Subgroups — #27289978

Order 13568.

822 conjugacy classes of subgroups.

The subgroup lattice has 394 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	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810383	C₂	1	2	yes	maximal
4	#20274411	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂×#13133176	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	#16810661	—	2	2	no	—
12	#16810661	—	2	2	no	—
13	C₂×#13133151	C₄	1	2	yes	complemented
14	C₂×#13133149	C₄	1	2	yes	complemented
15	#16810383	—	2	4	no	—
16	#16810377	—	2	4	no	—
17	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	—	2	2	no	—
21	C₂×C₄×C₈×C₅₃	—	2	2	no	—
22	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	complemented
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	C₂×C₄	1	4	yes	—
29	C₂²×#88322	—	2	2	no	—
30	#13133176	—	2	2	no	—
31	#13133176	—	2	2	no	—
32	#13133176	—	2	2	no	—
33	#13133176	—	2	2	no	—
34	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
36	C₂×C₄×Dic₅₃	—	2	2	no	—
37	#13133153	—	2	8	no	—
38	#13133151	—	2	4	no	—
39	#13133149	—	2	4	no	—
40	C₂³×C₄×C₅₃	—	2	1	no	—
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	—	2	1	no	—
43	C₂²×C₈×C₅₃	—	2	1	no	—
44	C₂²×C₈×C₅₃	—	2	1	no	—
45	C₂²×C₈×C₅₃	—	2	1	no	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₅₃	D₄	1	2	yes	—
52	C₂²×C₈×C₅₃	D₄	1	2	yes	—
53	C₂²×C₈×C₅₃	D₄	1	2	yes	—
54	C₂²×C₈×C₅₃	D₄	1	2	yes	—
55	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
56	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
57	C₂×C₄²×C₅₃	—	2	1	no	—
58	C₂×C₄²×C₅₃	—	2	1	no	—
59	C₂×C₄²×C₅₃	—	2	1	no	—
60	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₅₃	D₄	1	2	yes	—
62	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
63	C₄×C₈×C₅₃	—	2	8	no	—
64	C₂⁴×C₅₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	8	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²×C₄	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×D₄	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	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
83	C₂²×Dic₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	—	2	2	no	—
85	C₂×C₈×C₅₃	—	2	2	no	—
86	C₂×C₈×C₅₃	—	2	2	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	C₄²	1	2	yes	—
91	C₂×C₈×C₅₃	C₄²	1	2	yes	—
92	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
93	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
94	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
95	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
96	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
97	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
99	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
100	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
101	C₄²×C₅₃	—	2	2	no	—
102	C₄²×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
105	C₂²⋊Dic₅₃	—	4	2	no	—
106	C₂²⋊Dic₅₃	—	4	2	no	—
107	C₄⋊Dic₅₃	—	2	4	no	—
108	C₄⋊Dic₅₃	—	2	4	no	—
109	C₄×Dic₅₃	—	4	2	no	—
110	C₄×Dic₅₃	—	4	2	no	—
111	#11719917	—	2	4	no	—
112	#11719917	—	2	4	no	—
113	#11719917	—	4	2	no	—
114	#11719917	—	4	2	no	—
115	C₂×#88322	—	2	4	no	—
116	C₂×#88322	—	4	2	no	—
117	C₂×#88322	—	4	2	no	—
118	C₂³×C₅₃	—	2	4	no	—
119	C₂³×C₅₃	#117	1	1	yes	characteristic
120	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
122	C₂³×C₅₃	#121	1	1	yes	characteristic
123	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂³×C₅₃	#124	1	1	yes	characteristic
125	C₂³×C₅₃	#125	1	1	yes	characteristic
126	C₂×C₄×C₅₃	—	2	8	no	—
127	C₂×C₄×C₅₃	—	2	8	no	—
128	C₂×C₄×C₅₃	—	2	8	no	—
129	C₂×C₄×C₅₃	—	2	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	—	4	8	no	—
160	C₈×C₅₃	—	2	4	no	—
161	C₈×C₅₃	#95	1	4	yes	—
162	C₈×C₅₃	#95	1	4	yes	—
163	#88322	—	4	4	no	—
164	#10859	—	53	1	no	maximal, Sylow
165	C₂²×C₅₃	—	2	2	no	—
166	C₂²×C₅₃	—	2	2	no	—
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	C₄×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₅₃	#380	1	1	yes	characteristic
