Subgroups — #27274591

Order 13568.

554 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810674	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×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272542	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272542	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
9	#16810674	—	2	2	no	—
10	#16810674	—	2	2	no	—
11	C₂×#13133146	C₄	1	2	yes	—
12	C₂×#13133129	C₄	1	2	yes	—
13	C₂×#13133129	C₄	1	2	yes	—
14	C₂×#13133129	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	#16810344	—	2	2	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#322	—	2	2	no	—
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	C₂²×#88322	C₂×C₄	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	2	yes	—
30	C₂²×#88322	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄²×D₅₃	—	2	1	no	—
35	C₄²×D₅₃	—	2	1	no	—
36	C₄²×D₅₃	—	2	1	no	—
37	C₄²×D₅₃	—	2	1	no	—
38	#13133146	—	2	4	no	—
39	#13133129	—	2	4	no	—
40	#13133129	—	2	4	no	—
41	#13133129	—	2	4	no	—
42	C₄×#88322	—	2	1	no	—
43	C₄×#88322	—	2	1	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	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₅₃	—	2	1	no	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₄×C₈×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	D₄	1	2	yes	—
57	C₄×C₈×C₅₃	D₄	1	2	yes	—
58	C₂³×D₅₃	C₄²	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
63	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
65	C₂×C₄×D₅₃	—	2	1	no	—
66	C₂×C₄×D₅₃	—	2	1	no	—
67	C₂×C₄×D₅₃	—	2	2	no	—
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×C₄×D₅₃	—	2	1	no	—
70	C₂×C₄×D₅₃	—	2	1	no	—
71	C₂×C₄×D₅₃	—	2	1	no	—
72	C₂×C₄×D₅₃	—	2	2	no	—
73	C₂×C₄×D₅₃	—	2	2	no	—
74	C₂×C₄×D₅₃	—	2	2	no	—
75	C₂×C₄×D₅₃	—	2	2	no	—
76	C₂×C₄×D₅₃	M₁₆	1	4	yes	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
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	C₂²⋊C₄	1	4	yes	—
85	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₅₃	#41	1	1	yes	characteristic
89	C₄²×C₅₃	#41	1	1	yes	characteristic
90	C₄×Dic₅₃	—	2	1	no	—
91	C₄×Dic₅₃	—	2	1	no	—
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	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	2	2	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	—	2	2	no	—
104	C₂×#88322	—	2	2	no	—
105	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
106	C₂²×D₅₃	—	2	1	no	—
107	C₂²×D₅₃	—	2	1	no	—
108	C₂²×D₅₃	—	2	1	no	—
109	C₂²×D₅₃	—	2	1	no	—
110	C₂²×D₅₃	—	2	1	no	—
111	C₂²×D₅₃	—	2	1	no	—
112	C₂²×D₅₃	#97	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
121	C₂×C₄×C₅₃	#131	1	2	yes	—
122	C₂×C₄×C₅₃	#131	1	2	yes	—
123	C₂×C₄×C₅₃	#131	1	2	yes	—
124	C₂×C₄×C₅₃	#131	1	2	yes	—
125	C₂×Dic₅₃	—	2	1	no	—
126	C₂×Dic₅₃	—	2	1	no	—
127	C₂×Dic₅₃	—	2	1	no	—
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	#97	1	2	yes	—
134	C₄×D₅₃	—	2	4	no	—
135	C₄×D₅₃	—	2	4	no	—
136	C₄×D₅₃	—	4	2	no	—
137	C₄×D₅₃	—	4	2	no	—
138	C₄×D₅₃	—	4	2	no	—
139	C₄×D₅₃	—	4	2	no	—
140	C₈×C₅₃	—	2	4	no	—
141	C₈×C₅₃	—	2	4	no	—
142	C₈×C₅₃	—	4	2	no	—
143	#88322	—	4	2	no	—
144	#88322	—	4	2	no	—
