Subgroups — C₄×C₅₃×#510 #27240965

Order 13568.

674 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	C₂×C₅₃×#510	C₂	1	1	yes	maximal, characteristic
2	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2176	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2175	C₂	1	2	yes	maximal, complemented
6	C₄×Q₃₂×C₅₃	C₂	1	2	yes	maximal, complemented
7	C₄×SD₃₂×C₅₃	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2139	C₂	1	1	yes	maximal, characteristic
11	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×#41×C₅₃	—	2	1	no	—
15	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
16	C₅₃×#510	C₄	1	8	yes	direct product factor
17	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
18	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	—
19	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
21	C₄×Q₁₆×C₅₃	—	2	1	no	—
22	C₄×Q₁₆×C₅₃	—	2	1	no	—
23	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₅₃	—	2	1	no	—
28	C₄×SD₁₆×C₅₃	—	2	1	no	—
29	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#427	C₂²	1	1	yes	characteristic
32	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#416	—	2	1	no	—
34	C₂×#103×C₅₃	—	2	1	no	—
35	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#367	C₂²	1	2	yes	—
37	C₅₃×#366	C₂²	1	2	yes	—
38	C₅₃×#358	C₂²	1	2	yes	—
39	C₅₃×#358	C₂²	1	2	yes	—
40	C₅₃×#358	C₂²	1	2	yes	—
41	C₅₃×#357	C₂²	1	2	yes	—
42	C₅₃×#346	C₂²	1	2	yes	—
43	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
44	C₂×#41×C₅₃	—	2	1	no	—
45	C₂²×Q₈×C₅₃	—	2	1	no	—
46	#135×C₅₃	—	2	2	no	—
47	#135×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×Q₁₆×C₅₃	—	2	1	no	—
49	C₂×Q₁₆×C₅₃	—	2	2	no	—
50	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
51	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
52	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×SD₁₆×C₅₃	—	2	1	no	—
54	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄×Q₈×C₅₃	—	2	2	no	—
58	C₄×Q₈×C₅₃	—	2	2	no	—
59	C₄×Q₈×C₅₃	—	2	1	no	—
60	C₄×Q₈×C₅₃	—	2	1	no	—
61	C₄×Q₈×C₅₃	—	4	1	no	—
62	C₄×Q₈×C₅₃	—	4	1	no	—
63	C₄×D₄×C₅₃	—	2	1	no	—
64	C₄×D₄×C₅₃	—	2	1	no	—
65	C₄×D₄×C₅₃	—	4	1	no	—
66	C₄×D₄×C₅₃	—	4	1	no	—
67	#117×C₅₃	—	2	1	no	—
68	#117×C₅₃	—	4	1	no	—
69	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
70	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
71	C₂×C₄²×C₅₃	—	4	1	no	—
72	C₂×C₄²×C₅₃	—	4	1	no	—
73	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
74	Q₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
75	SD₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
76	M₃₂×C₅₃	—	2	2	no	—
77	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
78	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
79	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
80	#107×C₅₃	—	2	1	no	—
81	#107×C₅₃	C₂³	1	2	yes	—
82	#107×C₅₃	C₂³	1	1	yes	characteristic
83	#107×C₅₃	C₂³	1	1	yes	characteristic
84	#106×C₅₃	—	2	1	no	—
85	#103×C₅₃	—	2	2	no	—
86	#103×C₅₃	—	2	2	no	—
87	#103×C₅₃	—	2	1	no	—
88	#103×C₅₃	—	2	1	no	—
89	#102×C₅₃	—	2	1	no	—
90	#102×C₅₃	—	2	1	no	—
91	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
92	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
93	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	#41×C₅₃	—	2	4	no	—
96	#41×C₅₃	—	4	2	no	—
97	C₂×Q₈×C₅₃	—	2	4	no	—
98	C₂×Q₈×C₅₃	—	2	1	no	—
99	C₂×Q₈×C₅₃	—	2	2	no	—
100	C₂×Q₈×C₅₃	—	4	1	no	—
101	C₂×Q₈×C₅₃	—	4	2	no	—
102	C₂×D₄×C₅₃	—	2	1	no	—
103	C₂×D₄×C₅₃	—	4	1	no	—
104	C₂²×C₄×C₅₃	—	4	1	no	—
105	C₂²×C₄×C₅₃	—	4	1	no	—
106	C₂²×C₄×C₅₃	—	4	1	no	—
107	C₂²×C₄×C₅₃	—	4	1	no	—
108	C₂²×C₄×C₅₃	—	4	1	no	—
109	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
