Subgroups — #29833428

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21458952	C₂	1	1	yes	maximal, characteristic
2	#21454731	C₂	1	2	yes	maximal, complemented
3	#21447705	C₂	1	1	yes	maximal, characteristic, complemented
4	#21447704	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447703	C₂	1	2	yes	maximal, complemented
6	#21447701	C₂	1	1	yes	maximal, characteristic, complemented
7	#21447697	C₂	1	2	yes	maximal, complemented
8	#21447612	C₂	1	1	yes	maximal, characteristic, complemented
9	#359×D₆₁	C₂	1	2	yes	maximal, complemented
10	#21447414	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2178	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₆₁	C₂²	1	2	yes	—
13	#16840527	—	2	2	no	—
14	#16840204	C₄	1	2	yes	complemented
15	#16840201	C₄	1	2	yes	complemented
16	#16840199	C₄	1	2	yes	complemented
17	#16840196	C₄	1	2	yes	complemented
18	#16840194	C₂²	1	1	yes	characteristic
19	#16840192	C₂²	1	2	yes	—
20	C₂×#14519716	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
22	#16839882	—	2	2	no	—
23	#16839881	—	2	2	no	—
24	#16839880	C₂²	1	2	yes	complemented
25	#16839879	C₂²	1	2	yes	complemented
26	#16839878	C₂²	1	2	yes	complemented
27	#108×D₆₁	C₂²	1	2	yes	—
28	#16839777	—	2	2	no	—
29	#16839716	C₂²	1	1	yes	characteristic
30	#16839592	C₂²	1	1	yes	characteristic, complemented
31	#16839591	C₂²	1	2	yes	complemented
32	#16839570	C₂²	1	2	yes	complemented
33	#16839569	C₂²	1	2	yes	complemented
34	#16839568	C₂²	1	2	yes	complemented
35	#16839538	C₂²	1	1	yes	characteristic, complemented
36	#16839537	C₂²	1	1	yes	characteristic, complemented
37	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic, complemented
38	C₆₁×#508	C₂²	1	1	yes	characteristic
39	C₆₁×#443	C₂²	1	2	yes	complemented
40	C₆₁×#368	C₂²	1	1	yes	characteristic, complemented
41	C₆₁×#359	C₂²	1	2	yes	complemented
42	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic, complemented
43	#41×D₆₁	—	2	2	no	—
44	#14519800	C₂×C₄	1	2	yes	—
45	#14519799	C₂³	1	2	yes	—
46	#14519798	—	2	2	no	—
47	#14519797	—	2	2	no	—
48	#14519796	C₂×C₄	1	2	yes	—
49	Q₁₆×D₆₁	C₂×C₄	1	2	yes	—
50	SD₁₆×D₆₁	—	2	2	no	—
51	D₈×D₆₁	C₂×C₄	1	2	yes	—
52	#14519785	—	2	2	no	—
53	#14519785	—	4	2	no	—
54	#14519784	—	2	2	no	—
55	#14519783	—	2	2	no	—
56	#14519783	—	4	2	no	—
57	C₂×#11758189	—	4	2	no	—
58	C₂×#11758189	D₄	1	1	yes	characteristic
59	M₁₆×D₆₁	—	2	2	no	—
60	M₁₆×D₆₁	—	4	2	no	—
61	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
62	#14519734	D₄	1	1	yes	characteristic
63	Dic₄₈₈	C₂×C₄	1	1	yes	characteristic
64	#14519723	C₂×C₄	1	2	yes	—
65	#14519722	C₂×C₄	1	2	yes	—
66	#14519721	C₂×C₄	1	2	yes	—