145	#88322	—	4	2	no	—
146	#8149	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
149	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
150	C₂²×C₅₃	#405	1	1	yes	characteristic
151	C₂²×C₅₃	#405	1	1	yes	characteristic
152	C₂²×C₅₃	#405	1	1	yes	characteristic
153	C₂²×C₅₃	#405	1	1	yes	characteristic
154	C₂×D₅₃	—	2	2	no	—
155	C₂×D₅₃	—	2	2	no	—
156	C₂×D₅₃	—	2	2	no	—
157	C₂×D₅₃	—	4	1	no	—
158	C₂×D₅₃	—	4	1	no	—
159	C₂×D₅₃	—	4	1	no	—
160	C₂×D₅₃	—	4	1	no	—
161	C₄×C₅₃	#408	1	2	yes	—
162	C₄×C₅₃	#433	1	2	yes	—
163	C₄×C₅₃	#435	1	2	yes	—
164	C₄×C₅₃	#435	1	2	yes	—
165	C₄×C₅₃	#435	1	2	yes	—
166	C₄×C₅₃	#436	1	2	yes	—
167	Dic₅₃	—	2	2	no	—
168	Dic₅₃	—	4	1	no	—
169	Dic₅₃	—	4	1	no	—
170	C₂×#431	—	53	1	no	—
171	C₄×C₂²⋊C₈	—	53	1	no	—
172	C₄×C₂²⋊C₈	—	53	1	no	—
173	C₂×#322	—	53	1	no	—
174	C₂×#322	D₅₃	1	1	yes	characteristic, complemented
175	#1273	—	53	1	no	—
176	#1273	—	53	1	no	—
177	C₂×C₅₃	#1451	1	1	yes	characteristic
178	C₂×C₅₃	#1452	1	1	yes	characteristic
179	C₂×C₅₃	#1453	1	1	yes	characteristic
180	C₂×C₅₃	#1454	1	1	yes	characteristic
181	C₂×C₅₃	#1750	1	1	yes	characteristic
182	C₂×C₅₃	#1752	1	1	yes	characteristic
183	C₂×C₅₃	#1753	1	1	yes	characteristic
184	D₅₃	—	4	2	no	—
185	C₂²×C₄²	—	53	1	no	—
186	#431	—	106	2	no	—
187	#431	—	106	2	no	—
188	C₂×C₄⋊C₈	—	53	2	no	—
189	C₂×C₂²⋊C₈	—	53	2	no	—
190	C₂×C₂²⋊C₈	—	53	2	no	—
191	C₂×C₂²⋊C₈	—	53	2	no	—
192	C₂×C₄×C₈	—	53	1	no	—
193	C₂×C₄×C₈	—	53	1	no	—
194	C₂×C₄×C₈	—	53	1	no	—
195	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
197	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
198	#336	—	53	2	no	—
199	#336	—	53	2	no	—
200	#322	—	2	2	no	—
201	#322	—	106	2	no	—
202	C₅₃	#8149	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	—	53	1	no	—
204	C₂³×C₄	—	53	1	no	—
205	C₂³×C₄	—	53	1	no	—
206	C₂²×C₈	—	53	2	no	—
207	C₂²×C₈	—	53	2	no	—
208	C₂²×C₈	—	53	2	no	—
209	C₂²×C₈	C₄×D₅₃	1	2	yes	—
210	C₂²×C₈	C₄×D₅₃	1	2	yes	—
211	C₂²×C₈	C₄×D₅₃	1	2	yes	—
212	C₂×C₄²	—	53	1	no	—
213	C₂×C₄²	—	53	1	no	—
214	C₂×C₄²	—	53	1	no	—
215	C₂×C₄²	—	106	1	no	—
216	C₂×C₄²	—	106	1	no	—
217	C₂×C₄²	—	106	1	no	—
218	C₂×C₄²	—	106	1	no	—
219	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
220	C₄⋊C₈	—	106	4	no	—
221	C₂²⋊C₈	—	106	4	no	—
222	C₂²⋊C₈	—	106	4	no	—
223	C₂²⋊C₈	—	106	4	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	106	1	no	—
229	C₄×C₈	—	106	1	no	—
230	C₄×C₈	—	106	1	no	—
231	C₄×C₈	—	106	1	no	—
232	C₄×C₈	—	106	1	no	—
233	C₄×C₈	—	106	1	no	—
234	C₄×C₈	D₂₁₂	1	2	yes	—
235	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
236	C₂⁴	—	53	1	no	—
237	C₂²×C₄	—	53	1	no	—
238	C₂²×C₄	—	53	4	no	—
239	C₂²×C₄	—	53	1	no	—
240	C₂²×C₄	—	53	1	no	—
241	C₂²×C₄	—	106	1	no	—
242	C₂²×C₄	—	106	1	no	—
243	C₂²×C₄	—	106	2	no	—
244	C₂²×C₄	—	106	1	no	—
245	C₂²×C₄	—	106	1	no	—
246	C₂²×C₄	—	106	1	no	—
247	C₂²×C₄	—	106	1	no	—
248	C₂²×C₄	—	106	2	no	—
249	C₂²×C₄	—	106	2	no	—
250	C₂²×C₄	—	106	2	no	—
251	C₂²×C₄	—	106	2	no	—
252	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	106	2	no	—
263	C₂×C₈	—	106	2	no	—
264	C₂×C₈	—	106	2	no	—
265	C₂×C₈	—	106	2	no	—
266	C₂×C₈	—	106	2	no	—
