Subgroups — #29833249

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#504×D₆₁	C₂	1	2	yes	maximal, complemented
2	#21454717	C₂	1	1	yes	maximal, characteristic
3	#21447661	C₂	1	1	yes	maximal, characteristic, complemented
4	#21447660	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447658	C₂	1	2	yes	maximal, complemented
6	#21447657	C₂	1	1	yes	maximal, characteristic, complemented
7	#21447654	C₂	1	2	yes	maximal, complemented
8	#21447583	C₂	1	1	yes	maximal, characteristic, complemented
9	#21447445	C₂	1	2	yes	maximal, complemented
10	#21447414	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#2170	C₂	1	1	yes	maximal, characteristic, complemented
12	#16840559	C₄	1	1	yes	characteristic
13	#131×D₆₁	C₂²	1	2	yes	—
14	#16840180	—	2	2	no	—
15	#16840180	C₂²	1	2	yes	complemented
16	#16840179	C₂²	1	2	yes	complemented
17	#16840178	—	2	2	no	—
18	#16840178	C₂²	1	2	yes	complemented
19	C₂×#14519716	C₂²	1	1	yes	characteristic
20	M₃₂×D₆₁	—	2	2	no	—
21	M₃₂×D₆₁	C₂²	1	2	yes	—
22	C₂×C₁₆×D₆₁	—	2	2	no	—
23	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
24	#16839884	C₄	1	1	yes	characteristic
25	#16839883	C₄	1	1	yes	characteristic
26	#16839881	C₄	1	2	yes	—
27	#16839880	C₂²	1	1	yes	characteristic
28	#16839877	C₂²	1	2	yes	—
29	#16839809	C₄	1	1	yes	characteristic
30	#16839777	C₄	1	2	yes	—
31	#16839716	C₂²	1	1	yes	characteristic
32	#16839583	C₂²	1	1	yes	characteristic, complemented
33	#16839582	C₂²	1	2	yes	complemented
34	#16839547	C₂²	1	2	yes	complemented
35	#16839546	C₂²	1	2	yes	complemented
36	#16839545	C₂²	1	2	yes	complemented
37	#16839538	C₂²	1	1	yes	characteristic, complemented
38	#16839537	C₂²	1	1	yes	characteristic, complemented
39	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic, complemented
40	C₆₁×#504	C₂²	1	2	yes	complemented
41	C₆₁×#443	C₂²	1	1	yes	characteristic
42	C₆₁×#364	C₂²	1	1	yes	characteristic, complemented
43	C₆₁×#350	C₂²	1	2	yes	complemented
44	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic, complemented
45	#41×D₆₁	C₂×C₄	1	2	yes	—
46	#14519809	C₈	1	2	yes	—
47	#14519806	C₈	1	2	yes	—
48	#14519804	C₈	1	2	yes	—
49	#14519801	C₈	1	2	yes	—
50	#14519799	C₂×C₄	1	1	yes	characteristic
51	#14519797	C₂×C₄	1	2	yes	—
52	#14519785	—	2	2	no	—
53	#14519785	C₂×C₄	1	2	yes	—
54	#14519784	C₂×C₄	1	2	yes	—
55	#14519783	—	2	2	no	—
56	#14519783	C₂³	1	2	yes	—
57	C₂×#11758189	C₂×C₄	1	1	yes	characteristic
58	M₁₆×D₆₁	—	2	2	no	—
59	M₁₆×D₆₁	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₆₁	—	2	2	no	—
61	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
62	#14519734	C₂×C₄	1	1	yes	characteristic
63	#14519717	—	2	2	no	—
64	#14519717	C₂³	1	2	yes	—
