Subgroups — #28316328

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291649	C₂	1	1	yes	maximal, characteristic
2	#20291645	C₂	1	1	yes	maximal, characteristic
3	#20290293	C₂	1	1	yes	maximal, characteristic, complemented
4	#20290292	C₂	1	1	yes	maximal, characteristic, complemented
5	#20290287	C₂	1	2	yes	maximal, complemented
6	#20290285	C₂	1	2	yes	maximal, complemented
7	#20290280	C₂	1	2	yes	maximal, complemented
8	#20290273	C₂	1	1	yes	maximal, characteristic, complemented
9	#20290267	C₂	1	2	yes	maximal, complemented
10	#20290263	C₂	1	1	yes	maximal, characteristic, complemented
11	#20283680	C₂	1	1	yes	maximal, characteristic
12	#16811674	—	2	1	no	—
13	Q₈×#88324	—	2	1	no	—
14	#16811657	—	2	1	no	—
15	#16811657	—	2	2	no	—
16	#16811653	—	2	2	no	—
17	#16811651	—	2	1	no	—
18	#16811645	C₂²	1	2	yes	—
19	#16811642	C₂²	1	1	yes	characteristic
20	#16811642	C₂²	1	1	yes	characteristic
21	#16811642	C₂²	1	2	yes	—
22	#16811627	C₂²	1	1	yes	characteristic
23	#16811627	C₂²	1	1	yes	characteristic
24	#16811627	C₂²	1	2	yes	—
25	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
26	#16811487	C₂²	1	1	yes	characteristic
27	#16811487	C₂²	1	1	yes	characteristic
28	#16811487	C₂²	1	2	yes	—
29	#16811486	C₂²	1	2	yes	—
30	#16811486	C₂²	1	2	yes	—
31	#16811486	C₂²	1	2	yes	—
32	#16811485	C₂²	1	2	yes	—
33	#16811484	C₂²	1	1	yes	characteristic
34	#16811484	C₂²	1	1	yes	characteristic
35	#16811484	C₂²	1	2	yes	—
36	C₄×#11719949	C₂²	1	1	yes	characteristic
37	#16810754	C₄	1	1	yes	characteristic
38	#16810702	C₄	1	1	yes	characteristic
39	#16810700	C₄	1	1	yes	characteristic
40	#16810696	C₄	1	2	yes	—
41	#16810693	C₄	1	2	yes	—
42	#16810675	C₂²	1	2	yes	—
43	#16810675	C₂²	1	1	yes	characteristic
44	#16810575	C₂²	1	2	yes	—
45	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
46	C₅₃×#446	C₄	1	1	yes	characteristic
47	C₂×#11719936	—	2	1	no	—
48	C₂×#11719936	—	2	2	no	—
49	C₂×#11719936	—	2	1	no	—
50	C₂×D₅₃⋊C₈	—	2	2	no	—
51	#13133284	—	2	2	no	—
52	C₄⋊#88324	—	2	1	no	—
53	C₄⋊#88324	—	2	1	no	—
54	C₄⋊#88324	—	2	2	no	—
55	C₄⋊#88324	C₂³	1	1	yes	characteristic
56	C₄⋊#88324	C₂³	1	2	yes	—
57	C₄⋊#88324	C₂³	1	2	yes	—
58	C₄⋊#88324	C₂³	1	1	yes	characteristic
59	C₄⋊#88324	C₂³	1	1	yes	characteristic
60	C₄⋊#88324	C₂³	1	1	yes	characteristic
61	#13133274	—	2	1	no	—
62	#13133274	—	2	2	no	—
63	#13133274	—	2	1	no	—
64	#13133274	—	2	2	no	—
65	#13133272	—	2	2	no	—
66	C₄×#88324	—	2	1	no	—
67	C₄×#88324	—	2	1	no	—
68	C₄×#88324	—	2	2	no	—
69	C₂×#11719949	Q₈	1	1	yes	characteristic
70	C₂×#11719912	C₂×C₄	1	2	yes	—
71	C₂×#11719912	C₂×C₄	1	2	yes	—
72	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
74	Q₈×Dic₅₃	D₄	1	1	yes	characteristic
75	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
76	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
77	#13133179	D₄	1	1	yes	characteristic
78	#13133179	D₄	1	2	yes	—
79	#13133175	Q₈	1	1	yes	characteristic
80	#13133175	Q₈	1	2	yes	—
81	C₄²×D₅₃	C₂³	1	1	yes	characteristic
82	C₄²×D₅₃	C₂³	1	2	yes	—
83	C₄⋊#88322	C₂×C₄	1	2	yes	—
84	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
