Subgroups — C₅₃×#19428 #27226940

Order 13568.

704 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3226	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#3036	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1701	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1693	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#1540	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₅₃	—	2	1	no	—
11	C₂²×SD₁₆×C₅₃	—	2	1	no	—
12	C₅₃×#532	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#500	C₂²	1	2	yes	complemented
16	C₅₃×#486	C₂²	1	2	yes	complemented
17	C₅₃×#480	—	2	2	no	—
18	C₅₃×#474	—	2	2	no	—
19	C₅₃×#463	—	2	2	no	—
20	C₅₃×#461	—	2	2	no	—
21	C₅₃×#461	C₂²	1	2	yes	—
22	C₅₃×#461	C₂²	1	4	yes	—
23	C₅₃×#459	C₂²	1	2	yes	—
24	C₅₃×#436	—	2	2	no	—
25	C₅₃×#436	C₂²	1	4	yes	—
26	C₅₃×#431	C₂²	1	1	yes	characteristic
27	C₅₃×#431	C₂²	1	1	yes	characteristic
28	C₅₃×#423	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#417	—	2	1	no	—
31	C₂×#103×C₅₃	—	2	1	no	—
32	C₅₃×#407	—	2	2	no	—
33	C₅₃×#334	C₂²	1	4	yes	complemented
34	C₅₃×#332	C₂²	1	2	yes	complemented
35	C₅₃×#332	C₂²	1	4	yes	complemented
36	C₅₃×#331	C₂²	1	2	yes	complemented
37	C₂×#41×C₅₃	—	2	1	no	—
38	C₂²×Q₈×C₅₃	—	2	1	no	—
39	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	—	2	1	no	—
41	AGL(1,ℤ/8)×C₅₃	—	4	4	no	—
42	C₂×SD₁₆×C₅₃	—	2	4	no	—
43	C₂×SD₁₆×C₅₃	—	2	2	no	—
44	C₂×SD₁₆×C₅₃	—	4	4	no	—
45	C₂×D₈×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	4	no	—
47	C₂×M₁₆×C₅₃	—	2	1	no	—
48	C₂²×C₈×C₅₃	—	2	1	no	—
49	C₄⋊Q₈×C₅₃	—	2	1	no	—
50	C₄⋊Q₈×C₅₃	—	2	1	no	—
51	C₄⋊Q₈×C₅₃	—	2	1	no	—
52	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
53	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
54	#127×C₅₃	C₂³	1	1	yes	characteristic
55	#124×C₅₃	—	2	1	no	—
56	#121×C₅₃	—	2	1	no	—
57	#121×C₅₃	—	2	1	no	—
58	C₄×D₄×C₅₃	—	2	1	no	—
59	C₄×D₄×C₅₃	—	2	1	no	—
60	C₄×D₄×C₅₃	—	2	1	no	—
61	C₄×D₄×C₅₃	—	2	1	no	—
62	C₄×D₄×C₅₃	D₄	1	2	yes	—
63	C₄×D₄×C₅₃	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	#107×C₅₃	—	2	4	no	—
70	C₄⋊C₈×C₅₃	—	2	1	no	—
71	C₄⋊C₈×C₅₃	—	2	1	no	—
72	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
73	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
74	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
75	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
76	#103×C₅₃	—	2	4	no	—
77	#103×C₅₃	—	2	2	no	—
78	#102×C₅₃	—	2	4	no	—
79	#102×C₅₃	—	2	2	no	—
80	C₂²⋊C₈×C₅₃	—	2	4	no	—
81	C₂²⋊C₈×C₅₃	—	2	1	no	—
82	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
83	C₄×C₈×C₅₃	—	2	1	no	—
84	C₄×C₈×C₅₃	C₂³	1	2	yes	—
85	C₂⁴×C₅₃	—	2	1	no	—
86	#41×C₅₃	—	4	2	no	—
87	#41×C₅₃	—	4	2	no	—
88	C₂×Q₈×C₅₃	—	2	1	no	—
89	C₂×Q₈×C₅₃	—	2	2	no	—
90	C₂×Q₈×C₅₃	—	4	1	no	—
91	C₂×Q₈×C₅₃	—	4	2	no	—
92	C₂×Q₈×C₅₃	—	4	2	no	—
93	C₂×D₄×C₅₃	—	2	1	no	—
94	C₂×D₄×C₅₃	—	2	2	no	—
95	C₂×D₄×C₅₃	—	2	2	no	—
96	C₂×D₄×C₅₃	—	2	1	no	—
97	C₂×D₄×C₅₃	—	2	1	no	—
98	C₂×D₄×C₅₃	—	2	1	no	—
99	C₂×D₄×C₅₃	—	4	1	no	—
100	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₅₃	—	2	1	no	—
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	4	1	no	—
105	C₂²×C₄×C₅₃	—	4	1	no	—
106	C₂²×C₄×C₅₃	—	8	2	no	—
107	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
110	SD₁₆×C₅₃	—	4	8	no	—
