Subgroups — #27349720

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 418 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	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#20273008	C₂	1	2	yes	maximal, complemented
4	#20273005	C₂	1	2	yes	maximal, complemented
5	#20273002	C₂	1	2	yes	maximal, complemented
6	#20272999	C₂	1	2	yes	maximal, complemented
7	#20272996	C₂	1	2	yes	maximal, complemented
8	M₁₆×#88322	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
10	C₂³×#88322	C₄	1	2	yes	—
11	#16810680	C₄	1	2	yes	—
12	#16810680	C₄	1	2	yes	—
13	#16810676	C₂²	1	2	yes	—
14	#16810676	C₂²	1	2	yes	—
15	#16810647	C₄	1	2	yes	—
16	#16810587	C₄	1	2	yes	—
17	C₄×#11719917	C₂²	1	2	yes	—
18	C₂×C₄×#88322	C₄	1	2	yes	—
19	C₂×C₄×#88322	C₄	1	2	yes	—
20	C₂×C₄×#88322	C₄	1	2	yes	—
21	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
24	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
25	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
26	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
27	#16810345	C₂²	1	4	yes	complemented
28	#16810345	C₂²	1	4	yes	complemented
29	#16810344	C₂²	1	4	yes	complemented
30	C₈×#88322	C₂²	1	4	yes	complemented
31	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#432	C₄	1	2	yes	—
33	C₅₃×#432	C₄	1	2	yes	—
34	C₅₃×#431	C₂²	1	2	yes	—
35	C₅₃×#431	C₂²	1	2	yes	—
36	C₅₃×#423	C₄	1	2	yes	—
37	C₅₃×#407	C₄	1	2	yes	—
38	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
39	C₂×#11719917	C₂×C₄	1	4	yes	—
40	C₂²×#88322	—	2	8	no	—
41	C₂²×#88322	C₂×C₄	1	2	yes	—
42	C₂²×#88322	C₂×C₄	1	2	yes	—
43	C₂²×#88322	C₂×C₄	1	2	yes	—
44	C₂²×#88322	C₂×C₄	1	2	yes	—
45	C₂²×#88322	C₂×C₄	1	2	yes	—
46	C₂²×#88322	C₂×C₄	1	2	yes	—
47	C₄⋊#88322	C₂×C₄	1	4	yes	—
48	C₄⋊#88322	C₂×C₄	1	4	yes	—
49	C₂²⋊#88322	C₂×C₄	1	4	yes	—
50	C₂²⋊#88322	C₂×C₄	1	4	yes	—
51	#13133128	C₂×C₄	1	4	yes	—
52	C₄×#88322	C₂×C₄	1	2	yes	—
53	C₄×#88322	C₂×C₄	1	2	yes	—
54	C₄×#88322	C₂×C₄	1	2	yes	—
55	C₄×#88322	C₂×C₄	1	2	yes	—
56	C₄×#88322	C₂×C₄	1	2	yes	—
57	C₄×#88322	C₂×C₄	1	2	yes	—
58	C₄×#88322	C₂³	1	1	yes	characteristic
59	C₄×#88322	C₂³	1	1	yes	characteristic
60	C₄×#88322	C₂³	1	1	yes	characteristic
61	C₄×#88322	C₂³	1	1	yes	characteristic
62	C₄×#88322	C₂³	1	4	yes	—
63	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
67	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
80	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
81	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
82	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
83	#97×C₅₃	C₂×C₄	1	4	yes	—
84	C₄×C₈×C₅₃	C₂³	1	4	yes	—
85	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	—	2	4	no	—
87	C₂²×C₄×C₅₃	—	2	4	no	—
88	C₂²×C₄×C₅₃	—	2	4	no	—
89	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	M₁₆×C₅₃	—	2	8	no	—
108	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
109	C₄²×C₅₃	C₄²	1	1	yes	characteristic
110	C₄²×C₅₃	C₄²	1	1	yes	characteristic
111	C₄²×C₅₃	C₄²	1	1	yes	characteristic
112	C₄²×C₅₃	C₄²	1	1	yes	characteristic
113	C₄²×C₅₃	C₄²	1	1	yes	characteristic
114	C₄²×C₅₃	C₄²	1	1	yes	characteristic
115	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
125	#11719917	—	2	8	no	—
126	C₂×#88322	—	2	4	no	—
127	C₂×#88322	—	2	4	no	—
128	C₂×#88322	—	2	4	no	—
129	C₂×#88322	—	2	4	no	—
130	C₂×#88322	—	2	4	no	—
131	C₂×#88322	M₁₆	1	8	yes	complemented
132	C₂×#88322	C₂²×C₄	1	2	yes	—
133	C₂×#88322	C₂²×C₄	1	2	yes	—
134	C₂×#88322	C₂²×C₄	1	8	yes	—
135	C₂×#88322	C₂²×C₄	1	2	yes	—
136	C₂×#88322	C₂²×C₄	1	2	yes	—
137	C₂³×C₅₃	—	2	2	no	—
138	C₂³×C₅₃	—	2	2	no	—
139	C₂³×C₅₃	—	2	2	no	—
140	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
141	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
142	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	2	2	no	—
145	C₂×C₄×C₅₃	—	2	2	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	2	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	2	2	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
165	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
168	C₂×C₄×C₅₃	#131	1	4	yes	—
169	C₂×C₄×C₅₃	#131	1	4	yes	—
170	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
173	C₈×C₅₃	—	2	8	no	—
174	#88322	—	2	2	no	—
175	#88322	—	2	2	no	—
176	#88322	—	2	8	no	—
177	#88322	C₂×M₁₆	1	4	yes	—
178	#88322	C₂×M₁₆	1	4	yes	—
