Subgroups — #27289165

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275521	C₂	1	1	yes	maximal, characteristic
2	#20275520	C₂	1	1	yes	maximal, characteristic
3	#20275520	C₂	1	1	yes	maximal, characteristic
4	#20274842	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274842	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272542	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133214	C₂²	1	1	yes	characteristic
9	C₂×#13133139	—	2	2	no	—
10	C₂×#13133137	—	2	2	no	—
11	C₂×#13133137	—	2	2	no	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	#16810493	C₂²	1	1	yes	characteristic
15	#16810385	—	2	2	no	—
16	#16810385	—	2	2	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#384	C₂²	1	1	yes	characteristic
19	C₅₃×#384	C₂²	1	1	yes	characteristic
20	C₅₃×#340	—	2	2	no	—
21	C₂²×D₂₁₂	—	2	1	no	—
22	C₂²×D₂₁₂	—	2	1	no	—
23	C₂²×D₂₁₂	—	2	1	no	—
24	C₂²×#88322	D₄	1	2	yes	—
25	C₂²×#88322	D₄	1	2	yes	—
26	#13133214	—	2	1	no	—
27	#13133214	—	2	1	no	—
28	#13133214	—	2	1	no	—
29	#13133214	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₅₃	—	2	1	no	—
31	C₂×C₄⋊Dic₅₃	—	2	1	no	—
32	C₂×C₄⋊Dic₅₃	—	2	1	no	—
33	#13133139	—	4	4	no	—
34	#13133137	—	4	4	no	—
35	#13133137	—	4	4	no	—
36	C₄×#88322	—	2	1	no	—
37	C₄×#88322	—	2	1	no	—
38	C₄×#88322	—	2	1	no	—
39	C₄×#88322	—	2	1	no	—
40	C₂²×C₈×C₅₃	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
46	C₂×C₄⋊C₄×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	C₂³×D₅₃	—	4	1	no	—
51	C₂²×C₄×C₅₃	—	4	1	no	—
52	C₂²×C₄×C₅₃	—	4	1	no	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	—	4	1	no	—
57	C₂×D₂₁₂	—	2	2	no	—
58	C₂×D₂₁₂	—	2	2	no	—
59	C₂×D₂₁₂	—	2	2	no	—
60	C₂×D₂₁₂	—	4	2	no	—
61	C₂×D₂₁₂	—	4	2	no	—
62	C₂×D₂₁₂	—	4	2	no	—
63	C₂×D₂₁₂	—	4	2	no	—
64	C₂×D₂₁₂	—	4	2	no	—
65	C₂×D₂₁₂	—	4	2	no	—
66	C₂×D₂₁₂	—	4	1	no	—
67	C₂×D₂₁₂	—	4	1	no	—
68	C₂×D₂₁₂	—	4	1	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₄⋊C₄×C₅₃	—	4	2	no	—
73	C₄⋊C₄×C₅₃	—	4	2	no	—
74	C₄⋊C₄×C₅₃	—	4	2	no	—
75	C₄⋊C₄×C₅₃	—	4	2	no	—
76	C₄⋊C₄×C₅₃	—	4	2	no	—
77	C₄⋊C₄×C₅₃	—	4	2	no	—
78	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₅₃	#41	1	1	yes	characteristic
80	C₄²×C₅₃	#41	1	1	yes	characteristic
81	C₄²×C₅₃	#41	1	1	yes	characteristic
82	C₄⋊Dic₅₃	—	4	2	no	—
83	C₄⋊Dic₅₃	—	4	2	no	—
84	C₄⋊Dic₅₃	—	4	2	no	—
85	C₂×#88322	—	2	2	no	—
86	C₂×#88322	—	2	2	no	—
87	C₂×#88322	—	2	2	no	—
88	C₂×#88322	—	2	2	no	—
89	C₂×#88322	—	2	2	no	—
90	C₂×#88322	—	2	2	no	—
91	C₂³×C₅₃	#127	1	1	yes	characteristic
92	C₂²×D₅₃	—	4	2	no	—
93	C₂²×D₅₃	—	8	1	no	—
94	C₂²×D₅₃	—	8	1	no	—
95	C₂²×D₅₃	—	8	1	no	—
96	C₂²×D₅₃	—	8	1	no	—
97	C₂²×D₅₃	—	8	1	no	—
98	C₂²×D₅₃	—	8	1	no	—
99	C₂×C₄×C₅₃	—	8	1	no	—
100	C₂×C₄×C₅₃	—	8	1	no	—
101	C₂×C₄×C₅₃	—	8	1	no	—
102	C₂×C₄×C₅₃	—	8	1	no	—
103	C₂×C₄×C₅₃	—	8	1	no	—
104	C₂×C₄×C₅₃	—	8	1	no	—
105	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#135	1	2	yes	—
112	C₂×C₄×C₅₃	#135	1	2	yes	—
113	C₂×C₄×C₅₃	#135	1	2	yes	—
114	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×Dic₅₃	—	8	1	no	—
