Subgroups — #28310541

Order 13568.

426 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#350	C₂	1	1	yes	maximal, characteristic
2	#20290923	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×#6157	C₂	1	1	yes	maximal, characteristic, complemented
4	#20290489	C₂	1	1	yes	maximal, characteristic
5	#20289771	C₂	1	1	yes	maximal, characteristic, complemented
6	#20289758	C₂	1	1	yes	maximal, characteristic, complemented
7	#504×D₅₃	C₂	1	1	yes	maximal, characteristic
8	D₅₃⋊M₃₂	C₂²	1	1	yes	characteristic
9	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
10	D₅₃⋊#104	C₄	1	1	yes	characteristic
11	#16811613	C₄	1	1	yes	characteristic
12	#16811586	C₂²	1	1	yes	characteristic
13	C₂×C₈×#6157	C₂²	1	1	yes	characteristic
14	#16811522	C₄	1	2	yes	complemented
15	#16811521	C₄	1	2	yes	complemented
16	#16811510	—	2	1	no	—
17	C₁₆×#6157	—	2	1	no	—
18	C₁₆×#6157	—	2	1	no	—
19	C₁₆×#6157	—	2	1	no	—
20	#16811498	C₄	1	1	yes	characteristic
21	#16811495	C₄	1	1	yes	characteristic
22	#131×D₅₃	C₂²	1	1	yes	characteristic
23	#16811041	—	2	1	no	—
24	#16811041	C₄	1	1	yes	characteristic, complemented
25	#16811040	C₄	1	1	yes	characteristic, complemented
26	#16811039	—	2	1	no	—
27	#16811039	C₄	1	1	yes	characteristic, complemented
28	M₃₂×D₅₃	—	2	1	no	—
29	M₃₂×D₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₁₆×D₅₃	—	2	1	no	—
31	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#504	C₄	1	1	yes	characteristic, complemented
33	D₅₃⋊M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×D₅₃⋊C₈	C₂×C₄	1	1	yes	characteristic
35	#13133296	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×#6157	C₂×C₄	1	1	yes	characteristic
37	#13133292	—	2	1	no	—
38	#13133291	C₂×C₄	1	2	yes	—
39	D₅₃⋊C₁₆	—	2	1	no	—
40	D₅₃⋊C₁₆	—	2	1	no	—
41	D₅₃⋊C₁₆	—	2	1	no	—
42	C₂×#11719910	C₂×C₄	1	2	yes	—
43	#13133282	C₈	1	2	yes	—
44	#13133281	C₈	1	2	yes	—
45	#13133280	C₈	1	2	yes	—
46	#13133279	C₈	1	2	yes	—
47	#13133271	—	2	1	no	—
48	C₈×#6157	—	2	1	no	—
49	C₈×#6157	—	2	1	no	—
50	C₈×#6157	—	2	1	no	—
51	#41×D₅₃	C₂×C₄	1	1	yes	characteristic
52	#13133225	—	2	1	no	—
53	#13133225	C₂×C₄	1	1	yes	characteristic
54	#13133224	C₂×C₄	1	1	yes	characteristic
55	#13133223	—	2	1	no	—
56	#13133223	C₂×C₄	1	1	yes	characteristic
57	M₁₆×D₅₃	—	2	1	no	—
58	M₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₈×D₅₃	—	2	1	no	—
60	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
61	#13133157	—	2	1	no	—
62	#13133157	C₂×C₄	1	1	yes	characteristic
63	#13133156	—	2	1	no	—
64	#13133156	—	2	1	no	—
65	#13133156	—	2	1	no	—
66	C₂×#11719908	—	2	1	no	—
67	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
68	C₁₆×D₅₃	—	2	1	no	—
69	C₁₆×D₅₃	—	2	1	no	—
70	C₁₆×D₅₃	—	2	1	no	—
71	C₁₆×D₅₃	D₄	1	1	yes	characteristic
72	C₁₆×D₅₃	D₄	1	1	yes	characteristic
73	C₁₆×D₅₃	D₄	1	1	yes	characteristic
74	C₁₆×D₅₃	D₄	1	1	yes	characteristic
75	#131×C₅₃	C₂×C₄	1	1	yes	characteristic
76	M₃₂×C₅₃	—	2	1	no	—
77	M₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₁₆×C₅₃	—	2	1	no	—
