Subgroups — #27274408

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 392 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	C₂×C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272617	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
6	C₂²×C₄×Dic₅₃	—	2	1	no	—
7	#16810680	—	2	1	no	—
8	#16810680	C₄	1	2	yes	complemented
9	#16810676	—	2	1	no	—
10	#16810676	C₄	1	2	yes	complemented
11	#16810647	C₂²	1	2	yes	—
12	#16810587	—	2	1	no	—
13	C₄×#11719917	—	2	1	no	—
14	C₄×C₄⋊Dic₅₃	—	2	1	no	—
15	C₄×C₄⋊Dic₅₃	—	2	1	no	—
16	C₄×C₄⋊Dic₅₃	C₄	1	2	yes	—
17	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
18	C₂×#13133124	C₄	1	2	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×#432	—	2	1	no	—
21	C₅₃×#432	C₄	1	2	yes	complemented
22	C₅₃×#431	—	2	1	no	—
23	C₅₃×#431	C₂²	1	2	yes	—
24	C₅₃×#423	C₄	1	2	yes	complemented
25	C₅₃×#407	—	2	1	no	—
26	C₄×M₁₆×C₅₃	—	2	1	no	—
27	C₂³×Dic₅₃	—	2	2	no	—
28	C₂×#11719917	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	—	2	8	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×C₄×Dic₅₃	—	2	2	no	—
33	C₂×C₄×Dic₅₃	—	2	2	no	—
34	C₂×C₄×Dic₅₃	—	4	4	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	C₂×C₄	1	4	yes	complemented
37	C₂²⋊#88322	—	2	2	no	—
38	C₂²⋊#88322	—	2	2	no	—
39	#13133128	—	2	2	no	—
40	C₄×#88322	—	2	2	no	—
41	#13133124	—	2	2	no	—
42	#13133124	C₂×C₄	1	4	yes	—
43	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
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	D₄	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	C₄⋊C₈×C₅₃	—	2	2	no	—
60	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	complemented
61	C₂²⋊C₈×C₅₃	—	2	2	no	—
62	C₂²⋊C₈×C₅₃	—	2	2	no	—
63	#97×C₅₃	—	2	2	no	—
64	C₄×C₈×C₅₃	—	2	2	no	—
65	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	—	2	4	no	—
67	C₂²×C₄×C₅₃	—	2	2	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×Dic₅₃	—	2	8	no	—
89	C₂²×Dic₅₃	—	2	4	no	—
90	C₂²×Dic₅₃	—	4	4	no	—
91	C₂²×Dic₅₃	—	4	4	no	—
92	M₁₆×C₅₃	—	4	4	no	—
93	C₂×C₈×C₅₃	—	2	4	no	—
94	C₄²×C₅₃	C₄²	1	1	yes	characteristic
95	C₄²×C₅₃	C₄²	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
110	C₄⋊Dic₅₃	—	2	8	no	—
111	C₄⋊Dic₅₃	—	2	8	no	—
112	C₄×Dic₅₃	—	4	4	no	—
113	C₄×Dic₅₃	—	4	4	no	—
114	#11719917	—	4	2	no	—
115	#11719917	—	4	2	no	—
116	C₂×#88322	—	2	4	no	—
117	C₂³×C₅₃	—	2	2	no	—
118	C₂³×C₅₃	—	2	2	no	—
119	C₂³×C₅₃	—	2	1	no	—
120	C₂³×C₅₃	—	2	1	no	—
121	C₂³×C₅₃	#95	1	1	yes	characteristic
122	C₂³×C₅₃	#95	1	1	yes	characteristic
123	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	1	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	2	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	—	4	1	no	—
139	C₂×C₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#104	1	4	yes	—
148	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
155	C₂×C₄×C₅₃	#137	1	2	yes	—
156	C₂×Dic₅₃	—	2	8	no	—
157	C₂×Dic₅₃	—	4	16	no	—
158	C₂×Dic₅₃	—	4	4	no	—
159	C₂×Dic₅₃	—	4	4	no	—
160	C₈×C₅₃	—	4	4	no	—
161	#88322	—	4	4	no	—
162	#8098	—	53	1	no	maximal, Sylow
163	C₂²×C₅₃	—	2	4	no	—
164	C₂²×C₅₃	—	2	4	no	—
165	C₂²×C₅₃	—	2	1	no	—
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	2	1	no	—
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	#339	1	2	yes	—
172	C₂²×C₅₃	#344	1	2	yes	—
173	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
174	C₂²×C₅₃	#418	1	1	yes	characteristic
