Subgroups — #28318452

Order 13568.

1078 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#431	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133296	C₂	1	1	yes	maximal, characteristic
3	#20290509	C₂	1	4	yes	maximal, complemented
4	#20290451	C₂	1	4	yes	maximal
5	#20290428	C₂	1	2	yes	maximal, complemented
6	C₄×#13133271	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133296	C₄	1	2	yes	—
9	#16811642	C₄	1	2	yes	complemented
10	#16811641	C₄	1	2	yes	complemented
11	#16811640	C₄	1	2	yes	complemented
12	#16811639	C₄	1	2	yes	complemented
13	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
14	D₅₃⋊(C₂²⋊C₈)	C₂²	1	2	yes	—
15	C₂×#13133271	C₂²	1	8	yes	—
16	C₄×D₅₃⋊C₈	C₂²	1	2	yes	—
17	C₄×C₄⋊#6157	C₄	1	2	yes	—
18	C₄×C₄⋊#6157	C₂²	1	2	yes	—
19	C₄×C₂²⋊#6157	C₄	1	2	yes	—
20	C₄×C₂²⋊#6157	C₂²	1	2	yes	—
21	#16811559	C₄	1	2	yes	—
22	C₄²×#6157	C₂²	1	2	yes	—
23	(C₂×#6157)⋊C₈	C₂²	1	8	yes	—
24	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
25	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×#88322	C₄	1	1	yes	characteristic, complemented
27	C₂×C₈×Dic₅₃	C₄	1	2	yes	complemented
28	C₄×C₈×D₅₃	C₄	1	2	yes	complemented
29	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
30	C₄×C₈×D₅₃	C₄	1	1	yes	characteristic, complemented
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₅₃	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
35	C₂×D₅₃⋊C₈	C₂³	1	4	yes	—
36	#13133296	—	2	8	no	—
37	C₂×C₄⋊#6157	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊#6157	C₂×C₄	1	4	yes	—
39	C₂×C₄×#6157	C₂×C₄	1	4	yes	—
40	C₂×C₄×#6157	C₂×C₄	1	4	yes	—
41	C₂×C₄×#6157	C₂³	1	4	yes	—
42	#13133284	C₂×C₄	1	4	yes	—
43	C₄⋊#88324	C₂×C₄	1	4	yes	—
44	#13133274	C₂×C₄	1	4	yes	—
45	C₂²⋊#88324	C₂×C₄	1	4	yes	—
46	#13133271	—	2	16	no	—
47	C₄×#88324	C₂×C₄	1	4	yes	—
48	C₄×#88324	C₂×C₄	1	4	yes	—
49	(C₂×#6157)⋊C₄	C₂×C₄	1	4	yes	—
50	(C₂×#6157)⋊C₄	C₂×C₄	1	4	yes	—
51	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
54	C₂²×#88322	C₂×C₄	1	2	yes	—
55	C₂×C₈×D₅₃	—	2	8	no	—
56	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
57	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₈×D₅₃	C₂³	1	2	yes	—
59	C₂×C₈×D₅₃	C₂³	1	2	yes	—
60	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
67	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄²×D₅₃	C₂³	1	1	yes	characteristic
70	C₄²×D₅₃	C₂³	1	1	yes	characteristic
71	C₄×#88322	C₂×C₄	1	2	yes	—
72	C₄×#88322	C₂×C₄	1	1	yes	characteristic
73	C₄×#88322	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
75	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
76	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
