Subgroups — #27349789

Order 13568.

734 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	#20283681	C₂	1	1	yes	maximal, characteristic
2	C₈×#11719945	C₂	1	1	yes	maximal, characteristic
3	#20283357	C₂	1	1	yes	maximal, characteristic
4	#20273016	C₂	1	2	yes	maximal, complemented
5	#20273013	C₂	1	2	yes	maximal, complemented
6	#20273006	C₂	1	2	yes	maximal, complemented
7	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
8	#20272998	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
10	#20272994	C₂	1	2	yes	maximal, complemented
11	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11719945	C₂²	1	1	yes	characteristic
13	#16810753	C₄	1	2	yes	—
14	C₈×Dic₁₀₆	C₄	1	2	yes	—
15	#16810702	C₄	1	2	yes	—
16	#16810694	C₄	1	4	yes	—
17	C₈×D₂₁₂	C₄	1	2	yes	—
18	C₈×C₂²⋊D₅₃	C₄	1	4	yes	—
19	#16810682	C₄	1	4	yes	—
20	#16810676	C₂²	1	1	yes	characteristic
21	#16810676	C₂²	1	1	yes	characteristic
22	#16810674	C₂²	1	2	yes	—
23	#16810674	C₂²	1	2	yes	—
24	#16810674	C₂²	1	2	yes	—
25	#16810674	C₂²	1	2	yes	—
26	#16810649	C₄	1	2	yes	—
27	#16810589	C₄	1	2	yes	—
28	C₄×#11719917	C₂²	1	1	yes	characteristic
29	C₄×#11719912	C₂²	1	2	yes	—
30	C₄×C₈×D₅₃	C₂²	1	2	yes	—
31	#16810345	C₂²	1	2	yes	—
32	#16810345	C₂²	1	2	yes	—
33	#16810344	C₂²	1	2	yes	—
34	#16810344	C₂²	1	2	yes	—
35	C₈×#88322	C₂²	1	2	yes	—
36	C₈×#88322	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#322	C₂²	1	1	yes	characteristic
41	C₅₃×#322	C₂²	1	1	yes	characteristic
42	C₅₃×#322	C₂²	1	1	yes	characteristic
43	C₅₃×#322	C₂²	1	1	yes	characteristic
44	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
45	C₂×#11719917	C₂×C₄	1	2	yes	—
46	C₂×#11719912	C₂×C₄	1	4	yes	—
47	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
48	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
50	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
51	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
52	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
53	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
54	#13133176	C₂×C₄	1	1	yes	characteristic
55	#13133174	C₂×C₄	1	2	yes	—
56	C₄²×D₅₃	C₂³	1	2	yes	—
57	C₄⋊#88322	C₂×C₄	1	2	yes	—
58	C₄⋊#88322	C₂×C₄	1	2	yes	—
59	#13133146	C₂×C₄	1	4	yes	—
60	#13133146	C₂×C₄	1	4	yes	—
61	#13133146	C₂×C₄	1	4	yes	—
62	#13133146	C₂×C₄	1	4	yes	—
63	C₂²⋊#88322	C₂×C₄	1	2	yes	—
64	C₂²⋊#88322	C₂×C₄	1	2	yes	—
65	#13133129	C₂×C₄	1	4	yes	—
66	#13133129	C₂×C₄	1	4	yes	—
67	#13133129	C₂×C₄	1	4	yes	—
68	#13133129	C₂×C₄	1	4	yes	—
69	#13133128	C₂×C₄	1	2	yes	—
70	#13133127	C₂×C₄	1	4	yes	—
71	C₄×#88322	C₂³	1	2	yes	—
72	C₄×#88322	C₂³	1	2	yes	—
73	C₄×#88322	C₂³	1	2	yes	—
74	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
75	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
76	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
77	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
80	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₂²⋊D₅₃	C₄²	1	2	yes	—
92	#11719945	—	2	4	no	—
93	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
94	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
98	C₂×C₈×C₅₃	—	2	4	no	—
99	C₂×C₈×C₅₃	—	2	4	no	—
100	C₂×C₈×C₅₃	—	4	2	no	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
111	C₂²⋊Dic₅₃	C₄²	1	2	yes	—
112	#11719921	C₄²	1	2	yes	—
113	#11719921	C₄²	1	2	yes	—
114	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
115	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
116	#11719919	C₄²	1	2	yes	—
117	#11719919	C₄²	1	2	yes	—
