Subgroups — #27349754

Order 13568.

750 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	#20283682	C₂	1	1	yes	maximal, characteristic
2	#41×#88322	C₂	1	1	yes	maximal, characteristic
3	#20273015	C₂	1	2	yes	maximal, complemented
4	#20273010	C₂	1	2	yes	maximal, complemented
5	#20273007	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273002	C₂	1	1	yes	maximal, characteristic, complemented
7	#20272999	C₂	1	2	yes	maximal, complemented
8	#20272996	C₂	1	2	yes	maximal, complemented
9	M₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2922	C₂	1	1	yes	maximal, characteristic
12	#16810754	C₄	1	2	yes	—
13	Q₈×#88322	C₄	1	2	yes	—
14	#16810701	C₄	1	2	yes	—
15	#16810695	C₄	1	4	yes	—
16	D₄×#88322	C₄	1	2	yes	—
17	D₄×#88322	C₄	1	4	yes	—
18	#16810676	C₂²	1	1	yes	characteristic
19	#16810676	C₂²	1	2	yes	—
20	#16810676	C₂²	1	1	yes	characteristic
21	#16810676	C₂²	1	2	yes	—
22	C₄×#11719917	C₂²	1	2	yes	—
23	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
24	C₂×C₄×#88322	C₂²	1	2	yes	—
25	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
26	#16810345	C₂²	1	2	yes	—
27	#16810345	C₂²	1	2	yes	—
28	#16810345	C₂²	1	2	yes	—
29	#16810345	C₂²	1	2	yes	—
30	#16810344	C₂²	1	2	yes	—
31	C₈×#88322	C₂²	1	1	yes	characteristic
32	C₈×#88322	C₂²	1	1	yes	characteristic
33	C₈×#88322	C₂²	1	1	yes	characteristic
34	C₈×#88322	C₂²	1	1	yes	characteristic
35	C₈×#88322	C₂²	1	2	yes	—
36	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×#433	C₄	1	4	yes	—
38	C₅₃×#431	C₂²	1	2	yes	—
39	C₅₃×#431	C₂²	1	2	yes	—
40	C₅₃×#424	C₄	1	2	yes	—
41	C₅₃×#408	C₄	1	2	yes	—
42	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×#11719917	C₂×C₄	1	4	yes	—
45	C₂²×#88322	C₂×C₄	1	2	yes	—
46	C₂²×#88322	C₂×C₄	1	4	yes	—
47	C₂²×#88322	C₂×C₄	1	2	yes	—
48	C₄⋊#88322	C₂×C₄	1	4	yes	—
49	C₄⋊#88322	C₂×C₄	1	2	yes	—
50	C₄⋊#88322	C₂×C₄	1	2	yes	—
51	C₄⋊#88322	C₂×C₄	1	4	yes	—
52	C₂²⋊#88322	C₂×C₄	1	2	yes	—
53	C₂²⋊#88322	C₂×C₄	1	4	yes	—
54	C₂²⋊#88322	C₂×C₄	1	2	yes	—
55	C₂²⋊#88322	C₂×C₄	1	4	yes	—
56	#13133128	C₂×C₄	1	4	yes	—
57	C₄×#88322	C₂×C₄	1	2	yes	—
58	C₄×#88322	C₂×C₄	1	2	yes	—
59	C₄×#88322	C₂×C₄	1	4	yes	—
60	C₄×#88322	C₂³	1	1	yes	characteristic
61	C₄×#88322	C₂³	1	1	yes	characteristic
62	C₄×#88322	C₂³	1	1	yes	characteristic
63	C₄×#88322	C₂³	1	1	yes	characteristic
64	C₄×#88322	C₂³	1	2	yes	—
65	C₄×#88322	C₂³	1	2	yes	—
66	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
68	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
73	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
74	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
75	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
76	#117×C₅₃	C₂×C₄	1	2	yes	—
77	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
79	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
80	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
81	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
82	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
83	#97×C₅₃	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₅₃	C₂³	1	2	yes	—
88	#41×C₅₃	—	2	4	no	—
89	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
90	C₂×D₄×C₅₃	C₄²	1	2	yes	—
91	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	M₁₆×C₅₃	—	2	4	no	—
99	C₂×C₈×C₅₃	—	2	4	no	—
100	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
103	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
104	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
105	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
106	C₄⋊C₄×C₅₃	C₄²	1	1	yes	characteristic
107	C₂²⋊C₄×C₅₃	C₄²	1	2	yes	—
108	C₂²⋊C₄×C₅₃	C₄²	1	2	yes	—
109	C₂²⋊C₄×C₅₃	C₄²	1	2	yes	—
110	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
111	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
