Subgroups — #27298973

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 354 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
2	C₂×#16810660	C₂	1	1	yes	maximal, characteristic
3	C₂×#107×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272807	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272806	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272806	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×#13133175	—	2	1	no	—
10	C₂×C₄⋊C₄×D₅₃	—	2	1	no	—
11	C₂×#13133156	—	2	2	no	—
12	C₂×#13133156	—	2	2	no	—
13	C₂×#13133156	C₄	1	4	yes	complemented
14	C₂²×#11719908	C₂²	1	1	yes	characteristic
15	#16810660	—	2	2	no	—
16	#16810660	C₄	1	4	yes	complemented
17	C₂×#13133151	C₂²	1	1	yes	characteristic
18	C₂×#13133151	C₂²	1	1	yes	characteristic
19	C₂×#13133150	—	2	1	no	—
20	#107×D₅₃	—	2	2	no	—
21	#107×D₅₃	C₄	1	4	yes	complemented
22	C₂×#13133148	—	2	1	no	—
23	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄×D₅₃	—	4	1	no	—
27	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
28	C₂²×#88322	D₄	1	1	yes	characteristic
29	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₅₃	D₄	1	2	yes	—
33	C₂×C₈×D₅₃	D₄	1	2	yes	—
34	C₂×C₈×D₅₃	Q₈	1	2	yes	—
35	#13133175	—	2	4	no	—
36	#13133175	—	4	2	no	—
37	C₂×C₄⋊Dic₅₃	—	2	1	no	—
38	C₂×C₄⋊Dic₅₃	—	2	1	no	—
39	C₂×#11719919	—	4	1	no	—
40	C₄⋊C₄×D₅₃	—	2	4	no	—
41	C₄⋊C₄×D₅₃	—	4	2	no	—
42	C₂×#11719921	—	4	1	no	—
43	C₂×C₄×Dic₅₃	—	4	1	no	—
44	#13133156	—	2	8	no	—
45	C₂×#11719908	—	2	1	no	—
46	C₂×#11719908	—	2	1	no	—
47	C₂×#11719908	C₂×C₄	1	2	yes	complemented
48	#13133151	—	2	1	no	—
49	#13133151	—	2	1	no	—
50	#13133151	C₂×C₄	1	2	yes	—
51	#13133151	C₂×C₄	1	2	yes	complemented
52	#13133150	—	2	4	no	—
53	#13133148	—	2	4	no	—
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₁₆×C₅₃	—	2	1	no	—
58	C₂×C₁₆×C₅₃	—	2	1	no	—
59	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
60	#107×C₅₃	—	2	1	no	—
61	#107×C₅₃	—	2	1	no	—
62	#107×C₅₃	C₂×C₄	1	2	yes	—
63	#107×C₅₃	C₂×C₄	1	2	yes	complemented
64	C₂³×D₅₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	4	1	no	—
66	C₂²×C₄×C₅₃	—	4	1	no	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₅₃	—	4	1	no	—
69	C₂²×Dic₅₃	—	4	1	no	—
70	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
71	C₂×C₄×D₅₃	—	4	4	no	—
72	C₂×C₄×D₅₃	—	8	2	no	—
73	C₂×C₄×D₅₃	—	8	2	no	—
74	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
77	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
78	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
79	C₂×C₄×D₅₃	Q₁₆	1	2	yes	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
86	C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	—	4	1	no	—
89	C₄⋊C₄×C₅₃	—	4	1	no	—
90	#11719921	—	4	4	no	—
91	C₄⋊Dic₅₃	—	2	2	no	—
92	C₄⋊Dic₅₃	—	2	2	no	—
93	C₄⋊Dic₅₃	—	4	1	no	—
94	C₄⋊Dic₅₃	—	4	1	no	—
95	#11719919	—	4	4	no	—
96	C₄×Dic₅₃	—	4	4	no	—
97	C₂×#88322	C₂²⋊C₄	1	2	yes	—
98	C₂×#88322	C₂²⋊C₄	1	2	yes	—
99	C₂×#88322	C₄⋊C₄	1	2	yes	—
100	C₈×D₅₃	C₄²	1	4	yes	—
101	C₈×D₅₃	C₂²⋊C₄	1	4	yes	—
102	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
103	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
104	C₁₆×C₅₃	—	2	2	no	—
105	#11719908	—	2	2	no	—
106	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₅₃	—	2	1	no	—
108	C₂²×D₅₃	—	2	1	no	—
109	C₂²×D₅₃	—	2	1	no	—
110	C₂²×D₅₃	—	2	1	no	—
111	C₂²×D₅₃	#102	1	2	yes	—
112	C₂²×D₅₃	#102	1	2	yes	—
113	C₂²×D₅₃	#107	1	2	yes	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	8	1	no	—
117	C₂×C₄×C₅₃	—	8	1	no	—
118	C₂×C₄×C₅₃	—	8	1	no	—
119	C₂×C₄×C₅₃	—	8	1	no	—
120	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×Dic₅₃	—	4	2	no	—
127	C₂×Dic₅₃	—	4	2	no	—
128	C₂×Dic₅₃	—	8	1	no	—