112	Q₁₆×C₅₃	—	2	4	no	—
113	Q₁₆×C₅₃	—	2	2	no	—
114	Q₁₆×C₅₃	—	2	1	no	—
115	Q₁₆×C₅₃	C₂²×C₄	1	4	yes	—
116	Q₁₆×C₅₃	C₂²×C₄	1	2	yes	—
117	SD₁₆×C₅₃	—	2	4	no	—
118	D₈×C₅₃	—	2	1	no	—
119	D₈×C₅₃	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₅₃	—	2	2	no	—
121	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₅₃	#41	1	2	yes	—
126	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₅₃	—	2	1	no	—
128	C₄⋊C₄×C₅₃	—	2	1	no	—
129	C₄⋊C₄×C₅₃	—	2	2	no	—
130	C₄⋊C₄×C₅₃	—	4	1	no	—
131	C₄⋊C₄×C₅₃	—	4	2	no	—
132	C₄⋊C₄×C₅₃	—	4	2	no	—
133	C₄⋊C₄×C₅₃	—	4	1	no	—
134	C₄⋊C₄×C₅₃	—	4	1	no	—
135	C₂²⋊C₄×C₅₃	—	4	1	no	—
136	C₂²⋊C₄×C₅₃	—	4	1	no	—
137	C₂²⋊C₄×C₅₃	—	4	1	no	—
138	C₄²×C₅₃	—	4	1	no	—
139	C₄²×C₅₃	—	4	1	no	—
140	C₄²×C₅₃	—	4	2	no	—
141	C₄²×C₅₃	—	4	1	no	—
142	C₄²×C₅₃	—	4	2	no	—
143	C₄²×C₅₃	D₈	1	1	yes	characteristic
144	C₄²×C₅₃	D₈	1	1	yes	characteristic
145	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
146	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
147	C₁₆×C₅₃	—	2	2	no	—
148	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
149	C₂³×C₅₃	—	4	1	no	—
150	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
151	Q₈×C₅₃	—	2	4	no	—
152	Q₈×C₅₃	—	2	2	no	—
153	Q₈×C₅₃	—	4	2	no	—
154	Q₈×C₅₃	—	4	1	no	—
155	Q₈×C₅₃	—	4	4	no	—
156	D₄×C₅₃	—	2	2	no	—
157	D₄×C₅₃	—	4	4	no	—
158	D₄×C₅₃	—	4	1	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	4	1	no	—
161	C₂×C₄×C₅₃	—	4	1	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	—	4	4	no	—
164	C₂×C₄×C₅₃	—	4	4	no	—
165	C₂×C₄×C₅₃	—	4	1	no	—
166	C₂×C₄×C₅₃	—	4	1	no	—
167	C₂×C₄×C₅₃	—	4	2	no	—
168	C₂×C₄×C₅₃	—	8	2	no	—
169	C₂×C₄×C₅₃	—	8	2	no	—
170	C₂×C₄×C₅₃	—	8	2	no	—
171	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
172	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
177	C₂×C₄×C₅₃	#135	1	2	yes	—
178	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
179	C₈×C₅₃	—	2	1	no	—
180	C₈×C₅₃	—	2	1	no	—
181	C₈×C₅₃	C₄×D₄	1	2	yes	—
182	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
183	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
184	C₄×#510	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂²×C₅₃	—	2	1	no	—
186	C₂²×C₅₃	—	2	1	no	—
187	C₂²×C₅₃	—	4	2	no	—
188	C₂²×C₅₃	—	8	1	no	—
189	C₂²×C₅₃	—	8	1	no	—
190	C₂²×C₅₃	C₄×D₈	1	2	yes	—
191	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₅₃	—	2	1	no	—
193	C₄×C₅₃	—	2	1	no	—
194	C₄×C₅₃	—	2	1	no	—
195	C₄×C₅₃	—	4	2	no	—
196	C₄×C₅₃	—	4	4	no	—
197	C₄×C₅₃	—	8	1	no	—
198	C₄×C₅₃	—	8	1	no	—
199	C₄×C₅₃	—	8	2	no	—
200	C₄×C₅₃	C₄×D₈	1	2	yes	—
201	C₄×C₅₃	#510	1	2	yes	direct product factor
202	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
203	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
204	C₂×#510	C₂×C₅₃	1	1	yes	characteristic
205	C₄×#135	C₂×C₅₃	1	1	yes	characteristic
206	C₂×C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#2176	C₂×C₅₃	1	2	yes	complemented
208	#2175	C₂×C₅₃	1	2	yes	complemented
209	C₄×Q₃₂	C₂×C₅₃	1	2	yes	complemented
210	C₄×SD₃₂	C₂×C₅₃	1	2	yes	complemented
211	#2159	C₂×C₅₃	1	1	yes	characteristic, complemented
212	#2141	C₂×C₅₃	1	1	yes	characteristic, complemented
213	#2139	C₂×C₅₃	1	1	yes	characteristic
214	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
215	C₂×C₅₃	—	2	2	no	—
216	C₂×C₅₃	—	8	2	no	—
217	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
218	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
219	C₂×C₅₃	#3414	1	1	yes	characteristic
220	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
221	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
222	C₄×#41	—	2	1	no	—
223	C₂×C₄×Q₈	—	2	1	no	—
