Subgroups — #29833124

Order 15616.

590 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	#131×Dic₆₁	C₂	1	1	yes	maximal, characteristic
2	#21447623	C₂	1	2	yes	maximal, complemented
3	#21447623	C₂	1	2	yes	maximal, complemented
4	#21447619	C₂	1	2	yes	maximal, complemented
5	#21447615	C₂	1	2	yes	maximal, complemented
6	#21447574	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16839580	C₂	1	1	yes	maximal, characteristic, complemented
8	#21447433	C₂	1	1	yes	maximal, characteristic
9	#21447418	C₂	1	1	yes	maximal, characteristic, complemented
10	#21447415	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2113	C₂	1	1	yes	maximal, characteristic
12	C₂×#14519784	C₄	1	1	yes	characteristic
13	#41×Dic₆₁	C₄	1	1	yes	characteristic
14	C₂×#14519717	C₂²	1	1	yes	characteristic
15	C₂²×#11758185	C₂²	1	1	yes	characteristic
16	#16839720	C₄	1	1	yes	characteristic
17	#16839720	C₄	1	2	yes	—
18	#16839717	C₂²	1	2	yes	—
19	#16839717	C₂²	1	1	yes	characteristic
20	M₁₆×Dic₆₁	C₄	1	2	yes	—
21	M₁₆×Dic₆₁	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
24	#16839580	C₂²	1	1	yes	characteristic
25	#16839580	C₂²	1	1	yes	characteristic
26	#16839580	C₂²	1	1	yes	characteristic
27	#16839580	C₂²	1	1	yes	characteristic
28	#16839580	C₂²	1	2	yes	—
29	#16839580	C₂²	1	2	yes	—
30	C₂²⋊#11758185	C₂²	1	2	yes	—
31	C₂²⋊#11758185	C₂²	1	2	yes	—
32	#16839537	C₂²	1	2	yes	—
33	#16839537	C₂²	1	2	yes	—
34	C₁₆×Dic₆₁	C₂²	1	2	yes	—
35	C₁₆×Dic₆₁	C₂²	1	2	yes	—
36	C₂×#131×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
38	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
39	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₆₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
41	#14519784	—	2	4	no	—
42	C₂×#11758194	C₂×C₄	1	2	yes	—
43	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
44	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
45	C₂²×#92418	C₂×C₄	1	2	yes	—
46	Q₈×Dic₆₁	C₈	1	1	yes	characteristic
47	Q₈×Dic₆₁	C₈	1	1	yes	characteristic
48	D₄×Dic₆₁	C₈	1	2	yes	—
49	D₄×Dic₆₁	C₈	1	1	yes	characteristic
50	D₄×Dic₆₁	C₈	1	2	yes	—
51	D₄×Dic₆₁	C₈	1	1	yes	characteristic
52	#14519736	C₂×C₄	1	2	yes	—
53	#14519736	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
55	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
56	#14519717	—	2	2	no	—
57	C₂×#11758185	—	2	2	no	—
58	C₂×#11758185	C₂³	1	1	yes	characteristic
59	C₂×#11758185	C₂³	1	2	yes	—
60	C₂×#11758185	C₂³	1	1	yes	characteristic
61	#14519687	C₂×C₄	1	2	yes	—
62	#14519687	C₂×C₄	1	2	yes	—
63	#14519687	C₂×C₄	1	2	yes	—
64	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
66	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
67	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
69	C₈×Dic₆₁	C₂³	1	2	yes	—
70	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
71	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
72	#131×C₆₁	—	2	2	no	—
73	#131×C₆₁	—	2	2	no	—
74	C₂×M₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
75	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
76	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
