Subgroups — C₅₃×#21241 #27228753

Order 13568.

1498 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	C₂×C₅₃×#575	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#104×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2879	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1975	C₂	1	4	yes	maximal, complemented
7	C₅₃×#1974	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#575	—	2	4	no	—
10	C₅₃×#575	—	2	4	no	—
11	C₂×#137×C₅₃	C₄	1	4	yes	complemented
12	C₂×AGL(1,ℤ/8)×C₅₃	C₄	1	4	yes	complemented
13	C₂×#135×C₅₃	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#430	—	2	4	no	—
19	C₂×#108×C₅₃	C₂²	1	4	yes	complemented
20	C₂×#104×C₅₃	—	2	8	no	—
21	C₂×#104×C₅₃	C₂²	1	4	yes	complemented
22	C₂×#104×C₅₃	C₂²	1	4	yes	complemented
23	C₅₃×#408	—	2	4	no	—
24	C₅₃×#408	C₂²	1	8	yes	complemented
25	C₅₃×#407	—	2	4	no	—
26	C₄×M₁₆×C₅₃	—	2	4	no	—
27	C₂×#97×C₅₃	C₂²	1	2	yes	complemented
28	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
29	C₂×#41×C₅₃	—	2	2	no	—
30	C₂×#41×C₅₃	—	2	2	no	—
31	C₂×#41×C₅₃	—	2	4	no	—
32	C₂×#41×C₅₃	—	2	4	no	—
33	C₂×#41×C₅₃	—	2	4	no	—
34	C₂×#41×C₅₃	—	2	4	no	—
35	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
36	C₂×#41×C₅₃	C₂³	1	4	yes	—
37	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×D₄×C₅₃	—	2	2	no	—
39	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₅₃	—	2	1	no	—
41	C₂³×C₄×C₅₃	—	2	2	no	—
42	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	#137×C₅₃	—	2	16	no	—
44	AGL(1,ℤ/8)×C₅₃	—	2	16	no	—
45	#135×C₅₃	—	2	8	no	—
46	#135×C₅₃	—	2	8	no	—
47	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
48	C₂×SD₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
49	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	complemented
50	C₂×M₁₆×C₅₃	—	2	2	no	—
51	C₂×M₁₆×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	—	2	4	no	—
53	C₂×M₁₆×C₅₃	—	2	4	no	—
54	C₂×M₁₆×C₅₃	—	2	2	no	—
55	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
56	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
57	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
58	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
59	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
60	C₂²×C₈×C₅₃	—	2	2	no	—
61	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
63	C₂×C₄²×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	—	2	1	no	—
65	C₂×C₄²×C₅₃	D₄	1	4	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#108×C₅₃	—	2	8	no	—
69	#104×C₅₃	—	2	8	no	—
70	#104×C₅₃	—	2	8	no	—
71	C₂²⋊C₈×C₅₃	—	2	16	no	—
72	#97×C₅₃	—	2	4	no	—
73	C₄×C₈×C₅₃	—	2	4	no	—
74	C₂⁴×C₅₃	—	2	2	no	—
75	C₂⁴×C₅₃	#41	1	1	yes	characteristic
76	#41×C₅₃	—	2	8	no	—
77	#41×C₅₃	—	2	8	no	—
78	#41×C₅₃	—	2	4	no	—
79	#41×C₅₃	—	2	4	no	—
80	#41×C₅₃	—	2	4	no	—
81	#41×C₅₃	—	2	4	no	—
82	#41×C₅₃	—	4	4	no	—
83	#41×C₅₃	—	4	4	no	—
84	#41×C₅₃	—	4	4	no	—
