Subgroups — #27273981

Order 13568.

650 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	C₂×#16810680	C₂	1	1	yes	maximal, characteristic
2	#20277113	C₂	1	1	yes	maximal, characteristic
3	#20272582	C₂	1	4	yes	maximal
4	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic
5	#16810680	—	2	1	no	—
6	#16810680	—	2	1	no	—
7	#16810680	—	2	1	no	—
8	#16810680	—	2	1	no	—
9	C₂×C₄⋊#88322	C₄	1	2	yes	complemented
10	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
11	C₂×#13133128	C₄	1	2	yes	complemented
12	#16810491	C₄	1	2	yes	complemented
13	#16810489	—	2	1	no	—
14	#16810486	—	2	1	no	—
15	#16810483	C₂²	1	2	yes	—
16	C₄×C₄⋊Dic₅₃	—	2	1	no	—
17	C₄×C₂²⋊Dic₅₃	—	2	1	no	—
18	#16810469	C₄	1	2	yes	complemented
19	C₂×#117×C₅₃	—	2	1	no	—
20	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#384	—	2	1	no	—
22	C₅₃×#382	C₄	1	2	yes	complemented
23	C₅₃×#380	—	2	1	no	—
24	C₅₃×#380	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
26	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	complemented
27	C₂²×#88322	C₂×C₄	1	4	yes	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
30	C₂×C₄×Dic₅₃	—	2	2	no	—
31	C₄⋊#88322	—	2	2	no	—
32	C₄⋊#88322	—	2	2	no	—
33	C₂²⋊#88322	—	2	2	no	—
34	C₂²⋊#88322	—	2	2	no	—
35	#13133128	—	2	2	no	—
36	#13133128	—	2	2	no	—
37	#13133124	—	2	2	no	—
38	#13133124	—	2	2	no	—
39	#13133124	—	2	2	no	—
40	#13133124	C₂×C₄	1	4	yes	—
41	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	#117×C₅₃	—	4	4	no	—
45	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
62	#95×C₅₃	—	2	2	no	—
63	#95×C₅₃	C₂×C₄	1	4	yes	—
64	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	4	no	—
68	C₂²×C₄×C₅₃	—	4	2	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²⋊C₄	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	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂²×Dic₅₃	—	2	4	no	—
89	C₄⋊C₄×C₅₃	—	4	4	no	—
90	C₄⋊C₄×C₅₃	—	4	4	no	—
91	C₄⋊C₄×C₅₃	—	4	4	no	—
92	C₂²⋊C₄×C₅₃	—	4	4	no	—
93	C₄²×C₅₃	—	4	4	no	—
94	C₄²×C₅₃	C₄²	1	1	yes	characteristic
95	C₄²×C₅₃	C₄²	1	1	yes	characteristic
96	C₄²×C₅₃	C₄²	1	1	yes	characteristic
97	C₄²×C₅₃	C₄²	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	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
110	C₂²⋊Dic₅₃	—	4	2	no	—
111	C₂²⋊Dic₅₃	—	4	2	no	—
112	C₄⋊Dic₅₃	—	4	4	no	—
113	C₄×Dic₅₃	—	4	4	no	—
114	C₂×#88322	—	2	4	no	—
115	C₂×#88322	—	4	4	no	—
116	C₂³×C₅₃	—	4	1	no	—
117	C₂³×C₅₃	—	4	1	no	—
118	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂³×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
122	C₂³×C₅₃	#137	1	2	yes	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	1	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	—	4	1	no	—
142	C₂×C₄×C₅₃	—	4	1	no	—
143	C₂×C₄×C₅₃	—	4	4	no	—
144	C₂×C₄×C₅₃	—	4	4	no	—
145	C₂×C₄×C₅₃	—	4	4	no	—
146	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×Dic₅₃	—	4	4	no	—
159	C₂×Dic₅₃	—	4	4	no	—
160	C₂×Dic₅₃	—	4	4	no	—
161	#88322	—	4	4	no	—
162	#7984	—	53	1	no	maximal, Sylow
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	—	2	1	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	—	4	2	no	—
170	C₂²×C₅₃	—	4	2	no	—
171	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
172	C₂²×C₅₃	#417	1	2	yes	—
173	C₂²×C₅₃	#418	1	1	yes	characteristic
