Subgroups — #27537734

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276623	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276622	C₂	1	2	yes	maximal, complemented
5	#20276621	C₂	1	2	yes	maximal, complemented
6	#20276620	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276517	C₂	1	2	yes	maximal
8	#20276516	C₂	1	2	yes	maximal
9	#20276513	C₂	1	1	yes	maximal, characteristic
10	#20276512	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×D₅₃	C₄	1	1	yes	characteristic
13	Q₈×#88322	C₄	1	1	yes	characteristic
14	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₀₆	C₂²	1	2	yes	—
16	#16810747	C₄	1	2	yes	—
17	#16810747	C₄	1	1	yes	characteristic
18	C₄⋊C₈×D₅₃	C₄	1	2	yes	—
19	C₄⋊C₈×D₅₃	C₄	1	1	yes	characteristic
20	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×D₅₃	C₂²	1	2	yes	—
22	C₄⋊#11719908	C₂²	1	1	yes	characteristic
23	C₄⋊#11719908	C₂²	1	2	yes	—
24	#16810441	C₂²	1	1	yes	characteristic
25	#16810441	C₂²	1	1	yes	characteristic
26	#16810441	C₂²	1	2	yes	—
27	#16810441	C₂²	1	2	yes	—
28	#16810402	C₂²	1	1	yes	characteristic
29	#16810402	C₂²	1	1	yes	characteristic
30	#16810402	C₂²	1	2	yes	—
31	#16810399	C₂²	1	2	yes	—
32	#16810398	C₂²	1	2	yes	—
33	#16810398	C₂²	1	2	yes	—
34	C₄×#11719908	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₅₃	C₂²	1	2	yes	—
36	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
39	C₅₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₅₃	C₈	1	1	yes	characteristic
42	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
45	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
46	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
48	#13133182	C₈	1	1	yes	characteristic
49	#13133182	C₈	1	2	yes	—
50	C₄⋊C₄×D₅₃	C₈	1	2	yes	—
51	C₄⋊C₄×D₅₃	C₈	1	1	yes	characteristic
52	C₄²×D₅₃	C₂×C₄	1	2	yes	—
53	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719908	C₂³	1	2	yes	—
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#88322	C₂×C₄	1	2	yes	—
59	#13133146	C₂×C₄	1	2	yes	—
60	#13133146	C₂×C₄	1	2	yes	—
61	#13133146	C₂×C₄	1	2	yes	—
62	C₄×#88322	C₂×C₄	1	1	yes	characteristic
63	C₄×#88322	C₂×C₄	1	2	yes	—
64	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
66	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₈×Dic₅₃	C₂³	1	2	yes	—
68	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
72	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
73	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	2	yes	—
76	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
77	Q₈×D₅₃	—	2	4	no	—
78	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
84	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
89	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
93	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
94	#11719919	C₂×C₈	1	2	yes	—
95	#11719919	C₂×C₈	1	2	yes	—
96	#11719919	C₂×C₈	1	2	yes	—
97	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
98	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
99	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
100	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
101	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
102	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
104	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
