Subgroups — C₅₃×#10055 #27217567

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#492	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2548	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1987	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#331	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₅₃	—	2	2	no	—
9	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×#124×C₅₃	—	2	1	no	—
12	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#492	—	2	2	no	—
14	C₅₃×#492	—	2	2	no	—
15	C₅₃×#477	—	2	2	no	—
16	C₅₃×#477	—	2	2	no	—
17	C₂×#107×C₅₃	—	2	1	no	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#103×C₅₃	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#394	—	2	2	no	—
23	C₅₃×#390	—	2	2	no	—
24	C₅₃×#390	—	2	1	no	—
25	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×#340	—	2	2	no	—
27	C₅₃×#331	—	2	2	no	—
28	C₅₃×Q₈⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₅₃	—	2	1	no	—
30	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₅₃	—	2	1	no	—
32	C₂²×D₄×C₅₃	—	2	1	no	—
33	C₂²×D₄×C₅₃	D₄	1	2	yes	—
34	C₂×SD₁₆×C₅₃	—	2	8	no	—
35	C₂×SD₁₆×C₅₃	—	4	8	no	—
36	C₂²×C₈×C₅₃	—	2	1	no	—
37	C₂²×C₈×C₅₃	—	2	1	no	—
38	C₂²×C₈×C₅₃	—	2	1	no	—
39	C₄⋊Q₈×C₅₃	—	2	1	no	—
40	C₄⋊Q₈×C₅₃	—	2	1	no	—
41	C₄⋊Q₈×C₅₃	—	2	1	no	—
42	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
43	#127×C₅₃	—	2	1	no	—
44	#127×C₅₃	—	2	1	no	—
45	#127×C₅₃	—	2	1	no	—
46	#127×C₅₃	—	2	1	no	—
47	#124×C₅₃	—	4	2	no	—
48	#124×C₅₃	—	4	2	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	—	2	1	no	—
53	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
58	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
59	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
60	C₂×C₄²×C₅₃	—	2	1	no	—
61	C₂×C₄²×C₅₃	—	2	1	no	—
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	#107×C₅₃	—	2	4	no	—
64	C₄⋊C₈×C₅₃	—	2	1	no	—
65	C₄⋊C₈×C₅₃	—	2	1	no	—
66	C₄⋊C₈×C₅₃	—	2	1	no	—
67	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
68	#103×C₅₃	—	2	4	no	—
69	C₄×C₈×C₅₃	—	2	1	no	—
70	C₄×C₈×C₅₃	D₄	1	2	yes	—
71	#95×C₅₃	—	2	1	no	—
72	C₂⁴×C₅₃	—	2	2	no	—
73	C₂×Q₈×C₅₃	—	2	1	no	—
74	C₂×Q₈×C₅₃	—	2	1	no	—
75	C₂×Q₈×C₅₃	—	2	2	no	—
76	C₂×Q₈×C₅₃	—	2	1	no	—
77	C₂×Q₈×C₅₃	—	2	1	no	—
78	C₂×Q₈×C₅₃	—	2	1	no	—
79	C₂×Q₈×C₅₃	—	4	4	no	—
80	C₂×Q₈×C₅₃	—	4	1	no	—
81	C₂×Q₈×C₅₃	SD₁₆	1	2	yes	complemented
82	C₂×D₄×C₅₃	—	2	2	no	—
83	C₂×D₄×C₅₃	—	2	2	no	—
84	C₂×D₄×C₅₃	—	4	4	no	—
85	C₂×D₄×C₅₃	—	4	1	no	—
86	C₂×D₄×C₅₃	—	4	1	no	—
87	C₂×D₄×C₅₃	—	4	1	no	—
88	C₂×D₄×C₅₃	—	4	1	no	—
89	C₂×D₄×C₅₃	—	4	1	no	—
90	C₂×D₄×C₅₃	—	4	1	no	—
91	C₂×D₄×C₅₃	—	4	2	no	—
92	C₂×D₄×C₅₃	—	4	1	no	—
93	C₂×D₄×C₅₃	—	4	1	no	—
94	C₂×D₄×C₅₃	—	4	2	no	—
95	C₂²×C₄×C₅₃	—	2	1	no	—
96	C₂²×C₄×C₅₃	—	2	2	no	—
97	C₂²×C₄×C₅₃	—	2	1	no	—
98	C₂²×C₄×C₅₃	—	4	1	no	—
99	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
102	SD₁₆×C₅₃	—	4	16	no	—
103	C₂×C₈×C₅₃	—	2	4	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	4	1	no	—
