Subgroups — #27537449

Order 13568.

718 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	C₂×D₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20283562	C₂	1	2	yes	maximal, complemented
3	#20276589	C₂	1	2	yes	maximal, complemented
4	#20276588	C₂	1	2	yes	maximal, complemented
5	C₄×D₈₄₈	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276561	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276496	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276429	C₂	1	2	yes	maximal, complemented
10	#20276310	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₅₃	C₂²	1	2	yes	—
13	#16811095	—	2	2	no	—
14	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
15	C₂×#13133158	C₂²	1	2	yes	—
16	D₁₆×D₅₃	C₄	1	8	yes	complemented
17	C₂×#13133156	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
19	#16810733	—	2	2	no	—
20	#16810732	—	2	2	no	—
21	C₄×D₄₂₄	C₂²	1	2	yes	complemented
22	D₈×Dic₅₃	C₂²	1	2	yes	complemented
23	#16810705	C₂²	1	2	yes	complemented
24	#16810660	C₂²	1	2	yes	—
25	#16810626	—	2	2	no	—
26	#16810577	C₂²	1	1	yes	characteristic
27	#16810450	C₂²	1	2	yes	complemented
28	#16810449	C₂²	1	1	yes	characteristic, complemented
29	#16810425	C₂²	1	2	yes	complemented
30	#16810424	C₂²	1	2	yes	complemented
31	#16810423	C₂²	1	2	yes	complemented
32	#16810399	C₂²	1	1	yes	characteristic, complemented
33	#16810398	C₂²	1	1	yes	characteristic, complemented
34	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₅₃	C₂²	1	2	yes	complemented
37	C₅₃×#366	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#357	C₂²	1	2	yes	complemented
39	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₂×D₄×D₅₃	—	2	2	no	—
41	C₂×D₄₂₄	C₂³	1	2	yes	—
42	C₂×#11719922	—	2	2	no	—
43	D₈×D₅₃	—	2	4	no	—
44	D₈×D₅₃	C₂×C₄	1	8	yes	complemented
45	C₂×#11719912	D₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	C₄×D₂₁₂	—	2	2	no	—
48	D₄×Dic₅₃	—	2	2	no	—
49	#13133179	—	2	2	no	—
50	C₄×C₂²⋊D₅₃	—	4	2	no	—
51	#13133177	—	4	2	no	—
52	#13133175	—	2	2	no	—
53	#13133174	D₄	1	1	yes	characteristic
54	C₂×#11719921	—	4	2	no	—
55	C₂²⋊C₄×D₅₃	—	4	2	no	—
56	D₈₄₈	C₂×C₄	1	4	yes	complemented
57	#13133158	C₂×C₄	1	8	yes	complemented
58	#13133156	—	2	2	no	—
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	C₂×#11719908	C₂³	1	1	yes	characteristic
61	C₁₆×D₅₃	C₂×C₄	1	4	yes	complemented
62	#13133151	C₂³	1	2	yes	—
63	#13133148	—	2	2	no	—
64	#13133139	—	2	2	no	—
65	#13133138	—	2	2	no	—
66	#13133137	—	2	2	no	—
67	#13133128	D₄	1	1	yes	characteristic
68	#13133127	D₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₂×D₈×C₅₃	C₂³	1	2	yes	—
71	C₄×D₄×C₅₃	—	2	2	no	—
72	D₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	2	yes	—
76	#102×C₅₃	—	2	2	no	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₂³×D₅₃	—	4	2	no	—
79	C₂×D₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	4	2	no	—
81	C₂×C₂²⋊D₅₃	—	4	2	no	—
82	C₂²×Dic₅₃	—	4	2	no	—
83	D₄×D₅₃	—	2	8	no	—
