Subgroups — D₆₁⋊#2171 #30624266

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₆₁⋊#504	C₂	1	2	yes	maximal, complemented
2	#21462211	C₂	1	2	yes	maximal, complemented
3	#21462210	C₂	1	4	yes	maximal, complemented
4	#21462209	C₂	1	2	yes	maximal, complemented
5	D₆₁⋊#364	C₂	1	1	yes	maximal, characteristic, complemented
6	D₆₁⋊#350	C₂	1	2	yes	maximal, complemented
7	#21462020	C₂	1	1	yes	maximal, characteristic, complemented
8	#443×D₆₁	C₂	1	1	yes	maximal, characteristic
9	#16840821	—	2	4	no	—
10	#16840821	C₂²	1	4	yes	complemented
11	#16840820	C₂²	1	4	yes	complemented
12	D₆₁⋊M₃₂	—	2	2	no	—
13	D₆₁⋊M₃₂	C₂²	1	2	yes	—
14	C₂×D₆₁⋊C₁₆	—	2	2	no	—
15	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
17	#16840685	C₂²	1	2	yes	complemented
18	#16840684	C₂²	1	2	yes	complemented
19	#16840661	C₂²	1	4	yes	complemented
20	#16840660	C₂²	1	4	yes	complemented
21	#16840651	C₂²	1	2	yes	complemented
22	#16840650	C₂²	1	2	yes	complemented
23	#135×D₆₁	C₄	1	1	yes	characteristic
24	#131×D₆₁	C₂²	1	2	yes	—
25	#16839880	C₄	1	1	yes	characteristic
26	#16839879	C₂²	1	1	yes	characteristic
27	#16839878	C₂²	1	1	yes	characteristic
28	#16839877	C₄	1	2	yes	—
29	#16839876	C₂²	1	2	yes	—
30	#108×D₆₁	C₄	1	1	yes	characteristic
31	#104×D₆₁	C₄	1	2	yes	—
32	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
33	C₆₁×#443	C₄	1	1	yes	characteristic
34	#14519852	—	2	2	no	—
35	#14519852	—	2	4	no	—
36	#14519851	—	2	4	no	—
37	#14519851	C₂³	1	4	yes	—
38	D₆₁⋊C₁₆	—	2	2	no	—
39	D₆₁⋊C₁₆	—	2	1	no	—
40	D₆₁⋊C₁₆	—	2	1	no	—
41	D₆₁⋊C₁₆	—	2	4	no	—
42	D₆₁⋊C₁₆	D₄	1	2	yes	—
43	D₆₁⋊C₁₆	D₄	1	2	yes	—
44	C₂×#11758187	—	2	4	no	—
45	C₂×#11758187	C₂³	1	2	yes	—
46	C₂×#11758187	C₂³	1	2	yes	—
47	#41×D₆₁	C₂×C₄	1	2	yes	—
48	#14519800	C₈	1	1	yes	characteristic
49	#14519799	C₂×C₄	1	1	yes	characteristic
50	#14519798	C₈	1	2	yes	—
51	#14519797	C₂×C₄	1	2	yes	—
52	#14519796	C₈	1	1	yes	characteristic
53	Q₁₆×D₆₁	C₈	1	1	yes	characteristic
54	SD₁₆×D₆₁	C₈	1	2	yes	—
55	D₈×D₆₁	C₈	1	1	yes	characteristic
56	#14519785	—	2	2	no	—
57	#14519785	C₂³	1	2	yes	—
58	#14519784	C₂³	1	2	yes	—
59	#14519783	—	2	2	no	—
60	#14519783	C₂×C₄	1	2	yes	—
61	M₁₆×D₆₁	—	2	2	no	—
62	M₁₆×D₆₁	C₂×C₄	1	2	yes	—
63	C₂×C₈×D₆₁	—	2	2	no	—
64	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
66	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
67	#14519713	C₂×C₄	1	1	yes	characteristic
68	#14519712	C₂×C₄	1	2	yes	—
69	#14519705	C₂×C₄	1	2	yes	—
70	#14519704	C₂×C₄	1	2	yes	—
