Subgroups — #27300990

Order 13568.

488 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133156	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283411	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810408	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810406	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276325	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272804	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₂²	1	1	yes	characteristic
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	#16811135	C₄	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₂×#13133156	—	2	2	no	—
13	C₂×#13133156	—	2	2	no	—
14	C₂×#13133156	—	2	2	no	—
15	C₂²×#11719908	C₂²	1	1	yes	characteristic
16	#16810696	—	2	2	no	—
17	C₈×D₂₁₂	—	2	2	no	—
18	#16810650	C₄	1	1	yes	characteristic
19	#16810591	C₄	1	1	yes	characteristic
20	#16810586	C₄	1	1	yes	characteristic
21	#16810408	—	2	2	no	—
22	#16810406	—	2	2	no	—
23	#16810402	—	2	2	no	—
24	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#405	C₂²	1	1	yes	characteristic
27	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	#13133223	—	2	1	no	—
30	#13133223	—	2	1	no	—
31	#13133223	—	2	1	no	—
32	#13133223	—	2	1	no	—
33	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
34	C₂×#11719912	—	2	1	no	—
35	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₅₃	—	2	1	no	—
37	C₂×C₈×D₅₃	—	2	1	no	—
38	C₂×C₈×D₅₃	—	2	1	no	—
39	C₂×C₈×D₅₃	—	2	1	no	—
40	C₂×C₈×D₅₃	—	2	1	no	—
41	C₂×C₈×D₅₃	D₄	1	2	yes	—
42	C₂×C₈×D₅₃	D₄	1	2	yes	—
43	#13133201	—	2	1	no	—
44	#13133197	C₈	1	2	yes	—
45	#13133192	C₈	1	2	yes	—
46	#13133187	—	2	1	no	—
47	C₄×D₂₁₂	—	2	2	no	—
48	#13133157	—	2	1	no	—
49	#13133157	—	2	1	no	—
50	#13133156	—	2	8	no	—
51	C₂×#11719908	—	2	1	no	—
52	C₂×#11719908	—	2	1	no	—
53	C₂×#11719908	—	2	1	no	—
54	C₂×#11719908	D₄	1	2	yes	complemented
55	C₄⋊#88322	—	2	2	no	—
56	#13133129	—	2	2	no	—
57	#13133129	—	2	2	no	—
58	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
61	M₃₂×C₅₃	—	2	1	no	—
62	M₃₂×C₅₃	—	2	1	no	—
63	C₂×C₁₆×C₅₃	—	2	1	no	—
64	C₂×C₁₆×C₅₃	—	2	1	no	—
65	C₂×C₁₆×C₅₃	—	2	1	no	—
66	C₂×C₁₆×C₅₃	D₄	1	2	yes	complemented
67	#97×C₅₃	—	2	1	no	—
68	C₄×C₈×C₅₃	—	2	1	no	—
69	C₂³×D₅₃	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊D₅₃	—	2	1	no	—
72	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
73	#11719945	—	2	1	no	—
74	#11719945	—	2	1	no	—
75	#11719945	—	2	1	no	—
76	#11719945	—	2	1	no	—
77	C₂×D₂₁₂	—	2	2	no	—
78	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₅₃	—	2	1	no	—
80	C₂×C₄×D₅₃	—	2	1	no	—
81	C₂×C₄×D₅₃	—	2	1	no	—
82	C₂×C₄×D₅₃	—	2	1	no	—
83	C₂×C₄×D₅₃	—	2	1	no	—
84	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
