Subgroups — #27353196

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	1	yes	maximal, characteristic
2	#20283348	C₂	1	1	yes	maximal, characteristic
3	#20273503	C₂	1	4	yes	maximal, complemented
4	#20273493	C₂	1	4	yes	maximal, complemented
5	#20273465	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
7	#16810680	—	2	1	no	—
8	#16810680	C₄	1	2	yes	—
9	#16810680	C₄	1	2	yes	—
10	#16810680	C₂²	1	2	yes	—
11	#16810676	—	2	1	no	—
12	#16810676	C₂²	1	2	yes	—
13	#16810676	C₂²	1	2	yes	—
14	#16810676	C₂²	1	2	yes	—
15	#16810647	C₄	1	2	yes	—
16	#16810647	C₂²	1	2	yes	—
17	#16810587	—	2	1	no	—
18	#16810587	C₄	1	2	yes	—
19	C₄×#11719917	—	2	1	no	—
20	C₄×#11719917	C₂²	1	2	yes	—
21	#16810375	C₂²	1	8	yes	complemented
22	#16810373	C₂²	1	8	yes	complemented
23	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#431	—	2	1	no	—
25	C₅₃×#431	—	2	1	no	—
26	C₅₃×#431	C₂²	1	2	yes	—
27	C₅₃×#431	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
30	C₂×C₄×C₈×C₅₃	—	2	1	no	—
31	C₂×#11719917	—	2	2	no	—
32	C₂×#11719917	C₂×C₄	1	4	yes	—
33	C₄⋊#88322	—	2	2	no	—
34	C₄⋊#88322	C₂×C₄	1	4	yes	—
35	C₄⋊#88322	C₂×C₄	1	4	yes	—
36	C₄⋊#88322	C₂³	1	4	yes	—
37	C₂²⋊#88322	—	2	2	no	—
38	C₂²⋊#88322	—	2	2	no	—
39	C₂²⋊#88322	C₂×C₄	1	4	yes	—
40	C₂²⋊#88322	C₂×C₄	1	4	yes	—
41	#13133128	—	2	2	no	—
42	#13133128	C₂×C₄	1	4	yes	—
43	C₄×#88322	—	2	2	no	—
44	C₄×#88322	C₂³	1	4	yes	—
45	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₅₃	—	2	2	no	—
62	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
63	C₂²⋊C₈×C₅₃	—	2	2	no	—
64	C₂²⋊C₈×C₅₃	—	2	2	no	—
65	C₄×C₈×C₅₃	—	2	2	no	—
66	C₄×C₈×C₅₃	—	2	2	no	—
67	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₅₃	—	2	4	no	—
90	C₂×C₈×C₅₃	—	4	4	no	—
91	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
93	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
105	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
106	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
107	#11719917	—	2	8	no	—
108	#11719917	—	4	4	no	—
109	C₂×#88322	—	2	4	no	—
110	C₂×#88322	C₂²×C₄	1	8	yes	—
111	C₂³×C₅₃	—	2	2	no	—
112	C₂³×C₅₃	—	4	1	no	—
113	C₂³×C₅₃	—	4	1	no	—
114	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	1	no	—
132	C₂×C₄×C₅₃	—	4	1	no	—
133	C₂×C₄×C₅₃	—	4	1	no	—
134	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#131	1	4	yes	—
144	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₅₃	—	4	4	no	—
148	#88322	—	2	8	no	—
149	#88322	—	4	4	no	—
150	#17808	—	53	1	no	maximal, Sylow
151	C₂²×C₅₃	—	2	1	no	—
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	2	1	no	—
154	C₂²×C₅₃	—	2	1	no	—
155	C₂²×C₅₃	—	2	1	no	—
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	4	4	no	—
160	C₂²×C₅₃	#424	1	2	yes	—
161	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	—	2	1	no	—
164	C₄×C₅₃	—	2	1	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	#424	1	2	yes	—
171	C₄×C₅₃	#430	1	2	yes	—
172	C₄×C₅₃	C₂×#131	1	2	yes	—
173	C₄×C₅₃	#575	1	2	yes	—
174	#2920	—	53	1	no	—
175	#2920	—	53	1	no	—
176	C₂×#431	D₅₃	1	1	yes	characteristic
177	#1574	—	53	4	no	—
178	#1572	—	53	4	no	—
179	#1565	—	53	4	no	—
180	C₂×C₅₃	—	2	2	no	—
181	C₂×C₅₃	—	4	1	no	—
182	C₂×C₅₃	—	4	1	no	—
183	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
184	C₂×C₅₃	#2911	1	1	yes	characteristic
185	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
186	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
187	#432	—	53	2	no	—
188	#432	—	53	2	no	—
189	#432	—	53	2	no	—
190	#432	—	106	1	no	—
191	#431	—	2	1	no	—
192	#431	—	2	1	no	—
193	#431	—	53	2	no	—
194	#431	—	53	2	no	—
195	#431	—	53	2	no	—
196	#431	—	106	1	no	—
197	#431	C₂×D₅₃	1	2	yes	—
198	#431	C₂×D₅₃	1	2	yes	—
199	#423	—	53	2	no	—
200	#423	—	53	2	no	—
201	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
202	#407	—	53	2	no	—
203	#407	—	106	1	no	—
204	C₂×C₂²⋊C₈	—	2	1	no	—
205	C₄×M₁₆	—	53	2	no	—
206	C₄×M₁₆	—	106	1	no	—
207	C₂×C₄×C₈	—	2	1	no	—
208	#335	—	53	8	no	—
209	#334	—	53	8	no	—
210	C₅₃	#17808	1	1	yes	characteristic, complemented, Sylow
211	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
212	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
213	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