71	#14519703	C₂×C₄	1	2	yes	—
72	C₄×#92418	C₂³	1	1	yes	characteristic
73	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
75	#135×C₆₁	C₂×C₄	1	1	yes	characteristic
76	#131×C₆₁	C₂×C₄	1	2	yes	—
77	#108×C₆₁	C₂×C₄	1	1	yes	characteristic
78	#104×C₆₁	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
80	#41×C₆₁	C₂²×C₄	1	2	yes	—
81	#11758226	C₂×C₈	1	2	yes	—
82	Q₈×D₆₁	C₂×C₈	1	2	yes	—
83	#11758224	—	2	2	no	—
84	#11758224	C₂×C₈	1	2	yes	—
85	D₄×D₆₁	—	2	2	no	—
86	D₄×D₆₁	C₂×C₈	1	2	yes	—
87	#11758222	—	2	2	no	—
88	#11758222	C₂²×C₄	1	2	yes	—
89	C₂×C₄×D₆₁	—	2	2	no	—
90	C₂×C₄×D₆₁	—	2	1	no	—
91	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
92	Q₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
93	SD₁₆×C₆₁	C₂×C₈	1	2	yes	—
94	D₈×C₆₁	C₂×C₈	1	1	yes	characteristic
95	M₁₆×C₆₁	—	2	2	no	—
96	M₁₆×C₆₁	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₆₁	—	2	2	no	—
98	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
101	#11758202	C₂×C₈	1	2	yes	—
102	#11758201	C₂×C₈	1	2	yes	—
103	#11758200	C₂×C₈	1	2	yes	—
104	#11758199	C₂×C₈	1	2	yes	—
105	C₄×Dic₆₁	—	2	1	no	—
106	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
107	#11758194	—	2	2	no	—
108	#11758194	C₂²×C₄	1	2	yes	—
109	C₂×#92418	—	2	2	no	—
110	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
111	C₂×#92418	C₂⁴	1	1	yes	characteristic
112	Dic₂₄₄	C₂×C₈	1	1	yes	characteristic
113	D₄₈₈	C₂×C₈	1	1	yes	characteristic
114	#11758190	C₂×C₈	1	2	yes	—
115	#11758189	—	2	2	no	—
116	#11758189	—	2	2	no	—
117	#11758189	—	2	2	no	—
118	C₈×D₆₁	—	2	2	no	—
119	C₈×D₆₁	—	2	2	no	—
120	C₈×D₆₁	—	2	2	no	—
121	C₈×D₆₁	C₂×C₈	1	1	yes	characteristic
122	C₈×D₆₁	C₂×C₈	1	1	yes	characteristic
123	C₈×D₆₁	C₂×C₈	1	1	yes	characteristic
124	C₈×D₆₁	C₂×C₈	1	1	yes	characteristic
125	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
126	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
127	C₈×D₆₁	#41	1	1	yes	characteristic
128	C₈×D₆₁	#41	1	1	yes	characteristic
129	#11758187	—	2	4	no	—
130	#11758187	—	2	2	no	—
131	#11758187	C₂×D₄	1	4	yes	—
132	C₂²×D₆₁	—	2	2	no	—
133	C₂²×D₆₁	#131	1	1	yes	characteristic
134	Q₈×C₆₁	C₂²×C₈	1	2	yes	—
135	D₄×C₆₁	—	2	2	no	—
136	D₄×C₆₁	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₆₁	—	2	2	no	—
138	C₂×C₄×C₆₁	—	2	1	no	—
139	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
140	C₂²⋊D₆₁	—	2	2	no	—
141	C₂²⋊D₆₁	—	2	2	no	—
142	C₂²⋊D₆₁	—	2	2	no	—
143	C₂×Dic₆₁	—	2	2	no	—
144	C₂×Dic₆₁	—	2	1	no	—
145	C₂×Dic₆₁	#131	1	1	yes	characteristic
