Subgroups — #28303055

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 346 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×#16811484	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	#16811484	—	2	1	no	—
10	#16811484	—	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	2	no	—
126	C₂×Dic₅₃	—	4	1	no	—
127	C₂×Dic₅₃	#121	1	1	yes	characteristic
128	C₂×Dic₅₃	#121	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
131	D₂₁₂	—	2	1	no	—
132	D₂₁₂	C₄⋊C₈	1	2	yes	—
133	C₄×D₅₃	—	2	2	no	—
134	C₄×D₅₃	—	2	2	no	—
135	C₄×D₅₃	—	4	2	no	—
136	C₄×D₅₃	—	4	2	no	—
137	Dic₁₀₆	—	2	1	no	—
138	Dic₁₀₆	C₄⋊C₈	1	2	yes	—
139	#88324	—	8	2	no	—
140	#88324	—	8	2	no	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
143	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
144	#88322	—	2	2	no	—
145	#88322	—	2	1	no	—
146	#88322	—	2	1	no	—
147	#8604	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	#380	1	1	yes	characteristic
151	C₂²×C₅₃	#431	1	1	yes	characteristic
152	C₂²×C₅₃	#432	1	1	yes	characteristic
153	C₂×D₅₃	—	2	2	no	—
154	C₂×D₅₃	—	4	1	no	—
155	C₂×D₅₃	—	4	1	no	—
156	C₄×C₅₃	—	4	1	no	—
157	C₄×C₅₃	—	4	1	no	—
158	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₄×C₅₃	#407	1	1	yes	characteristic
160	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
161	C₄×C₅₃	#423	1	1	yes	characteristic
162	Dic₅₃	—	2	2	no	—
163	Dic₅₃	—	2	1	no	—
164	Dic₅₃	—	2	1	no	—
165	Dic₅₃	—	2	1	no	—
166	Dic₅₃	#471	1	2	yes	—
167	C₄×#131	—	53	1	no	—
168	#1564	—	53	1	no	—
169	C₂×#334	—	53	1	no	—
170	#1477	—	53	2	no	—
171	#1283	—	53	2	no	—
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	#1797	1	1	yes	characteristic
175	C₂×C₅₃	#1814	1	1	yes	characteristic
176	C₂×C₅₃	#1815	1	1	yes	characteristic
177	D₅₃	—	4	2	no	—
178	C₂×#131	—	53	1	no	—
179	C₂×#131	—	53	1	no	—
180	C₄×#41	—	53	1	no	—
181	#431	—	53	2	no	—
182	C₂×C₄⋊C₈	—	53	2	no	—
183	#405	—	53	1	no	—
184	#405	—	106	1	no	—
185	#405	—	106	1	no	—
186	#405	#6157	1	1	yes	characteristic
187	C₄×M₁₆	—	53	1	no	—
188	C₄×M₁₆	—	106	1	no	—
189	C₂×C₄×C₈	—	53	1	no	—
190	C₂×C₄×C₈	—	106	1	no	—
191	#335	—	106	1	no	—
192	#334	—	106	1	no	—
193	#334	—	106	1	no	—
194	#334	—	106	1	no	—
195	Q₈⋊C₈	—	106	2	no	—
196	D₄⋊C₈	—	106	2	no	—
197	C₅₃	#8604	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	53	1	no	—
199	#131	—	53	4	no	—
200	#131	—	106	2	no	—
201	#131	—	106	2	no	—
202	#131	—	106	2	no	—
203	C₂×M₁₆	—	53	1	no	—
204	C₂×M₁₆	—	106	1	no	—
205	C₂×M₁₆	—	106	1	no	—
206	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
207	C₂²×C₈	—	53	1	no	—
208	C₂²×C₈	—	106	1	no	—
209	C₂²×C₈	—	106	1	no	—
210	C₂²×C₈	—	106	2	no	—
211	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
212	C₄×Q₈	—	53	1	no	—
213	C₄×Q₈	—	53	1	no	—
214	C₄×D₄	—	53	1	no	—
215	C₄×D₄	—	53	1	no	—
216	C₄×D₄	—	106	1	no	—
217	C₄×D₄	—	106	1	no	—
218	#117	—	106	1	no	—
219	#117	C₂×#6157	1	1	yes	characteristic
220	C₂×C₄²	—	53	1	no	—
221	C₂×C₄²	—	106	1	no	—
222	C₄⋊C₈	—	106	2	no	—
223	C₄⋊C₈	—	106	2	no	—
224	C₄⋊C₈	—	106	2	no	—
225	C₂²⋊C₈	—	106	2	no	—
226	#97	—	2	1	no	—
227	#97	—	106	1	no	—
228	#97	—	106	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	53	1	no	—
231	C₄×C₈	—	53	1	no	—
232	C₄×C₈	—	53	1	no	—
233	C₄×C₈	—	53	1	no	—
234	C₄×C₈	—	106	1	no	—
235	C₄×C₈	—	106	1	no	—