77	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
78	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
79	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×#6157	C₂²×C₄	1	8	yes	—
85	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
89	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
90	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×D₅₃	—	2	4	no	—
92	C₂×C₄×D₅₃	—	2	4	no	—
93	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
94	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
95	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
96	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
97	C₂×C₄×D₅₃	M₁₆	1	8	yes	—
98	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
106	C₂²⋊#6157	—	2	8	no	—
107	C₂×#88324	C₂²×C₄	1	8	yes	—
108	C₄⋊#6157	—	2	8	no	—
109	C₄×#6157	—	2	8	no	—
110	C₄×#6157	—	2	8	no	—
111	C₄×#6157	—	2	8	no	—
112	D₅₃⋊C₈	—	2	8	no	—
113	C₂×C₈×C₅₃	—	2	4	no	—
114	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
116	C₄²×C₅₃	C₄²	1	1	yes	characteristic
117	C₄²×C₅₃	C₄²	1	1	yes	characteristic
118	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
121	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
122	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
123	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
124	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
125	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
126	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×#88322	—	2	4	no	—
133	C₂×#88322	C₂²×C₄	1	2	yes	—
134	C₂×#88322	C₂²×C₄	1	2	yes	—
135	C₈×D₅₃	—	2	4	no	—
136	C₈×D₅₃	—	2	4	no	—
137	C₈×D₅₃	—	2	4	no	—
138	C₈×D₅₃	#41	1	8	yes	—
139	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
140	C₂²×D₅₃	—	2	2	no	—
141	C₂²×D₅₃	—	2	2	no	—
142	C₂²×D₅₃	C₂×C₄²	1	1	yes	characteristic
143	C₂²×D₅₃	C₂×M₁₆	1	4	yes	—
144	C₂²×D₅₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×#6157	—	2	8	no	—
146	C₂×#6157	—	2	8	no	—
147	C₂×#6157	—	2	8	no	—
148	C₂×C₄×C₅₃	—	2	2	no	—
149	C₂×C₄×C₅₃	—	2	2	no	—
150	C₂×C₄×C₅₃	#97	1	4	yes	complemented
151	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
157	C₂×Dic₅₃	—	2	2	no	—
158	C₂×Dic₅₃	—	2	2	no	—
159	C₂×Dic₅₃	#97	1	4	yes	complemented
160	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
161	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
162	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
163	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
164	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
165	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
166	C₄×D₅₃	—	2	2	no	—
