Subgroups — #27274194

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810675	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810645	C₂	1	1	yes	maximal, characteristic
3	#384×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#20272599	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272599	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272598	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719912	—	2	1	no	—
9	C₂×C₄⋊C₄×D₅₃	—	2	1	no	—
10	C₂×C₄⋊C₄×D₅₃	—	2	1	no	—
11	C₂×C₄⋊C₄×D₅₃	C₄	1	2	yes	—
12	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
13	#16810675	—	2	2	no	—
14	#16810675	C₄	1	4	yes	complemented
15	#16810645	—	2	2	no	—
16	#16810645	C₄	1	4	yes	complemented
17	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×#13133146	—	2	1	no	—
20	C₂×#13133129	—	2	1	no	—
21	C₂×C₈×Dic₅₃	—	2	1	no	—
22	#16810493	C₂²	1	1	yes	characteristic
23	#16810492	—	2	1	no	—
24	#16810492	—	2	1	no	—
25	#16810492	C₄	1	2	yes	—
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#384	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	—	2	2	no	—
30	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
33	C₂×#11719912	—	2	4	no	—
34	C₂×#11719912	—	4	2	no	—
35	C₂×#11719912	—	4	2	no	—
36	C₂²×#88322	—	2	1	no	—
37	C₂²×#88322	—	2	1	no	—
38	C₂×C₄⋊Dic₅₃	—	2	1	no	—
39	C₂×C₄⋊Dic₅₃	—	2	1	no	—
40	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
41	C₂×#11719919	—	2	2	no	—
42	C₂×#11719919	—	2	2	no	—
43	C₂×#11719919	—	2	2	no	—
44	C₄⋊C₄×D₅₃	—	4	2	no	—
45	C₄⋊C₄×D₅₃	—	4	2	no	—
46	C₄⋊C₄×D₅₃	—	4	4	no	—
47	C₄⋊C₄×D₅₃	—	4	2	no	—
48	C₄⋊C₄×D₅₃	—	4	2	no	—
49	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
51	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
52	C₄²×D₅₃	C₂×C₄	1	2	yes	—
53	C₄²×D₅₃	C₂×C₄	1	2	yes	—
54	C₄²×D₅₃	D₄	1	2	yes	—
55	C₄²×D₅₃	Q₈	1	2	yes	—
56	C₄⋊#88322	—	2	1	no	—
57	C₄⋊#88322	—	2	1	no	—
58	C₄⋊#88322	C₂×C₄	1	2	yes	complemented
59	C₄⋊#88322	C₂×C₄	1	2	yes	complemented
60	#13133146	—	2	4	no	—
61	#13133129	—	2	4	no	—
62	C₈×Dic₅₃	—	2	4	no	—
63	C₂²×C₈×C₅₃	—	2	1	no	—
64	C₂²×C₈×C₅₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	—	2	1	no	—
70	C₄⋊C₈×C₅₃	—	2	1	no	—
71	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	complemented
72	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	complemented
73	C₂³×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	—	2	2	no	—
75	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₅₃	—	2	2	no	—
79	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₅₃	—	2	1	no	—
83	C₂×C₄×D₅₃	—	2	1	no	—
84	C₂×C₄×D₅₃	—	2	1	no	—
85	C₂×C₄×D₅₃	—	2	1	no	—
86	C₂×C₄×D₅₃	—	2	2	no	—
87	C₂×C₄×D₅₃	—	2	2	no	—
88	C₂×C₄×D₅₃	—	4	4	no	—
89	C₂×C₄×D₅₃	—	4	4	no	—
90	C₂×C₄×D₅₃	—	4	4	no	—
91	C₂×C₄×D₅₃	C₄²	1	2	yes	—
92	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
