Subgroups — D₆₁⋊#2170 #30624262

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 384 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	#21462208	C₂	1	2	yes	maximal, complemented
3	#21462207	C₂	1	4	yes	maximal, complemented
4	#21462206	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	C₄×D₆₁⋊C₁₆	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	#16840648	C₂²	1	2	yes	complemented
22	C₄×#11758187	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	1	yes	characteristic
43	D₆₁⋊C₁₆	D₄	1	1	yes	characteristic
44	D₆₁⋊C₁₆	D₄	1	2	yes	—
45	C₂×#11758187	—	2	4	no	—
46	C₂×#11758187	C₂³	1	2	yes	—
47	C₂×#11758187	C₂³	1	2	yes	—
48	#41×D₆₁	C₂×C₄	1	2	yes	—
49	#14519800	C₈	1	1	yes	characteristic
50	#14519799	C₂×C₄	1	1	yes	characteristic
51	#14519798	C₈	1	2	yes	—
52	#14519797	C₂×C₄	1	2	yes	—
53	#14519796	C₈	1	1	yes	characteristic
54	Q₁₆×D₆₁	C₈	1	1	yes	characteristic
55	SD₁₆×D₆₁	C₈	1	2	yes	—
56	D₈×D₆₁	C₈	1	1	yes	characteristic
57	#14519785	—	2	2	no	—
58	#14519785	C₂³	1	2	yes	—
59	#14519784	C₂³	1	2	yes	—
60	#14519783	—	2	2	no	—
61	#14519783	C₂×C₄	1	2	yes	—
62	M₁₆×D₆₁	—	2	2	no	—
63	M₁₆×D₆₁	C₂×C₄	1	2	yes	—
64	C₂×C₈×D₆₁	—	2	2	no	—
65	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
67	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
68	#14519713	C₂×C₄	1	1	yes	characteristic
69	#14519712	C₂×C₄	1	2	yes	—
70	#14519705	C₂×C₄	1	2	yes	—
71	#14519704	C₂×C₄	1	2	yes	—
72	#14519703	C₂×C₄	1	2	yes	—
73	C₄×#92418	C₂³	1	1	yes	characteristic
74	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
76	#135×C₆₁	C₂×C₄	1	1	yes	characteristic
77	#131×C₆₁	C₂×C₄	1	2	yes	—
78	#108×C₆₁	C₂×C₄	1	1	yes	characteristic
79	#104×C₆₁	C₂×C₄	1	2	yes	—
80	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
81	#41×C₆₁	C₂²×C₄	1	2	yes	—
82	#11758226	C₂×C₈	1	2	yes	—
83	Q₈×D₆₁	C₂×C₈	1	2	yes	—
84	#11758224	—	2	2	no	—
85	#11758224	C₂×C₈	1	2	yes	—
86	D₄×D₆₁	—	2	2	no	—
87	D₄×D₆₁	C₂×C₈	1	2	yes	—
88	#11758222	—	2	2	no	—
89	#11758222	C₂²×C₄	1	2	yes	—
90	C₂×C₄×D₆₁	—	2	1	no	—
91	C₂×C₄×D₆₁	—	2	2	no	—
92	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
93	Q₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
94	SD₁₆×C₆₁	C₂×C₈	1	2	yes	—
95	D₈×C₆₁	C₂×C₈	1	1	yes	characteristic
96	M₁₆×C₆₁	—	2	2	no	—
97	M₁₆×C₆₁	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₆₁	—	2	2	no	—
99	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
102	#11758202	C₂×C₈	1	2	yes	—
103	#11758201	C₂×C₈	1	2	yes	—
104	#11758200	C₂×C₈	1	2	yes	—
105	#11758199	C₂×C₈	1	2	yes	—
106	C₄×Dic₆₁	—	2	1	no	—
107	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
108	#11758194	—	2	2	no	—
109	#11758194	C₂²×C₄	1	2	yes	—