167	C₄×D₅₃	—	2	2	no	—
168	C₄×D₅₃	—	2	2	no	—
169	C₄×D₅₃	—	2	2	no	—
170	C₄×D₅₃	—	2	2	no	—
171	C₄×D₅₃	—	2	2	no	—
172	C₄×D₅₃	#97	1	4	yes	complemented
173	C₄×D₅₃	#97	1	4	yes	complemented
174	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
175	C₄×D₅₃	C₂×M₁₆	1	4	yes	—
176	C₄×D₅₃	#131	1	4	yes	—
177	C₄×D₅₃	C₂×#41	1	4	yes	—
178	#88324	—	2	8	no	—
179	C₈×C₅₃	—	2	2	no	—
180	C₈×C₅₃	#117	1	4	yes	complemented
181	#88322	—	2	2	no	—
182	#88322	#117	1	4	yes	complemented
183	#20473	—	53	1	no	maximal, Sylow
184	C₂²×C₅₃	—	2	1	no	—
185	C₂²×C₅₃	—	2	1	no	—
186	C₂²×C₅₃	C₂×#97	1	2	yes	—
187	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
188	C₂×D₅₃	—	2	1	no	—
189	C₂×D₅₃	—	2	1	no	—
190	C₂×D₅₃	—	2	8	no	—
191	C₂×D₅₃	C₂×#97	1	2	yes	—
192	C₂×D₅₃	C₄×#41	1	2	yes	—
193	C₂×D₅₃	C₂²×M₁₆	1	2	yes	—
194	C₂×D₅₃	C₂×#131	1	2	yes	—
195	#6157	—	4	8	no	—
196	C₄×C₅₃	—	2	1	no	—
197	C₄×C₅₃	—	2	1	no	—
198	C₄×C₅₃	C₂×#97	1	2	yes	—
199	C₄×C₅₃	C₂×#97	1	2	yes	—
200	C₄×C₅₃	#405	1	2	yes	—
201	C₄×C₅₃	C₂×#117	1	2	yes	—
202	Dic₅₃	—	2	1	no	—
203	Dic₅₃	—	2	1	no	—
204	Dic₅₃	C₂×#97	1	2	yes	—
205	Dic₅₃	C₂×#97	1	2	yes	—
206	Dic₅₃	#405	1	2	yes	—
207	Dic₅₃	C₄×#41	1	2	yes	—
208	C₂×#431	—	53	1	no	—
209	C₂²×C₄×C₈	—	53	1	no	—
210	C₄×#117	—	53	1	no	—
211	#1770	—	53	4	no	—
212	#1752	—	53	4	no	—
213	#1744	—	53	2	no	—
214	C₄×#97	—	53	2	no	—
215	C₂×C₅₃	—	2	2	no	—
216	C₂×C₅₃	C₄×#117	1	1	yes	characteristic
217	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
218	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
219	D₅₃	—	2	2	no	—
220	D₅₃	#2965	1	4	yes	—
221	C₂³×C₈	—	53	2	no	—
222	C₂×#117	—	53	2	no	—
223	C₂²×C₄²	—	53	1	no	—
224	#431	—	53	2	no	—
225	#431	—	53	2	no	—
226	#431	—	53	2	no	—
227	#431	—	53	2	no	—
228	C₂×C₄⋊C₈	—	53	2	no	—
229	C₂×C₂²⋊C₈	—	53	2	no	—
230	C₂×#97	—	53	8	no	—
231	C₂×C₄×C₈	—	53	2	no	—
232	C₂×C₄×C₈	—	53	2	no	—
233	C₂×C₄×C₈	—	53	1	no	—
234	C₂×C₄×C₈	—	53	2	no	—
235	C₂×C₄×C₈	—	53	1	no	—
236	C₂×C₄×C₈	—	53	1	no	—
237	C₂×C₄×C₈	—	53	1	no	—
238	C₂×C₄×C₈	—	53	1	no	—
239	C₂×C₄×C₈	—	53	2	no	—
240	C₂×C₄×C₈	#6157	1	1	yes	characteristic, complemented
241	C₄×C₄⋊C₄	—	53	2	no	—
242	C₄×C₄⋊C₄	—	53	2	no	—
243	C₄×C₂²⋊C₄	—	53	2	no	—
244	C₄×C₂²⋊C₄	—	53	2	no	—
245	#376	—	53	2	no	—
246	C₄³	—	53	2	no	—
247	#336	—	53	8	no	—
248	C₅₃	#20473	1	1	yes	characteristic, complemented, Sylow
249	C₂³×C₄	—	53	1	no	—
250	C₂³×C₄	—	53	1	no	—
251	C₂³×C₄	—	53	1	no	—
252	C₂²×C₈	—	53	2	no	—
253	C₂²×C₈	—	53	4	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	—	106	8	no	—
259	C₂²×C₈	C₂×#6157	1	2	yes	—
260	#117	—	106	8	no	—
261	C₂×C₄⋊C₄	—	53	4	no	—
262	C₂×C₂²⋊C₄	—	53	4	no	—
263	C₂×C₄²	—	53	1	no	—