95	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
96	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
97	C₂×C₄×D₅₃	D₈	1	2	yes	—
98	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
99	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
100	C₂×C₄×D₅₃	Q₁₆	1	2	yes	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	2	no	—
103	C₂×C₈×C₅₃	—	4	1	no	—
104	C₂×C₈×C₅₃	—	4	1	no	—
105	C₂×C₈×C₅₃	—	4	1	no	—
106	C₂×C₈×C₅₃	—	4	1	no	—
107	C₄⋊C₄×C₅₃	—	4	2	no	—
108	C₄⋊C₄×C₅₃	—	4	1	no	—
109	C₄⋊C₄×C₅₃	—	4	1	no	—
110	C₄⋊C₄×C₅₃	—	4	1	no	—
111	C₄⋊C₄×C₅₃	—	4	1	no	—
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
116	C₄⋊Dic₅₃	—	4	1	no	—
117	C₄⋊Dic₅₃	—	4	1	no	—
118	C₄⋊Dic₅₃	—	4	1	no	—
119	C₄⋊Dic₅₃	—	4	1	no	—
120	C₄⋊Dic₅₃	—	4	2	no	—
121	#11719919	—	4	4	no	—
122	#11719919	—	4	4	no	—
123	#11719919	—	4	4	no	—
124	C₄×Dic₅₃	C₄²	1	2	yes	—
125	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
126	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
127	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
128	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
129	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
130	C₂×#88322	—	2	2	no	—
131	C₂×#88322	—	2	2	no	—
132	C₂×#88322	—	4	1	no	—
133	C₂×#88322	—	4	1	no	—
134	C₂×#88322	—	4	1	no	—
135	C₂×#88322	—	4	1	no	—
136	#11719912	—	4	8	no	—
137	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×D₅₃	—	2	1	no	—
139	C₂²×D₅₃	—	2	1	no	—
140	C₂²×D₅₃	—	2	1	no	—
141	C₂²×D₅₃	—	2	1	no	—
142	C₂²×D₅₃	#95	1	2	yes	—
143	C₂²×D₅₃	#102	1	2	yes	—
144	C₂²×D₅₃	#103	1	2	yes	—
145	C₂×C₄×C₅₃	—	2	1	no	—
146	C₂×C₄×C₅₃	—	2	1	no	—
147	C₂×C₄×C₅₃	—	4	2	no	—
148	C₂×C₄×C₅₃	—	4	2	no	—
149	C₂×C₄×C₅₃	—	4	2	no	—
150	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
160	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
163	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
164	C₂×Dic₅₃	—	2	1	no	—
165	C₂×Dic₅₃	—	2	1	no	—
166	C₂×Dic₅₃	—	4	2	no	—
167	C₂×Dic₅₃	—	4	2	no	—
168	C₂×Dic₅₃	—	4	2	no	—
169	C₂×Dic₅₃	#95	1	2	yes	—
170	C₂×Dic₅₃	#95	1	2	yes	—
171	C₂×Dic₅₃	#95	1	2	yes	—
172	C₂×Dic₅₃	#100	1	2	yes	—
173	C₂×Dic₅₃	#101	1	2	yes	—
174	C₂×Dic₅₃	#102	1	2	yes	—
175	C₂×Dic₅₃	#103	1	2	yes	—
176	C₄×D₅₃	—	2	2	no	—
177	C₄×D₅₃	—	2	2	no	—
178	C₄×D₅₃	—	2	2	no	—
179	C₄×D₅₃	—	2	2	no	—
180	C₄×D₅₃	—	2	4	no	—
181	C₄×D₅₃	—	2	4	no	—
182	C₄×D₅₃	—	8	8	no	—
183	C₄×D₅₃	#102	1	4	yes	—
184	C₄×D₅₃	#103	1	4	yes	—
185	C₄×D₅₃	#106	1	4	yes	—
186	C₄×D₅₃	#107	1	4	yes	—
187	C₈×C₅₃	—	4	2	no	—
188	C₈×C₅₃	—	4	2	no	—
189	#88322	—	4	2	no	—
190	#88322	—	4	2	no	—
191	#8043	—	53	1	no	maximal, Sylow
192	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
193	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
194	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
195	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
