Subgroups — D₆₁⋊#2164 #30624145

Order 15616.

610 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462187	C₂	1	2	yes	maximal, complemented
2	#21462186	C₂	1	2	yes	maximal, complemented
3	#21462185	C₂	1	2	yes	maximal, complemented
4	#21462184	C₂	1	2	yes	maximal, complemented
5	D₆₁⋊(C₄⋊C₁₆)	C₂	1	2	yes	maximal, complemented
6	D₆₁⋊(C₂²⋊C₁₆)	C₂	1	2	yes	maximal
7	#21462020	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₆₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#405×D₆₁	C₂	1	1	yes	maximal, characteristic
10	C₂×D₆₁⋊C₁₆	C₂²	1	2	yes	—
11	C₂×D₆₁⋊C₁₆	C₂²	1	2	yes	—
12	#16840683	C₂²	1	4	yes	complemented
13	C₄⋊#11758187	C₂²	1	4	yes	complemented
14	#16840657	C₂²	1	4	yes	—
15	C₂²⋊#11758187	C₂²	1	4	yes	—
16	#16840651	C₂²	1	2	yes	complemented
17	#16840650	C₂²	1	2	yes	complemented
18	#16840648	C₂²	1	2	yes	complemented
19	C₄×#11758187	C₂²	1	2	yes	complemented
20	C₂×M₁₆×D₆₁	C₄	1	1	yes	characteristic
21	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
22	#117×D₆₁	C₄	1	1	yes	characteristic
23	#16839720	C₂²	1	1	yes	characteristic
24	#16839719	C₂²	1	1	yes	characteristic
25	#16839719	C₂²	1	1	yes	characteristic
26	#16839718	C₂²	1	1	yes	characteristic
27	#16839717	C₄	1	1	yes	characteristic
28	#16839716	C₄	1	1	yes	characteristic
29	#16839716	C₄	1	1	yes	characteristic
30	#97×D₆₁	C₄	1	1	yes	characteristic
31	#97×D₆₁	C₄	1	1	yes	characteristic
32	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
34	C₆₁×#405	C₄	1	1	yes	characteristic
35	D₆₁⋊C₁₆	—	2	4	no	—
36	D₆₁⋊C₁₆	—	2	4	no	—
37	C₂×#11758187	C₂³	1	4	yes	—
38	C₂×#11758187	C₂³	1	4	yes	—
39	C₂²×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
41	C₂×#11758189	C₂×C₄	1	1	yes	characteristic
42	C₂×#11758189	C₂×C₄	1	1	yes	characteristic
43	M₁₆×D₆₁	—	2	4	no	—
44	C₂²×#92418	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₆₁	—	2	4	no	—
46	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
52	#14519736	C₂×C₄	1	1	yes	characteristic
53	#14519735	C₈	1	2	yes	—
54	#14519734	C₂×C₄	1	1	yes	characteristic
55	#14519734	C₂×C₄	1	1	yes	characteristic
56	#14519733	C₈	1	2	yes	—
57	C₄⋊C₄×D₆₁	C₈	1	2	yes	—
58	C₂²⋊C₄×D₆₁	C₈	1	2	yes	—
59	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
60	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
61	#14519688	C₂×C₄	1	1	yes	characteristic
62	#14519688	C₂×C₄	1	1	yes	characteristic
63	#14519687	C₂×C₄	1	1	yes	characteristic
64	#14519687	C₂×C₄	1	1	yes	characteristic
65	#14519687	C₂×C₄	1	1	yes	characteristic
66	#14519687	C₂×C₄	1	1	yes	characteristic
67	C₄×#92418	C₂³	1	1	yes	characteristic
68	C₄×#92418	C₂³	1	1	yes	characteristic
69	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
72	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
75	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
76	#97×C₆₁	C₂×C₄	1	1	yes	characteristic
77	#97×C₆₁	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
79	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