116	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
117	#11719917	—	2	8	no	—
118	C₂×#88322	—	2	4	no	—
119	C₂×#88322	—	2	8	no	—
120	C₂×#88322	—	2	4	no	—
121	C₂×#88322	C₂²×C₄	1	2	yes	—
122	C₂×#88322	C₂²×C₄	1	2	yes	—
123	C₂×#88322	C₂²×C₄	1	4	yes	—
124	C₂×#88322	C₂²×C₄	1	4	yes	—
125	C₂×#88322	C₂²×C₄	1	2	yes	—
126	C₂×#88322	C₂²×C₄	1	2	yes	—
127	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
128	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
129	Q₈×C₅₃	—	2	2	no	—
130	D₄×C₅₃	—	2	4	no	—
131	D₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	4	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	4	no	—
136	C₂×C₄×C₅₃	—	2	4	no	—
137	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
138	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
141	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
142	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#131	1	4	yes	—
144	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
147	C₈×C₅₃	—	2	2	no	—
148	C₈×C₅₃	—	2	2	no	—
149	C₈×C₅₃	—	2	4	no	—
150	#88322	—	2	2	no	—
151	#88322	—	2	4	no	—
152	#88322	—	2	2	no	—
153	#88322	—	2	4	no	—
154	#88322	#131	1	4	yes	—
155	#88322	#131	1	4	yes	—
156	#17328	—	53	1	no	maximal, Sylow
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	2	2	no	—
160	C₂²×C₅₃	—	2	2	no	—
161	C₂²×C₅₃	—	2	2	no	—
162	C₂²×C₅₃	#405	1	2	yes	—
163	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
164	C₄×C₅₃	—	2	2	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	2	2	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	#405	1	2	yes	—
170	C₄×C₅₃	C₂×#131	1	2	yes	—
171	C₄×C₅₃	C₂×#131	1	2	yes	—
172	C₄×C₅₃	C₂×#131	1	2	yes	—
173	#2965	—	53	1	no	—
174	C₈×#41	—	53	1	no	—
175	#2922	D₅₃	1	1	yes	characteristic
176	#1454	—	53	2	no	—
177	#1453	—	53	2	no	—
178	#1452	—	53	1	no	—
179	#1451	—	53	1	no	—
180	#1450	—	53	2	no	—
181	C₈×M₁₆	—	53	2	no	—
182	C₈×M₁₆	—	53	1	no	—
183	C₂×C₈²	—	53	1	no	—
184	C₂×C₅₃	—	2	2	no	—
185	C₂×C₅₃	—	4	2	no	—
186	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
187	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
188	C₂×C₅₃	C₄×#131	1	1	yes	characteristic
189	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
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	2	no	—
195	C₈×D₄	—	53	4	no	—
196	#433	Dic₅₃	1	4	yes	—
197	#431	—	53	1	no	—
198	#431	—	53	2	no	—
199	#431	—	53	1	no	—
200	#431	—	53	2	no	—
201	#431	C₂×D₅₃	1	2	yes	—
202	#431	C₂×D₅₃	1	2	yes	—
203	#424	Dic₅₃	1	2	yes	—
204	#408	Dic₅₃	1	2	yes	—
205	C₄×M₁₆	—	53	2	no	—
206	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
207	C₂×C₄×C₈	—	53	1	no	—
208	C₂×C₄×C₈	—	53	2	no	—
209	C₂×C₄×C₈	—	53	1	no	—
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	—	53	2	no	—
216	C₈²	—	53	1	no	—
217	C₈²	—	53	1	no	—
218	C₈²	—	53	1	no	—
219	C₈²	—	53	1	no	—
220	C₈²	—	53	2	no	—
221	C₅₃	#17328	1	1	yes	characteristic, complemented, Sylow
222	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
223	C₂×M₁₆	—	53	4	no	—
224	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
225	C₂²×C₈	—	53	2	no	—
226	C₂²×C₈	—	53	4	no	—
227	C₂²×C₈	—	53	2	no	—
228	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
229	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
230	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
231	C₄×D₄	C₄×D₅₃	1	2	yes	—
232	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
233	C₄×D₄	C₄×D₅₃	1	2	yes	—
234	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
235	#117	C₂×Dic₅₃	1	1	yes	characteristic
236	#117	C₂×Dic₅₃	1	2	yes	—
237	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
239	C₄⋊C₈	—	53	2	no	—
240	C₄⋊C₈	—	53	4	no	—
241	C₄⋊C₈	—	53	4	no	—
242	C₄⋊C₈	—	53	2	no	—
243	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
