Subgroups — #27274569

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719912	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133129	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133129	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272542	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719912	C₄	1	2	yes	—
9	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×#13133129	C₄	1	2	yes	—
11	C₂×#13133129	C₄	1	2	yes	—
12	C₄×#11719912	—	2	2	no	—
13	C₄×#11719912	—	2	2	no	—
14	C₂×#13133127	C₄	1	2	yes	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	#16810377	C₄	1	2	yes	—
19	#16810377	C₄	1	2	yes	—
20	#16810345	—	2	2	no	—
21	C₈×#88322	—	2	2	no	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719912	—	2	4	no	—
29	C₂×#11719912	—	2	4	no	—
30	C₂×#11719912	C₈	1	8	yes	complemented
31	C₂²×#88322	C₂×C₄	1	2	yes	—
32	C₂²×#88322	C₂×C₄	1	2	yes	—
33	C₂²×#88322	C₂×C₄	1	2	yes	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×D₅₃	—	2	1	no	—
38	C₄²×D₅₃	—	2	1	no	—
39	C₄²×D₅₃	—	2	1	no	—
40	C₄²×D₅₃	—	2	1	no	—
41	#13133129	—	2	4	no	—
42	#13133129	—	2	4	no	—
43	#13133127	—	2	4	no	—
44	C₄×#88322	—	2	1	no	—
45	C₄×#88322	—	2	1	no	—
46	C₄×#88322	—	2	1	no	—
47	C₄×#88322	—	2	1	no	—
48	C₄×#88322	D₄	1	2	yes	—
49	C₄×#88322	D₄	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₂³×D₅₃	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
65	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
67	C₂×C₄×D₅₃	—	2	1	no	—
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×C₄×D₅₃	—	2	2	no	—
70	C₂×C₄×D₅₃	—	2	1	no	—
71	C₂×C₄×D₅₃	—	2	1	no	—
72	C₂×C₄×D₅₃	—	2	1	no	—
73	C₂×C₄×D₅₃	—	2	1	no	—
74	C₂×C₄×D₅₃	—	2	2	no	—
75	C₂×C₄×D₅₃	—	2	2	no	—
76	C₂×C₄×D₅₃	—	2	2	no	—
77	C₂×C₄×D₅₃	—	2	2	no	—
78	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
87	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	#41	1	1	yes	characteristic
91	C₄²×C₅₃	#41	1	1	yes	characteristic
92	C₄×Dic₅₃	—	2	1	no	—
93	C₄×Dic₅₃	—	2	1	no	—
94	C₄×Dic₅₃	—	2	1	no	—
95	C₄×Dic₅₃	—	2	1	no	—
96	C₄×Dic₅₃	—	2	1	no	—
97	C₄×Dic₅₃	—	2	1	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	—	2	2	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	2	2	no	—
103	C₂×#88322	C₂²⋊C₄	1	4	yes	—
104	C₂×#88322	C₂×C₈	1	4	yes	—
105	C₂×#88322	M₁₆	1	4	yes	—
106	C₂×#88322	M₁₆	1	4	yes	complemented
107	#11719912	—	2	16	no	—
108	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
109	C₂²×D₅₃	—	2	1	no	—
110	C₂²×D₅₃	—	2	1	no	—
111	C₂²×D₅₃	—	2	1	no	—
112	C₂²×D₅₃	—	2	1	no	—
113	C₂²×D₅₃	—	2	1	no	—
114	C₂²×D₅₃	—	2	1	no	—
115	C₂²×D₅₃	C₄×C₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₅₃	#131	1	2	yes	—
127	C₂×C₄×C₅₃	#131	1	2	yes	—
128	C₂×Dic₅₃	—	2	1	no	—
129	C₂×Dic₅₃	—	2	1	no	—
130	C₂×Dic₅₃	—	2	1	no	—
131	C₂×Dic₅₃	—	2	1	no	—
132	C₂×Dic₅₃	—	2	1	no	—
133	C₂×Dic₅₃	—	2	1	no	—
134	C₂×Dic₅₃	—	2	1	no	—
135	C₂×Dic₅₃	—	2	1	no	—
136	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
137	C₄×D₅₃	—	2	4	no	—
138	C₄×D₅₃	—	2	4	no	—
139	C₄×D₅₃	—	4	2	no	—
140	C₄×D₅₃	—	4	2	no	—
141	C₄×D₅₃	—	4	2	no	—
142	C₄×D₅₃	—	4	2	no	—
143	C₈×C₅₃	—	2	4	no	—
144	C₈×C₅₃	—	2	4	no	—
145	C₈×C₅₃	—	4	2	no	—
146	#88322	—	2	4	no	—
147	#88322	—	4	2	no	—
148	#88322	C₂²⋊C₈	1	8	yes	complemented
149	#8146	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
152	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