110	C₂×#92418	—	2	2	no	—
111	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
112	C₂×#92418	C₂⁴	1	1	yes	characteristic
113	Dic₂₄₄	C₂×C₈	1	1	yes	characteristic
114	D₄₈₈	C₂×C₈	1	1	yes	characteristic
115	#11758190	C₂×C₈	1	2	yes	—
116	#11758189	—	2	2	no	—
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	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×C₈	1	1	yes	characteristic
126	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
127	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
128	C₈×D₆₁	#41	1	1	yes	characteristic
129	C₈×D₆₁	#41	1	1	yes	characteristic
130	#11758187	—	2	4	no	—
131	#11758187	—	2	2	no	—
132	#11758187	C₂×D₄	1	2	yes	—
133	#11758187	C₂×D₄	1	2	yes	—
134	C₂²×D₆₁	—	2	2	no	—
135	C₂²×D₆₁	#131	1	1	yes	characteristic
136	Q₈×C₆₁	C₂²×C₈	1	2	yes	—
137	D₄×C₆₁	—	2	2	no	—
138	D₄×C₆₁	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₆₁	—	2	1	no	—
140	C₂×C₄×C₆₁	—	2	2	no	—
141	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
142	C₂²⋊D₆₁	—	2	2	no	—
143	C₂²⋊D₆₁	—	2	2	no	—
144	C₂²⋊D₆₁	—	2	2	no	—
145	C₂×Dic₆₁	—	2	1	no	—
146	C₂×Dic₆₁	—	2	2	no	—
147	C₂×Dic₆₁	#131	1	1	yes	characteristic
148	D₂₄₄	—	2	2	no	—
149	D₂₄₄	C₂²×C₈	1	2	yes	—
150	C₄×D₆₁	—	2	4	no	—
151	C₄×D₆₁	—	2	2	no	—
152	C₄×D₆₁	—	2	2	no	—
153	C₄×D₆₁	—	2	2	no	—
154	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
155	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
156	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
157	C₄×D₆₁	C₂²×C₈	1	1	yes	characteristic
158	Dic₁₂₂	—	2	2	no	—
159	Dic₁₂₂	C₂²×C₈	1	2	yes	—
160	C₈×C₆₁	—	2	2	no	—
161	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
162	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
163	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
164	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
165	#92418	—	2	2	no	—
166	#92418	C₂²×C₈	1	1	yes	characteristic
167	#92418	C₂²×C₈	1	1	yes	characteristic
168	#92418	C₂²×D₄	1	1	yes	characteristic
169	#92418	C₂×#41	1	1	yes	characteristic
170	C₂×#2170	—	61	1	no	maximal, Sylow
171	C₂²×C₆₁	—	2	2	no	—
172	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
173	C₂×D₆₁	—	2	2	no	—
174	C₂×D₆₁	—	2	2	no	—
175	C₂×D₆₁	—	4	4	no	—
176	C₂×D₆₁	C₈×D₄	1	1	yes	characteristic
177	C₂×D₆₁	C₈×D₄	1	1	yes	characteristic
178	C₄×C₆₁	—	2	2	no	—
179	C₄×C₆₁	—	2	2	no	—
180	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
182	Dic₆₁	—	2	2	no	—
183	Dic₆₁	—	2	2	no	—
184	Dic₆₁	C₈×D₄	1	1	yes	characteristic
185	Dic₆₁	C₈×D₄	1	1	yes	characteristic
186	C₂×#504	—	61	2	no	—
187	C₂×#443	—	61	1	no	—
188	#2170	—	61	2	no	—
189	#2170	—	61	4	no	—
190	#2170	—	61	2	no	—
191	C₂×#364	—	61	1	no	—