85	#13133129	C₂×C₄	1	2	yes	—
86	#13133129	C₂×C₄	1	2	yes	—
87	#13133129	C₂×C₄	1	2	yes	—
88	#13133128	C₂×C₄	1	2	yes	—
89	C₄×#88322	C₂×C₄	1	1	yes	characteristic
90	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
91	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
92	C₈×Dic₅₃	C₂³	1	2	yes	—
93	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
95	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
96	#97×C₅₃	C₂×C₄	1	2	yes	—
97	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	#11719949	—	4	2	no	—
100	C₂×D₂₁₂	C₄⋊C₄	1	1	yes	characteristic
101	C₂×D₂₁₂	C₄⋊C₄	1	2	yes	—
102	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
104	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
107	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
108	C₂×#88324	—	2	1	no	—
109	C₂×#88324	—	2	1	no	—
110	C₂×#88324	—	2	1	no	—
111	C₂×#88324	—	2	2	no	—
112	C₂×#88324	—	2	1	no	—
113	C₂×#88324	—	2	2	no	—
114	#11719936	—	8	1	no	—
115	#11719936	—	8	2	no	—
116	#11719936	—	8	1	no	—
117	D₅₃⋊C₈	—	8	2	no	—
118	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
122	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
123	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
125	#11719921	C₄⋊C₄	1	2	yes	—
126	#11719921	C₄⋊C₄	1	2	yes	—
127	#11719921	C₄⋊C₄	1	2	yes	—
128	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
129	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
130	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
131	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
132	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
133	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
134	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
135	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
136	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
137	C₂×#88322	C₂²×C₄	1	2	yes	—
138	#11719912	—	2	4	no	—
139	#11719912	—	4	2	no	—
140	C₈×D₅₃	—	2	2	no	—
141	C₈×D₅₃	—	4	1	no	—
142	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
144	Q₈×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
154	D₂₁₂	—	2	2	no	—
155	D₂₁₂	—	4	2	no	—
156	C₄×D₅₃	—	2	1	no	—
157	C₄×D₅₃	—	2	2	no	—
158	C₄×D₅₃	—	2	1	no	—
159	C₄×D₅₃	—	2	2	no	—
160	C₄×D₅₃	—	2	2	no	—
161	C₄×D₅₃	—	2	1	no	—
162	C₄×D₅₃	—	2	2	no	—
163	C₄×D₅₃	—	2	2	no	—
164	C₄×D₅₃	—	2	1	no	—
165	C₄×D₅₃	—	4	2	no	—
166	#88324	—	4	2	no	—
167	#88324	—	4	1	no	—
168	#88324	—	4	2	no	—
169	#88324	—	4	1	no	—
170	#88324	—	4	1	no	—
171	#88324	—	4	1	no	—
172	C₈×C₅₃	—	2	1	no	—
173	C₈×C₅₃	—	2	2	no	—
174	C₈×C₅₃	#131	1	2	yes	—
175	#88322	—	2	1	no	—
176	#88322	—	2	2	no	—
177	#88322	—	2	1	no	—
178	#17839	—	53	1	no	maximal, Sylow
179	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
180	C₂×D₅₃	—	2	1	no	—
181	C₂×D₅₃	—	2	2	no	—
182	C₂×D₅₃	—	2	1	no	—
183	C₂×D₅₃	—	2	2	no	—
184	C₂×D₅₃	—	2	1	no	—
185	C₂×D₅₃	—	2	2	no	—
186	C₄×C₅₃	—	2	1	no	—
187	C₄×C₅₃	—	2	2	no	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	#430	1	2	yes	—
190	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
191	C₄×C₅₃	#568	1	1	yes	characteristic
192	Dic₅₃	—	2	1	no	—
