Subgroups — #27274279

Order 13568.

542 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
3	#20272604	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
5	#16810680	C₄	1	2	yes	—
6	#16810680	C₄	1	2	yes	—
7	#16810676	—	2	1	no	—
8	#16810676	—	2	1	no	—
9	#16810676	—	2	1	no	—
10	#16810676	—	2	1	no	—
11	#16810676	C₄	1	2	yes	—
12	#16810676	C₂²	1	2	yes	—
13	#16810647	—	2	1	no	—
14	C₂×C₄⋊#88322	C₄	1	2	yes	—
15	#16810587	—	2	1	no	—
16	C₂×C₂²⋊#88322	—	2	1	no	—
17	C₄×#11719917	C₂²	1	2	yes	—
18	C₂×C₄×#88322	—	2	1	no	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#432	—	2	1	no	—
21	C₅₃×#432	—	2	1	no	—
22	C₅₃×#431	—	2	1	no	—
23	C₅₃×#431	—	2	1	no	—
24	C₅₃×#423	C₂²	1	2	yes	—
25	C₅₃×#407	C₄	1	2	yes	—
26	C₄×M₁₆×C₅₃	C₄	1	2	yes	—
27	C₂×#11719917	—	2	2	no	—
28	C₂²×#88322	—	2	2	no	—
29	C₄⋊#88322	—	2	2	no	—
30	C₄⋊#88322	—	2	2	no	—
31	C₄⋊#88322	—	2	2	no	—
32	C₄⋊#88322	C₂×C₄	1	4	yes	—
33	C₂²⋊#88322	—	2	2	no	—
34	C₂²⋊#88322	—	2	2	no	—
35	C₂²⋊#88322	—	2	2	no	—
36	C₂²⋊#88322	—	2	2	no	—
37	#13133128	C₂×C₄	1	4	yes	—
38	C₄×#88322	—	2	2	no	—
39	C₄×#88322	—	2	2	no	—
40	C₄×#88322	—	2	2	no	—
41	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
44	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₅₃	—	2	2	no	—
58	C₄⋊C₈×C₅₃	—	2	2	no	—
59	C₂²⋊C₈×C₅₃	—	2	2	no	—
60	C₂²⋊C₈×C₅₃	—	2	2	no	—
61	#97×C₅₃	—	2	2	no	—
62	C₄×C₈×C₅₃	—	2	2	no	—
63	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	—	2	4	no	—
65	C₂²×C₄×C₅₃	—	4	2	no	—
66	C₂²×C₄×C₅₃	—	4	2	no	—
67	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
85	M₁₆×C₅₃	—	4	4	no	—
86	C₂×C₈×C₅₃	—	2	4	no	—
87	C₄²×C₅₃	C₄²	1	1	yes	characteristic
88	C₄²×C₅₃	C₄²	1	1	yes	characteristic
89	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	#11719917	—	4	4	no	—
104	C₂×#88322	—	2	4	no	—
105	C₂×#88322	—	2	4	no	—
106	C₂×#88322	—	4	4	no	—
107	C₂³×C₅₃	—	2	2	no	—
108	C₂³×C₅₃	—	4	1	no	—
109	C₂³×C₅₃	—	4	1	no	—
110	C₂³×C₅₃	#95	1	1	yes	characteristic
111	C₂³×C₅₃	#95	1	1	yes	characteristic
112	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	—	2	2	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	4	1	no	—
125	C₂×C₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
136	C₂×C₄×C₅₃	#108	1	4	yes	—
137	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₈×C₅₃	—	4	4	no	—
144	#88322	—	4	4	no	—
145	#88322	—	4	4	no	—
146	#8066	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	2	1	no	—
154	C₂²×C₅₃	—	2	1	no	—
155	C₂²×C₅₃	—	4	4	no	—
156	C₂²×C₅₃	#341	1	2	yes	—
157	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	—	2	1	no	—
164	C₄×C₅₃	—	2	1	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	#341	1	2	yes	—
167	C₄×C₅₃	#413	1	2	yes	—
168	C₄×C₅₃	#421	1	2	yes	—
169	C₄×C₅₃	C₂×#108	1	2	yes	—
170	#2920	—	53	1	no	—
171	#2920	D₅₃	1	1	yes	characteristic
172	C₂×#431	—	53	1	no	—
173	#1299	—	53	4	no	—
174	C₂×C₅₃	—	2	2	no	—
175	C₂×C₅₃	—	4	1	no	—
176	C₂×C₅₃	—	4	1	no	—
177	C₂×C₅₃	#1733	1	1	yes	characteristic
178	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
179	C₂×C₅₃	#1742	1	1	yes	characteristic
180	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
181	#432	—	2	1	no	—
182	#432	—	2	1	no	—
183	#432	—	53	2	no	—
184	#432	—	53	2	no	—
185	#431	—	2	1	no	—
186	#431	—	2	1	no	—
187	#431	—	53	2	no	—
188	#431	—	53	2	no	—
189	#431	—	106	1	no	—
190	#431	—	106	1	no	—
191	#431	—	106	1	no	—
192	#431	—	106	1	no	—
193	#423	—	106	1	no	—
194	#423	C₂×D₅₃	1	2	yes	—
195	C₂×C₄⋊C₈	—	53	2	no	—
196	#407	—	106	1	no	—
197	#407	Dic₅₃	1	2	yes	—
198	C₂×C₂²⋊C₈	—	106	1	no	—
199	C₄×M₁₆	—	53	2	no	—
200	C₄×M₁₆	Dic₅₃	1	2	yes	—
201	C₂×C₄×C₈	—	106	1	no	—
202	C₅₃	#8066	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
204	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
205	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
206	C₂×M₁₆	—	106	2	no	—
207	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
208	C₂²×C₈	—	106	2	no	—
209	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
