Subgroups — #27367189

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285843	C₂	1	1	yes	maximal, characteristic
2	#20285640	C₂	1	1	yes	maximal, characteristic
3	#20283428	C₂	1	1	yes	maximal, characteristic
4	#20274211	C₂	1	4	yes	maximal, complemented
5	#20274087	C₂	1	4	yes	maximal, complemented
6	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273400	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273396	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133223	—	2	1	no	—
11	#16811173	C₂²	1	1	yes	characteristic
12	#16811019	—	2	1	no	—
13	#16810997	—	2	1	no	—
14	#16810973	—	2	1	no	—
15	#16810971	—	2	1	no	—
16	#16810941	—	2	2	no	—
17	#16810910	C₂²	1	4	yes	—
18	#16810825	C₂²	1	4	yes	—
19	#16810762	—	2	2	no	—
20	#16810702	—	2	1	no	—
21	#16810702	C₂²	1	2	yes	—
22	C₈×D₂₁₂	—	2	1	no	—
23	C₈×D₂₁₂	C₂²	1	2	yes	—
24	#16810676	C₂²	1	1	yes	characteristic
25	#16810676	C₂²	1	1	yes	characteristic
26	#16810647	C₂²	1	1	yes	characteristic
27	#16810589	—	2	1	no	—
28	#16810376	C₂²	1	2	yes	—
29	#16810374	C₂²	1	2	yes	—
30	#16810372	C₂²	1	4	yes	—
31	#16810369	C₂²	1	4	yes	—
32	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₅₃×#498	—	2	1	no	—
35	C₅₃×#498	—	2	1	no	—
36	C₅₃×#498	C₂²	1	2	yes	—
37	C₅₃×#498	C₂²	1	2	yes	—
38	C₂×#106×C₅₃	—	2	1	no	—
39	C₂×#106×C₅₃	—	2	1	no	—
40	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719945	—	2	1	no	—
42	C₂×#11719924	—	2	4	no	—
43	#13133223	—	4	4	no	—
44	C₂×#11719917	—	2	1	no	—
45	C₂×#11719912	—	2	2	no	—
46	C₂×C₈×D₅₃	—	2	2	no	—
47	#13133217	D₄	1	1	yes	characteristic
48	#13133214	D₄	1	1	yes	characteristic
49	#13133205	—	2	1	no	—
50	#13133191	—	2	1	no	—
51	#13133191	D₄	1	2	yes	—
52	C₄×D₂₁₂	—	2	1	no	—
53	C₄×D₂₁₂	C₂³	1	2	yes	—
54	#13133148	—	2	4	no	—
55	C₄⋊#88322	—	2	1	no	—
56	C₄⋊#88322	C₂³	1	2	yes	—
57	C₄⋊#88322	C₂³	1	2	yes	—
58	C₄⋊#88322	C₂³	1	2	yes	—
59	#13133141	—	2	4	no	—
60	#13133137	—	2	4	no	—
61	C₂²⋊#88322	—	2	1	no	—
62	C₂²⋊#88322	—	2	1	no	—
63	#13133129	—	2	2	no	—
64	#13133129	—	2	2	no	—
65	C₄×#88322	—	2	1	no	—
66	C₄×#88322	—	2	1	no	—
67	C₂²×Q₈×C₅₃	—	2	1	no	—
68	C₂²×Q₈×C₅₃	—	2	1	no	—
69	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
70	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
71	C₄⋊Q₈×C₅₃	—	2	1	no	—
72	C₄⋊Q₈×C₅₃	—	2	1	no	—
73	C₄⋊Q₈×C₅₃	—	2	1	no	—
74	C₄⋊Q₈×C₅₃	—	2	1	no	—
75	C₄⋊Q₈×C₅₃	—	2	1	no	—
76	C₄⋊Q₈×C₅₃	—	2	1	no	—
77	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
78	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
83	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
84	#106×C₅₃	—	2	2	no	—
85	#106×C₅₃	—	2	2	no	—
86	#106×C₅₃	—	2	2	no	—
87	#106×C₅₃	—	2	2	no	—
88	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
90	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₅₃	—	2	2	no	—
93	C₂×Q₈×C₅₃	—	2	2	no	—
94	C₂×Q₈×C₅₃	—	4	1	no	—
95	C₂×Q₈×C₅₃	—	4	1	no	—
96	C₂×Q₈×C₅₃	—	8	2	no	—