85	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
86	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
87	M₁₆×C₅₃	—	2	1	no	—
88	M₁₆×C₅₃	—	2	1	no	—
89	C₂×C₈×C₅₃	—	2	1	no	—
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
97	C₂²⋊C₄×C₅₃	—	2	1	no	—
98	C₄²×C₅₃	—	2	1	no	—
99	#11719921	—	2	2	no	—
100	#11719921	—	2	2	no	—
101	C₄⋊Dic₅₃	—	2	2	no	—
102	#11719917	—	2	1	no	—
103	#11719917	—	2	1	no	—
104	C₂×#88322	—	2	1	no	—
105	C₂×#88322	—	2	1	no	—
106	C₂×#88322	—	2	1	no	—
107	C₂×#88322	C₂²⋊C₄	1	2	yes	—
108	#11719912	—	4	1	no	—
109	#11719912	—	4	1	no	—
110	C₈×D₅₃	—	2	2	no	—
111	C₈×D₅₃	—	2	2	no	—
112	C₈×D₅₃	—	2	2	no	—
113	C₈×D₅₃	—	2	2	no	—
114	C₈×D₅₃	—	4	1	no	—
115	C₈×D₅₃	—	4	1	no	—
116	C₁₆×C₅₃	—	2	2	no	—
117	C₁₆×C₅₃	—	4	1	no	—
118	#11719908	—	2	2	no	—
119	#11719908	—	4	1	no	—
120	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂²×D₅₃	—	2	1	no	—
122	C₂²×D₅₃	—	2	1	no	—
123	C₂²×D₅₃	—	2	1	no	—
124	C₂²×D₅₃	—	4	2	no	—
125	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
133	C₂²⋊D₅₃	—	4	1	no	—
134	C₂²⋊D₅₃	—	4	1	no	—
135	C₂×Dic₅₃	—	2	1	no	—
136	C₂×Dic₅₃	—	2	1	no	—
137	C₂×Dic₅₃	—	2	1	no	—
138	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
139	D₂₁₂	—	2	1	no	—
140	D₂₁₂	—	2	1	no	—
141	D₂₁₂	—	4	4	no	—
142	C₄×D₅₃	—	2	2	no	—
143	C₄×D₅₃	—	2	2	no	—
144	C₄×D₅₃	—	2	2	no	—
145	C₄×D₅₃	—	2	2	no	—
146	C₄×D₅₃	—	4	1	no	—
147	C₄×D₅₃	—	4	1	no	—
148	Dic₁₀₆	—	2	1	no	—
149	Dic₁₀₆	—	2	1	no	—
150	C₈×C₅₃	—	4	1	no	—
151	C₈×C₅₃	#120	1	1	yes	characteristic
152	C₈×C₅₃	#121	1	1	yes	characteristic
153	C₈×C₅₃	#121	1	1	yes	characteristic
154	C₈×C₅₃	#123	1	1	yes	characteristic
155	#88322	—	2	2	no	—
156	#88322	—	4	1	no	—
157	#12984	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	—	2	2	no	—
159	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
160	C₂²×C₅₃	#435	1	1	yes	characteristic
161	C₂²×C₅₃	#435	1	1	yes	characteristic
162	C₂×D₅₃	—	2	2	no	—
163	C₂×D₅₃	—	2	2	no	—
164	C₂×D₅₃	—	2	2	no	—
165	C₂×D₅₃	—	4	2	no	—
166	C₂×D₅₃	—	4	2	no	—
167	C₂×D₅₃	—	4	2	no	—
168	C₂×D₅₃	—	4	1	no	—
169	C₄×C₅₃	—	4	2	no	—
170	C₄×C₅₃	#386	1	1	yes	characteristic
171	C₄×C₅₃	#408	1	1	yes	characteristic
172	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
173	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
174	Dic₅₃	—	2	2	no	—
175	Dic₅₃	—	4	1	no	—
176	C₂²×M₃₂	—	53	1	no	—
177	#2929	—	53	1	no	—
178	C₂×#350	—	53	1	no	—
179	C₂×#350	—	53	1	no	—
180	#2112	—	53	1	no	—
181	#1379	—	53	1	no	—
182	#1379	D₅₃	1	1	yes	characteristic, complemented
183	C₂×C₅₃	—	2	1	no	—
184	C₂×C₅₃	—	2	1	no	—
185	C₂×C₅₃	#1869	1	1	yes	characteristic
186	C₂×C₅₃	#2117	1	2	yes	—
187	D₅₃	—	4	2	no	—
188	D₅₃	—	8	1	no	—
189	C₂×#131	—	53	1	no	—
190	C₂³×C₈	—	53	1	no	—
191	#525	—	53	1	no	—
192	C₂×M₃₂	—	53	1	no	—