84	D₄×D₅₃	—	4	4	no	—
85	C₂×D₂₁₂	—	2	2	no	—
86	C₂×C₄×D₅₃	—	4	2	no	—
87	C₂×C₄×D₅₃	—	4	2	no	—
88	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
89	D₈×C₅₃	—	2	2	no	—
90	D₈×C₅₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₂²⋊C₄×C₅₃	—	4	2	no	—
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²⋊Dic₅₃	—	4	2	no	—
97	#11719922	—	2	8	no	—
98	#11719921	—	4	2	no	—
99	#11719921	—	4	2	no	—
100	#11719921	—	4	2	no	—
101	#11719921	D₈	1	1	yes	characteristic
102	#11719921	D₈	1	1	yes	characteristic
103	C₄⋊Dic₅₃	—	2	2	no	—
104	#11719919	D₈	1	1	yes	characteristic
105	#11719919	D₈	1	1	yes	characteristic
106	C₄×Dic₅₃	—	4	2	no	—
107	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂×#88322	C₂×D₄	1	1	yes	characteristic
109	C₂×#88322	C₂×D₄	1	1	yes	characteristic
110	D₄₂₄	—	2	2	no	—
111	D₄₂₄	C₂²×C₄	1	4	yes	—
112	#11719912	—	2	2	no	—
113	C₈×D₅₃	C₂²×C₄	1	2	yes	—
114	C₈×D₅₃	#41	1	2	yes	—
115	C₁₆×C₅₃	—	2	1	no	—
116	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
117	#11719908	—	2	1	no	—
118	#11719908	C₂²×C₄	1	2	yes	—
119	C₂³×C₅₃	—	4	2	no	—
120	C₂²×D₅₃	—	4	2	no	—
121	C₂²×D₅₃	—	4	8	no	—
122	C₂²×D₅₃	—	16	2	no	—
123	C₂²×D₅₃	—	16	2	no	—
124	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
125	D₄×C₅₃	—	2	4	no	—
126	D₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	16	2	no	—
129	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₅₃	—	4	8	no	—
133	C₂×Dic₅₃	—	4	2	no	—
134	C₂×Dic₅₃	—	16	2	no	—
135	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
136	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
137	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
138	D₂₁₂	—	2	4	no	—
139	D₂₁₂	—	4	2	no	—
140	C₄×D₅₃	—	16	2	no	—
141	C₄×D₅₃	—	16	2	no	—
142	C₄×D₅₃	C₄×D₄	1	2	yes	—
143	C₄×D₅₃	#135	1	2	yes	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
146	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
147	#88322	—	2	1	no	—
148	#88322	C₄×D₄	1	2	yes	—
149	#33456	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	4	4	no	—
151	C₂²×C₅₃	—	8	2	no	—
152	C₂²×C₅₃	—	8	2	no	—
153	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₅₃	—	2	1	no	—
155	C₂×D₅₃	—	2	1	no	—
156	C₂×D₅₃	—	4	4	no	—
157	C₂×D₅₃	—	8	2	no	—
158	C₂×D₅₃	—	8	2	no	—
159	C₂×D₅₃	—	16	8	no	—
160	C₂×D₅₃	C₄×D₈	1	2	yes	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	—	8	2	no	—
164	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
166	Dic₅₃	—	2	1	no	—
167	Dic₅₃	—	2	1	no	—
168	Dic₅₃	—	8	2	no	—
169	Dic₅₃	C₄×D₈	1	2	yes	—
170	C₂²×D₁₆	—	53	1	no	—
171	#2948	—	53	2	no	—
172	#2177	—	53	2	no	—
173	#2177	—	53	2	no	—
174	C₄×D₁₆	—	53	1	no	—
175	C₄×D₁₆	—	53	1	no	—
176	C₄×D₁₆	—	53	1	no	—
177	C₄×D₁₆	D₅₃	1	1	yes	characteristic, complemented
178	#2158	—	53	1	no	—
179	#2140	—	53	2	no	—
180	#2108	—	53	1	no	—
181	C₂×C₅₃	—	8	4	no	—
182	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
183	C₂×C₅₃	#3415	1	1	yes	characteristic
