Subgroups — #27401890

Order 13568.

744 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275434	C₂	1	1	yes	maximal, characteristic, complemented
2	#20275434	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275281	C₂	1	2	yes	maximal, complemented
4	#20275274	C₂	1	2	yes	maximal, complemented
5	#20274699	C₂	1	2	yes	maximal
6	#20274695	C₂	1	1	yes	maximal, characteristic
7	#20274584	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274481	C₂	1	2	yes	maximal
10	C₄×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	C₂×C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
17	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×#13133128	C₂²	1	2	yes	—
20	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
21	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
22	C₂×C₄×#88322	C₂²	1	2	yes	—
23	#16810378	C₂²	1	2	yes	—
24	#16810378	C₂²	1	2	yes	—
25	#16810378	C₂²	1	2	yes	—
26	#16810378	C₂²	1	1	yes	characteristic
27	#16810378	C₂²	1	1	yes	characteristic
28	#16810378	C₂²	1	2	yes	—
29	#16810378	C₂²	1	2	yes	—
30	C₂×#117×C₅₃	C₂²	1	2	yes	—
31	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#384	C₄	1	1	yes	characteristic
33	C₅₃×#384	C₄	1	1	yes	characteristic
34	C₅₃×#382	C₄	1	2	yes	—
35	C₅₃×#380	C₄	1	1	yes	characteristic
36	C₅₃×#380	C₄	1	1	yes	characteristic
37	C₅₃×#380	C₄	1	2	yes	—
38	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
39	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
41	C₂²×#88322	C₂³	1	1	yes	characteristic
42	C₂²×#88322	C₂³	1	1	yes	characteristic
43	C₂²×#88322	C₂³	1	2	yes	—
44	C₂²×#88322	C₂³	1	2	yes	—
45	C₂²×#88322	C₂³	1	1	yes	characteristic
46	C₂²×#88322	C₂³	1	1	yes	characteristic
47	C₄⋊#88322	—	2	2	no	—
48	C₄⋊#88322	—	2	2	no	—
49	C₄⋊#88322	—	4	2	no	—
50	C₄⋊#88322	—	4	2	no	—
51	C₂²⋊#88322	—	2	4	no	—
52	C₂²⋊#88322	—	2	2	no	—
53	C₂²⋊#88322	Q₈	1	4	yes	—
54	#13133128	—	4	2	no	—
55	C₄×#88322	—	2	2	no	—
56	C₄×#88322	—	2	2	no	—
57	C₄×#88322	—	4	2	no	—
58	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	#117×C₅₃	—	4	2	no	—
62	#117×C₅₃	—	4	2	no	—
63	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
72	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
73	C₂×C₄²×C₅₃	—	2	2	no	—
74	C₂×C₄²×C₅₃	—	2	2	no	—
75	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
79	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	#95×C₅₃	C₂×C₄	1	2	yes	—
82	#95×C₅₃	C₂×C₄	1	2	yes	—
83	#95×C₅₃	C₂×C₄	1	2	yes	—
84	#95×C₅₃	C₂×C₄	1	2	yes	—
85	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	—	2	1	no	—
87	C₂²×C₄×C₅₃	—	2	1	no	—
88	C₂²×C₄×C₅₃	—	2	1	no	—
89	C₂²×C₄×C₅₃	—	2	1	no	—
90	C₂²×C₄×C₅₃	—	2	2	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	2	2	no	—
93	C₂²×C₄×C₅₃	—	2	2	no	—
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	4	1	no	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
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	C₂×Q₈	1	2	yes	—
106	C₂²×C₄×C₅₃	#41	1	2	yes	—
107	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₅₃	—	2	4	no	—
109	C₄⋊C₄×C₅₃	—	2	2	no	—
110	C₄⋊C₄×C₅₃	—	2	2	no	—
111	C₄⋊C₄×C₅₃	—	2	2	no	—
112	C₄⋊C₄×C₅₃	—	2	2	no	—
113	C₄⋊C₄×C₅₃	—	4	2	no	—
114	C₄⋊C₄×C₅₃	—	4	2	no	—
115	C₄⋊C₄×C₅₃	—	4	2	no	—
116	C₂²⋊C₄×C₅₃	—	4	2	no	—
117	C₂²⋊C₄×C₅₃	—	4	2	no	—
118	C₄²×C₅₃	—	2	2	no	—
119	C₄²×C₅₃	—	2	2	no	—
120	C₄²×C₅₃	—	2	1	no	—
121	C₄²×C₅₃	—	2	1	no	—
122	C₄²×C₅₃	—	2	2	no	—
123	C₄²×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	#41	1	2	yes	—