146	D₂₄₄	—	2	2	no	—
147	D₂₄₄	C₂²×C₈	1	2	yes	—
148	C₄×D₆₁	—	2	2	no	—
149	C₄×D₆₁	—	2	2	no	—
150	C₄×D₆₁	—	2	2	no	—
151	C₄×D₆₁	—	4	2	no	—
152	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
153	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
154	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
155	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
156	Dic₁₂₂	—	2	2	no	—
157	Dic₁₂₂	C₂²×C₈	1	2	yes	—
158	C₈×C₆₁	—	2	2	no	—
159	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
160	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
161	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
162	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
163	#92418	—	2	2	no	—
164	#92418	C₂²×C₈	1	1	yes	characteristic
165	#92418	C₂²×C₈	1	1	yes	characteristic
166	#92418	C₂²×D₄	1	1	yes	characteristic
167	#92418	C₂×#41	1	1	yes	characteristic
168	C₂×#2171	—	61	1	no	maximal, Sylow
169	C₂²×C₆₁	—	2	2	no	—
170	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
171	C₂×D₆₁	—	2	2	no	—
172	C₂×D₆₁	—	2	2	no	—
173	C₂×D₆₁	—	4	4	no	—
174	C₂×D₆₁	C₈×D₄	1	1	yes	characteristic
175	C₂×D₆₁	C₈×D₄	1	1	yes	characteristic
176	C₄×C₆₁	—	2	2	no	—
177	C₄×C₆₁	—	4	1	no	—
178	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
179	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
180	Dic₆₁	—	2	2	no	—
181	Dic₆₁	—	4	1	no	—
182	Dic₆₁	C₈×D₄	1	1	yes	characteristic
183	Dic₆₁	C₈×D₄	1	1	yes	characteristic
184	C₂×#504	—	61	2	no	—
185	C₂×#443	—	61	1	no	—
186	#2171	—	61	2	no	—
187	#2171	—	61	4	no	—
188	#2171	—	61	2	no	—
189	C₂×#364	—	61	1	no	—
190	C₂×#350	—	61	2	no	—
191	C₂×#346	—	61	1	no	—
192	C₂×C₆₁	—	2	1	no	—
193	C₂×C₆₁	—	4	2	no	—
194	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
195	D₆₁	—	2	1	no	—
196	D₆₁	—	4	2	no	—
197	D₆₁	#2171	1	2	yes	complemented
198	C₂×#135	—	61	1	no	—
199	C₂×#131	—	61	2	no	—
200	#504	—	61	4	no	—
201	#504	—	61	4	no	—
202	#504	—	122	4	no	—
203	C₂×M₃₂	—	61	2	no	—
204	C₂×M₃₂	—	122	2	no	—
205	C₂²×C₁₆	—	61	1	no	—
206	C₂²×C₁₆	—	61	1	no	—
207	C₂²×C₁₆	—	122	2	no	—
208	#443	—	61	1	no	—
209	#443	—	61	2	no	—
210	#443	—	61	1	no	—
211	#443	—	61	2	no	—
212	#443	—	61	1	no	—
213	#443	#6923	1	1	yes	characteristic
214	C₂×#108	—	61	1	no	—
215	C₂×#104	—	61	2	no	—
216	C₂×C₄×C₈	—	61	1	no	—
217	#364	—	61	2	no	—
218	#364	—	61	2	no	—
219	#350	—	61	4	no	—
220	#350	—	61	4	no	—
221	#346	—	61	2	no	—
222	#346	—	61	2	no	—
223	C₆₁	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	—	61	2	no	—