80	C₂³×D₆₁	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×Dic₆₁	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₆₁	—	2	2	no	—
84	C₂×C₄×D₆₁	—	2	2	no	—
85	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
88	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
89	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
91	M₁₆×C₆₁	—	2	2	no	—
92	C₂×C₈×C₆₁	—	2	2	no	—
93	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
98	C₂²⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
99	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²⋊Dic₆₁	C₂×C₈	1	2	yes	—
102	#11758198	C₂×C₈	1	2	yes	—
103	#11758198	C₂×C₈	1	2	yes	—
104	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
105	#11758196	C₂×C₈	1	2	yes	—
106	#11758196	C₂×C₈	1	2	yes	—
107	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
108	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
109	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
110	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
111	#11758194	—	2	2	no	—
112	C₂×#92418	—	2	2	no	—
113	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
114	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
115	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
116	C₂×#92418	C₂⁴	1	1	yes	characteristic
117	#11758189	—	2	2	no	—
118	#11758189	—	2	2	no	—
119	C₈×D₆₁	—	2	2	no	—
120	C₈×D₆₁	—	2	2	no	—
121	C₈×D₆₁	—	2	2	no	—
122	C₈×D₆₁	—	2	2	no	—
123	C₈×D₆₁	M₁₆	1	4	yes	—
124	C₈×D₆₁	#41	1	4	yes	—
125	#11758187	—	2	4	no	—
126	#11758187	—	2	4	no	—
127	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
128	C₂²×D₆₁	—	2	2	no	—
129	C₂²×D₆₁	—	2	4	no	—
130	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
131	C₂²×D₆₁	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₆₁	—	2	1	no	—
133	C₂×C₄×C₆₁	—	2	1	no	—
134	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
138	C₂×Dic₆₁	—	2	2	no	—
139	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
140	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
141	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
142	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
143	C₄×D₆₁	—	2	2	no	—
144	C₄×D₆₁	—	2	4	no	—
145	C₄×D₆₁	—	2	4	no	—
146	C₄×D₆₁	—	2	1	no	—
147	C₄×D₆₁	—	2	1	no	—
148	C₄×D₆₁	—	2	2	no	—
149	C₄×D₆₁	#117	1	2	yes	—
150	C₄×D₆₁	C₂×M₁₆	1	2	yes	—
151	C₈×C₆₁	—	2	1	no	—
152	C₈×C₆₁	—	2	1	no	—
153	C₈×C₆₁	#117	1	2	yes	—
154	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
155	#92418	—	2	1	no	—
156	#92418	—	2	1	no	—
157	#92418	C₂×M₁₆	1	2	yes	—
158	#92418	C₂×#41	1	2	yes	—
159	C₂×#2164	—	61	1	no	maximal, Sylow
160	C₂²×C₆₁	—	2	2	no	—
161	C₂²×C₆₁	—	2	1	no	—
162	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
163	C₂×D₆₁	—	2	2	no	—
164	C₂×D₆₁	—	2	1	no	—
165	C₂×D₆₁	—	4	4	no	—
166	C₂×D₆₁	#431	1	2	yes	—
167	C₂×D₆₁	#504	1	4	yes	—
168	C₄×C₆₁	—	2	2	no	—
169	C₄×C₆₁	—	2	2	no	—
170	C₄×C₆₁	—	2	1	no	—
171	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
172	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
173	Dic₆₁	—	2	2	no	—