264	C₂×C₄²	—	53	1	no	—
265	C₂×C₄²	—	53	1	no	—
266	C₂×C₄²	—	53	1	no	—
267	C₂×C₄²	—	53	1	no	—
268	C₂×C₄²	—	53	1	no	—
269	C₂×C₄²	—	53	1	no	—
270	C₂×C₄²	—	53	1	no	—
271	C₂×C₄²	—	53	1	no	—
272	C₂×C₄²	—	53	1	no	—
273	C₂×C₄²	—	53	1	no	—
274	C₂×C₄²	—	53	4	no	—
275	C₂×C₄²	—	53	4	no	—
276	C₂×C₄²	—	53	4	no	—
277	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
278	C₄⋊C₈	—	53	4	no	—
279	C₄⋊C₈	—	53	4	no	—
280	C₂²⋊C₈	—	53	4	no	—
281	C₂²⋊C₈	—	53	4	no	—
282	#97	—	106	16	no	—
283	C₄×C₈	—	53	4	no	—
284	C₄×C₈	—	53	4	no	—
285	C₄×C₈	—	53	2	no	—
286	C₄×C₈	—	53	2	no	—
287	C₄×C₈	—	53	2	no	—
288	C₄×C₈	—	53	2	no	—
289	C₄×C₈	—	53	2	no	—
290	C₄×C₈	—	53	1	no	—
291	C₄×C₈	—	53	1	no	—
292	C₄×C₈	C₂×#6157	1	2	yes	—
293	C₄×C₈	C₂×#6157	1	1	yes	characteristic
294	C₄×C₈	C₂×#6157	1	1	yes	characteristic
295	#95	—	53	4	no	—
296	#95	—	53	4	no	—
297	C₂⁴	—	53	1	no	—
298	C₂²×C₄	—	53	1	no	—
299	C₂²×C₄	—	53	8	no	—
300	C₂²×C₄	—	53	1	no	—
301	C₂²×C₄	—	53	1	no	—
302	C₂²×C₄	—	53	1	no	—
303	C₂²×C₄	—	53	1	no	—
304	C₂²×C₄	—	53	1	no	—
305	C₂²×C₄	—	53	1	no	—
306	C₂²×C₄	—	53	1	no	—
307	C₂²×C₄	—	53	1	no	—
308	C₂²×C₄	—	53	1	no	—
309	C₂²×C₄	—	53	1	no	—
310	C₂²×C₄	—	53	1	no	—
311	C₂²×C₄	—	53	1	no	—
312	C₂²×C₄	—	53	1	no	—
313	C₂²×C₄	—	53	1	no	—
314	C₂²×C₄	—	53	8	no	—
315	C₂²×C₄	—	106	4	no	—
316	C₂²×C₄	—	106	4	no	—
317	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
318	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
319	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
320	C₂×C₈	—	2	4	no	—
321	C₂×C₈	—	53	2	no	—
322	C₂×C₈	—	53	8	no	—
323	C₂×C₈	—	53	8	no	—
324	C₂×C₈	—	53	2	no	—
325	C₂×C₈	—	106	4	no	—
326	C₂×C₈	—	106	4	no	—
327	C₂×C₈	—	106	4	no	—
328	C₂×C₈	—	106	8	no	—
329	C₂×C₈	—	106	4	no	—
330	C₂×C₈	C₂²×#6157	1	2	yes	—
331	C₂×C₈	C₂²×#6157	1	2	yes	—
332	C₄⋊C₄	—	106	8	no	—
333	C₂²⋊C₄	—	106	8	no	—
334	C₄²	—	53	1	no	—
335	C₄²	—	53	1	no	—
336	C₄²	—	53	1	no	—
337	C₄²	—	53	1	no	—
338	C₄²	—	53	1	no	—
339	C₄²	—	53	1	no	—
340	C₄²	—	53	1	no	—
341	C₄²	—	53	1	no	—
342	C₄²	—	53	1	no	—
343	C₄²	—	53	1	no	—
344	C₄²	—	53	1	no	—
345	C₄²	—	53	1	no	—
346	C₄²	—	106	8	no	—
347	C₄²	—	106	8	no	—
348	C₄²	—	106	8	no	—
349	C₄²	C₄×#6157	1	1	yes	characteristic
350	C₄²	C₄×#6157	1	1	yes	characteristic
351	C₄²	C₂²×#6157	1	1	yes	characteristic
352	C₄²	C₂²×#6157	1	1	yes	characteristic
353	C₂³	—	53	1	no	—
354	C₂³	—	53	4	no	—
355	C₂³	—	53	1	no	—
356	C₂³	—	106	2	no	—
357	C₂³	—	106	2	no	—
358	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	53	4	no	—
362	C₂×C₄	—	53	1	no	—
363	C₂×C₄	—	53	1	no	—
364	C₂×C₄	—	53	1	no	—
365	C₂×C₄	—	53	1	no	—
366	C₂×C₄	—	53	4	no	—
367	C₂×C₄	—	53	4	no	—