118	C₂×Dic₅₃	—	8	1	no	—
119	C₂×Dic₅₃	—	8	1	no	—
120	D₂₁₂	—	4	4	no	—
121	D₂₁₂	—	4	4	no	—
122	D₂₁₂	—	4	4	no	—
123	D₂₁₂	—	8	2	no	—
124	D₂₁₂	—	8	2	no	—
125	D₂₁₂	—	8	2	no	—
126	C₈×C₅₃	—	4	2	no	—
127	#88322	—	4	2	no	—
128	#88322	—	4	2	no	—
129	#10678	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	#399	1	1	yes	characteristic
131	C₂²×C₅₃	#495	1	1	yes	characteristic
132	C₂²×C₅₃	#495	1	1	yes	characteristic
133	C₂²×C₅₃	#495	1	1	yes	characteristic
134	C₂²×C₅₃	#496	1	1	yes	characteristic
135	C₂²×C₅₃	#496	1	1	yes	characteristic
136	C₂²×C₅₃	#496	1	1	yes	characteristic
137	C₂×D₅₃	—	8	2	no	—
138	C₂×D₅₃	—	8	2	no	—
139	C₂×D₅₃	—	8	2	no	—
140	C₂×D₅₃	—	16	1	no	—
141	C₂×D₅₃	—	16	1	no	—
142	C₂×D₅₃	—	16	1	no	—
143	C₂×D₅₃	—	16	1	no	—
144	C₄×C₅₃	—	16	1	no	—
145	C₄×C₅₃	—	16	1	no	—
146	C₄×C₅₃	#478	1	2	yes	—
147	C₄×C₅₃	#478	1	2	yes	—
148	C₄×C₅₃	#478	1	2	yes	—
149	C₄×C₅₃	#482	1	2	yes	—
150	C₄×C₅₃	#482	1	2	yes	—
151	C₄×C₅₃	#482	1	2	yes	—
152	Dic₅₃	—	16	1	no	—
153	#1957	—	53	1	no	—
154	#1957	—	53	1	no	—
155	#1957	—	53	1	no	—
156	#1834	—	53	1	no	—
157	#1834	—	53	1	no	—
158	#1834	D₅₃	1	1	yes	characteristic, complemented
159	#1273	—	53	1	no	—
160	C₂×C₅₃	#1711	1	1	yes	characteristic
161	C₂×C₅₃	#1717	1	1	yes	characteristic
162	C₂×C₅₃	#1717	1	1	yes	characteristic
163	C₂×C₅₃	#1717	1	1	yes	characteristic
164	C₂×C₅₃	#2092	1	1	yes	characteristic
165	C₂×C₅₃	#2092	1	1	yes	characteristic
166	C₂×C₅₃	#2092	1	1	yes	characteristic
167	D₅₃	—	16	2	no	—
168	C₂×#127	—	53	1	no	—
169	C₂×#102	—	106	2	no	—
170	C₂×#102	—	106	2	no	—
171	C₂×#102	—	106	2	no	—
172	C₂×C₄×C₈	—	53	1	no	—
173	C₂×C₄×C₈	—	53	1	no	—
174	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
175	#384	—	53	1	no	—
176	#384	C₂×D₅₃	1	1	yes	characteristic
177	#384	C₂×D₅₃	1	1	yes	characteristic
178	#340	—	2	2	no	—
179	#340	—	106	2	no	—
180	#340	—	106	2	no	—
181	C₅₃	#10678	1	1	yes	characteristic, complemented, Sylow
182	C₂²×D₄	—	106	1	no	—
183	C₂²×D₄	—	106	1	no	—
184	C₂²×D₄	—	106	1	no	—
185	C₂²×C₈	—	53	2	no	—
186	C₂²×C₈	—	53	2	no	—
187	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
188	#127	—	53	2	no	—
189	#127	—	106	1	no	—
190	#127	—	106	1	no	—
191	#127	—	106	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	106	1	no	—
199	C₂×C₄⋊C₄	—	106	1	no	—
200	C₂×C₄⋊C₄	—	106	1	no	—
201	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
202	#102	—	212	4	no	—
203	#102	—	212	4	no	—
204	#102	—	212	4	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	106	1	no	—
208	C₄×C₈	—	106	1	no	—
209	C₄×C₈	—	106	1	no	—
210	C₄×C₈	—	106	1	no	—
211	C₂⁴	—	212	1	no	—
212	C₂×D₄	—	106	2	no	—
213	C₂×D₄	—	106	2	no	—
214	C₂×D₄	—	106	2	no	—
215	C₂×D₄	—	212	1	no	—
216	C₂×D₄	—	212	1	no	—
217	C₂×D₄	—	212	1	no	—
218	C₂×D₄	—	212	2	no	—
219	C₂×D₄	—	212	2	no	—
220	C₂×D₄	—	212	2	no	—
221	C₂×D₄	—	212	2	no	—
222	C₂×D₄	—	212	2	no	—
223	C₂×D₄	—	212	2	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	212	1	no	—
227	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
