Subgroups — #27298933

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 328 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	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
4	#20272793	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272792	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272792	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719912	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×#13133174	C₄	1	1	yes	characteristic
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	C₂²×#11719908	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₅₃	—	2	2	no	—
17	C₂×C₁₆×D₅₃	—	2	2	no	—
18	C₂×C₁₆×D₅₃	C₄	1	4	yes	complemented
19	#16810577	—	2	2	no	—
20	#16810577	—	2	2	no	—
21	C₂×#13133128	C₄	1	1	yes	characteristic
22	C₂×#13133127	C₄	1	1	yes	characteristic
23	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719912	—	2	2	no	—
29	C₂×#11719912	—	2	2	no	—
30	C₂×#11719912	C₈	1	4	yes	—
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₅₃	D₄	1	2	yes	—
36	C₂×C₈×D₅₃	D₄	1	2	yes	—
37	C₂×C₈×D₅₃	D₄	1	2	yes	—
38	C₂×C₈×D₅₃	Q₈	1	2	yes	—
39	#13133174	—	2	2	no	—
40	#13133174	—	2	2	no	—
41	C₂×#11719919	C₈	1	2	yes	—
42	C₂×#11719921	C₈	1	2	yes	—
43	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
44	#13133156	—	2	8	no	—
45	C₂×#11719908	—	2	1	no	—
46	C₂×#11719908	—	2	1	no	—
47	C₂×#11719908	—	2	1	no	—
48	C₂×#11719908	—	2	1	no	—
49	C₂×#11719908	C₂×C₄	1	2	yes	—
50	C₂×#11719908	C₂×C₄	1	2	yes	—
51	C₁₆×D₅₃	—	2	8	no	—
52	#13133128	—	2	2	no	—
53	#13133127	—	2	2	no	—
54	C₈×Dic₅₃	—	2	1	no	—
55	C₈×Dic₅₃	—	2	1	no	—
56	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₅₃	—	2	1	no	—
60	C₂×C₁₆×C₅₃	—	2	1	no	—
61	C₂×C₁₆×C₅₃	—	2	1	no	—
62	C₂×C₁₆×C₅₃	—	2	1	no	—
63	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₅₃	—	2	1	no	—
66	C₄×C₈×C₅₃	—	2	1	no	—
67	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
71	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₅₃	C₄²	1	2	yes	—
73	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
76	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
77	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	2	1	no	—
80	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
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×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
87	C₄²×C₅₃	—	2	1	no	—
88	C₄²×C₅₃	—	2	1	no	—
89	#11719921	—	2	4	no	—
90	#11719919	—	2	4	no	—
91	C₄×Dic₅₃	—	2	1	no	—
92	C₄×Dic₅₃	—	2	1	no	—
93	C₂×#88322	—	2	1	no	—
94	C₂×#88322	—	2	1	no	—
95	C₂×#88322	C₄²	1	2	yes	—
96	C₂×#88322	C₂²⋊C₄	1	2	yes	—
97	C₂×#88322	C₄⋊C₄	1	2	yes	—
98	C₂×#88322	C₂×C₈	1	2	yes	—
99	#11719912	—	2	8	no	—
100	C₈×D₅₃	C₂²⋊C₄	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	C₁₆×C₅₃	—	2	2	no	—
106	#11719908	—	2	2	no	—
107	#11719908	—	2	2	no	—
108	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂²×D₅₃	—	2	2	no	—
110	C₂²×D₅₃	—	2	2	no	—
111	C₂²×D₅₃	C₄×C₈	1	2	yes	—
112	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
113	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
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	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂×Dic₅₃	—	2	2	no	—