77	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	M₃₂×C₆₁	—	2	2	no	—
79	C₂×C₁₆×C₆₁	—	2	2	no	—
80	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
82	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
83	#41×C₆₁	—	2	2	no	—
84	#41×C₆₁	—	2	2	no	—
85	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
86	C₂×D₄×C₆₁	C₂×C₈	1	2	yes	—
87	C₂×D₄×C₆₁	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
91	C₂²×Dic₆₁	C₂×C₈	1	1	yes	characteristic
92	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
93	C₂²×Dic₆₁	C₂×C₈	1	1	yes	characteristic
94	M₁₆×C₆₁	—	2	2	no	—
95	M₁₆×C₆₁	—	2	4	no	—
96	C₂×C₈×C₆₁	—	2	2	no	—
97	C₂×C₈×C₆₁	—	2	2	no	—
98	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₁	#41	1	2	yes	—
101	C₂×C₈×C₆₁	#41	1	2	yes	—
102	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
103	C₂²⋊Dic₆₁	C₂×C₈	1	2	yes	—
104	C₂²⋊Dic₆₁	C₂×C₈	1	2	yes	—
105	C₂²⋊Dic₆₁	C₂×C₈	1	2	yes	—
106	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
107	C₄⋊Dic₆₁	C₂×C₈	1	2	yes	—
108	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
109	C₄⋊Dic₆₁	C₂×C₈	1	1	yes	characteristic
110	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
111	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
112	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
113	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
114	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
116	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
117	#11758194	—	2	4	no	—
118	#11758194	M₁₆	1	4	yes	—
119	C₂×#92418	—	2	2	no	—
120	C₂×#92418	—	2	4	no	—
121	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
122	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
123	C₂×#92418	C₂²×C₄	1	2	yes	—
124	C₁₆×C₆₁	—	2	1	no	—
125	C₁₆×C₆₁	—	2	1	no	—
126	C₁₆×C₆₁	—	2	2	no	—
127	#11758185	—	2	1	no	—
128	#11758185	—	2	1	no	—
129	#11758185	—	2	2	no	—
130	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
131	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
132	Q₈×C₆₁	—	2	2	no	—
133	D₄×C₆₁	—	2	4	no	—
134	D₄×C₆₁	—	2	2	no	—
135	C₂×C₄×C₆₁	—	2	2	no	—
136	C₂×C₄×C₆₁	—	2	2	no	—
137	C₂×C₄×C₆₁	#117	1	2	yes	—
138	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
141	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
142	C₂×Dic₆₁	—	2	2	no	—
143	C₂×Dic₆₁	—	2	4	no	—
144	C₂×Dic₆₁	—	4	1	no	—
145	C₂×Dic₆₁	—	4	2	no	—
146	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
147	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
148	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
149	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
150	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
151	C₂×Dic₆₁	C₂²×C₈	1	1	yes	characteristic
152	C₈×C₆₁	—	2	2	no	—
153	C₈×C₆₁	#117	1	2	yes	—
154	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
155	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
156	#92418	—	2	1	no	—
157	#92418	—	2	1	no	—
158	#92418	C₂×M₁₆	1	4	yes	—
159	#33423	—	61	1	no	maximal, Sylow
160	C₂²×C₆₁	—	2	1	no	—
161	C₂²×C₆₁	—	2	1	no	—
162	C₂²×C₆₁	—	2	2	no	—
