Subgroups — #29700013

Order 15616.

616 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	#21446760	C₂	1	2	yes	maximal
2	#21446385	C₂	1	2	yes	maximal
3	#21446378	C₂	1	2	yes	maximal
4	#21445964	C₂	1	2	yes	maximal
5	#21445936	C₂	1	1	yes	maximal, characteristic
6	#21445694	C₂	1	2	yes	maximal
7	#21445672	C₂	1	1	yes	maximal, characteristic
8	#21445571	C₂	1	2	yes	maximal
9	C₆₁×#2295	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊#92418	C₂²	1	2	yes	—
11	C₂×C₄⋊#92418	C₂²	1	2	yes	—
12	C₂×#14519688	C₂²	1	2	yes	—
13	C₂×#14519688	C₂²	1	2	yes	—
14	C₂×C₄×#92418	C₂²	1	2	yes	—
15	#16839517	C₂²	1	1	yes	characteristic
16	#16839517	C₂²	1	1	yes	characteristic
17	#16839517	C₂²	1	2	yes	—
18	#16839517	C₂²	1	1	yes	characteristic
19	#16839517	C₂²	1	1	yes	characteristic
20	#16839517	C₂²	1	2	yes	—
21	#16839517	C₂²	1	2	yes	—
22	#16839517	C₂²	1	2	yes	—
23	#16839517	C₂²	1	2	yes	—
24	#16839517	C₂²	1	2	yes	—
25	#16839517	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
27	C₆₁×#391	C₄	1	1	yes	characteristic
28	C₆₁×#391	C₄	1	1	yes	characteristic
29	C₆₁×#389	C₄	1	1	yes	characteristic
30	C₆₁×#389	C₄	1	1	yes	characteristic
31	C₆₁×#387	C₄	1	2	yes	—
32	C₆₁×#387	C₄	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
35	C₆₁×#376	C₂²	1	2	yes	—
36	C₄³×C₆₁	C₂²	1	1	yes	characteristic
37	C₂²×#92418	C₂³	1	2	yes	—
38	C₂²×#92418	C₂³	1	2	yes	—
39	C₂²×#92418	C₂³	1	2	yes	—
40	C₂²×#92418	C₂³	1	2	yes	—
41	C₄⋊#92418	—	2	4	no	—
42	C₄⋊#92418	—	2	4	no	—
43	#14519688	—	2	4	no	—
44	#14519688	—	2	4	no	—
45	C₄×#92418	—	2	4	no	—
46	C₂²×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈×C₆₁	—	2	2	no	—
48	C₄×Q₈×C₆₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
60	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₆₁	C₂³	1	2	yes	—
62	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
64	#95×C₆₁	C₂×C₄	1	2	yes	—
65	#95×C₆₁	C₂×C₄	1	2	yes	—
66	#95×C₆₁	C₂×C₄	1	2	yes	—
67	#95×C₆₁	C₂×C₄	1	2	yes	—
68	#95×C₆₁	C₂×C₄	1	2	yes	—
69	#95×C₆₁	C₂×C₄	1	2	yes	—
70	#95×C₆₁	C₂×C₄	1	2	yes	—
71	C₂×Q₈×C₆₁	—	2	2	no	—
72	C₂×Q₈×C₆₁	—	4	1	no	—
73	C₂×Q₈×C₆₁	—	4	1	no	—
74	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₆₁	—	2	2	no	—
86	C₄⋊C₄×C₆₁	—	2	4	no	—
87	C₄⋊C₄×C₆₁	—	2	4	no	—
88	C₄⋊C₄×C₆₁	—	4	1	no	—
89	C₄⋊C₄×C₆₁	—	4	1	no	—
90	C₄⋊C₄×C₆₁	—	4	2	no	—
91	C₄²×C₆₁	—	2	2	no	—
92	C₄²×C₆₁	—	2	2	no	—
93	C₄²×C₆₁	—	2	2	no	—
94	C₄²×C₆₁	—	2	2	no	—
95	C₄²×C₆₁	—	2	2	no	—
96	C₄²×C₆₁	—	2	2	no	—
97	C₄²×C₆₁	—	2	2	no	—
98	C₄²×C₆₁	—	4	1	no	—
99	C₄²×C₆₁	—	4	1	no	—
100	C₄²×C₆₁	—	4	1	no	—
101	C₄²×C₆₁	#41	1	2	yes	—
102	C₄²×C₆₁	#41	1	2	yes	—
103	C₂×#92418	—	2	2	no	—
104	C₂×#92418	—	2	2	no	—
105	C₂×#92418	—	2	2	no	—
106	C₂×#92418	—	2	2	no	—
107	C₂×#92418	—	2	2	no	—
108	C₂×#92418	—	2	2	no	—
109	C₂×#92418	—	2	2	no	—
110	C₂×#92418	—	2	2	no	—
111	C₂×#92418	—	2	2	no	—
112	C₂×#92418	—	2	2	no	—
113	C₂×#92418	—	2	2	no	—
114	C₂×#92418	#41	1	4	yes	—
115	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
116	Q₈×C₆₁	—	4	4	no	—
117	C₂×C₄×C₆₁	—	2	1	no	—
118	C₂×C₄×C₆₁	—	2	1	no	—
119	C₂×C₄×C₆₁	—	2	1	no	—
120	C₂×C₄×C₆₁	—	2	1	no	—
121	C₂×C₄×C₆₁	—	2	1	no	—
122	C₂×C₄×C₆₁	—	2	1	no	—
123	C₂×C₄×C₆₁	—	2	2	no	—
124	C₂×C₄×C₆₁	—	2	2	no	—
125	C₂×C₄×C₆₁	—	2	2	no	—
126	C₂×C₄×C₆₁	—	2	1	no	—