174	Dic₆₁	—	2	2	no	—
175	Dic₆₁	—	2	1	no	—
176	Dic₆₁	#431	1	2	yes	—
177	C₂×#405	—	61	1	no	—
178	#2164	—	61	2	no	—
179	#2164	—	61	2	no	—
180	#2164	—	61	2	no	—
181	#2164	—	61	2	no	—
182	C₂×C₄⋊C₁₆	—	61	2	no	—
183	C₂×C₂²⋊C₁₆	—	61	2	no	—
184	C₂×#346	—	61	1	no	—
185	C₂×C₄×C₁₆	—	61	1	no	—
186	C₂×C₆₁	—	4	1	no	—
187	C₂×C₆₁	C₂×#431	1	1	yes	characteristic
188	C₂×C₆₁	C₂×#504	1	2	yes	—
189	D₆₁	—	4	1	no	—
190	D₆₁	#2164	1	4	yes	complemented
191	C₂²×M₁₆	—	61	1	no	—
192	C₂³×C₈	—	61	1	no	—
193	C₂×#117	—	61	1	no	—
194	C₂²×C₁₆	—	61	2	no	—
195	C₂²×C₁₆	—	61	2	no	—
196	#405	—	61	1	no	—
197	#405	—	61	1	no	—
198	#405	—	61	1	no	—
199	#405	—	61	1	no	—
200	#405	—	61	1	no	—
201	#405	—	61	1	no	—
202	#405	—	61	1	no	—
203	#405	#6923	1	1	yes	characteristic
204	C₂×#97	—	61	1	no	—
205	C₂×#97	—	61	1	no	—
206	C₂×C₄×C₈	—	61	1	no	—
207	C₂×C₄×C₈	—	61	1	no	—
208	C₄⋊C₁₆	—	61	4	no	—
209	C₄⋊C₁₆	—	61	4	no	—
210	C₂²⋊C₁₆	—	61	4	no	—
211	C₂²⋊C₁₆	—	61	4	no	—
212	#346	—	61	2	no	—
213	#346	—	61	2	no	—
214	C₄×C₁₆	—	61	2	no	—
215	C₄×C₁₆	—	61	2	no	—
216	C₆₁	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
217	C₂³×C₄	—	61	1	no	—
218	C₂×M₁₆	—	61	1	no	—
219	C₂×M₁₆	—	61	1	no	—
220	C₂×M₁₆	—	61	1	no	—
221	C₂×M₁₆	—	122	4	no	—
222	C₂×M₁₆	C₂×#6923	1	1	yes	characteristic
223	C₂²×C₈	—	61	1	no	—
224	C₂²×C₈	—	61	1	no	—
225	C₂²×C₈	—	61	1	no	—
226	C₂²×C₈	—	61	1	no	—
227	C₂²×C₈	—	61	1	no	—
228	C₂²×C₈	—	61	1	no	—
229	C₂²×C₈	—	61	1	no	—
230	C₂²×C₈	—	122	4	no	—
231	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
232	#117	—	61	1	no	—
233	#117	—	61	2	no	—
234	#117	—	61	2	no	—
235	#117	—	61	1	no	—
236	#117	—	61	1	no	—
237	#117	C₂×#6923	1	1	yes	characteristic
238	C₂×C₄⋊C₄	—	61	2	no	—
239	C₂×C₂²⋊C₄	—	61	2	no	—
240	C₂×C₄²	—	61	1	no	—
241	C₂×C₄²	—	61	1	no	—
242	C₂×C₁₆	—	61	4	no	—
243	C₂×C₁₆	—	61	4	no	—
244	C₂×C₁₆	—	122	4	no	—
245	C₂×C₁₆	—	122	4	no	—
246	#97	—	61	1	no	—
247	#97	—	61	1	no	—
248	#97	—	61	1	no	—
249	#97	—	61	1	no	—
250	#97	—	61	1	no	—
251	#97	—	61	1	no	—
252	#97	C₂×#6923	1	1	yes	characteristic
253	#97	C₂×#6923	1	1	yes	characteristic
254	C₄×C₈	—	61	1	no	—
255	C₄×C₈	—	61	1	no	—
256	C₄×C₈	—	61	1	no	—
257	C₄×C₈	—	61	1	no	—
258	C₄×C₈	—	61	1	no	—
259	C₄×C₈	—	61	1	no	—
260	C₄×C₈	C₂×#6923	1	1	yes	characteristic
261	C₄×C₈	C₂×#6923	1	1	yes	characteristic
262	C₂⁴	—	61	1	no	—
263	C₂²×C₄	—	61	1	no	—
264	C₂²×C₄	—	61	1	no	—
265	C₂²×C₄	—	61	2	no	—
266	C₂²×C₄	—	61	2	no	—
267	C₂²×C₄	—	61	1	no	—
268	C₂²×C₄	—	61	1	no	—
269	C₂²×C₄	—	61	1	no	—
270	C₂²×C₄	—	122	2	no	—
271	C₂²×C₄	—	122	2	no	—
272	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
273	M₁₆	—	2	2	no	—
274	M₁₆	—	122	2	no	—
275	M₁₆	—	122	2	no	—
276	M₁₆	—	122	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	61	1	no	—
279	C₂×C₈	—	61	1	no	—
280	C₂×C₈	—	61	4	no	—
281	C₂×C₈	—	61	4	no	—
282	C₂×C₈	—	61	1	no	—
283	C₂×C₈	—	61	1	no	—
284	C₂×C₈	—	122	2	no	—
285	C₂×C₈	—	122	2	no	—
286	C₂×C₈	—	122	2	no	—
287	C₂×C₈	—	122	2	no	—
288	C₂×C₈	—	122	2	no	—
