Subgroups — C₂×C₅₃×#1592 #27225399

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×SD₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1592	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₅₃	C₄	1	1	yes	characteristic
10	C₂²×M₁₆×C₅₃	—	2	1	no	—
11	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#446	—	2	2	no	—
14	C₅₃×#446	C₂²	1	4	yes	complemented
15	C₄×SD₁₆×C₅₃	C₄	1	4	yes	—
16	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	—
17	C₅₃×#436	—	2	2	no	—
18	C₅₃×#436	C₂²	1	4	yes	—
19	C₂×#106×C₅₃	C₄	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#103×C₅₃	C₄	1	1	yes	characteristic
24	C₂×#102×C₅₃	C₄	1	1	yes	characteristic
25	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
26	C₂×#97×C₅₃	—	2	1	no	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#335	C₂²	1	4	yes	complemented
31	C₅₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
32	C₅₃×D₄⋊C₈	C₂²	1	4	yes	—
33	C₅₃×#322	C₂²	1	4	yes	complemented
34	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	—	2	1	no	—
37	C₂×SD₁₆×C₅₃	—	2	4	no	—
38	C₂×SD₁₆×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×M₁₆×C₅₃	—	2	4	no	—
40	C₂×M₁₆×C₅₃	—	4	4	no	—
41	C₂²×C₈×C₅₃	—	2	1	no	—
42	C₂²×C₈×C₅₃	—	2	1	no	—
43	C₂²×C₈×C₅₃	—	2	1	no	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×Q₈×C₅₃	—	2	1	no	—
49	C₄×Q₈×C₅₃	—	2	1	no	—
50	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
52	C₄×D₄×C₅₃	—	2	1	no	—
53	C₄×D₄×C₅₃	—	2	1	no	—
54	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
55	C₄×D₄×C₅₃	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
60	C₂×C₄²×C₅₃	—	2	1	no	—
61	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	#106×C₅₃	C₂×C₄	1	4	yes	—
63	C₄⋊C₈×C₅₃	—	2	4	no	—
64	C₄⋊C₈×C₅₃	—	2	1	no	—
65	C₄⋊C₈×C₅₃	—	2	1	no	—
66	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
67	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
68	#103×C₅₃	C₂×C₄	1	4	yes	—
69	#102×C₅₃	C₂×C₄	1	4	yes	—
70	C₂²⋊C₈×C₅₃	—	2	4	no	—
71	#97×C₅₃	—	2	4	no	—
72	C₄×C₈×C₅₃	—	2	1	no	—
73	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
74	C₄×C₈×C₅₃	D₄	1	2	yes	—
75	C₄×C₈×C₅₃	C₂³	1	2	yes	—
76	C₄×C₈×C₅₃	C₂³	1	2	yes	—
77	C₄×C₈×C₅₃	C₂³	1	2	yes	—
78	C₂⁴×C₅₃	—	2	1	no	—
79	C₂×Q₈×C₅₃	—	2	1	no	—
80	C₂×Q₈×C₅₃	—	2	2	no	—
81	C₂×Q₈×C₅₃	—	2	2	no	—
82	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×D₄×C₅₃	—	2	1	no	—
84	C₂×D₄×C₅₃	—	2	2	no	—
85	C₂×D₄×C₅₃	—	2	2	no	—
86	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅₃	—	2	1	no	—
88	C₂²×C₄×C₅₃	—	2	1	no	—
89	C₂²×C₄×C₅₃	—	2	1	no	—
90	C₂²×C₄×C₅₃	—	2	2	no	—
91	C₂²×C₄×C₅₃	—	2	2	no	—
92	C₂²×C₄×C₅₃	—	4	2	no	—
93	C₂²×C₄×C₅₃	—	4	2	no	—
94	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
97	SD₁₆×C₅₃	—	2	8	no	—
98	M₁₆×C₅₃	—	4	8	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	—	2	2	no	—
103	C₂×C₈×C₅₃	—	2	2	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	4	2	no	—
106	C₂×C₈×C₅₃	—	4	2	no	—
107	C₂×C₈×C₅₃	—	4	2	no	—
108	C₂×C₈×C₅₃	M₁₆	1	4	yes	complemented
109	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
113	C₄⋊C₄×C₅₃	—	2	2	no	—
114	C₄⋊C₄×C₅₃	—	2	1	no	—
115	C₄⋊C₄×C₅₃	—	2	1	no	—
116	C₄⋊C₄×C₅₃	—	2	2	no	—
117	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
118	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
119	C₂²⋊C₄×C₅₃	—	2	2	no	—
120	C₂²⋊C₄×C₅₃	—	2	2	no	—
121	C₄²×C₅₃	—	2	2	no	—
122	C₄²×C₅₃	—	2	2	no	—
123	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₅₃	#41	1	1	yes	characteristic
126	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
127	C₂³×C₅₃	—	2	2	no	—
128	C₂³×C₅₃	—	4	2	no	—
129	C₂³×C₅₃	—	4	2	no	—
130	C₂³×C₅₃	—	4	2	no	—
131	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
132	Q₈×C₅₃	—	2	2	no	—
133	Q₈×C₅₃	—	2	2	no	—
134	Q₈×C₅₃	—	4	2	no	—
135	D₄×C₅₃	—	2	2	no	—
136	D₄×C₅₃	—	2	2	no	—
137	D₄×C₅₃	—	4	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	—	4	2	no	—
147	C₂×C₄×C₅₃	—	4	4	no	—
148	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
152	C₂×C₄×C₅₃	#131	1	2	yes	—
