor
181	C₂×C₈²	—	61	1	no	—
182	C₂×C₆₁	C₂×#492	1	1	yes	characteristic
183	C₂×C₆₁	C₂×#494	1	1	yes	characteristic
184	C₂×C₆₁	C₂×#495	1	1	yes	characteristic
185	D₆₁	#1707	1	4	yes	direct product factor
186	C₂²×Q₁₆	—	122	2	no	—
187	C₂×C₄⋊Q₈	—	61	2	no	—
188	C₂×C₄⋊Q₈	—	61	2	no	—
189	#500	—	61	2	no	—
190	#500	—	61	1	no	—
191	#500	—	122	2	no	—
192	#500	C₂×D₆₁	1	1	yes	characteristic
193	#498	—	61	2	no	—
194	#498	—	61	2	no	—
195	#498	—	61	2	no	—
196	#498	—	122	4	no	—
197	#498	C₂×D₆₁	1	2	yes	—
198	#494	—	61	2	no	—
199	#494	—	61	1	no	—
200	#494	—	122	2	no	—
201	#494	C₂×D₆₁	1	1	yes	characteristic
202	#487	—	61	2	no	—
203	#487	—	61	2	no	—
204	#487	—	61	2	no	—
205	#487	—	122	4	no	—
206	#487	C₂×D₆₁	1	2	yes	—
207	C₂×#107	—	122	2	no	—
208	C₂×#106	—	122	4	no	—
209	C₂×#103	—	122	4	no	—
210	C₂×C₄×C₈	—	61	1	no	—
211	C₂×C₄×C₈	—	61	1	no	—
212	C₂×C₄×C₈	—	61	1	no	—
213	C₈²	—	61	1	no	—
214	C₈²	—	61	1	no	—
215	C₈²	—	61	1	no	—
216	C₈²	C₂×D₆₁	1	1	yes	characteristic
217	C₆₁	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
218	C₂²×Q₈	—	122	2	no	—
219	C₂²×Q₈	—	122	2	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×Q₁₆	—	122	4	no	—
222	C₂×Q₁₆	—	122	2	no	—
223	C₂×Q₁₆	—	244	8	no	—
224	C₂²×C₈	—	61	2	no	—
225	C₂²×C₈	—	61	2	no	—
226	C₂²×C₈	—	61	2	no	—
227	C₄⋊Q₈	—	61	2	no	—
228	C₄⋊Q₈	—	61	2	no	—
229	C₄⋊Q₈	—	122	2	no	—
230	C₄⋊Q₈	—	122	2	no	—
231	C₄⋊Q₈	—	122	2	no	—
232	C₄⋊Q₈	—	122	2	no	—
233	C₄⋊Q₈	—	122	2	no	—
234	C₄⋊Q₈	—	122	2	no	—
235	C₄⋊Q₈	C₂²×D₆₁	1	2	yes	—
236	C₄⋊Q₈	C₂²×D₆₁	1	2	yes	—
237	C₂×C₄⋊C₄	—	122	2	no	—
238	C₂×C₄⋊C₄	—	122	2	no	—
239	C₂×C₄⋊C₄	—	122	2	no	—
240	C₂×C₄⋊C₄	—	122	2	no	—
241	C₂×C₄²	—	61	1	no	—
242	#107	—	2	2	no	—
243	#107	—	122	4	no	—
244	#107	—	122	2	no	—
245	#106	—	2	4	no	—
246	#106	—	122	4	no	—
247	#106	—	122	4	no	—
248	#106	—	122	4	no	—
249	#103	—	2	4	no	—
250	#103	—	122	4	no	—
251	#103	—	122	4	no	—
252	#103	—	122	4	no	—
253	C₄×C₈	—	61	2	no	—
254	C₄×C₈	—	61	2	no	—
255	C₄×C₈	—	61	2	no	—
256	C₄×C₈	—	61	1	no	—
257	C₄×C₈	—	61	1	no	—
258	C₄×C₈	—	61	1	no	—
259	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	122	2	no	—
265	C₂×Q₈	—	122	2	no	—
266	C₂×Q₈	—	244	2	no	—
267	C₂×Q₈	—	244	2	no	—
268	C₂×Q₈	—	244	8	no	—
269	C₂×Q₈	—	244	8	no	—
270	C₂²×C₄	—	61	1	no	—
271	C₂²×C₄	—	61	1	no	—
272	C₂²×C₄	—	61	1	no	—
273	C₂²×C₄	—	244	2	no	—
274	C₂²×C₄	—	244	2	no	—
275	Q₁₆	—	4	4	no	—
276	Q₁₆	—	244	8	no	—
277	Q₁₆	—	244	4	no	—
278	C₂×C₈	—	61	2	no	—
279	C₂×C₈	—	61	2	no	—
280	C₂×C₈	—	61	8	no	—
281	C₂×C₈	—	61	8	no	—
282	C₂×C₈	—	61	2	no	—
