Subgroups — #27274835

Order 13568.

460 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283681	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133145	C₂	1	1	yes	maximal, characteristic
3	#20273098	C₂	1	1	yes	maximal, characteristic
4	#20273004	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272552	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272548	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11719945	C₂²	1	1	yes	characteristic
9	#16810753	C₄	1	2	yes	complemented
10	#16810702	C₄	1	2	yes	complemented
11	#16810694	—	2	2	no	—
12	#16810676	C₂²	1	1	yes	characteristic
13	#16810676	C₂²	1	1	yes	characteristic
14	#16810674	—	2	1	no	—
15	C₂×#13133145	C₄	1	2	yes	—
16	C₄×#11719917	C₂²	1	1	yes	characteristic
17	C₄×#11719912	—	2	1	no	—
18	#16810383	C₄	1	2	yes	—
19	#16810355	C₄	1	2	yes	complemented
20	#16810352	C₄	1	2	yes	complemented
21	#16810345	—	2	1	no	—
22	#16810344	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₄³×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#336	—	2	2	no	—
27	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
28	C₂×#11719917	C₂×C₄	1	2	yes	—
29	C₂×#11719912	—	2	2	no	—
30	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
32	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
33	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
34	C₄×C₂²⋊D₅₃	—	2	1	no	—
35	C₄×C₂²⋊D₅₃	—	2	1	no	—
36	#13133176	C₂×C₄	1	1	yes	characteristic
37	#13133174	—	2	1	no	—
38	C₄²×D₅₃	—	2	1	no	—
39	C₄⋊#88322	C₂×C₄	1	2	yes	—
40	C₄⋊#88322	C₂×C₄	1	2	yes	—
41	#13133146	—	2	2	no	—
42	#13133145	—	2	4	no	—
43	C₂²⋊#88322	—	2	1	no	—
44	C₂²⋊#88322	—	2	1	no	—
45	#13133129	—	2	2	no	—
46	#13133128	—	2	1	no	—
47	#13133127	—	2	2	no	—
48	C₄×#88322	—	2	1	no	—
49	C₄×#88322	—	2	1	no	—
50	C₄×#88322	—	2	1	no	—
51	C₂²×C₈×C₅₃	—	2	1	no	—
52	C₂²×C₈×C₅₃	D₄	1	2	yes	—
53	C₂×C₄²×C₅₃	—	2	1	no	—
54	C₂×C₄²×C₅₃	—	2	1	no	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
66	C₂×C₂²⋊D₅₃	—	2	1	no	—
67	#11719945	—	2	2	no	—
68	#11719945	—	2	2	no	—
69	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
70	C₂×C₄×D₅₃	—	2	1	no	—
71	C₂×C₄×D₅₃	—	2	1	no	—
72	C₂×C₄×D₅₃	—	2	1	no	—
73	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	4	2	no	—
76	C₂×C₈×C₅₃	—	4	2	no	—
77	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₅₃	—	2	2	no	—
80	C₄²×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	—	4	2	no	—
82	C₄²×C₅₃	M₁₆	1	4	yes	complemented
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	#41	1	1	yes	characteristic
87	C₂²⋊Dic₅₃	—	2	1	no	—
88	#11719921	—	2	1	no	—
89	#11719921	—	2	1	no	—
90	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
91	C₄⋊Dic₅₃	C₄²	1	1	yes	characteristic
92	#11719919	—	2	1	no	—
93	#11719919	—	2	1	no	—
94	C₄×Dic₅₃	—	2	1	no	—
95	C₄×Dic₅₃	—	2	1	no	—
96	C₄×Dic₅₃	—	2	1	no	—
97	#11719917	—	2	2	no	—
98	#11719917	—	2	2	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	—	2	2	no	—
101	C₂×#88322	—	2	2	no	—
102	#11719912	—	4	4	no	—
103	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂²×D₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₅₃	#131	1	2	yes	—
