Subgroups — #27293793

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283422	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133156	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272671	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272669	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	—	2	1	no	—
9	#16811170	C₄	1	1	yes	characteristic
10	C₂×#13133156	—	2	1	no	—
11	C₂×C₁₆×D₅₃	—	2	1	no	—
12	#16810701	C₄	1	1	yes	characteristic
13	#16810700	C₄	1	1	yes	characteristic
14	#16810698	—	2	1	no	—
15	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
16	C₈×C₂²⋊D₅₃	—	2	1	no	—
17	#16810645	C₄	1	1	yes	characteristic
18	#16810588	—	2	1	no	—
19	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	C₄⋊#11719908	C₂²	1	1	yes	characteristic
21	#16810441	—	2	1	no	—
22	#16810398	—	2	1	no	—
23	C₄×#11719908	C₂²	1	1	yes	characteristic
24	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#11719945	—	2	1	no	—
28	#13133223	—	4	2	no	—
29	#13133223	—	4	2	no	—
30	C₂×#11719917	—	2	1	no	—
31	C₂×#11719912	—	2	1	no	—
32	C₂×#11719912	—	2	1	no	—
33	C₂×C₈×D₅₃	—	2	1	no	—
34	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
35	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
36	#13133216	C₈	1	1	yes	characteristic
37	#13133213	C₈	1	1	yes	characteristic
38	#13133203	—	2	1	no	—
39	#13133192	C₈	1	1	yes	characteristic
40	#13133190	—	2	1	no	—
41	#13133188	C₈	1	1	yes	characteristic
42	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
43	C₄×C₂²⋊D₅₃	—	2	1	no	—
44	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
45	#13133156	—	2	4	no	—
46	C₂×#11719908	—	2	1	no	—
47	C₂×#11719908	—	2	1	no	—
48	C₁₆×D₅₃	—	4	2	no	—
49	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
50	#13133146	—	2	1	no	—
51	C₂²⋊#88322	—	2	1	no	—
52	#13133129	—	2	1	no	—
53	#13133129	—	2	1	no	—
54	C₄×#88322	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₅₃	D₄	1	1	yes	characteristic
56	C₈×Dic₅₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	—	2	1	no	—
58	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₅₃	—	2	1	no	—
60	C₂×C₁₆×C₅₃	—	2	1	no	—
61	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₈×C₅₃	—	2	1	no	—
63	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	1	no	—
66	C₂×C₂²⋊D₅₃	—	2	1	no	—
67	C₂×C₂²⋊D₅₃	—	2	1	no	—
68	C₂×C₂²⋊D₅₃	—	2	1	no	—
69	#11719945	—	4	2	no	—
70	#11719945	—	4	2	no	—
71	C₂×D₂₁₂	—	2	1	no	—
72	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
73	C₂×C₄×D₅₃	—	2	1	no	—
74	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₅₃	M₁₆	1	1	yes	characteristic
77	C₂×Dic₁₀₆	—	2	1	no	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	4	1	no	—
80	C₂×C₈×C₅₃	—	4	1	no	—
81	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂²⋊C₄×C₅₃	—	2	1	no	—
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²⋊Dic₅₃	—	2	1	no	—
87	#11719921	—	2	1	no	—
88	#11719921	—	2	1	no	—
89	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
90	#11719919	—	2	1	no	—
91	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
93	C₄×Dic₅₃	M₁₆	1	1	yes	characteristic
94	#11719917	—	4	2	no	—
95	C₂×#88322	—	2	1	no	—
96	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
98	#11719912	—	4	2	no	—
99	#11719912	—	4	2	no	—
100	C₈×D₅₃	—	2	2	no	—
101	C₈×D₅₃	—	4	1	no	—
102	C₈×D₅₃	—	4	1	no	—
103	C₈×D₅₃	D₈	1	2	yes	—
104	C₈×D₅₃	SD₁₆	1	2	yes	—
105	C₁₆×C₅₃	—	2	2	no	—
106	C₁₆×C₅₃	—	4	1	no	—
107	#11719908	—	2	2	no	—