65	#14519716	—	2	2	no	—
66	#14519716	—	2	2	no	—
67	#14519716	—	2	2	no	—
68	#14519716	—	2	2	no	—
69	C₂×#11758185	—	2	2	no	—
70	C₂×#11758185	C₂³	1	1	yes	characteristic
71	C₂×#11758185	C₂³	1	1	yes	characteristic
72	C₁₆×D₆₁	—	2	2	no	—
73	C₁₆×D₆₁	—	2	2	no	—
74	C₁₆×D₆₁	—	2	2	no	—
75	C₁₆×D₆₁	D₄	1	2	yes	—
76	C₁₆×D₆₁	D₄	1	2	yes	—
77	#14519713	C₂×C₄	1	1	yes	characteristic
78	#14519712	C₂×C₄	1	2	yes	—
79	#14519705	C₂×C₄	1	2	yes	—
80	#14519704	C₂×C₄	1	2	yes	—
81	#14519703	C₂×C₄	1	2	yes	—
82	#14519688	C₂×C₄	1	1	yes	characteristic
83	#14519687	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
85	#135×C₆₁	C₂×C₄	1	1	yes	characteristic
86	#131×C₆₁	C₂³	1	2	yes	—
87	M₃₂×C₆₁	—	2	2	no	—
88	M₃₂×C₆₁	C₂³	1	2	yes	—
89	C₂×C₁₆×C₆₁	—	2	2	no	—
90	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
91	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
92	#108×C₆₁	C₂×C₄	1	1	yes	characteristic
93	#104×C₆₁	C₂×C₄	1	2	yes	—
94	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
95	#41×C₆₁	C₂²×C₄	1	2	yes	—
96	#11758226	C₂×C₈	1	2	yes	—
97	Q₈×D₆₁	C₂×C₈	1	2	yes	—
98	#11758224	—	2	2	no	—
99	#11758224	C₂×C₈	1	2	yes	—
100	D₄×D₆₁	—	2	2	no	—
101	D₄×D₆₁	C₂×C₈	1	2	yes	—
102	#11758222	—	2	2	no	—
103	#11758222	C₂²×C₄	1	2	yes	—
104	C₂×C₄×D₆₁	—	2	2	no	—
105	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
106	Q₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
107	SD₁₆×C₆₁	C₂×C₈	1	2	yes	—
108	D₈×C₆₁	C₂×C₈	1	1	yes	characteristic
109	M₁₆×C₆₁	—	2	2	no	—
110	M₁₆×C₆₁	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₆₁	—	2	2	no	—
112	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
114	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
115	#11758202	C₂×C₈	1	2	yes	—
116	#11758201	C₂×C₈	1	2	yes	—
117	#11758200	C₂×C₈	1	2	yes	—
118	#11758199	C₂×C₈	1	2	yes	—
119	#11758198	—	2	1	no	—
120	#11758196	—	2	1	no	—
121	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
122	#11758194	—	2	2	no	—
123	#11758194	C₂²×C₄	1	2	yes	—
124	C₂×#92418	—	2	2	no	—
125	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
126	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
127	Dic₂₄₄	C₂×C₈	1	1	yes	characteristic
128	D₄₈₈	C₂×C₈	1	1	yes	characteristic
129	#11758190	C₂×C₈	1	2	yes	—
130	#11758189	—	2	2	no	—
131	#11758189	—	2	2	no	—
132	#11758189	—	2	2	no	—
133	#11758189	C₂×C₈	1	2	yes	—
134	#11758189	C₂×C₈	1	2	yes	—
135	C₈×D₆₁	—	2	2	no	—
136	C₈×D₆₁	—	2	2	no	—
137	C₈×D₆₁	—	2	2	no	—
138	C₈×D₆₁	C₂×D₄	1	2	yes	—
139	C₈×D₆₁	#41	1	2	yes	—
140	C₁₆×C₆₁	—	2	2	no	—
141	C₁₆×C₆₁	—	2	1	no	—
142	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
143	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
144	#11758185	—	2	2	no	—
