Subgroups — #27354751

Order 13568.

640 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285291	C₂	1	1	yes	maximal, characteristic
2	#20283792	C₂	1	2	yes	maximal, complemented
3	#20273627	C₂	1	2	yes	maximal, complemented
4	#20273601	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273536	C₂	1	2	yes	maximal, complemented
6	D₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273509	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273477	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273182	C₂	1	2	yes	maximal, complemented
10	#20272997	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1589	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811170	C₂²	1	2	yes	—
13	#16811027	C₄	1	1	yes	characteristic
14	#16810998	C₄	1	1	yes	characteristic
15	#16810968	C₄	1	2	yes	—
16	#16810860	C₄	1	1	yes	characteristic
17	#16810858	C₄	1	2	yes	—
18	#16810856	C₄	1	1	yes	characteristic
19	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
20	C₄×#11719922	C₂²	1	2	yes	—
21	#16810702	C₂²	1	2	yes	—
22	#16810692	—	2	2	no	—
23	#16810691	—	2	2	no	—
24	D₄×#88322	C₂²	1	2	yes	—
25	#16810688	C₂²	1	2	yes	—
26	#16810679	C₂²	1	2	yes	—
27	#16810674	C₂²	1	1	yes	characteristic
28	#16810674	C₂²	1	1	yes	characteristic
29	#16810591	—	2	2	no	—
30	#16810588	—	2	2	no	—
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	#16810376	C₂²	1	1	yes	characteristic
33	#16810373	C₂²	1	2	yes	—
34	#16810352	C₂²	1	2	yes	—
35	D₄⋊#88322	C₂²	1	2	yes	—
36	#16810350	C₂²	1	2	yes	—
37	#16810345	C₂²	1	1	yes	characteristic
38	#16810345	C₂²	1	1	yes	characteristic
39	C₈×#88322	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#436	C₂²	1	2	yes	—
42	C₅₃×#335	C₂²	1	1	yes	characteristic
43	C₅₃×D₄⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×#11719945	—	2	2	no	—
46	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
47	C₂×#11719922	C₂×C₄	1	2	yes	—
48	C₂×#11719912	—	2	2	no	—
49	C₂²×#88322	—	2	2	no	—
50	C₂×C₈×D₅₃	—	2	2	no	—
51	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
53	#13133216	C₂×C₄	1	2	yes	—
54	#13133213	C₂×C₄	1	2	yes	—
55	#13133203	—	2	2	no	—
56	#13133192	C₂×C₄	1	2	yes	—
57	#13133190	—	2	2	no	—
58	#13133188	C₂×C₄	1	2	yes	—
59	C₄×D₂₁₂	C₂³	1	2	yes	—
60	C₄×C₂²⋊D₅₃	—	2	2	no	—
61	C₄²×D₅₃	C₂³	1	1	yes	characteristic
62	#13133151	C₂×C₄	1	1	yes	characteristic
63	#13133150	C₂×C₄	1	2	yes	—
64	C₄⋊#88322	C₂³	1	2	yes	—
65	#13133146	—	2	1	no	—
66	#13133146	D₄	1	1	yes	characteristic
67	#13133146	D₄	1	1	yes	characteristic
68	#13133139	C₂×C₄	1	2	yes	—
69	#13133138	C₂×C₄	1	2	yes	—
70	#13133137	C₂×C₄	1	2	yes	—
71	C₂²⋊#88322	—	2	2	no	—
72	#13133129	—	2	2	no	—
73	#13133129	—	2	2	no	—
74	#13133129	—	2	2	no	—
75	#13133129	—	2	1	no	—
76	#13133129	D₄	1	1	yes	characteristic
77	#13133129	D₄	1	1	yes	characteristic
78	#13133127	—	2	2	no	—
79	C₄×#88322	C₂³	1	1	yes	characteristic
80	C₄×#88322	C₂³	1	1	yes	characteristic
81	C₄×#88322	C₂³	1	1	yes	characteristic
82	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×M₁₆×C₅₃	—	2	2	no	—
86	C₄×D₄×C₅₃	C₂³	1	2	yes	—
87	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
89	#102×C₅₃	C₂×C₄	1	2	yes	—
90	C₂²⋊C₈×C₅₃	—	2	2	no	—