211	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
214	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
215	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
220	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	53	4	no	—
224	C₄⋊C₈	—	106	2	no	—
225	C₄⋊C₈	—	106	2	no	—
226	C₄⋊C₈	—	106	2	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₂²⋊C₈	—	2	2	no	—
229	C₂²⋊C₈	—	106	2	no	—
230	C₂²⋊C₈	—	106	2	no	—
231	C₂²⋊C₈	—	106	2	no	—
232	C₂²⋊C₈	—	106	2	no	—
233	#97	—	2	2	no	—
234	#97	—	53	4	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	—	106	2	no	—
237	C₄×C₈	—	106	2	no	—
238	C₄×C₈	—	106	2	no	—
239	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
244	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
245	C₂²×C₄	#11719919	1	1	yes	characteristic
246	C₂²×C₄	#11719919	1	1	yes	characteristic
247	C₂²×C₄	#11719919	1	1	yes	characteristic
248	C₂²×C₄	#11719919	1	1	yes	characteristic
249	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
250	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
251	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
252	C₂²×C₄	#11719921	1	1	yes	characteristic
253	C₂²×C₄	#11719921	1	1	yes	characteristic
254	C₂²×C₄	#11719921	1	1	yes	characteristic
255	C₂²×C₄	#11719921	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
261	M₁₆	—	4	4	no	—
262	M₁₆	—	212	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	106	4	no	—
265	C₂×C₈	—	106	4	no	—
266	C₂×C₈	—	212	4	no	—
267	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
268	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
269	C₄²	#11719919	1	1	yes	characteristic
270	C₄²	#11719919	1	1	yes	characteristic
271	C₄²	#11719919	1	1	yes	characteristic
272	C₄²	#11719919	1	1	yes	characteristic
273	C₄²	#11719921	1	1	yes	characteristic
274	C₄²	#11719921	1	1	yes	characteristic
275	C₄²	#11719921	1	1	yes	characteristic
276	C₄²	#11719921	1	1	yes	characteristic
277	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
278	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
279	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
280	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
281	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
283	C₂³	—	2	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	#13133124	1	1	yes	characteristic
287	C₂³	#13133124	1	1	yes	characteristic
288	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	#13133124	1	1	yes	characteristic
307	C₂×C₄	#13133124	1	1	yes	characteristic
308	C₂×C₄	#13133124	1	1	yes	characteristic
309	C₂×C₄	#13133124	1	1	yes	characteristic
310	C₂×C₄	#13133124	1	1	yes	characteristic
311	C₂×C₄	#13133124	1	1	yes	characteristic
312	C₂×C₄	#13133152	1	4	yes	—
313	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
314	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
315	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
316	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
317	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
318	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
319	C₈	—	4	4	no	—
320	C₈	—	212	4	no	—
321	C₈	—	212	4	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	4	4	no	—
331	C₂²	#16810387	1	2	yes	—
332	C₂²	C₂×#13133124	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	#16810387	1	2	yes	—
342	C₄	#16810607	1	2	yes	—
343	C₄	#16810639	1	2	yes	—
344	C₄	C₂×#13133152	1	2	yes	—
345	C₂	—	2	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	4	1	no	—
348	C₂	#20274426	1	1	yes	characteristic
349	C₂	C₂×#16810387	1	1	yes	characteristic
350	C₂	#20274454	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2920	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2920	D₅₃
hypocenter	C₅₃	#8066
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #20272604	C₂	semidirect
202. C₅₃	#8066	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂×#16810676	2
#20272604	2
C₅₃×#2920	2
#8066	53