111	SD₁₆×C₅₃	—	4	4	no	—
112	D₈×C₅₃	—	4	4	no	—
113	M₁₆×C₅₃	—	4	2	no	—
114	M₁₆×C₅₃	—	8	4	no	—
115	C₂×C₈×C₅₃	—	2	2	no	—
116	C₂×C₈×C₅₃	—	2	2	no	—
117	C₂×C₈×C₅₃	—	2	2	no	—
118	C₂×C₈×C₅₃	—	4	2	no	—
119	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
120	C₄⋊C₄×C₅₃	—	2	1	no	—
121	C₄⋊C₄×C₅₃	—	2	2	no	—
122	C₄⋊C₄×C₅₃	—	2	1	no	—
123	C₄⋊C₄×C₅₃	—	2	2	no	—
124	C₄⋊C₄×C₅₃	—	2	1	no	—
125	C₄⋊C₄×C₅₃	—	4	1	no	—
126	C₄⋊C₄×C₅₃	—	4	1	no	—
127	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
128	C₂²⋊C₄×C₅₃	—	2	2	no	—
129	C₂²⋊C₄×C₅₃	—	2	2	no	—
130	C₂²⋊C₄×C₅₃	—	4	1	no	—
131	C₂²⋊C₄×C₅₃	—	4	1	no	—
132	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₅₃	#41	1	1	yes	characteristic
135	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
136	C₂³×C₅₃	—	2	2	no	—
137	C₂³×C₅₃	—	4	2	no	—
138	C₂³×C₅₃	—	4	1	no	—
139	C₂³×C₅₃	—	8	1	no	—
140	C₂³×C₅₃	—	8	1	no	—
141	C₂³×C₅₃	#121	1	1	yes	characteristic
142	Q₈×C₅₃	—	4	2	no	—
143	Q₈×C₅₃	—	4	2	no	—
144	Q₈×C₅₃	—	4	1	no	—
145	Q₈×C₅₃	—	4	1	no	—
146	Q₈×C₅₃	—	8	2	no	—
147	D₄×C₅₃	—	2	2	no	—
148	D₄×C₅₃	—	2	2	no	—
149	D₄×C₅₃	—	4	1	no	—
150	D₄×C₅₃	—	4	1	no	—
151	D₄×C₅₃	—	4	2	no	—
152	D₄×C₅₃	—	4	2	no	—
153	D₄×C₅₃	—	8	1	no	—
154	D₄×C₅₃	—	8	2	no	—
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	4	1	no	—
158	C₂×C₄×C₅₃	—	4	4	no	—
159	C₂×C₄×C₅₃	—	4	1	no	—
160	C₂×C₄×C₅₃	—	4	2	no	—
161	C₂×C₄×C₅₃	—	8	2	no	—
162	C₂×C₄×C₅₃	—	8	1	no	—
163	C₂×C₄×C₅₃	—	8	2	no	—
164	C₂×C₄×C₅₃	—	8	2	no	—
165	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
166	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
167	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
168	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
169	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
170	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
173	C₈×C₅₃	—	2	4	no	—
174	C₈×C₅₃	—	4	2	no	—
175	C₈×C₅₃	—	4	2	no	—
176	C₈×C₅₃	—	4	2	no	—
177	#19428	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	—	2	1	no	—
180	C₂²×C₅₃	—	4	2	no	—
181	C₂²×C₅₃	—	4	2	no	—
182	C₂²×C₅₃	—	4	2	no	—
183	C₂²×C₅₃	—	8	4	no	—
184	C₂²×C₅₃	—	8	1	no	—
185	C₂²×C₅₃	—	8	1	no	—
186	C₂²×C₅₃	—	8	1	no	—
187	C₂²×C₅₃	#469	1	2	yes	—
188	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	2	1	no	—
191	C₄×C₅₃	—	2	1	no	—
192	C₄×C₅₃	—	4	2	no	—
193	C₄×C₅₃	—	8	1	no	—
194	C₄×C₅₃	—	8	2	no	—
195	C₄×C₅₃	#469	1	2	yes	—
196	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
197	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₄×C₅₃	#577	1	2	yes	—
199	#3226	C₂×C₅₃	1	1	yes	characteristic, complemented
200	#3036	C₂×C₅₃	1	1	yes	characteristic
201	C₂×#461	C₂×C₅₃	1	1	yes	characteristic, complemented
202	#2975	C₂×C₅₃	1	1	yes	characteristic, complemented
203	#1701	C₂×C₅₃	1	4	yes	complemented
204	#1693	C₂×C₅₃	1	4	yes	complemented
205	#1570	C₂×C₅₃	1	1	yes	characteristic, complemented
206	#1546	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#1540	C₂×C₅₃	1	1	yes	characteristic, complemented
208	C₂×C₅₃	—	2	2	no	—
209	C₂×C₅₃	—	8	2	no	—
210	C₂×C₅₃	—	16	1	no	—
211	C₂×C₅₃	#3042	1	1	yes	characteristic
212	C₂×C₅₃	C₂×#469	1	1	yes	characteristic
213	C₂×C₅₃	#3069	1	1	yes	characteristic
214	C₂×AGL(1,ℤ/8)	—	2	1	no	—
215	C₂²×SD₁₆	—	2	1	no	—