214	C₂×M₁₆	—	53	4	no	—
215	C₂×M₁₆	—	106	2	no	—
216	C₂²×C₈	—	2	2	no	—
217	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
218	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
219	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
220	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
221	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
222	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
227	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	53	4	no	—
231	C₄⋊C₈	—	53	4	no	—
232	C₄⋊C₈	—	53	4	no	—
233	C₄⋊C₈	—	106	2	no	—
234	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
235	C₂²⋊C₈	—	2	2	no	—
236	C₂²⋊C₈	—	2	2	no	—
237	C₂²⋊C₈	—	53	4	no	—
238	C₂²⋊C₈	—	53	4	no	—
239	C₂²⋊C₈	—	106	2	no	—
240	C₂²⋊C₈	—	106	2	no	—
241	#97	—	53	4	no	—
242	#97	—	106	2	no	—
243	C₄×C₈	—	2	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	53	4	no	—
246	C₄×C₈	—	106	2	no	—
247	C₂⁴	#11719919	1	1	yes	characteristic
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	#11719919	1	1	yes	characteristic
252	C₂²×C₄	#11719919	1	1	yes	characteristic
253	C₂²×C₄	#11719919	1	1	yes	characteristic
254	C₂²×C₄	#11719919	1	1	yes	characteristic
255	C₂²×C₄	#11719919	1	1	yes	characteristic
256	C₂²×C₄	#11719919	1	1	yes	characteristic
257	C₂²×C₄	#11719919	1	1	yes	characteristic
258	C₂²×C₄	#11719919	1	1	yes	characteristic
259	C₂²×C₄	#11719919	1	1	yes	characteristic
260	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
261	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
262	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
269	M₁₆	—	106	8	no	—
270	M₁₆	—	212	4	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	4	4	no	—
273	C₂×C₈	—	53	8	no	—
274	C₂×C₈	—	106	4	no	—
275	C₄²	#11719919	1	1	yes	characteristic
276	C₄²	#11719919	1	1	yes	characteristic
277	C₄²	#11719919	1	1	yes	characteristic
278	C₄²	#11719919	1	1	yes	characteristic
279	C₄²	#11719919	1	1	yes	characteristic
280	C₄²	#11719919	1	1	yes	characteristic
281	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
282	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
283	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
284	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
286	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
289	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
290	C₄²	C₂³×D₅₃	1	1	yes	characteristic
291	C₂³	—	2	2	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	4	1	no	—
294	C₂³	C₂×#11719919	1	1	yes	characteristic
295	C₂³	C₂×#11719919	1	1	yes	characteristic
296	C₂³	C₂×#11719919	1	1	yes	characteristic
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
315	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
316	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
317	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
318	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
319	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
320	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
321	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
322	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
323	C₂×C₄	#13133223	1	4	yes	—
324	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
326	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
327	C₈	—	4	4	no	—
328	C₈	—	106	8	no	—
329	C₈	—	212	4	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	4	no	—
339	C₂²	#16810649	1	2	yes	—
340	C₂²	C₂²×#11719919	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	#16810649	1	2	yes	—
350	C₄	#16810671	1	2	yes	—
351	C₄	C₂×#13133223	1	2	yes	—
352	C₄	#16811421	1	2	yes	—
353	C₂	—	2	2	no	—
354	C₂	—	4	1	no	—
355	C₂	—	4	1	no	—
356	C₂	C₂×#16810649	1	1	yes	characteristic
357	C₂	#20283304	1	1	yes	characteristic
358	C₂	C₂×#16810671	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#17808
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20273503	C₂	semidirect
4. #20273493	C₂	semidirect
5. #20273465	C₂	semidirect
21. #16810375	C₂²	semidirect
22. #16810373	C₂²	semidirect
210. C₅₃	#17808	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
#20273503	2
#20273493	2
#20273465	2
C₂×C₅₃×#431	2
#17808	53