Subgroups — #27537172

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 298 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	#20276516	C₂	1	4	yes	maximal, complemented
3	#20276512	C₂	1	4	yes	maximal, complemented
4	#20276470	C₂	1	2	yes	maximal, complemented
5	#20276325	C₂	1	2	yes	maximal
6	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₃₂×C₅₃	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	#16810441	C₂²	1	8	yes	complemented
20	#16810402	C₂²	1	8	yes	—
21	#16810399	C₂²	1	2	yes	complemented
22	C₄×#11719908	C₂²	1	2	yes	complemented
23	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#346	C₂²	1	2	yes	complemented
26	C₄×C₁₆×C₅₃	C₂²	1	2	yes	complemented
27	C₂²×C₄×D₅₃	C₈	1	2	yes	—
28	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₅₃	—	2	4	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	2	yes	—
48	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
50	C₈×Dic₅₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₅₃	—	2	4	no	—
55	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
56	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
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	M₁₆	1	4	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	#41	1	4	yes	—
79	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×C₈	1	2	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
84	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
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	C₂×C₈	1	2	yes	—
89	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
90	C₂×#88322	—	2	2	no	—
91	C₂×#88322	—	2	2	no	—
92	C₂×#88322	C₂²×C₄	1	2	yes	—
93	C₂×#88322	C₂²×C₄	1	2	yes	—
94	C₈×D₅₃	—	2	4	no	—
95	C₈×D₅₃	—	2	4	no	—
96	C₈×D₅₃	—	2	4	no	—
97	C₈×D₅₃	—	2	4	no	—
98	C₁₆×C₅₃	—	2	4	no	—
99	#11719908	—	2	4	no	—
100	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₂²×D₅₃	—	2	2	no	—
102	C₂²×D₅₃	—	2	4	no	—
103	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	#117	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×Dic₅₃	—	2	2	no	—
112	C₂×Dic₅₃	—	2	4	no	—
113	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
114	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
115	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
116	C₄×D₅₃	—	2	4	no	—
117	C₄×D₅₃	—	2	2	no	—
118	C₄×D₅₃	—	2	2	no	—
119	C₄×D₅₃	—	2	4	no	—
120	C₄×D₅₃	—	2	4	no	—
121	C₄×D₅₃	—	2	4	no	—
122	C₄×D₅₃	M₃₂	1	8	yes	complemented
123	C₈×C₅₃	#117	1	2	yes	—
124	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
125	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
126	C₈×C₅₃	C₂×#41	1	2	yes	—
127	#88322	—	2	2	no	—
128	#88322	—	2	2	no	—
129	#33418	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	#431	1	2	yes	—
132	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
133	C₂×D₅₃	—	2	8	no	—
134	C₂×D₅₃	—	2	2	no	—
135	C₂×D₅₃	—	2	2	no	—
136	C₂×D₅₃	C₂×M₃₂	1	4	yes	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	#431	1	2	yes	—
139	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
140	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
141	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
142	Dic₅₃	—	2	2	no	—
143	Dic₅₃	—	2	2	no	—
144	Dic₅₃	C₂×M₃₂	1	4	yes	—
145	C₂²×C₄×C₈	—	53	1	no	—
146	#2163	—	53	4	no	—
147	#2162	—	53	4	no	—
148	#2150	—	53	2	no	—
149	#2112	—	53	2	no	—
150	C₄×M₃₂	—	53	1	no	—
151	C₄×M₃₂	D₅₃	1	1	yes	characteristic, complemented
152	C₂×C₅₃	—	2	2	no	—
153	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
154	C₂×C₅₃	C₂²×M₃₂	1	2	yes	—
155	D₅₃	—	2	4	no	—
156	C₂³×C₈	—	53	1	no	—
157	C₂³×C₈	—	53	1	no	—
158	C₂²×C₄²	—	53	1	no	—
159	C₂×M₃₂	—	53	2	no	—
160	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
161	C₂×C₄×C₈	—	53	1	no	—
162	C₂×C₄×C₈	—	53	1	no	—
163	C₂×C₄×C₈	—	53	1	no	—
164	C₂×C₄×C₈	—	53	2	no	—
165	C₂×C₄×C₈	—	53	2	no	—
166	C₂×C₄×C₈	—	53	2	no	—
167	C₂×C₄×C₈	—	53	2	no	—
168	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
169	C₄⋊C₁₆	—	53	8	no	—
170	C₂²⋊C₁₆	—	53	8	no	—
171	#346	—	53	2	no	—
172	#346	C₂×D₅₃	1	2	yes	complemented