79	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂²×#6157	—	2	1	no	—
81	#41×C₅₃	C₄²	1	1	yes	characteristic
82	#11719949	C₂×C₈	1	1	yes	characteristic
83	Q₈×D₅₃	C₂×C₈	1	1	yes	characteristic
84	#11719947	—	2	1	no	—
85	#11719947	C₂×C₈	1	1	yes	characteristic
86	D₄×D₅₃	—	2	1	no	—
87	D₄×D₅₃	C₂×C₈	1	1	yes	characteristic
88	#11719945	—	2	1	no	—
89	#11719945	C₄²	1	1	yes	characteristic
90	C₂×C₄×D₅₃	—	2	1	no	—
91	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²⋊#6157	—	2	1	no	—
93	#11719940	—	2	1	no	—
94	C₂×#88324	—	2	1	no	—
95	C₄⋊#6157	C₂×C₈	1	2	yes	—
96	C₄×#6157	—	2	1	no	—
97	C₄×#6157	—	2	1	no	—
98	C₄×#6157	C₂×C₈	1	2	yes	—
99	#11719936	C₂×C₈	1	2	yes	—
100	D₅₃⋊C₈	—	2	1	no	—
101	D₅₃⋊C₈	—	2	1	no	—
102	D₅₃⋊C₈	C₂×C₈	1	2	yes	—
103	M₁₆×C₅₃	—	2	1	no	—
104	M₁₆×C₅₃	C₄²	1	1	yes	characteristic
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	#11719917	—	2	1	no	—
108	#11719917	C₄²	1	1	yes	characteristic
109	C₂×#88322	—	2	1	no	—
110	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
111	#11719912	—	2	1	no	—
112	#11719912	—	2	1	no	—
113	#11719912	—	2	1	no	—
114	C₈×D₅₃	—	2	1	no	—
115	C₈×D₅₃	—	2	1	no	—
116	C₈×D₅₃	—	2	1	no	—
117	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
118	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
119	C₈×D₅₃	#41	1	1	yes	characteristic
120	C₈×D₅₃	#41	1	1	yes	characteristic
121	#11719910	—	2	2	no	—
122	#11719910	—	2	2	no	—
123	C₁₆×C₅₃	—	2	1	no	—
124	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
125	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
126	#11719908	—	2	1	no	—
127	#11719908	C₂²⋊C₄	1	1	yes	characteristic
128	#11719908	C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×D₅₃	—	2	1	no	—
130	C₂²×D₅₃	#131	1	1	yes	characteristic
131	C₂×#6157	—	2	4	no	—
132	C₂×#6157	—	2	2	no	—
133	C₂×#6157	—	2	2	no	—
134	Q₈×C₅₃	C₄×C₈	1	1	yes	characteristic
135	D₄×C₅₃	—	2	1	no	—
136	D₄×C₅₃	C₄×C₈	1	1	yes	characteristic
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
139	C₂²⋊D₅₃	—	2	1	no	—
140	C₂²⋊D₅₃	—	2	1	no	—
141	C₂²⋊D₅₃	—	2	1	no	—
142	C₂×Dic₅₃	—	2	1	no	—
143	C₂×Dic₅₃	#131	1	1	yes	characteristic
144	D₂₁₂	—	2	1	no	—
145	D₂₁₂	C₄×C₈	1	1	yes	characteristic
146	C₄×D₅₃	—	2	1	no	—
147	C₄×D₅₃	—	2	1	no	—
148	C₄×D₅₃	—	2	1	no	—
149	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
150	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
151	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
152	C₄×D₅₃	C₂²×C₈	1	1	yes	characteristic
153	Dic₁₀₆	—	2	1	no	—
154	Dic₁₀₆	C₄×C₈	1	1	yes	characteristic
155	#88324	—	2	2	no	—
156	#88324	—	2	2	no	—
157	C₈×C₅₃	—	2	1	no	—
158	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₈×C₅₃	#117	1	1	yes	characteristic
160	#88322	—	2	1	no	—
161	#88322	C₄×D₄	1	1	yes	characteristic
162	#88322	C₄×D₄	1	1	yes	characteristic
163	#12614	—	53	1	no	maximal, Sylow
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	#405	1	1	yes	characteristic