175	C₂²×C₅₃	#419	1	1	yes	characteristic
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	#339	1	2	yes	—
185	C₄×C₅₃	#417	1	2	yes	—
186	C₄×C₅₃	C₂×#104	1	2	yes	—
187	C₄×C₅₃	#428	1	2	yes	—
188	Dic₅₃	—	4	8	no	—
189	#2920	—	53	1	no	—
190	#2920	D₅₃	1	1	yes	characteristic
191	C₂×C₄×C₄⋊C₄	—	53	1	no	—
192	#1303	—	53	4	no	—
193	C₂×C₅₃	—	2	1	no	—
194	C₂×C₅₃	—	2	2	no	—
195	C₂×C₅₃	—	2	2	no	—
196	C₂×C₅₃	—	2	1	no	—
197	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
198	C₂×C₅₃	#1736	1	1	yes	characteristic
199	C₂×C₅₃	C₂×#344	1	1	yes	characteristic
200	C₂²×C₄⋊C₄	—	106	1	no	—
201	C₂²×C₄²	—	106	1	no	—
202	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
203	#432	—	2	1	no	—
204	#432	—	53	2	no	—
205	#432	—	106	1	no	—
206	#432	Dic₅₃	1	2	yes	complemented
207	#431	—	2	1	no	—
208	#431	—	53	2	no	—
209	#431	—	106	1	no	—
210	#431	C₂×D₅₃	1	2	yes	—
211	#423	—	53	2	no	—
212	#423	Dic₅₃	1	2	yes	complemented
213	#407	—	2	1	no	—
214	#407	—	106	1	no	—
215	C₄×M₁₆	—	2	1	no	—
216	C₄×M₁₆	—	106	1	no	—
217	C₄×C₄⋊C₄	—	53	2	no	—
218	C₄×C₄⋊C₄	—	53	2	no	—
219	C₄×C₄⋊C₄	—	106	1	no	—
220	C₄×C₄⋊C₄	—	106	1	no	—
221	C₂×#95	—	53	2	no	—
222	C₅₃	#8098	1	1	yes	characteristic, complemented, Sylow
223	C₂³×C₄	—	106	2	no	—
224	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
225	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
226	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	—	106	2	no	—
229	C₂×C₄⋊C₄	—	106	8	no	—
230	C₂×C₄⋊C₄	—	106	2	no	—
231	C₂×C₄⋊C₄	—	106	2	no	—
232	C₂×C₄²	—	106	2	no	—
233	C₂×C₄²	—	106	2	no	—
234	C₂×C₄²	—	212	4	no	—
235	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
236	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
237	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
239	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
240	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
241	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
243	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
244	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
245	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
246	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
247	C₄⋊C₈	—	2	2	no	—
248	C₄⋊C₈	—	53	4	no	—
249	C₄⋊C₈	—	106	2	no	—
250	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	complemented
251	C₂²⋊C₈	—	2	2	no	—
252	C₂²⋊C₈	—	2	2	no	—
253	C₂²⋊C₈	—	106	2	no	—
254	C₂²⋊C₈	—	106	2	no	—
255	#97	—	2	2	no	—
256	#97	—	106	2	no	—
257	C₄×C₈	—	2	2	no	—
258	C₄×C₈	—	106	2	no	—
259	#95	—	53	4	no	—
260	#95	—	106	2	no	—
261	C₂⁴	#11719921	1	1	yes	characteristic
262	C₂²×C₄	—	2	4	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	—	106	8	no	—
267	C₂²×C₄	—	106	4	no	—
268	C₂²×C₄	—	212	4	no	—
269	C₂²×C₄	—	212	4	no	—
270	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
272	C₂²×C₄	#11719919	1	1	yes	characteristic
273	C₂²×C₄	#11719919	1	1	yes	characteristic
274	C₂²×C₄	#11719919	1	1	yes	characteristic
275	C₂²×C₄	#11719919	1	1	yes	characteristic
276	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
278	C₂²×C₄	#11719921	1	1	yes	characteristic
279	C₂²×C₄	#11719921	1	1	yes	characteristic
280	C₂²×C₄	#11719921	1	1	yes	characteristic
281	C₂²×C₄	#11719921	1	1	yes	characteristic
282	C₂²×C₄	#11719921	1	1	yes	characteristic
283	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
