Subgroups — #27537173

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20276517	C₂	1	4	yes	maximal, complemented
3	#20276513	C₂	1	4	yes	maximal, complemented
4	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276325	C₂	1	2	yes	maximal
6	M₃₂×Dic₅₃	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄²×D₅₃	C₄	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	2	yes	—
12	C₂×C₄×#88322	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×D₅₃	C₄	1	2	yes	—
16	C₄×C₈×D₅₃	C₄	1	2	yes	—
17	C₄×C₈×D₅₃	C₂²	1	2	yes	—
18	C₄×C₈×D₅₃	C₂²	1	2	yes	—
19	C₄⋊#11719908	C₂²	1	4	yes	complemented
20	#16810402	C₂²	1	8	yes	—
21	#16810398	C₂²	1	4	yes	complemented
22	C₁₆×Dic₅₃	C₂²	1	4	yes	complemented
23	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
26	C₂²×C₄×D₅₃	C₈	1	2	yes	—
27	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×D₅₃	—	2	2	no	—
31	C₂×C₈×D₅₃	—	2	2	no	—
32	C₂×C₈×D₅₃	—	2	4	no	—
33	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₅₃	C₂³	1	2	yes	—
37	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
38	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×D₅₃	C₈	1	2	yes	—
40	C₄²×D₅₃	C₈	1	2	yes	—
41	C₄²×D₅₃	C₂×C₄	1	2	yes	—
42	C₄²×D₅₃	C₂×C₄	1	2	yes	—
43	#13133157	—	2	4	no	—
44	C₂×#11719908	C₂³	1	4	yes	—
45	C₄×#88322	C₂×C₄	1	2	yes	—
46	C₄×#88322	C₂×C₄	1	2	yes	—
47	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
48	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
50	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
51	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
52	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	M₃₂×C₅₃	—	2	4	no	—
57	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
58	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₅₃	C₂³	1	2	yes	—
60	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₅₃	—	2	2	no	—
66	C₂×C₄×D₅₃	—	2	2	no	—
67	C₂×C₄×D₅₃	—	2	8	no	—
68	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	#41	1	2	yes	—
78	C₂×C₈×C₅₃	#41	1	2	yes	—
79	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
83	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
84	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
85	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
86	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
87	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
88	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×#88322	—	2	2	no	—
91	C₂×#88322	M₁₆	1	4	yes	—
92	C₂×#88322	C₂²×C₄	1	2	yes	—
93	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
94	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
95	C₈×D₅₃	—	2	4	no	—
96	C₈×D₅₃	—	2	2	no	—
97	C₈×D₅₃	—	2	2	no	—
98	C₈×D₅₃	—	2	4	no	—
99	C₈×D₅₃	—	2	4	no	—
100	C₁₆×C₅₃	—	2	4	no	—
101	#11719908	—	2	4	no	—
102	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₅₃	—	2	2	no	—
104	C₂²×D₅₃	—	2	4	no	—
105	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	#117	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×Dic₅₃	—	2	2	no	—
114	C₂×Dic₅₃	—	2	4	no	—
115	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
116	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
117	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
118	C₄×D₅₃	—	2	4	no	—
119	C₄×D₅₃	—	2	2	no	—
120	C₄×D₅₃	—	2	2	no	—
121	C₄×D₅₃	—	2	4	no	—
122	C₄×D₅₃	—	2	4	no	—
123	C₄×D₅₃	—	2	4	no	—
124	C₄×D₅₃	M₃₂	1	8	yes	complemented
125	C₈×C₅₃	—	2	2	no	—
126	C₈×C₅₃	#117	1	2	yes	—
127	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
128	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
129	#88322	—	2	2	no	—
130	#88322	C₂×M₁₆	1	2	yes	—
131	#88322	C₂×M₁₆	1	2	yes	—
132	#33418	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	#431	1	2	yes	—
135	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
136	C₂×D₅₃	—	2	8	no	—
137	C₂×D₅₃	—	2	2	no	—
138	C₂×D₅₃	—	2	2	no	—
139	C₂×D₅₃	C₂×M₃₂	1	4	yes	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	#431	1	2	yes	—
142	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
143	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
144	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
145	Dic₅₃	—	2	2	no	—
146	Dic₅₃	—	2	2	no	—
147	Dic₅₃	C₂×M₃₂	1	4	yes	—
148	C₂²×C₄×C₈	—	53	1	no	—
149	#2163	—	53	4	no	—
150	#2162	—	53	4	no	—
151	#2150	—	53	1	no	—
152	#2150	D₅₃	1	1	yes	characteristic, complemented
153	#2112	—	53	2	no	—
154	C₄×M₃₂	—	53	2	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
157	C₂×C₅₃	C₂²×M₃₂	1	2	yes	—
158	D₅₃	—	2	4	no	—
159	C₂³×C₈	—	53	1	no	—
160	C₂³×C₈	—	53	1	no	—
161	C₂²×C₄²	—	53	1	no	—