166	C₂×D₅₃	—	2	2	no	—
167	C₂×D₅₃	—	2	1	no	—
168	C₂×D₅₃	—	4	2	no	—
169	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
170	C₂×D₅₃	C₈×D₄	1	1	yes	characteristic
171	#6157	—	2	4	no	—
172	#6157	—	4	2	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
175	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
176	Dic₅₃	—	2	1	no	—
177	Dic₅₃	C₈×D₄	1	1	yes	characteristic
178	Dic₅₃	C₈×D₄	1	1	yes	characteristic
179	C₂×#504	—	53	1	no	—
180	C₂×#350	—	53	1	no	—
181	#2108	—	53	1	no	—
182	C₂×C₄×C₁₆	—	53	1	no	—
183	#1764	—	53	1	no	—
184	#1379	—	53	1	no	—
185	#1376	—	53	1	no	—
186	C₂×C₅₃	—	2	1	no	—
187	C₂×C₅₃	—	4	1	no	—
188	C₂×C₅₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
189	D₅₃	—	2	1	no	—
190	D₅₃	—	4	1	no	—
191	D₅₃	#2170	1	2	yes	—
192	C₂×#131	—	53	1	no	—
193	#504	—	53	1	no	—
194	#504	—	53	1	no	—
195	#504	—	53	1	no	—
196	#504	—	106	1	no	—
197	#504	—	106	1	no	—
198	#504	#6157	1	1	yes	characteristic, complemented
199	C₂×M₃₂	—	53	1	no	—
200	C₂×M₃₂	—	53	1	no	—
201	C₂×M₃₂	—	106	1	no	—
202	C₂²×C₁₆	—	53	1	no	—
203	C₂²×C₁₆	—	53	1	no	—
204	C₂²×C₁₆	—	106	1	no	—
205	C₂×#104	—	53	1	no	—
206	#416	—	53	1	no	—
207	#405	—	53	1	no	—
208	C₂×C₄×C₈	—	53	1	no	—
209	#350	—	53	2	no	—
210	#350	—	53	2	no	—
211	#346	—	106	1	no	—
212	C₄×C₁₆	—	106	1	no	—
213	C₄×C₁₆	—	106	1	no	—
214	C₄×C₁₆	—	106	1	no	—
215	#341	—	53	1	no	—
216	#339	—	53	1	no	—
217	C₅₃	#12614	1	1	yes	characteristic, complemented, Sylow
218	C₂×#41	—	53	1	no	—
219	#131	—	53	1	no	—
220	#131	—	53	1	no	—
221	#131	—	53	1	no	—
222	#131	—	106	1	no	—
223	#131	—	106	1	no	—
224	#131	C₂×#6157	1	1	yes	characteristic
225	C₂×M₁₆	—	53	1	no	—
226	C₂×M₁₆	—	53	1	no	—
227	C₂×M₁₆	—	106	1	no	—
228	C₂²×C₈	—	53	1	no	—
229	C₂²×C₈	—	53	1	no	—
230	C₂²×C₈	—	106	1	no	—
231	#117	—	53	1	no	—
232	C₂×C₄²	—	53	1	no	—
233	M₃₂	—	2	1	no	—
234	M₃₂	—	53	2	no	—
235	M₃₂	—	53	1	no	—
236	M₃₂	—	106	1	no	—
237	M₃₂	—	106	1	no	—
238	M₃₂	—	106	1	no	—
239	M₃₂	—	106	1	no	—
240	M₃₂	—	106	1	no	—
241	M₃₂	C₂×#6157	1	1	yes	characteristic
242	C₂×C₁₆	—	2	1	no	—
243	C₂×C₁₆	—	53	1	no	—
244	C₂×C₁₆	—	53	1	no	—
245	C₂×C₁₆	—	53	1	no	—
246	C₂×C₁₆	—	53	1	no	—
247	C₂×C₁₆	—	53	1	no	—
248	C₂×C₁₆	—	53	2	no	—
249	C₂×C₁₆	—	106	1	no	—
250	C₂×C₁₆	—	106	1	no	—
251	C₂×C₁₆	—	106	1	no	—
252	C₂×C₁₆	—	106	1	no	—
253	C₂×C₁₆	—	106	1	no	—
254	C₂×C₁₆	—	106	1	no	—
255	C₂×C₁₆	—	106	1	no	—
256	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
257	#104	—	53	2	no	—
258	#104	—	53	2	no	—
259	#103	—	53	2	no	—
260	#102	—	53	2	no	—
261	#97	—	106	1	no	—
262	C₄×C₈	—	106	1	no	—
263	C₄×C₈	—	106	1	no	—
264	C₄×C₈	—	106	1	no	—
265	#41	—	53	1	no	—
266	#41	—	53	1	no	—
267	#41	—	53	1	no	—
268	#41	—	106	1	no	—
269	#41	—	106	1	no	—
270	#41	C₄×#6157	1	1	yes	characteristic
271	C₂×Q₈	—	53	1	no	—
272	C₂×D₄	—	53	1	no	—
273	C₂×D₄	—	106	1	no	—
274	C₂²×C₄	—	53	1	no	—
275	C₂²×C₄	—	106	1	no	—
276	C₂²×C₄	—	106	1	no	—
277	M₁₆	—	2	1	no	—
278	M₁₆	—	53	2	no	—
279	M₁₆	—	53	1	no	—
280	M₁₆	—	106	1	no	—
281	M₁₆	—	106	1	no	—