193	Dic₅₃	—	2	2	no	—
194	Dic₅₃	—	2	1	no	—
195	Dic₅₃	#576	1	1	yes	characteristic
196	Dic₅₃	#578	1	1	yes	characteristic
197	#2991	—	53	1	no	—
198	#2990	—	53	1	no	—
199	#2965	—	53	1	no	—
200	#1574	—	53	1	no	—
201	#1574	—	53	1	no	—
202	#1572	—	53	2	no	—
203	#1571	—	53	2	no	—
204	#1569	—	53	2	no	—
205	#1567	—	53	1	no	—
206	#1565	—	53	2	no	—
207	#1564	—	53	1	no	—
208	C₂×C₅₃	#2906	1	1	yes	characteristic
209	C₂×C₅₃	#2913	1	1	yes	characteristic
210	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
211	D₅₃	—	4	1	no	—
212	D₅₃	—	4	2	no	—
213	C₄×#41	—	53	1	no	—
214	#446	—	53	1	no	—
215	#446	—	106	1	no	—
216	#446	#6157	1	1	yes	characteristic
217	C₈×Q₈	—	106	1	no	—
218	#436	—	53	1	no	—
219	#436	—	53	1	no	—
220	#436	—	106	1	no	—
221	#436	—	106	2	no	—
222	#435	—	53	2	no	—
223	#435	—	53	2	no	—
224	#435	—	106	2	no	—
225	C₈×D₄	—	106	1	no	—
226	#432	—	53	2	no	—
227	#431	—	53	2	no	—
228	#431	—	53	1	no	—
229	#431	—	53	1	no	—
230	#431	—	53	1	no	—
231	#431	—	53	2	no	—
232	#423	—	53	1	no	—
233	#423	—	53	1	no	—
234	#423	—	53	2	no	—
235	C₂×C₄⋊C₈	—	53	2	no	—
236	C₄×M₁₆	—	53	2	no	—
237	C₂×C₄×C₈	—	53	1	no	—
238	#335	—	53	1	no	—
239	#335	—	53	1	no	—
240	#335	—	53	2	no	—
241	#335	—	53	2	no	—
242	#335	—	53	2	no	—
243	#335	—	53	2	no	—
244	#334	—	53	1	no	—
245	#334	—	53	1	no	—
246	#334	—	53	2	no	—
247	#334	—	53	2	no	—
248	C₅₃	#17839	1	1	yes	characteristic, complemented, Sylow
249	C₂×#41	—	53	1	no	—
250	C₂×M₁₆	—	53	2	no	—
251	C₂×M₁₆	—	53	2	no	—
252	C₂×M₁₆	—	106	1	no	—
253	C₂×M₁₆	—	106	2	no	—
254	C₂×M₁₆	—	106	1	no	—
255	C₂²×C₈	—	53	1	no	—
256	C₂²×C₈	—	53	1	no	—
257	C₂²×C₈	—	106	2	no	—
258	C₄×Q₈	—	53	1	no	—
259	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
260	C₄×D₄	—	53	1	no	—
261	C₄×D₄	—	53	2	no	—
262	C₄×D₄	—	53	2	no	—
263	C₄×D₄	—	53	1	no	—
264	#117	—	53	1	no	—
265	#117	—	53	2	no	—
266	C₂×C₄²	—	53	1	no	—
267	C₂×C₄²	—	53	2	no	—
268	C₄⋊C₈	—	53	1	no	—
269	C₄⋊C₈	—	53	1	no	—
270	C₄⋊C₈	—	53	1	no	—
271	C₄⋊C₈	—	53	2	no	—
272	C₄⋊C₈	—	53	1	no	—
273	C₄⋊C₈	—	53	2	no	—
274	C₄⋊C₈	—	53	2	no	—
275	C₄⋊C₈	—	53	1	no	—
276	C₄⋊C₈	—	106	2	no	—
277	C₄⋊C₈	—	106	1	no	—
278	C₄⋊C₈	—	106	1	no	—
279	C₄⋊C₈	—	106	2	no	—
280	C₄⋊C₈	C₂×#6157	1	2	yes	—
281	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
282	C₂²⋊C₈	—	53	2	no	—
283	C₂²⋊C₈	—	53	2	no	—
284	C₂²⋊C₈	—	53	2	no	—
285	C₂²⋊C₈	—	106	1	no	—
286	C₂²⋊C₈	—	106	2	no	—
287	C₂²⋊C₈	—	106	1	no	—
288	C₂²⋊C₈	—	106	2	no	—
289	#97	—	53	2	no	—
290	#97	—	106	2	no	—
291	#97	C₂×#6157	1	2	yes	—
292	C₄×C₈	—	53	1	no	—
293	C₄×C₈	—	53	2	no	—
294	C₄×C₈	—	53	1	no	—
295	C₄×C₈	—	53	1	no	—
296	C₄×C₈	—	106	1	no	—
297	C₄×C₈	—	106	1	no	—
298	C₄×C₈	—	106	2	no	—
299	C₄×C₈	C₂×#6157	1	1	yes	characteristic
300	#41	—	212	2	no	—
301	C₂×Q₈	C₄⋊#6157	1	1	yes	characteristic
302	C₂×D₄	—	53	1	no	—
303	C₂×D₄	—	53	2	no	—
304	C₂²×C₄	—	53	2	no	—
305	C₂²×C₄	—	53	1	no	—
306	C₂²×C₄	—	53	1	no	—
307	C₂²×C₄	—	53	1	no	—
308	C₂²×C₄	—	53	2	no	—
309	C₂²×C₄	—	53	2	no	—
310	M₁₆	—	106	4	no	—
311	M₁₆	—	424	1	no	—
312	M₁₆	—	424	2	no	—
313	M₁₆	—	424	1	no	—
314	M₁₆	—	212	2	no	—
315	C₂×C₈	—	53	1	no	—
316	C₂×C₈	—	53	1	no	—