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂×C₂²⋊D₅₃	—	2	2	no	—
97	C₂×C₂²⋊D₅₃	—	2	2	no	—
98	C₂×C₂²⋊D₅₃	—	2	2	no	—
99	#11719945	—	8	2	no	—
100	#11719945	—	8	2	no	—
101	C₂×D₂₁₂	—	2	2	no	—
102	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
103	C₂×C₄×D₅₃	—	2	2	no	—
104	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
107	C₂×Dic₁₀₆	—	2	2	no	—
108	D₈×C₅₃	—	2	2	no	—
109	M₁₆×C₅₃	—	8	2	no	—
110	C₂×C₈×C₅₃	—	2	2	no	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
116	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
117	C₂²⋊C₄×C₅₃	—	2	2	no	—
118	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
119	C₂²⋊Dic₅₃	—	2	2	no	—
120	#11719922	—	2	4	no	—
121	#11719921	—	2	2	no	—
122	#11719921	—	2	2	no	—
123	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
124	#11719919	—	2	2	no	—
125	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
127	C₄×Dic₅₃	#41	1	1	yes	characteristic
128	C₂×#88322	—	2	2	no	—
129	C₂×#88322	—	2	1	no	—
130	C₂×#88322	—	8	2	no	—
131	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
132	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
133	C₂×#88322	C₂×D₄	1	1	yes	characteristic
134	C₂×#88322	C₂×D₄	1	1	yes	characteristic
135	D₄₂₄	—	2	2	no	—
136	#11719912	—	8	2	no	—
137	C₈×D₅₃	—	2	2	no	—
138	C₈×D₅₃	—	2	2	no	—
139	C₈×D₅₃	—	8	2	no	—
140	C₂³×C₅₃	—	2	2	no	—
141	C₂²×D₅₃	—	2	2	no	—
142	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
143	D₄×C₅₃	—	2	2	no	—
144	D₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	4	2	no	—
147	C₂×C₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
151	C₂²⋊D₅₃	—	8	2	no	—
152	C₂²⋊D₅₃	—	8	2	no	—
153	C₂²⋊D₅₃	—	8	2	no	—
154	C₂×Dic₅₃	—	2	2	no	—
155	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
156	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
157	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
158	D₂₁₂	—	2	2	no	—
159	D₂₁₂	—	2	2	no	—
160	D₂₁₂	—	8	2	no	—
161	C₄×D₅₃	—	2	1	no	—
162	C₄×D₅₃	—	2	1	no	—
163	C₄×D₅₃	—	2	1	no	—
164	C₄×D₅₃	—	2	1	no	—
165	C₄×D₅₃	—	4	1	no	—
166	C₄×D₅₃	—	4	2	no	—
167	C₄×D₅₃	—	4	2	no	—
168	Dic₁₀₆	—	8	2	no	—
169	C₈×C₅₃	—	2	1	no	—
170	C₈×C₅₃	—	2	1	no	—
171	C₈×C₅₃	—	2	1	no	—
172	C₈×C₅₃	—	4	1	no	—
173	C₈×C₅₃	—	4	2	no	—
174	C₈×C₅₃	#131	1	2	yes	—
175	#88322	—	2	1	no	—
176	#88322	—	2	1	no	—
177	#88322	—	2	1	no	—
178	#88322	—	4	1	no	—
179	#88322	—	4	2	no	—
180	#88322	#131	1	2	yes	—
181	#17970	—	53	1	no	maximal, Sylow
182	C₂²×C₅₃	—	4	2	no	—
183	C₂²×C₅₃	—	4	2	no	—
184	C₂²×C₅₃	—	4	2	no	—
185	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
186	C₂×D₅₃	—	2	1	no	—
187	C₂×D₅₃	—	2	1	no	—
188	C₂×D₅₃	—	2	1	no	—
189	C₂×D₅₃	—	4	2	no	—
190	C₂×D₅₃	—	4	2	no	—
191	C₂×D₅₃	—	4	2	no	—
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	—	4	2	no	—
194	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
195	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
196	C₄×C₅₃	#576	1	1	yes	characteristic
197	C₄×C₅₃	#576	1	1	yes	characteristic
198	Dic₅₃	—	2	1	no	—
199	Dic₅₃	—	2	1	no	—
200	Dic₅₃	—	2	1	no	—
201	Dic₅₃	—	4	2	no	—
202	#3169	—	53	1	no	—
203	#2981	—	53	2	no	—
204	#1594	—	53	2	no	—
205	#1589	—	53	1	no	—