179	#17323	—	53	1	no	maximal, Sylow
180	C₂²×C₅₃	—	2	8	no	—
181	C₂²×C₅₃	—	2	1	no	—
182	C₂²×C₅₃	—	2	1	no	—
183	C₂²×C₅₃	—	2	1	no	—
184	C₂²×C₅₃	—	2	1	no	—
185	C₂²×C₅₃	—	2	1	no	—
186	C₂²×C₅₃	—	2	1	no	—
187	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
188	C₂²×C₅₃	#405	1	2	yes	—
189	C₂²×C₅₃	#405	1	2	yes	—
190	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	—	2	1	no	—
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	2	1	no	—
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
198	C₄×C₅₃	#405	1	2	yes	—
199	C₄×C₅₃	#405	1	2	yes	—
200	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
201	C₄×C₅₃	C₂×#131	1	2	yes	—
202	C₄×C₅₃	C₂×#131	1	2	yes	—
203	#2920	—	53	1	no	—
204	#2920	D₅₃	1	1	yes	characteristic
205	C₂²×C₄×C₈	—	53	1	no	—
206	#1452	—	53	2	no	—
207	#1451	—	53	2	no	—
208	#1451	—	53	2	no	—
209	#1450	—	53	2	no	—
210	C₈×M₁₆	—	53	2	no	—
211	C₈×M₁₆	—	53	2	no	—
212	C₂×C₅₃	—	2	2	no	—
213	C₂×C₅₃	—	2	2	no	—
214	C₂×C₅₃	—	2	2	no	—
215	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
216	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
217	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
218	C₂³×C₈	—	53	2	no	—
219	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
220	#432	—	53	2	no	—
221	#432	—	53	2	no	—
222	#432	Dic₅₃	1	2	yes	—
223	#432	Dic₅₃	1	2	yes	—
224	#431	—	53	2	no	—
225	#431	—	53	2	no	—
226	#431	C₂×D₅₃	1	2	yes	—
227	#431	C₂×D₅₃	1	2	yes	—
228	#423	—	53	2	no	—
229	#423	Dic₅₃	1	2	yes	—
230	#407	—	53	2	no	—
231	#407	Dic₅₃	1	2	yes	—
232	C₄×M₁₆	—	53	2	no	—
233	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
234	C₂×C₄×C₈	—	53	2	no	—
235	C₂×C₄×C₈	—	53	2	no	—
236	C₂×C₄×C₈	—	53	2	no	—
237	C₂×C₄×C₈	—	53	1	no	—
238	C₂×C₄×C₈	—	53	1	no	—
239	C₂×C₄×C₈	—	53	1	no	—
240	C₂×C₄×C₈	—	53	1	no	—
241	C₂×C₄×C₈	—	53	1	no	—
242	C₂×C₄×C₈	—	53	1	no	—
243	#322	—	53	4	no	—
244	#322	—	53	4	no	—
245	#322	—	53	4	no	—
246	C₈²	—	53	4	no	—
247	C₅₃	#17323	1	1	yes	characteristic, complemented, Sylow
248	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
249	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
250	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
251	C₂×M₁₆	—	53	4	no	—
252	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
253	C₂²×C₈	—	53	2	no	—
254	C₂²×C₈	—	53	2	no	—
255	C₂²×C₈	—	53	2	no	—
256	C₂²×C₈	—	53	2	no	—
257	C₂²×C₈	—	53	2	no	—
258	C₂²×C₈	—	53	2	no	—
259	C₂²×C₈	—	106	8	no	—
260	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
261	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
262	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
263	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
264	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
265	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
266	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
267	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
268	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
269	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
270	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
271	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
272	C₄⋊C₈	—	53	4	no	—
273	C₄⋊C₈	—	53	4	no	—
274	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
275	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
276	C₂²⋊C₈	—	53	4	no	—
277	C₂²⋊C₈	—	53	4	no	—
278	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
279	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
280	#97	—	53	4	no	—
281	#97	C₂×Dic₅₃	1	4	yes	—
282	C₄×C₈	—	53	2	no	—
283	C₄×C₈	—	53	2	no	—
284	C₄×C₈	—	53	2	no	—
285	C₄×C₈	—	53	2	no	—
286	C₄×C₈	—	53	2	no	—
287	C₄×C₈	—	53	2	no	—
288	C₄×C₈	—	53	1	no	—
289	C₄×C₈	—	53	4	no	—
290	C₄×C₈	—	53	1	no	—
291	C₄×C₈	—	53	1	no	—
292	C₄×C₈	—	53	1	no	—
293	C₄×C₈	C₂²×D₅₃	1	4	yes	—
294	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	4	no	—
298	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
299	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
300	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
301	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
302	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
303	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
304	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