163	C₂²×C₆₁	—	2	2	no	—
164	C₂²×C₆₁	—	2	2	no	—
165	C₂²×C₆₁	#431	1	2	yes	—
166	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
167	C₄×C₆₁	—	2	2	no	—
168	C₄×C₆₁	#431	1	2	yes	—
169	C₄×C₆₁	C₂×#117	1	1	yes	characteristic
170	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
171	Dic₆₁	—	2	1	no	—
172	Dic₆₁	—	2	2	no	—
173	Dic₆₁	—	2	1	no	—
174	Dic₆₁	—	2	1	no	—
175	Dic₆₁	—	2	2	no	—
176	Dic₆₁	#504	1	2	yes	—
177	C₄×#131	—	61	1	no	—
178	#2164	—	61	2	no	—
179	#2164	—	61	2	no	—
180	#2163	—	61	2	no	—
181	#2162	—	61	2	no	—
182	#2150	—	61	1	no	—
183	C₂×C₄⋊C₁₆	—	61	1	no	—
184	#2113	—	61	1	no	—
185	#2113	D₆₁	1	1	yes	characteristic
186	#2108	—	61	1	no	—
187	#2108	—	61	1	no	—
188	C₂×C₆₁	—	2	2	no	—
189	C₂×C₆₁	—	4	2	no	—
190	C₂×C₆₁	C₂×#431	1	1	yes	characteristic
191	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
192	C₂×C₆₁	#3407	1	1	yes	characteristic
193	C₂×#131	—	61	1	no	—
194	C₂×#131	C₂×D₆₁	1	1	yes	characteristic
195	C₄×#41	—	61	1	no	—
196	C₂×M₃₂	—	61	1	no	—
197	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
198	C₂²×C₁₆	—	61	1	no	—
199	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
200	#405	—	61	1	no	—
201	#405	—	61	2	no	—
202	#405	—	61	1	no	—
203	#405	—	61	2	no	—
204	C₄×M₁₆	—	61	1	no	—
205	C₄×M₁₆	—	61	2	no	—
206	C₂×C₄×C₈	—	61	1	no	—
207	C₂×C₄×C₈	—	61	2	no	—
208	C₄⋊C₁₆	—	61	1	no	—
209	C₄⋊C₁₆	—	61	1	no	—
210	C₄⋊C₁₆	—	61	1	no	—
211	C₄⋊C₁₆	—	61	1	no	—
212	C₄⋊C₁₆	—	61	2	no	—
213	C₄⋊C₁₆	—	61	2	no	—
214	C₂²⋊C₁₆	—	61	2	no	—
215	C₂²⋊C₁₆	—	61	2	no	—
216	C₂²⋊C₁₆	C₂×D₆₁	1	2	yes	—
217	C₂²⋊C₁₆	C₂×D₆₁	1	2	yes	—
218	#346	—	61	2	no	—
219	#346	—	61	2	no	—
220	C₄×C₁₆	—	61	2	no	—
221	C₄×C₁₆	—	61	2	no	—
222	C₆₁	#33423	1	1	yes	characteristic, complemented, Sylow
223	C₂×#41	C₄×D₆₁	1	1	yes	characteristic
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	122	4	no	—
227	C₂×M₁₆	—	61	1	no	—
228	C₂×M₁₆	—	61	2	no	—
229	C₂×M₁₆	C₄×D₆₁	1	1	yes	characteristic
230	C₂×M₁₆	C₄×D₆₁	1	2	yes	—
231	C₂²×C₈	—	61	2	no	—
232	C₂²×C₈	—	61	1	no	—
233	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
234	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
235	C₄×Q₈	—	61	1	no	—
236	C₄×Q₈	—	61	1	no	—
237	C₄×D₄	—	61	2	no	—
238	C₄×D₄	—	61	1	no	—
239	C₄×D₄	—	61	2	no	—
240	C₄×D₄	—	61	1	no	—
241	#117	—	61	2	no	—
242	#117	—	61	1	no	—
243	C₂×C₄²	—	61	2	no	—
244	C₂×C₄²	—	61	1	no	—
245	M₃₂	—	2	2	no	—
246	M₃₂	—	122	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	—	61	1	no	—
249	C₂×C₁₆	—	61	2	no	—
250	C₂×C₁₆	—	61	1	no	—
251	C₂×C₁₆	—	122	2	no	—
252	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
253	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
254	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
255	#97	—	61	2	no	—
256	#97	—	61	2	no	—
257	#97	—	61	2	no	—
258	C₄×C₈	—	61	2	no	—
259	C₄×C₈	—	61	1	no	—
260	C₄×C₈	—	61	2	no	—
261	C₄×C₈	—	61	1	no	—
262	C₄×C₈	—	61	2	no	—
263	C₄×C₈	—	61	1	no	—
264	C₄×C₈	—	61	1	no	—
265	#41	—	2	2	no	—
266	#41	—	2	2	no	—
267	C₂×Q₈	C₈×D₆₁	1	1	yes	characteristic
268	C₂×D₄	C₈×D₆₁	1	2	yes	—
269	C₂×D₄	C₈×D₆₁	1	1	yes	characteristic
