Subgroups — #27277256

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274735	C₂	1	2	yes	maximal
2	C₄×#13133149	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273471	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272617	C₂	1	2	yes	maximal, complemented
5	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133149	—	2	1	no	—
7	#16810647	C₂²	1	2	yes	—
8	C₂×#13133144	—	2	2	no	—
9	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	#16810386	—	2	1	no	—
11	#16810382	—	2	2	no	—
12	#16810374	—	2	1	no	—
13	#16810374	—	2	1	no	—
14	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
16	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₈×D₄×C₅₃	—	2	1	no	—
18	C₈×D₄×C₅₃	—	2	1	no	—
19	C₈×D₄×C₅₃	C₄	1	1	yes	characteristic, complemented
20	C₈×D₄×C₅₃	C₄	1	1	yes	characteristic, complemented
21	C₅₃×#431	—	2	1	no	—
22	C₅₃×#431	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	—	2	1	no	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#11719917	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄×Dic₅₃	—	2	2	no	—
28	#13133149	—	2	4	no	—
29	C₄⋊#88322	—	2	2	no	—
30	C₄⋊#88322	—	2	2	no	—
31	#13133144	—	4	4	no	—
32	#13133124	—	2	2	no	—
33	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	—	2	1	no	—
35	C₂²×C₈×C₅₃	—	2	1	no	—
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
40	C₄×D₄×C₅₃	—	2	1	no	—
41	C₄×D₄×C₅₃	—	2	1	no	—
42	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
44	#117×C₅₃	—	2	1	no	—
45	#117×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₃	—	2	1	no	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₅₃	—	2	1	no	—
49	C₄⋊C₈×C₅₃	—	2	1	no	—
50	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²⋊C₈×C₅₃	—	2	1	no	—
53	C₂²⋊C₈×C₅₃	—	2	1	no	—
54	C₂²⋊C₈×C₅₃	—	2	1	no	—
55	C₄×C₈×C₅₃	—	2	1	no	—
56	C₄×C₈×C₅₃	—	2	1	no	—
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	#41×C₅₃	—	2	2	no	—
63	#41×C₅₃	SD₁₆	1	4	yes	—
64	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×D₄×C₅₃	—	2	1	no	—
66	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	4	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	—	2	1	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	—	4	2	no	—
79	C₂×C₈×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₅₃	—	2	1	no	—
85	C₄⋊C₄×C₅₃	—	2	1	no	—
86	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₅₃	—	2	1	no	—
89	C₂²⋊C₄×C₅₃	—	2	1	no	—
90	C₂²⋊C₄×C₅₃	—	2	1	no	—
91	C₄²×C₅₃	—	2	1	no	—
92	C₄²×C₅₃	—	2	1	no	—
93	C₄²×C₅₃	—	2	1	no	—
94	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
97	C₄²×C₅₃	#41	1	1	yes	characteristic
98	C₄⋊Dic₅₃	—	2	4	no	—
99	C₄⋊Dic₅₃	—	4	2	no	—
100	C₄×Dic₅₃	—	4	2	no	—
101	C₄×Dic₅₃	—	4	2	no	—
102	#11719917	—	4	2	no	—
103	#11719917	—	4	2	no	—
104	C₂×#88322	—	4	2	no	—
105	C₂³×C₅₃	—	2	1	no	—
106	C₂³×C₅₃	#121	1	1	yes	characteristic
107	Q₈×C₅₃	—	2	1	no	—
108	Q₈×C₅₃	#106	1	2	yes	complemented
109	D₄×C₅₃	—	2	4	no	—
110	D₄×C₅₃	—	2	1	no	—
111	D₄×C₅₃	#106	1	2	yes	complemented
112	C₂×C₄×C₅₃	—	2	1	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	2	1	no	—
115	C₂×C₄×C₅₃	—	2	4	no	—
116	C₂×C₄×C₅₃	—	2	1	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	4	1	no	—
119	C₂×C₄×C₅₃	—	4	2	no	—
120	C₂×C₄×C₅₃	—	4	1	no	—
121	C₂×C₄×C₅₃	—	4	1	no	—
122	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₅₃	#135	1	2	yes	—
130	C₂×Dic₅₃	—	4	4	no	—
131	C₂×Dic₅₃	—	8	2	no	—
132	C₂×Dic₅₃	—	8	2	no	—
133	C₈×C₅₃	—	2	1	no	—
134	C₈×C₅₃	—	2	1	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	—	4	1	no	—
137	C₈×C₅₃	—	4	1	no	—
138	C₈×C₅₃	#104	1	2	yes	complemented
139	#88322	—	8	2	no	—
140	#8666	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	—	2	1	no	—
142	C₂²×C₅₃	—	2	1	no	—
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	4	1	no	—
145	C₂²×C₅₃	—	4	1	no	—
146	C₂²×C₅₃	#380	1	1	yes	characteristic
147	C₂²×C₅₃	#465	1	2	yes	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	2	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	4	1	no	—
153	C₄×C₅₃	—	4	1	no	—
154	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
155	C₄×C₅₃	#421	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
157	C₄×C₅₃	#427	1	1	yes	characteristic
158	C₄×C₅₃	#465	1	2	yes	—
159	Dic₅₃	—	8	4	no	—
160	C₈×#41	D₅₃	1	1	yes	characteristic
161	#1806	—	53	2	no	—
162	C₄×#106	—	53	1	no	—
163	#1566	—	53	1	no	—
164	#1303	—	53	2	no	—
165	C₂×C₅₃	—	2	2	no	—
166	C₂×C₅₃	—	4	2	no	—
167	C₂×C₅₃	#1806	1	1	yes	characteristic
168	C₂×C₅₃	#1807	1	1	yes	characteristic
169	C₂×C₅₃	#1814	1	1	yes	characteristic
170	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
171	C₈×Q₈	Dic₅₃	1	1	yes	characteristic, complemented