97	C₂×Q₈×C₅₃	—	8	2	no	—
98	C₂²×C₄×C₅₃	—	4	1	no	—
99	C₂²×C₄×C₅₃	—	4	1	no	—
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
103	C₂×C₂²⋊D₅₃	—	2	2	no	—
104	#11719945	—	4	2	no	—
105	#11719945	—	4	2	no	—
106	C₂×D₂₁₂	—	2	1	no	—
107	C₂×D₂₁₂	—	2	1	no	—
108	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
109	C₂×C₄×D₅₃	—	2	2	no	—
110	C₂×Dic₁₀₆	—	2	1	no	—
111	C₂×C₈×C₅₃	—	2	2	no	—
112	C₂×C₈×C₅₃	SD₁₆	1	4	yes	complemented
113	C₂×C₈×C₅₃	C₂×D₄	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	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₅₃	—	2	2	no	—
118	C₄⋊C₄×C₅₃	—	2	2	no	—
119	C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₄⋊C₄×C₅₃	—	2	2	no	—
121	C₄⋊C₄×C₅₃	—	4	1	no	—
122	C₄⋊C₄×C₅₃	—	4	1	no	—
123	C₄⋊C₄×C₅₃	—	4	1	no	—
124	C₄⋊C₄×C₅₃	—	4	1	no	—
125	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₅₃	#41	1	1	yes	characteristic
128	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
129	#11719924	—	8	4	no	—
130	#11719921	—	2	2	no	—
131	#11719921	—	2	2	no	—
132	C₄⋊Dic₅₃	—	2	1	no	—
133	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
134	#11719917	—	4	2	no	—
135	C₂×#88322	—	2	2	no	—
136	C₂×#88322	—	2	2	no	—
137	C₂×#88322	—	2	2	no	—
138	#11719912	—	4	4	no	—
139	C₈×D₅₃	—	4	4	no	—
140	C₂³×C₅₃	#121	1	1	yes	characteristic
141	C₂²×D₅₃	—	2	2	no	—
142	Q₈×C₅₃	—	4	4	no	—
143	Q₈×C₅₃	—	8	2	no	—
144	Q₈×C₅₃	—	8	2	no	—
145	Q₈×C₅₃	—	8	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	4	2	no	—
149	C₂×C₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	—	8	2	no	—
151	C₂×C₄×C₅₃	—	8	2	no	—
152	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
154	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
156	C₂×C₄×C₅₃	#137	1	2	yes	—
157	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
160	C₂²⋊D₅₃	—	4	4	no	—
161	C₂×Dic₅₃	—	2	2	no	—
162	D₂₁₂	—	2	4	no	—
163	D₂₁₂	—	4	2	no	—
164	D₂₁₂	—	4	2	no	—
165	C₄×D₅₃	—	4	2	no	—
166	C₄×D₅₃	—	4	2	no	—
167	Dic₁₀₆	—	4	2	no	—
168	C₈×C₅₃	—	2	1	no	—
169	C₈×C₅₃	—	2	1	no	—
170	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
171	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
172	#88322	—	4	2	no	—
173	#88322	—	4	2	no	—
174	#88322	—	4	2	no	—
175	#19191	—	53	1	no	maximal, Sylow
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	#461	1	2	yes	—
179	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
180	C₂×D₅₃	—	4	2	no	—
181	C₂×D₅₃	—	4	2	no	—
182	C₂×D₅₃	—	4	2	no	—
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	—	2	1	no	—
185	C₄×C₅₃	—	8	2	no	—
186	C₄×C₅₃	—	8	2	no	—
187	C₄×C₅₃	#461	1	2	yes	—
188	C₄×C₅₃	#471	1	2	yes	—
189	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
190	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
191	C₄×C₅₃	#577	1	2	yes	—
192	Dic₅₃	—	4	2	no	—
193	#3227	—	53	1	no	—
194	#3206	—	53	1	no	—
195	C₂×#498	D₅₃	1	1	yes	characteristic, complemented
196	#2931	—	53	1	no	—
197	#1691	—	53	4	no	—
