Subgroups — #27276971

Order 13568.

462 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	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273102	C₂	1	2	yes	maximal
4	#20272566	C₂	1	2	yes	maximal, complemented
5	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	#16810376	—	2	1	no	—
9	#16810376	—	2	1	no	—
10	#16810376	—	2	1	no	—
11	#16810374	—	2	1	no	—
12	Q₈⋊#88322	—	2	2	no	—
13	D₄⋊#88322	—	2	2	no	—
14	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
16	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#405	—	2	1	no	—
18	C₅₃×#405	—	2	1	no	—
19	C₅₃×#405	C₄	1	1	yes	characteristic
20	C₅₃×#405	C₄	1	1	yes	characteristic
21	C₄×M₁₆×C₅₃	—	2	1	no	—
22	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	—	2	1	no	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₂²⋊#88322	—	2	2	no	—
30	C₄×#88322	—	2	2	no	—
31	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
32	#131×C₅₃	—	2	2	no	—
33	#131×C₅₃	—	2	2	no	—
34	#131×C₅₃	—	2	2	no	—
35	#131×C₅₃	C₈	1	4	yes	complemented
36	C₂×M₁₆×C₅₃	—	2	1	no	—
37	C₂×M₁₆×C₅₃	—	2	1	no	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	—	2	1	no	—
41	C₂²×C₈×C₅₃	—	2	1	no	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
46	C₄×D₄×C₅₃	—	2	1	no	—
47	C₄×D₄×C₅₃	—	2	1	no	—
48	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
49	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
50	#117×C₅₃	—	2	1	no	—
51	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	—	2	1	no	—
53	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	#97×C₅₃	—	2	1	no	—
55	#97×C₅₃	—	2	1	no	—
56	#97×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
64	#41×C₅₃	—	2	1	no	—
65	#41×C₅₃	—	2	1	no	—
66	#41×C₅₃	C₂×C₈	1	2	yes	—
67	#41×C₅₃	M₁₆	1	2	yes	—
68	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
69	C₂×D₄×C₅₃	—	2	1	no	—
70	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	—	2	1	no	—
73	C₂²×C₄×C₅₃	—	2	1	no	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
77	M₁₆×C₅₃	—	2	4	no	—
78	M₁₆×C₅₃	—	2	1	no	—
79	M₁₆×C₅₃	—	2	1	no	—
80	M₁₆×C₅₃	—	2	4	no	—
81	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
82	M₁₆×C₅₃	M₁₆	1	2	yes	—
83	C₂×C₈×C₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	—	2	1	no	—
85	C₂×C₈×C₅₃	—	2	1	no	—
86	C₂×C₈×C₅₃	—	2	1	no	—
87	C₂×C₈×C₅₃	—	2	1	no	—
88	C₂×C₈×C₅₃	—	2	4	no	—
89	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
94	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
95	C₄⋊C₄×C₅₃	—	2	1	no	—
96	C₄⋊C₄×C₅₃	—	2	1	no	—
97	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂²⋊C₄×C₅₃	—	2	1	no	—
100	C₂²⋊C₄×C₅₃	—	2	1	no	—
101	C₂²⋊C₄×C₅₃	—	2	1	no	—
102	C₄²×C₅₃	—	2	1	no	—
103	C₄²×C₅₃	—	2	1	no	—
104	C₄²×C₅₃	—	2	1	no	—
105	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
109	C₂×#88322	—	4	2	no	—
110	C₂×#88322	—	4	2	no	—
111	C₂×#88322	—	8	2	no	—
112	C₂³×C₅₃	—	2	1	no	—
113	C₂³×C₅₃	#117	1	1	yes	characteristic
114	Q₈×C₅₃	—	2	1	no	—
115	Q₈×C₅₃	C₄⋊C₈	1	2	yes	—
116	D₄×C₅₃	—	2	2	no	—
117	D₄×C₅₃	—	2	2	no	—
118	D₄×C₅₃	—	2	1	no	—
119	D₄×C₅₃	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	1	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	2	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	2	2	no	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
146	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
147	#88322	—	8	2	no	—
148	#88322	—	8	2	no	—
149	#8605	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	—	2	2	no	—
153	C₂²×C₅₃	—	4	1	no	—
154	C₂²×C₅₃	—	4	1	no	—
155	C₂²×C₅₃	#380	1	1	yes	characteristic
156	C₂²×C₅₃	#431	1	1	yes	characteristic
157	C₂²×C₅₃	#432	1	1	yes	characteristic
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	4	1	no	—
163	C₄×C₅₃	—	4	1	no	—
164	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
165	C₄×C₅₃	#407	1	1	yes	characteristic
166	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
167	C₄×C₅₃	#423	1	1	yes	characteristic
168	C₄×C₅₃	#472	1	1	yes	characteristic
169	C₄×C₅₃	#473	1	1	yes	characteristic
170	C₄×#131	D₅₃	1	1	yes	characteristic
171	#1564	—	53	1	no	—
172	C₂×#335	—	53	1	no	—
173	#1477	—	53	2	no	—
174	#1283	—	53	2	no	—
175	C₂×C₅₃	—	2	1	no	—
176	C₂×C₅₃	—	2	1	no	—
177	C₂×C₅₃	—	4	2	no	—
178	C₂×C₅₃	#1797	1	1	yes	characteristic
179	C₂×C₅₃	#1814	1	1	yes	characteristic
180	C₂×C₅₃	#1816	1	1	yes	characteristic
181	C₂×#131	Dic₅₃	1	1	yes	characteristic
182	C₂×#131	Dic₅₃	1	1	yes	characteristic
183	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