192	C₂×#350	—	61	2	no	—
193	C₂×C₄×C₁₆	—	61	1	no	—
194	C₂×C₆₁	—	2	1	no	—
195	C₂×C₆₁	—	4	2	no	—
196	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
197	D₆₁	—	2	1	no	—
198	D₆₁	—	4	2	no	—
199	D₆₁	#2170	1	2	yes	complemented
200	C₂×#135	—	61	1	no	—
201	C₂×#131	—	61	2	no	—
202	#504	—	61	4	no	—
203	#504	—	61	4	no	—
204	#504	—	122	4	no	—
205	C₂×M₃₂	—	61	2	no	—
206	C₂×M₃₂	—	122	2	no	—
207	C₂²×C₁₆	—	61	1	no	—
208	C₂²×C₁₆	—	61	1	no	—
209	C₂²×C₁₆	—	122	2	no	—
210	#443	—	61	1	no	—
211	#443	—	61	2	no	—
212	#443	—	61	1	no	—
213	#443	—	61	2	no	—
214	#443	—	61	1	no	—
215	#443	#6923	1	1	yes	characteristic
216	C₂×#108	—	61	1	no	—
217	C₂×#104	—	61	2	no	—
218	C₂×C₄×C₈	—	61	1	no	—
219	#364	—	61	2	no	—
220	#364	—	61	2	no	—
221	#350	—	61	4	no	—
222	#350	—	61	4	no	—
223	C₄×C₁₆	—	61	2	no	—
224	C₄×C₁₆	—	61	2	no	—
225	C₆₁	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
226	C₂×#41	—	61	2	no	—
227	#135	—	61	2	no	—
228	#135	—	61	1	no	—
229	#135	—	61	2	no	—
230	#135	—	61	1	no	—
231	#135	—	61	1	no	—
232	#135	C₂×#6923	1	1	yes	characteristic
233	C₂×Q₁₆	—	61	1	no	—
234	C₂×SD₁₆	—	61	2	no	—
235	C₂×D₈	—	61	1	no	—
236	#131	—	61	2	no	—
237	#131	—	61	2	no	—
238	#131	—	61	2	no	—
239	#131	—	122	2	no	—
240	#131	—	122	2	no	—
241	#131	C₂×#6923	1	2	yes	—
242	C₂×M₁₆	—	61	2	no	—
243	C₂×M₁₆	—	122	2	no	—
244	C₂²×C₈	—	61	1	no	—
245	C₂²×C₈	—	61	1	no	—
246	C₂²×C₈	—	122	2	no	—
247	C₂×C₄²	—	61	1	no	—
248	M₃₂	—	61	4	no	—
249	M₃₂	—	122	4	no	—
250	M₃₂	—	122	4	no	—
251	M₃₂	—	122	2	no	—
252	C₂×C₁₆	—	61	2	no	—
253	C₂×C₁₆	—	61	1	no	—
254	C₂×C₁₆	—	61	1	no	—
255	C₂×C₁₆	—	61	2	no	—
256	C₂×C₁₆	—	61	2	no	—
257	C₂×C₁₆	—	122	4	no	—
258	C₂×C₁₆	—	122	4	no	—
259	C₂×C₁₆	—	122	1	no	—
260	C₂×C₁₆	—	122	1	no	—
261	C₂×C₁₆	—	122	2	no	—
262	#108	—	61	2	no	—
263	#108	—	61	1	no	—
264	#108	C₂×#6923	1	1	yes	characteristic
265	#104	—	61	2	no	—
266	#104	—	61	2	no	—
267	#104	—	61	2	no	—
268	#104	C₂×#6923	1	2	yes	—
269	C₄×C₈	—	61	1	no	—
270	C₄×C₈	—	61	1	no	—
271	C₄×C₈	—	61	1	no	—
272	C₄×C₈	C₂×#6923	1	1	yes	characteristic
273	#41	—	61	2	no	—
274	#41	—	61	2	no	—
275	#41	—	61	2	no	—
276	#41	—	122	2	no	—
277	#41	—	122	2	no	—
278	#41	C₂²×#6923	1	2	yes	—
279	C₂×Q₈	—	61	2	no	—
280	C₂×D₄	—	61	2	no	—
281	C₂×D₄	—	122	2	no	—
282	C₂²×C₄	—	61	1	no	—
283	C₂²×C₄	—	122	2	no	—
284	C₂²×C₄	—	122	1	no	—
285	Q₁₆	—	61	2	no	—
286	Q₁₆	—	61	1	no	—
287	Q₁₆	D₆₁⋊C₈	1	1	yes	characteristic
288	SD₁₆	—	61	2	no	—
289	SD₁₆	—	61	2	no	—
290	SD₁₆	—	61	2	no	—
291	SD₁₆	D₆₁⋊C₈	1	2	yes	—
292	D₈	—	61	2	no	—
293	D₈	—	61	1	no	—
294	D₈	D₆₁⋊C₈	1	1	yes	characteristic
295	M₁₆	—	2	2	no	—
296	M₁₆	—	61	2	no	—
297	M₁₆	—	122	2	no	—
298	M₁₆	—	122	2	no	—
299	M₁₆	—	122	2	no	—