196	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
197	C₂²×C₅₃	#418	1	1	yes	characteristic
198	C₂²×C₅₃	#419	1	1	yes	characteristic
199	C₂×D₅₃	—	2	2	no	—
200	C₂×D₅₃	—	2	2	no	—
201	C₂×D₅₃	—	2	2	no	—
202	C₂×D₅₃	—	2	2	no	—
203	C₂×D₅₃	—	2	2	no	—
204	C₂×D₅₃	—	2	2	no	—
205	C₂×D₅₃	#340	1	4	yes	—
206	C₄×C₅₃	—	2	1	no	—
207	C₄×C₅₃	—	2	1	no	—
208	C₄×C₅₃	—	8	2	no	—
209	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
210	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
211	C₄×C₅₃	#417	1	1	yes	characteristic
212	C₄×C₅₃	#417	1	1	yes	characteristic
213	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
214	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
215	C₄×C₅₃	#428	1	1	yes	characteristic
216	C₄×C₅₃	#428	1	1	yes	characteristic
217	Dic₅₃	—	2	2	no	—
218	Dic₅₃	—	8	2	no	—
219	Dic₅₃	#340	1	4	yes	—
220	Dic₅₃	#343	1	4	yes	—
221	C₂×#431	—	53	1	no	—
222	C₂×#423	—	53	1	no	—
223	C₂×#384	—	53	1	no	—
224	#1296	—	53	1	no	—
225	#1296	—	53	1	no	—
226	#1296	—	53	1	no	—
227	#1296	D₅₃	1	1	yes	characteristic, complemented
228	C₂×C₅₃	#1545	1	1	yes	characteristic
229	C₂×C₅₃	#1546	1	1	yes	characteristic
230	C₂×C₅₃	#1548	1	1	yes	characteristic
231	C₂×C₅₃	#1549	1	1	yes	characteristic
232	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
233	C₂×C₅₃	#1736	1	1	yes	characteristic
234	C₂×C₅₃	C₂×#343	1	1	yes	characteristic
235	D₅₃	—	2	4	no	—
236	C₂²×M₁₆	—	106	1	no	—
237	C₂²×C₄⋊C₄	—	53	2	no	—
238	C₂²×C₄⋊C₄	—	106	1	no	—
239	C₂²×C₄⋊C₄	—	106	1	no	—
240	C₂²×C₄²	—	53	1	no	—
241	#431	—	53	4	no	—
242	#431	—	106	2	no	—
243	#423	—	53	4	no	—
244	#423	—	106	2	no	—
245	C₂×C₄⋊C₈	—	53	1	no	—
246	C₂×C₄⋊C₈	—	53	1	no	—
247	C₂×C₄⋊C₈	—	106	1	no	—
248	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
249	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
250	C₂×C₂²⋊C₈	—	106	1	no	—
251	C₂×C₄×C₈	—	106	1	no	—
252	#384	—	53	2	no	—
253	#384	—	53	1	no	—
254	#384	—	106	1	no	—
255	#384	—	106	1	no	—
256	#384	C₂×D₅₃	1	1	yes	characteristic
257	C₅₃	#8043	1	1	yes	characteristic, complemented, Sylow
258	C₂³×C₄	—	53	1	no	—
259	C₂³×C₄	—	53	1	no	—
260	C₂³×C₄	—	53	1	no	—
261	C₂³×C₄	—	106	2	no	—
262	C₂×M₁₆	—	106	4	no	—
263	C₂×M₁₆	—	212	2	no	—
264	C₂×M₁₆	—	212	2	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	—	106	1	no	—
268	C₂²×C₈	—	106	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	53	2	no	—
272	C₂×C₄⋊C₄	—	106	2	no	—
273	C₂×C₄⋊C₄	—	106	2	no	—
274	C₂×C₄⋊C₄	—	106	2	no	—
275	C₂×C₄⋊C₄	—	106	1	no	—
276	C₂×C₄⋊C₄	—	106	1	no	—
277	C₂×C₄⋊C₄	—	212	2	no	—
278	C₂×C₄⋊C₄	—	212	2	no	—
279	C₂×C₄⋊C₄	—	212	2	no	—
280	C₂×C₄⋊C₄	—	212	2	no	—
281	C₂×C₄⋊C₄	—	212	4	no	—
282	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
283	C₂×C₄²	—	53	1	no	—
284	C₂×C₄²	—	53	1	no	—
285	C₂×C₄²	—	53	1	no	—
286	C₂×C₄²	—	53	2	no	—
287	C₂×C₄²	—	53	2	no	—
288	C₂×C₄²	—	53	2	no	—