184	#431	—	53	2	no	—
185	C₂×C₄⋊C₈	—	53	2	no	—
186	#405	—	2	1	no	—
187	#405	—	2	1	no	—
188	#405	Dic₅₃	1	1	yes	characteristic
189	#405	Dic₅₃	1	1	yes	characteristic
190	C₄×M₁₆	—	2	1	no	—
191	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
192	C₂×C₄×C₈	—	2	1	no	—
193	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
194	#335	—	106	1	no	—
195	#335	—	106	1	no	—
196	#335	—	106	1	no	—
197	#334	—	106	1	no	—
198	Q₈⋊C₈	—	106	2	no	—
199	D₄⋊C₈	—	106	2	no	—
200	C₅₃	#8605	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
202	#131	—	2	2	no	—
203	#131	—	2	2	no	—
204	#131	—	2	2	no	—
205	#131	#88322	1	4	yes	complemented
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
209	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	—	106	2	no	—
213	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
214	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
215	C₄×Q₈	Dic₁₀₆	1	1	yes	characteristic
216	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
217	C₄×D₄	—	2	1	no	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	Dic₁₀₆	1	1	yes	characteristic
220	C₄×D₄	D₂₁₂	1	1	yes	characteristic
221	#117	—	2	1	no	—
222	#117	C₂×Dic₅₃	1	1	yes	characteristic
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
225	C₄⋊C₈	—	106	2	no	—
226	C₄⋊C₈	—	106	2	no	—
227	C₄⋊C₈	—	106	2	no	—
228	C₂²⋊C₈	—	106	2	no	—
229	#97	—	2	1	no	—
230	#97	—	2	1	no	—
231	#97	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	106	2	no	—
236	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
237	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
238	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
239	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
240	#41	—	2	1	no	—
241	#41	—	2	1	no	—
242	#41	C₂×#88322	1	2	yes	—
243	#41	#11719917	1	2	yes	—
244	C₂×Q₈	C₄⋊Dic₅₃	1	1	yes	characteristic
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	C₄⋊Dic₅₃	1	1	yes	characteristic
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	#11719945	1	1	yes	characteristic
251	C₂²×C₄	#11719945	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
253	M₁₆	—	2	4	no	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	C₂×#88322	1	2	yes	—
258	M₁₆	#11719917	1	2	yes	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	424	2	no	—
266	C₂×C₈	—	212	2	no	—
267	C₂×C₈	—	212	2	no	—
268	C₂×C₈	C₂×#88322	1	2	yes	—
269	C₂×C₈	#11719917	1	2	yes	—
270	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
271	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
273	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
277	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
285	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
288	C₂³	—	2	1	no	—
289	C₂³	#13133176	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	C₄⋊#88322	1	2	yes	—
292	D₄	—	2	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	2	1	no	—
295	D₄	C₄⋊#88322	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
309	C₂×C₄	#13133191	1	1	yes	characteristic
310	C₂×C₄	#13133191	1	1	yes	characteristic
311	C₂×C₄	#13133196	1	1	yes	characteristic
312	C₂×C₄	#13133196	1	1	yes	characteristic
313	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
314	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
315	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
316	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	—	424	2	no	—
322	C₈	—	424	2	no	—
323	C₈	C₂²⋊#88322	1	2	yes	—
324	C₈	C₂²⋊#88322	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	#16810483	1	1	yes	characteristic
331	C₂²	#16810676	1	1	yes	characteristic
332	C₂²	#16810680	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
340	C₄	#16810587	1	1	yes	characteristic
341	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
342	C₄	#16810647	1	1	yes	characteristic
343	C₄	#16810893	1	1	yes	characteristic
344	C₄	#16810898	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	4	2	no	—
348	C₂	#20274695	1	1	yes	characteristic
349	C₂	#20274769	1	1	yes	characteristic
350	C₂	#20274779	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#8605
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #20273463	C₂	semidirect
2. C₂×#16810376	C₂	semidirect
4. #20272566	C₂	semidirect
35. #131×C₅₃	C₈	semidirect
200. C₅₃	#8605	semidirect
205. #131	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20273463	2
C₂×#16810376	2
#20273102	2
#20272566	2
C₄×#131×C₅₃	2
#8605	53