172	C₈×Q₈	Dic₅₃	1	1	yes	characteristic, complemented
173	C₈×D₄	—	2	1	no	—
174	C₈×D₄	—	2	1	no	—
175	C₈×D₄	Dic₅₃	1	1	yes	characteristic, complemented
176	C₈×D₄	Dic₅₃	1	1	yes	characteristic, complemented
177	#431	—	2	1	no	—
178	#431	C₂×D₅₃	1	1	yes	characteristic
179	C₂×#106	—	106	1	no	—
180	#423	—	53	2	no	—
181	C₂×#104	—	106	2	no	—
182	C₂×C₄×C₈	—	2	1	no	—
183	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
184	C₄×C₄⋊C₄	—	53	2	no	—
185	#340	—	106	1	no	—
186	#339	—	106	2	no	—
187	#334	—	106	1	no	—
188	#334	—	106	1	no	—
189	C₅₃	#8666	1	1	yes	characteristic, complemented, Sylow
190	C₂×#41	D₂₁₂	1	1	yes	characteristic
191	C₂×M₁₆	—	106	2	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	1	no	—
194	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
196	C₄×Q₈	Dic₁₀₆	1	1	yes	characteristic
197	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	Dic₁₀₆	1	1	yes	characteristic
201	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
202	#117	—	2	1	no	—
203	#117	D₂₁₂	1	1	yes	characteristic
204	C₂×C₄⋊C₄	—	106	2	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	106	2	no	—
207	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
208	#106	—	106	4	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	106	2	no	—
212	C₄⋊C₈	—	106	2	no	—
213	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
214	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
215	#104	—	212	4	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	—	2	1	no	—
218	C₂²⋊C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
223	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
226	#95	—	106	2	no	—
227	#41	—	2	2	no	—
228	#41	#11719913	1	4	yes	—
229	C₂×Q₈	C₄⋊Dic₅₃	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₄⋊Dic₅₃	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	212	2	no	—
236	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
237	C₂²×C₄	#11719945	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
239	M₁₆	—	212	2	no	—
240	M₁₆	—	212	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	212	2	no	—
248	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	106	4	no	—
255	C₄⋊C₄	—	212	2	no	—
256	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
257	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	212	2	no	—
265	C₄²	—	212	2	no	—
266	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
267	C₄²	#11719945	1	1	yes	characteristic
268	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	#13133191	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	#13133149	1	2	yes	complemented
274	D₄	—	2	4	no	—
275	D₄	—	2	1	no	—
276	D₄	#13133149	1	2	yes	complemented
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	424	2	no	—
288	C₂×C₄	—	424	2	no	—
289	C₂×C₄	—	212	4	no	—
290	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
292	C₂×C₄	#13133176	1	1	yes	characteristic
293	C₂×C₄	#13133191	1	1	yes	characteristic
294	C₂×C₄	#13133196	1	1	yes	characteristic
295	C₂×C₄	#13133196	1	1	yes	characteristic
296	C₂×C₄	C₂×#11719913	1	2	yes	—
297	C₂×C₄	#13133239	1	2	yes	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	1	no	—
303	C₈	—	424	2	no	—
304	C₈	#13133144	1	2	yes	complemented
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	#16810483	1	1	yes	characteristic
311	C₂²	#16810852	1	2	yes	—
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	424	4	no	—
319	C₄	C₂×#13133144	1	1	yes	characteristic
320	C₄	#16810640	1	1	yes	characteristic
321	C₄	C₂×#13133149	1	1	yes	characteristic
322	C₄	#16810661	1	1	yes	characteristic
323	C₄	#16810852	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	4	2	no	—
326	C₂	#20274735	1	1	yes	characteristic
327	C₂	#20274740	1	1	yes	characteristic
328	C₂	#20274769	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₈×#41×C₅₃	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₈×#41	D₅₃
hypocenter	C₅₃	#8666
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#13133149	C₂	semidirect
3. #20273471	C₂	semidirect
4. #20272617	C₂	semidirect
15. C₈×Q₈×C₅₃	C₄	semidirect
16. C₈×Q₈×C₅₃	C₄	semidirect
19. C₈×D₄×C₅₃	C₄	semidirect
20. C₈×D₄×C₅₃	C₄	semidirect
108. Q₈×C₅₃	#106	semidirect
111. D₄×C₅₃	#106	semidirect
138. C₈×C₅₃	#104	semidirect
171. C₈×Q₈	Dic₅₃	semidirect
172. C₈×Q₈	Dic₅₃	semidirect
175. C₈×D₄	Dic₅₃	semidirect
176. C₈×D₄	Dic₅₃	semidirect
189. C₅₃	#8666	semidirect
273. Q₈	#13133149	semidirect
276. D₄	#13133149	semidirect
304. C₈	#13133144	semidirect

Maximal subgroups

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

subgroup	index
#20274735	2
C₄×#13133149	2
#20273471	2
#20272617	2
C₈×#41×C₅₃	2
#8666	53