Subgroups — #27537715

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283422	C₂	1	1	yes	maximal, characteristic
2	#20276548	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276547	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276544	C₂	1	2	yes	maximal, complemented
5	#20276534	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×C₂²⋊D₅₃	C₂	1	2	yes	maximal, complemented
7	#20276513	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276512	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276328	C₂	1	2	yes	maximal, complemented
10	#20276327	C₂	1	2	yes	maximal, complemented
11	D₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133223	C₂²	1	2	yes	—
13	#16811170	C₄	1	1	yes	characteristic
14	C₂×#13133157	C₂²	1	2	yes	—
15	C₂×#13133156	C₂²	1	2	yes	—
16	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
17	#16810701	C₄	1	1	yes	characteristic
18	#16810700	C₄	1	1	yes	characteristic
19	#16810698	C₄	1	2	yes	—
20	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
21	C₈×C₂²⋊D₅₃	C₂²	1	2	yes	—
22	#16810645	C₄	1	1	yes	characteristic
23	#16810588	C₄	1	2	yes	—
24	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊#11719908	C₂²	1	1	yes	characteristic
26	#16810441	C₂²	1	1	yes	characteristic
27	#16810441	C₂²	1	1	yes	characteristic
28	C₂²⋊#11719908	C₂²	1	2	yes	—
29	#16810402	C₂²	1	1	yes	characteristic
30	#16810402	C₂²	1	1	yes	characteristic
31	#16810402	C₂²	1	2	yes	—
32	#16810402	C₂²	1	2	yes	—
33	#16810402	C₂²	1	2	yes	—
34	C₄×#11719908	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₅₃	C₂²	1	2	yes	—
36	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
37	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719945	C₂×C₄	1	2	yes	—
42	#13133223	—	4	2	no	—
43	#13133223	—	4	2	no	—
44	C₂×#11719917	C₂×C₄	1	2	yes	—
45	C₂×#11719912	C₂×C₄	1	2	yes	—
46	C₂×#11719912	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₅₃	C₂³	1	2	yes	—
49	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
50	#13133216	C₈	1	1	yes	characteristic
51	#13133213	C₈	1	1	yes	characteristic
52	#13133203	C₈	1	2	yes	—
53	#13133192	C₈	1	1	yes	characteristic
54	#13133190	C₈	1	2	yes	—
55	#13133188	C₈	1	1	yes	characteristic
56	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
57	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
58	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
59	#13133157	—	4	2	no	—
60	#13133156	—	4	2	no	—
61	C₂×#11719908	C₂³	1	1	yes	characteristic
62	C₂×#11719908	C₂³	1	2	yes	—
63	C₂×#11719908	C₂³	1	1	yes	characteristic
64	C₁₆×D₅₃	—	4	2	no	—
65	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
66	#13133146	C₂×C₄	1	2	yes	—
67	C₂²⋊#88322	C₂×C₄	1	2	yes	—
68	#13133129	C₂×C₄	1	2	yes	—
69	#13133129	C₂×C₄	1	2	yes	—
70	C₄×#88322	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
72	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
74	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₁₆×C₅₃	—	4	2	no	—
76	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
78	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
