Subgroups — #29650586

Order 15616.

614 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456389	C₂	1	1	yes	maximal, characteristic
2	#21454788	C₂	1	2	yes	maximal, complemented
3	#21444732	C₂	1	1	yes	maximal, characteristic, complemented
4	#21444727	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444720	C₂	1	2	yes	maximal, complemented
6	#21444642	C₂	1	1	yes	maximal, characteristic, complemented
7	#21444637	C₂	1	1	yes	maximal, characteristic, complemented
8	#21444630	C₂	1	2	yes	maximal, complemented
9	#21444277	C₂	1	2	yes	maximal, complemented
10	C₂×#16839483	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840296	C₂²	1	2	yes	—
13	#16840164	C₄	1	1	yes	characteristic
14	#16840146	C₄	1	1	yes	characteristic
15	#16840117	C₄	1	2	yes	—
16	#16840003	C₄	1	1	yes	characteristic
17	#16839998	C₄	1	1	yes	characteristic
18	#16839991	C₄	1	2	yes	—
19	#16839889	C₂²	1	2	yes	—
20	#16839886	C₂²	1	2	yes	—
21	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
22	C₄×#11758190	C₂²	1	2	yes	—
23	C₄×D₄₈₈	C₂²	1	1	yes	characteristic
24	#16839839	C₂²	1	2	yes	—
25	#16839830	—	2	2	no	—
26	#16839827	C₂²	1	2	yes	—
27	#16839825	—	2	2	no	—
28	#16839824	—	2	2	no	—
29	#16839719	—	2	2	no	—
30	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
31	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
32	#16839492	C₂²	1	2	yes	—
33	#16839492	C₂²	1	2	yes	—
34	#16839489	C₂²	1	2	yes	—
35	#16839489	C₂²	1	2	yes	—
36	#16839483	C₂²	1	1	yes	characteristic
37	#16839483	C₂²	1	1	yes	characteristic
38	#16839483	C₂²	1	1	yes	characteristic
39	#16839483	C₂²	1	1	yes	characteristic
40	C₆₁×#432	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
42	C₆₁×#335	C₂²	1	2	yes	—
43	C₆₁×#334	C₂²	1	1	yes	characteristic
44	C₆₁×#334	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
46	C₂×#11758190	C₂×C₄	1	2	yes	—
47	C₂×D₄₈₈	C₂×C₄	1	1	yes	characteristic
48	C₂×#11758189	—	2	2	no	—
49	C₂²×#92418	—	2	1	no	—
50	C₂²×#92418	D₄	1	2	yes	—
51	C₂×C₈×D₆₁	—	2	2	no	—
52	#14519771	—	2	2	no	—
53	#14519769	C₂×C₄	1	2	yes	—
54	#14519765	C₂×C₄	1	2	yes	—
55	#14519760	—	2	2	no	—
56	#14519756	C₂×C₄	1	2	yes	—
57	#14519751	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₂₂	C₂³	1	2	yes	—
59	C₄×D₂₄₄	C₂³	1	2	yes	—
60	C₄×C₂²⋊D₆₁	—	2	2	no	—
61	#14519734	—	2	2	no	—
62	#14519711	C₂×C₄	1	1	yes	characteristic
63	#14519711	C₂×C₄	1	1	yes	characteristic
64	#14519709	C₂×C₄	1	2	yes	—
65	#14519706	—	2	2	no	—
66	#14519706	—	2	2	no	—
67	#14519702	C₂×C₄	1	2	yes	—
68	#14519702	C₂×C₄	1	2	yes	—
69	#14519699	C₂×C₄	1	2	yes	—
70	#14519699	C₂×C₄	1	2	yes	—
71	#14519689	—	2	2	no	—
72	#14519689	—	2	2	no	—
73	#14519687	—	2	2	no	—
74	C₄×#92418	—	2	1	no	—
75	C₄×#92418	D₄	1	2	yes	—
76	C₄×#92418	C₂³	1	1	yes	characteristic
77	C₄×#92418	C₂³	1	1	yes	characteristic
78	C₄×#92418	C₂³	1	1	yes	characteristic
79	C₄×#92418	C₂³	1	1	yes	characteristic