118	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
119	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
120	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
121	#11719917	—	2	4	no	—
122	C₂×#88322	C₂²×C₄	1	4	yes	—
123	C₂×#88322	C₂²×C₄	1	4	yes	—
124	C₂×#88322	C₂²×C₄	1	4	yes	—
125	#11719912	—	2	8	no	—
126	C₈×D₅₃	—	4	4	no	—
127	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
128	C₂²×D₅₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#131	1	4	yes	—
135	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₂²⋊D₅₃	—	2	4	no	—
139	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
140	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
141	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
142	D₂₁₂	—	2	2	no	—
143	C₄×D₅₃	—	2	4	no	—
144	C₄×D₅₃	—	2	4	no	—
145	C₄×D₅₃	—	2	4	no	—
146	Dic₁₀₆	—	2	2	no	—
147	C₈×C₅₃	—	2	2	no	—
148	C₈×C₅₃	—	2	2	no	—
149	C₈×C₅₃	—	2	2	no	—
150	C₈×C₅₃	—	2	2	no	—
151	C₈×C₅₃	—	2	2	no	—
152	C₈×C₅₃	#131	1	4	yes	—
153	#88322	—	2	4	no	—
154	#88322	—	2	4	no	—
155	#88322	—	2	4	no	—
156	#17331	—	53	1	no	maximal, Sylow
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	#405	1	2	yes	—
160	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
161	C₂×D₅₃	—	2	2	no	—
162	C₂×D₅₃	—	2	2	no	—
163	C₂×D₅₃	—	2	2	no	—
164	C₄×C₅₃	—	2	1	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	#405	1	2	yes	—
167	C₄×C₅₃	C₂×#131	1	2	yes	—
168	C₄×C₅₃	C₂×#131	1	2	yes	—
169	C₄×C₅₃	#568	1	2	yes	—
170	Dic₅₃	—	2	2	no	—
171	Dic₅₃	—	2	2	no	—
172	Dic₅₃	—	2	2	no	—
173	#2965	—	53	1	no	—
174	C₈×#41	—	53	1	no	—
175	#2922	—	53	1	no	—
176	#1454	—	53	2	no	—
177	#1453	—	53	2	no	—
178	#1452	—	53	2	no	—
179	#1451	—	53	1	no	—
180	#1450	—	53	1	no	—
181	C₈×M₁₆	—	53	1	no	—
182	C₈×M₁₆	—	53	2	no	—
183	C₂×#322	D₅₃	1	1	yes	characteristic, complemented
184	C₂×C₅₃	—	2	2	no	—
185	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
186	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
187	C₂×C₅₃	#2875	1	1	yes	characteristic
188	D₅₃	—	4	2	no	—
189	C₄×#41	—	53	1	no	—
190	#446	—	53	2	no	—
191	C₈×Q₈	—	53	2	no	—
192	#436	—	53	2	no	—
193	#435	—	53	4	no	—
194	C₈×D₄	—	53	4	no	—
195	C₈×D₄	—	53	2	no	—
196	#433	—	53	4	no	—
197	#431	—	53	1	no	—
198	#431	—	53	1	no	—
199	#431	—	53	2	no	—
200	#431	—	53	2	no	—
201	#431	—	53	2	no	—
202	#431	—	53	2	no	—
203	#424	—	53	2	no	—
204	#408	—	53	2	no	—
205	C₄×M₁₆	—	53	1	no	—
206	C₄×M₁₆	—	53	2	no	—
207	C₂×C₄×C₈	—	53	2	no	—
208	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
209	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
210	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
211	#322	—	53	2	no	—
212	#322	—	53	2	no	—
213	#322	—	53	2	no	—
214	#322	—	53	2	no	—
215	#322	C₂×D₅₃	1	1	yes	characteristic
216	#322	C₂×D₅₃	1	1	yes	characteristic
217	#322	C₂×D₅₃	1	1	yes	characteristic
218	#322	C₂×D₅₃	1	1	yes	characteristic
219	C₈²	—	53	2	no	—
220	C₈²	—	53	2	no	—
221	C₅₃	#17331	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	—	53	1	no	—
223	C₂×M₁₆	—	53	2	no	—
224	C₂×M₁₆	—	53	4	no	—
225	C₂²×C₈	—	53	4	no	—
226	C₂²×C₈	C₄×D₅₃	1	2	yes	—
227	C₂²×C₈	C₄×D₅₃	1	2	yes	—
228	C₂²×C₈	C₄×D₅₃	1	2	yes	—
229	C₄×Q₈	—	53	1	no	—
230	C₄×Q₈	—	53	1	no	—
231	C₄×D₄	—	53	2	no	—
232	C₄×D₄	—	53	2	no	—
233	C₄×D₄	—	53	1	no	—
234	C₄×D₄	—	53	1	no	—
235	#117	—	53	1	no	—
236	#117	—	53	2	no	—
237	C₂×C₄²	—	53	2	no	—
238	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
239	C₄⋊C₈	—	53	4	no	—
240	C₄⋊C₈	—	53	4	no	—
241	C₄⋊C₈	—	53	4	no	—
242	C₄⋊C₈	—	53	4	no	—