198	#1671	—	53	4	no	—
199	#1564	—	53	1	no	—
200	#1549	—	53	1	no	—
201	#1548	—	53	1	no	—
202	C₂×C₅₃	—	2	2	no	—
203	C₂×C₅₃	C₂×#461	1	1	yes	characteristic
204	C₂×C₅₃	#3057	1	1	yes	characteristic
205	C₂×C₅₃	C₂×#471	1	1	yes	characteristic
206	D₅₃	—	8	2	no	—
207	C₂×#131	—	106	1	no	—
208	#532	—	53	1	no	—
209	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
210	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
211	#498	—	2	1	no	—
212	#498	—	2	1	no	—
213	#498	—	106	1	no	—
214	#498	C₂×D₅₃	1	2	yes	—
215	#498	C₂×D₅₃	1	2	yes	—
216	#492	—	106	1	no	—
217	#487	—	106	1	no	—
218	#486	—	106	1	no	—
219	#481	—	106	2	no	—
220	#476	—	53	4	no	—
221	#461	—	53	4	no	—
222	#448	—	106	2	no	—
223	#436	—	53	2	no	—
224	#436	—	106	1	no	—
225	C₈×D₄	—	53	2	no	—
226	C₈×D₄	—	106	1	no	—
227	#431	—	53	1	no	—
228	#431	—	53	1	no	—
229	C₂×#106	—	2	1	no	—
230	C₂×#106	—	2	1	no	—
231	#423	—	53	1	no	—
232	#408	—	106	1	no	—
233	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
234	#335	—	53	2	no	—
235	#334	—	53	2	no	—
236	#333	—	53	4	no	—
237	#332	—	53	4	no	—
238	C₅₃	#19191	1	1	yes	characteristic, complemented, Sylow
239	C₂×#41	—	106	1	no	—
240	C₂²×Q₈	—	2	1	no	—
241	C₂²×Q₈	—	2	1	no	—
242	C₂×SD₁₆	—	106	4	no	—
243	#131	—	212	4	no	—
244	C₂×M₁₆	—	106	1	no	—
245	C₂×M₁₆	—	106	2	no	—
246	C₂²×C₈	—	106	2	no	—
247	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
248	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	C₄⋊Q₈	—	53	1	no	—
256	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
257	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
258	#127	—	53	1	no	—
259	#124	—	106	1	no	—
260	#121	—	53	2	no	—
261	#121	—	106	1	no	—
262	C₄×D₄	—	53	2	no	—
263	C₄×D₄	—	106	1	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
269	#106	—	2	2	no	—
270	#106	—	2	2	no	—
271	#106	—	2	2	no	—
272	#106	—	2	2	no	—
273	#106	—	106	4	no	—
274	C₄⋊C₈	—	53	2	no	—
275	C₄⋊C₈	—	53	2	no	—
276	C₄⋊C₈	—	53	2	no	—
277	C₄⋊C₈	—	106	1	no	—
278	#103	—	106	4	no	—
279	#102	—	106	4	no	—
280	C₂²⋊C₈	—	106	1	no	—
281	C₂²⋊C₈	—	106	1	no	—
282	C₂²⋊C₈	—	106	2	no	—
283	C₂²⋊C₈	—	106	2	no	—
284	C₄×C₈	—	106	1	no	—
285	C₄×C₈	—	106	1	no	—
286	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
287	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
290	#41	—	212	2	no	—
291	#41	—	212	2	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	4	1	no	—
295	C₂×Q₈	—	4	1	no	—
296	C₂×Q₈	—	8	2	no	—
297	C₂×Q₈	—	8	2	no	—
298	C₂×Q₈	—	106	1	no	—
299	C₂×D₄	—	53	2	no	—
300	C₂×D₄	—	106	2	no	—
301	C₂×D₄	—	106	1	no	—
302	C₂×D₄	—	106	1	no	—
303	C₂²×C₄	—	4	1	no	—
304	C₂²×C₄	—	4	1	no	—
305	C₂²×C₄	—	106	2	no	—
306	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
307	C₂²×C₄	#11719947	1	1	yes	characteristic
308	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
309	SD₁₆	—	424	4	no	—
310	M₁₆	—	212	2	no	—
311	M₁₆	—	212	4	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	106	2	no	—