206	#1589	D₅₃	1	1	yes	characteristic, complemented
207	#1580	—	53	2	no	—
208	C₈×D₈	—	53	1	no	—
209	C₈×D₈	—	53	1	no	—
210	#1568	—	53	1	no	—
211	#1496	—	53	2	no	—
212	#1450	—	53	1	no	—
213	C₂×C₅₃	—	8	2	no	—
214	C₂×C₅₃	#2931	1	1	yes	characteristic
215	C₂×C₅₃	#2948	1	1	yes	characteristic
216	C₂×C₅₃	#2957	1	1	yes	characteristic
217	D₅₃	—	4	1	no	—
218	D₅₃	—	8	2	no	—
219	#532	—	53	2	no	—
220	#500	—	53	1	no	—
221	#493	—	53	1	no	—
222	#486	—	53	2	no	—
223	#466	—	53	1	no	—
224	#466	—	53	1	no	—
225	#466	—	53	2	no	—
226	C₄×D₈	—	53	2	no	—
227	C₄×D₈	—	53	1	no	—
228	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
229	#436	—	53	2	no	—
230	#436	C₂×D₅₃	1	2	yes	—
231	#435	—	106	2	no	—
232	#435	—	106	2	no	—
233	C₈×D₄	—	53	2	no	—
234	C₈×D₄	—	53	2	no	—
235	#432	—	53	2	no	—
236	#431	—	53	1	no	—
237	#431	—	53	1	no	—
238	#408	—	106	2	no	—
239	#408	—	106	2	no	—
240	C₂×C₄×C₈	—	53	1	no	—
241	#335	—	53	1	no	—
242	#335	C₂×D₅₃	1	1	yes	characteristic
243	#334	—	53	2	no	—
244	D₄⋊C₈	—	53	2	no	—
245	D₄⋊C₈	—	53	2	no	—
246	D₄⋊C₈	—	53	2	no	—
247	D₄⋊C₈	C₂×D₅₃	1	2	yes	—
248	#322	—	53	1	no	—
249	#322	—	53	1	no	—
250	C₈²	—	53	1	no	—
251	C₈²	C₂×D₅₃	1	1	yes	characteristic
252	C₅₃	#17970	1	1	yes	characteristic, complemented, Sylow
253	C₂×#41	—	106	2	no	—
254	C₂×D₈	—	53	2	no	—
255	C₂×D₈	—	53	1	no	—
256	C₂×D₈	C₄×D₅₃	1	1	yes	characteristic
257	C₂×M₁₆	—	2	2	no	—
258	C₂×M₁₆	—	106	2	no	—
259	C₂²×C₈	—	53	1	no	—
260	C₂²×C₈	—	53	1	no	—
261	C₂²×C₈	—	106	2	no	—
262	C₂²×C₈	—	106	2	no	—
263	C₄⋊Q₈	—	53	2	no	—
264	#127	—	53	2	no	—
265	#124	—	106	2	no	—
266	#121	—	53	2	no	—
267	#121	—	53	2	no	—
268	#121	—	106	2	no	—
269	C₄×D₄	—	53	2	no	—
270	C₄×D₄	—	106	2	no	—
271	C₄×D₄	C₂²×D₅₃	1	2	yes	—
272	C₂×C₄²	—	53	1	no	—
273	#107	—	53	1	no	—
274	#107	—	53	2	no	—
275	#107	C₄×D₅₃	1	1	yes	characteristic
276	C₄⋊C₈	—	53	1	no	—
277	C₄⋊C₈	—	53	1	no	—
278	C₄⋊C₈	—	53	2	no	—
279	C₄⋊C₈	—	106	1	no	—
280	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
281	#102	—	53	2	no	—
282	#102	—	53	2	no	—
283	#102	—	53	2	no	—
284	#102	C₄×D₅₃	1	2	yes	—
285	C₂²⋊C₈	—	2	2	no	—
286	C₂²⋊C₈	—	53	1	no	—
287	C₂²⋊C₈	—	53	1	no	—
288	C₂²⋊C₈	—	106	2	no	—
289	C₂²⋊C₈	—	106	2	no	—
290	C₂²⋊C₈	—	106	1	no	—
291	C₂²⋊C₈	—	106	2	no	—
292	C₂²⋊C₈	—	106	2	no	—
293	#97	—	106	2	no	—
294	C₄×C₈	—	53	1	no	—
295	C₄×C₈	—	53	1	no	—
296	C₄×C₈	—	53	1	no	—
297	C₄×C₈	—	53	1	no	—
298	C₄×C₈	—	53	1	no	—
299	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
300	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
301	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
302	#41	—	424	2	no	—
303	#41	—	424	2	no	—
304	C₂×Q₈	—	106	2	no	—
305	C₂×D₄	—	53	2	no	—
306	C₂×D₄	—	106	2	no	—
307	C₂×D₄	—	106	2	no	—
308	C₂×D₄	—	106	2	no	—
309	C₂×D₄	—	106	2	no	—
310	C₂×D₄	C₂×C₄×D₅₃	1	2	yes	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	53	1	no	—
313	C₂²×C₄	—	53	1	no	—
314	C₂²×C₄	—	53	1	no	—
315	C₂²×C₄	—	106	2	no	—
316	D₈	—	2	2	no	—
317	D₈	—	106	4	no	—
318	D₈	—	106	2	no	—
319	M₁₆	—	8	2	no	—
320	M₁₆	—	424	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	53	1	no	—
324	C₂×C₈	—	53	1	no	—
325	C₂×C₈	—	53	1	no	—
326	C₂×C₈	—	53	1	no	—
327	C₂×C₈	—	106	2	no	—
328	C₂×C₈	—	106	2	no	—