129	C₂×Dic₅₃	—	8	1	no	—
130	C₂×Dic₅₃	—	8	1	no	—
131	C₂×Dic₅₃	—	8	1	no	—
132	C₂×Dic₅₃	#102	1	2	yes	—
133	C₂×Dic₅₃	#102	1	2	yes	—
134	C₂×Dic₅₃	#107	1	2	yes	—
135	C₄×D₅₃	—	8	8	no	—
136	C₄×D₅₃	#95	1	4	yes	—
137	C₄×D₅₃	#103	1	4	yes	—
138	C₄×D₅₃	#103	1	4	yes	—
139	C₄×D₅₃	#106	1	4	yes	—
140	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
141	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
144	#88322	#95	1	4	yes	—
145	#12557	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
148	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
149	C₂²×C₅₃	#509	1	1	yes	characteristic
150	C₂²×C₅₃	#509	1	1	yes	characteristic
151	C₂²×C₅₃	#510	1	1	yes	characteristic
152	C₂²×C₅₃	#510	1	1	yes	characteristic
153	C₂×D₅₃	—	2	2	no	—
154	C₂×D₅₃	—	2	2	no	—
155	C₂×D₅₃	—	2	2	no	—
156	C₂×D₅₃	—	2	2	no	—
157	C₂×D₅₃	—	2	2	no	—
158	C₂×D₅₃	—	2	2	no	—
159	C₂×D₅₃	#340	1	4	yes	—
160	C₄×C₅₃	—	8	2	no	—
161	C₄×C₅₃	—	8	2	no	—
162	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
163	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
164	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
165	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
166	Dic₅₃	—	8	2	no	—
167	Dic₅₃	—	8	2	no	—
168	Dic₅₃	#340	1	4	yes	—
169	C₂²×M₃₂	—	53	1	no	—
170	C₂×#427	—	53	1	no	—
171	C₂²×#107	—	53	1	no	—
172	#1380	—	53	1	no	—
173	#1380	—	53	1	no	—
174	#1380	—	53	1	no	—
175	#1380	D₅₃	1	1	yes	characteristic, complemented
176	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
177	C₂×C₅₃	#2139	1	1	yes	characteristic
178	C₂×C₅₃	#2139	1	1	yes	characteristic
179	C₂×C₅₃	#2140	1	1	yes	characteristic
180	C₂×C₅₃	#2141	1	1	yes	characteristic
181	C₂×C₅₃	#2159	1	1	yes	characteristic
182	C₂×C₅₃	#2159	1	1	yes	characteristic
183	D₅₃	—	2	4	no	—
184	C₂³×C₈	—	53	1	no	—
185	C₂×#117	—	106	1	no	—
186	C₂²×C₄⋊C₄	—	106	1	no	—
187	C₂×M₃₂	—	53	4	no	—
188	C₂×M₃₂	—	106	2	no	—
189	C₂×M₃₂	—	106	2	no	—
190	C₂²×C₁₆	—	53	1	no	—
191	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
192	#427	—	53	4	no	—
193	#427	—	106	2	no	—
194	C₂×#107	—	53	1	no	—
195	C₂×#107	—	53	1	no	—
196	C₂×#107	—	53	4	no	—
197	C₂×#107	—	106	2	no	—
198	C₂×#107	—	106	1	no	—
199	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
200	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
201	C₂×#106	—	106	1	no	—
202	C₅₃	#12557	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₄	—	53	1	no	—
204	C₂³×C₄	—	212	1	no	—
205	C₂²×C₈	—	53	1	no	—
206	C₂²×C₈	—	53	2	no	—
207	C₂²×C₈	—	53	2	no	—
208	C₂²×C₈	—	53	2	no	—
209	C₂²×C₈	—	53	2	no	—
210	C₂²×C₈	—	53	2	no	—
211	C₂²×C₈	—	53	2	no	—
212	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
213	#117	—	106	4	no	—
214	#117	—	212	2	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	106	4	no	—
218	C₂×C₄⋊C₄	—	106	1	no	—
219	C₂×C₄⋊C₄	—	106	1	no	—
220	C₂×C₄⋊C₄	—	212	2	no	—
221	C₂×C₄⋊C₄	—	212	1	no	—
222	C₂×C₂²⋊C₄	—	212	1	no	—
223	C₂×C₄²	—	212	1	no	—
224	M₃₂	—	106	8	no	—
225	C₂×C₁₆	—	2	1	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	53	2	no	—
228	C₂×C₁₆	—	106	1	no	—
229	C₂×C₁₆	—	106	1	no	—
230	C₂×C₁₆	C₄×D₅₃	1	2	yes	complemented
231	#107	—	2	1	no	—
232	#107	—	2	1	no	—
233	#107	—	53	2	no	—
234	#107	—	53	2	no	—
235	#107	—	106	4	no	—
236	#107	—	106	1	no	—
237	#107	—	106	1	no	—
238	#107	C₄×D₅₃	1	2	yes	—
239	#107	C₄×D₅₃	1	2	yes	complemented
240	#106	—	106	4	no	—
241	C₂⁴	—	53	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	53	1	no	—
245	C₂²×C₄	—	53	2	no	—
246	C₂²×C₄	—	53	2	no	—
247	C₂²×C₄	—	53	2	no	—
248	C₂²×C₄	—	53	2	no	—
249	C₂²×C₄	—	53	2	no	—
250	C₂²×C₄	—	53	2	no	—
251	C₂²×C₄	—	424	2	no	—
252	C₂²×C₄	—	424	2	no	—
253	C₂²×C₄	—	212	4	no	—
254	C₂²×C₄	—	212	1	no	—
255	C₂²×C₄	—	212	1	no	—
