Subgroups — #27290022

Order 13568.

774 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133149	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810391	C₂	1	2	yes	maximal
4	#20274436	C₂	1	2	yes	maximal
5	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	C₂×#13133153	—	2	2	no	—
9	C₂×#13133153	—	2	2	no	—
10	C₂×#13133153	C₄	1	4	yes	complemented
11	C₂×#13133149	—	2	2	no	—
12	C₂×#13133149	—	2	1	no	—
13	C₂×#13133149	—	2	1	no	—
14	C₂×#13133149	C₄	1	4	yes	complemented
15	#16810391	—	2	4	no	—
16	#16810385	—	2	4	no	—
17	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#405	—	2	2	no	—
21	C₅₃×#405	C₄	1	4	yes	complemented
22	C₂×#97×C₅₃	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	—	2	1	no	—
24	C₂³×Dic₅₃	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	2	no	—
29	C₂×#11719917	—	2	2	no	—
30	C₂×#11719917	C₂×C₄	1	4	yes	—
31	C₂²×#88322	—	2	2	no	—
32	C₂×C₄⋊Dic₅₃	—	2	2	no	—
33	C₂×C₄⋊Dic₅₃	—	2	2	no	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×C₄⋊Dic₅₃	—	2	2	no	—
36	C₂×C₄⋊Dic₅₃	—	2	2	no	—
37	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
38	#13133153	—	2	8	no	—
39	#13133149	—	2	4	no	—
40	#13133149	—	2	4	no	—
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₅₃	—	2	1	no	—
43	C₂×M₁₆×C₅₃	—	2	1	no	—
44	C₂×M₁₆×C₅₃	—	2	1	no	—
45	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
47	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
48	C₂²×C₈×C₅₃	—	2	1	no	—
49	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₅₃	D₄	1	2	yes	—
53	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	D₄	1	2	yes	—
56	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
57	#117×C₅₃	—	2	1	no	—
58	#117×C₅₃	—	2	1	no	—
59	#117×C₅₃	C₂×C₄	1	2	yes	—
60	#117×C₅₃	Q₈	1	2	yes	—
61	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
63	C₂×C₄²×C₅₃	—	2	1	no	—
64	#97×C₅₃	—	2	4	no	—
65	C₄×C₈×C₅₃	—	2	4	no	—
66	C₂⁴×C₅₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
82	C₂²×Dic₅₃	—	2	4	no	—
83	C₂²×Dic₅₃	—	4	2	no	—
84	C₂²×Dic₅₃	—	4	2	no	—
85	C₂²×Dic₅₃	—	8	4	no	—
86	M₁₆×C₅₃	—	2	2	no	—
87	M₁₆×C₅₃	—	2	2	no	—
88	M₁₆×C₅₃	—	2	2	no	—
89	M₁₆×C₅₃	C₄⋊C₄	1	4	yes	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	C₄²	1	4	yes	—
94	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
97	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
100	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
101	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
103	C₄⋊C₄×C₅₃	—	2	2	no	—
104	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
105	C₂²⋊C₄×C₅₃	—	2	2	no	—
106	C₂²⋊C₄×C₅₃	—	2	2	no	—
107	C₄²×C₅₃	—	2	2	no	—
108	C₄²×C₅₃	—	2	2	no	—
109	C₄⋊Dic₅₃	—	2	4	no	—
110	C₄⋊Dic₅₃	—	2	4	no	—
111	C₄⋊Dic₅₃	—	4	2	no	—
112	C₄⋊Dic₅₃	—	4	2	no	—
113	#11719917	—	2	4	no	—
114	#11719917	—	2	4	no	—
115	#11719917	—	4	2	no	—
116	#11719917	—	4	2	no	—
117	C₂×#88322	—	2	4	no	—
118	C₂×#88322	—	4	2	no	—
119	C₂×#88322	—	4	2	no	—
120	C₂³×C₅₃	—	2	4	no	—
121	C₂³×C₅₃	#117	1	1	yes	characteristic
122	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
124	C₂³×C₅₃	#121	1	1	yes	characteristic
125	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂³×C₅₃	#124	1	1	yes	characteristic
127	C₂³×C₅₃	#125	1	1	yes	characteristic
128	C₂×C₄×C₅₃	—	2	4	no	—
129	C₂×C₄×C₅₃	—	4	4	no	—
130	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#117	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
154	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
158	C₂×Dic₅₃	—	4	4	no	—
159	C₂×Dic₅₃	—	8	4	no	—
160	C₂×Dic₅₃	—	8	4	no	—
161	C₂×Dic₅₃	—	8	4	no	—
162	C₈×C₅₃	—	2	4	no	—
163	C₈×C₅₃	#95	1	4	yes	—
164	C₈×C₅₃	#95	1	4	yes	—
165	#88322	—	4	4	no	—
166	#10869	—	53	1	no	maximal, Sylow
167	C₂²×C₅₃	—	2	2	no	—
168	C₂²×C₅₃	—	2	2	no	—
169	C₂²×C₅₃	—	2	2	no	—
170	C₂²×C₅₃	—	2	2	no	—
171	C₂²×C₅₃	—	2	2	no	—
172	C₂²×C₅₃	—	2	2	no	—
