Subgroups — #29594930

Order 15616.

586 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839797	C₂	1	1	yes	maximal, characteristic
3	#427×D₆₁	C₂	1	1	yes	maximal, characteristic
4	#21443926	C₂	1	1	yes	maximal, characteristic, complemented
5	#21443925	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443924	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1386	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
9	C₂×#14519735	—	2	1	no	—
10	#117×D₆₁	—	2	1	no	—
11	C₂×#14519717	C₂²	1	1	yes	characteristic
12	C₂×#14519716	—	2	1	no	—
13	M₃₂×D₆₁	—	2	2	no	—
14	M₃₂×D₆₁	C₄	1	4	yes	complemented
15	C₂×C₁₆×D₆₁	—	2	1	no	—
16	#16839800	C₂²	1	1	yes	characteristic
17	#16839799	C₄	1	2	yes	complemented
18	#16839798	—	2	1	no	—
19	#16839797	—	2	1	no	—
20	#16839797	—	2	2	no	—
21	#16839797	C₄	1	4	yes	complemented
22	C₂×#14519710	—	2	1	no	—
23	#107×D₆₁	C₄	1	2	yes	complemented
24	C₂×#14519709	C₂²	1	1	yes	characteristic
25	#106×D₆₁	—	2	1	no	—
26	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#427	C₂²	1	1	yes	characteristic
28	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₆₁	D₄	1	1	yes	characteristic
30	C₂²×#92418	D₄	1	1	yes	characteristic
31	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
36	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
37	C₂×C₈×D₆₁	D₄	1	2	yes	—
38	C₂×C₈×D₆₁	Q₈	1	2	yes	—
39	#14519736	—	2	1	no	—
40	#14519735	—	2	2	no	—
41	#14519735	—	2	4	no	—
42	#14519735	—	4	2	no	—
43	#14519734	—	4	1	no	—
44	#14519733	—	4	1	no	—
45	C₂×C₄⋊Dic₆₁	—	2	1	no	—
46	C₄⋊C₄×D₆₁	—	2	2	no	—
47	C₂×#11758198	—	4	1	no	—
48	C₂²⋊C₄×D₆₁	—	4	1	no	—
49	C₂×C₄×Dic₆₁	—	4	1	no	—
50	C₄²×D₆₁	—	4	1	no	—
51	#14519717	—	2	1	no	—
52	#14519717	C₂×C₄	1	2	yes	complemented
53	#14519716	—	2	4	no	—
54	C₂×#11758185	—	2	1	no	—
55	C₁₆×D₆₁	—	2	4	no	—
56	#14519711	C₂×C₄	1	2	yes	—
57	#14519710	—	2	2	no	—
58	#14519710	—	2	4	no	—
59	#14519709	—	2	1	no	—
60	#14519709	—	2	1	no	—
61	#14519709	C₂×C₄	1	2	yes	complemented
62	#14519708	—	2	2	no	—
63	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
64	#117×C₆₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
66	M₃₂×C₆₁	—	2	1	no	—
67	M₃₂×C₆₁	C₂×C₄	1	2	yes	complemented
68	C₂×C₁₆×C₆₁	—	2	1	no	—
69	#107×C₆₁	C₂×C₄	1	2	yes	—
70	#106×C₆₁	—	2	1	no	—
71	#106×C₆₁	—	2	1	no	—
72	#106×C₆₁	C₂×C₄	1	2	yes	complemented
73	C₂³×D₆₁	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₁	—	4	1	no	—
75	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×Dic₆₁	—	4	1	no	—
77	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
78	C₂×C₄×D₆₁	—	4	2	no	—
79	C₂×C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×D₆₁	C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×D₆₁	C₄⋊C₄	1	2	yes	—
83	C₂×C₄×D₆₁	SD₁₆	1	1	yes	characteristic
84	C₂×C₄×D₆₁	SD₁₆	1	1	yes	characteristic
85	C₂×C₄×D₆₁	SD₁₆	1	2	yes	—
86	C₂×C₄×D₆₁	Q₁₆	1	2	yes	—
87	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
93	C₄⋊C₄×C₆₁	—	2	2	no	—