193	C₂×M₃₂	—	106	2	no	—
194	C₂×M₃₂	—	106	2	no	—
195	C₂×M₃₂	—	106	2	no	—
196	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
197	C₂²×C₁₆	—	53	1	no	—
198	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
199	#435	—	106	2	no	—
200	C₈×D₄	—	106	2	no	—
201	#424	—	53	1	no	—
202	#408	—	53	1	no	—
203	#407	—	53	1	no	—
204	#405	C₂×D₅₃	1	1	yes	characteristic
205	#350	—	106	2	no	—
206	#350	—	106	2	no	—
207	C₂²⋊C₁₆	—	106	2	no	—
208	C₅₃	#12984	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	53	1	no	—
210	C₂³×C₄	—	53	1	no	—
211	#131	—	106	1	no	—
212	#131	—	106	1	no	—
213	#131	—	106	1	no	—
214	#131	—	106	1	no	—
215	C₂×M₁₆	—	53	1	no	—
216	C₂×M₁₆	—	106	1	no	—
217	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	2	no	—
220	C₂²×C₈	—	53	2	no	—
221	C₂²×C₈	—	106	1	no	—
222	C₂²×C₈	—	106	1	no	—
223	C₂²×C₈	—	106	1	no	—
224	C₂²×C₈	—	106	1	no	—
225	C₂²×C₈	—	106	1	no	—
226	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
227	#123	—	106	1	no	—
228	C₂²⋊Q₈	—	53	2	no	—
229	#121	—	53	2	no	—
230	#120	—	106	1	no	—
231	C₄×D₄	—	106	2	no	—
232	#117	C₄×D₅₃	1	1	yes	characteristic
233	M₃₂	—	2	1	no	—
234	M₃₂	—	2	1	no	—
235	M₃₂	—	106	8	no	—
236	M₃₂	—	106	1	no	—
237	M₃₂	—	106	1	no	—
238	C₂×C₁₆	—	2	1	no	—
239	C₂×C₁₆	—	2	1	no	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	—	53	2	no	—
242	C₂×C₁₆	—	106	1	no	—
243	C₂×C₁₆	—	106	1	no	—
244	C₂×C₁₆	—	106	1	no	—
245	C₂×C₁₆	D₂₁₂	1	2	yes	complemented
246	C₄⋊C₈	—	106	2	no	—
247	C₂²⋊C₈	—	106	2	no	—
248	C₂²⋊C₈	—	106	2	no	—
249	#97	—	2	1	no	—
250	C₄×C₈	—	2	1	no	—
251	C₂⁴	—	53	1	no	—
252	#41	—	106	1	no	—
253	#41	—	106	1	no	—
254	#41	—	106	1	no	—
255	#41	—	106	1	no	—
256	C₂×Q₈	—	53	1	no	—
257	C₂×D₄	—	53	1	no	—
258	C₂×D₄	—	106	2	no	—
259	C₂×D₄	—	106	1	no	—
260	C₂²×C₄	—	53	1	no	—
261	C₂²×C₄	—	53	2	no	—
262	C₂²×C₄	—	53	2	no	—
263	C₂²×C₄	—	106	1	no	—
264	C₂²×C₄	—	106	1	no	—
265	C₂²×C₄	—	106	1	no	—
266	C₂²×C₄	—	106	1	no	—
267	C₂²×C₄	—	106	1	no	—
268	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	106	1	no	—
272	M₁₆	—	106	1	no	—
273	M₁₆	—	212	1	no	—
274	M₁₆	—	212	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	53	2	no	—
277	C₂×C₈	—	106	1	no	—
278	C₂×C₈	—	106	1	no	—
279	C₂×C₈	—	106	2	no	—
280	C₂×C₈	—	106	2	no	—
281	C₂×C₈	—	106	2	no	—
282	C₂×C₈	—	106	2	no	—
283	C₂×C₈	—	106	1	no	—
284	C₂×C₈	—	212	1	no	—
285	C₂×C₈	—	212	1	no	—
286	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
287	C₂×C₈	#11719945	1	1	yes	characteristic
288	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
289	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
290	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
291	C₂×C₈	#11719949	1	1	yes	characteristic
292	C₄⋊C₄	—	106	2	no	—