127	C₄²×C₅₃	#41	1	2	yes	—
128	C₂×#88322	—	2	1	no	—
129	C₂×#88322	—	2	1	no	—
130	C₂×#88322	—	2	1	no	—
131	C₂×#88322	—	2	1	no	—
132	C₂×#88322	—	2	1	no	—
133	C₂×#88322	—	2	1	no	—
134	C₂×#88322	—	2	1	no	—
135	C₂×#88322	—	2	1	no	—
136	C₂×#88322	—	2	2	no	—
137	C₂×#88322	—	2	2	no	—
138	C₂×#88322	—	2	2	no	—
139	C₂×#88322	—	2	2	no	—
140	C₂×#88322	—	2	2	no	—
141	C₂×#88322	—	2	2	no	—
142	C₂×#88322	—	2	1	no	—
143	C₂×#88322	—	2	1	no	—
144	C₂×#88322	C₂×Q₈	1	2	yes	—
145	C₂×#88322	C₂×Q₈	1	2	yes	—
146	C₂³×C₅₃	—	2	1	no	—
147	C₂³×C₅₃	—	2	1	no	—
148	C₂³×C₅₃	—	2	1	no	—
149	C₂³×C₅₃	—	2	1	no	—
150	C₂³×C₅₃	—	2	1	no	—
151	C₂³×C₅₃	—	2	1	no	—
152	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
153	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	—	2	1	no	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	1	no	—
157	C₂×C₄×C₅₃	—	2	1	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	2	1	no	—
160	C₂×C₄×C₅₃	—	2	2	no	—
161	C₂×C₄×C₅₃	—	2	2	no	—
162	C₂×C₄×C₅₃	—	2	2	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	1	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	2	no	—
167	C₂×C₄×C₅₃	—	2	2	no	—
168	C₂×C₄×C₅₃	—	2	2	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	2	no	—
176	C₂×C₄×C₅₃	—	2	2	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	4	4	no	—
180	C₂×C₄×C₅₃	—	4	4	no	—
181	C₂×C₄×C₅₃	#117	1	2	yes	—
182	C₂×C₄×C₅₃	#117	1	2	yes	—
183	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
184	C₂×C₄×C₅₃	#131	1	2	yes	—
185	C₂×C₄×C₅₃	#131	1	2	yes	—
186	C₂×C₄×C₅₃	#131	1	2	yes	—
187	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
188	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
192	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
193	#88322	—	2	2	no	—
194	#88322	—	2	2	no	—
195	#88322	—	4	1	no	—
196	#88322	—	4	1	no	—
197	#88322	—	4	2	no	—
198	#88322	—	4	2	no	—
199	#22626	—	53	1	no	maximal, Sylow
200	C₂²×C₅₃	—	2	2	no	—
201	C₂²×C₅₃	—	2	2	no	—
202	C₂²×C₅₃	—	2	2	no	—
203	C₂²×C₅₃	—	4	1	no	—
204	C₂²×C₅₃	—	4	1	no	—
205	C₂²×C₅₃	—	4	1	no	—
206	C₂²×C₅₃	—	4	1	no	—
207	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
208	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
209	C₂²×C₅₃	#518	1	1	yes	characteristic
210	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
211	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
212	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
213	C₂²×C₅₃	#568	1	1	yes	characteristic
214	C₄×C₅₃	—	2	2	no	—
215	C₄×C₅₃	—	2	2	no	—
216	C₄×C₅₃	—	2	2	no	—
217	C₄×C₅₃	—	4	1	no	—
218	C₄×C₅₃	—	4	2	no	—
219	C₄×C₅₃	—	4	2	no	—
220	C₄×C₅₃	—	4	2	no	—
221	C₄×C₅₃	—	4	2	no	—
222	C₄×C₅₃	#433	1	4	yes	—
223	C₄×C₅₃	#553	1	1	yes	characteristic
224	C₄×C₅₃	#553	1	1	yes	characteristic
225	C₄×C₅₃	#554	1	1	yes	characteristic
226	C₄×C₅₃	#554	1	1	yes	characteristic
227	#2285	D₅₃	1	1	yes	characteristic
228	#1939	—	53	1	no	—
229	#1939	—	53	1	no	—
230	#1917	—	53	2	no	—
231	#1916	—	53	2	no	—
232	#1798	—	53	2	no	—
233	#1797	—	53	1	no	—
234	#1770	—	53	1	no	—
235	C₈×C₄⋊C₄	—	53	1	no	—
236	#1750	—	53	2	no	—
237	C₄×C₂²⋊C₈	—	53	1	no	—
238	C₂×C₅₃	—	4	2	no	—
239	C₂×C₅₃	#2345	1	1	yes	characteristic
240	C₂×C₅₃	C₂×#433	1	1	yes	characteristic
241	C₂×C₅₃	#2924	1	1	yes	characteristic
242	C₂×C₅₃	#2960	1	1	yes	characteristic
243	C₂×C₅₃	#2962	1	1	yes	characteristic
244	C₂×C₅₃	#2979	1	1	yes	characteristic
245	C₂×C₅₃	#2981	1	1	yes	characteristic