94	C₄⋊C₄×C₆₁	—	2	2	no	—
95	C₄⋊C₄×C₆₁	—	4	1	no	—
96	C₂²⋊C₄×C₆₁	—	4	1	no	—
97	C₄²×C₆₁	—	4	1	no	—
98	C₂²⋊Dic₆₁	—	4	1	no	—
99	#11758198	—	4	2	no	—
100	#11758198	—	4	4	no	—
101	C₄⋊Dic₆₁	—	2	2	no	—
102	C₄⋊Dic₆₁	—	2	2	no	—
103	C₄⋊Dic₆₁	—	4	1	no	—
104	#11758196	—	4	2	no	—
105	C₄×Dic₆₁	—	4	1	no	—
106	C₄×Dic₆₁	—	4	2	no	—
107	C₄×Dic₆₁	—	4	4	no	—
108	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
110	C₂×#92418	C₂²⋊C₄	1	2	yes	—
111	C₂×#92418	C₄⋊C₄	1	2	yes	—
112	C₈×D₆₁	C₄²	1	2	yes	—
113	C₈×D₆₁	C₄²	1	2	yes	—
114	C₈×D₆₁	C₂²⋊C₄	1	2	yes	—
115	C₈×D₆₁	C₂²⋊C₄	1	2	yes	—
116	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
117	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
118	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
119	C₈×D₆₁	C₄⋊C₄	1	2	yes	—
120	C₁₆×C₆₁	—	2	2	no	—
121	#11758185	—	2	2	no	—
122	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₆₁	—	4	2	no	—
124	C₂²×D₆₁	#102	1	1	yes	characteristic
125	C₂²×D₆₁	#102	1	1	yes	characteristic
126	C₂²×D₆₁	#102	1	2	yes	—
127	C₂²×D₆₁	#107	1	2	yes	—
128	C₂×C₄×C₆₁	—	4	2	no	—
129	C₂×C₄×C₆₁	—	4	2	no	—
130	C₂×C₄×C₆₁	—	8	2	no	—
131	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×Dic₆₁	—	4	2	no	—
138	C₂×Dic₆₁	—	4	2	no	—
139	C₂×Dic₆₁	—	8	2	no	—
140	C₂×Dic₆₁	#102	1	2	yes	—
141	C₂×Dic₆₁	#102	1	1	yes	characteristic
142	C₂×Dic₆₁	#102	1	1	yes	characteristic
143	C₂×Dic₆₁	#107	1	2	yes	—
144	C₄×D₆₁	—	8	4	no	—
145	C₄×D₆₁	#95	1	2	yes	—
146	C₄×D₆₁	#95	1	2	yes	—
147	C₄×D₆₁	#103	1	2	yes	—
148	C₄×D₆₁	#103	1	2	yes	—
149	C₄×D₆₁	#103	1	2	yes	—
150	C₄×D₆₁	#103	1	2	yes	—
151	C₄×D₆₁	#106	1	2	yes	—
152	C₄×D₆₁	#106	1	2	yes	—
153	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
154	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
157	#92418	#95	1	2	yes	—
158	#92418	#95	1	2	yes	—
159	#12567	—	61	1	no	maximal, Sylow
160	C₂²×C₆₁	—	2	2	no	—
161	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
162	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
163	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
164	C₂×D₆₁	—	2	2	no	—
165	C₂×D₆₁	—	2	2	no	—
166	C₂×D₆₁	—	4	2	no	—
167	C₂×D₆₁	—	4	2	no	—
168	C₂×D₆₁	#340	1	2	yes	—
169	C₂×D₆₁	#340	1	2	yes	—
170	C₄×C₆₁	—	8	2	no	—
171	C₄×C₆₁	—	8	2	no	—
172	C₄×C₆₁	C₂×#95	1	1	yes	characteristic
173	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
174	C₄×C₆₁	C₂×#103	1	1	yes	characteristic
175	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
176	Dic₆₁	—	8	2	no	—
177	Dic₆₁	—	8	2	no	—
178	Dic₆₁	#340	1	2	yes	—
179	Dic₆₁	#340	1	2	yes	—
180	C₂²×M₃₂	—	61	1	no	—
181	C₂×#427	—	61	1	no	—
182	C₂×#427	—	61	1	no	—
183	#1386	—	61	1	no	—
184	#1386	—	61	1	no	—
185	#1386	—	61	1	no	—
186	#1386	D₆₁	1	1	yes	characteristic, complemented
187	C₂×C₆₁	—	2	1	no	—
188	C₂×C₆₁	—	2	1	no	—
189	C₂×C₆₁	C₂×#340	1	1	yes	characteristic
190	C₂×C₆₁	#2145	1	2	yes	—
191	D₆₁	—	2	2	no	—
192	D₆₁	—	2	2	no	—
193	C₂³×C₈	—	61	1	no	—
194	C₂×#117	—	122	1	no	—