85	#41×C₅₃	—	4	4	no	—
86	C₂×Q₈×C₅₃	—	2	2	no	—
87	C₂×Q₈×C₅₃	—	2	4	no	—
88	C₂×Q₈×C₅₃	—	2	4	no	—
89	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
90	C₂×D₄×C₅₃	—	2	4	no	—
91	C₂×D₄×C₅₃	—	2	4	no	—
92	C₂×D₄×C₅₃	—	2	4	no	—
93	C₂×D₄×C₅₃	—	2	4	no	—
94	C₂×D₄×C₅₃	—	2	2	no	—
95	C₂×D₄×C₅₃	—	4	4	no	—
96	C₂×D₄×C₅₃	—	4	4	no	—
97	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
98	C₂²×C₄×C₅₃	—	2	1	no	—
99	C₂²×C₄×C₅₃	—	2	2	no	—
100	C₂²×C₄×C₅₃	—	2	4	no	—
101	C₂²×C₄×C₅₃	—	2	1	no	—
102	C₂²×C₄×C₅₃	—	2	4	no	—
103	C₂²×C₄×C₅₃	—	2	4	no	—
104	C₂²×C₄×C₅₃	—	4	4	no	—
105	C₂²×C₄×C₅₃	—	4	4	no	—
106	C₂²×C₄×C₅₃	—	4	4	no	—
107	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	#41	1	2	yes	—
114	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
115	Q₁₆×C₅₃	—	2	8	no	—
116	SD₁₆×C₅₃	—	2	16	no	—
117	D₈×C₅₃	—	2	8	no	—
118	M₁₆×C₅₃	—	2	8	no	—
119	M₁₆×C₅₃	—	2	4	no	—
120	M₁₆×C₅₃	—	2	4	no	—
121	M₁₆×C₅₃	—	4	4	no	—
122	M₁₆×C₅₃	C₂²⋊C₄	1	16	yes	—
123	C₂×C₈×C₅₃	—	2	2	no	—
124	C₂×C₈×C₅₃	—	2	2	no	—
125	C₂×C₈×C₅₃	—	2	4	no	—
126	C₂×C₈×C₅₃	—	4	4	no	—
127	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
128	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
129	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
130	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
131	C₄²×C₅₃	—	2	4	no	—
132	C₄²×C₅₃	—	2	1	no	—
133	C₄²×C₅₃	—	2	1	no	—
134	C₄²×C₅₃	C₂×D₄	1	2	yes	—
135	C₄²×C₅₃	C₂×D₄	1	2	yes	—
136	C₂³×C₅₃	—	2	4	no	—
137	C₂³×C₅₃	—	4	2	no	—
138	C₂³×C₅₃	—	4	4	no	—
139	C₂³×C₅₃	—	8	4	no	—
140	C₂³×C₅₃	C₄×D₄	1	2	yes	—
141	C₂³×C₅₃	#121	1	2	yes	—
142	C₂³×C₅₃	#124	1	2	yes	—
143	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
144	Q₈×C₅₃	—	2	4	no	—
145	Q₈×C₅₃	—	2	4	no	—
146	Q₈×C₅₃	—	4	4	no	—
147	D₄×C₅₃	—	2	4	no	—
148	D₄×C₅₃	—	2	4	no	—
149	D₄×C₅₃	—	4	4	no	—
150	D₄×C₅₃	—	4	4	no	—
151	D₄×C₅₃	—	4	8	no	—
152	D₄×C₅₃	—	4	4	no	—
153	C₂×C₄×C₅₃	—	2	4	no	—
154	C₂×C₄×C₅₃	—	2	2	no	—
155	C₂×C₄×C₅₃	—	2	4	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	4	8	no	—
158	C₂×C₄×C₅₃	—	4	2	no	—
159	C₂×C₄×C₅₃	—	4	4	no	—
160	C₂×C₄×C₅₃	—	4	4	no	—
161	C₂×C₄×C₅₃	—	4	4	no	—
162	C₂×C₄×C₅₃	—	4	8	no	—
163	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
164	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
165	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₅₃	#121	1	4	yes	—
167	C₂×C₄×C₅₃	#121	1	2	yes	—
168	C₂×C₄×C₅₃	#124	1	2	yes	—
169	C₂×C₄×C₅₃	#127	1	4	yes	—
170	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
176	C₈×C₅₃	—	2	4	no	—
177	C₈×C₅₃	—	4	4	no	—
178	C₈×C₅₃	C₂×C₂²⋊C₄	1	8	yes	—
179	#21241	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂²×C₅₃	—	2	2	no	—
181	C₂²×C₅₃	—	2	4	no	—
182	C₂²×C₅₃	—	4	2	no	—
183	C₂²×C₅₃	—	4	2	no	—
184	C₂²×C₅₃	—	4	8	no	—
185	C₂²×C₅₃	—	4	4	no	—
186	C₂²×C₅₃	—	8	8	no	—
187	C₂²×C₅₃	#390	1	4	yes	—