153	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₄×C₅₃	#137	1	2	yes	—
155	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
158	C₈×C₅₃	—	2	2	no	—
159	C₈×C₅₃	—	2	2	no	—
160	C₈×C₅₃	—	2	2	no	—
161	C₈×C₅₃	—	4	2	no	—
162	C₈×C₅₃	—	4	2	no	—
163	C₈×C₅₃	—	4	2	no	—
164	C₈×C₅₃	C₂×M₁₆	1	4	yes	complemented
165	C₂×#1592	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂²×C₅₃	—	4	2	no	—
167	C₂²×C₅₃	—	4	2	no	—
168	C₂²×C₅₃	—	4	2	no	—
169	C₂²×C₅₃	—	8	4	no	—
170	C₂²×C₅₃	#436	1	2	yes	—
171	C₂²×C₅₃	#440	1	2	yes	—
172	C₂²×C₅₃	#444	1	2	yes	—
173	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	4	2	no	—
177	C₄×C₅₃	—	4	2	no	—
178	C₄×C₅₃	—	4	2	no	—
179	C₄×C₅₃	#436	1	2	yes	—
180	C₄×C₅₃	#440	1	2	yes	—
181	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
182	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
183	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
185	C₂×#446	C₂×C₅₃	1	1	yes	characteristic, complemented
186	C₂×C₄×SD₁₆	C₂×C₅₃	1	1	yes	characteristic
187	C₂×#436	C₂×C₅₃	1	1	yes	characteristic
188	#1592	C₂×C₅₃	1	8	yes	direct product factor
189	C₂×#335	C₂×C₅₃	1	1	yes	characteristic, complemented
190	C₂×Q₈⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
191	C₂×D₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic
192	C₂×#322	C₂×C₅₃	1	1	yes	characteristic, complemented
193	C₂×C₅₃	—	8	2	no	—
194	C₂×C₅₃	#1592	1	4	yes	direct product factor
195	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
196	C₂×C₅₃	C₂×#440	1	1	yes	characteristic
197	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
198	C₂²×SD₁₆	C₄×C₅₃	1	1	yes	characteristic
199	C₂²×M₁₆	—	2	1	no	—
200	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
201	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
202	#446	—	2	2	no	—
203	#446	C₂²×C₅₃	1	4	yes	complemented
204	C₄×SD₁₆	C₄×C₅₃	1	4	yes	—
205	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	—
206	#436	—	2	2	no	—
207	#436	C₂²×C₅₃	1	4	yes	—
208	C₂×#106	C₄×C₅₃	1	1	yes	characteristic
209	C₂×C₄⋊C₈	—	2	1	no	—
210	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
211	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
212	C₂×#103	C₄×C₅₃	1	1	yes	characteristic
213	C₂×#102	C₄×C₅₃	1	1	yes	characteristic
214	C₂×C₂²⋊C₈	—	2	1	no	—
215	C₂×#97	—	2	1	no	—
216	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
217	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
218	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
219	#335	C₂²×C₅₃	1	4	yes	complemented
220	Q₈⋊C₈	C₂²×C₅₃	1	4	yes	complemented
221	D₄⋊C₈	C₂²×C₅₃	1	4	yes	—
222	#322	C₂²×C₅₃	1	4	yes	complemented
223	C₅₃	C₂×#1592	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
225	C₂²×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
226	C₂³×C₄	—	2	1	no	—
227	C₂×SD₁₆	—	2	4	no	—
228	C₂×SD₁₆	C₂×C₄×C₅₃	1	4	yes	—
229	C₂×M₁₆	—	2	4	no	—
230	C₂×M₁₆	—	4	4	no	—
231	C₂²×C₈	—	2	1	no	—
232	C₂²×C₈	—	2	1	no	—
233	C₂²×C₈	—	2	1	no	—
234	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
235	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
236	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
237	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
241	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
242	C₄×D₄	—	2	1	no	—
243	C₄×D₄	—	2	1	no	—
244	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
245	C₄×D₄	C₂³×C₅₃	1	2	yes	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
248	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₂²⋊C₄	—	2	1	no	—
250	C₂×C₄²	—	2	1	no	—
251	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
252	#106	C₂×C₄×C₅₃	1	4	yes	—
253	C₄⋊C₈	—	2	4	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	—	2	1	no	—
256	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
257	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
258	#103	C₂×C₄×C₅₃	1	4	yes	—
259	#102	C₂×C₄×C₅₃	1	4	yes	—
260	C₂²⋊C₈	—	2	4	no	—
261	#97	—	2	4	no	—
262	C₄×C₈	—	2	1	no	—
263	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