174	C₂²×C₅₃	#419	1	1	yes	characteristic
175	C₂²×C₅₃	#428	1	2	yes	—
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	—	2	1	no	—
185	C₄×C₅₃	—	2	1	no	—
186	C₄×C₅₃	—	8	4	no	—
187	C₄×C₅₃	H(3,ℤ/4)	1	2	yes	—
188	C₄×C₅₃	#410	1	2	yes	—
189	Dic₅₃	—	8	4	no	—
190	C₂×#432	—	53	1	no	—
191	#2285	D₅₃	1	1	yes	characteristic
192	#2283	—	53	1	no	—
193	#1289	—	53	4	no	—
194	C₂×C₅₃	—	2	2	no	—
195	C₂×C₅₃	—	2	1	no	—
196	C₂×C₅₃	—	2	1	no	—
197	C₂×C₅₃	—	2	2	no	—
198	C₂×C₅₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
199	C₂×C₅₃	#1736	1	1	yes	characteristic
200	C₂×C₅₃	#1740	1	1	yes	characteristic
201	C₂×#117	—	2	1	no	—
202	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
203	#432	—	106	1	no	—
204	#432	—	106	1	no	—
205	#432	—	106	1	no	—
206	#432	—	106	1	no	—
207	C₂×C₄⋊C₈	—	53	2	no	—
208	C₂×C₂²⋊C₈	—	53	2	no	—
209	C₂×#97	—	53	2	no	—
210	#384	—	2	1	no	—
211	#383	—	53	2	no	—
212	#382	—	106	1	no	—
213	#382	Dic₅₃	1	2	yes	complemented
214	#381	—	106	1	no	—
215	#380	—	2	1	no	—
216	#380	—	53	2	no	—
217	#380	C₂×D₅₃	1	2	yes	—
218	C₄×C₄⋊C₄	—	2	1	no	—
219	C₄×C₄⋊C₄	—	106	1	no	—
220	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	complemented
221	C₄×C₂²⋊C₄	—	106	1	no	—
222	#376	—	53	2	no	—
223	C₅₃	#7984	1	1	yes	characteristic, complemented, Sylow
224	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
226	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
227	C₂²×C₈	—	53	4	no	—
228	#117	—	4	4	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	—	2	2	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	106	2	no	—
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₂²⋊C₄	—	106	2	no	—
235	C₂×C₄²	—	2	2	no	—
236	C₂×C₄²	—	106	2	no	—
237	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
238	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
239	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
244	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
245	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
246	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
247	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
248	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
249	C₄⋊C₈	—	106	2	no	—
250	C₄⋊C₈	—	106	2	no	—
251	C₂²⋊C₈	—	106	2	no	—
252	C₂²⋊C₈	—	106	2	no	—
253	#97	—	106	2	no	—
254	#97	—	106	2	no	—
255	#95	—	2	2	no	—
256	#95	—	53	4	no	—
257	#95	—	106	2	no	—
258	#95	—	106	2	no	—
259	#95	—	106	2	no	—
260	#95	C₂×Dic₅₃	1	4	yes	—
261	C₂⁴	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	—	4	2	no	—
267	C₂²×C₄	—	106	4	no	—
268	C₂²×C₄	#11719919	1	1	yes	characteristic
269	C₂²×C₄	#11719919	1	1	yes	characteristic
270	C₂²×C₄	#11719919	1	1	yes	characteristic
271	C₂²×C₄	#11719919	1	1	yes	characteristic
272	C₂²×C₄	#11719921	1	1	yes	characteristic
273	C₂²×C₄	#11719921	1	1	yes	characteristic
274	C₂²×C₄	#11719921	1	1	yes	characteristic
275	C₂²×C₄	#11719921	1	1	yes	characteristic
276	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
281	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
284	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
286	C₂×C₈	—	106	4	no	—
287	C₂×C₈	—	212	4	no	—
288	C₄⋊C₄	—	4	4	no	—
289	C₄⋊C₄	—	4	4	no	—
290	C₄⋊C₄	—	4	4	no	—
291	C₄⋊C₄	—	212	4	no	—
292	C₂²⋊C₄	—	4	4	no	—
293	C₂²⋊C₄	—	212	2	no	—