195	C₂×#117	—	122	1	no	—
196	C₂×M₃₂	—	61	1	no	—
197	C₂×M₃₂	—	61	4	no	—
198	C₂×M₃₂	—	122	1	no	—
199	C₂×M₃₂	—	122	2	no	—
200	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
201	C₂²×C₁₆	—	122	1	no	—
202	#427	—	61	4	no	—
203	#427	—	61	1	no	—
204	#427	—	61	2	no	—
205	#427	—	122	1	no	—
206	#427	—	122	2	no	—
207	#427	—	122	1	no	—
208	#427	C₂×D₆₁	1	1	yes	characteristic
209	C₂×#107	—	61	2	no	—
210	C₂×#107	—	122	1	no	—
211	C₂×#106	—	61	1	no	—
212	C₂×#106	—	122	1	no	—
213	C₂×#106	C₂×D₆₁	1	1	yes	characteristic
214	C₆₁	#12567	1	1	yes	characteristic, complemented, Sylow
215	C₂³×C₄	—	61	1	no	—
216	C₂²×C₈	—	61	1	no	—
217	C₂²×C₈	—	61	1	no	—
218	C₂²×C₈	—	61	1	no	—
219	C₂²×C₈	—	61	2	no	—
220	C₂²×C₈	—	61	2	no	—
221	C₂²×C₈	—	61	1	no	—
222	C₂²×C₈	—	61	1	no	—
223	C₂²×C₈	—	61	2	no	—
224	C₂²×C₈	—	61	2	no	—
225	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
226	#117	—	2	1	no	—
227	#117	—	122	2	no	—
228	#117	—	122	4	no	—
229	#117	—	122	1	no	—
230	#117	—	244	1	no	—
231	#117	—	244	1	no	—
232	#117	—	244	2	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	122	2	no	—
235	C₂×C₄⋊C₄	—	122	1	no	—
236	C₂×C₂²⋊C₄	—	244	1	no	—
237	C₂×C₂²⋊C₄	—	244	1	no	—
238	C₂×C₄²	—	244	1	no	—
239	C₂×C₄²	—	244	1	no	—
240	M₃₂	—	2	1	no	—
241	M₃₂	—	61	2	no	—
242	M₃₂	—	122	4	no	—
243	M₃₂	—	122	1	no	—
244	M₃₂	C₄×D₆₁	1	2	yes	complemented
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	122	4	no	—
247	C₂×C₁₆	—	122	1	no	—
248	#107	—	61	2	no	—
249	#107	—	122	2	no	—
250	#107	—	122	4	no	—
251	#107	C₄×D₆₁	1	2	yes	—
252	#106	—	2	1	no	—
253	#106	—	2	1	no	—
254	#106	—	61	2	no	—
255	#106	—	122	2	no	—
256	#106	—	122	1	no	—
257	#106	—	122	1	no	—
258	#106	C₄×D₆₁	1	2	yes	complemented
259	C₂⁴	—	61	1	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	61	1	no	—
262	C₂²×C₄	—	61	1	no	—
263	C₂²×C₄	—	61	2	no	—
264	C₂²×C₄	—	61	1	no	—
265	C₂²×C₄	—	61	2	no	—
266	C₂²×C₄	—	61	1	no	—
267	C₂²×C₄	—	61	2	no	—
268	C₂²×C₄	—	61	1	no	—
269	C₂²×C₄	—	61	2	no	—
270	C₂²×C₄	—	244	2	no	—
271	C₂²×C₄	—	244	1	no	—
272	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
273	C₂×C₈	—	61	1	no	—
274	C₂×C₈	—	61	2	no	—
275	C₂×C₈	—	61	1	no	—
276	C₂×C₈	—	61	2	no	—
277	C₂×C₈	—	61	2	no	—
278	C₂×C₈	—	61	2	no	—
279	C₂×C₈	—	61	2	no	—
280	C₂×C₈	—	61	2	no	—
281	C₂×C₈	—	61	2	no	—
282	C₂×C₈	—	61	2	no	—
283	C₂×C₈	—	61	2	no	—
284	C₂×C₈	—	61	2	no	—
285	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
286	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
287	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
288	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
289	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
290	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	122	2	no	—
295	C₄⋊C₄	—	122	2	no	—
296	C₄⋊C₄	—	244	2	no	—
297	C₄⋊C₄	—	244	1	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	244	4	no	—
300	C₂²⋊C₄	—	244	2	no	—