107	C₂×C₈×C₅₃	—	4	1	no	—
108	C₂×C₈×C₅₃	—	4	1	no	—
109	C₂×C₈×C₅₃	—	4	2	no	—
110	C₄⋊C₄×C₅₃	—	2	2	no	—
111	C₄⋊C₄×C₅₃	—	2	1	no	—
112	C₄⋊C₄×C₅₃	—	2	2	no	—
113	C₄⋊C₄×C₅₃	—	2	1	no	—
114	C₄⋊C₄×C₅₃	—	2	2	no	—
115	C₄⋊C₄×C₅₃	—	2	2	no	—
116	C₄⋊C₄×C₅₃	—	4	1	no	—
117	C₄⋊C₄×C₅₃	—	4	1	no	—
118	C₄⋊C₄×C₅₃	—	4	1	no	—
119	C₄⋊C₄×C₅₃	—	4	1	no	—
120	C₂²⋊C₄×C₅₃	—	4	8	no	—
121	C₂²⋊C₄×C₅₃	—	8	4	no	—
122	C₄²×C₅₃	—	4	1	no	—
123	C₄²×C₅₃	—	4	1	no	—
124	C₄²×C₅₃	—	4	1	no	—
125	C₄²×C₅₃	—	4	1	no	—
126	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₅₃	#41	1	1	yes	characteristic
130	C₂³×C₅₃	—	4	2	no	—
131	C₂³×C₅₃	—	4	2	no	—
132	C₂³×C₅₃	—	4	2	no	—
133	C₂³×C₅₃	—	4	4	no	—
134	C₂³×C₅₃	—	4	4	no	—
135	C₂³×C₅₃	#120	1	1	yes	characteristic
136	Q₈×C₅₃	—	2	2	no	—
137	Q₈×C₅₃	—	2	2	no	—
138	Q₈×C₅₃	—	4	4	no	—
139	Q₈×C₅₃	—	4	1	no	—
140	Q₈×C₅₃	—	4	1	no	—
141	Q₈×C₅₃	—	8	2	no	—
142	D₄×C₅₃	—	4	4	no	—
143	D₄×C₅₃	—	8	2	no	—
144	D₄×C₅₃	—	8	1	no	—
145	D₄×C₅₃	—	8	1	no	—
146	D₄×C₅₃	—	8	2	no	—
147	D₄×C₅₃	—	8	1	no	—
148	D₄×C₅₃	—	8	1	no	—
149	D₄×C₅₃	—	8	1	no	—
150	D₄×C₅₃	—	8	1	no	—
151	C₂×C₄×C₅₃	—	2	2	no	—
152	C₂×C₄×C₅₃	—	2	2	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	—	4	1	no	—
155	C₂×C₄×C₅₃	—	4	1	no	—
156	C₂×C₄×C₅₃	—	4	2	no	—
157	C₂×C₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	—	4	2	no	—
161	C₂×C₄×C₅₃	—	8	2	no	—
162	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
163	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
164	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
165	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
166	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
167	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
168	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
169	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
170	C₂×C₄×C₅₃	#135	1	2	yes	—
171	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
173	C₂×C₄×C₅₃	#137	1	2	yes	—
174	C₈×C₅₃	—	2	4	no	—
175	C₈×C₅₃	—	4	2	no	—
176	C₈×C₅₃	—	8	1	no	—
177	#10055	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂²×C₅₃	—	8	4	no	—
179	C₂²×C₅₃	—	8	4	no	—
180	C₂²×C₅₃	—	8	2	no	—
181	C₂²×C₅₃	—	8	2	no	—
182	C₂²×C₅₃	—	8	2	no	—
183	C₂²×C₅₃	#392	1	1	yes	characteristic
184	C₂²×C₅₃	#450	1	1	yes	characteristic
185	C₂²×C₅₃	#452	1	1	yes	characteristic
186	C₂²×C₅₃	#452	1	2	yes	—
187	C₂²×C₅₃	#453	1	2	yes	—
188	C₄×C₅₃	—	2	1	no	—
189	C₄×C₅₃	—	2	1	no	—
190	C₄×C₅₃	—	4	2	no	—
191	C₄×C₅₃	—	8	2	no	—
192	C₄×C₅₃	—	8	1	no	—
193	C₄×C₅₃	—	8	2	no	—
194	C₄×C₅₃	#448	1	1	yes	characteristic
195	C₄×C₅₃	#449	1	1	yes	characteristic
196	C₄×C₅₃	#464	1	2	yes	—
197	C₄×C₅₃	#465	1	1	yes	characteristic
198	C₄×C₅₃	#468	1	1	yes	characteristic
199	C₄×C₅₃	#495	1	1	yes	characteristic
200	C₄×C₅₃	#497	1	1	yes	characteristic
201	C₂×#492	C₂×C₅₃	1	1	yes	characteristic
202	C₂×#477	C₂×C₅₃	1	1	yes	characteristic, complemented
203	#2548	C₂×C₅₃	1	1	yes	characteristic
204	#1987	C₂×C₅₃	1	1	yes	characteristic, complemented