188	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
190	C₂²×C₅₃	C₂×#124	1	1	yes	characteristic
191	C₄×C₅₃	—	2	4	no	—
192	C₄×C₅₃	—	4	2	no	—
193	C₄×C₅₃	—	4	4	no	—
194	C₄×C₅₃	#390	1	4	yes	—
195	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
196	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
197	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
198	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
199	C₂×#575	C₂×C₅₃	1	1	yes	characteristic
200	C₂×#430	C₂×C₅₃	1	1	yes	characteristic, complemented
201	C₂²×#104	C₂×C₅₃	1	2	yes	complemented
202	#2879	C₂×C₅₃	1	2	yes	complemented
203	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
204	#1975	C₂×C₅₃	1	4	yes	complemented
205	#1974	C₂×C₅₃	1	4	yes	complemented
206	C₂×C₅₃	—	2	1	no	—
207	C₂×C₅₃	—	2	4	no	—
208	C₂×C₅₃	—	2	1	no	—
209	C₂×C₅₃	—	8	4	no	—
210	C₂×C₅₃	C₂×#390	1	1	yes	characteristic
211	C₂×C₅₃	#2956	1	2	yes	—
212	C₂²×#41	C₂²×C₅₃	1	2	yes	—
213	#575	—	2	4	no	—
214	#575	—	2	4	no	—
215	C₂×#137	C₄×C₅₃	1	4	yes	complemented
216	C₂×AGL(1,ℤ/8)	C₄×C₅₃	1	4	yes	complemented
217	C₂×#135	C₂²×C₅₃	1	4	yes	—
218	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
219	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
220	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
221	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
222	#430	—	2	4	no	—
223	C₂×#108	C₂²×C₅₃	1	4	yes	complemented
224	C₂×#104	—	2	8	no	—
225	C₂×#104	C₂²×C₅₃	1	4	yes	complemented
226	C₂×#104	C₂²×C₅₃	1	4	yes	complemented
227	#408	—	2	4	no	—
228	#408	C₂²×C₅₃	1	8	yes	complemented
229	#407	—	2	4	no	—
230	C₄×M₁₆	—	2	4	no	—
231	C₂×#97	C₂²×C₅₃	1	2	yes	complemented
232	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	complemented
233	C₅₃	#21241	1	1	yes	characteristic, direct product factor, Sylow
234	C₂×#41	—	2	2	no	—
235	C₂×#41	—	2	2	no	—
236	C₂×#41	—	2	4	no	—
237	C₂×#41	—	2	4	no	—
238	C₂×#41	—	2	4	no	—
239	C₂×#41	—	2	4	no	—
240	C₂×#41	C₂×C₄×C₅₃	1	4	yes	—
241	C₂×#41	C₂³×C₅₃	1	4	yes	—
242	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
243	C₂²×D₄	—	2	2	no	—
244	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
245	C₂³×C₄	—	2	1	no	—
246	C₂³×C₄	—	2	2	no	—
247	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
248	#137	—	2	16	no	—
249	AGL(1,ℤ/8)	—	2	16	no	—
250	#135	—	2	8	no	—
251	#135	—	2	8	no	—
252	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
253	C₂×SD₁₆	C₂×C₄×C₅₃	1	8	yes	complemented
254	C₂×D₈	C₂×C₄×C₅₃	1	4	yes	complemented
255	C₂×M₁₆	—	2	2	no	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	—	2	2	no	—
258	C₂×M₁₆	—	2	4	no	—
259	C₂×M₁₆	—	2	4	no	—
260	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
261	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
262	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
263	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
