Subgroups — #27537629

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283692	C₂	1	1	yes	maximal, characteristic
2	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276546	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276539	C₂	1	2	yes	maximal, complemented
5	C₁₆×D₂₁₂	C₂	1	2	yes	maximal, complemented
6	C₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276521	C₂	1	2	yes	maximal, complemented
8	#20276517	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276310	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811160	C₄	1	1	yes	characteristic
13	C₂×#13133156	C₂²	1	2	yes	—
14	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
16	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
17	#16810754	C₄	1	1	yes	characteristic
18	#16810700	C₄	1	1	yes	characteristic
19	#16810693	C₄	1	2	yes	—
20	C₈×D₂₁₂	C₂²	1	2	yes	—
21	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
22	#16810683	C₄	1	2	yes	—
23	#16810679	C₄	1	1	yes	characteristic
24	#16810577	C₂²	1	2	yes	—
25	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
26	C₄⋊#11719908	C₂²	1	1	yes	characteristic
27	C₄⋊#11719908	C₂²	1	2	yes	—
28	#16810402	C₂²	1	2	yes	—
29	#16810402	C₂²	1	2	yes	—
30	#16810402	C₂²	1	2	yes	—
31	#16810399	C₂²	1	2	yes	—
32	#16810398	C₂²	1	2	yes	—
33	C₄×#11719908	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
36	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
41	C₂×#11719912	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₅₃	C₂³	1	2	yes	—
45	#13133212	C₈	1	2	yes	—
46	#13133208	C₈	1	1	yes	characteristic
47	#13133206	C₈	1	1	yes	characteristic
48	#13133202	C₈	1	2	yes	—
49	#13133193	C₈	1	1	yes	characteristic
50	#13133192	C₈	1	1	yes	characteristic
51	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
52	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
53	#13133174	C₂×C₄	1	2	yes	—
54	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
55	#13133156	—	2	4	no	—
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	C₂×#11719908	C₂³	1	2	yes	—
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₁₆×D₅₃	—	2	2	no	—
60	C₁₆×D₅₃	—	2	4	no	—
61	C₁₆×D₅₃	—	2	2	no	—
62	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
63	C₄⋊#88322	C₂×C₄	1	2	yes	—
64	#13133129	C₂×C₄	1	2	yes	—
65	#13133129	C₂×C₄	1	2	yes	—
66	#13133129	C₂×C₄	1	2	yes	—
67	#13133128	C₂×C₄	1	2	yes	—
68	#13133127	C₂×C₄	1	2	yes	—
69	C₄×#88322	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
75	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂³	1	2	yes	—
80	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
81	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
82	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
92	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
94	#11719921	C₂×C₈	1	2	yes	—
95	#11719921	C₂×C₈	1	2	yes	—
96	#11719921	C₂×C₈	1	2	yes	—
97	#11719921	C₂×C₈	1	2	yes	—
98	#11719921	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
100	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
101	#11719919	C₂×C₈	1	2	yes	—
102	#11719919	C₂×C₈	1	2	yes	—
103	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
107	C₂×#88322	C₂²×C₄	1	2	yes	—
108	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
109	#11719912	—	2	4	no	—
110	C₈×D₅₃	—	2	1	no	—
111	C₈×D₅₃	—	2	1	no	—
112	C₈×D₅₃	—	2	2	no	—
113	C₈×D₅₃	—	2	2	no	—
114	C₈×D₅₃	—	2	2	no	—
115	C₁₆×C₅₃	—	2	1	no	—