289	C₂×C₄²	—	53	2	no	—
290	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
291	C₄⋊C₈	—	2	1	no	—
292	C₄⋊C₈	—	2	1	no	—
293	C₄⋊C₈	—	53	2	no	—
294	C₄⋊C₈	—	53	2	no	—
295	C₄⋊C₈	—	106	4	no	—
296	C₄⋊C₈	—	106	1	no	—
297	C₄⋊C₈	—	106	1	no	—
298	C₄⋊C₈	C₄×D₅₃	1	2	yes	complemented
299	C₄⋊C₈	C₄×D₅₃	1	2	yes	complemented
300	C₂²⋊C₈	—	106	4	no	—
301	C₄×C₈	—	106	4	no	—
302	C₂⁴	—	53	1	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	53	1	no	—
305	C₂²×C₄	—	53	1	no	—
306	C₂²×C₄	—	53	2	no	—
307	C₂²×C₄	—	53	2	no	—
308	C₂²×C₄	—	53	2	no	—
309	C₂²×C₄	—	53	2	no	—
310	C₂²×C₄	—	53	2	no	—
311	C₂²×C₄	—	53	2	no	—
312	C₂²×C₄	—	53	1	no	—
313	C₂²×C₄	—	53	2	no	—
314	C₂²×C₄	—	53	2	no	—
315	C₂²×C₄	—	53	2	no	—
316	C₂²×C₄	—	53	2	no	—
317	C₂²×C₄	—	106	2	no	—
318	C₂²×C₄	—	106	2	no	—
319	C₂²×C₄	—	106	1	no	—
320	C₂²×C₄	—	106	1	no	—
321	C₂²×C₄	—	106	1	no	—
322	C₂²×C₄	—	106	1	no	—
323	C₂²×C₄	—	106	2	no	—
324	C₂²×C₄	—	212	4	no	—
325	C₂²×C₄	—	212	4	no	—
326	C₂²×C₄	—	212	4	no	—
327	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
328	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
329	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
330	M₁₆	—	212	8	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	4	1	no	—
334	C₂×C₈	—	4	1	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	—	106	2	no	—
338	C₂×C₈	—	106	2	no	—
339	C₂×C₈	—	212	1	no	—
340	C₂×C₈	—	212	1	no	—
341	C₂×C₈	—	212	1	no	—
342	C₂×C₈	—	212	1	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	—	4	1	no	—
345	C₄⋊C₄	—	4	1	no	—
346	C₄⋊C₄	—	4	1	no	—
347	C₄⋊C₄	—	4	1	no	—
348	C₄⋊C₄	—	212	2	no	—
349	C₄⋊C₄	—	212	4	no	—
350	C₄⋊C₄	—	212	4	no	—
351	C₄⋊C₄	—	212	4	no	—
352	C₄⋊C₄	—	212	1	no	—
353	C₄⋊C₄	—	212	1	no	—
354	C₄⋊C₄	—	212	1	no	—
355	C₄⋊C₄	—	212	1	no	—
356	C₄²	—	53	2	no	—
357	C₄²	—	53	2	no	—
358	C₄²	—	53	2	no	—
359	C₄²	—	53	2	no	—
360	C₄²	—	53	2	no	—
361	C₄²	—	53	2	no	—
362	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
363	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
364	C₄²	D₄×D₅₃	1	1	yes	characteristic
365	C₄²	Q₈×D₅₃	1	1	yes	characteristic
366	C₂³	—	53	2	no	—
367	C₂³	—	53	2	no	—
368	C₂³	—	53	2	no	—
369	C₂³	—	106	1	no	—
370	C₂³	—	106	1	no	—
371	C₂³	—	106	1	no	—
372	C₂³	—	106	1	no	—
373	C₂³	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	53	2	no	—
380	C₂×C₄	—	53	2	no	—
381	C₂×C₄	—	53	2	no	—
382	C₂×C₄	—	53	2	no	—
383	C₂×C₄	—	53	2	no	—
384	C₂×C₄	—	53	2	no	—
385	C₂×C₄	—	53	4	no	—
386	C₂×C₄	—	53	4	no	—
387	C₂×C₄	—	53	4	no	—
388	C₂×C₄	—	53	4	no	—
389	C₂×C₄	—	53	2	no	—
390	C₂×C₄	—	106	4	no	—
391	C₂×C₄	—	106	4	no	—
392	C₂×C₄	—	106	1	no	—
393	C₂×C₄	—	106	1	no	—
394	C₂×C₄	—	106	2	no	—
395	C₂×C₄	—	106	2	no	—
396	C₂×C₄	—	106	2	no	—
397	C₂×C₄	—	106	2	no	—
398	C₂×C₄	—	424	8	no	—
399	C₂×C₄	—	212	2	no	—
400	C₂×C₄	—	212	2	no	—
401	C₂×C₄	—	212	2	no	—