108	#11719908	—	4	1	no	—
109	C₂³×C₅₃	—	2	1	no	—
110	C₂²×D₅₃	—	2	1	no	—
111	C₂²×D₅₃	C₂²⋊C₈	1	1	yes	characteristic
112	D₄×C₅₃	—	2	1	no	—
113	D₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	4	1	no	—
116	C₂×C₄×C₅₃	—	4	1	no	—
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂²⋊D₅₃	—	4	2	no	—
121	C₂²⋊D₅₃	—	4	2	no	—
122	C₂²⋊D₅₃	—	4	2	no	—
123	C₂×Dic₅₃	—	2	1	no	—
124	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
127	D₂₁₂	—	2	1	no	—
128	D₂₁₂	—	2	1	no	—
129	D₂₁₂	—	4	2	no	—
130	C₄×D₅₃	—	2	1	no	—
131	C₄×D₅₃	—	2	1	no	—
132	C₄×D₅₃	—	4	1	no	—
133	C₄×D₅₃	—	4	1	no	—
134	C₄×D₅₃	—	4	1	no	—
135	C₄×D₅₃	#102	1	2	yes	—
136	C₄×D₅₃	#104	1	2	yes	—
137	Dic₁₀₆	—	4	2	no	—
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	—	4	1	no	—
140	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
141	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
142	#88322	—	2	1	no	—
143	#88322	—	4	1	no	—
144	#88322	#102	1	2	yes	—
145	#11582	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	—	4	1	no	—
147	C₂²×C₅₃	—	4	1	no	—
148	C₂²×C₅₃	—	4	1	no	—
149	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₂×D₅₃	—	2	1	no	—
151	C₂×D₅₃	—	2	1	no	—
152	C₂×D₅₃	—	4	1	no	—
153	C₂×D₅₃	—	4	1	no	—
154	C₂×D₅₃	—	4	1	no	—
155	C₂×D₅₃	D₄⋊C₈	1	2	yes	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	4	1	no	—
158	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
159	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
160	C₄×C₅₃	#504	1	1	yes	characteristic
161	C₄×C₅₃	#504	1	1	yes	characteristic
162	Dic₅₃	—	2	1	no	—
163	Dic₅₃	—	2	1	no	—
164	Dic₅₃	—	4	1	no	—
165	Dic₅₃	D₄⋊C₈	1	2	yes	—
166	#2931	—	53	1	no	—
167	C₂×C₄⋊C₁₆	—	53	1	no	—
168	C₄×M₃₂	—	53	1	no	—
169	D₄⋊C₁₆	—	53	1	no	—
170	D₄⋊C₁₆	—	53	1	no	—
171	D₄⋊C₁₆	—	53	1	no	—
172	D₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
173	C₂×C₅₃	—	8	1	no	—
174	C₂×C₅₃	C₂×D₄⋊C₈	1	1	yes	characteristic
175	C₂×C₅₃	#2113	1	1	yes	characteristic
176	C₂×C₅₃	#2117	1	1	yes	characteristic
177	D₅₃	—	2	2	no	—
178	D₅₃	—	8	1	no	—
179	C₂×#131	—	106	1	no	—
180	#532	—	53	1	no	—
181	C₂×M₃₂	—	106	1	no	—
182	C₂²×C₁₆	—	106	1	no	—
183	#436	—	53	1	no	—
184	#436	—	53	1	no	—
185	#436	—	106	1	no	—
186	C₈×D₄	—	53	1	no	—
187	C₈×D₄	—	106	1	no	—
188	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
189	#423	—	53	1	no	—
190	#408	—	106	1	no	—
191	C₂×C₄×C₈	—	53	1	no	—
192	C₄⋊C₁₆	—	53	1	no	—
193	C₄⋊C₁₆	—	106	1	no	—
194	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
195	#346	—	106	1	no	—
196	C₄×C₁₆	—	53	1	no	—
197	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
198	C₅₃	#11582	1	1	yes	characteristic, complemented, Sylow
199	C₂×#41	—	106	1	no	—
200	#131	—	212	2	no	—
201	#131	—	212	2	no	—
202	C₂×M₁₆	—	106	1	no	—
203	C₂×M₁₆	—	106	1	no	—
204	C₂×M₁₆	—	106	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	53	1	no	—
207	C₂²×C₈	—	53	1	no	—
208	C₂²×C₈	—	106	1	no	—
209	C₄⋊Q₈	—	53	1	no	—
210	#127	—	53	1	no	—
211	#124	—	106	1	no	—
212	#121	—	53	1	no	—
213	#121	—	53	1	no	—
214	#121	—	106	1	no	—
215	C₄×D₄	—	53	1	no	—
216	C₄×D₄	—	106	1	no	—
217	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
218	C₂×C₄²	—	53	1	no	—
219	M₃₂	—	106	4	no	—
220	C₂×C₁₆	—	2	1	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	—	106	1	no	—
223	C₂×C₁₆	—	106	1	no	—
224	C₂×C₁₆	—	212	2	no	—
225	C₄⋊C₈	—	53	1	no	—
226	C₄⋊C₈	—	106	1	no	—
227	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