116	C₁₆×C₅₃	—	2	2	no	—
117	C₁₆×C₅₃	#41	1	2	yes	—
118	#11719908	—	2	1	no	—
119	#11719908	—	2	2	no	—
120	#11719908	#41	1	2	yes	—
121	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
122	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
133	D₂₁₂	—	2	4	no	—
134	D₂₁₂	—	2	2	no	—
135	C₄×D₅₃	—	2	2	no	—
136	C₄×D₅₃	—	2	1	no	—
137	C₄×D₅₃	—	2	2	no	—
138	C₄×D₅₃	—	2	1	no	—
139	C₄×D₅₃	—	2	2	no	—
140	C₄×D₅₃	—	2	2	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	2	no	—
143	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
144	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
145	#88322	—	2	1	no	—
146	#88322	—	2	2	no	—
147	#88322	#131	1	2	yes	—
148	#33480	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
150	C₂×D₅₃	—	2	1	no	—
151	C₂×D₅₃	—	2	1	no	—
152	C₂×D₅₃	—	2	2	no	—
153	C₂×D₅₃	—	2	2	no	—
154	C₂×D₅₃	—	2	2	no	—
155	C₂×D₅₃	#504	1	2	yes	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	#504	1	2	yes	—
160	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
161	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
162	Dic₅₃	—	2	1	no	—
163	Dic₅₃	—	2	1	no	—
164	Dic₅₃	—	2	1	no	—
165	Dic₅₃	—	2	2	no	—
166	#2966	—	53	1	no	—
167	#2183	—	53	1	no	—
168	Q₈×C₁₆	D₅₃	1	1	yes	characteristic, complemented
169	#2169	—	53	1	no	—
170	#2168	—	53	2	no	—
171	D₄×C₁₆	—	53	1	no	—
172	D₄×C₁₆	—	53	2	no	—
173	#2164	—	53	2	no	—
174	#2163	—	53	1	no	—
175	#2108	—	53	2	no	—
176	C₂×C₄×C₁₆	—	53	1	no	—
177	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
178	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
179	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
180	D₅₃	—	2	2	no	—
181	D₅₃	—	4	2	no	—
182	#529	—	53	1	no	—
183	C₂×M₃₂	—	53	2	no	—
184	C₂²×C₁₆	—	53	1	no	—
185	C₂²×C₁₆	—	53	1	no	—
186	C₂²×C₁₆	—	53	2	no	—
187	#446	—	53	1	no	—
188	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
189	#436	—	53	1	no	—
190	#435	—	53	2	no	—
191	C₈×D₄	—	53	2	no	—
192	C₈×D₄	—	53	1	no	—
193	#433	—	53	2	no	—
194	#432	—	53	1	no	—
195	#405	—	53	2	no	—
196	C₂×C₄×C₈	—	53	1	no	—
197	C₄⋊C₁₆	—	53	1	no	—
198	C₄⋊C₁₆	—	53	2	no	—
199	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
200	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
201	C₂²⋊C₁₆	—	53	2	no	—
202	C₂²⋊C₁₆	—	53	2	no	—
203	C₂²⋊C₁₆	—	53	2	no	—
204	#346	—	53	2	no	—
205	#346	—	53	2	no	—
206	C₄×C₁₆	—	53	1	no	—
207	C₄×C₁₆	—	53	1	no	—
208	C₄×C₁₆	—	53	1	no	—
209	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
210	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
211	C₅₃	#33480	1	1	yes	characteristic, complemented, Sylow
212	C₂×M₁₆	—	53	2	no	—
213	C₂²×C₈	—	53	1	no	—
214	C₂²×C₈	—	53	1	no	—
215	C₂²×C₈	—	53	2	no	—
216	#126	—	53	2	no	—
217	#125	—	53	1	no	—
218	#124	—	53	1	no	—
219	#123	—	53	2	no	—
220	C₂²⋊Q₈	—	53	1	no	—
221	#121	—	53	1	no	—
222	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
223	C₄×D₄	—	53	2	no	—
224	C₄×D₄	—	53	1	no	—
225	#117	—	53	2	no	—
226	C₂×C₄²	—	53	1	no	—
227	M₃₂	—	106	4	no	—
228	C₂×C₁₆	—	53	1	no	—
229	C₂×C₁₆	—	53	1	no	—
230	C₂×C₁₆	—	53	2	no	—
231	C₂×C₁₆	—	106	2	no	—
232	C₂×C₁₆	—	106	2	no	—
233	C₂×C₁₆	—	106	4	no	—
234	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
235	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
236	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
237	C₄⋊C₈	—	53	1	no	—
238	C₄⋊C₈	—	53	2	no	—
239	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
