Subgroups — #27382706

Order 13568.

886 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	C₂×#16810754	C₂	1	1	yes	maximal, characteristic
2	#20277193	C₂	1	1	yes	maximal, characteristic
3	#20274592	C₂	1	2	yes	maximal
4	#20274585	C₂	1	1	yes	maximal, characteristic
5	C₈×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
6	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	#20274574	C₂	1	2	yes	maximal
8	#20274570	C₂	1	1	yes	maximal, characteristic
9	#20274401	C₂	1	2	yes	maximal
10	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
11	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
12	#16810754	C₄	1	8	yes	complemented
13	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	C₂×#13133128	C₂²	1	2	yes	—
16	C₂×#13133127	C₂²	1	2	yes	—
17	C₂×#13133127	C₂²	1	2	yes	—
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
22	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
24	#16810530	C₄	1	2	yes	—
25	#16810522	C₄	1	2	yes	—
26	#16810509	C₄	1	4	yes	—
27	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
29	#16810469	C₂²	1	2	yes	—
30	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
31	#16810377	C₂²	1	2	yes	—
32	#16810377	C₂²	1	2	yes	—
33	#16810377	C₂²	1	2	yes	—
34	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₈×Q₈×C₅₃	C₄	1	8	yes	complemented
36	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
40	C₂²×#88322	C₂³	1	1	yes	characteristic
41	C₂²×#88322	C₂³	1	2	yes	—
42	C₂²×#88322	C₂³	1	1	yes	characteristic
43	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
49	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
50	C₄⋊#88322	C₂×C₄	1	8	yes	—
51	C₄⋊#88322	C₂×C₄	1	4	yes	—
52	#13133128	C₂×C₄	1	8	yes	—
53	#13133127	—	2	4	no	—
54	#13133127	—	2	4	no	—
55	C₄×#88322	C₂×C₄	1	4	yes	—
56	C₈×Dic₅₃	—	2	2	no	—
57	C₈×Dic₅₃	—	2	4	no	—
58	C₈×Dic₅₃	—	2	2	no	—
59	C₈×Dic₅₃	—	2	2	no	—
60	C₈×Dic₅₃	Q₈	1	4	yes	complemented
61	#13133124	C₂×C₄	1	2	yes	—
62	#13133124	C₂×C₄	1	4	yes	—
63	#13133124	C₂×C₄	1	4	yes	—
64	#13133124	C₂×C₄	1	4	yes	—
65	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
70	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
75	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
76	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
77	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
78	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
79	C₂×Q₈×C₅₃	—	2	4	no	—
80	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
87	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
88	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₅₃	—	2	1	no	—
90	C₂×C₈×C₅₃	—	2	1	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₅₃	#41	1	2	yes	—
98	C₄⋊C₄×C₅₃	C₄²	1	8	yes	—
99	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
100	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
101	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
102	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
103	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₅₃	—	2	4	no	—
105	C₄⋊Dic₅₃	—	2	8	no	—
106	C₄×Dic₅₃	—	2	4	no	—
107	C₄×Dic₅₃	—	2	1	no	—
108	C₄×Dic₅₃	—	2	1	no	—
109	C₄×Dic₅₃	—	2	2	no	—
110	C₄×Dic₅₃	—	2	2	no	—
111	C₄×Dic₅₃	—	2	4	no	—
112	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
113	C₄×Dic₅₃	#41	1	2	yes	—
114	C₂×#88322	—	2	1	no	—
115	C₂×#88322	—	2	1	no	—
116	C₂×#88322	—	2	2	no	—
117	C₂×#88322	—	2	2	no	—
118	C₂×#88322	C₂²×C₄	1	2	yes	—
119	C₂×#88322	C₂²×C₄	1	4	yes	—
120	C₂×#88322	C₂²×C₄	1	2	yes	—
121	C₂×#88322	C₂×Q₈	1	2	yes	—
122	C₂×#88322	#41	1	2	yes	—
123	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	Q₈×C₅₃	—	2	8	no	—