127	C₂×C₄×C₆₁	—	2	2	no	—
128	C₂×C₄×C₆₁	—	2	1	no	—
129	C₂×C₄×C₆₁	—	2	1	no	—
130	C₂×C₄×C₆₁	—	2	1	no	—
131	C₂×C₄×C₆₁	—	2	2	no	—
132	C₂×C₄×C₆₁	—	2	2	no	—
133	C₂×C₄×C₆₁	—	2	1	no	—
134	C₂×C₄×C₆₁	—	2	1	no	—
135	C₂×C₄×C₆₁	—	2	1	no	—
136	C₂×C₄×C₆₁	—	2	1	no	—
137	C₂×C₄×C₆₁	—	2	1	no	—
138	C₂×C₄×C₆₁	—	2	2	no	—
139	C₂×C₄×C₆₁	—	2	2	no	—
140	C₂×C₄×C₆₁	—	2	2	no	—
141	C₂×C₄×C₆₁	—	2	2	no	—
142	C₂×C₄×C₆₁	—	2	2	no	—
143	C₂×C₄×C₆₁	—	2	2	no	—
144	C₂×C₄×C₆₁	#117	1	2	yes	—
145	C₂×C₄×C₆₁	#117	1	2	yes	—
146	C₂×C₄×C₆₁	#131	1	2	yes	—
147	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₆₁	C₂×#41	1	2	yes	—
149	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
151	#92418	—	2	4	no	—
152	#92418	—	4	2	no	—
153	#92418	—	4	2	no	—
154	#92418	—	4	2	no	—
155	#22831	—	61	1	no	maximal, Sylow
156	C₂²×C₆₁	C₂×#117	1	1	yes	characteristic
157	C₂²×C₆₁	C₄×#41	1	1	yes	characteristic
158	C₂²×C₆₁	#519	1	1	yes	characteristic
159	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
160	C₂²×C₆₁	#568	1	1	yes	characteristic
161	C₂²×C₆₁	#568	1	1	yes	characteristic
162	C₂²×C₆₁	#568	1	1	yes	characteristic
163	C₄×C₆₁	—	2	2	no	—
164	C₄×C₆₁	—	2	2	no	—
165	C₄×C₆₁	—	2	2	no	—
166	C₄×C₆₁	—	4	2	no	—
167	C₄×C₆₁	—	4	2	no	—
168	C₄×C₆₁	—	4	1	no	—
169	C₄×C₆₁	—	4	2	no	—
170	C₄×C₆₁	—	4	2	no	—
171	C₄×C₆₁	—	4	1	no	—
172	C₄×C₆₁	—	4	1	no	—
173	C₄×C₆₁	#552	1	2	yes	—
174	C₄×C₆₁	#556	1	2	yes	—
175	#2295	D₆₁	1	1	yes	characteristic
176	#1985	—	61	2	no	—
177	#1917	—	61	2	no	—
178	#1916	—	61	2	no	—
179	#1840	—	61	2	no	—
180	#1831	—	61	1	no	—
181	#1771	—	61	2	no	—
182	#1767	—	61	1	no	—
183	#1746	—	61	2	no	—
184	C₂×C₆₁	#2356	1	1	yes	characteristic
185	C₂×C₆₁	#2921	1	1	yes	characteristic
186	C₂×C₆₁	#2925	1	1	yes	characteristic
187	C₂×C₆₁	#2967	1	1	yes	characteristic
188	C₂×C₆₁	#2968	1	1	yes	characteristic
189	C₂×C₆₁	#2985	1	1	yes	characteristic
190	C₂×C₆₁	#2987	1	1	yes	characteristic
191	C₂×C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
192	C₂×C₄⋊C₈	—	61	2	no	—
193	C₂×C₄⋊C₈	—	61	2	no	—
194	C₂×#97	—	61	2	no	—
195	C₂×#97	—	61	2	no	—
196	C₂×C₄×C₈	—	61	2	no	—
197	#391	Dic₆₁	1	1	yes	characteristic
198	#391	Dic₆₁	1	1	yes	characteristic
199	#389	Dic₆₁	1	1	yes	characteristic
200	#389	Dic₆₁	1	1	yes	characteristic
201	#387	Dic₆₁	1	2	yes	—
202	#387	Dic₆₁	1	2	yes	—
203	C₄×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
204	C₄×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
205	#376	C₂×D₆₁	1	2	yes	—
206	C₄³	C₂×D₆₁	1	1	yes	characteristic
207	#336	—	61	2	no	—
208	#336	—	61	1	no	—
209	#336	—	61	1	no	—
210	#336	—	61	1	no	—
211	#336	—	61	2	no	—
212	#336	—	61	2	no	—
213	#336	—	61	2	no	—
214	#336	—	61	2	no	—
215	#336	—	61	2	no	—
216	#336	—	61	2	no	—
217	#336	—	61	1	no	—
218	C₆₁	#22831	1	1	yes	characteristic, complemented, Sylow
219	C₂²×Q₈	C₂×Dic₆₁	1	1	yes	characteristic
220	C₂²×C₈	—	61	2	no	—
221	C₂²×C₈	—	61	2	no	—
222	C₂²×C₈	—	61	2	no	—
223	C₂²×C₈	—	61	2	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
228	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
229	C₂×C₄⋊C₄	C₂×Dic₆₁	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
231	C₂×C₄⋊C₄	C₂×Dic₆₁	1	1	yes	characteristic
232	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