173	C₄×C₁₆	—	53	2	no	—
174	C₄×C₁₆	C₂×D₅₃	1	2	yes	complemented
175	C₅₃	#33418	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	—	53	1	no	—
177	C₂³×C₄	—	53	2	no	—
178	C₂²×C₈	—	53	1	no	—
179	C₂²×C₈	—	53	1	no	—
180	C₂²×C₈	—	53	2	no	—
181	C₂²×C₈	—	53	2	no	—
182	C₂²×C₈	—	53	2	no	—
183	C₂²×C₈	—	53	2	no	—
184	C₂²×C₈	—	106	4	no	—
185	C₂²×C₈	—	106	4	no	—
186	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
188	C₂×C₄²	—	53	1	no	—
189	C₂×C₄²	—	53	2	no	—
190	C₂×C₄²	—	53	2	no	—
191	C₂×C₄²	—	53	2	no	—
192	C₂×C₄²	—	53	2	no	—
193	C₂×C₄²	—	53	2	no	—
194	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
195	M₃₂	—	2	4	no	—
196	M₃₂	—	106	4	no	—
197	C₂×C₁₆	—	53	4	no	—
198	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
199	C₄×C₈	—	53	2	no	—
200	C₄×C₈	—	53	2	no	—
201	C₄×C₈	—	53	2	no	—
202	C₄×C₈	—	53	2	no	—
203	C₄×C₈	—	53	2	no	—
204	C₄×C₈	—	53	2	no	—
205	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
206	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	C₂⁴	—	53	1	no	—
210	C₂²×C₄	—	53	2	no	—
211	C₂²×C₄	—	53	2	no	—
212	C₂²×C₄	—	53	1	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	—	53	2	no	—
216	C₂²×C₄	—	53	2	no	—
217	C₂²×C₄	—	53	2	no	—
218	C₂²×C₄	—	106	8	no	—
219	C₂²×C₄	—	106	2	no	—
220	C₂²×C₄	—	106	2	no	—
221	C₂²×C₄	C₈×D₅₃	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
223	C₂×C₈	—	53	2	no	—
224	C₂×C₈	—	53	2	no	—
225	C₂×C₈	—	106	2	no	—
226	C₂×C₈	—	106	2	no	—
227	C₂×C₈	—	106	4	no	—
228	C₂×C₈	—	106	4	no	—
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	106	4	no	—
231	C₂×C₈	#11719912	1	4	yes	—
232	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
235	C₂×C₈	#11719945	1	4	yes	—
236	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
237	C₄²	—	53	1	no	—
238	C₄²	—	53	2	no	—
239	C₄²	—	53	2	no	—
240	C₄²	—	53	2	no	—
241	C₄²	—	53	2	no	—
242	C₄²	—	53	1	no	—
243	C₄²	—	53	2	no	—
244	C₄²	C₈×D₅₃	1	2	yes	—
245	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
247	C₁₆	—	2	4	no	—
248	C₁₆	—	106	4	no	—
249	C₂³	—	53	2	no	—
250	C₂³	—	106	4	no	—
251	C₂³	—	106	2	no	—
252	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	53	2	no	—
255	C₂×C₄	—	53	8	no	—
256	C₂×C₄	—	53	2	no	—
257	C₂×C₄	—	53	2	no	—
258	C₂×C₄	—	106	4	no	—
259	C₂×C₄	—	106	4	no	—
260	C₂×C₄	—	106	4	no	—
261	C₂×C₄	—	106	2	no	—
262	C₂×C₄	—	106	4	no	—
263	C₂×C₄	—	106	2	no	—
264	C₂×C₄	—	106	2	no	—
265	C₂×C₄	—	106	4	no	—
266	C₂×C₄	#13133174	1	2	yes	—
267	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
268	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
269	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719912	1	2	yes	—
271	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
272	C₈	—	106	2	no	—
273	C₈	—	106	2	no	—
274	C₈	#13133174	1	2	yes	—
275	C₈	C₂×#11719912	1	2	yes	—
276	C₈	C₂×#11719912	1	2	yes	—
277	C₈	C₂×#11719945	1	2	yes	—
278	C₂²	—	2	2	no	—
279	C₂²	—	53	4	no	—
280	C₂²	—	106	8	no	—
281	C₂²	—	106	2	no	—
282	C₂²	—	106	2	no	—
283	C₂²	#16810674	1	2	yes	—
284	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	53	4	no	—
287	C₄	—	106	2	no	—
288	C₄	—	106	2	no	—
289	C₄	#16810674	1	2	yes	—
290	C₄	M₃₂×D₅₃	1	4	yes	—
291	C₄	C₂×#13133174	1	1	yes	characteristic
292	C₄	C₂²×#11719912	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	106	4	no	—
295	C₂	C₂×#16810674	1	1	yes	characteristic
296	C₂	C₂×M₃₂×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#33418
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #20276516	C₂	semidirect
3. #20276512	C₂	semidirect
4. #20276470	C₂	semidirect
6. C₄×#13133157	C₂	semidirect
7. C₄×M₃₂×C₅₃	C₂	semidirect
19. #16810441	C₂²	semidirect
21. #16810399	C₂²	semidirect
22. C₄×#11719908	C₂²	semidirect
25. C₅₃×#346	C₂²	semidirect
26. C₄×C₁₆×C₅₃	C₂²	semidirect
122. C₄×D₅₃	M₃₂	semidirect
151. C₄×M₃₂	D₅₃	semidirect
172. #346	C₂×D₅₃	semidirect
174. C₄×C₁₆	C₂×D₅₃	semidirect
175. 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
#20276516	2
#20276512	2
#20276470	2
#20276325	2
C₄×#13133157	2
C₄×M₃₂×C₅₃	2
#33418	53