264	C₂×M₁₆	C₂³×C₅₃	1	4	yes	—
265	C₂²×C₈	—	2	2	no	—
266	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
267	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
268	C₂×C₄²	—	2	2	no	—
269	C₂×C₄²	—	2	1	no	—
270	C₂×C₄²	D₄×C₅₃	1	4	yes	—
271	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
272	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
273	#108	—	2	8	no	—
274	#104	—	2	8	no	—
275	#104	—	2	8	no	—
276	C₂²⋊C₈	—	2	16	no	—
277	#97	—	2	4	no	—
278	C₄×C₈	—	2	4	no	—
279	C₂⁴	—	2	2	no	—
280	C₂⁴	#41×C₅₃	1	1	yes	characteristic
281	#41	—	2	8	no	—
282	#41	—	2	8	no	—
283	#41	—	2	4	no	—
284	#41	—	2	4	no	—
285	#41	—	2	4	no	—
286	#41	—	2	4	no	—
287	#41	—	4	4	no	—
288	#41	—	4	4	no	—
289	#41	—	4	4	no	—
290	#41	—	4	4	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	4	no	—
293	C₂×Q₈	—	2	4	no	—
294	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
295	C₂×D₄	—	2	4	no	—
296	C₂×D₄	—	2	4	no	—
297	C₂×D₄	—	2	4	no	—
298	C₂×D₄	—	2	4	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	4	4	no	—
301	C₂×D₄	—	4	4	no	—
302	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	4	no	—
308	C₂²×C₄	—	2	4	no	—
309	C₂²×C₄	—	4	4	no	—
310	C₂²×C₄	—	4	4	no	—
311	C₂²×C₄	—	4	4	no	—
312	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
313	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
317	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
318	C₂²×C₄	#41×C₅₃	1	2	yes	—
319	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
320	Q₁₆	—	2	8	no	—
321	SD₁₆	—	2	16	no	—
322	D₈	—	2	8	no	—
323	M₁₆	—	2	8	no	—
324	M₁₆	—	2	4	no	—
325	M₁₆	—	2	4	no	—
326	M₁₆	—	4	4	no	—
327	M₁₆	C₂²⋊C₄×C₅₃	1	16	yes	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	4	no	—
331	C₂×C₈	—	4	4	no	—
332	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
333	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
334	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
335	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
336	C₄²	—	2	4	no	—
337	C₄²	—	2	1	no	—
338	C₄²	—	2	1	no	—
339	C₄²	C₂×D₄×C₅₃	1	2	yes	—
340	C₄²	C₂×D₄×C₅₃	1	2	yes	—
341	C₂³	—	2	4	no	—
342	C₂³	—	4	2	no	—
343	C₂³	—	4	4	no	—
344	C₂³	—	8	4	no	—
345	C₂³	C₄×D₄×C₅₃	1	2	yes	—
346	C₂³	#121×C₅₃	1	2	yes	—
347	C₂³	#124×C₅₃	1	2	yes	—
348	C₂³	C₂×#41×C₅₃	1	1	yes	characteristic
349	Q₈	—	2	4	no	—
350	Q₈	—	2	4	no	—
351	Q₈	—	4	4	no	—
352	D₄	—	2	4	no	—
353	D₄	—	2	4	no	—
354	D₄	—	4	4	no	—
355	D₄	—	4	4	no	—
356	D₄	—	4	8	no	—
357	D₄	—	4	4	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	4	8	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	4	no	—
365	C₂×C₄	—	4	4	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	—	4	8	no	—
368	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	4	yes	—
369	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
370	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
