Subgroups — #29650692

Order 15616.

560 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21456402	C₂	1	1	yes	maximal, characteristic
2	#21454915	C₂	1	2	yes	maximal
3	#21444726	C₂	1	2	yes	maximal, complemented
4	#21444717	C₂	1	1	yes	maximal, characteristic, complemented
5	#21444635	C₂	1	2	yes	maximal, complemented
6	Q₁₆×#92418	C₂	1	1	yes	maximal, characteristic, complemented
7	#21444625	C₂	1	1	yes	maximal, characteristic, complemented
8	#21444581	C₂	1	1	yes	maximal, characteristic
9	#21444290	C₂	1	2	yes	maximal
10	#21444101	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840316	C₂²	1	2	yes	—
13	#16840166	C₄	1	1	yes	characteristic
14	#16840140	C₄	1	1	yes	characteristic
15	#16840111	C₄	1	2	yes	—
16	#16840004	C₄	1	1	yes	characteristic
17	#16840002	C₄	1	2	yes	—
18	#16840000	C₄	1	1	yes	characteristic
19	#16839892	C₂²	1	2	yes	—
20	#16839890	—	2	2	no	—
21	#16839889	—	2	2	no	—
22	Q₈×#92418	C₂²	1	2	yes	—
23	#16839886	C₂²	1	2	yes	—
24	C₄×Dic₂₄₄	C₂²	1	1	yes	characteristic
25	C₄×#11758202	C₂²	1	2	yes	—
26	#16839824	—	2	2	no	—
27	#16839822	—	2	2	no	—
28	#16839818	C₂²	1	2	yes	—
29	#16839813	C₂²	1	1	yes	characteristic
30	#16839813	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
32	#16839515	C₂²	1	1	yes	characteristic
33	#16839512	C₂²	1	2	yes	—
34	#16839494	C₂²	1	2	yes	—
35	Q₈⋊#92418	C₂²	1	2	yes	—
36	#16839492	C₂²	1	2	yes	—
37	#16839484	C₂²	1	1	yes	characteristic
38	#16839484	C₂²	1	1	yes	characteristic
39	C₈×#92418	C₂²	1	1	yes	characteristic
40	C₆₁×#446	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
42	C₆₁×#335	C₂²	1	1	yes	characteristic
43	C₆₁×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₆₁	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
46	C₂×#11758202	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
49	#14519776	C₂×C₄	1	2	yes	—
50	#14519775	C₂×C₄	1	2	yes	—
51	#14519767	—	2	2	no	—
52	#14519757	C₂×C₄	1	2	yes	—
53	#14519755	—	2	2	no	—
54	#14519753	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₂₂	—	2	2	no	—
56	C₄×Dic₁₂₂	C₂³	1	2	yes	—
57	#14519734	—	2	2	no	—
58	C₄²×D₆₁	C₂³	1	1	yes	characteristic
59	#14519711	C₂×C₄	1	1	yes	characteristic
60	#14519710	C₂×C₄	1	2	yes	—
61	C₄⋊#92418	—	2	2	no	—
62	C₄⋊#92418	C₂³	1	2	yes	—
63	#14519706	—	2	2	no	—
64	#14519706	—	2	1	no	—
65	#14519706	—	2	2	no	—
66	#14519706	D₄	1	1	yes	characteristic
67	#14519706	D₄	1	1	yes	characteristic
68	#14519702	C₂×C₄	1	2	yes	—
69	#14519701	C₂×C₄	1	2	yes	—
70	#14519700	C₂×C₄	1	2	yes	—
71	#14519689	—	2	2	no	—
72	#14519689	—	2	1	no	—
73	#14519689	D₄	1	1	yes	characteristic
74	#14519689	D₄	1	1	yes	characteristic
75	#14519687	—	2	2	no	—
76	#14519687	—	2	2	no	—
77	C₄×#92418	—	2	2	no	—
78	C₄×#92418	C₂³	1	1	yes	characteristic