289	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
290	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
291	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
292	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
293	C₄⋊C₄	—	61	2	no	—
294	C₄⋊C₄	—	61	2	no	—
295	C₄⋊C₄	—	61	2	no	—
296	C₄⋊C₄	D₆₁⋊C₈	1	2	yes	—
297	C₂²⋊C₄	—	61	2	no	—
298	C₂²⋊C₄	—	61	2	no	—
299	C₂²⋊C₄	—	61	2	no	—
300	C₂²⋊C₄	D₆₁⋊C₈	1	2	yes	—
301	C₄²	—	61	2	no	—
302	C₄²	—	61	2	no	—
303	C₄²	—	61	1	no	—
304	C₄²	—	61	1	no	—
305	C₄²	C₂²×#6923	1	1	yes	characteristic
306	C₄²	C₂²×#6923	1	1	yes	characteristic
307	C₁₆	—	122	4	no	—
308	C₁₆	—	122	4	no	—
309	C₂³	—	61	1	no	—
310	C₂³	—	61	1	no	—
311	C₂³	—	122	2	no	—
312	C₂³	—	122	4	no	—
313	C₂³	C₂×D₆₁⋊C₈	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	61	1	no	—
317	C₂×C₄	—	61	2	no	—
318	C₂×C₄	—	61	2	no	—
319	C₂×C₄	—	61	2	no	—
320	C₂×C₄	—	61	2	no	—
321	C₂×C₄	—	61	1	no	—
322	C₂×C₄	—	122	2	no	—
323	C₂×C₄	—	122	4	no	—
324	C₂×C₄	—	122	4	no	—
325	C₂×C₄	—	122	2	no	—
326	C₂×C₄	—	122	2	no	—
327	C₂×C₄	—	122	1	no	—
328	C₂×C₄	—	122	1	no	—
329	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
330	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
331	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
332	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	61	2	no	—
336	C₈	—	61	2	no	—
337	C₈	—	122	1	no	—
338	C₈	—	122	1	no	—
339	C₈	#14519856	1	2	yes	—
340	C₈	D₆₁⋊M₁₆	1	2	yes	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	61	4	no	—
344	C₂²	—	61	2	no	—
345	C₂²	—	122	2	no	—
346	C₂²	—	122	1	no	—
347	C₂²	—	244	4	no	—
348	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	1	no	—
352	C₄	—	61	2	no	—
353	C₄	—	122	2	no	—
354	C₄	—	122	2	no	—
355	C₄	—	122	1	no	—
356	C₄	C₂×#14519856	1	1	yes	characteristic
357	C₄	C₂×D₆₁⋊M₁₆	1	1	yes	characteristic
358	C₂	—	4	1	no	—
359	C₂	—	61	4	no	—
360	C₂	—	244	1	no	—
361	C₂	D₆₁⋊#431	1	1	yes	characteristic
362	C₂	D₆₁⋊#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#6923
commutator, socle	C₂²×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#405	C₄
prime radical	C₂³×D₆₁	C₂×C₈
cosocle	C₂×#92418	C₂⁴
hypercenter	#405	#6923
hypocenter	C₆₁	C₂×#2164
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #21462187	C₂	semidirect
2. #21462186	C₂	semidirect
3. #21462185	C₂	semidirect
4. #21462184	C₂	semidirect
5. D₆₁⋊(C₄⋊C₁₆)	C₂	semidirect
7. #21462020	C₂	semidirect
8. C₄×D₆₁⋊C₁₆	C₂	semidirect
12. #16840683	C₂²	semidirect
13. C₄⋊#11758187	C₂²	semidirect
16. #16840651	C₂²	semidirect
17. #16840650	C₂²	semidirect
18. #16840648	C₂²	semidirect
19. C₄×#11758187	C₂²	semidirect
190. D₆₁	#2164	semidirect
216. C₆₁	C₂×#2164	semidirect

Maximal subgroups

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

subgroup	index
#21462187	2
#21462186	2
#21462185	2
#21462184	2
D₆₁⋊(C₄⋊C₁₆)	2
D₆₁⋊(C₂²⋊C₁₆)	2
#21462020	2
C₄×D₆₁⋊C₁₆	2
#405×D₆₁	2
C₂×#2164	61

Subgroups — D61⋊#2164 #30624145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁⋊#2164 #30624145