300	M₁₆	—	122	2	no	—
301	M₁₆	C₂²×#6923	1	2	yes	—
302	C₂×C₈	—	2	2	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	—	61	1	no	—
311	C₂×C₈	—	61	1	no	—
312	C₂×C₈	—	61	1	no	—
313	C₂×C₈	—	122	2	no	—
314	C₂×C₈	—	122	2	no	—
315	C₂×C₈	—	122	2	no	—
316	C₂×C₈	—	122	2	no	—
317	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
318	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
319	C₄²	—	61	1	no	—
320	C₄²	—	122	1	no	—
321	C₄²	C₂²×#6923	1	1	yes	characteristic
322	C₁₆	—	61	2	no	—
323	C₁₆	—	61	2	no	—
324	C₁₆	—	122	2	no	—
325	C₁₆	—	122	4	no	—
326	C₂³	—	61	1	no	—
327	C₂³	—	122	2	no	—
328	Q₈	—	61	2	no	—
329	Q₈	—	122	2	no	—
330	Q₈	C₂×D₆₁⋊C₈	1	2	yes	—
331	D₄	—	2	2	no	—
332	D₄	—	61	2	no	—
333	D₄	—	122	2	no	—
334	D₄	—	122	2	no	—
335	D₄	—	122	2	no	—
336	D₄	—	122	2	no	—
337	D₄	C₂×D₆₁⋊C₈	1	2	yes	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	61	1	no	—
341	C₂×C₄	—	61	1	no	—
342	C₂×C₄	—	61	1	no	—
343	C₂×C₄	—	61	1	no	—
344	C₂×C₄	—	61	1	no	—
345	C₂×C₄	—	122	2	no	—
346	C₂×C₄	—	122	2	no	—
347	C₂×C₄	—	122	4	no	—
348	C₂×C₄	—	122	2	no	—
349	C₂×C₄	—	122	1	no	—
350	C₂×C₄	—	122	2	no	—
351	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
352	C₈	—	2	2	no	—
353	C₈	—	61	1	no	—
354	C₈	—	61	1	no	—
355	C₈	—	61	1	no	—
356	C₈	—	61	1	no	—
357	C₈	—	122	2	no	—
358	C₈	D₄×#6923	1	1	yes	characteristic
359	C₈	D₄×#6923	1	1	yes	characteristic
360	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
361	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
362	C₂²	—	2	2	no	—
363	C₂²	—	61	1	no	—
364	C₂²	—	61	1	no	—
365	C₂²	—	122	2	no	—
366	C₂²	—	122	2	no	—
367	C₂²	—	244	4	no	—
368	C₂²	D₆₁⋊#131	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	2	2	no	—
371	C₄	—	61	1	no	—
372	C₄	—	61	1	no	—
373	C₄	—	122	2	no	—
374	C₄	—	122	2	no	—
375	C₄	C₂×D₄×#6923	1	1	yes	characteristic
376	C₄	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
377	C₂	—	2	1	no	—
378	C₂	—	4	2	no	—
379	C₂	—	61	2	no	—
380	C₂	—	122	1	no	—
381	C₂	—	244	2	no	—
382	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×#2170
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. #21462208	C₂	semidirect
3. #21462207	C₂	semidirect
4. #21462206	C₂	semidirect
5. D₆₁⋊#364	C₂	semidirect
6. D₆₁⋊#350	C₂	semidirect
7. C₄×D₆₁⋊C₁₆	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. #16840648	C₂²	semidirect
22. C₄×#11758187	C₂²	semidirect
199. D₆₁	#2170	semidirect
225. C₆₁	C₂×#2170	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
#21462208	2
#21462207	2
#21462206	2
D₆₁⋊#364	2
D₆₁⋊#350	2
C₄×D₆₁⋊C₁₆	2
#443×D₆₁	2
C₂×#2170	61

Subgroups — D61⋊#2170 #30624262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₆₁⋊#2170 #30624262