Subgroups — #28303059

Order 13568.

462 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	#20290262	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811486	C₂	1	1	yes	maximal, characteristic, complemented
3	#20290012	C₂	1	2	yes	maximal
4	#20289561	C₂	1	2	yes	maximal, complemented
5	#20283137	C₂	1	1	yes	maximal, characteristic
6	#16811639	C₂²	1	2	yes	—
7	C₂×C₄⋊#88324	C₂²	1	2	yes	—
8	#16811486	—	2	1	no	—
9	#16811486	—	2	1	no	—
10	#16811486	—	2	1	no	—
11	#16811484	—	2	1	no	—
12	#16811473	—	2	2	no	—
13	#16811471	—	2	2	no	—
14	C₂×#13133225	C₄	1	1	yes	characteristic
15	C₂×#13133223	C₄	1	1	yes	characteristic
16	#16811091	C₂²	1	1	yes	characteristic
17	#16810580	—	2	1	no	—
18	#16810579	C₄	1	1	yes	characteristic
19	#16810577	—	2	1	no	—
20	#16810575	—	2	1	no	—
21	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₈×D₅₃	—	2	1	no	—
24	C₅₃×#405	C₄	1	1	yes	characteristic
25	C₂²×#88324	—	2	2	no	—
26	C₄⋊#88324	—	2	2	no	—
27	C₄⋊#88324	—	2	2	no	—
28	C₄⋊#88324	—	2	2	no	—
29	C₂²⋊#88324	—	2	2	no	—
30	C₄×#88324	—	2	2	no	—
31	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
32	#13133225	—	2	2	no	—
33	#13133225	—	2	2	no	—
34	#13133223	—	2	2	no	—
35	#13133223	C₈	1	4	yes	complemented
36	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719912	—	2	1	no	—
38	C₂×#11719912	—	2	1	no	—
39	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₅₃	—	2	1	no	—
41	C₂×C₈×D₅₃	—	2	1	no	—
42	#13133182	D₄	1	1	yes	characteristic
43	#13133182	Q₈	1	1	yes	characteristic
44	#13133179	D₄	1	1	yes	characteristic
45	#13133179	Q₈	1	1	yes	characteristic
46	#13133177	—	2	1	no	—
47	#13133177	—	2	1	no	—
48	#13133174	—	2	1	no	—
49	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
50	C₄²×D₅₃	—	2	1	no	—
51	#13133128	—	2	1	no	—
52	#13133127	—	2	1	no	—
53	C₄×#88322	—	2	1	no	—
54	C₈×Dic₅₃	—	2	1	no	—
55	C₈×Dic₅₃	D₄	1	1	yes	characteristic
56	C₈×Dic₅₃	D₄	1	1	yes	characteristic
57	C₈×Dic₅₃	D₄	1	1	yes	characteristic
58	C₈×Dic₅₃	D₄	1	1	yes	characteristic
59	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
62	#97×C₅₃	—	2	1	no	—
63	C₄×C₈×C₅₃	—	2	1	no	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₅₃	—	2	1	no	—
66	C₂²×Dic₅₃	#41	1	1	yes	characteristic
67	C₂²×Dic₅₃	#41	1	1	yes	characteristic
68	#11719945	—	2	1	no	—
69	#11719945	—	2	1	no	—
70	#11719945	C₂×C₈	1	2	yes	—
71	#11719945	M₁₆	1	2	yes	—
72	C₂×D₂₁₂	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×D₅₃	—	2	1	no	—
74	C₂×C₄×D₅₃	—	2	1	no	—
75	C₂×C₄×D₅₃	—	2	1	no	—
76	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
77	C₂×#88324	—	4	2	no	—
78	C₂×#88324	—	4	2	no	—
79	C₂×#88324	—	8	2	no	—
80	M₁₆×C₅₃	—	2	1	no	—
81	M₁₆×C₅₃	M₁₆	1	2	yes	—
82	C₂×C₈×C₅₃	—	2	1	no	—
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
86	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
87	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
88	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂²⋊C₄×C₅₃	—	2	1	no	—
90	C₄²×C₅₃	—	2	1	no	—
91	#11719921	—	2	1	no	—
92	#11719921	—	2	1	no	—
93	#11719919	—	2	1	no	—
94	#11719919	—	2	1	no	—
95	C₄×Dic₅₃	—	2	1	no	—
96	C₄×Dic₅₃	—	2	1	no	—
97	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
99	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
100	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
101	#11719917	—	2	1	no	—
102	#11719917	C₂×C₈	1	2	yes	—
103	C₂×#88322	—	2	1	no	—
104	C₂×#88322	—	2	1	no	—
105	C₂×#88322	—	2	1	no	—
106	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
108	#11719912	—	2	4	no	—
109	#11719912	—	2	4	no	—
110	C₈×D₅₃	—	2	4	no	—
