Subgroups — C₅₃×#21169 #27228681

Order 13568.

1026 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#427	C₂	1	2	yes	maximal
2	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic
4	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1949	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1948	C₂	1	2	yes	maximal, complemented
7	C₅₃×#1947	C₂	1	2	yes	maximal, complemented
8	C₅₃×#1735	C₂	1	2	yes	maximal
9	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×#117×C₅₃	C₂²	1	2	yes	—
11	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#427	—	2	8	no	—
15	C₂×#107×C₅₃	C₂²	1	4	yes	—
16	C₂×#106×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈×C₅₃	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#384	C₂²	1	2	yes	—
22	C₅₃×#382	C₂²	1	2	yes	—
23	C₅₃×#380	—	2	1	no	—
24	C₅₃×#380	—	2	1	no	—
25	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
26	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
30	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
31	C₅₃×#376	—	2	1	no	—
32	C₄³×C₅₃	—	2	1	no	—
33	C₅₃×#340	C₂²	1	4	yes	—
34	C₅₃×#340	C₂²	1	4	yes	—
35	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	—	2	4	no	—
39	C₂²×C₈×C₅₃	—	2	4	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
41	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
43	#117×C₅₃	—	2	4	no	—
44	#117×C₅₃	—	2	2	no	—
45	#117×C₅₃	—	2	2	no	—
46	#117×C₅₃	—	2	4	no	—
47	#117×C₅₃	D₄	1	4	yes	—
48	#117×C₅₃	Q₈	1	4	yes	—
49	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
54	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
55	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
57	C₂×C₄²×C₅₃	—	2	2	no	—
58	C₂×C₄²×C₅₃	—	2	2	no	—
59	C₂×C₄²×C₅₃	—	2	1	no	—
60	C₂×C₄²×C₅₃	—	2	1	no	—
61	C₂×C₄²×C₅₃	—	2	2	no	—
62	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
63	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	Q₈	1	4	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#107×C₅₃	—	2	8	no	—
69	#106×C₅₃	—	2	8	no	—
70	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
71	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
72	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
73	#95×C₅₃	—	2	2	no	—
74	#95×C₅₃	—	2	2	no	—
75	#95×C₅₃	—	2	2	no	—
76	#95×C₅₃	—	2	2	no	—
77	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	—	2	4	no	—
79	C₂²×C₄×C₅₃	—	2	2	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	—	2	4	no	—
82	C₂²×C₄×C₅₃	—	4	2	no	—
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
84	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
93	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₅₃	#41	1	2	yes	—
95	C₂²×C₄×C₅₃	#41	1	2	yes	—
96	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₂×C₈×C₅₃	—	2	16	no	—
98	C₂×C₈×C₅₃	—	2	2	no	—
99	C₂×C₈×C₅₃	—	2	2	no	—
100	C₂×C₈×C₅₃	—	2	2	no	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
103	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
104	C₄⋊C₄×C₅₃	—	2	2	no	—
105	C₄⋊C₄×C₅₃	—	2	2	no	—
106	C₄⋊C₄×C₅₃	—	2	4	no	—
107	C₄⋊C₄×C₅₃	—	2	4	no	—
108	C₄⋊C₄×C₅₃	—	4	4	no	—
109	C₄⋊C₄×C₅₃	—	4	4	no	—
110	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
111	C₄⋊C₄×C₅₃	C₂×Q₈	1	4	yes	—
112	C₂²⋊C₄×C₅₃	—	2	4	no	—
113	C₂²⋊C₄×C₅₃	—	2	4	no	—
114	C₂²⋊C₄×C₅₃	—	4	4	no	—
115	C₄²×C₅₃	—	2	4	no	—
116	C₄²×C₅₃	—	2	4	no	—
117	C₄²×C₅₃	—	4	4	no	—
118	C₄²×C₅₃	—	4	4	no	—
119	C₄²×C₅₃	—	4	4	no	—
120	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
121	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
122	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
123	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
124	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
125	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
126	C₂³×C₅₃	—	2	2	no	—
127	C₂³×C₅₃	—	2	2	no	—
128	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂³×C₅₃	#121	1	2	yes	—
130	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
131	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	8	no	—
136	C₂×C₄×C₅₃	—	2	4	no	—
137	C₂×C₄×C₅₃	—	2	4	no	—
138	C₂×C₄×C₅₃	—	4	4	no	—
139	C₂×C₄×C₅₃	—	4	2	no	—
140	C₂×C₄×C₅₃	—	4	2	no	—
141	C₂×C₄×C₅₃	—	4	2	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	—	4	4	no	—
144	C₂×C₄×C₅₃	—	4	4	no	—