79	C₄×#92418	C₂³	1	1	yes	characteristic
80	C₄×#92418	C₂³	1	1	yes	characteristic
81	C₈×Dic₆₁	—	2	2	no	—
82	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₆₁	C₂³	1	2	yes	—
86	#107×C₆₁	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₆₁	—	2	2	no	—
88	C₄⋊C₈×C₆₁	C₂³	1	2	yes	—
89	#103×C₆₁	C₂×C₄	1	2	yes	—
90	#97×C₆₁	—	2	2	no	—
91	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
98	C₂×Dic₁₂₂	—	2	2	no	—
99	C₂×Dic₁₂₂	C₂²×C₄	1	2	yes	—
100	Q₁₆×C₆₁	—	2	2	no	—
101	C₂×C₈×C₆₁	—	2	2	no	—
102	C₂×C₈×C₆₁	—	2	1	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	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₆₁	—	2	2	no	—
108	C₄⋊C₄×C₆₁	C₂²×C₄	1	2	yes	—
109	C₄²×C₆₁	—	2	2	no	—
110	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
111	#11758202	—	2	4	no	—
112	#11758198	—	2	2	no	—
113	#11758198	—	2	2	no	—
114	C₄⋊Dic₆₁	—	2	2	no	—
115	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
116	#11758196	—	2	2	no	—
117	#11758196	—	2	2	no	—
118	#11758196	—	2	2	no	—
119	#11758196	—	2	2	no	—
120	#11758196	—	2	2	no	—
121	C₄×Dic₆₁	—	2	2	no	—
122	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₆₁	#41	1	1	yes	characteristic
125	C₂×#92418	—	2	2	no	—
126	C₂×#92418	—	2	1	no	—
127	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
128	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
129	C₂×#92418	C₂×D₄	1	1	yes	characteristic
130	C₂×#92418	C₂×D₄	1	1	yes	characteristic
131	Dic₂₄₄	—	2	2	no	—
132	C₈×D₆₁	—	2	2	no	—
133	C₈×D₆₁	—	2	2	no	—
134	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
135	Q₈×C₆₁	—	2	2	no	—
136	Q₈×C₆₁	—	2	2	no	—
137	C₂×C₄×C₆₁	—	2	2	no	—
138	C₂×C₄×C₆₁	—	2	2	no	—
139	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₆₁	—	2	2	no	—
143	C₂×Dic₆₁	—	2	2	no	—
144	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
147	C₄×D₆₁	—	2	1	no	—
148	C₄×D₆₁	—	2	1	no	—
149	C₄×D₆₁	—	2	1	no	—
150	C₄×D₆₁	—	2	1	no	—
151	C₄×D₆₁	—	4	1	no	—
152	Dic₁₂₂	—	2	2	no	—
153	Dic₁₂₂	—	2	2	no	—
154	Dic₁₂₂	—	8	2	no	—
155	C₈×C₆₁	—	2	1	no	—
156	C₈×C₆₁	—	2	1	no	—
157	C₈×C₆₁	—	2	1	no	—
158	C₈×C₆₁	—	4	1	no	—
159	C₈×C₆₁	—	4	2	no	—
160	C₈×C₆₁	#131	1	2	yes	—
161	#92418	—	2	1	no	—
162	#92418	—	2	1	no	—
163	#92418	—	2	1	no	—
164	#92418	—	4	1	no	—
165	#92418	—	4	2	no	—
166	#92418	#131	1	2	yes	—
167	#17972	—	61	1	no	maximal, Sylow
168	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂×D₆₁	—	2	1	no	—
170	C₂×D₆₁	—	2	1	no	—
171	C₂×D₆₁	—	2	1	no	—
172	C₄×C₆₁	—	2	1	no	—
173	C₄×C₆₁	—	4	2	no	—
174	C₄×C₆₁	—	4	2	no	—
175	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
176	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
177	C₄×C₆₁	#578	1	1	yes	characteristic
178	C₄×C₆₁	#578	1	1	yes	characteristic
179	Dic₆₁	—	2	1	no	—