224	#510	C₄×C₅₃	1	8	yes	direct product factor
225	C₂×Q₃₂	C₂²×C₅₃	1	2	yes	—
226	C₂×SD₃₂	C₂²×C₅₃	1	2	yes	—
227	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
228	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
229	C₄×Q₁₆	—	2	1	no	—
230	C₄×Q₁₆	—	2	1	no	—
231	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
232	C₄×Q₁₆	C₂²×C₅₃	1	2	yes	—
233	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
234	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
235	C₄×SD₁₆	—	2	1	no	—
236	C₄×SD₁₆	—	2	1	no	—
237	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
238	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic
239	#427	C₂²×C₅₃	1	1	yes	characteristic
240	C₂×#107	C₂²×C₅₃	1	1	yes	characteristic
241	#416	—	2	1	no	—
242	C₂×#103	—	2	1	no	—
243	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
244	#367	C₂²×C₅₃	1	2	yes	—
245	#366	C₂²×C₅₃	1	2	yes	—
246	#358	C₂²×C₅₃	1	2	yes	—
247	#358	C₂²×C₅₃	1	2	yes	—
248	#358	C₂²×C₅₃	1	2	yes	—
249	#357	C₂²×C₅₃	1	2	yes	—
250	#346	C₂²×C₅₃	1	2	yes	—
251	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
252	C₅₃	C₄×#510	1	1	yes	characteristic, direct product factor, Sylow
253	C₂×#41	—	2	1	no	—
254	C₂²×Q₈	—	2	1	no	—
255	#135	—	2	2	no	—
256	#135	C₂×C₄×C₅₃	1	4	yes	—
257	C₂×Q₁₆	—	2	1	no	—
258	C₂×Q₁₆	—	2	2	no	—
259	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
260	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
261	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
262	C₂×SD₁₆	—	2	1	no	—
263	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
264	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
266	C₄×Q₈	—	2	2	no	—
267	C₄×Q₈	—	2	2	no	—
268	C₄×Q₈	—	2	1	no	—
269	C₄×Q₈	—	2	1	no	—
270	C₄×Q₈	—	4	1	no	—
271	C₄×Q₈	—	4	1	no	—
272	C₄×D₄	—	2	1	no	—
273	C₄×D₄	—	2	1	no	—
274	C₄×D₄	—	4	1	no	—
275	C₄×D₄	—	4	1	no	—
276	#117	—	2	1	no	—
277	#117	—	4	1	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄⋊C₄	—	4	1	no	—
280	C₂×C₄²	—	4	1	no	—
281	C₂×C₄²	—	4	1	no	—
282	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
283	Q₃₂	C₂×C₄×C₅₃	1	8	yes	complemented
284	SD₃₂	C₂×C₄×C₅₃	1	8	yes	complemented
285	M₃₂	—	2	2	no	—
286	M₃₂	C₂×C₄×C₅₃	1	4	yes	complemented
287	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
288	C₂×C₁₆	C₂³×C₅₃	1	2	yes	—
289	#107	—	2	1	no	—
290	#107	C₂³×C₅₃	1	2	yes	—
291	#107	C₂³×C₅₃	1	1	yes	characteristic
292	#107	C₂³×C₅₃	1	1	yes	characteristic
293	#106	—	2	1	no	—
294	#103	—	2	2	no	—
295	#103	—	2	2	no	—
296	#103	—	2	1	no	—
297	#103	—	2	1	no	—
298	#102	—	2	1	no	—
299	#102	—	2	1	no	—
300	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
301	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
302	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
303	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
304	#41	—	2	4	no	—
305	#41	—	4	2	no	—
306	C₂×Q₈	—	2	4	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	4	1	no	—
310	C₂×Q₈	—	4	2	no	—
311	C₂×D₄	—	2	1	no	—
312	C₂×D₄	—	4	1	no	—
313	C₂²×C₄	—	4	1	no	—
314	C₂²×C₄	—	4	1	no	—
315	C₂²×C₄	—	4	1	no	—
316	C₂²×C₄	—	4	1	no	—
317	C₂²×C₄	—	4	1	no	—
318	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
319	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
321	Q₁₆	—	2	4	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	2	1	no	—
324	Q₁₆	C₂²×C₄×C₅₃	1	4	yes	—
325	Q₁₆	C₂²×C₄×C₅₃	1	2	yes	—
326	SD₁₆	—	2	4	no	—
327	D₈	—	2	1	no	—
328	D₈	C₂²×C₄×C₅₃	1	2	yes	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
331	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
332	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
333	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