173	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
174	C₂²×C₅₃	#380	1	1	yes	characteristic
175	C₂²×C₅₃	#382	1	1	yes	characteristic
176	C₂²×C₅₃	#389	1	1	yes	characteristic
177	C₂²×C₅₃	#389	1	1	yes	characteristic
178	C₂²×C₅₃	#390	1	1	yes	characteristic
179	C₂²×C₅₃	#391	1	1	yes	characteristic
180	C₂²×C₅₃	#471	1	4	yes	—
181	C₄×C₅₃	—	4	4	no	—
182	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
183	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
184	C₄×C₅₃	#380	1	1	yes	characteristic
185	C₄×C₅₃	#384	1	1	yes	characteristic
186	C₄×C₅₃	#389	1	1	yes	characteristic
187	C₄×C₅₃	#389	1	1	yes	characteristic
188	C₄×C₅₃	#390	1	1	yes	characteristic
189	C₄×C₅₃	#391	1	1	yes	characteristic
190	Dic₅₃	—	8	4	no	—
191	C₂²×#108	—	53	1	no	—
192	C₂²×#106	—	53	1	no	—
193	C₂×#405	D₅₃	1	1	yes	characteristic
194	C₂×#343	—	53	2	no	—
195	#1736	—	53	2	no	—
196	C₂×C₅₃	—	2	1	no	—
197	C₂×C₅₃	—	2	1	no	—
198	C₂×C₅₃	—	2	1	no	—
199	C₂×C₅₃	—	2	1	no	—
200	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
201	C₂×C₅₃	#1815	1	2	yes	—
202	C₂×C₅₃	#1951	1	2	yes	—
203	C₂×C₅₃	#1977	1	2	yes	—
204	C₂²×M₁₆	—	53	2	no	—
205	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
206	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
207	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
208	C₂²×C₄⋊C₄	—	53	2	no	—
209	C₂×#108	—	53	4	no	—
210	C₂×#108	—	106	2	no	—
211	C₂×#108	—	106	2	no	—
212	C₂×#106	—	53	4	no	—
213	C₂×#106	—	106	2	no	—
214	C₂×#106	—	106	1	no	—
215	C₂×#106	—	106	1	no	—
216	#405	—	2	2	no	—
217	#405	Dic₅₃	1	4	yes	complemented
218	C₂×#97	—	2	1	no	—
219	C₂×C₄×C₈	—	2	1	no	—
220	#343	—	106	4	no	—
221	#340	—	106	4	no	—
222	C₅₃	#10869	1	1	yes	characteristic, complemented, Sylow
223	C₂³×C₄	—	106	2	no	—
224	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
225	C₂×M₁₆	—	2	1	no	—
226	C₂×M₁₆	—	2	1	no	—
227	C₂×M₁₆	—	2	1	no	—
228	C₂×M₁₆	—	53	4	no	—
229	C₂×M₁₆	—	106	2	no	—
230	C₂×M₁₆	—	106	2	no	—
231	C₂×M₁₆	—	106	2	no	—
232	C₂×M₁₆	—	106	2	no	—
233	C₂×M₁₆	—	106	2	no	—
234	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
235	C₂×M₁₆	D₂₁₂	1	2	yes	—
236	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	—	106	2	no	—
239	C₂²×C₈	Dic₁₀₆	1	2	yes	—
240	C₂²×C₈	C₄×D₅₃	1	2	yes	—
241	C₂²×C₈	C₄×D₅₃	1	2	yes	—
242	C₂²×C₈	D₂₁₂	1	2	yes	—
243	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
244	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
245	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
246	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
247	#117	—	2	1	no	—
248	#117	—	2	1	no	—
249	#117	Dic₁₀₆	1	2	yes	—
250	#117	C₂×Dic₅₃	1	2	yes	—
251	C₂×C₄⋊C₄	—	53	4	no	—
252	C₂×C₄⋊C₄	—	106	2	no	—
253	C₂×C₄⋊C₄	—	106	2	no	—
254	C₂×C₄⋊C₄	—	106	2	no	—
255	C₂×C₄⋊C₄	—	106	2	no	—
256	C₂×C₄⋊C₄	—	106	2	no	—
257	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
258	C₂×C₂²⋊C₄	—	2	1	no	—
259	C₂×C₄²	—	2	1	no	—
260	#108	—	106	8	no	—
261	#106	—	106	4	no	—
262	#106	—	106	4	no	—
263	#97	—	2	4	no	—
264	C₄×C₈	—	2	4	no	—
265	C₂⁴	#11719945	1	1	yes	characteristic
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	106	4	no	—
268	C₂²×C₄	—	424	4	no	—
269	C₂²×C₄	—	212	2	no	—
270	C₂²×C₄	—	212	2	no	—
271	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
272	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
273	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
279	C₂²×C₄	#11719945	1	1	yes	characteristic
280	C₂²×C₄	#11719945	1	1	yes	characteristic
281	C₂²×C₄	#11719945	1	1	yes	characteristic
282	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	106	4	no	—
289	M₁₆	—	106	4	no	—
290	M₁₆	—	212	2	no	—
291	M₁₆	—	212	2	no	—
292	M₁₆	C₄⋊Dic₅₃	1	4	yes	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	106	4	no	—
297	C₂×C₈	—	212	2	no	—
298	C₂×C₈	—	212	2	no	—
299	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