145	#11758185	—	2	1	no	—
146	#11758185	C₂×D₄	1	2	yes	—
147	C₂²×D₆₁	—	2	2	no	—
148	C₂²×D₆₁	#131	1	1	yes	characteristic
149	Q₈×C₆₁	C₂²×C₈	1	2	yes	—
150	D₄×C₆₁	—	2	2	no	—
151	D₄×C₆₁	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₆₁	—	2	2	no	—
153	C₂×C₄×C₆₁	—	2	1	no	—
154	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₆₁	—	2	2	no	—
156	C₂²⋊D₆₁	—	2	2	no	—
157	C₂²⋊D₆₁	—	2	2	no	—
158	C₂×Dic₆₁	—	2	2	no	—
159	C₂×Dic₆₁	—	2	1	no	—
160	C₂×Dic₆₁	#131	1	1	yes	characteristic
161	D₂₄₄	—	2	2	no	—
162	D₂₄₄	C₂²×C₈	1	2	yes	—
163	C₄×D₆₁	—	2	2	no	—
164	C₄×D₆₁	—	2	2	no	—
165	C₄×D₆₁	—	2	2	no	—
166	C₄×D₆₁	C₄×D₄	1	2	yes	—
167	C₄×D₆₁	C₂²×C₈	1	2	yes	—
168	Dic₁₂₂	—	2	2	no	—
169	Dic₁₂₂	C₂²×C₈	1	2	yes	—
170	C₈×C₆₁	—	2	2	no	—
171	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
172	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
175	#92418	—	2	2	no	—
176	#92418	C₄×D₄	1	2	yes	—
177	#92418	C₂²×C₈	1	2	yes	—
178	#33441	—	61	1	no	maximal, Sylow
179	C₂²×C₆₁	—	2	2	no	—
180	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
181	C₂×D₆₁	—	2	2	no	—
182	C₂×D₆₁	—	4	1	no	—
183	C₂×D₆₁	—	4	4	no	—
184	C₂×D₆₁	C₈×D₄	1	2	yes	—
185	C₄×C₆₁	—	2	2	no	—
186	C₄×C₆₁	—	4	1	no	—
187	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
188	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
189	Dic₆₁	—	2	2	no	—
190	Dic₆₁	—	4	1	no	—
191	Dic₆₁	C₈×D₄	1	2	yes	—
192	C₂×#504	—	61	2	no	—
193	#2955	—	61	1	no	—
194	#2171	—	61	1	no	—
195	#2171	—	61	1	no	—
196	#2171	—	61	2	no	—
197	#2170	—	61	2	no	—
198	#2170	—	61	1	no	—
199	#2170	D₆₁	1	1	yes	characteristic, complemented
200	#2153	—	61	1	no	—
201	#2117	—	61	2	no	—
202	#2108	—	61	1	no	—
203	C₂×C₆₁	—	2	1	no	—
204	C₂×C₆₁	—	4	2	no	—
205	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
206	D₆₁	—	2	2	no	—
207	D₆₁	—	4	2	no	—
208	#575	—	61	1	no	—
209	C₂×#131	—	61	2	no	—
210	#504	—	61	2	no	—
211	#504	—	61	2	no	—
212	#504	—	61	2	no	—
213	#504	—	122	2	no	—
214	#504	—	122	2	no	—
215	#504	C₂×D₆₁	1	2	yes	complemented
216	C₂×M₃₂	—	61	1	no	—
217	C₂×M₃₂	—	61	2	no	—
218	C₂×M₃₂	—	122	2	no	—
219	C₂²×C₁₆	—	61	1	no	—
220	C₂²×C₁₆	—	122	2	no	—
221	#444	—	61	1	no	—
222	#444	—	61	1	no	—
223	#444	—	61	2	no	—
224	#443	—	61	2	no	—
225	#443	—	61	1	no	—
226	#443	C₂×D₆₁	1	1	yes	characteristic
227	#430	—	61	1	no	—
228	#421	—	61	2	no	—
229	#405	—	61	1	no	—
230	#364	—	61	2	no	—
231	#364	—	61	1	no	—
232	#364	C₂×D₆₁	1	1	yes	characteristic, complemented
233	#350	—	61	2	no	—
234	#350	—	61	2	no	—
235	#350	—	61	2	no	—
236	#350	C₂×D₆₁	1	2	yes	complemented
237	#346	—	61	1	no	—
238	#346	—	61	1	no	—
239	C₄×C₁₆	—	61	1	no	—
240	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic, complemented