314	C₂×C₈	—	106	2	no	—
315	C₂×C₈	—	106	2	no	—
316	C₂×C₈	—	212	4	no	—
317	C₂×C₈	#11719924	1	4	yes	complemented
318	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
319	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
320	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
321	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	—	53	2	no	—
331	C₄⋊C₄	—	106	1	no	—
332	C₂²⋊C₄	—	106	2	no	—
333	C₂²⋊C₄	—	106	2	no	—
334	C₄²	D₄×D₅₃	1	1	yes	characteristic
335	C₄²	#11719947	1	1	yes	characteristic
336	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
337	C₄²	C₂³×D₅₃	1	1	yes	characteristic
338	C₂³	—	106	2	no	—
339	C₂³	#13133188	1	1	yes	characteristic
340	Q₈	—	4	4	no	—
341	Q₈	—	8	2	no	—
342	Q₈	—	8	2	no	—
343	Q₈	—	8	2	no	—
344	Q₈	—	212	2	no	—
345	D₄	—	106	4	no	—
346	D₄	—	212	4	no	—
347	D₄	—	212	2	no	—
348	D₄	—	212	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	8	2	no	—
354	C₂×C₄	—	8	2	no	—
355	C₂×C₄	—	106	2	no	—
356	C₂×C₄	—	212	2	no	—
357	C₂×C₄	—	212	2	no	—
358	C₂×C₄	#13133188	1	1	yes	characteristic
359	C₂×C₄	#13133188	1	1	yes	characteristic
360	C₂×C₄	#13133188	1	1	yes	characteristic
361	C₂×C₄	C₂×#11719924	1	2	yes	—
362	C₂×C₄	#13133248	1	2	yes	—
363	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
364	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
365	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	212	2	no	—
369	C₈	—	212	2	no	—
370	C₈	—	212	2	no	—
371	C₈	C₂×#11719924	1	2	yes	—
372	C₈	C₂×#11719924	1	2	yes	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	212	2	no	—
376	C₂²	—	212	2	no	—
377	C₂²	—	212	2	no	—
378	C₂²	#16810825	1	2	yes	—
379	C₂²	C₂×#13133188	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	8	2	no	—
383	C₄	—	8	2	no	—
384	C₄	—	212	2	no	—
385	C₄	#16810825	1	2	yes	—
386	C₄	#16810883	1	2	yes	—
387	C₄	C₂²×#11719924	1	1	yes	characteristic
388	C₄	C₂×#13133248	1	1	yes	characteristic
389	C₄	#16811429	1	2	yes	—
390	C₂	—	2	2	no	—
391	C₂	—	424	2	no	—
392	C₂	C₂×#16810825	1	1	yes	characteristic
393	C₂	#20284350	1	1	yes	characteristic
394	C₂	C₂×#16810883	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133188
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#498	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	#16811173	C₂²
hypercenter	C₂×#498	D₅₃
hypocenter	C₅₃	#19191
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
4. #20274211	C₂	semidirect
5. #20274087	C₂	semidirect
6. #20273463	C₂	semidirect
7. #20273400	C₂	semidirect
8. #20273396	C₂	semidirect
9. C₂×C₅₃×#498	C₂	semidirect
112. C₂×C₈×C₅₃	SD₁₆	semidirect
195. C₂×#498	D₅₃	semidirect
238. C₅₃	#19191	semidirect
317. C₂×C₈	#11719924	semidirect

Maximal subgroups

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

subgroup	index
#20285843	2
#20285640	2
#20283428	2
#20274211	2
#20274087	2
#20273463	2
#20273400	2
#20273396	2
C₂×C₅₃×#498	2
#19191	53