118	C₂²⋊D₅₃	—	4	2	no	—
119	C₂×Dic₅₃	—	2	1	no	—
120	C₂×Dic₅₃	—	2	1	no	—
121	C₂×Dic₅₃	—	2	1	no	—
122	D₂₁₂	—	2	1	no	—
123	D₂₁₂	—	2	1	no	—
124	C₄×D₅₃	—	4	2	no	—
125	C₄×D₅₃	—	4	2	no	—
126	C₄×D₅₃	—	4	2	no	—
127	Dic₁₀₆	—	2	1	no	—
128	Dic₁₀₆	—	2	1	no	—
129	C₈×C₅₃	—	2	2	no	—
130	C₈×C₅₃	—	2	2	no	—
131	C₈×C₅₃	—	4	2	no	—
132	#88322	—	4	2	no	—
133	#88322	—	4	2	no	—
134	#88322	—	4	2	no	—
135	#8195	—	53	1	no	maximal, Sylow
136	C₂²×C₅₃	—	2	1	no	—
137	C₂²×C₅₃	—	2	1	no	—
138	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₅₃	#435	1	2	yes	—
140	C₂×D₅₃	—	4	1	no	—
141	C₂×D₅₃	—	4	1	no	—
142	C₂×D₅₃	—	4	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	4	4	no	—
146	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
147	C₄×C₅₃	#405	1	1	yes	characteristic
148	C₄×C₅₃	#421	1	2	yes	—
149	C₄×C₅₃	#436	1	2	yes	—
150	C₄×C₅₃	#444	1	2	yes	—
151	Dic₅₃	—	4	1	no	—
152	Dic₅₃	—	4	1	no	—
153	Dic₅₃	—	4	1	no	—
154	#2965	—	53	1	no	—
155	C₄×#104	—	53	1	no	—
156	#1744	D₅₃	1	1	yes	characteristic, complemented
157	#1476	—	53	1	no	—
158	#1451	—	53	1	no	—
159	#1278	—	53	1	no	—
160	#1276	—	53	1	no	—
161	C₂×C₅₃	—	2	2	no	—
162	C₂×C₅₃	#1753	1	1	yes	characteristic
163	C₂×C₅₃	#1759	1	1	yes	characteristic
164	C₂×C₅₃	#1765	1	1	yes	characteristic
165	D₅₃	—	8	1	no	—
166	C₄×#41	—	53	1	no	—
167	#446	—	53	2	no	—
168	#436	—	53	2	no	—
169	#435	—	106	2	no	—
170	#431	—	53	1	no	—
171	#431	—	53	1	no	—
172	#431	—	106	1	no	—
173	C₂×#104	—	53	2	no	—
174	C₄×M₁₆	—	53	1	no	—
175	C₄×M₁₆	—	106	1	no	—
176	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
177	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
178	C₄³	C₂×D₅₃	1	1	yes	characteristic
179	#339	—	53	2	no	—
180	#336	—	2	2	no	—
181	Q₈⋊C₈	—	53	2	no	—
182	D₄⋊C₈	—	53	2	no	—
183	#322	—	106	1	no	—
184	#322	—	106	1	no	—
185	C₅₃	#8195	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	—	53	1	no	—
187	C₂×M₁₆	—	53	2	no	—
188	C₂×M₁₆	—	106	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
191	C₄×Q₈	—	53	1	no	—
192	C₄×Q₈	—	53	1	no	—
193	C₄×D₄	—	53	1	no	—
194	C₄×D₄	—	53	1	no	—
195	C₄×D₄	—	106	1	no	—
196	C₄×D₄	—	106	1	no	—
197	#117	—	53	1	no	—
198	#117	—	106	1	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	106	1	no	—
202	C₂×C₄²	C₄×D₅₃	1	2	yes	—
203	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
204	C₄⋊C₈	—	53	2	no	—
205	C₄⋊C₈	—	53	2	no	—
206	C₄⋊C₈	—	106	2	no	—
207	#104	—	106	4	no	—
208	C₂²⋊C₈	—	106	1	no	—
209	C₂²⋊C₈	—	106	1	no	—
210	C₂²⋊C₈	—	106	2	no	—
211	#97	—	106	2	no	—
212	#97	—	106	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	106	1	no	—
215	C₄×C₈	—	106	1	no	—
216	C₄×C₈	—	106	1	no	—
217	C₄×C₈	C₄×D₅₃	1	2	yes	—
218	C₄×C₈	C₄×D₅₃	1	2	yes	—
219	C₄×C₈	D₂₁₂	1	1	yes	characteristic
220	C₄×C₈	D₂₁₂	1	1	yes	characteristic
221	#41	—	106	2	no	—
222	#41	—	106	2	no	—
223	C₂×Q₈	—	53	1	no	—
224	C₂×D₄	—	53	1	no	—
225	C₂×D₄	—	106	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	106	1	no	—
228	C₂²×C₄	—	106	1	no	—
229	C₂²×C₄	—	106	1	no	—
230	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