241	C₆₁	#33441	1	1	yes	characteristic, complemented, Sylow
242	C₂×#41	—	61	2	no	—
243	#137	—	61	2	no	—
244	#137	—	61	2	no	—
245	AGL(1,ℤ/8)	—	61	2	no	—
246	AGL(1,ℤ/8)	—	61	2	no	—
247	#135	—	61	2	no	—
248	#135	—	61	1	no	—
249	#135	C₄×D₆₁	1	1	yes	characteristic
250	#131	—	61	2	no	—
251	#131	—	61	2	no	—
252	#131	—	61	2	no	—
253	#131	—	122	2	no	—
254	#131	—	122	2	no	—
255	#131	C₂²×D₆₁	1	2	yes	—
256	C₂×M₁₆	—	61	1	no	—
257	C₂×M₁₆	—	61	2	no	—
258	C₂×M₁₆	—	122	2	no	—
259	C₂²×C₈	—	61	1	no	—
260	C₂²×C₈	—	122	2	no	—
261	#117	—	61	1	no	—
262	M₃₂	—	2	2	no	—
263	M₃₂	—	61	2	no	—
264	M₃₂	—	122	2	no	—
265	M₃₂	—	122	2	no	—
266	M₃₂	—	122	2	no	—
267	M₃₂	—	122	2	no	—
268	M₃₂	—	122	2	no	—
269	M₃₂	C₂²×D₆₁	1	2	yes	—
270	C₂×C₁₆	—	2	2	no	—
271	C₂×C₁₆	—	61	1	no	—
272	C₂×C₁₆	—	61	2	no	—
273	C₂×C₁₆	—	61	2	no	—
274	C₂×C₁₆	—	61	1	no	—
275	C₂×C₁₆	—	122	2	no	—
276	C₂×C₁₆	—	122	2	no	—
277	C₂×C₁₆	—	122	2	no	—
278	C₂×C₁₆	—	122	2	no	—
279	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
280	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
281	#108	—	61	2	no	—
282	#108	—	61	1	no	—
283	#108	C₄×D₆₁	1	1	yes	characteristic
284	#104	—	61	2	no	—
285	#104	—	61	2	no	—
286	#104	—	61	2	no	—
287	#104	C₄×D₆₁	1	2	yes	—
288	#97	—	61	1	no	—
289	#97	—	61	1	no	—
290	C₄×C₈	—	61	1	no	—
291	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
292	#41	—	61	2	no	—
293	#41	—	61	2	no	—
294	#41	—	61	2	no	—
295	#41	—	122	2	no	—
296	#41	—	122	2	no	—
297	#41	C₂×C₄×D₆₁	1	2	yes	—
298	C₂×Q₈	—	61	2	no	—
299	C₂×D₄	—	61	2	no	—
300	C₂×D₄	—	122	2	no	—
301	C₂²×C₄	—	61	1	no	—
302	C₂²×C₄	—	122	2	no	—
303	Q₁₆	—	61	2	no	—
304	Q₁₆	—	61	1	no	—
305	Q₁₆	C₈×D₆₁	1	1	yes	characteristic
306	SD₁₆	—	61	2	no	—
307	SD₁₆	—	61	2	no	—
308	SD₁₆	—	61	2	no	—
309	SD₁₆	C₈×D₆₁	1	2	yes	—
310	D₈	—	61	2	no	—
311	D₈	—	61	1	no	—
312	D₈	C₈×D₆₁	1	1	yes	characteristic
313	M₁₆	—	2	2	no	—
314	M₁₆	—	61	2	no	—
315	M₁₆	—	61	2	no	—
316	M₁₆	—	61	2	no	—
317	M₁₆	—	122	2	no	—
318	M₁₆	—	122	2	no	—
319	M₁₆	—	122	2	no	—
320	M₁₆	—	122	2	no	—
321	M₁₆	C₂×C₄×D₆₁	1	2	yes	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	61	1	no	—
324	C₂×C₈	—	61	1	no	—
325	C₂×C₈	—	61	2	no	—
326	C₂×C₈	—	61	2	no	—
327	C₂×C₈	—	122	2	no	—
328	C₂×C₈	—	122	2	no	—
329	C₂×C₈	—	122	2	no	—
330	C₂×C₈	—	122	2	no	—
331	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
332	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
333	C₄⋊C₄	—	122	1	no	—
334	C₂²⋊C₄	—	122	1	no	—
335	C₄²	—	61	1	no	—
336	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
337	C₁₆	—	2	2	no	—
338	C₁₆	—	2	1	no	—
339	C₁₆	—	61	2	no	—
340	C₁₆	—	122	2	no	—
341	C₁₆	—	122	1	no	—
342	C₁₆	D₄×D₆₁	1	1	yes	characteristic