233	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
234	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
235	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
237	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
239	C₂×C₄²	C₂²×D₆₁	1	2	yes	—
240	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
241	C₄⋊C₈	—	122	4	no	—
242	C₄⋊C₈	—	122	4	no	—
243	#97	—	122	4	no	—
244	#97	—	122	4	no	—
245	C₄×C₈	—	122	4	no	—
246	#95	C₂×Dic₆₁	1	2	yes	—
247	#95	C₂×Dic₆₁	1	2	yes	—
248	#95	C₂×Dic₆₁	1	2	yes	—
249	#95	C₂×Dic₆₁	1	2	yes	—
250	#95	C₂×Dic₆₁	1	2	yes	—
251	#95	C₂×Dic₆₁	1	2	yes	—
252	#95	C₂×Dic₆₁	1	2	yes	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
258	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
259	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
260	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
262	C₂²×C₄	C₂²×Dic₆₁	1	2	yes	—
263	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
264	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
265	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
266	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
267	C₂×C₈	—	61	4	no	—
268	C₂×C₈	—	122	2	no	—
269	C₂×C₈	—	122	2	no	—
270	C₂×C₈	—	122	2	no	—
271	C₂×C₈	—	122	2	no	—
272	C₂×C₈	—	122	2	no	—
273	C₂×C₈	—	122	2	no	—
274	C₂×C₈	—	122	2	no	—
275	C₂×C₈	—	122	2	no	—
276	C₂×C₈	—	122	2	no	—
277	C₂×C₈	—	122	2	no	—
278	C₂×C₈	—	122	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	1	no	—
294	C₄²	—	4	1	no	—
295	C₄²	#11758222	1	2	yes	—
296	C₄²	#11758222	1	2	yes	—
297	C₂³	C₂³×Dic₆₁	1	1	yes	characteristic
298	Q₈	—	4	4	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	#14519736	1	2	yes	—
327	C₂×C₄	#14519736	1	2	yes	—
328	C₂×C₄	#14519784	1	2	yes	—
329	C₂×C₄	#41×D₆₁	1	2	yes	—
330	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
331	C₂×C₄	C₂×#11758222	1	1	yes	characteristic
332	C₂×C₄	#14519825	1	2	yes	—
333	C₈	—	122	4	no	—
334	C₈	—	244	2	no	—
335	C₈	—	244	2	no	—
336	C₈	—	244	2	no	—
337	C₂²	C₂×#14519736	1	1	yes	characteristic
338	C₂²	#41×Dic₆₁	1	1	yes	characteristic
339	C₂²	#16840240	1	1	yes	characteristic
340	C₂²	C₂×#14519784	1	1	yes	characteristic
341	C₂²	#16840529	1	1	yes	characteristic
342	C₂²	#16840529	1	1	yes	characteristic
343	C₂²	#16840529	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	—	2	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	1	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	#16840453	1	2	yes	—
355	C₄	#16840484	1	2	yes	—
356	C₂	#21448784	1	1	yes	characteristic
357	C₂	#21454457	1	1	yes	characteristic
358	C₂	#21454485	1	1	yes	characteristic
359	C₂	#21454806	1	1	yes	characteristic
360	C₂	#21454815	1	1	yes	characteristic
361	C₂	#21454914	1	1	yes	characteristic
362	C₂	#21454925	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂³×C₄
Fitting	C₆₁×#2295	C₂
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#2295	D₆₁
hypocenter	C₆₁	#22831
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
218. C₆₁	#22831	semidirect

Maximal subgroups

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

subgroup	index
#21446760	2
#21446385	2
#21446378	2
#21445964	2
#21445936	2
#21445694	2
#21445672	2
#21445571	2
C₆₁×#2295	2
#22831	61