180	Dic₆₁	—	2	1	no	—
181	Dic₆₁	—	2	1	no	—
182	Dic₆₁	—	4	2	no	—
183	Dic₆₁	—	4	2	no	—
184	#3170	—	61	1	no	—
185	#2985	—	61	2	no	—
186	#1593	—	61	2	no	—
187	#1591	—	61	1	no	—
188	#1591	D₆₁	1	1	yes	characteristic, complemented
189	#1579	—	61	2	no	—
190	C₈×Q₁₆	—	61	1	no	—
191	C₈×Q₁₆	—	61	1	no	—
192	#1568	—	61	1	no	—
193	#1497	—	61	2	no	—
194	#1450	—	61	1	no	—
195	C₂×C₆₁	#2931	1	1	yes	characteristic
196	C₂×C₆₁	#2950	1	1	yes	characteristic
197	C₂×C₆₁	#2957	1	1	yes	characteristic
198	D₆₁	—	4	1	no	—
199	#533	—	61	2	no	—
200	#500	—	61	1	no	—
201	#494	—	61	1	no	—
202	#487	—	61	2	no	—
203	#467	—	61	1	no	—
204	#467	—	61	1	no	—
205	#467	—	61	2	no	—
206	#446	—	61	2	no	—
207	#446	—	122	2	no	—
208	#446	—	122	2	no	—
209	#446	C₂×D₆₁	1	2	yes	—
210	C₈×Q₈	—	61	2	no	—
211	C₈×Q₈	—	61	2	no	—
212	C₄×Q₁₆	—	61	2	no	—
213	C₄×Q₁₆	—	61	1	no	—
214	C₄×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
215	#433	—	122	2	no	—
216	#433	—	122	2	no	—
217	#432	—	61	2	no	—
218	#431	—	61	1	no	—
219	#431	—	61	1	no	—
220	C₂×C₄×C₈	—	61	1	no	—
221	#335	—	61	1	no	—
222	#335	C₂×D₆₁	1	1	yes	characteristic
223	#334	—	61	2	no	—
224	Q₈⋊C₈	—	61	2	no	—
225	Q₈⋊C₈	—	61	2	no	—
226	Q₈⋊C₈	—	61	2	no	—
227	Q₈⋊C₈	C₂×D₆₁	1	2	yes	—
228	#322	—	61	1	no	—
229	#322	—	61	1	no	—
230	C₈²	—	61	1	no	—
231	C₈²	C₂×D₆₁	1	1	yes	characteristic
232	C₆₁	#17972	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₁₆	—	61	2	no	—
234	C₂×Q₁₆	—	61	1	no	—
235	C₂×Q₁₆	C₄×D₆₁	1	1	yes	characteristic
236	C₂²×C₈	—	61	1	no	—
237	C₂²×C₈	—	61	1	no	—
238	C₄⋊Q₈	—	61	2	no	—
239	C₄⋊Q₈	—	61	2	no	—
240	#125	—	122	2	no	—
241	C₂²⋊Q₈	—	61	2	no	—
242	C₂²⋊Q₈	—	61	2	no	—
243	C₂²⋊Q₈	—	122	2	no	—
244	C₄×Q₈	—	61	2	no	—
245	C₄×Q₈	—	122	2	no	—
246	C₄×Q₈	C₂²×D₆₁	1	2	yes	—
247	#117	—	122	2	no	—
248	C₂×C₄²	—	61	1	no	—
249	#107	—	61	1	no	—
250	#107	—	61	2	no	—
251	#107	C₄×D₆₁	1	1	yes	characteristic
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	61	1	no	—
254	C₄⋊C₈	—	61	1	no	—
255	C₄⋊C₈	—	61	2	no	—
256	C₄⋊C₈	—	122	2	no	—
257	C₄⋊C₈	—	122	1	no	—
258	C₄⋊C₈	—	122	2	no	—
259	C₄⋊C₈	—	122	2	no	—
260	C₄⋊C₈	C₂²×D₆₁	1	2	yes	—
261	#103	—	61	2	no	—
262	#103	—	61	2	no	—
263	#103	—	61	2	no	—
264	#103	C₄×D₆₁	1	2	yes	—
265	C₂²⋊C₈	—	61	1	no	—
266	C₂²⋊C₈	—	61	1	no	—
267	C₂²⋊C₈	—	122	1	no	—
268	C₂²⋊C₈	—	122	2	no	—
269	#97	—	2	2	no	—
270	#97	—	122	2	no	—
271	#97	—	122	2	no	—
272	C₄×C₈	—	61	1	no	—
273	C₄×C₈	—	61	1	no	—
274	C₄×C₈	—	61	1	no	—
275	C₄×C₈	—	61	1	no	—
276	C₄×C₈	—	61	1	no	—
277	C₄×C₈	—	122	2	no	—
278	C₄×C₈	—	122	2	no	—
279	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
282	C₂×Q₈	—	61	2	no	—
283	C₂×Q₈	—	122	2	no	—
284	C₂×Q₈	C₂×C₄×D₆₁	1	2	yes	—
285	C₂²×C₄	—	61	1	no	—
286	C₂²×C₄	—	61	1	no	—