301	C₂²⋊C₄	—	244	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	—	244	4	no	—
304	C₄²	—	244	2	no	—
305	C₄²	—	244	1	no	—
306	C₁₆	—	2	2	no	—
307	C₁₆	—	122	2	no	—
308	C₂³	—	61	1	no	—
309	C₂³	—	61	1	no	—
310	C₂³	—	61	2	no	—
311	C₂³	—	61	2	no	—
312	C₂³	—	244	2	no	—
313	C₂³	D₈×D₆₁	1	1	yes	characteristic
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	61	2	no	—
318	C₂×C₄	—	61	1	no	—
319	C₂×C₄	—	61	1	no	—
320	C₂×C₄	—	61	2	no	—
321	C₂×C₄	—	61	2	no	—
322	C₂×C₄	—	61	2	no	—
323	C₂×C₄	—	61	2	no	—
324	C₂×C₄	—	61	2	no	—
325	C₂×C₄	—	61	2	no	—
326	C₂×C₄	—	61	2	no	—
327	C₂×C₄	—	61	2	no	—
328	C₂×C₄	—	61	2	no	—
329	C₂×C₄	—	488	4	no	—
330	C₂×C₄	—	488	2	no	—
331	C₂×C₄	—	244	2	no	—
332	C₂×C₄	—	244	2	no	—
333	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
334	C₂×C₄	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
335	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
336	C₂×C₄	SD₁₆×D₆₁	1	1	yes	characteristic
337	C₂×C₄	SD₁₆×D₆₁	1	1	yes	characteristic
338	C₂×C₄	Q₁₆×D₆₁	1	1	yes	characteristic
339	C₈	—	61	2	no	—
340	C₈	—	61	2	no	—
341	C₈	C₄²×D₆₁	1	1	yes	characteristic
342	C₈	C₂²⋊C₄×D₆₁	1	1	yes	characteristic
343	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
344	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
345	C₂²	—	2	2	no	—
346	C₂²	—	61	2	no	—
347	C₂²	—	61	2	no	—
348	C₂²	—	122	2	no	—
349	C₂²	—	122	2	no	—
350	C₂²	—	244	2	no	—
351	C₂²	—	244	2	no	—
352	C₂²	#102×D₆₁	1	1	yes	characteristic
353	C₂²	#102×D₆₁	1	1	yes	characteristic
354	C₂²	#107×D₆₁	1	1	yes	characteristic
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	—	61	2	no	—
358	C₄	—	61	2	no	—
359	C₄	—	488	2	no	—
360	C₄	—	488	2	no	—
361	C₄	#95×D₆₁	1	1	yes	characteristic
362	C₄	#103×D₆₁	1	1	yes	characteristic
363	C₄	#103×D₆₁	1	1	yes	characteristic
364	C₄	#106×D₆₁	1	1	yes	characteristic
365	C₂	—	2	1	no	—
366	C₂	—	2	1	no	—
367	C₂	—	122	2	no	—
368	C₂	—	122	2	no	—
369	C₂	#340×D₆₁	1	1	yes	characteristic
370	C₂	#21447553	1	2	yes	—

	subgroup	G/H
center	C₂²	#102×D₆₁
commutator	C₈×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₆₁×#1386	C₂
socle	C₂²×C₆₁	C₂×#102
prime radical	C₂³×D₆₁	D₈
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1386	D₆₁
hypocenter	C₆₁	#12567
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1386

Complements

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

N	H ≅ G/N	type
4. #21443926	C₂	semidirect
5. #21443925	C₂	semidirect
6. #21443924	C₂	semidirect
7. C₆₁×#1386	C₂	semidirect
14. M₃₂×D₆₁	C₄	semidirect
17. #16839799	C₄	semidirect
21. #16839797	C₄	semidirect
23. #107×D₆₁	C₄	semidirect
52. #14519717	C₂×C₄	semidirect
61. #14519709	C₂×C₄	semidirect
67. M₃₂×C₆₁	C₂×C₄	semidirect
72. #106×C₆₁	C₂×C₄	semidirect
186. #1386	D₆₁	semidirect
214. C₆₁	#12567	semidirect
244. M₃₂	C₄×D₆₁	semidirect
258. #106	C₄×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₆₁	2
C₂×#16839797	2
#427×D₆₁	2
#21443926	2
#21443925	2
#21443924	2
C₆₁×#1386	2
#12567	61