246	C₂×#117	C₂×D₅₃	1	2	yes	—
247	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
248	C₂×C₄⋊C₈	—	53	1	no	—
249	C₂×C₄⋊C₈	—	53	1	no	—
250	C₂×C₄⋊C₈	—	53	2	no	—
251	C₂×C₄⋊C₈	—	53	2	no	—
252	C₂×C₂²⋊C₈	—	53	1	no	—
253	C₂×C₂²⋊C₈	—	53	2	no	—
254	C₂×C₂²⋊C₈	—	53	1	no	—
255	C₂×#97	—	53	2	no	—
256	C₂×C₄×C₈	—	53	1	no	—
257	C₂×C₄×C₈	—	53	1	no	—
258	C₂×C₄×C₈	—	53	2	no	—
259	#384	Dic₅₃	1	1	yes	characteristic
260	#384	Dic₅₃	1	1	yes	characteristic
261	#382	Dic₅₃	1	2	yes	—
262	#380	Dic₅₃	1	1	yes	characteristic
263	#380	Dic₅₃	1	1	yes	characteristic
264	#380	Dic₅₃	1	2	yes	—
265	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
266	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
267	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
268	#336	—	53	2	no	—
269	#336	—	53	2	no	—
270	#336	—	53	2	no	—
271	#336	—	53	1	no	—
272	#336	—	53	2	no	—
273	#336	—	53	1	no	—
274	#336	—	53	2	no	—
275	C₅₃	#22626	1	1	yes	characteristic, complemented, Sylow
276	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
277	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
278	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
279	C₂²×C₈	—	53	1	no	—
280	C₂²×C₈	—	53	1	no	—
281	C₂²×C₈	—	53	1	no	—
282	C₂²×C₈	—	53	1	no	—
283	C₂²×C₈	—	53	2	no	—
284	C₂²×C₈	—	53	2	no	—
285	#117	—	4	2	no	—
286	#117	—	4	2	no	—
287	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
288	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
291	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
292	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
293	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
294	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
295	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
296	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
297	C₂×C₄²	—	2	2	no	—
298	C₂×C₄²	—	2	2	no	—
299	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
300	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
301	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
302	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
303	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
304	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
305	C₄⋊C₈	—	106	2	no	—
306	C₄⋊C₈	—	106	2	no	—
307	C₄⋊C₈	—	212	2	no	—
308	C₄⋊C₈	—	212	2	no	—
309	C₂²⋊C₈	—	53	4	no	—
310	C₂²⋊C₈	—	106	4	no	—
311	C₂²⋊C₈	—	106	2	no	—
312	#97	—	212	2	no	—
313	C₄×C₈	—	106	2	no	—
314	C₄×C₈	—	106	2	no	—
315	C₄×C₈	—	212	2	no	—
316	#95	C₂×Dic₅₃	1	2	yes	—
317	#95	C₂×Dic₅₃	1	2	yes	—
318	#95	C₂×Dic₅₃	1	2	yes	—
319	#95	C₂×Dic₅₃	1	2	yes	—
320	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	4	1	no	—
331	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
332	C₂²×C₄	#11719949	1	2	yes	—
333	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
334	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
335	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
336	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
337	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
338	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
339	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
340	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
341	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
342	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
343	C₂×C₈	—	53	2	no	—
344	C₂×C₈	—	53	2	no	—
345	C₂×C₈	—	106	1	no	—