145	C₂×C₄×C₅₃	—	4	4	no	—
146	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
152	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
153	C₂×C₄×C₅₃	#121	1	2	yes	—
154	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
155	C₂×C₄×C₅₃	#125	1	4	yes	—
156	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
157	C₂×C₄×C₅₃	#135	1	4	yes	—
158	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
164	C₈×C₅₃	—	2	4	no	—
165	C₈×C₅₃	—	2	4	no	—
166	#21169	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂²×C₅₃	—	2	4	no	—
168	C₂²×C₅₃	—	2	2	no	—
169	C₂²×C₅₃	—	2	2	no	—
170	C₂²×C₅₃	—	2	4	no	—
171	C₂²×C₅₃	#389	1	4	yes	—
172	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
174	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
175	C₂²×C₅₃	C₂×#135	1	2	yes	—
176	C₂²×C₅₃	#575	1	2	yes	—
177	C₄×C₅₃	—	2	2	no	—
178	C₄×C₅₃	—	2	2	no	—
179	C₄×C₅₃	—	2	2	no	—
180	C₄×C₅₃	—	4	4	no	—
181	C₄×C₅₃	—	4	4	no	—
182	C₄×C₅₃	—	8	4	no	—
183	C₄×C₅₃	#389	1	4	yes	—
184	C₄×C₅₃	#427	1	4	yes	complemented
185	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
187	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
188	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
189	C₂×#427	C₂×C₅₃	1	2	yes	—
190	C₂²×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
191	#2285	C₂×C₅₃	1	1	yes	characteristic
192	C₄×#117	C₂×C₅₃	1	1	yes	characteristic
193	#1949	C₂×C₅₃	1	4	yes	complemented
194	#1948	C₂×C₅₃	1	2	yes	complemented
195	#1947	C₂×C₅₃	1	2	yes	complemented
196	#1735	C₂×C₅₃	1	2	yes	—
197	C₂×C₅₃	—	2	4	no	—
198	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
199	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
200	C₂×C₅₃	#2911	1	1	yes	characteristic
201	C₂×C₅₃	#3035	1	2	yes	—
202	C₂×C₅₃	#3076	1	2	yes	—
203	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
204	C₂×#117	C₂²×C₅₃	1	2	yes	—
205	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
206	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
207	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
208	#427	—	2	8	no	—
209	C₂×#107	C₂²×C₅₃	1	4	yes	—
210	C₂×#106	C₂²×C₅₃	1	4	yes	—
211	C₂×C₄⋊C₈	C₄×C₅₃	1	8	yes	complemented
212	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
213	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
214	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
215	#384	C₂²×C₅₃	1	2	yes	—
216	#382	C₂²×C₅₃	1	2	yes	—
217	#380	—	2	1	no	—
218	#380	—	2	1	no	—
219	C₄×C₄⋊C₄	—	2	1	no	—
220	C₄×C₄⋊C₄	—	2	1	no	—
221	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
222	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
223	C₄×C₂²⋊C₄	—	2	1	no	—
224	C₄×C₂²⋊C₄	—	2	1	no	—
225	#376	—	2	1	no	—
226	C₄³	—	2	1	no	—
227	#340	C₂²×C₅₃	1	4	yes	—
228	#340	C₂²×C₅₃	1	4	yes	—
229	C₅₃	#21169	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
231	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
232	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
233	C₂²×C₈	—	2	4	no	—
234	C₂²×C₈	—	2	4	no	—
235	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
236	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
237	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
238	#117	—	2	2	no	—
239	#117	—	2	2	no	—
240	#117	—	2	4	no	—
241	#117	—	2	4	no	—
242	#117	D₄×C₅₃	1	4	yes	—
243	#117	Q₈×C₅₃	1	4	yes	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	2	no	—
246	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
247	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
248	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
249	C₂×C₂²⋊C₄	—	2	1	no	—
250	C₂×C₂²⋊C₄	—	2	1	no	—
251	C₂×C₂²⋊C₄	—	2	2	no	—
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	—	2	1	no	—
254	C₂×C₄²	—	2	2	no	—
255	C₂×C₄²	—	2	2	no	—
256	C₂×C₄²	—	2	2	no	—
257	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
258	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
259	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
260	C₂×C₄²	Q₈×C₅₃	1	4	yes	—
261	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
263	#107	—	2	8	no	—
264	#106	—	2	8	no	—
265	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
266	C₄⋊C₈	C₂×C₄×C₅₃	1	4	yes	—
267	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