228	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	106	2	no	—
234	C₂×C₈	—	106	2	no	—
235	C₂×C₈	—	106	2	no	—
236	C₂×C₈	—	106	2	no	—
237	C₂×C₈	—	106	2	no	—
238	C₂×C₈	—	106	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	212	2	no	—
246	C₄⋊C₄	—	212	2	no	—
247	C₄⋊C₄	—	212	2	no	—
248	C₄²	#11719945	1	1	yes	characteristic
249	C₄²	#11719945	1	1	yes	characteristic
250	C₄²	Q₈×D₅₃	1	1	yes	characteristic
251	C₄²	#11719949	1	1	yes	characteristic
252	C₂³	—	424	1	no	—
253	C₂³	—	424	1	no	—
254	C₂³	—	424	1	no	—
255	C₂³	—	424	1	no	—
256	C₂³	—	424	1	no	—
257	C₂³	—	424	1	no	—
258	C₂³	—	212	2	no	—
259	C₂³	#13133213	1	1	yes	characteristic
260	D₄	—	424	2	no	—
261	D₄	—	424	2	no	—
262	D₄	—	424	2	no	—
263	D₄	—	212	4	no	—
264	D₄	—	212	4	no	—
265	D₄	—	212	4	no	—
266	C₂×C₄	—	8	1	no	—
267	C₂×C₄	—	8	1	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	424	1	no	—
273	C₂×C₄	—	424	1	no	—
274	C₂×C₄	—	424	1	no	—
275	C₂×C₄	#13133193	1	1	yes	characteristic
276	C₂×C₄	#13133195	1	1	yes	characteristic
277	C₂×C₄	#13133195	1	1	yes	characteristic
278	C₂×C₄	#13133200	1	1	yes	characteristic
279	C₂×C₄	#13133202	1	1	yes	characteristic
280	C₂×C₄	#13133202	1	1	yes	characteristic
281	C₂×C₄	#13133237	1	2	yes	—
282	C₂×C₄	#13133238	1	2	yes	—
283	C₂×C₄	#13133240	1	2	yes	—
284	C₂×C₄	#13133241	1	2	yes	—
285	C₂×C₄	#13133242	1	2	yes	—
286	C₂×C₄	#13133244	1	2	yes	—
287	C₈	—	4	2	no	—
288	C₈	—	212	2	no	—
289	C₈	—	212	2	no	—
290	C₂²	—	848	1	no	—
291	C₂²	—	848	1	no	—
292	C₂²	—	848	1	no	—
293	C₂²	—	848	1	no	—
294	C₂²	—	424	2	no	—
295	C₂²	—	424	2	no	—
296	C₂²	—	424	2	no	—
297	C₂²	#16810558	1	1	yes	characteristic
298	C₂²	#16811004	1	1	yes	characteristic
299	C₂²	#16811005	1	1	yes	characteristic
300	C₂²	#16811008	1	1	yes	characteristic
301	C₂²	#16811009	1	1	yes	characteristic
302	C₂²	#16811011	1	1	yes	characteristic
303	C₂²	#16811013	1	1	yes	characteristic
304	C₄	—	16	1	no	—
305	C₄	—	16	1	no	—
306	C₄	—	848	1	no	—
307	C₄	#16810920	1	2	yes	—
308	C₄	#16810922	1	2	yes	—
309	C₄	#16810926	1	2	yes	—
310	C₄	#16810947	1	2	yes	—
311	C₄	#16810949	1	2	yes	—
312	C₄	#16810950	1	2	yes	—
313	C₂	—	848	2	no	—
314	C₂	#20274328	1	1	yes	characteristic
315	C₂	#20274355	1	1	yes	characteristic
316	C₂	#20274356	1	1	yes	characteristic
317	C₂	#20274357	1	1	yes	characteristic
318	C₂	#20276233	1	1	yes	characteristic
319	C₂	#20276237	1	1	yes	characteristic
320	C₂	#20276239	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133213
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1834	C₂
socle	C₂³×C₅₃	#127
prime radical	C₂×#13133214	C₂²
hypercenter	#1834	D₅₃
hypocenter	C₅₃	#10678
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
4. #20274842	C₂	semidirect
5. #20274842	C₂	semidirect
6. #20272542	C₂	semidirect
7. C₅₃×#1834	C₂	semidirect
158. #1834	D₅₃	semidirect
181. C₅₃	#10678	semidirect

Maximal subgroups

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

subgroup	index
#20275521	2
#20275520	2
#20274842	2
#20272542	2
C₅₃×#1834	2
#10678	53