153	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
154	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
155	C₂²×C₅₃	#405	1	1	yes	characteristic
156	C₂²×C₅₃	#405	1	1	yes	characteristic
157	C₂×D₅₃	—	2	2	no	—
158	C₂×D₅₃	—	2	2	no	—
159	C₂×D₅₃	—	2	2	no	—
160	C₂×D₅₃	—	4	1	no	—
161	C₂×D₅₃	—	4	1	no	—
162	C₂×D₅₃	—	4	1	no	—
163	C₂×D₅₃	—	4	1	no	—
164	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
165	C₄×C₅₃	#431	1	2	yes	—
166	C₄×C₅₃	#435	1	2	yes	—
167	C₄×C₅₃	#435	1	2	yes	—
168	C₄×C₅₃	#436	1	2	yes	—
169	C₄×C₅₃	#436	1	2	yes	—
170	Dic₅₃	—	2	2	no	—
171	Dic₅₃	—	4	1	no	—
172	Dic₅₃	—	4	1	no	—
173	C₂×C₄×M₁₆	—	53	1	no	—
174	C₄×C₂²⋊C₈	—	53	1	no	—
175	C₄×C₂²⋊C₈	—	53	1	no	—
176	C₂×#322	—	53	1	no	—
177	C₂×C₈²	—	53	1	no	—
178	#1273	—	53	1	no	—
179	#1273	D₅₃	1	1	yes	characteristic, complemented
180	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
181	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
182	C₂×C₅₃	#1453	1	1	yes	characteristic
183	C₂×C₅₃	#1455	1	1	yes	characteristic
184	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
185	C₂×C₅₃	#1753	1	1	yes	characteristic
186	C₂×C₅₃	#1753	1	1	yes	characteristic
187	D₅₃	—	4	2	no	—
188	C₂²×M₁₆	—	53	2	no	—
189	C₂²×C₄²	—	53	1	no	—
190	C₂×C₂²⋊C₈	—	53	2	no	—
191	C₂×C₂²⋊C₈	—	53	2	no	—
192	C₄×M₁₆	—	106	2	no	—
193	C₄×M₁₆	—	106	2	no	—
194	C₂×#97	—	53	2	no	—
195	C₂×C₄×C₈	—	53	1	no	—
196	C₂×C₄×C₈	—	53	1	no	—
197	C₂×C₄×C₈	—	53	1	no	—
198	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
199	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
200	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
201	#336	—	53	2	no	—
202	#336	—	53	2	no	—
203	#322	—	106	2	no	—
204	C₈²	—	106	2	no	—
205	C₅₃	#8146	1	1	yes	characteristic, complemented, Sylow
206	C₂³×C₄	—	53	1	no	—
207	C₂³×C₄	—	53	1	no	—
208	C₂³×C₄	—	53	1	no	—
209	C₂×M₁₆	—	53	8	no	—
210	C₂×M₁₆	—	106	4	no	—
211	C₂×M₁₆	—	106	4	no	—
212	C₂²×C₈	—	53	2	no	—
213	C₂²×C₈	—	53	2	no	—
214	C₂²×C₈	—	53	2	no	—
215	C₂²×C₈	C₄×D₅₃	1	2	yes	—
216	C₂²×C₈	C₄×D₅₃	1	2	yes	—
217	C₂²×C₈	C₄×D₅₃	1	2	yes	—
218	C₂×C₄²	—	53	1	no	—
219	C₂×C₄²	—	53	1	no	—
220	C₂×C₄²	—	53	1	no	—
221	C₂×C₄²	—	106	1	no	—
222	C₂×C₄²	—	106	1	no	—
223	C₂×C₄²	—	106	1	no	—
224	C₂×C₄²	—	106	1	no	—
225	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
226	C₂²⋊C₈	—	106	4	no	—
227	C₂²⋊C₈	—	106	4	no	—
228	#97	—	106	4	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	53	2	no	—
236	C₄×C₈	—	53	2	no	—
237	C₄×C₈	—	106	1	no	—
238	C₄×C₈	—	106	1	no	—
239	C₄×C₈	—	106	1	no	—
240	C₄×C₈	—	106	1	no	—
241	C₂⁴	—	53	1	no	—
242	C₂²×C₄	—	53	1	no	—
243	C₂²×C₄	—	53	4	no	—
244	C₂²×C₄	—	53	1	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	106	1	no	—
247	C₂²×C₄	—	106	1	no	—
248	C₂²×C₄	—	106	2	no	—
249	C₂²×C₄	—	106	1	no	—
250	C₂²×C₄	—	106	1	no	—
251	C₂²×C₄	—	106	1	no	—
252	C₂²×C₄	—	106	1	no	—
253	C₂²×C₄	—	106	2	no	—
254	C₂²×C₄	—	106	2	no	—
255	C₂²×C₄	—	106	2	no	—
256	C₂²×C₄	—	106	2	no	—
257	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
260	M₁₆	—	106	16	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	53	4	no	—
269	C₂×C₈	—	53	4	no	—
270	C₂×C₈	—	53	4	no	—
271	C₂×C₈	—	53	4	no	—
272	C₂×C₈	—	106	2	no	—
273	C₂×C₈	—	106	2	no	—
274	C₂×C₈	—	106	2	no	—
275	C₂×C₈	—	106	2	no	—
276	C₂×C₈	—	106	2	no	—
277	C₂×C₈	C₈×D₅₃	1	4	yes	—