268	#95	—	2	2	no	—
269	#95	—	2	2	no	—
270	#95	—	2	2	no	—
271	#95	—	2	2	no	—
272	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	—	4	2	no	—
278	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
279	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
280	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
281	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
287	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
288	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
289	C₂²×C₄	#41×C₅₃	1	2	yes	—
290	C₂²×C₄	#41×C₅₃	1	2	yes	—
291	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
292	C₂×C₈	—	2	16	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
298	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	4	no	—
302	C₄⋊C₄	—	2	4	no	—
303	C₄⋊C₄	—	4	4	no	—
304	C₄⋊C₄	—	4	4	no	—
305	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
306	C₄⋊C₄	C₂×Q₈×C₅₃	1	4	yes	—
307	C₂²⋊C₄	—	2	4	no	—
308	C₂²⋊C₄	—	2	4	no	—
309	C₂²⋊C₄	—	4	4	no	—
310	C₄²	—	2	4	no	—
311	C₄²	—	2	4	no	—
312	C₄²	—	4	4	no	—
313	C₄²	—	4	4	no	—
314	C₄²	—	4	4	no	—
315	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
316	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
317	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
318	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
319	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
320	C₄²	C₂×Q₈×C₅₃	1	2	yes	—
321	C₂³	—	2	2	no	—
322	C₂³	—	2	2	no	—
323	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
324	C₂³	#121×C₅₃	1	2	yes	—
325	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
326	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	8	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	4	no	—
339	C₂×C₄	—	4	4	no	—
340	C₂×C₄	—	4	4	no	—
341	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
342	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
343	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
344	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
345	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
346	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
347	C₂×C₄	C₄×Q₈×C₅₃	1	4	yes	—
348	C₂×C₄	#121×C₅₃	1	2	yes	—
349	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
350	C₂×C₄	#125×C₅₃	1	4	yes	—
351	C₂×C₄	C₄⋊Q₈×C₅₃	1	4	yes	—
352	C₂×C₄	#135×C₅₃	1	4	yes	—
353	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
357	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
358	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
359	C₈	—	2	4	no	—
360	C₈	—	2	4	no	—
361	C₂²	—	2	4	no	—
362	C₂²	—	2	4	no	—
363	C₂²	—	2	2	no	—
364	C₂²	—	2	2	no	—
365	C₂²	C₅₃×#389	1	4	yes	—
366	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
367	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
368	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
369	C₂²	C₂×#135×C₅₃	1	2	yes	—
370	C₂²	C₅₃×#575	1	2	yes	—
371	C₄	—	2	2	no	—
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	4	4	no	—
375	C₄	—	4	4	no	—
376	C₄	—	8	4	no	—
377	C₄	C₅₃×#389	1	4	yes	—
378	C₄	C₅₃×#427	1	4	yes	complemented
379	C₄	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
380	C₄	C₂×C₄×Q₈×C₅₃	1	1	yes	characteristic
381	C₄	C₂×#125×C₅₃	1	1	yes	characteristic
382	C₄	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
383	C₂	—	2	4	no	—
384	C₂	C₂×C₅₃×#389	1	1	yes	characteristic
385	C₂	C₂×C₅₃×#427	1	1	yes	characteristic
386	C₂	C₅₃×#2911	1	1	yes	characteristic
387	C₂	C₅₃×#3035	1	2	yes	—
388	C₂	C₅₃×#3076	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂⁴×C₅₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅₃×#1949	C₂	semidirect
6. C₅₃×#1948	C₂	semidirect
7. C₅₃×#1947	C₂	semidirect
17. C₂×C₄⋊C₈×C₅₃	C₄	semidirect
166. #21169	C₅₃	direct (N × H)
184. C₄×C₅₃	#427	semidirect
193. #1949	C₂×C₅₃	semidirect
194. #1948	C₂×C₅₃	semidirect
195. #1947	C₂×C₅₃	semidirect
211. C₂×C₄⋊C₈	C₄×C₅₃	semidirect
229. C₅₃	#21169	direct (N × H)
378. C₄	C₅₃×#427	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#427	2
C₂²×C₄⋊C₈×C₅₃	2
C₅₃×#2285	2
C₄×#117×C₅₃	2
C₅₃×#1949	2
C₅₃×#1948	2
C₅₃×#1947	2
C₅₃×#1735	2
#21169	53

Subgroups — C53×#21169 #27228681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#21169 #27228681