124	C₂×Dic₅₃	—	2	2	no	—
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
127	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
128	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
129	C₄×D₅₃	#95	1	4	yes	—
130	C₄×D₅₃	#97	1	4	yes	—
131	C₄×D₅₃	C₂²⋊C₈	1	4	yes	—
132	C₄×D₅₃	C₄⋊C₈	1	4	yes	—
133	C₈×C₅₃	—	2	2	no	—
134	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	#88322	—	2	2	no	—
139	#88322	#95	1	4	yes	—
140	#12545	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
142	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₂²×C₅₃	#504	1	2	yes	—
145	C₂²×C₅₃	#504	1	2	yes	—
146	C₂×D₅₃	—	2	2	no	—
147	C₂×D₅₃	—	2	2	no	—
148	C₂×D₅₃	—	2	4	no	—
149	C₂×D₅₃	—	2	4	no	—
150	C₂×D₅₃	#336	1	4	yes	—
151	C₄×C₅₃	—	4	2	no	—
152	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
153	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
154	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
155	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
156	Dic₅₃	—	4	2	no	—
157	Dic₅₃	#336	1	4	yes	—
158	C₂²×M₃₂	—	53	1	no	—
159	C₂³×C₁₆	—	53	1	no	—
160	C₂×#405	—	53	1	no	—
161	#1374	—	53	1	no	—
162	#1374	—	53	1	no	—
163	#1374	—	53	1	no	—
164	#1374	D₅₃	1	1	yes	characteristic, complemented
165	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
166	C₂×C₅₃	#2108	1	1	yes	characteristic
167	C₂×C₅₃	#2108	1	1	yes	characteristic
168	C₂×C₅₃	#2113	1	2	yes	—
169	C₂×C₅₃	#2151	1	2	yes	—
170	D₅₃	—	2	4	no	—
171	C₂²×M₁₆	—	53	1	no	—
172	C₂³×C₈	—	53	1	no	—
173	C₂×#117	—	53	1	no	—
174	C₂×M₃₂	—	53	4	no	—
175	C₂×M₃₂	—	106	2	no	—
176	C₂×M₃₂	—	106	2	no	—
177	C₂²×C₁₆	—	53	1	no	—
178	C₂²×C₁₆	—	53	1	no	—
179	C₂²×C₁₆	—	53	4	no	—
180	C₂²×C₁₆	—	106	2	no	—
181	C₂²×C₁₆	—	106	2	no	—
182	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
183	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
184	#405	—	106	2	no	—
185	#405	—	106	2	no	—
186	C₂×#97	—	53	1	no	—
187	C₂×#97	—	53	1	no	—
188	C₂×C₄×C₈	—	53	1	no	—
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	C₅₃	#12545	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	—	53	1	no	—
192	C₂×M₁₆	—	53	4	no	—
193	C₂×M₁₆	—	106	2	no	—
194	C₂×M₁₆	—	106	2	no	—
195	C₂²×C₈	—	53	1	no	—
196	C₂²×C₈	—	53	2	no	—
197	C₂²×C₈	—	53	2	no	—
198	C₂²×C₈	—	53	2	no	—
199	C₂²×C₈	—	53	2	no	—
200	C₂²×C₈	—	53	2	no	—
201	C₂²×C₈	—	53	2	no	—
202	C₂²×C₈	—	53	1	no	—
203	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
205	#117	—	106	2	no	—
206	#117	—	106	2	no	—
207	C₂×C₄⋊C₄	—	53	2	no	—
208	C₂×C₂²⋊C₄	—	53	2	no	—
209	C₂×C₄²	—	53	1	no	—
210	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
211	M₃₂	—	106	8	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	53	2	no	—
217	C₂×C₁₆	—	53	2	no	—
218	C₂×C₁₆	—	106	8	no	—
219	C₂×C₁₆	—	106	1	no	—
220	C₂×C₁₆	—	106	1	no	—
221	C₂×C₁₆	—	106	1	no	—
222	C₂×C₁₆	—	106	1	no	—
223	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
224	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
225	#97	—	106	2	no	—
226	#97	—	106	2	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	106	1	no	—
230	C₄×C₈	—	106	1	no	—
231	C₂⁴	—	53	1	no	—
232	C₂²×C₄	—	53	1	no	—
233	C₂²×C₄	—	53	2	no	—
234	C₂²×C₄	—	53	2	no	—
235	C₂²×C₄	—	53	2	no	—
236	C₂²×C₄	—	53	2	no	—