343	C₁₆	D₄×D₆₁	1	1	yes	characteristic
344	C₂³	—	61	1	no	—
345	C₂³	—	122	2	no	—
346	Q₈	—	61	2	no	—
347	Q₈	—	122	2	no	—
348	Q₈	C₂×C₈×D₆₁	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	61	2	no	—
351	D₄	—	122	2	no	—
352	D₄	—	122	2	no	—
353	D₄	—	122	2	no	—
354	D₄	—	122	2	no	—
355	D₄	C₂×C₈×D₆₁	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	61	1	no	—
359	C₂×C₄	—	61	2	no	—
360	C₂×C₄	—	61	2	no	—
361	C₂×C₄	—	122	2	no	—
362	C₂×C₄	—	122	2	no	—
363	C₂×C₄	—	122	2	no	—
364	C₂×C₄	—	122	2	no	—
365	C₂×C₄	—	122	1	no	—
366	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	61	2	no	—
369	C₈	—	61	2	no	—
370	C₈	—	122	2	no	—
371	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
372	C₈	C₂×C₈×D₆₁	1	1	yes	characteristic
373	C₈	C₂×D₄×D₆₁	1	1	yes	characteristic
374	C₈	#41×D₆₁	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	61	2	no	—
377	C₂²	—	122	2	no	—
378	C₂²	—	244	1	no	—
379	C₂²	—	244	4	no	—
380	C₂²	#131×D₆₁	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	61	2	no	—
384	C₄	—	122	2	no	—
385	C₄	—	244	1	no	—
386	C₄	C₄×D₄×D₆₁	1	1	yes	characteristic
387	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	122	2	no	—
391	C₂	—	244	2	no	—
392	C₂	C₈×D₄×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₆₁×#2170	C₂
socle	C₂×C₆₁	C₂×C₈×D₄
prime radical	#16840559	C₄
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	#2170	D₆₁
hypocenter	C₆₁	#33441
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

Complements

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

N	H ≅ G/N	type
1. #504×D₆₁	C₂	semidirect
3. #21447661	C₂	semidirect
4. #21447660	C₂	semidirect
5. #21447658	C₂	semidirect
6. #21447657	C₂	semidirect
7. #21447654	C₂	semidirect
8. #21447583	C₂	semidirect
9. #21447445	C₂	semidirect
10. #21447414	C₂	semidirect
11. C₆₁×#2170	C₂	semidirect
15. #16840180	C₂²	semidirect
16. #16840179	C₂²	semidirect
18. #16840178	C₂²	semidirect
32. #16839583	C₂²	semidirect
33. #16839582	C₂²	semidirect
34. #16839547	C₂²	semidirect
35. #16839546	C₂²	semidirect
36. #16839545	C₂²	semidirect
37. #16839538	C₂²	semidirect
38. #16839537	C₂²	semidirect
39. C₁₆×Dic₆₁	C₂²	semidirect
40. C₆₁×#504	C₂²	semidirect
42. C₆₁×#364	C₂²	semidirect
43. C₆₁×#350	C₂²	semidirect
44. C₄×C₁₆×C₆₁	C₂²	semidirect
199. #2170	D₆₁	semidirect
215. #504	C₂×D₆₁	semidirect
232. #364	C₂×D₆₁	semidirect
236. #350	C₂×D₆₁	semidirect
240. C₄×C₁₆	C₂×D₆₁	semidirect
241. C₆₁	#33441	semidirect

Maximal subgroups

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

subgroup	index
#504×D₆₁	2
#21454717	2
#21447661	2
#21447660	2
#21447658	2
#21447657	2
#21447654	2
#21447583	2
#21447445	2
#21447414	2
C₆₁×#2170	2
#33441	61