205	C₂×#331	C₂×C₅₃	1	1	yes	characteristic
206	C₂×Q₈⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
207	#1279	C₂×C₅₃	1	1	yes	characteristic, complemented
208	C₂×C₅₃	—	16	2	no	—
209	C₂×C₅₃	#1630	1	2	yes	—
210	C₂×C₅₃	#1631	1	2	yes	—
211	C₂×C₅₃	#2000	1	1	yes	characteristic
212	C₂×C₅₃	#2002	1	1	yes	characteristic
213	C₂×C₅₃	#2015	1	1	yes	characteristic
214	C₂²×SD₁₆	—	2	2	no	—
215	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
216	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
217	C₂×#124	—	2	1	no	—
218	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
219	#492	—	2	2	no	—
220	#492	—	2	2	no	—
221	#477	—	2	2	no	—
222	#477	—	2	2	no	—
223	C₂×#107	—	2	1	no	—
224	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
225	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
226	C₂×#103	—	2	1	no	—
227	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
228	#394	—	2	2	no	—
229	#390	—	2	2	no	—
230	#390	—	2	1	no	—
231	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
232	#340	—	2	2	no	—
233	#331	—	2	2	no	—
234	Q₈⋊C₈	—	2	2	no	—
235	C₅₃	#10055	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²×Q₈	—	2	1	no	—
237	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
238	C₂²×D₄	—	2	1	no	—
239	C₂²×D₄	—	2	1	no	—
240	C₂²×D₄	D₄×C₅₃	1	2	yes	—
241	C₂×SD₁₆	—	2	8	no	—
242	C₂×SD₁₆	—	4	8	no	—
243	C₂²×C₈	—	2	1	no	—
244	C₂²×C₈	—	2	1	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
250	#127	—	2	1	no	—
251	#127	—	2	1	no	—
252	#127	—	2	1	no	—
253	#127	—	2	1	no	—
254	#124	—	4	2	no	—
255	#124	—	4	2	no	—
256	C₄×Q₈	—	2	1	no	—
257	C₄×Q₈	—	2	1	no	—
258	C₄×Q₈	—	2	1	no	—
259	C₄×Q₈	—	2	1	no	—
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
265	C₂×C₂²⋊C₄	—	2	4	no	—
266	C₂×C₂²⋊C₄	—	2	4	no	—
267	C₂×C₄²	—	2	1	no	—
268	C₂×C₄²	—	2	1	no	—
269	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
270	#107	—	2	4	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	D₄×C₅₃	1	2	yes	—
275	#103	—	2	4	no	—
276	C₄×C₈	—	2	1	no	—
277	C₄×C₈	D₄×C₅₃	1	2	yes	—
278	#95	—	2	1	no	—
279	C₂⁴	—	2	2	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	2	1	no	—
282	C₂×Q₈	—	2	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	4	4	no	—
287	C₂×Q₈	—	4	1	no	—
288	C₂×Q₈	SD₁₆×C₅₃	1	2	yes	complemented
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	4	4	no	—
292	C₂×D₄	—	4	1	no	—
293	C₂×D₄	—	4	1	no	—
294	C₂×D₄	—	4	1	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	2	no	—
298	C₂×D₄	—	4	1	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	4	2	no	—
301	C₂×D₄	—	4	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	4	1	no	—
306	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
309	SD₁₆	—	4	16	no	—
310	C₂×C₈	—	2	4	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	4	1	no	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	4	1	no	—