80	C₈×Dic₆₁	—	2	2	no	—
81	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
85	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
86	C₂²⋊C₈×C₆₁	—	2	2	no	—
87	#97×C₆₁	—	2	2	no	—
88	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
89	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
90	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
95	C₂×C₂²⋊D₆₁	—	2	2	no	—
96	C₂×D₂₄₄	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₆₁	—	2	2	no	—
98	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₆₁	—	2	2	no	—
100	C₂×C₈×C₆₁	—	2	2	no	—
101	C₂×C₈×C₆₁	—	2	2	no	—
102	C₂×C₈×C₆₁	—	2	2	no	—
103	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₆₁	#41	1	1	yes	characteristic
110	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
111	C₂²⋊Dic₆₁	—	2	2	no	—
112	#11758198	—	2	2	no	—
113	#11758198	—	2	2	no	—
114	#11758198	—	2	2	no	—
115	#11758198	—	2	2	no	—
116	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
117	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
118	#11758196	—	2	2	no	—
119	#11758196	—	2	2	no	—
120	#11758196	—	2	2	no	—
121	#11758196	—	2	2	no	—
122	C₄×Dic₆₁	—	2	2	no	—
123	C₂×#92418	—	2	1	no	—
124	C₂×#92418	—	2	1	no	—
125	C₂×#92418	—	4	2	no	—
126	C₂×#92418	—	4	2	no	—
127	C₂×#92418	C₂×D₄	1	2	yes	—
128	C₂×#92418	C₂×D₄	1	2	yes	—
129	Dic₂₄₄	—	2	2	no	—
130	D₄₈₈	—	2	2	no	—
131	#11758190	—	2	4	no	—
132	#11758189	—	4	4	no	—
133	C₈×D₆₁	—	4	4	no	—
134	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
135	C₂²×D₆₁	—	2	2	no	—
136	C₂×C₄×C₆₁	—	2	2	no	—
137	C₂×C₄×C₆₁	—	2	1	no	—
138	C₂×C₄×C₆₁	—	2	1	no	—
139	C₂×C₄×C₆₁	—	2	2	no	—
140	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
146	C₂²⋊D₆₁	—	4	4	no	—
147	C₂×Dic₆₁	—	2	2	no	—
148	C₂×Dic₆₁	—	2	2	no	—
149	C₂×Dic₆₁	—	2	2	no	—
150	D₂₄₄	—	2	2	no	—
151	D₂₄₄	—	2	2	no	—
152	C₄×D₆₁	—	4	2	no	—
153	C₄×D₆₁	—	4	2	no	—
154	Dic₁₂₂	—	2	2	no	—
155	Dic₁₂₂	—	2	2	no	—
156	C₈×C₆₁	—	2	1	no	—
157	C₈×C₆₁	—	2	1	no	—
158	C₈×C₆₁	—	4	2	no	—
159	C₈×C₆₁	—	4	2	no	—
160	C₈×C₆₁	#131	1	2	yes	—
161	C₈×C₆₁	#131	1	2	yes	—
162	#92418	—	2	2	no	—
163	#92418	—	2	2	no	—
164	#92418	—	4	1	no	—
165	#92418	—	4	1	no	—
166	#17963	—	61	1	no	maximal, Sylow
167	C₂²×C₆₁	—	2	1	no	—
168	C₂²×C₆₁	—	2	1	no	—
169	C₂²×C₆₁	#444	1	2	yes	—
170	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₆₁	—	4	2	no	—
172	C₂×D₆₁	—	4	2	no	—
173	C₂×D₆₁	—	4	2	no	—
174	C₄×C₆₁	—	2	1	no	—
175	C₄×C₆₁	—	2	1	no	—
176	C₄×C₆₁	—	2	1	no	—
177	C₄×C₆₁	—	2	1	no	—
178	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
179	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
180	C₄×C₆₁	#575	1	2	yes	—
181	Dic₆₁	—	4	2	no	—
182	Dic₆₁	—	4	2	no	—
183	Dic₆₁	—	4	2	no	—
184	#3168	—	61	1	no	—
185	#2966	—	61	2	no	—
186	#1594	—	61	1	no	—
187	#1593	—	61	1	no	—
188	#1592	—	61	2	no	—
189	#1580	—	61	1	no	—
190	#1579	—	61	1	no	—
191	SD₁₆⋊C₈	—	61	2	no	—