368	C₂×C₄	—	53	4	no	—
369	C₂×C₄	—	53	4	no	—
370	C₂×C₄	—	53	4	no	—
371	C₂×C₄	—	53	4	no	—
372	C₂×C₄	—	53	1	no	—
373	C₂×C₄	—	53	1	no	—
374	C₂×C₄	—	106	2	no	—
375	C₂×C₄	—	106	2	no	—
376	C₂×C₄	—	106	2	no	—
377	C₂×C₄	—	106	2	no	—
378	C₂×C₄	—	106	2	no	—
379	C₂×C₄	—	106	2	no	—
380	C₂×C₄	—	106	2	no	—
381	C₂×C₄	—	106	2	no	—
382	C₂×C₄	—	106	8	no	—
383	C₂×C₄	—	106	8	no	—
384	C₂×C₄	—	106	8	no	—
385	C₂×C₄	#13133271	1	4	yes	complemented
386	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
387	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
388	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
389	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
390	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
391	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
392	C₈	—	2	2	no	—
393	C₈	—	53	4	no	—
394	C₈	—	106	8	no	—
395	C₈	—	106	2	no	—
396	C₈	#13133296	1	4	yes	complemented
397	C₂²	—	2	1	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	53	2	no	—
400	C₂²	—	53	2	no	—
401	C₂²	—	53	2	no	—
402	C₂²	—	53	2	no	—
403	C₂²	—	106	8	no	—
404	C₂²	—	106	1	no	—
405	C₂²	—	106	1	no	—
406	C₂²	C₂×#13133271	1	2	yes	—
407	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
408	C₄	—	2	1	no	—
409	C₄	—	2	1	no	—
410	C₄	—	53	2	no	—
411	C₄	—	53	2	no	—
412	C₄	—	53	2	no	—
413	C₄	—	53	2	no	—
414	C₄	—	106	1	no	—
415	C₄	—	106	1	no	—
416	C₄	—	212	8	no	—
417	C₄	C₂×#13133271	1	2	yes	—
418	C₄	C₂×#13133271	1	2	yes	—
419	C₄	#16811586	1	2	yes	—
420	C₄	C₂×#13133296	1	2	yes	—
421	C₂	—	2	2	no	—
422	C₂	—	53	4	no	—
423	C₂	—	106	2	no	—
424	C₂	C₄×#13133296	1	1	yes	characteristic
425	C₂	C₂²×#13133271	1	1	yes	characteristic
426	C₂	C₂×#16811586	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#20473
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #20290509	C₂	semidirect
5. #20290428	C₂	semidirect
6. C₄×#13133271	C₂	semidirect
9. #16811642	C₄	semidirect
10. #16811641	C₄	semidirect
11. #16811640	C₄	semidirect
12. #16811639	C₄	semidirect
26. C₂×C₄×#88322	C₄	semidirect
27. C₂×C₈×Dic₅₃	C₄	semidirect
28. C₄×C₈×D₅₃	C₄	semidirect
29. C₄×C₈×D₅₃	C₄	semidirect
30. C₄×C₈×D₅₃	C₄	semidirect
34. C₂×C₄×C₈×C₅₃	C₄	semidirect
150. C₂×C₄×C₅₃	#97	semidirect
159. C₂×Dic₅₃	#97	semidirect
172. C₄×D₅₃	#97	semidirect
173. C₄×D₅₃	#97	semidirect
180. C₈×C₅₃	#117	semidirect
182. #88322	#117	semidirect
240. C₂×C₄×C₈	#6157	semidirect
248. C₅₃	#20473	semidirect
385. C₂×C₄	#13133271	semidirect
396. C₈	#13133296	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊#431	2
C₄×#13133296	2
#20290509	2
#20290451	2
#20290428	2
C₄×#13133271	2
C₂×C₄×C₈×D₅₃	2
#20473	53