67	#14519720	C₂×C₄	1	2	yes	—
68	D₉₇₆	C₂×C₄	1	1	yes	characteristic
69	#14519718	C₂×C₄	1	2	yes	—
70	#14519716	C₂×C₄	1	2	yes	—
71	#14519716	C₂×C₄	1	2	yes	—
72	C₂×#11758185	C₂³	1	1	yes	characteristic
73	C₂×#11758185	C₂³	1	1	yes	characteristic
74	C₁₆×D₆₁	—	2	2	no	—
75	#14519713	C₂³	1	2	yes	—
76	#14519712	—	2	2	no	—
77	#14519705	—	2	2	no	—
78	#14519704	—	2	2	no	—
79	#14519703	—	2	2	no	—
80	#14519688	D₄	1	1	yes	characteristic
81	#14519687	D₄	1	1	yes	characteristic
82	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
83	#135×C₆₁	C₂³	1	2	yes	—
84	#131×C₆₁	—	2	2	no	—
85	Q₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
86	SD₃₂×C₆₁	C₂×C₄	1	2	yes	—
87	D₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
88	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
89	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
90	#108×C₆₁	C₂³	1	2	yes	—
91	#104×C₆₁	—	2	2	no	—
92	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
93	#41×C₆₁	—	2	2	no	—
94	#11758226	—	2	2	no	—
95	Q₈×D₆₁	—	2	2	no	—
96	#11758224	—	2	2	no	—
97	#11758224	—	4	2	no	—
98	D₄×D₆₁	—	2	2	no	—
99	D₄×D₆₁	—	4	2	no	—
100	#11758222	—	2	2	no	—
101	#11758222	—	4	2	no	—
102	C₂×C₄×D₆₁	—	4	2	no	—
103	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
104	Q₁₆×C₆₁	C₂²×C₄	1	2	yes	—
105	SD₁₆×C₆₁	—	2	2	no	—
106	D₈×C₆₁	C₂²×C₄	1	2	yes	—
107	M₁₆×C₆₁	—	2	2	no	—
108	M₁₆×C₆₁	—	4	2	no	—
109	C₂×C₈×C₆₁	—	4	2	no	—
110	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
112	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
113	#11758202	—	2	2	no	—
114	#11758201	—	2	2	no	—
115	#11758200	—	2	2	no	—
116	#11758199	—	2	2	no	—
117	#11758198	—	2	1	no	—
118	#11758196	—	2	1	no	—
119	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
120	#11758194	—	2	2	no	—
121	#11758194	—	4	2	no	—
122	C₂×#92418	—	4	2	no	—
123	C₂×#92418	C₂×D₄	1	1	yes	characteristic
124	C₂×#92418	C₂×D₄	1	1	yes	characteristic
125	Dic₂₄₄	C₂²×C₄	1	2	yes	—
126	D₄₈₈	C₂²×C₄	1	2	yes	—
127	#11758190	—	2	2	no	—
128	#11758189	—	4	2	no	—
129	#11758189	—	4	2	no	—
130	#11758189	—	4	2	no	—
131	#11758189	D₈	1	2	yes	—
132	#11758189	D₈	1	2	yes	—
133	C₈×D₆₁	—	4	2	no	—
134	C₈×D₆₁	—	4	2	no	—
135	C₈×D₆₁	—	4	2	no	—
136	C₈×D₆₁	C₂²×C₄	1	2	yes	—
137	C₈×D₆₁	#41	1	2	yes	—
138	C₁₆×C₆₁	—	2	1	no	—
139	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
140	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
141	#11758185	—	2	1	no	—
142	#11758185	C₂²×C₄	1	2	yes	—
143	C₂²×D₆₁	—	4	2	no	—
144	C₂²×D₆₁	#135	1	1	yes	characteristic
145	Q₈×C₆₁	—	2	2	no	—
146	D₄×C₆₁	—	2	2	no	—
147	D₄×C₆₁	—	4	2	no	—
148	C₂×C₄×C₆₁	—	2	1	no	—