287	C₂²×C₄	—	61	1	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	122	4	no	—
290	Q₁₆	—	122	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	61	1	no	—
294	C₂×C₈	—	61	1	no	—
295	C₂×C₈	—	61	1	no	—
296	C₂×C₈	—	61	1	no	—
297	C₂×C₈	—	122	2	no	—
298	C₂×C₈	—	122	2	no	—
299	C₂×C₈	—	122	2	no	—
300	C₂×C₈	—	122	1	no	—
301	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
303	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
304	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	61	2	no	—
307	C₄⋊C₄	—	122	2	no	—
308	C₄⋊C₄	—	122	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	C₄⋊C₄	C₂×C₄×D₆₁	1	2	yes	—
314	C₂²⋊C₄	—	122	2	no	—
315	C₂²⋊C₄	—	122	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	61	1	no	—
318	C₄²	—	61	1	no	—
319	C₄²	—	61	1	no	—
320	C₄²	—	122	2	no	—
321	C₄²	C₂³×D₆₁	1	1	yes	characteristic
322	C₂³	—	61	1	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	2	2	no	—
325	Q₈	—	122	2	no	—
326	Q₈	—	122	2	no	—
327	Q₈	—	488	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	61	1	no	—
331	C₂×C₄	—	61	1	no	—
332	C₂×C₄	—	61	1	no	—
333	C₂×C₄	—	122	1	no	—
334	C₂×C₄	—	122	1	no	—
335	C₂×C₄	—	122	1	no	—
336	C₂×C₄	—	122	1	no	—
337	C₂×C₄	—	122	2	no	—
338	C₂×C₄	—	122	2	no	—
339	C₂×C₄	—	244	1	no	—
340	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
342	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	2	no	—
348	C₈	—	61	2	no	—
349	C₈	—	122	1	no	—
350	C₈	—	122	1	no	—
351	C₈	—	122	1	no	—
352	C₈	—	244	1	no	—
353	C₈	—	244	2	no	—
354	C₈	#14519785	1	2	yes	—
355	C₂²	—	122	1	no	—
356	C₂²	—	122	1	no	—
357	C₂²	—	122	1	no	—
358	C₂²	C₄×D₄×D₆₁	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	122	1	no	—
363	C₄	—	122	1	no	—
364	C₄	—	122	1	no	—
365	C₄	—	244	2	no	—
366	C₄	—	244	2	no	—
367	C₄	#131×D₆₁	1	1	yes	characteristic
368	C₄	C₂×#14519785	1	1	yes	characteristic
369	C₄	#16840575	1	1	yes	characteristic
370	C₄	#16840576	1	1	yes	characteristic
371	C₂	—	244	1	no	—
372	C₂	#21454528	1	1	yes	characteristic
373	C₂	#21454682	1	1	yes	characteristic
374	C₂	#21454731	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×#1591	C₂
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂²×D₆₁	C₄×D₄
cosocle	C₄²×C₆₁	C₂⁴
hypercenter	#1591	D₆₁
hypocenter	C₆₁	#17972
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #21444726	C₂	semidirect
4. #21444717	C₂	semidirect
5. #21444635	C₂	semidirect
6. Q₁₆×#92418	C₂	semidirect
7. #21444625	C₂	semidirect
11. C₆₁×#1591	C₂	semidirect
188. #1591	D₆₁	semidirect
232. C₆₁	#17972	semidirect

Maximal subgroups

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

subgroup	index
#21456402	2
#21454915	2
#21444726	2
#21444717	2
#21444635	2
Q₁₆×#92418	2
#21444625	2
#21444581	2
#21444290	2
#21444101	2
C₆₁×#1591	2
#17972	61