231	C₂²×C₄	#11719945	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
233	M₁₆	—	106	2	no	—
234	M₁₆	—	106	2	no	—
235	M₁₆	—	212	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	106	2	no	—
240	C₂×C₈	—	106	2	no	—
241	C₂×C₈	—	106	2	no	—
242	C₂×C₈	#11719921	1	2	yes	—
243	C₂×C₈	#11719921	1	2	yes	—
244	C₄⋊C₄	—	53	1	no	—
245	C₄⋊C₄	—	53	1	no	—
246	C₄⋊C₄	—	106	1	no	—
247	C₄⋊C₄	—	106	1	no	—
248	C₂²⋊C₄	—	106	1	no	—
249	C₂²⋊C₄	—	106	1	no	—
250	C₂²⋊C₄	—	106	1	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	4	2	no	—
253	C₄²	—	4	2	no	—
254	C₄²	—	106	1	no	—
255	C₄²	—	106	1	no	—
256	C₄²	—	106	1	no	—
257	C₄²	#11719912	1	4	yes	complemented
258	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
260	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
261	C₄²	#11719945	1	1	yes	characteristic
262	C₂³	—	106	1	no	—
263	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
264	Q₈	—	106	1	no	—
265	Q₈	—	106	1	no	—
266	D₄	—	106	1	no	—
267	D₄	—	106	1	no	—
268	D₄	—	212	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	106	1	no	—
275	C₂×C₄	—	106	1	no	—
276	C₂×C₄	—	106	1	no	—
277	C₂×C₄	—	212	2	no	—
278	C₂×C₄	—	212	2	no	—
279	C₂×C₄	—	212	2	no	—
280	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
282	C₂×C₄	#13133174	1	1	yes	characteristic
283	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
284	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
285	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
286	C₂×C₄	C₂×#11719912	1	2	yes	—
287	C₂×C₄	#13133223	1	2	yes	—
288	C₈	—	2	2	no	—
289	C₈	—	2	2	no	—
290	C₈	—	4	2	no	—
291	C₈	—	212	2	no	—
292	C₈	—	212	2	no	—
293	C₈	—	212	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	212	1	no	—
297	C₂²	—	212	1	no	—
298	C₂²	—	212	1	no	—
299	C₂²	C₄×#11719921	1	1	yes	characteristic
300	C₂²	#16810694	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	4	no	—
304	C₄	—	212	1	no	—
305	C₄	—	212	1	no	—
306	C₄	—	212	1	no	—
307	C₄	C₄×#11719912	1	1	yes	characteristic
308	C₄	#16810577	1	1	yes	characteristic
309	C₄	#16810641	1	2	yes	—
310	C₄	#16810702	1	2	yes	—
311	C₄	#16810746	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	424	1	no	—
314	C₂	#20274498	1	1	yes	characteristic
315	C₂	#20274526	1	1	yes	characteristic
316	C₂	#20274559	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1744	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#8195
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ #1744

Complements

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

N	H ≅ G/N	type
4. #20273004	C₂	semidirect
5. #20272552	C₂	semidirect
6. #20272548	C₂	semidirect
7. C₅₃×#1744	C₂	semidirect
9. #16810753	C₄	semidirect
10. #16810702	C₄	semidirect
19. #16810355	C₄	semidirect
20. #16810352	C₄	semidirect
82. C₄²×C₅₃	M₁₆	semidirect
156. #1744	D₅₃	semidirect
185. C₅₃	#8195	semidirect
257. C₄²	#11719912	semidirect

Maximal subgroups

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

subgroup	index
#20283681	2
C₄×#13133145	2
#20273098	2
#20273004	2
#20272552	2
#20272548	2
C₅₃×#1744	2
#8195	53