85	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
86	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
87	#11719945	—	4	2	no	—
88	#11719945	—	4	2	no	—
89	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
90	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
91	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
92	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
93	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
96	C₂×C₈×C₅₃	—	2	2	no	—
97	C₂×C₈×C₅₃	—	2	2	no	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
102	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
103	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
105	#11719921	C₂×C₈	1	2	yes	—
106	#11719921	C₂×C₈	1	2	yes	—
107	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
108	#11719919	C₂×C₈	1	2	yes	—
109	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
110	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
111	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
112	#11719917	—	4	2	no	—
113	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
114	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
115	C₂×#88322	C₂²×C₄	1	2	yes	—
116	#11719912	—	4	2	no	—
117	#11719912	—	4	2	no	—
118	C₈×D₅₃	—	2	2	no	—
119	C₈×D₅₃	—	2	2	no	—
120	C₈×D₅₃	—	2	1	no	—
121	C₈×D₅₃	—	2	1	no	—
122	C₈×D₅₃	—	4	1	no	—
123	C₁₆×C₅₃	—	2	1	no	—
124	C₁₆×C₅₃	—	2	2	no	—
125	C₁₆×C₅₃	—	2	1	no	—
126	#11719908	—	2	1	no	—
127	#11719908	—	2	2	no	—
128	#11719908	—	2	1	no	—
129	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
130	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
131	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
132	D₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
139	C₂²⋊D₅₃	—	2	4	no	—
140	C₂²⋊D₅₃	—	4	2	no	—
141	C₂²⋊D₅₃	—	4	2	no	—
142	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
143	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
144	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
145	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
146	D₂₁₂	—	2	2	no	—
147	D₂₁₂	—	4	2	no	—
148	C₄×D₅₃	—	2	1	no	—
149	C₄×D₅₃	—	2	2	no	—
150	C₄×D₅₃	—	2	2	no	—
151	C₄×D₅₃	—	2	1	no	—
152	C₄×D₅₃	—	2	2	no	—
153	C₄×D₅₃	—	2	2	no	—
154	Dic₁₀₆	—	4	2	no	—
155	C₈×C₅₃	—	2	2	no	—
156	C₈×C₅₃	—	2	1	no	—
157	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
158	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
159	#88322	—	2	1	no	—
160	#88322	—	2	2	no	—
161	#88322	—	2	1	no	—
162	#33493	—	53	1	no	maximal, Sylow
163	C₂²×C₅₃	—	2	2	no	—
164	C₂²×C₅₃	—	2	2	no	—
165	C₂²×C₅₃	—	2	2	no	—
166	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
167	C₂×D₅₃	—	2	1	no	—
168	C₂×D₅₃	—	2	1	no	—
169	C₂×D₅₃	—	2	2	no	—
170	C₂×D₅₃	—	2	2	no	—
171	C₂×D₅₃	—	2	1	no	—
172	C₂×D₅₃	—	2	2	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	2	no	—
175	C₄×C₅₃	#504	1	2	yes	—
176	C₄×C₅₃	#518	1	1	yes	characteristic
177	C₄×C₅₃	#568	1	1	yes	characteristic
178	Dic₅₃	—	2	1	no	—
179	Dic₅₃	—	2	1	no	—
180	Dic₅₃	—	2	2	no	—
181	Dic₅₃	#504	1	2	yes	—