283	C₂×C₈	—	122	4	no	—
284	C₂×C₈	D₄×D₆₁	1	2	yes	—
285	C₂×C₈	D₄×D₆₁	1	2	yes	—
286	C₂×C₈	D₄×D₆₁	1	2	yes	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	122	2	no	—
292	C₄⋊C₄	—	122	2	no	—
293	C₄⋊C₄	—	122	2	no	—
294	C₄⋊C₄	—	122	2	no	—
295	C₄⋊C₄	—	244	2	no	—
296	C₄⋊C₄	—	244	2	no	—
297	C₄⋊C₄	—	244	2	no	—
298	C₄⋊C₄	—	244	2	no	—
299	C₄²	—	61	1	no	—
300	C₄²	—	61	1	no	—
301	C₄²	—	61	1	no	—
302	C₄²	C₂³×D₆₁	1	1	yes	characteristic
303	C₂³	—	61	1	no	—
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	Q₈	—	488	4	no	—
307	Q₈	—	488	4	no	—
308	Q₈	—	244	4	no	—
309	Q₈	—	244	4	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	61	1	no	—
313	C₂×C₄	—	61	1	no	—
314	C₂×C₄	—	61	1	no	—
315	C₂×C₄	—	61	4	no	—
316	C₂×C₄	—	61	4	no	—
317	C₂×C₄	—	61	4	no	—
318	C₂×C₄	—	488	4	no	—
319	C₂×C₄	—	488	4	no	—
320	C₂×C₄	—	244	2	no	—
321	C₂×C₄	—	244	2	no	—
322	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
324	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	61	4	no	—
327	C₈	—	61	4	no	—
328	C₈	—	122	2	no	—
329	C₈	SD₁₆×D₆₁	1	4	yes	—
330	C₈	Q₁₆×D₆₁	1	4	yes	complemented
331	C₂²	—	61	2	no	—
332	C₂²	—	61	2	no	—
333	C₂²	—	61	2	no	—
334	C₂²	#127×D₆₁	1	1	yes	characteristic
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	61	2	no	—
338	C₄	—	61	2	no	—
339	C₄	—	61	2	no	—
340	C₄	—	488	2	no	—
341	C₄	—	488	2	no	—
342	C₄	C₂×SD₁₆×D₆₁	1	2	yes	—
343	C₄	C₂×Q₁₆×D₆₁	1	2	yes	—
344	C₄	#135×D₆₁	1	2	yes	—
345	C₂	—	61	4	no	—
346	C₂	#492×D₆₁	1	1	yes	characteristic
347	C₂	#494×D₆₁	1	1	yes	characteristic
348	C₂	#495×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₆₁
commutator, cosocle	C₄²×C₆₁	C₂⁴
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1707	C₂
socle	C₂²×C₆₁	C₂×#127
prime radical	C₂²×D₆₁	#127
hypercenter	#1707	D₆₁
hypocenter	C₆₁	C₂×#1707
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #21445417	C₂	semidirect
6. #21445416	C₂	semidirect
7. #21445415	C₂	semidirect
8. #21445414	C₂	semidirect
9. #21445413	C₂	semidirect
11. C₆₁×#1707	C₂	semidirect
127. C₈×D₆₁	Q₁₆	semidirect
152. C₈×C₆₁	C₂×Q₁₆	semidirect
155. #92418	C₂×Q₁₆	semidirect
180. #1707	D₆₁	direct (N × H)
185. D₆₁	#1707	direct (N × H)
217. C₆₁	C₂×#1707	semidirect
330. C₈	Q₁₆×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
#498×D₆₁	2
#494×D₆₁	2
#487×D₆₁	2
#21445417	2
#21445416	2
#21445415	2
#21445414	2
#21445413	2
C₈²×D₆₁	2
C₆₁×#1707	2
C₂×#1707	61

Subgroups — D61×#1707 #29667658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁×#1707 #29667658