111	C₈×D₅₃	—	2	4	no	—
112	C₂³×C₅₃	#117	1	1	yes	characteristic
113	C₂²×D₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	1	no	—
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₄⋊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	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂²⋊D₅₃	—	2	2	no	—
123	C₂²⋊D₅₃	—	2	2	no	—
124	C₂×Dic₅₃	—	2	1	no	—
125	C₂×Dic₅₃	—	2	1	no	—
126	C₂×Dic₅₃	—	2	1	no	—
127	C₂×Dic₅₃	—	4	1	no	—
128	C₂×Dic₅₃	#121	1	1	yes	characteristic
129	C₂×Dic₅₃	#121	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
132	D₂₁₂	—	2	1	no	—
133	D₂₁₂	C₄⋊C₈	1	2	yes	—
134	C₄×D₅₃	—	2	2	no	—
135	C₄×D₅₃	—	2	2	no	—
136	C₄×D₅₃	—	4	2	no	—
137	C₄×D₅₃	—	4	1	no	—
138	C₄×D₅₃	—	4	1	no	—
139	Dic₁₀₆	—	2	1	no	—
140	Dic₁₀₆	C₄⋊C₈	1	2	yes	—
141	#88324	—	8	2	no	—
142	#88324	—	8	2	no	—
143	C₈×C₅₃	—	2	2	no	—
144	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
145	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
146	#88322	—	2	2	no	—
147	#88322	—	2	1	no	—
148	#88322	—	2	1	no	—
149	#8605	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	#380	1	1	yes	characteristic
153	C₂²×C₅₃	#431	1	1	yes	characteristic
154	C₂²×C₅₃	#432	1	1	yes	characteristic
155	C₂×D₅₃	—	2	2	no	—
156	C₂×D₅₃	—	4	1	no	—
157	C₂×D₅₃	—	4	1	no	—
158	C₄×C₅₃	—	4	1	no	—
159	C₄×C₅₃	—	4	1	no	—
160	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₄×C₅₃	#407	1	1	yes	characteristic
162	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
163	C₄×C₅₃	#423	1	1	yes	characteristic
164	Dic₅₃	—	2	2	no	—
165	Dic₅₃	—	2	1	no	—
166	Dic₅₃	—	2	1	no	—
167	Dic₅₃	—	2	1	no	—
168	Dic₅₃	#472	1	1	yes	characteristic
169	Dic₅₃	#473	1	1	yes	characteristic
170	C₄×#131	—	53	1	no	—
171	#1564	—	53	1	no	—
172	C₂×#335	—	53	1	no	—
173	#1477	—	53	2	no	—
174	#1283	—	53	2	no	—
175	C₂×C₅₃	—	2	1	no	—
176	C₂×C₅₃	—	2	1	no	—
177	C₂×C₅₃	#1797	1	1	yes	characteristic
178	C₂×C₅₃	#1814	1	1	yes	characteristic
179	C₂×C₅₃	#1816	1	1	yes	characteristic
180	D₅₃	—	4	2	no	—
181	C₂×#131	—	53	1	no	—
182	C₂×#131	—	53	1	no	—
183	C₄×#41	—	53	1	no	—
184	#431	—	53	2	no	—
185	C₂×C₄⋊C₈	—	53	2	no	—
186	#405	—	53	1	no	—
187	#405	—	106	1	no	—
188	#405	—	106	1	no	—
189	#405	#6157	1	1	yes	characteristic
190	C₄×M₁₆	—	53	1	no	—
191	C₄×M₁₆	—	106	1	no	—
192	C₂×C₄×C₈	—	53	1	no	—
193	C₂×C₄×C₈	—	106	1	no	—
194	#335	—	106	1	no	—
195	#335	—	106	1	no	—
196	#335	—	106	1	no	—
197	#334	—	106	1	no	—
198	Q₈⋊C₈	—	106	2	no	—
199	D₄⋊C₈	—	106	2	no	—
200	C₅₃	#8605	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	53	1	no	—
202	#131	—	53	4	no	—
203	#131	—	106	2	no	—
204	#131	—	106	2	no	—
205	#131	—	106	2	no	—
206	C₂×M₁₆	—	53	1	no	—
207	C₂×M₁₆	—	106	1	no	—
208	C₂×M₁₆	—	106	1	no	—
209	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
210	C₂²×C₈	—	53	1	no	—
211	C₂²×C₈	—	106	1	no	—
212	C₂²×C₈	—	106	1	no	—
213	C₂²×C₈	—	106	2	no	—
214	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
215	C₄×Q₈	—	53	1	no	—
216	C₄×Q₈	—	53	1	no	—
217	C₄×D₄	—	53	1	no	—
218	C₄×D₄	—	53	1	no	—
219	C₄×D₄	—	106	1	no	—
220	C₄×D₄	—	106	1	no	—
221	#117	—	106	1	no	—
222	#117	C₂×#6157	1	1	yes	characteristic
223	C₂×C₄²	—	53	1	no	—
224	C₂×C₄²	—	106	1	no	—
225	C₄⋊C₈	—	106	2	no	—
226	C₄⋊C₈	—	106	2	no	—
227	C₄⋊C₈	—	106	2	no	—
228	C₂²⋊C₈	—	106	2	no	—
229	#97	—	2	1	no	—
230	#97	—	106	1	no	—
231	#97	—	106	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	53	1	no	—
234	C₄×C₈	—	53	1	no	—
235	C₄×C₈	—	53	1	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	—	106	1	no	—
238	C₄×C₈	—	106	1	no	—
239	C₄×C₈	—	106	2	no	—
240	#41	—	53	2	no	—
241	#41	—	53	2	no	—