225	#135	—	61	2	no	—
226	#135	—	61	1	no	—
227	#135	—	61	2	no	—
228	#135	—	61	1	no	—
229	#135	—	61	1	no	—
230	#135	C₂×#6923	1	1	yes	characteristic
231	C₂×Q₁₆	—	61	1	no	—
232	C₂×SD₁₆	—	61	2	no	—
233	C₂×D₈	—	61	1	no	—
234	#131	—	61	2	no	—
235	#131	—	61	2	no	—
236	#131	—	61	2	no	—
237	#131	—	122	2	no	—
238	#131	—	122	2	no	—
239	#131	C₂×#6923	1	2	yes	—
240	C₂×M₁₆	—	61	2	no	—
241	C₂×M₁₆	—	122	2	no	—
242	C₂²×C₈	—	61	1	no	—
243	C₂²×C₈	—	61	1	no	—
244	C₂²×C₈	—	122	2	no	—
245	C₂×C₄²	—	61	1	no	—
246	M₃₂	—	61	4	no	—
247	M₃₂	—	122	4	no	—
248	M₃₂	—	122	4	no	—
249	M₃₂	—	122	2	no	—
250	C₂×C₁₆	—	61	2	no	—
251	C₂×C₁₆	—	61	2	no	—
252	C₂×C₁₆	—	61	2	no	—
253	C₂×C₁₆	—	61	2	no	—
254	C₂×C₁₆	—	122	4	no	—
255	C₂×C₁₆	—	122	4	no	—
256	C₂×C₁₆	—	122	1	no	—
257	C₂×C₁₆	—	122	1	no	—
258	C₂×C₁₆	—	122	2	no	—
259	#108	—	61	2	no	—
260	#108	—	61	1	no	—
261	#108	C₂×#6923	1	1	yes	characteristic
262	#104	—	61	2	no	—
263	#104	—	61	2	no	—
264	#104	—	61	2	no	—
265	#104	C₂×#6923	1	2	yes	—
266	C₄×C₈	—	61	1	no	—
267	C₄×C₈	—	61	1	no	—
268	C₄×C₈	—	61	1	no	—
269	C₄×C₈	C₂×#6923	1	1	yes	characteristic
270	#41	—	61	2	no	—
271	#41	—	61	2	no	—
272	#41	—	61	2	no	—
273	#41	—	122	2	no	—
274	#41	—	122	2	no	—
275	#41	C₂²×#6923	1	2	yes	—
276	C₂×Q₈	—	61	2	no	—
277	C₂×D₄	—	61	2	no	—
278	C₂×D₄	—	122	2	no	—
279	C₂²×C₄	—	61	1	no	—
280	C₂²×C₄	—	122	2	no	—
281	C₂²×C₄	—	122	1	no	—
282	Q₁₆	—	61	2	no	—
283	Q₁₆	—	61	1	no	—
284	Q₁₆	D₆₁⋊C₈	1	1	yes	characteristic
285	SD₁₆	—	61	2	no	—
286	SD₁₆	—	61	2	no	—
287	SD₁₆	—	61	2	no	—
288	SD₁₆	D₆₁⋊C₈	1	2	yes	—
289	D₈	—	61	2	no	—
290	D₈	—	61	1	no	—
291	D₈	D₆₁⋊C₈	1	1	yes	characteristic
292	M₁₆	—	2	2	no	—
293	M₁₆	—	61	2	no	—
294	M₁₆	—	122	2	no	—
295	M₁₆	—	122	2	no	—
296	M₁₆	—	122	2	no	—
297	M₁₆	—	122	2	no	—
298	M₁₆	C₂²×#6923	1	2	yes	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	61	1	no	—
301	C₂×C₈	—	61	1	no	—
302	C₂×C₈	—	61	1	no	—
303	C₂×C₈	—	61	1	no	—
304	C₂×C₈	—	61	1	no	—
305	C₂×C₈	—	61	1	no	—
306	C₂×C₈	—	61	1	no	—
307	C₂×C₈	—	61	1	no	—
308	C₂×C₈	—	61	1	no	—
309	C₂×C₈	—	61	1	no	—
310	C₂×C₈	—	122	2	no	—
311	C₂×C₈	—	122	2	no	—
312	C₂×C₈	—	122	2	no	—
313	C₂×C₈	—	122	2	no	—
314	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
315	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
316	C₄²	—	61	1	no	—
317	C₄²	—	122	1	no	—
318	C₄²	C₂²×#6923	1	1	yes	characteristic
319	C₁₆	—	61	4	no	—
320	C₁₆	—	122	2	no	—
321	C₁₆	—	122	4	no	—
322	C₂³	—	61	1	no	—