317	C₂×C₈	—	53	2	no	—
318	C₂×C₈	—	106	2	no	—
319	C₂×C₈	—	106	1	no	—
320	C₂×C₈	—	106	1	no	—
321	C₂×C₈	—	106	1	no	—
322	C₂×C₈	—	106	1	no	—
323	C₂×C₈	—	106	2	no	—
324	C₂×C₈	—	106	2	no	—
325	C₂×C₈	—	424	2	no	—
326	C₂×C₈	—	212	1	no	—
327	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
328	C₂×C₈	C₂²×#6157	1	2	yes	—
329	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
330	C₄⋊C₄	—	53	2	no	—
331	C₄⋊C₄	—	53	1	no	—
332	C₄⋊C₄	C₄⋊#6157	1	2	yes	—
333	C₄⋊C₄	C₄⋊#6157	1	1	yes	characteristic
334	C₂²⋊C₄	—	53	2	no	—
335	C₂²⋊C₄	—	53	2	no	—
336	C₂²⋊C₄	—	53	2	no	—
337	C₄²	—	53	1	no	—
338	C₄²	—	53	2	no	—
339	C₄²	—	53	1	no	—
340	C₄²	—	53	2	no	—
341	C₄²	—	53	1	no	—
342	C₄²	C₂²×#6157	1	1	yes	characteristic
343	C₄²	C₂²×#6157	1	2	yes	—
344	C₂³	—	53	1	no	—
345	C₂³	—	53	2	no	—
346	Q₈	—	2	2	no	—
347	D₄	—	106	2	no	—
348	D₄	—	212	2	no	—
349	C₂×C₄	—	53	1	no	—
350	C₂×C₄	—	53	1	no	—
351	C₂×C₄	—	53	2	no	—
352	C₂×C₄	—	53	1	no	—
353	C₂×C₄	—	106	2	no	—
354	C₂×C₄	—	106	1	no	—
355	C₂×C₄	—	106	2	no	—
356	C₂×C₄	—	106	1	no	—
357	C₂×C₄	—	106	2	no	—
358	C₂×C₄	—	106	1	no	—
359	C₂×C₄	—	106	2	no	—
360	C₂×C₄	—	106	1	no	—
361	C₂×C₄	—	106	2	no	—
362	C₂×C₄	—	212	2	no	—
363	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
364	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
365	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
366	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
367	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
368	C₈	—	2	1	no	—
369	C₈	—	2	2	no	—
370	C₈	—	106	1	no	—
371	C₈	—	106	2	no	—
372	C₈	—	106	1	no	—
373	C₈	—	212	1	no	—
374	C₈	—	212	1	no	—
375	C₈	—	212	1	no	—
376	C₈	—	212	2	no	—
377	C₈	—	212	1	no	—
378	C₈	—	212	2	no	—
379	C₈	#13133305	1	2	yes	—
380	C₂²	—	106	1	no	—
381	C₂²	—	106	2	no	—
382	C₂²	—	106	2	no	—
383	C₂²	—	106	1	no	—
384	C₂²	—	106	1	no	—
385	C₂²	—	106	2	no	—
386	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
387	C₄	—	2	2	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	2	no	—
390	C₄	—	53	1	no	—
391	C₄	—	53	1	no	—
392	C₄	—	106	1	no	—
393	C₄	—	106	2	no	—
394	C₄	—	106	1	no	—
395	C₄	#16811638	1	2	yes	—
396	C₄	C₂×#13133305	1	1	yes	characteristic
397	C₄	#16811702	1	1	yes	characteristic
398	C₂	—	212	1	no	—
399	C₂	—	212	2	no	—
400	C₂	#20291361	1	1	yes	characteristic
401	C₂	#20291370	1	1	yes	characteristic
402	C₂	C₂×#16811638	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#446	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#11719949	Q₈
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	#446	#6157
hypocenter	C₅₃	#17839
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
3. #20290293	C₂	semidirect
4. #20290292	C₂	semidirect
5. #20290287	C₂	semidirect
6. #20290285	C₂	semidirect
7. #20290280	C₂	semidirect
8. #20290273	C₂	semidirect
9. #20290267	C₂	semidirect
10. #20290263	C₂	semidirect
248. C₅₃	#17839	semidirect

Maximal subgroups

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

subgroup	index
#20291649	2
#20291645	2
#20290293	2
#20290292	2
#20290287	2
#20290285	2
#20290280	2
#20290273	2
#20290267	2
#20290263	2
#20283680	2
#17839	53