264	C₄×C₈	D₄×C₅₃	1	2	yes	—
265	C₄×C₈	C₂³×C₅₃	1	2	yes	—
266	C₄×C₈	C₂³×C₅₃	1	2	yes	—
267	C₄×C₈	C₂³×C₅₃	1	2	yes	—
268	C₂⁴	—	2	1	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
277	C₂²×C₄	—	2	1	no	—
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	—	4	2	no	—
284	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
286	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
287	SD₁₆	—	2	8	no	—
288	M₁₆	—	4	8	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	4	2	no	—
296	C₂×C₈	—	4	2	no	—
297	C₂×C₈	—	4	2	no	—
298	C₂×C₈	M₁₆×C₅₃	1	4	yes	complemented
299	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
300	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
301	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
302	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	1	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
308	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	2	no	—
313	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
314	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
315	C₄²	#41×C₅₃	1	1	yes	characteristic
316	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
317	C₂³	—	2	2	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	4	2	no	—
320	C₂³	—	4	2	no	—
321	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
322	Q₈	—	2	2	no	—
323	Q₈	—	2	2	no	—
324	Q₈	—	4	2	no	—
325	D₄	—	2	2	no	—
326	D₄	—	2	2	no	—
327	D₄	—	4	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	4	no	—
338	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
339	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
340	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
341	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
342	C₂×C₄	#131×C₅₃	1	2	yes	—
343	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
344	C₂×C₄	#137×C₅₃	1	2	yes	—
345	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
346	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
347	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	2	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	4	2	no	—
353	C₈	—	4	2	no	—
354	C₈	C₂×M₁₆×C₅₃	1	4	yes	complemented
355	C₂²	—	4	2	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	8	4	no	—
359	C₂²	C₅₃×#436	1	2	yes	—
360	C₂²	C₅₃×#440	1	2	yes	—
361	C₂²	C₅₃×#444	1	2	yes	—
362	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	4	2	no	—
366	C₄	—	4	2	no	—
367	C₄	—	4	2	no	—
368	C₄	C₅₃×#436	1	2	yes	—
369	C₄	C₅₃×#440	1	2	yes	—
370	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
371	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
372	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
373	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
374	C₂	—	8	2	no	—
375	C₂	C₅₃×#1592	1	4	yes	direct product factor
376	C₂	C₂×C₅₃×#436	1	1	yes	characteristic
377	C₂	C₂×C₅₃×#440	1	1	yes	characteristic
378	C₂	C₂×C₅₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#446	C₂	semidirect
4. C₅₃×#1592	C₂	direct (N × H)
5. C₂×C₅₃×#335	C₂	semidirect
6. C₂×C₅₃×Q₈⋊C₈	C₂	semidirect
8. C₂×C₅₃×#322	C₂	semidirect
14. C₅₃×#446	C₂²	semidirect
30. C₅₃×#335	C₂²	semidirect
31. C₅₃×Q₈⋊C₈	C₂²	semidirect
33. C₅₃×#322	C₂²	semidirect
108. C₂×C₈×C₅₃	M₁₆	semidirect
164. C₈×C₅₃	C₂×M₁₆	semidirect
165. C₂×#1592	C₅₃	direct (N × H)
185. C₂×#446	C₂×C₅₃	semidirect
188. #1592	C₂×C₅₃	direct (N × H)
189. C₂×#335	C₂×C₅₃	semidirect
190. C₂×Q₈⋊C₈	C₂×C₅₃	semidirect
192. C₂×#322	C₂×C₅₃	semidirect
194. C₂×C₅₃	#1592	direct (N × H)
203. #446	C₂²×C₅₃	semidirect
219. #335	C₂²×C₅₃	semidirect
220. Q₈⋊C₈	C₂²×C₅₃	semidirect
222. #322	C₂²×C₅₃	semidirect
223. C₅₃	C₂×#1592	direct (N × H)
298. C₂×C₈	M₁₆×C₅₃	semidirect
354. C₈	C₂×M₁₆×C₅₃	semidirect
375. C₂	C₅₃×#1592	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×#446	2
C₂×C₄×SD₁₆×C₅₃	2
C₂×C₅₃×#436	2
C₅₃×#1592	2
C₂×C₅₃×#335	2
C₂×C₅₃×Q₈⋊C₈	2
C₂×C₅₃×D₄⋊C₈	2
C₂×C₅₃×#322	2
C₂×#1592	53

Subgroups — C2×C53×#1592 #27225399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1592 #27225399