125	C₂×C₄×C₅₃	—	2	4	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	4	no	—
128	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
129	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
131	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₅₃	#131	1	4	yes	—
133	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₅₃	—	2	2	no	—
140	C₂×Dic₅₃	—	2	2	no	—
141	C₂×Dic₅₃	—	2	2	no	—
142	C₂×Dic₅₃	—	2	2	no	—
143	C₂×Dic₅₃	—	2	4	no	—
144	C₂×Dic₅₃	—	2	4	no	—
145	C₂×Dic₅₃	—	2	4	no	—
146	C₂×Dic₅₃	C₄×Q₈	1	4	yes	—
147	C₂×Dic₅₃	#131	1	4	yes	—
148	C₈×C₅₃	—	2	2	no	—
149	C₈×C₅₃	—	2	4	no	—
150	C₈×C₅₃	C₄×Q₈	1	4	yes	—
151	#88322	—	2	2	no	—
152	#88322	—	2	4	no	—
153	#88322	C₄×Q₈	1	4	yes	—
154	#20516	—	53	1	no	maximal, Sylow
155	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
156	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
158	C₂²×C₅₃	C₂×#131	1	2	yes	—
159	C₂²×C₅₃	C₂×#131	1	2	yes	—
160	C₄×C₅₃	—	2	4	no	—
161	C₄×C₅₃	—	2	2	no	—
162	C₄×C₅₃	—	2	4	no	—
163	C₄×C₅₃	#405	1	4	yes	—
164	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
166	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
167	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
168	Dic₅₃	—	2	4	no	—
169	Dic₅₃	—	2	4	no	—
170	Dic₅₃	—	4	4	no	—
171	C₂×#446	—	53	1	no	—
172	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
173	#2295	—	53	1	no	—
174	#1771	—	53	2	no	—
175	#1770	—	53	1	no	—
176	C₈×C₄⋊C₄	—	53	2	no	—
177	C₈×C₄⋊C₄	—	53	1	no	—
178	#1768	—	53	2	no	—
179	#1767	—	53	1	no	—
180	#1726	—	53	2	no	—
181	C₄²×C₈	—	53	1	no	—
182	C₂×C₅₃	C₄²×Q₈	1	1	yes	characteristic
183	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
184	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
185	C₂×C₅₃	#2960	1	2	yes	—
186	C₂×C₅₃	#2966	1	2	yes	—
187	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
188	#446	—	53	8	no	—
189	C₈×Q₈	Dic₅₃	1	8	yes	complemented
190	C₂×C₄⋊C₈	—	53	1	no	—
191	C₂×C₄⋊C₈	—	53	2	no	—
192	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
193	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
194	C₂×#97	—	53	2	no	—
195	C₂×#97	—	53	2	no	—
196	C₂×#97	—	53	2	no	—
197	C₂×C₄×C₈	—	53	1	no	—
198	C₂×C₄×C₈	—	53	1	no	—
199	C₂×C₄×C₈	—	53	1	no	—
200	C₂×C₄×C₈	—	53	1	no	—
201	C₂×C₄×C₈	—	53	1	no	—
202	C₂×C₄×C₈	—	53	2	no	—
203	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
205	#391	—	53	2	no	—
206	#389	—	53	2	no	—
207	#387	—	53	4	no	—
208	C₄×C₄⋊C₄	—	53	1	no	—
209	C₄×C₄⋊C₄	—	53	2	no	—
210	#376	—	53	2	no	—
211	C₄³	—	53	1	no	—
212	#336	—	53	2	no	—
213	#336	—	53	2	no	—
214	#336	—	53	2	no	—
215	C₅₃	#20516	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₈	—	53	1	no	—
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	2	no	—
220	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
221	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
222	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
223	C₄×Q₈	C₄×D₅₃	1	4	yes	—
224	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
225	C₂×C₄⋊C₄	—	53	4	no	—
226	C₂×C₄⋊C₄	—	53	2	no	—
227	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
228	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
229	C₂×C₄²	—	53	1	no	—
230	C₂×C₄²	—	53	2	no	—
231	C₂×C₄²	—	53	2	no	—
232	C₂×C₄²	—	53	1	no	—
233	C₂×C₄²	—	53	2	no	—
234	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
236	C₄⋊C₈	—	53	4	no	—
237	C₄⋊C₈	—	53	8	no	—
238	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
239	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
240	#97	—	53	8	no	—
241	#97	—	106	4	no	—
242	#97	—	106	4	no	—
243	C₄×C₈	—	53	4	no	—
244	C₄×C₈	—	53	4	no	—
245	C₄×C₈	—	106	2	no	—
246	C₄×C₈	—	106	2	no	—