270	C₂²×C₄	—	61	2	no	—
271	C₂²×C₄	—	61	1	no	—
272	C₂²×C₄	—	61	2	no	—
273	C₂²×C₄	—	61	1	no	—
274	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
275	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
276	M₁₆	—	2	2	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	61	4	no	—
279	M₁₆	—	122	4	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	61	2	no	—
283	C₂×C₈	—	61	1	no	—
284	C₂×C₈	—	61	1	no	—
285	C₂×C₈	—	122	4	no	—
286	C₂×C₈	—	122	2	no	—
287	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
288	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
289	C₂×C₈	#11758224	1	2	yes	—
290	C₂×C₈	#11758224	1	2	yes	—
291	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
292	C₄⋊C₄	—	61	2	no	—
293	C₄⋊C₄	—	61	2	no	—
294	C₄⋊C₄	—	61	1	no	—
295	C₄⋊C₄	—	61	1	no	—
296	C₂²⋊C₄	—	61	2	no	—
297	C₂²⋊C₄	—	61	2	no	—
298	C₂²⋊C₄	—	61	2	no	—
299	C₄²	—	61	1	no	—
300	C₄²	—	61	2	no	—
301	C₄²	—	61	1	no	—
302	C₄²	—	61	2	no	—
303	C₄²	—	61	1	no	—
304	C₄²	—	61	2	no	—
305	C₄²	—	61	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	122	1	no	—
310	C₁₆	—	122	1	no	—
311	C₁₆	—	122	2	no	—
312	C₂³	C₂×C₈×D₆₁	1	2	yes	—
313	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
314	Q₈	—	2	2	no	—
315	D₄	—	2	4	no	—
316	D₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	61	1	no	—
320	C₂×C₄	—	61	1	no	—
321	C₂×C₄	—	61	2	no	—
322	C₂×C₄	—	61	1	no	—
323	C₂×C₄	—	61	2	no	—
324	C₂×C₄	—	61	1	no	—
325	C₂×C₄	—	122	2	no	—
326	C₂×C₄	—	122	4	no	—
327	C₂×C₄	—	244	1	no	—
328	C₂×C₄	—	244	2	no	—
329	C₂×C₄	#14519733	1	2	yes	—
330	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
331	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
332	C₂×C₄	M₁₆×D₆₁	1	2	yes	—
333	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	61	4	no	—
336	C₈	—	122	1	no	—
337	C₈	—	122	1	no	—
338	C₈	#14519733	1	2	yes	—
339	C₈	C₂×#11758224	1	1	yes	characteristic
340	C₈	C₂×#11758224	1	1	yes	characteristic
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	#16839812	1	2	yes	—
347	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	61	2	no	—
350	C₄	—	122	1	no	—
351	C₄	—	122	2	no	—
352	C₄	—	122	1	no	—
353	C₄	—	122	2	no	—
354	C₄	—	122	1	no	—
355	C₄	#16839812	1	2	yes	—
356	C₄	C₂×#14519733	1	1	yes	characteristic
357	C₄	C₂×M₁₆×D₆₁	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	4	2	no	—
360	C₂	C₂×#16839812	1	1	yes	characteristic
361	C₂	#504×D₆₁	1	1	yes	characteristic
362	C₂	#21458920	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₈
Fitting	C₆₁×#2113	C₂
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2113	D₆₁
hypocenter	C₆₁	#33423
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
2. #21447623	C₂	semidirect
3. #21447623	C₂	semidirect
4. #21447619	C₂	semidirect
5. #21447615	C₂	semidirect
6. #21447574	C₂	semidirect
7. C₂×#16839580	C₂	semidirect
9. #21447418	C₂	semidirect
10. #21447415	C₂	semidirect
222. C₆₁	#33423	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₆₁	2
#21447623	2
#21447619	2
#21447615	2
#21447574	2
C₂×#16839580	2
#21447433	2
#21447418	2
#21447415	2
C₆₁×#2113	2
#33423	61