282	M₁₆	—	106	1	no	—
283	M₁₆	—	106	1	no	—
284	M₁₆	—	106	1	no	—
285	M₁₆	C₄×#6157	1	1	yes	characteristic
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	53	1	no	—
288	C₂×C₈	—	53	1	no	—
289	C₂×C₈	—	53	1	no	—
290	C₂×C₈	—	53	1	no	—
291	C₂×C₈	—	53	1	no	—
292	C₂×C₈	—	53	2	no	—
293	C₂×C₈	—	106	1	no	—
294	C₂×C₈	—	106	1	no	—
295	C₂×C₈	—	106	1	no	—
296	C₂×C₈	—	106	1	no	—
297	C₂×C₈	—	106	1	no	—
298	C₂×C₈	—	106	1	no	—
299	C₂×C₈	—	106	1	no	—
300	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
301	C₄⋊C₄	—	53	2	no	—
302	C₂²⋊C₄	—	106	1	no	—
303	C₄²	—	53	2	no	—
304	C₄²	—	106	1	no	—
305	C₄²	—	106	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	53	1	no	—
308	C₁₆	—	53	1	no	—
309	C₁₆	—	106	2	no	—
310	C₁₆	—	106	2	no	—
311	C₁₆	—	106	1	no	—
312	C₁₆	C₂²⋊#6157	1	1	yes	characteristic
313	C₁₆	C₂²⋊#6157	1	1	yes	characteristic
314	C₂³	—	53	1	no	—
315	C₂³	—	106	1	no	—
316	Q₈	—	53	1	no	—
317	Q₈	—	106	1	no	—
318	Q₈	C₈×#6157	1	1	yes	characteristic
319	D₄	—	2	1	no	—
320	D₄	—	53	1	no	—
321	D₄	—	106	1	no	—
322	D₄	—	106	1	no	—
323	D₄	—	106	1	no	—
324	D₄	—	106	1	no	—
325	D₄	C₈×#6157	1	1	yes	characteristic
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	53	1	no	—
328	C₂×C₄	—	53	1	no	—
329	C₂×C₄	—	53	1	no	—
330	C₂×C₄	—	53	1	no	—
331	C₂×C₄	—	53	1	no	—
332	C₂×C₄	—	106	1	no	—
333	C₂×C₄	—	106	1	no	—
334	C₂×C₄	—	106	1	no	—
335	C₂×C₄	—	106	2	no	—
336	C₂×C₄	—	106	4	no	—
337	C₂×C₄	—	106	1	no	—
338	C₂×C₄	—	106	2	no	—
339	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
340	C₈	—	2	1	no	—
341	C₈	—	53	1	no	—
342	C₈	—	53	1	no	—
343	C₈	—	106	2	no	—
344	C₈	—	106	2	no	—
345	C₈	—	106	1	no	—
346	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
347	C₈	#13133296	1	1	yes	characteristic
348	C₂²	—	2	1	no	—
349	C₂²	—	53	1	no	—
350	C₂²	—	53	1	no	—
351	C₂²	—	106	2	no	—
352	C₂²	—	106	1	no	—
353	C₂²	—	212	2	no	—
354	C₂²	#16811586	1	1	yes	characteristic
355	C₄	—	2	1	no	—
356	C₄	—	53	1	no	—
357	C₄	—	53	1	no	—
358	C₄	—	106	4	no	—
359	C₄	—	106	1	no	—
360	C₄	—	212	2	no	—
361	C₄	C₄×C₂²⋊#6157	1	1	yes	characteristic
362	C₄	C₂×C₈×#6157	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	4	1	no	—
365	C₂	—	53	2	no	—
366	C₂	—	106	1	no	—
367	C₂	—	212	1	no	—
368	C₂	C₈×C₂²⋊#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#504	C₄
socle	C₂×C₅₃	C₈×C₂²⋊C₄
prime radical	#41×D₅₃	C₂×C₄
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	#504	#6157
hypocenter	C₅₃	#12614
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
2. #20290923	C₂	semidirect
3. C₂×C₁₆×#6157	C₂	semidirect
5. #20289771	C₂	semidirect
6. #20289758	C₂	semidirect
14. #16811522	C₄	semidirect
15. #16811521	C₄	semidirect
24. #16811041	C₄	semidirect
25. #16811040	C₄	semidirect
27. #16811039	C₄	semidirect
32. C₅₃×#504	C₄	semidirect
198. #504	#6157	semidirect
217. C₅₃	#12614	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊#350	2
#20290923	2
C₂×C₁₆×#6157	2
#20290489	2
#20289771	2
#20289758	2
#504×D₅₃	2
#12614	53