240	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
241	C₂²⋊C₈	—	53	2	no	—
242	C₂²⋊C₈	—	53	2	no	—
243	C₂²⋊C₈	—	53	2	no	—
244	#97	—	53	2	no	—
245	#97	—	53	2	no	—
246	C₄×C₈	—	53	1	no	—
247	C₄×C₈	—	53	1	no	—
248	C₄×C₈	—	53	1	no	—
249	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
250	C₄×C₈	C₂²×D₅₃	1	2	yes	—
251	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
252	C₂×D₄	—	53	2	no	—
253	C₂×D₄	—	53	1	no	—
254	C₂²×C₄	—	53	1	no	—
255	C₂²×C₄	—	53	1	no	—
256	C₂²×C₄	—	53	1	no	—
257	C₂²×C₄	—	53	2	no	—
258	M₁₆	—	106	4	no	—
259	C₂×C₈	—	53	1	no	—
260	C₂×C₈	—	53	1	no	—
261	C₂×C₈	—	53	2	no	—
262	C₂×C₈	—	106	2	no	—
263	C₂×C₈	—	106	1	no	—
264	C₂×C₈	—	106	2	no	—
265	C₂×C₈	—	106	1	no	—
266	C₂×C₈	—	106	2	no	—
267	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
269	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
270	C₄⋊C₄	—	53	2	no	—
271	C₄⋊C₄	—	53	2	no	—
272	C₄⋊C₄	—	53	1	no	—
273	C₄⋊C₄	—	53	2	no	—
274	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
275	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
276	C₂²⋊C₄	—	53	2	no	—
277	C₂²⋊C₄	—	53	2	no	—
278	C₂²⋊C₄	—	53	2	no	—
279	C₂²⋊C₄	—	53	2	no	—
280	C₂²⋊C₄	—	53	2	no	—
281	C₄²	—	53	1	no	—
282	C₄²	—	53	1	no	—
283	C₄²	—	53	1	no	—
284	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₅₃	1	2	yes	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	2	2	no	—
288	C₁₆	—	53	2	no	—
289	C₁₆	—	106	1	no	—
290	C₁₆	—	106	2	no	—
291	C₁₆	#11719949	1	2	yes	—
292	C₂³	—	53	1	no	—
293	C₂³	—	53	2	no	—
294	Q₈	—	2	2	no	—
295	D₄	—	106	4	no	—
296	D₄	—	106	2	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	53	1	no	—
300	C₂×C₄	—	53	2	no	—
301	C₂×C₄	—	106	2	no	—
302	C₂×C₄	—	106	2	no	—
303	C₂×C₄	—	106	1	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	106	1	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
309	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
310	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	53	2	no	—
315	C₈	—	106	1	no	—
316	C₈	—	106	2	no	—
317	C₈	#41×D₅₃	1	1	yes	characteristic
318	C₈	C₂×#11719949	1	1	yes	characteristic
319	C₂²	—	53	2	no	—
320	C₂²	—	106	1	no	—
321	C₂²	—	106	1	no	—
322	C₂²	—	106	2	no	—
323	C₂²	—	106	2	no	—
324	C₂²	—	106	2	no	—
325	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	106	1	no	—
330	C₄	—	106	2	no	—
331	C₄	—	106	1	no	—
332	C₄	—	106	1	no	—
333	C₄	#16811039	1	2	yes	—
334	C₄	C₄×#11719949	1	1	yes	characteristic
335	C₄	#131×D₅₃	1	1	yes	characteristic
336	C₂	—	106	2	no	—
337	C₂	—	212	2	no	—
338	C₂	C₈×#11719949	1	1	yes	characteristic
339	C₂	#504×D₅₃	1	1	yes	characteristic
340	C₂	C₂×#16811039	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	Q₈×C₁₆×C₅₃	C₂
prime radical	#13133192	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	Q₈×C₁₆	D₅₃
hypocenter	C₅₃	#33480
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276624	C₂	semidirect
3. #20276546	C₂	semidirect
4. #20276539	C₂	semidirect
5. C₁₆×D₂₁₂	C₂	semidirect
6. C₁₆×D₂₁₂	C₂	semidirect
7. #20276521	C₂	semidirect
8. #20276517	C₂	semidirect
9. #20276310	C₂	semidirect
10. C₄×C₁₆×D₅₃	C₂	semidirect
11. Q₈×C₁₆×C₅₃	C₂	semidirect
168. Q₈×C₁₆	D₅₃	semidirect
211. C₅₃	#33480	semidirect

Maximal subgroups

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

subgroup	index
#20283692	2
#20276624	2
#20276546	2
#20276539	2
C₁₆×D₂₁₂	2
#20276521	2
#20276517	2
#20276310	2
C₄×C₁₆×D₅₃	2
Q₈×C₁₆×C₅₃	2
#33480	53