162	C₂×M₃₂	—	53	2	no	—
163	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
164	C₂×C₄×C₈	—	53	1	no	—
165	C₂×C₄×C₈	—	53	1	no	—
166	C₂×C₄×C₈	—	53	1	no	—
167	C₂×C₄×C₈	—	53	2	no	—
168	C₂×C₄×C₈	—	53	2	no	—
169	C₂×C₄×C₈	—	53	2	no	—
170	C₂×C₄×C₈	—	53	2	no	—
171	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
172	C₄⋊C₁₆	—	53	4	no	—
173	C₄⋊C₁₆	C₂×D₅₃	1	4	yes	complemented
174	C₂²⋊C₁₆	—	53	8	no	—
175	#346	—	53	4	no	—
176	C₄×C₁₆	—	53	4	no	—
177	C₅₃	#33418	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	53	1	no	—
179	C₂³×C₄	—	53	2	no	—
180	C₂²×C₈	—	53	1	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	2	no	—
183	C₂²×C₈	—	53	2	no	—
184	C₂²×C₈	—	53	2	no	—
185	C₂²×C₈	—	53	2	no	—
186	C₂²×C₈	—	106	4	no	—
187	C₂²×C₈	—	106	2	no	—
188	C₂²×C₈	—	106	2	no	—
189	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
191	C₂×C₄²	—	53	1	no	—
192	C₂×C₄²	—	53	2	no	—
193	C₂×C₄²	—	53	2	no	—
194	C₂×C₄²	—	53	2	no	—
195	C₂×C₄²	—	53	2	no	—
196	C₂×C₄²	—	53	2	no	—
197	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
198	M₃₂	—	2	4	no	—
199	M₃₂	—	106	4	no	—
200	C₂×C₁₆	—	53	4	no	—
201	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
202	C₄×C₈	—	53	1	no	—
203	C₄×C₈	—	53	1	no	—
204	C₄×C₈	—	53	2	no	—
205	C₄×C₈	—	53	2	no	—
206	C₄×C₈	—	53	1	no	—
207	C₄×C₈	—	53	1	no	—
208	C₄×C₈	—	53	2	no	—
209	C₄×C₈	—	53	2	no	—
210	C₄×C₈	C₄×D₅₃	1	2	yes	—
211	C₄×C₈	C₂²×D₅₃	1	2	yes	—
212	C₂⁴	—	53	1	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	—	53	1	no	—
216	C₂²×C₄	—	53	2	no	—
217	C₂²×C₄	—	53	2	no	—
218	C₂²×C₄	—	53	2	no	—
219	C₂²×C₄	—	53	2	no	—
220	C₂²×C₄	—	53	2	no	—
221	C₂²×C₄	—	106	8	no	—
222	C₂²×C₄	—	106	2	no	—
223	C₂²×C₄	—	106	2	no	—
224	C₂²×C₄	C₈×D₅₃	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	53	4	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	1	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	—	106	2	no	—
234	C₂×C₈	—	106	4	no	—
235	C₂×C₈	—	106	2	no	—
236	C₂×C₈	—	106	2	no	—
237	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
238	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
239	C₂×C₈	#11719947	1	2	yes	—
240	C₂×C₈	#11719949	1	2	yes	—
241	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
242	C₄²	—	53	2	no	—
243	C₄²	—	53	2	no	—
244	C₄²	—	53	2	no	—
245	C₄²	—	53	2	no	—
246	C₄²	—	53	2	no	—
247	C₄²	—	53	1	no	—
248	C₄²	—	53	1	no	—
249	C₄²	C₈×D₅₃	1	1	yes	characteristic
250	C₄²	C₈×D₅₃	1	1	yes	characteristic
251	C₄²	C₂×C₄×D₅₃	1	2	yes	—
252	C₁₆	—	2	4	no	—
253	C₁₆	—	106	4	no	—
254	C₂³	—	53	2	no	—
255	C₂³	—	106	4	no	—
256	C₂³	—	106	2	no	—
257	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	53	2	no	—
260	C₂×C₄	—	53	8	no	—
261	C₂×C₄	—	53	2	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	106	4	no	—
264	C₂×C₄	—	106	4	no	—
265	C₂×C₄	—	106	4	no	—
266	C₂×C₄	—	106	2	no	—
267	C₂×C₄	—	106	4	no	—
268	C₂×C₄	—	106	2	no	—
269	C₂×C₄	—	106	2	no	—
270	C₂×C₄	—	106	4	no	—
271	C₂×C₄	#13133175	1	2	yes	—
272	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
273	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
274	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
275	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
276	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	53	2	no	—
279	C₈	—	53	2	no	—
280	C₈	—	106	2	no	—
281	C₈	#13133175	1	2	yes	—
282	C₈	C₂×#11719947	1	1	yes	characteristic
283	C₈	C₂×#11719949	1	1	yes	characteristic
284	C₂²	—	2	2	no	—
285	C₂²	—	53	4	no	—
286	C₂²	—	106	8	no	—
287	C₂²	—	106	2	no	—
288	C₂²	—	106	2	no	—
289	C₂²	#16810675	1	2	yes	—
290	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	53	4	no	—
293	C₄	—	106	2	no	—
294	C₄	—	106	2	no	—
295	C₄	#16810675	1	2	yes	—
296	C₄	M₃₂×D₅₃	1	4	yes	—
297	C₄	C₂×#13133175	1	1	yes	characteristic
298	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	106	4	no	—
301	C₂	C₂×#16810675	1	1	yes	characteristic
302	C₂	C₂×M₃₂×D₅₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2150	C₂
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#33418
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
2. #20276517	C₂	semidirect
3. #20276513	C₂	semidirect
4. #20276471	C₂	semidirect
6. M₃₂×Dic₅₃	C₂	semidirect
7. C₅₃×#2150	C₂	semidirect
19. C₄⋊#11719908	C₂²	semidirect
21. #16810398	C₂²	semidirect
22. C₁₆×Dic₅₃	C₂²	semidirect
25. C₅₃×C₄⋊C₁₆	C₂²	semidirect
124. C₄×D₅₃	M₃₂	semidirect
152. #2150	D₅₃	semidirect
173. C₄⋊C₁₆	C₂×D₅₃	semidirect
177. C₅₃	#33418	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₅₃	2
#20276517	2
#20276513	2
#20276471	2
#20276325	2
M₃₂×Dic₅₃	2
C₅₃×#2150	2
#33418	53