371	C₂×C₄	#121×C₅₃	1	4	yes	—
372	C₂×C₄	#121×C₅₃	1	2	yes	—
373	C₂×C₄	#124×C₅₃	1	2	yes	—
374	C₂×C₄	#127×C₅₃	1	4	yes	—
375	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
377	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
378	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
379	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
380	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
381	C₈	—	2	4	no	—
382	C₈	—	4	4	no	—
383	C₈	C₂×C₂²⋊C₄×C₅₃	1	8	yes	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	4	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	4	8	no	—
389	C₂²	—	4	4	no	—
390	C₂²	—	8	8	no	—
391	C₂²	C₅₃×#390	1	4	yes	—
392	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
393	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
394	C₂²	C₂×#124×C₅₃	1	1	yes	characteristic
395	C₄	—	2	4	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	4	no	—
398	C₄	C₅₃×#390	1	4	yes	—
399	C₄	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
400	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
401	C₄	C₂×#121×C₅₃	1	1	yes	characteristic
402	C₄	C₂×#127×C₅₃	1	1	yes	characteristic
403	C₂	—	2	1	no	—
404	C₂	—	2	4	no	—
405	C₂	—	2	1	no	—
406	C₂	—	8	4	no	—
407	C₂	C₂×C₅₃×#390	1	1	yes	characteristic
408	C₂	C₅₃×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×C₅₃×#575	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂²×M₁₆×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#430	C₂	semidirect
3. C₂²×#104×C₅₃	C₂	semidirect
4. C₅₃×#2879	C₂	semidirect
5. C₂×C₄×M₁₆×C₅₃	C₂	semidirect
6. C₅₃×#1975	C₂	semidirect
7. C₅₃×#1974	C₂	semidirect
11. C₂×#137×C₅₃	C₄	semidirect
12. C₂×AGL(1,ℤ/8)×C₅₃	C₄	semidirect
19. C₂×#108×C₅₃	C₂²	semidirect
21. C₂×#104×C₅₃	C₂²	semidirect
22. C₂×#104×C₅₃	C₂²	semidirect
24. C₅₃×#408	C₂²	semidirect
27. C₂×#97×C₅₃	C₂²	semidirect
28. C₂×C₄×C₈×C₅₃	C₂²	semidirect
47. C₂×Q₁₆×C₅₃	C₂×C₄	semidirect
48. C₂×SD₁₆×C₅₃	C₂×C₄	semidirect
49. C₂×D₈×C₅₃	C₂×C₄	semidirect
179. #21241	C₅₃	direct (N × H)
200. C₂×#430	C₂×C₅₃	semidirect
201. C₂²×#104	C₂×C₅₃	semidirect
202. #2879	C₂×C₅₃	semidirect
203. C₂×C₄×M₁₆	C₂×C₅₃	semidirect
204. #1975	C₂×C₅₃	semidirect
205. #1974	C₂×C₅₃	semidirect
215. C₂×#137	C₄×C₅₃	semidirect
216. C₂×AGL(1,ℤ/8)	C₄×C₅₃	semidirect
223. C₂×#108	C₂²×C₅₃	semidirect
225. C₂×#104	C₂²×C₅₃	semidirect
226. C₂×#104	C₂²×C₅₃	semidirect
228. #408	C₂²×C₅₃	semidirect
231. C₂×#97	C₂²×C₅₃	semidirect
232. C₂×C₄×C₈	C₂²×C₅₃	semidirect
233. C₅₃	#21241	direct (N × H)
252. C₂×Q₁₆	C₂×C₄×C₅₃	semidirect
253. C₂×SD₁₆	C₂×C₄×C₅₃	semidirect
254. C₂×D₈	C₂×C₄×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×#575	2
C₂×C₅₃×#430	2
C₂²×#104×C₅₃	2
C₅₃×#2879	2
C₂×C₄×M₁₆×C₅₃	2
C₅₃×#1975	2
C₅₃×#1974	2
#21241	53

Subgroups — C53×#21241 #27228753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21241 #27228753