105	C₂×#88322	C₂²×C₄	1	2	yes	—
106	C₈×D₅₃	—	2	1	no	—
107	C₈×D₅₃	—	2	1	no	—
108	C₈×D₅₃	—	4	1	no	—
109	C₈×D₅₃	—	4	2	no	—
110	C₁₆×C₅₃	—	2	1	no	—
111	C₁₆×C₅₃	—	2	2	no	—
112	C₁₆×C₅₃	—	2	1	no	—
113	#11719908	—	2	1	no	—
114	#11719908	—	2	2	no	—
115	#11719908	—	2	1	no	—
116	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
117	Q₈×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
124	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
125	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₄×D₅₃	—	2	4	no	—
129	C₄×D₅₃	—	2	1	no	—
130	C₄×D₅₃	—	2	1	no	—
131	C₄×D₅₃	—	4	2	no	—
132	C₄×D₅₃	—	4	1	no	—
133	C₄×D₅₃	M₃₂	1	4	yes	—
134	Dic₁₀₆	—	2	2	no	—
135	Dic₁₀₆	—	2	4	no	—
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	—	2	2	no	—
138	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
139	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
140	#88322	—	2	1	no	—
141	#88322	—	2	1	no	—
142	#88322	—	2	2	no	—
143	#33497	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
145	C₂×D₅₃	—	2	1	no	—
146	C₂×D₅₃	—	2	1	no	—
147	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	2	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
152	C₄×C₅₃	#519	1	1	yes	characteristic
153	C₄×C₅₃	#568	1	1	yes	characteristic
154	Dic₅₃	—	2	2	no	—
155	Dic₅₃	—	2	2	no	—
156	Dic₅₃	—	2	1	no	—
157	Dic₅₃	—	2	1	no	—
158	Dic₅₃	C₂×M₃₂	1	2	yes	—
159	C₂×C₈×Q₈	—	53	1	no	—
160	#2183	—	53	1	no	—
161	#2183	—	53	1	no	—
162	#2183	—	53	2	no	—
163	#2183	—	53	2	no	—
164	#2183	—	53	1	no	—
165	#2183	D₅₃	1	1	yes	characteristic, complemented
166	#2163	—	53	2	no	—
167	#2163	—	53	2	no	—
168	#2162	—	53	1	no	—
169	#2162	—	53	1	no	—
170	C₂×C₅₃	#2982	1	1	yes	characteristic
171	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
172	C₂×C₅₃	#3407	1	1	yes	characteristic
173	D₅₃	—	2	2	no	—
174	C₂×C₄×Q₈	—	53	1	no	—
175	C₈×Q₈	—	53	1	no	—
176	C₈×Q₈	—	53	2	no	—
177	C₈×Q₈	—	53	1	no	—
178	C₈×Q₈	—	53	2	no	—
179	C₈×Q₈	—	53	1	no	—
180	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
181	C₂×C₄⋊C₈	—	53	1	no	—
182	C₂×C₄⋊C₈	—	53	2	no	—
183	C₂×C₄×C₈	—	53	1	no	—
184	C₂×C₄×C₈	—	53	2	no	—
185	C₄⋊C₁₆	—	53	1	no	—
186	C₄⋊C₁₆	—	53	1	no	—
187	C₄⋊C₁₆	—	53	2	no	—
188	C₄⋊C₁₆	—	53	2	no	—
189	C₄⋊C₁₆	—	53	1	no	—
190	C₄⋊C₁₆	—	53	2	no	—
191	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
192	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
193	C₂²⋊C₁₆	—	53	2	no	—
194	C₂²⋊C₁₆	—	53	1	no	—
195	C₂²⋊C₁₆	—	53	1	no	—
196	#346	—	53	2	no	—
197	#346	—	53	2	no	—
198	#346	—	53	2	no	—
199	#346	C₂×D₅₃	1	2	yes	—
200	C₄×C₁₆	—	53	2	no	—
201	C₄×C₁₆	—	53	1	no	—
202	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
203	C₅₃	#33497	1	1	yes	characteristic, complemented, Sylow
204	C₂²×Q₈	—	53	1	no	—
205	C₂²×C₈	—	53	1	no	—
206	C₂²×C₈	—	53	1	no	—
207	C₂²×C₈	—	53	2	no	—
208	C₄×Q₈	—	53	1	no	—
209	C₄×Q₈	—	53	2	no	—
210	C₄×Q₈	—	53	1	no	—
211	C₄×Q₈	—	53	2	no	—
212	C₄×Q₈	—	53	1	no	—
213	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
214	C₂×C₄⋊C₄	—	53	1	no	—
215	C₂×C₄⋊C₄	—	53	2	no	—
216	C₂×C₄²	—	53	1	no	—
217	C₂×C₄²	—	53	2	no	—
218	C₂×C₁₆	—	53	1	no	—
219	C₂×C₁₆	—	53	2	no	—
220	C₂×C₁₆	—	53	1	no	—
221	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
222	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