192	#1568	D₆₁	1	1	yes	characteristic, complemented
193	#1494	—	61	2	no	—
194	C₂×#322	—	61	1	no	—
195	C₂×C₆₁	—	2	2	no	—
196	C₂×C₆₁	#2931	1	1	yes	characteristic
197	C₂×C₆₁	#2943	1	1	yes	characteristic
198	C₂×C₆₁	C₂×#444	1	1	yes	characteristic
199	D₆₁	—	8	2	no	—
200	#529	—	61	2	no	—
201	#499	—	61	1	no	—
202	#495	—	61	1	no	—
203	#488	—	61	2	no	—
204	#467	—	61	1	no	—
205	#466	—	61	1	no	—
206	#465	—	61	2	no	—
207	#446	—	61	2	no	—
208	C₈×Q₈	—	61	2	no	—
209	C₄×Q₁₆	—	61	1	no	—
210	C₄×SD₁₆	—	61	2	no	—
211	C₄×D₈	—	61	1	no	—
212	#436	—	61	2	no	—
213	#435	—	122	2	no	—
214	C₈×D₄	—	61	2	no	—
215	C₈×D₄	—	122	2	no	—
216	#433	—	122	2	no	—
217	#432	C₂×D₆₁	1	2	yes	—
218	#405	—	122	2	no	—
219	C₂×C₄×C₈	—	61	1	no	—
220	C₂×C₄×C₈	—	61	1	no	—
221	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
222	#335	C₂×D₆₁	1	2	yes	—
223	#334	C₂×D₆₁	1	1	yes	characteristic
224	#334	C₂×D₆₁	1	1	yes	characteristic
225	Q₈⋊C₈	—	61	2	no	—
226	Q₈⋊C₈	—	61	2	no	—
227	D₄⋊C₈	—	61	2	no	—
228	D₄⋊C₈	—	61	2	no	—
229	#322	—	61	1	no	—
230	#322	—	61	1	no	—
231	#322	—	61	1	no	—
232	#322	—	61	1	no	—
233	C₆₁	#17963	1	1	yes	characteristic, complemented, Sylow
234	C₂×Q₁₆	—	61	1	no	—
235	C₂×SD₁₆	—	61	2	no	—
236	C₂×D₈	—	61	1	no	—
237	C₂×M₁₆	—	122	2	no	—
238	C₂²×C₈	—	61	2	no	—
239	C₂²×C₈	—	122	2	no	—
240	C₂²×C₈	—	122	1	no	—
241	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
242	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
243	#126	—	122	2	no	—
244	#125	—	61	2	no	—
245	#124	—	61	2	no	—
246	#123	—	122	2	no	—
247	C₂²⋊Q₈	—	61	2	no	—
248	#121	—	61	2	no	—
249	C₄×Q₈	—	61	2	no	—
250	C₄×D₄	—	61	2	no	—
251	C₄×D₄	—	122	2	no	—
252	#117	—	122	2	no	—
253	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
254	#107	—	61	1	no	—
255	#107	—	61	1	no	—
256	#106	—	61	2	no	—
257	C₄⋊C₈	—	122	2	no	—
258	C₄⋊C₈	—	122	2	no	—
259	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
260	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
261	#103	—	61	2	no	—
262	#103	—	61	2	no	—
263	#102	—	61	2	no	—
264	#102	—	61	2	no	—
265	C₂²⋊C₈	—	2	2	no	—
266	C₂²⋊C₈	—	122	2	no	—
267	C₂²⋊C₈	—	122	2	no	—
268	#97	—	2	2	no	—
269	#97	—	122	2	no	—
270	C₄×C₈	—	61	2	no	—
271	C₄×C₈	—	61	1	no	—
272	C₄×C₈	—	61	1	no	—
273	C₄×C₈	—	61	1	no	—
274	C₄×C₈	—	61	1	no	—
275	C₄×C₈	—	122	2	no	—
276	C₄×C₈	—	122	1	no	—
277	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
278	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
281	C₂×Q₈	—	61	2	no	—
282	C₂×D₄	—	61	2	no	—
283	C₂×D₄	—	122	2	no	—
284	C₂²×C₄	—	122	2	no	—
285	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
286	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
287	C₂²×C₄	#11758226	1	1	yes	characteristic
288	Q₁₆	—	122	2	no	—
289	SD₁₆	—	122	4	no	—
290	D₈	—	122	2	no	—
291	M₁₆	—	244	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	61	2	no	—