182	#2931	—	53	1	no	—
183	#2169	—	53	1	no	—
184	#2169	—	53	2	no	—
185	#2169	—	53	1	no	—
186	D₄×C₁₆	—	53	2	no	—
187	D₄×C₁₆	—	53	1	no	—
188	D₄×C₁₆	D₅₃	1	1	yes	characteristic, complemented
189	#2162	—	53	1	no	—
190	#2162	—	53	1	no	—
191	#2113	—	53	2	no	—
192	#2113	—	53	2	no	—
193	C₂×C₅₃	—	4	2	no	—
194	C₂×C₅₃	#2972	1	1	yes	characteristic
195	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
196	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
197	D₅₃	—	4	1	no	—
198	D₅₃	—	4	2	no	—
199	C₂×#131	—	53	2	no	—
200	#532	—	53	1	no	—
201	C₂×M₃₂	—	53	2	no	—
202	C₂×M₃₂	—	53	2	no	—
203	C₂²×C₁₆	—	53	2	no	—
204	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
205	#436	—	53	1	no	—
206	#436	—	53	1	no	—
207	#436	—	53	2	no	—
208	C₈×D₄	—	53	2	no	—
209	C₈×D₄	—	53	1	no	—
210	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
211	#423	—	53	1	no	—
212	#408	—	53	2	no	—
213	C₂×C₄×C₈	—	53	1	no	—
214	C₄⋊C₁₆	—	53	1	no	—
215	C₄⋊C₁₆	—	53	1	no	—
216	C₄⋊C₁₆	—	53	1	no	—
217	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
218	C₂²⋊C₁₆	—	53	2	no	—
219	C₂²⋊C₁₆	—	53	1	no	—
220	C₂²⋊C₁₆	—	53	2	no	—
221	C₂²⋊C₁₆	—	53	1	no	—
222	C₂²⋊C₁₆	—	53	2	no	—
223	C₂²⋊C₁₆	—	53	2	no	—
224	C₂²⋊C₁₆	C₂×D₅₃	1	2	yes	—
225	C₄×C₁₆	—	53	2	no	—
226	C₄×C₁₆	—	53	1	no	—
227	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
228	C₅₃	#33493	1	1	yes	characteristic, complemented, Sylow
229	C₂×#41	—	53	2	no	—
230	#131	—	212	2	no	—
231	#131	—	212	2	no	—
232	C₂×M₁₆	—	53	2	no	—
233	C₂×M₁₆	—	53	2	no	—
234	C₂×M₁₆	—	53	2	no	—
235	C₂²×C₈	—	53	1	no	—
236	C₂²×C₈	—	53	2	no	—
237	C₂²×C₈	—	53	1	no	—
238	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
239	C₄⋊Q₈	—	53	1	no	—
240	#127	—	53	1	no	—
241	#124	—	53	2	no	—
242	#121	—	53	1	no	—
243	#121	—	53	1	no	—
244	#121	—	53	2	no	—
245	C₄×D₄	—	53	2	no	—
246	C₄×D₄	—	53	1	no	—
247	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
248	C₂×C₄²	—	53	1	no	—
249	M₃₂	—	212	2	no	—
250	M₃₂	—	212	2	no	—
251	C₂×C₁₆	—	4	2	no	—
252	C₂×C₁₆	—	53	1	no	—
253	C₂×C₁₆	—	53	2	no	—
254	C₂×C₁₆	—	53	1	no	—
255	C₂×C₁₆	—	212	2	no	—
256	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
257	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
258	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
259	C₄⋊C₈	—	53	2	no	—
260	C₄⋊C₈	—	53	1	no	—
261	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
262	C₂²⋊C₈	—	53	2	no	—
263	C₂²⋊C₈	—	53	2	no	—
264	C₂²⋊C₈	—	53	2	no	—
265	C₂²⋊C₈	C₄×D₅₃	1	2	yes	—
266	C₄×C₈	—	53	1	no	—
267	C₄×C₈	—	53	1	no	—
268	C₄×C₈	—	53	1	no	—
269	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
270	#41	—	212	2	no	—
271	#41	—	212	2	no	—
272	C₂×Q₈	—	53	2	no	—
273	C₂×D₄	—	53	2	no	—
274	C₂×D₄	—	53	2	no	—
275	C₂×D₄	—	53	2	no	—
276	C₂×D₄	—	53	1	no	—
277	C₂×D₄	—	53	2	no	—
278	C₂×D₄	C₈×D₅₃	1	1	yes	characteristic
279	C₂²×C₄	—	53	1	no	—
280	C₂²×C₄	—	53	1	no	—
281	C₂²×C₄	—	53	2	no	—
282	C₂²×C₄	—	53	1	no	—
283	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
284	M₁₆	—	212	2	no	—
285	M₁₆	—	212	2	no	—