329	C₂×C₈	—	106	2	no	—
330	C₂×C₈	—	106	1	no	—
331	C₂×C₈	—	424	2	no	—
332	C₂×C₈	—	424	2	no	—
333	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
334	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
335	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
336	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
337	C₄⋊C₄	—	53	2	no	—
338	C₄⋊C₄	—	106	2	no	—
339	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
340	C₂²⋊C₄	—	2	2	no	—
341	C₂²⋊C₄	—	106	2	no	—
342	C₂²⋊C₄	—	106	2	no	—
343	C₂²⋊C₄	—	106	2	no	—
344	C₄²	—	53	1	no	—
345	C₄²	—	53	1	no	—
346	C₄²	—	53	1	no	—
347	C₄²	C₂³×D₅₃	1	1	yes	characteristic
348	C₂³	—	2	2	no	—
349	C₂³	—	53	1	no	—
350	C₂³	—	106	2	no	—
351	Q₈	—	424	2	no	—
352	D₄	—	2	2	no	—
353	D₄	—	2	2	no	—
354	D₄	—	106	2	no	—
355	D₄	—	106	2	no	—
356	D₄	—	424	2	no	—
357	D₄	—	424	2	no	—
358	D₄	—	424	2	no	—
359	D₄	—	424	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	53	1	no	—
364	C₂×C₄	—	53	1	no	—
365	C₂×C₄	—	53	1	no	—
366	C₂×C₄	—	106	1	no	—
367	C₂×C₄	—	106	1	no	—
368	C₂×C₄	—	106	1	no	—
369	C₂×C₄	—	106	1	no	—
370	C₂×C₄	—	106	2	no	—
371	C₂×C₄	—	212	1	no	—
372	C₂×C₄	—	212	2	no	—
373	C₂×C₄	—	212	2	no	—
374	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
376	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
377	C₈	—	2	1	no	—
378	C₈	—	2	1	no	—
379	C₈	—	2	1	no	—
380	C₈	—	4	1	no	—
381	C₈	—	4	2	no	—
382	C₈	—	53	2	no	—
383	C₈	—	106	1	no	—
384	C₈	—	106	1	no	—
385	C₈	—	106	1	no	—
386	C₈	—	212	1	no	—
387	C₈	—	212	2	no	—
388	C₈	#13133225	1	2	yes	—
389	C₂²	—	4	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	106	1	no	—
393	C₂²	—	106	1	no	—
394	C₂²	—	106	1	no	—
395	C₂²	—	212	2	no	—
396	C₂²	—	212	2	no	—
397	C₂²	—	212	2	no	—
398	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
399	C₄	—	2	1	no	—
400	C₄	—	4	2	no	—
401	C₄	—	106	1	no	—
402	C₄	—	106	1	no	—
403	C₄	—	106	1	no	—
404	C₄	—	212	2	no	—
405	C₄	#131×D₅₃	1	1	yes	characteristic
406	C₄	C₂×#13133225	1	1	yes	characteristic
407	C₄	#16811424	1	1	yes	characteristic
408	C₄	#16811425	1	1	yes	characteristic
409	C₂	—	8	2	no	—
410	C₂	—	424	2	no	—
411	C₂	—	212	1	no	—
412	C₂	#20283424	1	1	yes	characteristic
413	C₂	#20283562	1	1	yes	characteristic
414	C₂	#20283627	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1589	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20285291	C₂
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1589	D₅₃
hypocenter	C₅₃	#17970
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1589

Complements

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

N	H ≅ G/N	type
2. #20283792	C₂	semidirect
3. #20273627	C₂	semidirect
4. #20273601	C₂	semidirect
5. #20273536	C₂	semidirect
6. D₈×#88322	C₂	semidirect
7. #20273509	C₂	semidirect
8. #20273477	C₂	semidirect
9. #20273182	C₂	semidirect
10. #20272997	C₂	semidirect
11. C₅₃×#1589	C₂	semidirect
206. #1589	D₅₃	semidirect
252. C₅₃	#17970	semidirect

Maximal subgroups

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

subgroup	index
#20285291	2
#20283792	2
#20273627	2
#20273601	2
#20273536	2
D₈×#88322	2
#20273509	2
#20273477	2
#20273182	2
#20272997	2
C₅₃×#1589	2
#17970	53