305	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
306	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
307	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
308	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
309	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
310	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
311	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
312	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
313	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
314	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
315	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
316	M₁₆	—	2	8	no	—
317	M₁₆	—	106	8	no	—
318	C₂×C₈	—	53	8	no	—
319	C₂×C₈	—	53	2	no	—
320	C₂×C₈	—	53	2	no	—
321	C₂×C₈	—	53	2	no	—
322	C₂×C₈	—	53	2	no	—
323	C₂×C₈	—	53	8	no	—
324	C₂×C₈	—	106	4	no	—
325	C₂×C₈	—	106	4	no	—
326	C₂×C₈	—	106	4	no	—
327	C₂×C₈	—	106	4	no	—
328	C₂×C₈	—	106	4	no	—
329	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
330	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
331	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
332	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
333	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
334	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
335	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
336	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
337	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
338	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
339	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
340	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
341	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
342	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
343	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
344	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
345	C₄²	C₂³×D₅₃	1	1	yes	characteristic
346	C₂³	—	2	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	2	no	—
349	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
350	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
351	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
369	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
370	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
371	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
372	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
373	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
374	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
376	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
377	C₂×C₄	#13133224	1	4	yes	—
378	C₂×C₄	#13133225	1	4	yes	—
379	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
380	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
381	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
382	C₈	—	2	8	no	—
383	C₈	—	53	4	no	—
384	C₈	—	53	4	no	—
385	C₈	—	106	2	no	—
386	C₈	—	106	2	no	—
387	C₈	—	106	8	no	—
388	C₂²	—	2	8	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	M₁₆×Dic₅₃	1	2	yes	—
396	C₂²	#16810579	1	2	yes	—
397	C₂²	#16810581	1	2	yes	—
398	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	2	1	no	—
403	C₄	—	2	1	no	—
404	C₄	—	2	1	no	—
405	C₄	M₁₆×Dic₅₃	1	2	yes	—
406	C₄	#16810579	1	2	yes	—
407	C₄	#16810581	1	2	yes	—
408	C₄	C₂×M₁₆×D₅₃	1	2	yes	—
409	C₄	C₂×#13133224	1	2	yes	—
410	C₄	C₂×#13133225	1	2	yes	—
411	C₂	—	2	2	no	—
412	C₂	—	2	2	no	—
413	C₂	—	2	2	no	—
414	C₂	C₂×M₁₆×Dic₅₃	1	1	yes	characteristic
415	C₂	C₂×#16810579	1	1	yes	characteristic
416	C₂	C₂×#16810581	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#2920	C₂
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2920	D₅₃
hypocenter	C₅₃	#17323
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #20273008	C₂	semidirect
4. #20273005	C₂	semidirect
5. #20273002	C₂	semidirect
6. #20272999	C₂	semidirect
7. #20272996	C₂	semidirect
8. M₁₆×#88322	C₂	semidirect
27. #16810345	C₂²	semidirect
28. #16810345	C₂²	semidirect
29. #16810344	C₂²	semidirect
30. C₈×#88322	C₂²	semidirect
131. C₂×#88322	M₁₆	semidirect
247. C₅₃	#17323	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂²×C₄×#88322	2
#20273008	2
#20273005	2
#20273002	2
#20272999	2
#20272996	2
M₁₆×#88322	2
C₅₃×#2920	2
#17323	53