236	C₄×C₈	—	106	2	no	—
237	#41	—	53	2	no	—
238	#41	—	53	2	no	—
239	#41	—	106	1	no	—
240	#41	—	106	1	no	—
241	C₂×Q₈	—	53	1	no	—
242	C₂×D₄	—	53	1	no	—
243	C₂×D₄	—	106	1	no	—
244	C₂²×C₄	—	53	1	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	106	1	no	—
247	C₂²×C₄	—	106	1	no	—
248	C₂²×C₄	—	106	1	no	—
249	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
250	M₁₆	—	2	1	no	—
251	M₁₆	—	53	2	no	—
252	M₁₆	—	106	4	no	—
253	M₁₆	—	106	1	no	—
254	M₁₆	—	106	4	no	—
255	M₁₆	#11719936	1	2	yes	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	106	4	no	—
260	C₂×C₈	—	106	1	no	—
261	C₂×C₈	—	106	1	no	—
262	C₂×C₈	—	106	1	no	—
263	C₂×C₈	—	106	4	no	—
264	C₂×C₈	—	424	2	no	—
265	C₂×C₈	—	212	2	no	—
266	C₂×C₈	—	212	2	no	—
267	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
268	C₂×C₈	#11719936	1	2	yes	—
269	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
271	C₄⋊C₄	—	106	1	no	—
272	C₄⋊C₄	—	106	1	no	—
273	C₄⋊C₄	C₄⋊#6157	1	1	yes	characteristic
274	C₄⋊C₄	C₄⋊#6157	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	106	1	no	—
277	C₂²⋊C₄	—	106	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	53	1	no	—
280	C₄²	—	53	1	no	—
281	C₄²	—	53	1	no	—
282	C₄²	—	53	1	no	—
283	C₄²	—	106	1	no	—
284	C₄²	—	106	1	no	—
285	C₂³	—	106	1	no	—
286	C₂³	#13133296	1	1	yes	characteristic
287	Q₈	—	53	2	no	—
288	Q₈	—	106	1	no	—
289	D₄	—	53	2	no	—
290	D₄	—	106	2	no	—
291	D₄	—	106	2	no	—
292	D₄	—	106	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	53	1	no	—
296	C₂×C₄	—	53	1	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	106	2	no	—
300	C₂×C₄	—	106	1	no	—
301	C₂×C₄	—	106	2	no	—
302	C₂×C₄	—	106	2	no	—
303	C₂×C₄	—	212	2	no	—
304	C₂×C₄	—	212	2	no	—
305	C₂×C₄	—	212	1	no	—
306	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
307	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
308	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
309	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
310	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
311	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	106	2	no	—
314	C₈	—	106	1	no	—
315	C₈	—	106	1	no	—
316	C₈	—	424	2	no	—
317	C₈	—	424	2	no	—
318	C₈	#13133274	1	2	yes	—
319	C₈	#13133274	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	106	2	no	—
323	C₂²	—	212	1	no	—
324	C₂²	—	212	1	no	—
325	C₂²	#16811565	1	1	yes	characteristic
326	C₂²	#16811642	1	1	yes	characteristic
327	C₂²	#16811645	1	1	yes	characteristic
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	53	2	no	—
331	C₄	—	106	2	no	—
332	C₄	—	106	1	no	—
333	C₄	—	106	1	no	—
334	C₄	—	106	1	no	—
335	C₄	C₂×#13133274	1	1	yes	characteristic
336	C₄	#16811594	1	1	yes	characteristic
337	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
338	C₄	#16811627	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	212	2	no	—
342	C₂	#20290570	1	1	yes	characteristic
343	C₂	#20290601	1	1	yes	characteristic
344	C₂	#20290603	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	#8604
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×#16811484	C₂	semidirect
4. #20289561	C₂	semidirect
35. #13133223	C₈	semidirect
197. C₅₃	#8604	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×#16811484	2
#20290012	2
#20289561	2
#20283137	2
#8604	53