149	C₂×C₄×C₆₁	—	4	2	no	—
150	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₆₁	—	4	2	no	—
152	C₂²⋊D₆₁	—	4	2	no	—
153	C₂²⋊D₆₁	—	4	2	no	—
154	C₂×Dic₆₁	—	2	1	no	—
155	C₂×Dic₆₁	—	4	2	no	—
156	C₂×Dic₆₁	#135	1	1	yes	characteristic
157	D₂₄₄	—	2	2	no	—
158	D₂₄₄	—	4	2	no	—
159	C₄×D₆₁	—	4	2	no	—
160	C₄×D₆₁	—	4	2	no	—
161	C₄×D₆₁	—	4	2	no	—
162	C₄×D₆₁	C₄×D₄	1	2	yes	—
163	C₄×D₆₁	C₂×D₈	1	2	yes	—
164	Dic₁₂₂	—	2	2	no	—
165	Dic₁₂₂	—	4	2	no	—
166	C₈×C₆₁	—	4	2	no	—
167	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
168	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
169	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
170	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
171	#92418	—	4	2	no	—
172	#92418	C₄×D₄	1	2	yes	—
173	#92418	C₂×D₈	1	2	yes	—
174	#33464	—	61	1	no	maximal, Sylow
175	C₂²×C₆₁	—	4	2	no	—
176	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
177	C₂×D₆₁	—	4	1	no	—
178	C₂×D₆₁	—	4	2	no	—
179	C₂×D₆₁	—	16	2	no	—
180	C₂×D₆₁	C₄×D₈	1	2	yes	—
181	C₄×C₆₁	—	4	1	no	—
182	C₄×C₆₁	—	4	2	no	—
183	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
185	Dic₆₁	—	4	1	no	—
186	Dic₆₁	—	4	2	no	—
187	Dic₆₁	C₄×D₈	1	2	yes	—
188	#3414	—	61	1	no	—
189	#2957	—	61	2	no	—
190	#2179	—	61	1	no	—
191	#2179	—	61	2	no	—
192	#2179	—	61	1	no	—
193	#2178	—	61	2	no	—
194	#2178	—	61	1	no	—
195	#2178	D₆₁	1	1	yes	characteristic, complemented
196	#2161	—	61	1	no	—
197	C₂×#359	—	61	2	no	—
198	#2108	—	61	1	no	—
199	C₂×C₆₁	—	2	1	no	—
200	C₂×C₆₁	—	8	2	no	—
201	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
202	D₆₁	—	2	2	no	—
203	D₆₁	—	8	2	no	—
204	C₂×#135	—	61	2	no	—
205	#568	—	122	2	no	—
206	#510	—	61	2	no	—
207	#510	—	61	2	no	—
208	#509	—	61	2	no	—
209	#509	—	61	2	no	—
210	#508	—	61	2	no	—
211	#508	—	61	1	no	—
212	#508	C₂×D₆₁	1	1	yes	characteristic
213	C₂×M₃₂	—	61	1	no	—
214	C₂²×C₁₆	—	61	1	no	—
215	#444	—	122	2	no	—
216	#444	—	122	2	no	—
217	#443	—	61	2	no	—
218	#443	—	61	2	no	—
219	#443	—	61	2	no	—
220	#443	C₂×D₆₁	1	2	yes	complemented
221	C₂×#108	—	61	2	no	—
222	#421	—	122	2	no	—
223	#405	—	61	1	no	—
224	#368	—	61	1	no	—
225	#368	—	61	2	no	—
226	#368	C₂×D₆₁	1	1	yes	characteristic, complemented
227	#359	—	61	2	no	—
228	#359	—	61	2	no	—
229	#359	—	61	2	no	—
230	#359	C₂×D₆₁	1	2	yes	complemented
231	#346	—	61	1	no	—
232	#346	—	61	1	no	—
233	C₄×C₁₆	—	61	1	no	—
234	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic, complemented
235	C₆₁	#33464	1	1	yes	characteristic, complemented, Sylow
236	C₂×#41	—	122	2	no	—
237	#135	—	61	2	no	—
238	#135	—	61	2	no	—
239	#135	—	61	2	no	—
240	#135	—	122	2	no	—