346	C₂×C₈	—	106	1	no	—
347	C₂×C₈	—	106	1	no	—
348	C₂×C₈	—	106	1	no	—
349	C₂×C₈	—	106	1	no	—
350	C₂×C₈	—	106	1	no	—
351	C₂×C₈	—	106	2	no	—
352	C₂×C₈	—	106	2	no	—
353	C₂×C₈	—	106	1	no	—
354	C₂×C₈	—	106	1	no	—
355	C₂×C₈	—	106	2	no	—
356	C₂×C₈	—	106	2	no	—
357	C₂×C₈	—	106	2	no	—
358	C₂×C₈	—	106	2	no	—
359	C₂×C₈	—	106	1	no	—
360	C₂×C₈	—	106	1	no	—
361	C₄⋊C₄	—	2	4	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	4	2	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₂²⋊C₄	—	4	2	no	—
370	C₂²⋊C₄	—	4	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	2	no	—
373	C₄²	—	2	1	no	—
374	C₄²	—	2	1	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	#11719945	1	2	yes	—
380	C₄²	#11719945	1	2	yes	—
381	C₂³	—	2	1	no	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	1	no	—
386	C₂³	—	2	1	no	—
387	C₂³	Q₈×Dic₅₃	1	2	yes	—
388	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	—	4	4	no	—
416	C₂×C₄	#13133176	1	2	yes	—
417	C₂×C₄	#13133176	1	2	yes	—
418	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
419	C₂×C₄	#13133224	1	2	yes	—
420	C₂×C₄	#13133224	1	2	yes	—
421	C₂×C₄	#13133224	1	2	yes	—
422	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
423	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
424	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
425	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
426	C₂×C₄	#13133262	1	1	yes	characteristic
427	C₂×C₄	#13133262	1	1	yes	characteristic
428	C₈	—	106	2	no	—
429	C₈	—	106	2	no	—
430	C₈	—	212	1	no	—
431	C₈	—	212	1	no	—
432	C₈	—	212	2	no	—
433	C₈	—	212	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	4	1	no	—
438	C₂²	—	4	1	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	4	1	no	—
441	C₂²	C₂×#13133176	1	1	yes	characteristic
442	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
443	C₂²	#16811098	1	1	yes	characteristic
444	C₂²	C₂×#13133224	1	1	yes	characteristic
445	C₂²	C₂×#13133224	1	1	yes	characteristic
446	C₂²	C₂×#13133224	1	1	yes	characteristic
447	C₂²	#16811390	1	1	yes	characteristic
448	C₄	—	2	2	no	—
449	C₄	—	2	2	no	—
450	C₄	—	2	2	no	—
451	C₄	—	4	2	no	—
452	C₄	—	4	2	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	1	no	—
456	C₄	#16810684	1	4	yes	—
457	C₄	#16811321	1	1	yes	characteristic
458	C₄	#16811321	1	1	yes	characteristic
459	C₄	#16811328	1	1	yes	characteristic
460	C₄	#16811328	1	1	yes	characteristic
461	C₂	—	4	2	no	—
462	C₂	#20277592	1	1	yes	characteristic
463	C₂	C₂×#16810684	1	1	yes	characteristic
464	C₂	#20283373	1	1	yes	characteristic
465	C₂	#20283650	1	1	yes	characteristic
466	C₂	#20283664	1	1	yes	characteristic
467	C₂	#20283782	1	1	yes	characteristic
468	C₂	#20283791	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#2285	C₂
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#2285	D₅₃
hypocenter	C₅₃	#22626
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2285

Complements

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

N	H ≅ G/N	type
1. #20275434	C₂	semidirect
2. #20275434	C₂	semidirect
3. #20275281	C₂	semidirect
4. #20275274	C₂	semidirect
7. #20274584	C₂	semidirect
8. C₄⋊C₄×#88322	C₂	semidirect
275. C₅₃	#22626	semidirect

Maximal subgroups

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

subgroup	index
#20275434	2
#20275281	2
#20275274	2
#20274699	2
#20274695	2
#20274584	2
C₄⋊C₄×#88322	2
#20274481	2
C₄×C₂²⋊#88322	2
C₅₃×#2285	2
#22626	53