267	C₂×C₈	—	106	2	no	—
268	C₂×C₈	—	106	2	no	—
269	C₂×C₈	—	106	2	no	—
270	C₂×C₈	—	106	2	no	—
271	C₂×C₈	#11719912	1	4	yes	—
272	C₂×C₈	#11719921	1	4	yes	—
273	C₄²	—	106	1	no	—
274	C₄²	—	106	1	no	—
275	C₄²	—	106	1	no	—
276	C₄²	—	106	1	no	—
277	C₄²	—	106	1	no	—
278	C₄²	—	106	1	no	—
279	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
280	C₄²	#11719945	1	1	yes	characteristic
281	C₄²	#11719945	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
283	C₂³	—	53	2	no	—
284	C₂³	—	106	1	no	—
285	C₂³	—	106	1	no	—
286	C₂³	—	106	1	no	—
287	C₂³	—	106	1	no	—
288	C₂³	—	106	1	no	—
289	C₂³	—	106	1	no	—
290	C₂³	C₄²×D₅₃	1	1	yes	characteristic
291	C₂×C₄	—	53	2	no	—
292	C₂×C₄	—	106	1	no	—
293	C₂×C₄	—	106	1	no	—
294	C₂×C₄	—	106	4	no	—
295	C₂×C₄	—	106	4	no	—
296	C₂×C₄	—	106	1	no	—
297	C₂×C₄	—	106	1	no	—
298	C₂×C₄	—	106	1	no	—
299	C₂×C₄	—	106	1	no	—
300	C₂×C₄	—	106	1	no	—
301	C₂×C₄	—	106	1	no	—
302	C₂×C₄	—	212	2	no	—
303	C₂×C₄	—	212	2	no	—
304	C₂×C₄	—	212	2	no	—
305	C₂×C₄	—	212	2	no	—
306	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
307	C₂×C₄	#13133174	1	1	yes	characteristic
308	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
309	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
311	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
312	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
313	C₂×C₄	C₂×#11719912	1	2	yes	—
314	C₂×C₄	#13133223	1	2	yes	—
315	C₂×C₄	#13133225	1	2	yes	—
316	C₂×C₄	#13133225	1	2	yes	—
317	C₂×C₄	#13133225	1	2	yes	—
318	C₈	—	2	4	no	—
319	C₈	—	2	4	no	—
320	C₈	—	4	2	no	—
321	C₈	—	212	2	no	—
322	C₈	—	212	2	no	—
323	C₈	—	212	2	no	—
324	C₂²	—	106	2	no	—
325	C₂²	—	106	2	no	—
326	C₂²	—	106	2	no	—
327	C₂²	—	212	1	no	—
328	C₂²	—	212	1	no	—
329	C₂²	—	212	1	no	—
330	C₂²	—	212	1	no	—
331	C₂²	C₄×#11719921	1	1	yes	characteristic
332	C₂²	#97×D₅₃	1	1	yes	characteristic
333	C₂²	C₄×#11719912	1	1	yes	characteristic
334	C₂²	#16810577	1	1	yes	characteristic
335	C₂²	#16810580	1	1	yes	characteristic
336	C₂²	#16810580	1	1	yes	characteristic
337	C₂²	#16810580	1	1	yes	characteristic
338	C₄	—	106	2	no	—
339	C₄	—	212	1	no	—
340	C₄	—	212	1	no	—
341	C₄	#16810591	1	2	yes	—
342	C₄	#16810685	1	2	yes	—
343	C₄	#16810692	1	2	yes	—
344	C₄	#16810694	1	2	yes	—
345	C₄	#16810696	1	2	yes	—
346	C₄	#16810702	1	2	yes	—
347	C₂	—	212	2	no	—
348	C₂	#20273001	1	1	yes	characteristic
349	C₂	#20273006	1	1	yes	characteristic
350	C₂	#20273013	1	1	yes	characteristic
351	C₂	#20273016	1	1	yes	characteristic
352	C₂	#20274484	1	1	yes	characteristic
353	C₂	#20274492	1	1	yes	characteristic
354	C₂	#20274498	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂×C₄²
Fitting	C₂×C₅₃×#322	C₂
prime radical	C₂³×D₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#8149
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. C₂×#16810344	C₂	semidirect
5. #20272542	C₂	semidirect
6. #20272542	C₂	semidirect
7. C₂×C₅₃×#322	C₂	semidirect
174. C₂×#322	D₅₃	semidirect
202. C₅₃	#8149	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810674	2
C₄×#13133129	2
C₂×#16810344	2
#20272542	2
C₂×C₅₃×#322	2
#8149	53