241	#135	—	122	2	no	—
242	#135	C₂²×D₆₁	1	2	yes	—
243	C₂×Q₁₆	—	61	2	no	—
244	C₂×SD₁₆	—	122	2	no	—
245	C₂×D₈	—	61	2	no	—
246	#131	—	2	2	no	—
247	#131	—	122	2	no	—
248	#131	—	122	2	no	—
249	#131	—	122	2	no	—
250	#131	—	244	2	no	—
251	#131	—	244	2	no	—
252	C₂×M₁₆	—	61	1	no	—
253	C₂×M₁₆	—	122	2	no	—
254	C₂×M₁₆	—	244	2	no	—
255	C₂×M₁₆	—	244	2	no	—
256	C₂²×C₈	—	61	1	no	—
257	#117	—	61	1	no	—
258	Q₃₂	—	61	2	no	—
259	Q₃₂	—	61	1	no	—
260	Q₃₂	C₄×D₆₁	1	1	yes	characteristic
261	SD₃₂	—	61	2	no	—
262	SD₃₂	—	61	2	no	—
263	SD₃₂	—	61	2	no	—
264	SD₃₂	C₄×D₆₁	1	2	yes	—
265	D₁₆	—	61	2	no	—
266	D₁₆	—	61	1	no	—
267	D₁₆	C₄×D₆₁	1	1	yes	characteristic
268	M₃₂	—	61	2	no	—
269	M₃₂	—	61	2	no	—
270	C₂×C₁₆	—	61	1	no	—
271	C₂×C₁₆	—	61	1	no	—
272	C₂×C₁₆	—	122	2	no	—
273	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
274	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
275	#108	—	61	2	no	—
276	#108	—	122	2	no	—
277	#108	C₂²×D₆₁	1	2	yes	—
278	#104	—	2	2	no	—
279	#104	—	122	2	no	—
280	#104	—	122	2	no	—
281	#104	—	122	2	no	—
282	#97	—	61	1	no	—
283	#97	—	61	1	no	—
284	C₄×C₈	—	61	1	no	—
285	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
286	#41	—	2	2	no	—
287	#41	—	122	2	no	—
288	#41	—	122	2	no	—
289	#41	—	122	2	no	—
290	#41	—	244	2	no	—
291	#41	—	244	2	no	—
292	C₂×Q₈	—	122	2	no	—
293	C₂×D₄	—	122	2	no	—
294	C₂×D₄	—	244	2	no	—
295	C₂²×C₄	—	61	1	no	—
296	C₂²×C₄	—	244	2	no	—
297	Q₁₆	—	61	2	no	—
298	Q₁₆	—	122	2	no	—
299	Q₁₆	C₂×C₄×D₆₁	1	2	yes	—
300	SD₁₆	—	2	2	no	—
301	SD₁₆	—	122	2	no	—
302	SD₁₆	—	122	2	no	—
303	SD₁₆	—	122	2	no	—
304	D₈	—	61	2	no	—
305	D₈	—	122	2	no	—
306	D₈	C₂×C₄×D₆₁	1	2	yes	—
307	M₁₆	—	2	2	no	—
308	M₁₆	—	4	2	no	—
309	M₁₆	—	61	2	no	—
310	M₁₆	—	61	2	no	—
311	M₁₆	—	122	2	no	—
312	M₁₆	—	244	2	no	—
313	M₁₆	—	244	2	no	—
314	M₁₆	—	244	2	no	—
315	M₁₆	—	244	2	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	—	61	1	no	—
318	C₂×C₈	—	61	1	no	—
319	C₂×C₈	—	61	2	no	—
320	C₂×C₈	—	61	2	no	—
321	C₂×C₈	—	244	2	no	—
322	C₂×C₈	—	244	2	no	—
323	C₂×C₈	—	244	2	no	—
324	C₂×C₈	—	244	2	no	—
325	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
326	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
327	C₄⋊C₄	—	122	1	no	—
328	C₂²⋊C₄	—	122	1	no	—
329	C₄²	—	61	1	no	—
330	C₄²	D₄×D₆₁	1	1	yes	characteristic
331	C₁₆	—	2	1	no	—
332	C₁₆	—	61	2	no	—
333	C₁₆	—	122	1	no	—
334	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
335	C₁₆	C₂×C₄×D₆₁	1	1	yes	characteristic
336	C₂³	—	61	1	no	—
337	C₂³	—	244	2	no	—
338	Q₈	—	2	2	no	—
339	Q₈	—	122	2	no	—
340	Q₈	—	244	2	no	—
341	D₄	—	2	2	no	—
342	D₄	—	4	2	no	—
343	D₄	—	122	2	no	—
344	D₄	—	244	2	no	—