278	C₂×C₈	#11719912	1	4	yes	complemented
279	C₄²	—	106	1	no	—
280	C₄²	—	106	1	no	—
281	C₄²	—	106	1	no	—
282	C₄²	—	106	1	no	—
283	C₄²	—	106	1	no	—
284	C₄²	—	106	1	no	—
285	C₄²	D₄×D₅₃	1	1	yes	characteristic
286	C₄²	D₄×D₅₃	1	1	yes	characteristic
287	C₄²	#11719947	1	1	yes	characteristic
288	C₄²	#11719949	1	1	yes	characteristic
289	C₂³	—	53	2	no	—
290	C₂³	—	106	1	no	—
291	C₂³	—	106	1	no	—
292	C₂³	—	106	1	no	—
293	C₂³	—	106	1	no	—
294	C₂³	—	106	1	no	—
295	C₂³	—	106	1	no	—
296	C₂³	C₄²×D₅₃	1	1	yes	characteristic
297	C₂×C₄	—	53	2	no	—
298	C₂×C₄	—	106	1	no	—
299	C₂×C₄	—	106	1	no	—
300	C₂×C₄	—	106	4	no	—
301	C₂×C₄	—	106	4	no	—
302	C₂×C₄	—	106	1	no	—
303	C₂×C₄	—	106	1	no	—
304	C₂×C₄	—	106	1	no	—
305	C₂×C₄	—	106	1	no	—
306	C₂×C₄	—	106	1	no	—
307	C₂×C₄	—	106	1	no	—
308	C₂×C₄	—	212	2	no	—
309	C₂×C₄	—	212	2	no	—
310	C₂×C₄	—	212	2	no	—
311	C₂×C₄	—	212	2	no	—
312	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
313	C₂×C₄	#13133175	1	1	yes	characteristic
314	C₂×C₄	#13133177	1	1	yes	characteristic
315	C₂×C₄	#13133177	1	1	yes	characteristic
316	C₂×C₄	#13133179	1	1	yes	characteristic
317	C₂×C₄	#13133179	1	1	yes	characteristic
318	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
319	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
320	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
321	C₂×C₄	C₂×#11719912	1	2	yes	—
322	C₂×C₄	#13133223	1	2	yes	—
323	C₂×C₄	#13133225	1	2	yes	—
324	C₈	—	2	4	no	—
325	C₈	—	2	4	no	—
326	C₈	—	4	2	no	—
327	C₈	—	53	8	no	—
328	C₈	—	106	4	no	—
329	C₈	—	212	2	no	—
330	C₂²	—	106	2	no	—
331	C₂²	—	106	2	no	—
332	C₂²	—	106	2	no	—
333	C₂²	—	212	1	no	—
334	C₂²	—	212	1	no	—
335	C₂²	—	212	1	no	—
336	C₂²	—	212	1	no	—
337	C₂²	#16810470	1	1	yes	characteristic
338	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
339	C₂²	C₄×#11719912	1	1	yes	characteristic
340	C₂²	#16810575	1	1	yes	characteristic
341	C₂²	#16810575	1	1	yes	characteristic
342	C₂²	#16810577	1	1	yes	characteristic
343	C₂²	#16810580	1	1	yes	characteristic
344	C₄	—	106	2	no	—
345	C₄	—	212	1	no	—
346	C₄	—	212	1	no	—
347	C₄	C₂²⋊C₈×D₅₃	1	2	yes	—
348	C₄	#16810675	1	2	yes	—
349	C₄	#16810691	1	2	yes	—
350	C₄	#16810693	1	2	yes	—
351	C₄	#16810698	1	2	yes	—
352	C₄	#16810700	1	2	yes	—
353	C₂	—	212	2	no	—
354	C₂	C₈×#11719912	1	1	yes	characteristic
355	C₂	#20272994	1	1	yes	characteristic
356	C₂	#20273011	1	1	yes	characteristic
357	C₂	#20273019	1	1	yes	characteristic
358	C₂	#20274472	1	1	yes	characteristic
359	C₂	#20274495	1	1	yes	characteristic
360	C₂	#20274497	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1273	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂³×D₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1273	D₅₃
hypocenter	C₅₃	#8146
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ #1273

Complements

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

N	H ≅ G/N	type
4. C₂×#16810345	C₂	semidirect
5. C₂×C₈×#88322	C₂	semidirect
6. #20272542	C₂	semidirect
7. C₅₃×#1273	C₂	semidirect
30. C₂×#11719912	C₈	semidirect
87. C₂×C₈×C₅₃	M₁₆	semidirect
106. C₂×#88322	M₁₆	semidirect
148. #88322	C₂²⋊C₈	semidirect
179. #1273	D₅₃	semidirect
205. C₅₃	#8146	semidirect
278. C₂×C₈	#11719912	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719912	2
C₄×#13133129	2
C₂×#16810345	2
C₂×C₈×#88322	2
#20272542	2
C₅₃×#1273	2
#8146	53