228	C₂²⋊C₈	—	2	1	no	—
229	C₂²⋊C₈	—	106	1	no	—
230	C₂²⋊C₈	—	106	1	no	—
231	C₂²⋊C₈	—	106	1	no	—
232	C₄×C₈	—	53	1	no	—
233	C₄×C₈	—	53	1	no	—
234	C₄×C₈	—	53	1	no	—
235	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
236	#41	—	212	2	no	—
237	#41	—	212	2	no	—
238	C₂×Q₈	—	106	1	no	—
239	C₂×D₄	—	53	1	no	—
240	C₂×D₄	—	106	1	no	—
241	C₂×D₄	—	106	1	no	—
242	C₂×D₄	—	106	1	no	—
243	C₂×D₄	—	106	1	no	—
244	C₂×D₄	C₈×D₅₃	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	53	1	no	—
248	C₂²×C₄	—	53	1	no	—
249	C₂²×C₄	—	106	1	no	—
250	M₁₆	—	212	2	no	—
251	M₁₆	—	212	2	no	—
252	M₁₆	—	212	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	53	1	no	—
257	C₂×C₈	—	53	2	no	—
258	C₂×C₈	—	53	2	no	—
259	C₂×C₈	—	53	1	no	—
260	C₂×C₈	—	106	2	no	—
261	C₂×C₈	—	106	1	no	—
262	C₂×C₈	—	212	1	no	—
263	C₂×C₈	—	212	1	no	—
264	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
265	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
266	C₄⋊C₄	—	53	1	no	—
267	C₄⋊C₄	—	106	1	no	—
268	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	106	1	no	—
271	C₂²⋊C₄	—	106	1	no	—
272	C₂²⋊C₄	—	106	1	no	—
273	C₄²	—	53	1	no	—
274	C₄²	—	53	1	no	—
275	C₄²	—	53	1	no	—
276	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
277	C₁₆	—	2	2	no	—
278	C₁₆	—	4	1	no	—
279	C₁₆	—	106	2	no	—
280	C₁₆	—	212	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	53	1	no	—
283	C₂³	—	106	1	no	—
284	Q₈	—	212	2	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	106	1	no	—
288	D₄	—	106	1	no	—
289	D₄	—	212	2	no	—
290	D₄	—	212	2	no	—
291	D₄	—	212	2	no	—
292	D₄	—	212	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	53	1	no	—
297	C₂×C₄	—	53	2	no	—
298	C₂×C₄	—	53	2	no	—
299	C₂×C₄	—	53	1	no	—
300	C₂×C₄	—	53	1	no	—
301	C₂×C₄	—	106	1	no	—
302	C₂×C₄	—	106	1	no	—
303	C₂×C₄	—	106	1	no	—
304	C₂×C₄	—	212	1	no	—
305	C₂×C₄	—	212	1	no	—
306	C₂×C₄	—	212	1	no	—
307	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	4	1	no	—
312	C₈	—	53	2	no	—
313	C₈	—	106	1	no	—
314	C₈	—	212	1	no	—
315	C₈	D₈×D₅₃	1	1	yes	characteristic
316	C₈	SD₁₆×D₅₃	1	1	yes	characteristic
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	53	2	no	—
321	C₂²	—	106	1	no	—
322	C₂²	—	106	1	no	—
323	C₂²	—	212	1	no	—
324	C₂²	—	212	1	no	—
325	C₂²	—	212	1	no	—
326	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	53	2	no	—
330	C₄	—	106	1	no	—
331	C₄	—	106	1	no	—
332	C₄	—	212	1	no	—
333	C₄	#102×D₅₃	1	1	yes	characteristic
334	C₄	#104×D₅₃	1	1	yes	characteristic
335	C₄	#16811039	1	1	yes	characteristic
336	C₄	#16811041	1	1	yes	characteristic
337	C₂	—	8	1	no	—
338	C₂	—	106	2	no	—
339	C₂	—	424	1	no	—
340	C₂	D₄⋊C₈×D₅₃	1	1	yes	characteristic
341	C₂	#20276327	1	1	yes	characteristic
342	C₂	#20276341	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×D₄⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	#16811170	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	D₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#11582
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₁₆×D₅₃	C₂	semidirect
3. C₄×#13133156	C₂	semidirect
4. #20272671	C₂	semidirect
5. D₄⋊#11719908	C₂	semidirect
6. #20272669	C₂	semidirect
7. C₅₃×D₄⋊C₁₆	C₂	semidirect
172. D₄⋊C₁₆	D₅₃	semidirect
198. C₅₃	#11582	semidirect

Maximal subgroups

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

subgroup	index
#20283422	2
C₄⋊C₁₆×D₅₃	2
C₄×#13133156	2
#20272671	2
D₄⋊#11719908	2
#20272669	2
C₅₃×D₄⋊C₁₆	2
#11582	53