256	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
257	C₂×C₈	—	53	2	no	—
258	C₂×C₈	—	53	2	no	—
259	C₂×C₈	—	53	2	no	—
260	C₂×C₈	—	53	4	no	—
261	C₂×C₈	—	53	4	no	—
262	C₂×C₈	—	53	4	no	—
263	C₂×C₈	—	53	4	no	—
264	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
267	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
268	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
269	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	106	2	no	—
275	C₄⋊C₄	—	106	2	no	—
276	C₄⋊C₄	—	212	4	no	—
277	C₄⋊C₄	—	212	1	no	—
278	C₄⋊C₄	—	212	1	no	—
279	C₂²⋊C₄	—	212	4	no	—
280	C₄²	—	212	4	no	—
281	C₁₆	—	2	2	no	—
282	C₁₆	—	106	2	no	—
283	C₂³	—	53	2	no	—
284	C₂³	—	53	2	no	—
285	C₂³	—	53	2	no	—
286	C₂³	—	106	1	no	—
287	C₂³	—	106	1	no	—
288	C₂³	—	106	1	no	—
289	C₂³	—	106	1	no	—
290	C₂³	D₈×D₅₃	1	1	yes	characteristic
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	53	2	no	—
298	C₂×C₄	—	53	2	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	53	4	no	—
301	C₂×C₄	—	53	4	no	—
302	C₂×C₄	—	53	4	no	—
303	C₂×C₄	—	53	4	no	—
304	C₂×C₄	—	424	8	no	—
305	C₂×C₄	—	424	1	no	—
306	C₂×C₄	—	424	1	no	—
307	C₂×C₄	—	424	1	no	—
308	C₂×C₄	—	424	1	no	—
309	C₂×C₄	—	212	2	no	—
310	C₂×C₄	—	212	2	no	—
311	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
312	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
313	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
314	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
315	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
316	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
317	C₈	—	53	4	no	—
318	C₈	C₄²×D₅₃	1	1	yes	characteristic
319	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
320	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
321	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
322	C₂²	—	53	4	no	—
323	C₂²	—	106	2	no	—
324	C₂²	—	106	2	no	—
325	C₂²	—	106	2	no	—
326	C₂²	—	106	2	no	—
327	C₂²	—	106	2	no	—
328	C₂²	—	106	2	no	—
329	C₂²	#102×D₅₃	1	1	yes	characteristic
330	C₂²	#102×D₅₃	1	1	yes	characteristic
331	C₂²	#107×D₅₃	1	1	yes	characteristic
332	C₂²	#16811057	1	1	yes	characteristic
333	C₂²	#16811060	1	1	yes	characteristic
334	C₂²	#16811062	1	1	yes	characteristic
335	C₂²	#16811065	1	1	yes	characteristic
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	—	53	4	no	—
339	C₄	—	424	2	no	—
340	C₄	—	424	2	no	—
341	C₄	#95×D₅₃	1	1	yes	characteristic
342	C₄	#103×D₅₃	1	1	yes	characteristic
343	C₄	#103×D₅₃	1	1	yes	characteristic
344	C₄	#106×D₅₃	1	1	yes	characteristic
345	C₂	—	106	4	no	—
346	C₂	#340×D₅₃	1	1	yes	characteristic
347	C₂	#20276424	1	1	yes	characteristic
348	C₂	#20276426	1	1	yes	characteristic
349	C₂	#20276429	1	1	yes	characteristic
350	C₂	#20276433	1	1	yes	characteristic
351	C₂	#20276500	1	1	yes	characteristic
352	C₂	#20276502	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1380	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂³×D₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1380	D₅₃
hypocenter	C₅₃	#12557
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
4. #20272807	C₂	semidirect
5. #20272806	C₂	semidirect
6. #20272806	C₂	semidirect
7. C₅₃×#1380	C₂	semidirect
13. C₂×#13133156	C₄	semidirect
16. #16810660	C₄	semidirect
21. #107×D₅₃	C₄	semidirect
47. C₂×#11719908	C₂×C₄	semidirect
51. #13133151	C₂×C₄	semidirect
59. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
63. #107×C₅₃	C₂×C₄	semidirect
175. #1380	D₅₃	semidirect
202. C₅₃	#12557	semidirect
230. C₂×C₁₆	C₄×D₅₃	semidirect
239. #107	C₄×D₅₃	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
C₂×#16810660	2
C₂×#107×D₅₃	2
#20272807	2
#20272806	2
C₅₃×#1380	2
#12557	53