216	#532	C₂²×C₅₃	1	1	yes	characteristic
217	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
218	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
219	#500	C₂²×C₅₃	1	2	yes	complemented
220	#486	C₂²×C₅₃	1	2	yes	complemented
221	#480	—	2	2	no	—
222	#474	—	2	2	no	—
223	#463	—	2	2	no	—
224	#461	—	2	2	no	—
225	#461	C₂²×C₅₃	1	2	yes	—
226	#461	C₂²×C₅₃	1	4	yes	—
227	#459	C₂²×C₅₃	1	2	yes	—
228	#436	—	2	2	no	—
229	#436	C₂²×C₅₃	1	4	yes	—
230	#431	C₂²×C₅₃	1	1	yes	characteristic
231	#431	C₂²×C₅₃	1	1	yes	characteristic
232	#423	C₂²×C₅₃	1	1	yes	characteristic
233	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
234	#417	—	2	1	no	—
235	C₂×#103	—	2	1	no	—
236	#407	—	2	2	no	—
237	#334	C₂²×C₅₃	1	4	yes	complemented
238	#332	C₂²×C₅₃	1	2	yes	complemented
239	#332	C₂²×C₅₃	1	4	yes	complemented
240	#331	C₂²×C₅₃	1	2	yes	complemented
241	C₅₃	#19428	1	1	yes	characteristic, direct product factor, Sylow
242	C₂×#41	—	2	1	no	—
243	C₂²×Q₈	—	2	1	no	—
244	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
245	C₂³×C₄	—	2	1	no	—
246	AGL(1,ℤ/8)	—	4	4	no	—
247	C₂×SD₁₆	—	2	4	no	—
248	C₂×SD₁₆	—	2	2	no	—
249	C₂×SD₁₆	—	4	4	no	—
250	C₂×D₈	—	2	2	no	—
251	C₂×M₁₆	—	2	1	no	—
252	C₂×M₁₆	—	2	4	no	—
253	C₂²×C₈	—	2	1	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	C₄⋊Q₈	—	2	1	no	—
256	C₄⋊Q₈	—	2	1	no	—
257	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
258	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
259	#127	C₂³×C₅₃	1	1	yes	characteristic
260	#124	—	2	1	no	—
261	#121	—	2	1	no	—
262	#121	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	2	1	no	—
265	C₄×D₄	—	2	1	no	—
266	C₄×D₄	—	2	1	no	—
267	C₄×D₄	D₄×C₅₃	1	2	yes	—
268	C₄×D₄	C₂³×C₅₃	1	2	yes	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
272	C₂×C₂²⋊C₄	—	2	1	no	—
273	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
274	#107	—	2	4	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	2	1	no	—
277	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
278	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
279	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
280	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
281	#103	—	2	4	no	—
282	#103	—	2	2	no	—
283	#102	—	2	4	no	—
284	#102	—	2	2	no	—
285	C₂²⋊C₈	—	2	1	no	—
286	C₂²⋊C₈	—	2	4	no	—
287	C₂²⋊C₈	D₄×C₅₃	1	2	yes	—
288	C₄×C₈	—	2	1	no	—
289	C₄×C₈	C₂³×C₅₃	1	2	yes	—
290	C₂⁴	—	2	1	no	—
291	#41	—	4	2	no	—
292	#41	—	4	2	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	1	no	—
296	C₂×Q₈	—	4	2	no	—
297	C₂×Q₈	—	4	2	no	—
298	C₂×D₄	—	2	1	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	2	no	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	1	no	—
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	4	1	no	—
305	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	4	1	no	—
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	8	2	no	—
312	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
314	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
315	SD₁₆	—	4	8	no	—
316	SD₁₆	—	4	4	no	—
317	D₈	—	4	4	no	—
318	M₁₆	—	4	2	no	—
319	M₁₆	—	8	4	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	4	2	no	—
324	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	4	1	no	—
336	C₂²⋊C₄	—	4	1	no	—