173	C₂²×C₅₃	#382	1	1	yes	characteristic
174	C₂²×C₅₃	#389	1	1	yes	characteristic
175	C₂²×C₅₃	#389	1	1	yes	characteristic
176	C₂²×C₅₃	#390	1	1	yes	characteristic
177	C₂²×C₅₃	#391	1	1	yes	characteristic
178	C₂²×C₅₃	#443	1	4	yes	—
179	C₄×C₅₃	—	2	8	no	—
180	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
181	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
182	C₄×C₅₃	#380	1	1	yes	characteristic
183	C₄×C₅₃	#384	1	1	yes	characteristic
184	C₄×C₅₃	#389	1	1	yes	characteristic
185	C₄×C₅₃	#389	1	1	yes	characteristic
186	C₄×C₅₃	#390	1	1	yes	characteristic
187	C₄×C₅₃	#391	1	1	yes	characteristic
188	Dic₅₃	—	8	4	no	—
189	C₂²×#108	—	53	1	no	—
190	C₂×#427	—	53	1	no	—
191	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
192	C₂×#339	—	53	2	no	—
193	#1729	—	53	2	no	—
194	C₂×C₅₃	—	2	1	no	—
195	C₂×C₅₃	—	2	1	no	—
196	C₂×C₅₃	—	2	1	no	—
197	C₂×C₅₃	—	2	1	no	—
198	C₂×C₅₃	C₄⋊#95	1	1	yes	characteristic
199	C₂×C₅₃	#1772	1	2	yes	—
200	C₂×C₅₃	#1945	1	2	yes	—
201	C₂×C₅₃	#1974	1	2	yes	—
202	C₂²×M₁₆	—	53	2	no	—
203	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
204	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
205	C₂×#117	—	53	2	no	—
206	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
207	C₂×#108	—	53	4	no	—
208	C₂×#108	—	106	2	no	—
209	C₂×#108	—	106	2	no	—
210	#427	—	106	2	no	—
211	#427	—	106	2	no	—
212	C₂×#107	—	53	2	no	—
213	C₂×#106	—	53	2	no	—
214	C₂×C₄×C₈	—	2	2	no	—
215	C₂×C₄×C₈	—	2	2	no	—
216	C₂×C₄×C₈	Dic₅₃	1	4	yes	complemented
217	#339	—	106	4	no	—
218	#336	—	106	4	no	—
219	C₅₃	#10859	1	1	yes	characteristic, complemented, Sylow
220	C₂³×C₄	—	2	1	no	—
221	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
222	C₂×M₁₆	—	53	4	no	—
223	C₂×M₁₆	—	106	2	no	—
224	C₂×M₁₆	—	106	2	no	—
225	C₂×M₁₆	—	106	2	no	—
226	C₂×M₁₆	—	106	2	no	—
227	C₂×M₁₆	—	106	2	no	—
228	C₂²×C₈	—	2	1	no	—
229	C₂²×C₈	—	2	1	no	—
230	C₂²×C₈	—	2	1	no	—
231	C₂²×C₈	—	2	1	no	—
232	C₂²×C₈	—	106	2	no	—
233	C₂²×C₈	Dic₁₀₆	1	2	yes	—
234	C₂²×C₈	Dic₁₀₆	1	2	yes	—
235	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
236	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
237	C₂²×C₈	C₄×D₅₃	1	2	yes	—
238	C₂²×C₈	D₂₁₂	1	2	yes	—
239	C₂²×C₈	D₂₁₂	1	2	yes	—
240	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
241	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
242	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
243	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
244	#117	—	106	2	no	—
245	#117	—	106	2	no	—
246	#117	—	106	2	no	—
247	#117	—	106	2	no	—
248	C₂×C₄⋊C₄	—	53	4	no	—
249	C₂×C₂²⋊C₄	—	106	2	no	—
250	C₂×C₄²	—	2	1	no	—
251	C₂×C₄²	—	2	1	no	—
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	—	106	2	no	—
254	C₂×C₄²	Dic₁₀₆	1	2	yes	—
255	C₂×C₄²	D₂₁₂	1	2	yes	—
256	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
257	#108	—	106	8	no	—
258	#107	—	106	4	no	—
259	#106	—	106	4	no	—
260	C₄×C₈	—	2	8	no	—
261	C₂⁴	#11719945	1	1	yes	characteristic
262	C₂²×C₄	—	2	8	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	106	4	no	—
267	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
268	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
269	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
274	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
275	C₂²×C₄	#11719945	1	1	yes	characteristic
276	C₂²×C₄	#11719945	1	1	yes	characteristic
277	C₂²×C₄	#11719945	1	1	yes	characteristic
278	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
281	M₁₆	—	106	4	no	—
282	M₁₆	—	106	4	no	—
283	M₁₆	—	212	2	no	—
284	M₁₆	—	212	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	106	4	no	—
292	C₂×C₈	—	212	2	no	—
293	C₂×C₈	—	212	2	no	—
294	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
295	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
296	C₂×C₈	#11719919	1	2	yes	—