316	C₂×C₈	—	4	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	1	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	—	4	1	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₂²⋊C₄	—	4	8	no	—
328	C₂²⋊C₄	—	8	4	no	—
329	C₄²	—	4	1	no	—
330	C₄²	—	4	1	no	—
331	C₄²	—	4	1	no	—
332	C₄²	—	4	1	no	—
333	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
334	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
335	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
336	C₄²	#41×C₅₃	1	1	yes	characteristic
337	C₂³	—	4	2	no	—
338	C₂³	—	4	2	no	—
339	C₂³	—	4	2	no	—
340	C₂³	—	4	4	no	—
341	C₂³	—	4	4	no	—
342	C₂³	#120×C₅₃	1	1	yes	characteristic
343	Q₈	—	2	2	no	—
344	Q₈	—	2	2	no	—
345	Q₈	—	4	4	no	—
346	Q₈	—	4	1	no	—
347	Q₈	—	4	1	no	—
348	Q₈	—	8	2	no	—
349	D₄	—	4	4	no	—
350	D₄	—	8	2	no	—
351	D₄	—	8	1	no	—
352	D₄	—	8	1	no	—
353	D₄	—	8	1	no	—
354	D₄	—	8	1	no	—
355	D₄	—	8	2	no	—
356	D₄	—	8	1	no	—
357	D₄	—	8	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	2	no	—
369	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
370	C₂×C₄	#120×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	#121×C₅₃	1	1	yes	characteristic
374	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
375	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
376	C₂×C₄	#135×C₅₃	1	2	yes	—
377	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
378	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
379	C₂×C₄	#137×C₅₃	1	2	yes	—
380	C₂×C₄	#137×C₅₃	1	1	yes	characteristic
381	C₈	—	2	4	no	—
382	C₈	—	4	2	no	—
383	C₈	—	8	1	no	—
384	C₂²	—	8	2	no	—
385	C₂²	—	8	2	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	8	4	no	—
388	C₂²	—	8	4	no	—
389	C₂²	C₅₃×#392	1	1	yes	characteristic
390	C₂²	C₅₃×#450	1	1	yes	characteristic
391	C₂²	C₅₃×#452	1	1	yes	characteristic
392	C₂²	C₅₃×#452	1	2	yes	—
393	C₂²	C₅₃×#453	1	2	yes	—
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	—	8	2	no	—
400	C₄	C₅₃×#448	1	1	yes	characteristic
401	C₄	C₅₃×#449	1	1	yes	characteristic
402	C₄	C₅₃×#464	1	2	yes	—
403	C₄	C₅₃×#465	1	1	yes	characteristic
404	C₄	C₅₃×#468	1	1	yes	characteristic
405	C₄	C₅₃×#495	1	1	yes	characteristic
406	C₄	C₅₃×#497	1	1	yes	characteristic
407	C₂	—	16	2	no	—
408	C₂	C₅₃×#1630	1	2	yes	—
409	C₂	C₅₃×#1631	1	2	yes	—
410	C₂	C₅₃×#2000	1	1	yes	characteristic
411	C₂	C₅₃×#2002	1	1	yes	characteristic
412	C₂	C₅₃×#2015	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#477	C₂	semidirect
4. C₅₃×#1987	C₂	semidirect
6. C₂×C₅₃×Q₈⋊C₈	C₂	semidirect
7. C₅₃×#1279	C₂	semidirect
81. C₂×Q₈×C₅₃	SD₁₆	semidirect
177. #10055	C₅₃	direct (N × H)
202. C₂×#477	C₂×C₅₃	semidirect
204. #1987	C₂×C₅₃	semidirect
206. C₂×Q₈⋊C₈	C₂×C₅₃	semidirect
207. #1279	C₂×C₅₃	semidirect
235. C₅₃	#10055	direct (N × H)
288. C₂×Q₈	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#492	2
C₂×C₅₃×#477	2
C₅₃×#2548	2
C₅₃×#1987	2
C₂×C₅₃×#331	2
C₂×C₅₃×Q₈⋊C₈	2
C₅₃×#1279	2
#10055	53

Subgroups — C53×#10055 #27217567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10055 #27217567