297	C₂×C₈	—	61	2	no	—
298	C₂×C₈	—	122	1	no	—
299	C₂×C₈	—	122	1	no	—
300	C₂×C₈	—	244	4	no	—
301	C₂×C₈	—	244	2	no	—
302	C₂×C₈	—	244	2	no	—
303	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
306	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
307	C₄⋊C₄	—	61	2	no	—
308	C₄⋊C₄	—	61	2	no	—
309	C₄⋊C₄	—	122	2	no	—
310	C₄⋊C₄	—	122	2	no	—
311	C₄⋊C₄	—	122	2	no	—
312	C₄⋊C₄	—	122	2	no	—
313	C₂²⋊C₄	—	122	2	no	—
314	C₂²⋊C₄	—	122	2	no	—
315	C₂²⋊C₄	—	122	2	no	—
316	C₂²⋊C₄	—	122	2	no	—
317	C₂²⋊C₄	—	122	2	no	—
318	C₄²	—	122	2	no	—
319	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
320	C₄²	D₄×D₆₁	1	1	yes	characteristic
321	C₄²	#11758226	1	1	yes	characteristic
322	C₄²	C₂³×D₆₁	1	1	yes	characteristic
323	C₂³	—	122	2	no	—
324	C₂³	#14519739	1	1	yes	characteristic
325	Q₈	—	122	2	no	—
326	Q₈	—	122	2	no	—
327	D₄	—	122	2	no	—
328	D₄	—	122	2	no	—
329	D₄	—	244	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	122	2	no	—
335	C₂×C₄	—	122	2	no	—
336	C₂×C₄	—	122	2	no	—
337	C₂×C₄	—	244	2	no	—
338	C₂×C₄	—	244	2	no	—
339	C₂×C₄	#14519739	1	1	yes	characteristic
340	C₂×C₄	#14519739	1	1	yes	characteristic
341	C₂×C₄	#14519739	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
344	C₂×C₄	C₂×#11758226	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	4	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	122	2	no	—
350	C₈	—	122	2	no	—
351	C₈	—	244	1	no	—
352	C₈	—	244	1	no	—
353	C₈	#14519785	1	2	yes	—
354	C₈	#14519785	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	244	2	no	—
358	C₂²	—	244	2	no	—
359	C₂²	—	244	2	no	—
360	C₂²	#16839883	1	2	yes	—
361	C₂²	C₂×#14519739	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	244	2	no	—
367	C₄	—	244	2	no	—
368	C₄	—	244	2	no	—
369	C₄	C₂×#14519785	1	1	yes	characteristic
370	C₄	C₂×#14519785	1	1	yes	characteristic
371	C₄	#16840559	1	2	yes	—
372	C₂	—	2	2	no	—
373	C₂	—	488	2	no	—
374	C₂	#21454530	1	1	yes	characteristic
375	C₂	#21454619	1	1	yes	characteristic
376	C₂	C₂×#16839883	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₄²	C₂³×D₆₁
Fitting	C₆₁×#1568	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	#16839998	C₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#1568	D₆₁
hypocenter	C₆₁	#17963
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
2. #21454788	C₂	semidirect
3. #21444732	C₂	semidirect
4. #21444727	C₂	semidirect
5. #21444720	C₂	semidirect
6. #21444642	C₂	semidirect
7. #21444637	C₂	semidirect
8. #21444630	C₂	semidirect
9. #21444277	C₂	semidirect
10. C₂×#16839483	C₂	semidirect
11. C₆₁×#1568	C₂	semidirect
192. #1568	D₆₁	semidirect
233. C₆₁	#17963	semidirect

Maximal subgroups

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

subgroup	index
#21456389	2
#21454788	2
#21444732	2
#21444727	2
#21444720	2
#21444642	2
#21444637	2
#21444630	2
#21444277	2
C₂×#16839483	2
C₆₁×#1568	2
#17963	61