300	C₂×C₈	#11719919	1	4	yes	—
301	C₂×C₈	#11719919	1	2	yes	—
302	C₂×C₈	#11719919	1	2	yes	—
303	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
304	C₂×C₈	#11719921	1	2	yes	—
305	C₂×C₈	#11719921	1	2	yes	—
306	C₂×C₈	#11719921	1	4	yes	—
307	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
308	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	106	4	no	—
311	C₄⋊C₄	—	106	4	no	—
312	C₄⋊C₄	—	212	2	no	—
313	C₄⋊C₄	—	212	2	no	—
314	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	—	2	2	no	—
319	C₂³	—	2	4	no	—
320	C₂³	#13133176	1	1	yes	characteristic
321	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
322	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
323	C₂³	#13133191	1	1	yes	characteristic
324	C₂³	#13133196	1	1	yes	characteristic
325	C₂³	#13133205	1	1	yes	characteristic
326	C₂³	#13133209	1	1	yes	characteristic
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	4	4	no	—
329	C₂×C₄	—	424	4	no	—
330	C₂×C₄	—	424	4	no	—
331	C₂×C₄	—	424	4	no	—
332	C₂×C₄	—	212	4	no	—
333	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
335	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
336	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
338	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
341	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
342	C₂×C₄	#13133176	1	1	yes	characteristic
343	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
344	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
345	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
346	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
347	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
348	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
349	C₂×C₄	#13133191	1	1	yes	characteristic
350	C₂×C₄	#13133191	1	1	yes	characteristic
351	C₂×C₄	#13133191	1	1	yes	characteristic
352	C₂×C₄	#13133196	1	1	yes	characteristic
353	C₂×C₄	#13133196	1	1	yes	characteristic
354	C₂×C₄	#13133196	1	1	yes	characteristic
355	C₂×C₄	#13133205	1	1	yes	characteristic
356	C₂×C₄	#13133209	1	1	yes	characteristic
357	C₂×C₄	#13133214	1	1	yes	characteristic
358	C₂×C₄	#13133217	1	1	yes	characteristic
359	C₂×C₄	#13133217	1	1	yes	characteristic
360	C₂×C₄	#13133217	1	1	yes	characteristic
361	C₈	—	2	4	no	—
362	C₈	—	212	4	no	—
363	C₈	#13133124	1	4	yes	—
364	C₈	#13133124	1	4	yes	—
365	C₂²	—	2	2	no	—
366	C₂²	—	2	2	no	—
367	C₂²	—	2	2	no	—
368	C₂²	—	2	2	no	—
369	C₂²	—	2	2	no	—
370	C₂²	—	2	2	no	—
371	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
372	C₂²	#16810483	1	1	yes	characteristic
373	C₂²	#16810489	1	1	yes	characteristic
374	C₂²	#16810521	1	1	yes	characteristic
375	C₂²	#16810521	1	1	yes	characteristic
376	C₂²	#16810525	1	1	yes	characteristic
377	C₂²	#16810529	1	1	yes	characteristic
378	C₂²	#16810886	1	4	yes	—
379	C₄	—	4	4	no	—
380	C₄	—	424	4	no	—
381	C₄	C₂×#13133124	1	1	yes	characteristic
382	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
383	C₄	#16810483	1	1	yes	characteristic
384	C₄	#16810493	1	1	yes	characteristic
385	C₄	#16810521	1	1	yes	characteristic
386	C₄	#16810521	1	1	yes	characteristic
387	C₄	#16810525	1	1	yes	characteristic
388	C₄	#16810529	1	1	yes	characteristic
389	C₂	—	2	1	no	—
390	C₂	—	2	1	no	—
391	C₂	—	2	1	no	—
392	C₂	—	2	1	no	—
393	C₂	#20272945	1	1	yes	characteristic
394	C₂	#20274774	1	2	yes	—
395	C₂	#20275495	1	2	yes	—
396	C₂	#20275618	1	2	yes	—

	subgroup	G/H
center	C₂³	#13133191
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#10869
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
10. C₂×#13133153	C₄	semidirect
14. C₂×#13133149	C₄	semidirect
21. C₅₃×#405	C₄	semidirect
217. #405	Dic₅₃	semidirect
222. C₅₃	#10869	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
C₂²×#13133149	2
C₂×#16810391	2
#20274436	2
C₂×C₅₃×#405	2
#10869	53