244	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
245	C₂²⋊C₈	—	53	2	no	—
246	C₂²⋊C₈	—	53	4	no	—
247	C₂²⋊C₈	—	53	2	no	—
248	C₂²⋊C₈	—	53	4	no	—
249	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
250	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
251	#97	—	53	4	no	—
252	#97	C₂×Dic₅₃	1	2	yes	—
253	C₄×C₈	—	53	2	no	—
254	C₄×C₈	—	53	4	no	—
255	C₄×C₈	—	53	2	no	—
256	C₄×C₈	—	53	1	no	—
257	C₄×C₈	—	53	1	no	—
258	C₄×C₈	—	53	1	no	—
259	C₄×C₈	—	53	1	no	—
260	C₄×C₈	—	53	2	no	—
261	C₄×C₈	—	53	2	no	—
262	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
263	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₅₃	1	2	yes	—
266	#41	—	2	4	no	—
267	C₂×Q₈	C₄×Dic₅₃	1	1	yes	characteristic
268	C₂×D₄	C₄×Dic₅₃	1	2	yes	—
269	C₂×D₄	C₄×Dic₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
271	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
274	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
275	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
276	M₁₆	—	2	4	no	—
277	M₁₆	—	106	8	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	53	2	no	—
280	C₂×C₈	—	53	2	no	—
281	C₂×C₈	—	53	2	no	—
282	C₂×C₈	—	53	4	no	—
283	C₂×C₈	—	53	2	no	—
284	C₂×C₈	—	53	4	no	—
285	C₂×C₈	—	106	4	no	—
286	C₂×C₈	—	106	8	no	—
287	C₂×C₈	—	106	4	no	—
288	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
289	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
290	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
291	C₄⋊C₄	C₄×Dic₅₃	1	2	yes	—
292	C₄⋊C₄	C₄×Dic₅₃	1	2	yes	—
293	C₄⋊C₄	C₄×Dic₅₃	1	1	yes	characteristic
294	C₄⋊C₄	C₄×Dic₅₃	1	1	yes	characteristic
295	C₂²⋊C₄	C₄×Dic₅₃	1	2	yes	—
296	C₂²⋊C₄	C₄×Dic₅₃	1	2	yes	—
297	C₂²⋊C₄	C₄×Dic₅₃	1	2	yes	—
298	C₄²	C₂×C₄×D₅₃	1	2	yes	—
299	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
300	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
301	C₄²	C₂×C₄×D₅₃	1	2	yes	—
302	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
303	C₄²	C₂²×Dic₅₃	1	2	yes	—
304	C₄²	C₂³×D₅₃	1	1	yes	characteristic
305	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
306	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
316	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
319	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
320	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
321	C₂×C₄	#13133224	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	C₂³×Dic₅₃	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	4	no	—
328	C₈	—	53	4	no	—
329	C₈	—	53	4	no	—
330	C₈	—	106	2	no	—
331	C₈	—	106	4	no	—
332	C₈	—	106	2	no	—
333	C₈	—	106	4	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	#16810579	1	2	yes	—
340	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	#16810579	1	2	yes	—
347	C₄	#131×D₅₃	1	2	yes	—
348	C₄	#131×D₅₃	1	2	yes	—
349	C₄	C₂×#13133224	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	4	2	no	—
352	C₂	C₂×#16810579	1	1	yes	characteristic
353	C₂	#131×Dic₅₃	1	1	yes	characteristic
354	C₂	#131×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#2922	C₂
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2922	D₅₃
hypocenter	C₅₃	#17328
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #2922

Complements

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

N	H ≅ G/N	type
3. #20273015	C₂	semidirect
4. #20273010	C₂	semidirect
5. #20273007	C₂	semidirect
6. #20273002	C₂	semidirect
7. #20272999	C₂	semidirect
8. #20272996	C₂	semidirect
9. M₁₆×#88322	C₂	semidirect
10. C₂×C₈×#88322	C₂	semidirect
221. C₅₃	#17328	semidirect

Maximal subgroups

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

subgroup	index
#20283682	2
#41×#88322	2
#20273015	2
#20273010	2
#20273007	2
#20273002	2
#20272999	2
#20272996	2
M₁₆×#88322	2
C₂×C₈×#88322	2
C₅₃×#2922	2
#17328	53