288	M₁₆	—	4	4	no	—
289	M₁₆	—	212	2	no	—
290	M₁₆	—	212	2	no	—
291	C₂×C₈	—	2	4	no	—
292	C₂×C₈	—	106	4	no	—
293	C₄⋊C₄	—	106	8	no	—
294	C₄⋊C₄	—	106	8	no	—
295	C₄²	—	212	4	no	—
296	C₄²	—	212	4	no	—
297	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
298	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
299	C₄²	#11719919	1	1	yes	characteristic
300	C₄²	#11719919	1	1	yes	characteristic
301	C₄²	#11719919	1	1	yes	characteristic
302	C₄²	#11719919	1	1	yes	characteristic
303	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
304	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
305	C₄²	#11719921	1	1	yes	characteristic
306	C₄²	#11719921	1	1	yes	characteristic
307	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
308	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
309	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
310	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
311	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
313	C₂³	—	2	1	no	—
314	C₂³	—	2	1	no	—
315	C₂³	—	2	2	no	—
316	C₂³	—	2	2	no	—
317	C₂³	#13133124	1	1	yes	characteristic
318	C₂³	#13133124	1	1	yes	characteristic
319	C₂³	C₂×#11719921	1	1	yes	characteristic
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	106	8	no	—
338	C₂×C₄	—	212	16	no	—
339	C₂×C₄	—	212	4	no	—
340	C₂×C₄	—	212	4	no	—
341	C₂×C₄	#13133124	1	1	yes	characteristic
342	C₂×C₄	#13133124	1	1	yes	characteristic
343	C₂×C₄	#13133124	1	1	yes	characteristic
344	C₂×C₄	#13133124	1	1	yes	characteristic
345	C₂×C₄	#13133124	1	1	yes	characteristic
346	C₂×C₄	#13133124	1	1	yes	characteristic
347	C₂×C₄	#13133144	1	4	yes	—
348	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
349	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
350	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
351	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
352	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
353	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
354	C₂×C₄	#13133245	1	2	yes	—
355	C₂×C₄	#13133250	1	2	yes	—
356	C₈	—	4	4	no	—
357	C₈	—	212	4	no	—
358	C₂²	—	2	4	no	—
359	C₂²	—	2	4	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	#16810382	1	2	yes	—
367	C₂²	#16810395	1	2	yes	—
368	C₂²	C₂×#13133124	1	1	yes	characteristic
369	C₂²	#16810629	1	1	yes	characteristic
370	C₂²	#16810633	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	212	8	no	—
380	C₄	#16810382	1	2	yes	—
381	C₄	#16810625	1	2	yes	—
382	C₄	C₂×#13133144	1	2	yes	—
383	C₄	#16810665	1	2	yes	—
384	C₂	—	2	1	no	—
385	C₂	—	2	1	no	—
386	C₂	—	2	2	no	—
387	C₂	—	2	2	no	—
388	C₂	C₂×#16810382	1	1	yes	characteristic
389	C₂	#20274435	1	1	yes	characteristic
390	C₂	C₂×#16810395	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2920	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2920	D₅₃
hypocenter	C₅₃	#8098
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #20272617	C₂	semidirect
8. #16810680	C₄	semidirect
10. #16810676	C₄	semidirect
21. C₅₃×#432	C₄	semidirect
24. C₅₃×#423	C₄	semidirect
36. C₄⋊#88322	C₂×C₄	semidirect
60. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
206. #432	Dic₅₃	semidirect
212. #423	Dic₅₃	semidirect
222. C₅₃	#8098	semidirect
250. C₄⋊C₈	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂×C₄×C₄⋊Dic₅₃	2
#20272617	2
C₅₃×#2920	2
#8098	53