297	C₂×C₈	#11719919	1	2	yes	—
298	C₂×C₈	#11719919	1	4	yes	—
299	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
300	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
301	C₂×C₈	#11719921	1	2	yes	—
302	C₂×C₈	#11719921	1	2	yes	—
303	C₂×C₈	#11719921	1	4	yes	—
304	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
305	C₄⋊C₄	—	106	4	no	—
306	C₄⋊C₄	—	106	4	no	—
307	C₂²⋊C₄	—	212	2	no	—
308	C₂²⋊C₄	—	212	2	no	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	212	2	no	—
313	C₄²	—	212	2	no	—
314	C₄²	C₄⋊Dic₅₃	1	4	yes	—
315	C₂³	—	2	4	no	—
316	C₂³	#13133176	1	1	yes	characteristic
317	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
318	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
319	C₂³	#13133191	1	1	yes	characteristic
320	C₂³	#13133196	1	1	yes	characteristic
321	C₂³	#13133205	1	1	yes	characteristic
322	C₂³	#13133209	1	1	yes	characteristic
323	C₂×C₄	—	2	8	no	—
324	C₂×C₄	—	2	8	no	—
325	C₂×C₄	—	2	8	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	212	4	no	—
328	C₂×C₄	—	212	8	no	—
329	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
330	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
331	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
332	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
334	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
335	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
338	C₂×C₄	#13133176	1	1	yes	characteristic
339	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
340	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
341	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
342	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
343	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
344	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
345	C₂×C₄	#13133191	1	1	yes	characteristic
346	C₂×C₄	#13133191	1	1	yes	characteristic
347	C₂×C₄	#13133191	1	1	yes	characteristic
348	C₂×C₄	#13133196	1	1	yes	characteristic
349	C₂×C₄	#13133196	1	1	yes	characteristic
350	C₂×C₄	#13133196	1	1	yes	characteristic
351	C₂×C₄	#13133205	1	1	yes	characteristic
352	C₂×C₄	#13133209	1	1	yes	characteristic
353	C₂×C₄	#13133214	1	1	yes	characteristic
354	C₂×C₄	#13133217	1	1	yes	characteristic
355	C₂×C₄	#13133217	1	1	yes	characteristic
356	C₂×C₄	#13133217	1	1	yes	characteristic
357	C₈	—	2	4	no	—
358	C₈	—	212	4	no	—
359	C₈	#13133124	1	4	yes	—
360	C₈	#13133124	1	4	yes	—
361	C₂²	—	2	2	no	—
362	C₂²	—	2	2	no	—
363	C₂²	—	2	2	no	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	2	2	no	—
367	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
368	C₂²	#16810483	1	1	yes	characteristic
369	C₂²	#16810489	1	1	yes	characteristic
370	C₂²	#16810521	1	1	yes	characteristic
371	C₂²	#16810521	1	1	yes	characteristic
372	C₂²	#16810525	1	1	yes	characteristic
373	C₂²	#16810529	1	1	yes	characteristic
374	C₂²	#16810741	1	4	yes	—
375	C₄	—	2	8	no	—
376	C₄	—	424	4	no	—
377	C₄	C₂×#13133124	1	1	yes	characteristic
378	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
379	C₄	#16810483	1	1	yes	characteristic
380	C₄	#16810493	1	1	yes	characteristic
381	C₄	#16810521	1	1	yes	characteristic
382	C₄	#16810521	1	1	yes	characteristic
383	C₄	#16810525	1	1	yes	characteristic
384	C₄	#16810529	1	1	yes	characteristic
385	C₂	—	2	1	no	—
386	C₂	—	2	1	no	—
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	#20272945	1	1	yes	characteristic
390	C₂	#20274594	1	2	yes	—
391	C₂	#20275463	1	2	yes	—
392	C₂	#20275603	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#10859
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
10. C₂×#13133153	C₄	semidirect
13. C₂×#13133151	C₄	semidirect
14. C₂×#13133149	C₄	semidirect
22. C₂×C₄×C₈×C₅₃	C₄	semidirect
216. C₂×C₄×C₈	Dic₅₃	semidirect
219. C₅₃	#10859	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
C₂×#16810661	2
C₂×#16810383	2
#20274411	2
C₂²×C₄×C₈×C₅₃	2
#10859	53