242	#41	—	106	1	no	—
243	#41	—	106	1	no	—
244	C₂×Q₈	—	53	1	no	—
245	C₂×D₄	—	53	1	no	—
246	C₂×D₄	—	106	1	no	—
247	C₂²×C₄	—	53	1	no	—
248	C₂²×C₄	—	53	1	no	—
249	C₂²×C₄	—	106	1	no	—
250	C₂²×C₄	—	106	1	no	—
251	C₂²×C₄	—	106	1	no	—
252	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
253	M₁₆	—	2	1	no	—
254	M₁₆	—	53	2	no	—
255	M₁₆	—	106	4	no	—
256	M₁₆	—	106	1	no	—
257	M₁₆	—	106	4	no	—
258	M₁₆	#11719936	1	2	yes	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	53	1	no	—
261	C₂×C₈	—	53	1	no	—
262	C₂×C₈	—	106	4	no	—
263	C₂×C₈	—	106	1	no	—
264	C₂×C₈	—	106	1	no	—
265	C₂×C₈	—	106	1	no	—
266	C₂×C₈	—	106	4	no	—
267	C₂×C₈	—	424	2	no	—
268	C₂×C₈	—	212	2	no	—
269	C₂×C₈	—	212	2	no	—
270	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
271	C₂×C₈	#11719936	1	2	yes	—
272	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
273	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
274	C₄⋊C₄	—	106	1	no	—
275	C₄⋊C₄	—	106	1	no	—
276	C₄⋊C₄	C₄⋊#6157	1	1	yes	characteristic
277	C₄⋊C₄	C₄⋊#6157	1	1	yes	characteristic
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	106	1	no	—
280	C₂²⋊C₄	—	106	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	53	1	no	—
283	C₄²	—	53	1	no	—
284	C₄²	—	53	1	no	—
285	C₄²	—	53	1	no	—
286	C₄²	—	106	1	no	—
287	C₄²	—	106	1	no	—
288	C₂³	—	106	1	no	—
289	C₂³	#13133296	1	1	yes	characteristic
290	Q₈	—	53	2	no	—
291	Q₈	—	106	1	no	—
292	D₄	—	53	2	no	—
293	D₄	—	106	2	no	—
294	D₄	—	106	2	no	—
295	D₄	—	106	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	53	1	no	—
300	C₂×C₄	—	53	1	no	—
301	C₂×C₄	—	53	1	no	—
302	C₂×C₄	—	106	1	no	—
303	C₂×C₄	—	106	1	no	—
304	C₂×C₄	—	106	1	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	—	212	1	no	—
308	C₂×C₄	—	212	1	no	—
309	C₂×C₄	—	212	2	no	—
310	C₂×C₄	—	212	1	no	—
311	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
313	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
314	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
315	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
316	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	106	2	no	—
319	C₈	—	106	1	no	—
320	C₈	—	106	1	no	—
321	C₈	—	424	2	no	—
322	C₈	—	424	2	no	—
323	C₈	#13133274	1	2	yes	—
324	C₈	#13133274	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	106	2	no	—
328	C₂²	—	212	1	no	—
329	C₂²	—	212	1	no	—
330	C₂²	#16811565	1	1	yes	characteristic
331	C₂²	#16811642	1	1	yes	characteristic
332	C₂²	#16811645	1	1	yes	characteristic
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	53	1	no	—
336	C₄	—	53	1	no	—
337	C₄	—	106	2	no	—
338	C₄	—	106	1	no	—
339	C₄	—	106	1	no	—
340	C₄	—	106	1	no	—
341	C₄	C₂×#13133274	1	1	yes	characteristic
342	C₄	#16811594	1	1	yes	characteristic
343	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
344	C₄	#16811627	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	212	2	no	—
348	C₂	#20290570	1	1	yes	characteristic
349	C₂	#20290602	1	1	yes	characteristic
350	C₂	#20290604	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16811565
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#405	C₄
socle	C₂²×C₅₃	#380
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	#405	#6157
hypocenter	C₅₃	#8605
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #20290262	C₂	semidirect
2. C₂×#16811486	C₂	semidirect
4. #20289561	C₂	semidirect
35. #13133223	C₈	semidirect
200. C₅₃	#8605	semidirect

Maximal subgroups

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

subgroup	index
#20290262	2
C₂×#16811486	2
#20290012	2
#20289561	2
#20283137	2
#8605	53