243	C₄⋊C₈	—	53	2	no	—
244	C₄⋊C₈	—	53	2	no	—
245	C₂²⋊C₈	—	53	2	no	—
246	C₂²⋊C₈	—	53	2	no	—
247	C₂²⋊C₈	—	53	4	no	—
248	C₂²⋊C₈	—	53	4	no	—
249	C₂²⋊C₈	—	53	4	no	—
250	C₂²⋊C₈	—	53	4	no	—
251	#97	—	53	4	no	—
252	#97	—	53	2	no	—
253	C₄×C₈	—	53	4	no	—
254	C₄×C₈	—	53	2	no	—
255	C₄×C₈	—	53	2	no	—
256	C₄×C₈	—	53	2	no	—
257	C₄×C₈	C₄×D₅₃	1	2	yes	—
258	C₄×C₈	C₄×D₅₃	1	2	yes	—
259	C₄×C₈	C₄×D₅₃	1	2	yes	—
260	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
266	#41	—	106	4	no	—
267	C₂×Q₈	—	53	1	no	—
268	C₂×D₄	—	53	2	no	—
269	C₂×D₄	—	53	1	no	—
270	C₂²×C₄	—	53	2	no	—
271	C₂²×C₄	—	53	2	no	—
272	C₂²×C₄	—	53	2	no	—
273	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
276	M₁₆	—	106	4	no	—
277	M₁₆	—	106	8	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	—	4	2	no	—
281	C₂×C₈	—	53	4	no	—
282	C₂×C₈	—	53	4	no	—
283	C₂×C₈	—	53	4	no	—
284	C₂×C₈	—	212	4	no	—
285	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
286	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
287	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
288	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
290	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
291	C₄⋊C₄	—	53	2	no	—
292	C₄⋊C₄	—	53	2	no	—
293	C₄⋊C₄	—	53	1	no	—
294	C₄⋊C₄	—	53	1	no	—
295	C₂²⋊C₄	—	53	2	no	—
296	C₂²⋊C₄	—	53	2	no	—
297	C₂²⋊C₄	—	53	2	no	—
298	C₄²	—	53	2	no	—
299	C₄²	—	53	2	no	—
300	C₄²	—	53	2	no	—
301	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
302	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
304	C₄²	C₂³×D₅₃	1	1	yes	characteristic
305	C₂³	—	53	2	no	—
306	C₂³	C₄²×D₅₃	1	1	yes	characteristic
307	Q₈	—	106	2	no	—
308	D₄	—	106	4	no	—
309	D₄	—	106	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	53	2	no	—
313	C₂×C₄	—	53	2	no	—
314	C₂×C₄	—	53	2	no	—
315	C₂×C₄	—	106	4	no	—
316	C₂×C₄	—	106	4	no	—
317	C₂×C₄	—	106	4	no	—
318	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
319	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
320	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
321	C₂×C₄	#13133225	1	4	yes	—
322	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	—	2	2	no	—
329	C₈	—	2	2	no	—
330	C₈	—	106	4	no	—
331	C₈	—	106	4	no	—
332	C₈	—	106	4	no	—
333	C₈	#13133223	1	4	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	106	2	no	—
337	C₂²	—	106	2	no	—
338	C₂²	—	106	2	no	—
339	C₂²	#16810580	1	2	yes	—
340	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	106	2	no	—
344	C₄	—	106	2	no	—
345	C₄	—	106	2	no	—
346	C₄	#16810580	1	2	yes	—
347	C₄	C₂×#13133223	1	2	yes	—
348	C₄	C₂×#13133225	1	2	yes	—
349	C₄	#16811389	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	212	2	no	—
352	C₂	C₂×#16810580	1	1	yes	characteristic
353	C₂	C₄×#13133223	1	1	yes	characteristic
354	C₂	#20283140	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₂×C₅₃×#322	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#17331
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. #20273016	C₂	semidirect
5. #20273013	C₂	semidirect
6. #20273006	C₂	semidirect
7. #20273004	C₂	semidirect
8. #20272998	C₂	semidirect
9. C₈×#11719917	C₂	semidirect
10. #20272994	C₂	semidirect
11. C₂×C₅₃×#322	C₂	semidirect
183. C₂×#322	D₅₃	semidirect
221. C₅₃	#17331	semidirect

Maximal subgroups

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

subgroup	index
#20283681	2
C₈×#11719945	2
#20283357	2
#20273016	2
#20273013	2
#20273006	2
#20273004	2
#20272998	2
C₈×#11719917	2
#20272994	2
C₂×C₅₃×#322	2
#17331	53