294	C₂²⋊C₄	—	212	2	no	—
295	C₄²	—	4	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	C₄×Dic₅₃	1	1	yes	characteristic
300	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
301	C₄²	#11719919	1	1	yes	characteristic
302	C₄²	#11719919	1	1	yes	characteristic
303	C₄²	#11719919	1	1	yes	characteristic
304	C₄²	#11719919	1	1	yes	characteristic
305	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
306	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
307	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
308	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
309	C₄²	#11719921	1	1	yes	characteristic
310	C₄²	#11719921	1	1	yes	characteristic
311	C₄²	#11719921	1	1	yes	characteristic
312	C₄²	#11719921	1	1	yes	characteristic
313	C₂³	—	4	1	no	—
314	C₂³	—	4	1	no	—
315	C₂³	C₂×#11719921	1	1	yes	characteristic
316	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
317	C₂³	C₂×#11719919	1	1	yes	characteristic
318	C₂³	#13133243	1	2	yes	—
319	C₂³	#13133248	1	2	yes	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	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	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	4	no	—
341	C₂×C₄	—	4	4	no	—
342	C₂×C₄	—	4	4	no	—
343	C₂×C₄	—	212	4	no	—
344	C₂×C₄	—	212	4	no	—
345	C₂×C₄	—	212	4	no	—
346	C₂×C₄	#13133124	1	1	yes	characteristic
347	C₂×C₄	#13133124	1	1	yes	characteristic
348	C₂×C₄	#13133124	1	1	yes	characteristic
349	C₂×C₄	#13133124	1	1	yes	characteristic
350	C₂×C₄	#13133124	1	1	yes	characteristic
351	C₂×C₄	#13133124	1	1	yes	characteristic
352	C₂×C₄	#13133124	1	1	yes	characteristic
353	C₂×C₄	#13133124	1	1	yes	characteristic
354	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
356	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
357	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
358	C₈	—	212	4	no	—
359	C₂²	—	2	1	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	—	4	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	C₂×#13133124	1	1	yes	characteristic
368	C₂²	#16810622	1	2	yes	—
369	C₂²	#16810628	1	1	yes	characteristic
370	C₂²	#16810632	1	1	yes	characteristic
371	C₂²	#16810663	1	2	yes	—
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	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	8	4	no	—
383	C₄	—	424	4	no	—
384	C₄	#16810379	1	2	yes	—
385	C₄	#16810596	1	2	yes	—
386	C₂	—	2	1	no	—
387	C₂	—	2	2	no	—
388	C₂	—	2	1	no	—
389	C₂	—	2	2	no	—
390	C₂	C₂×#16810379	1	1	yes	characteristic
391	C₂	#20274434	1	1	yes	characteristic
392	C₂	#20274447	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133124
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#2285	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×D₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#2285	D₅₃
hypocenter	C₅₃	#7984
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #2285

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊#88322	C₄	semidirect
11. C₂×#13133128	C₄	semidirect
12. #16810491	C₄	semidirect
18. #16810469	C₄	semidirect
22. C₅₃×#382	C₄	semidirect
26. C₄×C₄⋊C₄×C₅₃	C₄	semidirect
213. #382	Dic₅₃	semidirect
220. C₄×C₄⋊C₄	Dic₅₃	semidirect
223. C₅₃	#7984	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810680	2
#20277113	2
#20272582	2
C₅₃×#2285	2
#7984	53