247	C₄×C₈	—	106	2	no	—
248	C₄×C₈	—	106	4	no	—
249	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
250	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
251	#95	—	53	4	no	—
252	#95	—	53	4	no	—
253	#95	—	53	2	no	—
254	#95	—	53	4	no	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	C₄×Dic₅₃	1	4	yes	—
257	C₂²×C₄	—	53	2	no	—
258	C₂²×C₄	—	53	2	no	—
259	C₂²×C₄	—	53	4	no	—
260	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
261	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
263	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	53	2	no	—
270	C₂×C₈	—	53	2	no	—
271	C₂×C₈	—	53	2	no	—
272	C₂×C₈	—	53	2	no	—
273	C₂×C₈	—	53	4	no	—
274	C₂×C₈	—	106	1	no	—
275	C₂×C₈	—	106	2	no	—
276	C₂×C₈	—	106	1	no	—
277	C₂×C₈	—	106	2	no	—
278	C₂×C₈	Q₈×D₅₃	1	2	yes	—
279	C₂×C₈	#11719949	1	2	yes	—
280	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
281	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
282	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
283	C₄⋊C₄	—	106	8	no	—
284	C₄⋊C₄	—	106	4	no	—
285	C₄⋊C₄	C₄×Dic₅₃	1	8	yes	—
286	C₄⋊C₄	C₄×Dic₅₃	1	4	yes	—
287	C₄²	—	53	2	no	—
288	C₄²	—	53	2	no	—
289	C₄²	—	106	4	no	—
290	C₄²	—	106	4	no	—
291	C₄²	—	106	1	no	—
292	C₄²	—	106	2	no	—
293	C₄²	—	106	1	no	—
294	C₄²	—	106	2	no	—
295	C₄²	C₂×C₄×D₅₃	1	2	yes	—
296	C₄²	C₂×C₄×D₅₃	1	4	yes	—
297	C₄²	C₂²×Dic₅₃	1	4	yes	—
298	C₄²	C₂²×Dic₅₃	1	2	yes	—
299	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
300	Q₈	—	2	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	53	4	no	—
305	C₂×C₄	—	53	4	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	—	106	2	no	—
308	C₂×C₄	—	106	2	no	—
309	C₂×C₄	—	106	2	no	—
310	C₂×C₄	—	106	4	no	—
311	C₂×C₄	—	106	4	no	—
312	C₂×C₄	—	106	4	no	—
313	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
314	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
315	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
316	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
317	C₂×C₄	#13133223	1	4	yes	—
318	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
319	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
320	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
321	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
322	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
323	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	4	no	—
326	C₈	—	53	4	no	—
327	C₈	—	106	2	no	—
328	C₈	—	106	4	no	—
329	C₈	Q₈×Dic₅₃	1	4	yes	—
330	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
331	C₂²	C₄×Q₈×D₅₃	1	1	yes	characteristic
332	C₂²	C₄×#11719949	1	1	yes	characteristic
333	C₂²	#131×D₅₃	1	2	yes	—
334	C₂²	C₂×#13133223	1	2	yes	—
335	C₄	—	2	2	no	—
336	C₄	—	2	4	no	—
337	C₄	—	2	4	no	—
338	C₄	—	106	4	no	—
339	C₄	—	106	4	no	—
340	C₄	—	212	4	no	—
341	C₄	#16810578	1	4	yes	—
342	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
343	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
344	C₄	C₄×#11719949	1	1	yes	characteristic
345	C₄	#41×Dic₅₃	1	1	yes	characteristic
346	C₂	C₄×Q₈×Dic₅₃	1	1	yes	characteristic
347	C₂	C₂×#16810578	1	1	yes	characteristic
348	C₂	#131×Dic₅₃	1	1	yes	characteristic
349	C₂	#20283644	1	2	yes	—
350	C₂	#20283692	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#20516
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
12. #16810754	C₄	semidirect
35. C₈×Q₈×C₅₃	C₄	semidirect
60. C₈×Dic₅₃	Q₈	semidirect
189. C₈×Q₈	Dic₅₃	semidirect
215. C₅₃	#20516	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810754	2
#20277193	2
#20274592	2
#20274585	2
C₈×C₄⋊Dic₅₃	2
#20274574	2
#20274570	2
#20274401	2
C₄×C₈×Dic₅₃	2
C₂×C₈×Q₈×C₅₃	2
#20516	53