286	M₁₆	—	212	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	53	1	no	—
290	C₂×C₈	—	53	1	no	—
291	C₂×C₈	—	53	2	no	—
292	C₂×C₈	—	106	2	no	—
293	C₂×C₈	—	106	1	no	—
294	C₂×C₈	—	106	2	no	—
295	C₂×C₈	—	106	1	no	—
296	C₂×C₈	—	212	1	no	—
297	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
299	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
300	C₄⋊C₄	—	53	2	no	—
301	C₄⋊C₄	—	53	1	no	—
302	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
303	C₂²⋊C₄	—	53	2	no	—
304	C₂²⋊C₄	—	53	2	no	—
305	C₂²⋊C₄	—	53	2	no	—
306	C₂²⋊C₄	C₈×D₅₃	1	2	yes	—
307	C₄²	—	53	1	no	—
308	C₄²	—	53	1	no	—
309	C₄²	—	53	1	no	—
310	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
311	C₁₆	—	2	1	no	—
312	C₁₆	—	2	2	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	—	106	1	no	—
315	C₁₆	—	106	2	no	—
316	C₁₆	—	106	1	no	—
317	C₂³	—	53	2	no	—
318	C₂³	—	53	1	no	—
319	C₂³	C₂×C₈×D₅₃	1	2	yes	—
320	Q₈	—	212	2	no	—
321	D₄	—	2	2	no	—
322	D₄	—	106	4	no	—
323	D₄	—	106	2	no	—
324	D₄	—	212	2	no	—
325	D₄	—	212	2	no	—
326	D₄	—	212	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	53	1	no	—
330	C₂×C₄	—	53	1	no	—
331	C₂×C₄	—	53	1	no	—
332	C₂×C₄	—	53	2	no	—
333	C₂×C₄	—	106	2	no	—
334	C₂×C₄	—	106	2	no	—
335	C₂×C₄	—	106	1	no	—
336	C₂×C₄	—	106	2	no	—
337	C₂×C₄	—	106	1	no	—
338	C₂×C₄	—	106	2	no	—
339	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
340	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
341	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
343	C₈	—	2	2	no	—
344	C₈	—	2	1	no	—
345	C₈	—	106	2	no	—
346	C₈	—	106	1	no	—
347	C₈	—	106	1	no	—
348	C₈	#13133262	1	1	yes	characteristic
349	C₈	#13133263	1	1	yes	characteristic
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	106	1	no	—
354	C₂²	—	106	2	no	—
355	C₂²	—	106	1	no	—
356	C₂²	—	106	2	no	—
357	C₂²	—	106	1	no	—
358	C₂²	—	106	2	no	—
359	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	2	no	—
362	C₄	—	53	2	no	—
363	C₄	—	106	1	no	—
364	C₄	—	106	1	no	—
365	C₄	—	106	2	no	—
366	C₄	#16811039	1	2	yes	—
367	C₄	#16811095	1	1	yes	characteristic
368	C₄	#16811388	1	1	yes	characteristic
369	C₂	—	4	2	no	—
370	C₂	—	212	1	no	—
371	C₂	—	212	2	no	—
372	C₂	#20283736	1	1	yes	characteristic
373	C₂	#504×D₅₃	1	1	yes	characteristic
374	C₂	C₂×#16811039	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	D₄×C₁₆×C₅₃	C₂
prime radical	#16811170	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	D₄×C₁₆	D₅₃
hypocenter	C₅₃	#33493
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276548	C₂	semidirect
3. #20276547	C₂	semidirect
4. #20276544	C₂	semidirect
5. #20276534	C₂	semidirect
6. C₁₆×C₂²⋊D₅₃	C₂	semidirect
7. #20276513	C₂	semidirect
8. #20276512	C₂	semidirect
9. #20276328	C₂	semidirect
10. #20276327	C₂	semidirect
11. D₄×C₁₆×C₅₃	C₂	semidirect
188. D₄×C₁₆	D₅₃	semidirect
228. C₅₃	#33493	semidirect

Maximal subgroups

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

subgroup	index
#20283422	2
#20276548	2
#20276547	2
#20276544	2
#20276534	2
C₁₆×C₂²⋊D₅₃	2
#20276513	2
#20276512	2
#20276328	2
#20276327	2
D₄×C₁₆×C₅₃	2
#33493	53