184	C₂×C₅₃	#3415	1	1	yes	characteristic
185	D₅₃	—	2	2	no	—
186	D₅₃	—	8	4	no	—
187	C₂²×D₈	—	53	2	no	—
188	#518	—	106	2	no	—
189	C₂×D₁₆	—	53	2	no	—
190	C₂×D₁₆	—	53	8	no	—
191	C₂×D₁₆	—	53	1	no	—
192	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic
193	C₂×M₃₂	—	53	1	no	—
194	C₂²×C₁₆	—	53	1	no	—
195	#442	—	106	2	no	—
196	#442	—	106	2	no	—
197	C₄×D₈	—	53	2	no	—
198	C₄×D₈	—	53	2	no	—
199	C₄×D₈	—	53	2	no	—
200	C₄×D₈	C₂×D₅₃	1	2	yes	complemented
201	#427	—	53	2	no	—
202	#418	—	106	2	no	—
203	#405	—	53	1	no	—
204	#367	—	53	2	no	—
205	#366	—	53	1	no	—
206	#366	C₂×D₅₃	1	1	yes	characteristic, complemented
207	#357	—	53	2	no	—
208	#357	—	53	2	no	—
209	#357	—	53	2	no	—
210	#357	C₂×D₅₃	1	2	yes	complemented
211	#346	—	53	1	no	—
212	#346	—	53	1	no	—
213	C₄×C₁₆	—	53	1	no	—
214	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
215	C₅₃	#33456	1	1	yes	characteristic, complemented, Sylow
216	C₂²×D₄	—	106	2	no	—
217	C₂×D₈	—	53	8	no	—
218	C₂×D₈	—	53	2	no	—
219	C₂×D₈	—	106	4	no	—
220	C₂×D₈	—	106	2	no	—
221	C₂×D₈	C₂²×D₅₃	1	2	yes	—
222	C₂×M₁₆	—	53	1	no	—
223	C₂²×C₈	—	53	1	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	106	2	no	—
226	C₄×D₄	—	106	2	no	—
227	C₄×D₄	—	106	2	no	—
228	C₄×D₄	—	212	2	no	—
229	C₄×D₄	—	212	2	no	—
230	#117	—	53	1	no	—
231	#117	—	106	2	no	—
232	C₂×C₂²⋊C₄	—	212	2	no	—
233	C₂×C₂²⋊C₄	—	212	2	no	—
234	D₁₆	—	53	8	no	—
235	D₁₆	—	53	4	no	—
236	D₁₆	C₄×D₅₃	1	4	yes	complemented
237	M₃₂	—	106	2	no	—
238	C₂×C₁₆	—	53	1	no	—
239	C₂×C₁₆	—	53	4	no	—
240	C₂×C₁₆	—	53	1	no	—
241	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
242	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
243	#107	—	53	2	no	—
244	#107	C₂²×D₅₃	1	2	yes	—
245	#106	—	106	2	no	—
246	#102	—	2	2	no	—
247	#102	—	106	2	no	—
248	#102	—	106	2	no	—
249	#102	—	106	2	no	—
250	#97	—	53	1	no	—
251	#97	—	53	1	no	—
252	C₄×C₈	—	53	1	no	—
253	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
254	C₂⁴	—	212	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	106	8	no	—
257	C₂×D₄	—	106	2	no	—
258	C₂×D₄	—	212	4	no	—
259	C₂×D₄	—	212	2	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	53	1	no	—
262	C₂²×C₄	—	212	2	no	—
263	C₂²×C₄	—	212	2	no	—
264	C₂²×C₄	—	212	2	no	—
265	D₈	—	2	2	no	—
266	D₈	—	53	4	no	—
267	D₈	—	106	8	no	—
268	D₈	—	106	2	no	—
269	D₈	C₂×C₄×D₅₃	1	4	yes	—
270	M₁₆	—	106	2	no	—
271	C₂×C₈	—	53	1	no	—
272	C₂×C₈	—	53	2	no	—
273	C₂×C₈	—	53	2	no	—
274	C₂×C₈	—	53	1	no	—
275	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	53	1	no	—
279	C₄⋊C₄	—	53	1	no	—
280	C₄⋊C₄	—	106	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	53	1	no	—
283	C₂²⋊C₄	—	53	1	no	—
284	C₂²⋊C₄	—	212	2	no	—
285	C₂²⋊C₄	—	212	2	no	—
286	C₂²⋊C₄	—	212	2	no	—
287	C₂²⋊C₄	—	212	2	no	—
288	C₄²	—	53	1	no	—
289	C₄²	—	212	2	no	—
290	C₄²	D₄×D₅₃	1	1	yes	characteristic
291	C₁₆	—	2	1	no	—
292	C₁₆	—	53	2	no	—
293	C₁₆	—	106	1	no	—
294	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