345	D₄	—	244	2	no	—
346	D₄	—	244	2	no	—
347	D₄	—	244	2	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	61	1	no	—
351	C₂×C₄	—	61	2	no	—
352	C₂×C₄	—	61	2	no	—
353	C₂×C₄	—	122	1	no	—
354	C₂×C₄	—	244	2	no	—
355	C₂×C₄	—	244	2	no	—
356	C₂×C₄	—	244	2	no	—
357	C₂×C₄	—	244	2	no	—
358	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	61	2	no	—
361	C₈	—	61	2	no	—
362	C₈	—	244	2	no	—
363	C₈	D₈×D₆₁	1	1	yes	characteristic
364	C₈	D₈×D₆₁	1	1	yes	characteristic
365	C₈	C₂²×C₄×D₆₁	1	1	yes	characteristic
366	C₈	#41×D₆₁	1	1	yes	characteristic
367	C₂²	—	4	2	no	—
368	C₂²	—	61	2	no	—
369	C₂²	—	976	2	no	—
370	C₂²	—	244	1	no	—
371	C₂²	—	244	2	no	—
372	C₂²	#135×D₆₁	1	1	yes	characteristic
373	C₄	—	4	1	no	—
374	C₄	—	4	2	no	—
375	C₄	—	61	2	no	—
376	C₄	—	244	1	no	—
377	C₄	—	244	2	no	—
378	C₄	C₄×D₄×D₆₁	1	1	yes	characteristic
379	C₄	C₂×D₈×D₆₁	1	1	yes	characteristic
380	C₂	—	2	1	no	—
381	C₂	—	8	2	no	—
382	C₂	—	122	2	no	—
383	C₂	—	488	2	no	—
384	C₂	C₄×D₈×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₆₁
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2178	C₂
socle	C₂×C₆₁	C₂×C₄×D₈
prime radical	#21458952	C₂
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2178	D₆₁
hypocenter	C₆₁	#33464
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

Complements

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

N	H ≅ G/N	type
2. #21454731	C₂	semidirect
3. #21447705	C₂	semidirect
4. #21447704	C₂	semidirect
5. #21447703	C₂	semidirect
6. #21447701	C₂	semidirect
7. #21447697	C₂	semidirect
8. #21447612	C₂	semidirect
9. #359×D₆₁	C₂	semidirect
10. #21447414	C₂	semidirect
11. C₆₁×#2178	C₂	semidirect
14. #16840204	C₄	semidirect
15. #16840201	C₄	semidirect
16. #16840199	C₄	semidirect
17. #16840196	C₄	semidirect
24. #16839880	C₂²	semidirect
25. #16839879	C₂²	semidirect
26. #16839878	C₂²	semidirect
30. #16839592	C₂²	semidirect
31. #16839591	C₂²	semidirect
32. #16839570	C₂²	semidirect
33. #16839569	C₂²	semidirect
34. #16839568	C₂²	semidirect
35. #16839538	C₂²	semidirect
36. #16839537	C₂²	semidirect
37. C₁₆×Dic₆₁	C₂²	semidirect
39. C₆₁×#443	C₂²	semidirect
40. C₆₁×#368	C₂²	semidirect
41. C₆₁×#359	C₂²	semidirect
42. C₄×C₁₆×C₆₁	C₂²	semidirect
195. #2178	D₆₁	semidirect
220. #443	C₂×D₆₁	semidirect
226. #368	C₂×D₆₁	semidirect
230. #359	C₂×D₆₁	semidirect
234. C₄×C₁₆	C₂×D₆₁	semidirect
235. C₆₁	#33464	semidirect

Maximal subgroups

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

subgroup	index
#21458952	2
#21454731	2
#21447705	2
#21447704	2
#21447703	2
#21447701	2
#21447697	2
#21447612	2
#359×D₆₁	2
#21447414	2
C₆₁×#2178	2
#33464	61