323	C₂³	—	122	2	no	—
324	Q₈	—	61	2	no	—
325	Q₈	—	122	2	no	—
326	Q₈	C₂×D₆₁⋊C₈	1	2	yes	—
327	D₄	—	2	2	no	—
328	D₄	—	61	2	no	—
329	D₄	—	122	2	no	—
330	D₄	—	122	2	no	—
331	D₄	—	122	2	no	—
332	D₄	—	122	2	no	—
333	D₄	C₂×D₆₁⋊C₈	1	2	yes	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	61	1	no	—
337	C₂×C₄	—	61	1	no	—
338	C₂×C₄	—	61	1	no	—
339	C₂×C₄	—	61	1	no	—
340	C₂×C₄	—	61	1	no	—
341	C₂×C₄	—	122	2	no	—
342	C₂×C₄	—	122	2	no	—
343	C₂×C₄	—	122	2	no	—
344	C₂×C₄	—	122	2	no	—
345	C₂×C₄	—	122	1	no	—
346	C₂×C₄	—	244	2	no	—
347	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	61	1	no	—
350	C₈	—	61	1	no	—
351	C₈	—	61	1	no	—
352	C₈	—	61	1	no	—
353	C₈	—	122	2	no	—
354	C₈	D₄×#6923	1	1	yes	characteristic
355	C₈	D₄×#6923	1	1	yes	characteristic
356	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
357	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
358	C₂²	—	2	2	no	—
359	C₂²	—	61	1	no	—
360	C₂²	—	61	1	no	—
361	C₂²	—	122	2	no	—
362	C₂²	—	122	2	no	—
363	C₂²	—	244	4	no	—
364	C₂²	D₆₁⋊#131	1	1	yes	characteristic
365	C₄	—	2	2	no	—
366	C₄	—	4	1	no	—
367	C₄	—	61	1	no	—
368	C₄	—	61	1	no	—
369	C₄	—	122	2	no	—
370	C₄	—	244	1	no	—
371	C₄	C₂×D₄×#6923	1	1	yes	characteristic
372	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
373	C₂	—	2	1	no	—
374	C₂	—	4	2	no	—
375	C₂	—	61	2	no	—
376	C₂	—	122	1	no	—
377	C₂	—	244	2	no	—
378	C₂	D₄×D₆₁⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₄×#6923
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₆₁×#443	C₄
socle	C₂×C₆₁	C₂×C₈×D₄
prime radical	#135×D₆₁	C₄
cosocle	C₂×#92418	C₂⁴
hypercenter	#443	#6923
hypocenter	C₆₁	C₂×#2171
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
1. D₆₁⋊#504	C₂	semidirect
2. #21462211	C₂	semidirect
3. #21462210	C₂	semidirect
4. #21462209	C₂	semidirect
5. D₆₁⋊#364	C₂	semidirect
6. D₆₁⋊#350	C₂	semidirect
7. #21462020	C₂	semidirect
10. #16840821	C₂²	semidirect
11. #16840820	C₂²	semidirect
17. #16840685	C₂²	semidirect
18. #16840684	C₂²	semidirect
19. #16840661	C₂²	semidirect
20. #16840660	C₂²	semidirect
21. #16840651	C₂²	semidirect
22. #16840650	C₂²	semidirect
197. D₆₁	#2171	semidirect
223. C₆₁	C₂×#2171	semidirect

Maximal subgroups

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

subgroup	index
D₆₁⋊#504	2
#21462211	2
#21462210	2
#21462209	2
D₆₁⋊#364	2
D₆₁⋊#350	2
#21462020	2
#443×D₆₁	2
C₂×#2171	61

Subgroups — D61⋊#2171 #30624266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁⋊#2171 #30624266