337	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
338	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
339	C₄²	#41×C₅₃	1	1	yes	characteristic
340	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
341	C₂³	—	2	2	no	—
342	C₂³	—	4	2	no	—
343	C₂³	—	4	1	no	—
344	C₂³	—	8	1	no	—
345	C₂³	—	8	1	no	—
346	C₂³	#121×C₅₃	1	1	yes	characteristic
347	Q₈	—	4	2	no	—
348	Q₈	—	4	1	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	4	1	no	—
351	Q₈	—	8	2	no	—
352	D₄	—	2	2	no	—
353	D₄	—	2	2	no	—
354	D₄	—	4	1	no	—
355	D₄	—	4	1	no	—
356	D₄	—	4	2	no	—
357	D₄	—	4	2	no	—
358	D₄	—	8	1	no	—
359	D₄	—	8	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	4	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	2	no	—
367	C₂×C₄	—	8	1	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	—	8	2	no	—
370	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
371	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
372	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
373	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
374	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
375	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
377	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
378	C₈	—	2	4	no	—
379	C₈	—	4	2	no	—
380	C₈	—	4	2	no	—
381	C₈	—	4	2	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	8	4	no	—
388	C₂²	—	8	1	no	—
389	C₂²	—	8	1	no	—
390	C₂²	—	8	1	no	—
391	C₂²	C₅₃×#469	1	2	yes	—
392	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
393	C₄	—	2	1	no	—
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	4	2	no	—
397	C₄	—	8	1	no	—
398	C₄	—	8	2	no	—
399	C₄	C₅₃×#469	1	2	yes	—
400	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
401	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
402	C₄	C₅₃×#577	1	2	yes	—
403	C₂	—	2	2	no	—
404	C₂	—	8	2	no	—
405	C₂	—	16	1	no	—
406	C₂	C₅₃×#3042	1	1	yes	characteristic
407	C₂	C₂×C₅₃×#469	1	1	yes	characteristic
408	C₂	C₅₃×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3036	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3226	C₂	semidirect
3. C₂×C₅₃×#461	C₂	semidirect
4. C₅₃×#2975	C₂	semidirect
5. C₅₃×#1701	C₂	semidirect
6. C₅₃×#1693	C₂	semidirect
7. C₅₃×#1570	C₂	semidirect
8. C₅₃×#1546	C₂	semidirect
9. C₅₃×#1540	C₂	semidirect
15. C₅₃×#500	C₂²	semidirect
16. C₅₃×#486	C₂²	semidirect
33. C₅₃×#334	C₂²	semidirect
34. C₅₃×#332	C₂²	semidirect
35. C₅₃×#332	C₂²	semidirect
36. C₅₃×#331	C₂²	semidirect
177. #19428	C₅₃	direct (N × H)
199. #3226	C₂×C₅₃	semidirect
201. C₂×#461	C₂×C₅₃	semidirect
202. #2975	C₂×C₅₃	semidirect
203. #1701	C₂×C₅₃	semidirect
204. #1693	C₂×C₅₃	semidirect
205. #1570	C₂×C₅₃	semidirect
206. #1546	C₂×C₅₃	semidirect
207. #1540	C₂×C₅₃	semidirect
219. #500	C₂²×C₅₃	semidirect
220. #486	C₂²×C₅₃	semidirect
237. #334	C₂²×C₅₃	semidirect
238. #332	C₂²×C₅₃	semidirect
239. #332	C₂²×C₅₃	semidirect
240. #331	C₂²×C₅₃	semidirect
241. C₅₃	#19428	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3226	2
C₅₃×#3036	2
C₂×C₅₃×#461	2
C₅₃×#2975	2
C₅₃×#1701	2
C₅₃×#1693	2
C₅₃×#1570	2
C₅₃×#1546	2
C₅₃×#1540	2
#19428	53

Subgroups — C53×#19428 #27226940

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#19428 #27226940