223	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
224	C₄⋊C₈	—	53	2	no	—
225	C₄⋊C₈	—	53	2	no	—
226	C₄⋊C₈	—	53	2	no	—
227	C₄⋊C₈	—	53	2	no	—
228	C₄⋊C₈	—	53	1	no	—
229	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
230	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
231	C₄×C₈	—	53	1	no	—
232	C₄×C₈	—	53	2	no	—
233	C₄×C₈	—	53	1	no	—
234	C₄×C₈	—	53	2	no	—
235	C₄×C₈	—	53	1	no	—
236	C₄×C₈	—	53	2	no	—
237	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₅₃	1	2	yes	—
239	C₂×Q₈	—	53	2	no	—
240	C₂×Q₈	—	53	1	no	—
241	C₂×Q₈	—	106	4	no	—
242	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
243	C₂²×C₄	—	53	1	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	53	2	no	—
248	C₂×C₈	—	53	1	no	—
249	C₂×C₈	—	53	1	no	—
250	C₂×C₈	—	53	2	no	—
251	C₂×C₈	—	106	1	no	—
252	C₂×C₈	—	106	1	no	—
253	C₂×C₈	—	212	1	no	—
254	C₂×C₈	—	212	2	no	—
255	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
257	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
258	C₄⋊C₄	—	53	2	no	—
259	C₄⋊C₄	—	53	2	no	—
260	C₄⋊C₄	—	53	2	no	—
261	C₄⋊C₄	—	53	2	no	—
262	C₄⋊C₄	—	53	1	no	—
263	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
264	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
265	C₄²	—	53	1	no	—
266	C₄²	—	53	2	no	—
267	C₄²	—	53	1	no	—
268	C₄²	—	53	2	no	—
269	C₄²	—	53	1	no	—
270	C₄²	—	53	2	no	—
271	C₄²	C₂×C₄×D₅₃	1	2	yes	—
272	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
273	C₁₆	—	2	2	no	—
274	C₁₆	—	2	1	no	—
275	C₁₆	—	2	1	no	—
276	C₁₆	—	106	1	no	—
277	C₁₆	—	106	2	no	—
278	C₁₆	—	106	1	no	—
279	C₂³	—	53	1	no	—
280	Q₈	—	2	2	no	—
281	Q₈	—	106	4	no	—
282	Q₈	—	106	2	no	—
283	C₂×C₄	—	53	1	no	—
284	C₂×C₄	—	53	4	no	—
285	C₂×C₄	—	53	1	no	—
286	C₂×C₄	—	53	2	no	—
287	C₂×C₄	—	53	1	no	—
288	C₂×C₄	—	53	2	no	—
289	C₂×C₄	—	106	4	no	—
290	C₂×C₄	—	106	1	no	—
291	C₂×C₄	—	106	1	no	—
292	C₂×C₄	—	212	1	no	—
293	C₂×C₄	—	212	2	no	—
294	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
296	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
297	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	—	106	1	no	—
302	C₈	—	106	1	no	—
303	C₈	—	106	2	no	—
304	C₈	#13133264	1	1	yes	characteristic
305	C₈	#13133265	1	1	yes	characteristic
306	C₂²	—	53	2	no	—
307	C₂²	—	106	1	no	—
308	C₂²	—	106	1	no	—
309	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	53	2	no	—
314	C₄	—	106	2	no	—
315	C₄	—	106	2	no	—
316	C₄	—	106	1	no	—
317	C₄	—	106	1	no	—
318	C₄	M₃₂×D₅₃	1	2	yes	—
319	C₄	#16811099	1	1	yes	characteristic
320	C₄	#16811388	1	1	yes	characteristic
321	C₂	—	106	2	no	—
322	C₂	#20283794	1	1	yes	characteristic
323	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
324	C₂	#20287816	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2183	C₂
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2183	D₅₃
hypocenter	C₅₃	#33497
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. #20276624	C₂	semidirect
3. #20276623	C₂	semidirect
4. #20276622	C₂	semidirect
5. #20276621	C₂	semidirect
6. #20276620	C₂	semidirect
11. C₅₃×#2183	C₂	semidirect
165. #2183	D₅₃	semidirect
203. C₅₃	#33497	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₅₃	2
#20276624	2
#20276623	2
#20276622	2
#20276621	2
#20276620	2
#20276517	2
#20276516	2
#20276513	2
#20276512	2
C₅₃×#2183	2
#33497	53