295	C₂³	—	4	2	no	—
296	C₂³	—	53	1	no	—
297	C₂³	—	848	2	no	—
298	C₂³	—	848	2	no	—
299	C₂³	—	212	8	no	—
300	C₂³	—	212	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	4	2	no	—
303	D₄	—	106	4	no	—
304	D₄	—	212	8	no	—
305	D₄	—	212	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	16	2	no	—
308	C₂×C₄	—	53	1	no	—
309	C₂×C₄	—	53	2	no	—
310	C₂×C₄	—	53	2	no	—
311	C₂×C₄	—	53	1	no	—
312	C₂×C₄	—	53	1	no	—
313	C₂×C₄	—	848	2	no	—
314	C₂×C₄	—	848	2	no	—
315	C₂×C₄	—	848	2	no	—
316	C₂×C₄	—	212	2	no	—
317	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
318	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	53	2	no	—
322	C₈	—	106	1	no	—
323	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
324	C₈	#41×D₅₃	1	1	yes	characteristic
325	C₂²	—	4	4	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	53	2	no	—
329	C₂²	—	848	8	no	—
330	C₂²	—	106	1	no	—
331	C₂²	—	106	1	no	—
332	C₂²	—	424	2	no	—
333	C₂²	—	424	2	no	—
334	C₂²	—	212	4	no	—
335	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	8	2	no	—
339	C₄	—	53	2	no	—
340	C₄	—	106	1	no	—
341	C₄	—	106	1	no	—
342	C₄	—	424	2	no	—
343	C₄	C₄×D₄×D₅₃	1	1	yes	characteristic
344	C₄	#135×D₅₃	1	1	yes	characteristic
345	C₂	—	8	4	no	—
346	C₂	—	106	2	no	—
347	C₂	—	424	4	no	—
348	C₂	C₄×D₈×D₅₃	1	1	yes	characteristic
349	C₂	#20287852	1	1	yes	characteristic
350	C₂	#20287853	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×D₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₁₆×D₅₃	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×D₁₆	D₅₃
hypocenter	C₅₃	#33456
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
2. #20283562	C₂	semidirect
3. #20276589	C₂	semidirect
4. #20276588	C₂	semidirect
5. C₄×D₈₄₈	C₂	semidirect
6. D₁₆×Dic₅₃	C₂	semidirect
7. #20276561	C₂	semidirect
8. #20276496	C₂	semidirect
9. #20276429	C₂	semidirect
10. #20276310	C₂	semidirect
11. C₄×D₁₆×C₅₃	C₂	semidirect
16. D₁₆×D₅₃	C₄	semidirect
21. C₄×D₄₂₄	C₂²	semidirect
22. D₈×Dic₅₃	C₂²	semidirect
23. #16810705	C₂²	semidirect
27. #16810450	C₂²	semidirect
28. #16810449	C₂²	semidirect
29. #16810425	C₂²	semidirect
30. #16810424	C₂²	semidirect
31. #16810423	C₂²	semidirect
32. #16810399	C₂²	semidirect
33. #16810398	C₂²	semidirect
34. C₁₆×Dic₅₃	C₂²	semidirect
36. C₄×D₈×C₅₃	C₂²	semidirect
37. C₅₃×#366	C₂²	semidirect
38. C₅₃×#357	C₂²	semidirect
39. C₄×C₁₆×C₅₃	C₂²	semidirect
44. D₈×D₅₃	C₂×C₄	semidirect
56. D₈₄₈	C₂×C₄	semidirect
57. #13133158	C₂×C₄	semidirect
61. C₁₆×D₅₃	C₂×C₄	semidirect
72. D₁₆×C₅₃	C₂×C₄	semidirect
177. C₄×D₁₆	D₅₃	semidirect
200. C₄×D₈	C₂×D₅₃	semidirect
206. #366	C₂×D₅₃	semidirect
210. #357	C₂×D₅₃	semidirect
214. C₄×C₁₆	C₂×D₅₃	semidirect
215. C₅₃	#33456	semidirect
236. D₁₆	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆×D₅₃	2
#20283562	2
#20276589	2
#20276588	2
C₄×D₈₄₈	2
D₁₆×Dic₅₃	2
#20276561	2
#20276496	2
#20276429	2
#20276310	2
C₄×D₁₆×C₅₃	2
#33456	53