334	C₂×C₈	#41×C₅₃	1	2	yes	—
335	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	4	1	no	—
340	C₄⋊C₄	—	4	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₄⋊C₄	—	4	1	no	—
344	C₂²⋊C₄	—	4	1	no	—
345	C₂²⋊C₄	—	4	1	no	—
346	C₂²⋊C₄	—	4	1	no	—
347	C₄²	—	4	1	no	—
348	C₄²	—	4	1	no	—
349	C₄²	—	4	1	no	—
350	C₄²	—	4	2	no	—
351	C₄²	—	4	2	no	—
352	C₄²	D₈×C₅₃	1	1	yes	characteristic
353	C₄²	D₈×C₅₃	1	1	yes	characteristic
354	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
355	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
356	C₁₆	—	2	2	no	—
357	C₁₆	C₂²×C₄×C₅₃	1	4	yes	—
358	C₂³	—	4	1	no	—
359	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
360	Q₈	—	2	2	no	—
361	Q₈	—	2	4	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	4	1	no	—
364	Q₈	—	4	4	no	—
365	D₄	—	2	2	no	—
366	D₄	—	4	4	no	—
367	D₄	—	4	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	4	no	—
373	C₂×C₄	—	4	4	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	8	2	no	—
378	C₂×C₄	—	8	2	no	—
379	C₂×C₄	—	8	2	no	—
380	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
381	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
382	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
383	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
384	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
385	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
386	C₂×C₄	#135×C₅₃	1	2	yes	—
387	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	2	1	no	—
390	C₈	C₄×D₄×C₅₃	1	2	yes	—
391	C₈	C₂³×C₄×C₅₃	1	1	yes	characteristic
392	C₈	C₂×#41×C₅₃	1	1	yes	characteristic
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	4	2	no	—
396	C₂²	—	8	1	no	—
397	C₂²	—	8	1	no	—
398	C₂²	C₄×D₈×C₅₃	1	2	yes	—
399	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	2	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	4	no	—
405	C₄	—	8	1	no	—
406	C₄	—	8	1	no	—
407	C₄	—	8	2	no	—
408	C₄	C₄×D₈×C₅₃	1	2	yes	—
409	C₄	C₅₃×#510	1	2	yes	direct product factor
410	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
411	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
412	C₂	—	2	2	no	—
413	C₂	—	8	2	no	—
414	C₂	C₂×C₄×D₈×C₅₃	1	1	yes	characteristic
415	C₂	C₂×C₅₃×#510	1	1	yes	characteristic
416	C₂	C₅₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂×D₈
commutator	C₈	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×Q₁₆×C₅₃	C₂	semidirect
4. C₅₃×#2176	C₂	semidirect
5. C₅₃×#2175	C₂	semidirect
6. C₄×Q₃₂×C₅₃	C₂	semidirect
7. C₄×SD₃₂×C₅₃	C₂	semidirect
8. C₅₃×#2159	C₂	semidirect
9. C₅₃×#2141	C₂	semidirect
11. C₄×M₃₂×C₅₃	C₂	semidirect
16. C₅₃×#510	C₄	direct (N × H)
50. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
74. Q₃₂×C₅₃	C₂×C₄	semidirect
75. SD₃₂×C₅₃	C₂×C₄	semidirect
77. M₃₂×C₅₃	C₂×C₄	semidirect
184. C₄×#510	C₅₃	direct (N × H)
201. C₄×C₅₃	#510	direct (N × H)
206. C₂×C₄×Q₁₆	C₂×C₅₃	semidirect
207. #2176	C₂×C₅₃	semidirect
208. #2175	C₂×C₅₃	semidirect
209. C₄×Q₃₂	C₂×C₅₃	semidirect
210. C₄×SD₃₂	C₂×C₅₃	semidirect
211. #2159	C₂×C₅₃	semidirect
212. #2141	C₂×C₅₃	semidirect
214. C₄×M₃₂	C₂×C₅₃	semidirect
224. #510	C₄×C₅₃	direct (N × H)
252. C₅₃	C₄×#510	direct (N × H)
259. C₂×Q₁₆	C₂×C₄×C₅₃	semidirect
283. Q₃₂	C₂×C₄×C₅₃	semidirect
284. SD₃₂	C₂×C₄×C₅₃	semidirect
286. M₃₂	C₂×C₄×C₅₃	semidirect
409. C₄	C₅₃×#510	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#510	2
C₄×#135×C₅₃	2
C₂×C₄×Q₁₆×C₅₃	2
C₅₃×#2176	2
C₅₃×#2175	2
C₄×Q₃₂×C₅₃	2
C₄×SD₃₂×C₅₃	2
C₅₃×#2159	2
C₅₃×#2141	2
C₅₃×#2139	2
C₄×M₃₂×C₅₃	2
C₄×#510	53

Subgroups — C4×C53×#510 #27240965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#510 #27240965