237	C₂²×C₄	—	53	2	no	—
238	C₂²×C₄	—	53	2	no	—
239	C₂²×C₄	—	53	2	no	—
240	C₂²×C₄	C₈×D₅₃	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
242	M₁₆	—	106	8	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	53	2	no	—
246	C₂×C₈	—	53	2	no	—
247	C₂×C₈	—	53	2	no	—
248	C₂×C₈	—	53	4	no	—
249	C₂×C₈	—	53	4	no	—
250	C₂×C₈	—	53	4	no	—
251	C₂×C₈	—	53	4	no	—
252	C₂×C₈	—	53	2	no	—
253	C₂×C₈	—	106	1	no	—
254	C₂×C₈	—	106	1	no	—
255	C₂×C₈	C₈×D₅₃	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
258	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
259	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
260	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
261	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
262	C₄⋊C₄	—	106	4	no	—
263	C₂²⋊C₄	—	106	4	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	106	1	no	—
267	C₄²	—	106	1	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	2	2	no	—
270	C₁₆	—	106	2	no	—
271	C₁₆	—	106	2	no	—
272	C₂³	—	53	2	no	—
273	C₂³	—	53	2	no	—
274	C₂³	—	53	2	no	—
275	C₂³	—	106	2	no	—
276	C₂³	—	106	2	no	—
277	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	53	2	no	—
282	C₂×C₄	—	53	2	no	—
283	C₂×C₄	—	53	2	no	—
284	C₂×C₄	—	53	4	no	—
285	C₂×C₄	—	53	4	no	—
286	C₂×C₄	—	53	4	no	—
287	C₂×C₄	—	53	4	no	—
288	C₂×C₄	—	106	2	no	—
289	C₂×C₄	—	106	2	no	—
290	C₂×C₄	—	106	2	no	—
291	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
293	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
295	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
296	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	53	4	no	—
299	C₈	—	106	2	no	—
300	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
301	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
302	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
303	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
304	C₂²	—	53	4	no	—
305	C₂²	—	106	2	no	—
306	C₂²	—	106	2	no	—
307	C₂²	—	106	4	no	—
308	C₂²	—	106	4	no	—
309	C₂²	C₄×C₈×D₅₃	1	1	yes	characteristic
310	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
311	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
312	C₂²	#16811039	1	2	yes	—
313	C₂²	#16811041	1	2	yes	—
314	C₄	—	4	2	no	—
315	C₄	—	53	4	no	—
316	C₄	—	212	2	no	—
317	C₄	#95×D₅₃	1	1	yes	characteristic
318	C₄	#97×D₅₃	1	1	yes	characteristic
319	C₄	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
320	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
321	C₂	—	106	4	no	—
322	C₂	#336×D₅₃	1	1	yes	characteristic
323	C₂	#20276310	1	1	yes	characteristic
324	C₂	#20276313	1	1	yes	characteristic
325	C₂	#20276327	1	2	yes	—
326	C₂	#20276472	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#1374	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1374	D₅₃
hypocenter	C₅₃	#12545
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
4. #20272793	C₂	semidirect
5. #20272792	C₂	semidirect
6. #20272792	C₂	semidirect
7. C₅₃×#1374	C₂	semidirect
13. C₂×#13133156	C₄	semidirect
18. C₂×C₁₆×D₅₃	C₄	semidirect
164. #1374	D₅₃	semidirect
190. C₅₃	#12545	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
C₂²×C₁₆×D₅₃	2
C₂×#16810577	2
#20272793	2
#20272792	2
C₅₃×#1374	2
#12545	53