293	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
294	C₂²⋊C₄	—	2	1	no	—
295	C₂²⋊C₄	—	106	2	no	—
296	C₂²⋊C₄	—	106	2	no	—
297	C₄²	—	2	1	no	—
298	C₁₆	—	2	2	no	—
299	C₁₆	—	4	1	no	—
300	C₁₆	—	106	2	no	—
301	C₁₆	—	212	1	no	—
302	C₂³	—	53	2	no	—
303	C₂³	—	106	1	no	—
304	C₂³	—	106	1	no	—
305	C₂³	—	106	1	no	—
306	C₂³	—	212	2	no	—
307	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
308	Q₈	—	106	1	no	—
309	Q₈	—	106	1	no	—
310	D₄	—	106	1	no	—
311	D₄	—	106	1	no	—
312	D₄	—	212	4	no	—
313	D₄	—	212	1	no	—
314	D₄	—	212	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	53	2	no	—
317	C₂×C₄	—	106	1	no	—
318	C₂×C₄	—	106	1	no	—
319	C₂×C₄	—	106	2	no	—
320	C₂×C₄	—	106	2	no	—
321	C₂×C₄	—	106	2	no	—
322	C₂×C₄	—	106	2	no	—
323	C₂×C₄	—	106	1	no	—
324	C₂×C₄	—	212	1	no	—
325	C₂×C₄	—	212	1	no	—
326	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
327	C₂×C₄	#13133179	1	1	yes	characteristic
328	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
329	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
330	C₂×C₄	#13133223	1	1	yes	characteristic
331	C₂×C₄	#13133225	1	1	yes	characteristic
332	C₈	—	4	1	no	—
333	C₈	—	106	2	no	—
334	C₈	—	212	1	no	—
335	C₈	#13133187	1	1	yes	characteristic
336	C₈	#13133190	1	1	yes	characteristic
337	C₈	#13133192	1	1	yes	characteristic
338	C₈	#13133202	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	106	2	no	—
341	C₂²	—	106	2	no	—
342	C₂²	—	106	2	no	—
343	C₂²	—	212	2	no	—
344	C₂²	—	212	2	no	—
345	C₂²	—	212	1	no	—
346	C₂²	—	212	2	no	—
347	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
348	C₂²	#16810693	1	1	yes	characteristic
349	C₂²	#16810696	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	106	2	no	—
352	C₄	—	212	1	no	—
353	C₄	#16810501	1	1	yes	characteristic
354	C₄	#16810588	1	1	yes	characteristic
355	C₄	#16810688	1	1	yes	characteristic
356	C₄	C₈×D₂₁₂	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	424	1	no	—
360	C₂	—	212	2	no	—
361	C₂	#20275006	1	1	yes	characteristic
362	C₂	#20276341	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1379	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	#16811135	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1379	D₅₃
hypocenter	C₅₃	#12984
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133156	C₂	semidirect
3. C₂×#16810408	C₂	semidirect
4. C₂×#16810406	C₂	semidirect
5. #20276325	C₂	semidirect
6. #20272804	C₂	semidirect
7. C₅₃×#1379	C₂	semidirect
54. C₂×#11719908	D₄	semidirect
66. C₂×C₁₆×C₅₃	D₄	semidirect
182. #1379	D₅₃	semidirect
208. C₅₃	#12984	semidirect
245. C₂×C₁₆	D₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133156	2
#20283411	2
C₂×#16810408	2
C₂×#16810406	2
#20276325	2
#20272804	2
C₅₃×#1379	2
#12984	53