402	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
403	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
404	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
405	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
406	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
407	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
408	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
409	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
410	C₂×C₄	SD₁₆×D₅₃	1	1	yes	characteristic
411	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
412	C₂×C₄	#13133241	1	1	yes	characteristic
413	C₂×C₄	#13133244	1	1	yes	characteristic
414	C₂×C₄	#13133246	1	1	yes	characteristic
415	C₂×C₄	#13133249	1	1	yes	characteristic
416	C₈	—	4	2	no	—
417	C₈	—	4	2	no	—
418	C₈	—	212	2	no	—
419	C₈	—	212	2	no	—
420	C₂²	—	53	4	no	—
421	C₂²	—	106	2	no	—
422	C₂²	—	106	2	no	—
423	C₂²	—	106	2	no	—
424	C₂²	—	106	2	no	—
425	C₂²	—	106	2	no	—
426	C₂²	—	106	2	no	—
427	C₂²	#95×D₅₃	1	1	yes	characteristic
428	C₂²	#100×D₅₃	1	1	yes	characteristic
429	C₂²	#101×D₅₃	1	1	yes	characteristic
430	C₂²	#102×D₅₃	1	1	yes	characteristic
431	C₂²	#103×D₅₃	1	1	yes	characteristic
432	C₂²	#16810626	1	1	yes	characteristic
433	C₂²	#16810630	1	1	yes	characteristic
434	C₄	—	2	1	no	—
435	C₄	—	2	1	no	—
436	C₄	—	8	2	no	—
437	C₄	—	53	4	no	—
438	C₄	—	53	4	no	—
439	C₄	—	106	2	no	—
440	C₄	—	424	2	no	—
441	C₄	#102×D₅₃	1	1	yes	characteristic
442	C₄	#103×D₅₃	1	1	yes	characteristic
443	C₄	#16810621	1	1	yes	characteristic
444	C₄	#16810623	1	1	yes	characteristic
445	C₄	#106×D₅₃	1	1	yes	characteristic
446	C₄	#107×D₅₃	1	1	yes	characteristic
447	C₄	#16810662	1	1	yes	characteristic
448	C₄	#16810664	1	1	yes	characteristic
449	C₂	—	106	4	no	—
450	C₂	#20273381	1	1	yes	characteristic
451	C₂	#20273385	1	1	yes	characteristic
452	C₂	#20273394	1	1	yes	characteristic
453	C₂	#20273398	1	1	yes	characteristic
454	C₂	#340×D₅₃	1	1	yes	characteristic
455	C₂	#20274433	1	1	yes	characteristic
456	C₂	#343×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1296	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂³×D₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1296	D₅₃
hypocenter	C₅₃	#8043
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
4. #20272599	C₂	semidirect
5. #20272599	C₂	semidirect
6. #20272598	C₂	semidirect
7. C₅₃×#1296	C₂	semidirect
14. #16810675	C₄	semidirect
16. #16810645	C₄	semidirect
58. C₄⋊#88322	C₂×C₄	semidirect
59. C₄⋊#88322	C₂×C₄	semidirect
71. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
72. C₄⋊C₈×C₅₃	C₂×C₄	semidirect
227. #1296	D₅₃	semidirect
257. C₅₃	#8043	semidirect
298. C₄⋊C₈	C₄×D₅₃	semidirect
299. C₄⋊C₈	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810675	2
C₂×#16810645	2
#384×D₅₃	2
#20272599	2
#20272598	2
C₅₃×#1296	2
#8043	53