Subgroups — C₅₃×#17403 #27224915

Order 13568.

1240 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1471	C₂	1	4	yes	maximal, complemented
5	C₅₃×#1469	C₂	1	4	yes	maximal, complemented
6	C₅₃×#1466	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₅₃	C₄	1	2	yes	—
8	C₅₃×#532	—	2	1	no	—
9	C₅₃×#532	—	2	1	no	—
10	C₅₃×#532	—	2	1	no	—
11	C₅₃×#532	—	2	1	no	—
12	C₅₃×#532	—	2	1	no	—
13	C₅₃×#532	—	2	1	no	—
14	C₅₃×#532	—	2	1	no	—
15	C₅₃×#532	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#127×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#124×C₅₃	C₂²	1	2	yes	—
19	C₂×#121×C₅₃	C₄	1	2	yes	—
20	C₂×#121×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₅₃	C₄	1	2	yes	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
23	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#432	—	2	1	no	—
25	C₅₃×#432	—	2	1	no	—
26	C₅₃×#432	—	2	1	no	—
27	C₅₃×#432	C₂²	1	2	yes	—
28	C₅₃×#432	C₂²	1	2	yes	—
29	C₅₃×#432	C₂²	1	2	yes	—
30	C₅₃×#431	—	2	1	no	—
31	C₅₃×#431	C₂²	1	2	yes	—
32	C₅₃×#423	—	2	1	no	—
33	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
34	C₅₃×#407	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
36	C₄×M₁₆×C₅₃	—	2	1	no	—
37	C₂×#97×C₅₃	—	2	1	no	—
38	C₅₃×#323	C₂²	1	16	yes	—
39	C₂×#41×C₅₃	—	2	2	no	—
40	C₂×#41×C₅₃	—	2	2	no	—
41	C₂×#41×C₅₃	—	2	2	no	—
42	C₂×#41×C₅₃	—	2	2	no	—
43	C₂×#41×C₅₃	—	4	8	no	—
44	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²×D₄×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₅₃	C₂×C₄	1	4	yes	—
51	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	—	2	2	no	—
54	C₂²×C₈×C₅₃	—	2	2	no	—
55	C₄⋊Q₈×C₅₃	—	2	1	no	—
56	C₄⋊Q₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	—	2	1	no	—
59	#127×C₅₃	—	2	1	no	—
60	#127×C₅₃	—	2	1	no	—
61	#127×C₅₃	—	2	1	no	—
62	#127×C₅₃	—	2	1	no	—
63	#124×C₅₃	—	2	2	no	—
64	#124×C₅₃	—	2	2	no	—
65	#124×C₅₃	—	2	2	no	—
66	#124×C₅₃	—	2	2	no	—
67	#121×C₅₃	—	2	2	no	—
68	#121×C₅₃	—	2	2	no	—
69	#121×C₅₃	—	2	2	no	—
70	#121×C₅₃	—	2	2	no	—
71	#121×C₅₃	—	2	2	no	—
72	#121×C₅₃	—	2	2	no	—
73	#121×C₅₃	—	2	2	no	—
74	#121×C₅₃	—	2	2	no	—
75	C₄×D₄×C₅₃	—	2	2	no	—
76	C₄×D₄×C₅₃	—	2	2	no	—
77	C₄×D₄×C₅₃	—	2	2	no	—
78	C₄×D₄×C₅₃	—	2	2	no	—
79	C₄×D₄×C₅₃	—	2	2	no	—
80	C₄×D₄×C₅₃	—	2	2	no	—
81	C₄×D₄×C₅₃	—	2	2	no	—
82	C₄×D₄×C₅₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
84	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
85	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
87	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
90	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₈×C₅₃	—	2	2	no	—
100	C₄⋊C₈×C₅₃	—	2	2	no	—
101	C₄⋊C₈×C₅₃	—	2	2	no	—
102	C₄⋊C₈×C₅₃	—	2	2	no	—
103	C₂²⋊C₈×C₅₃	—	2	2	no	—
104	C₂²⋊C₈×C₅₃	—	2	2	no	—
105	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
106	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
107	#97×C₅₃	—	2	2	no	—
108	#97×C₅₃	—	2	2	no	—
109	#97×C₅₃	—	2	2	no	—
110	C₄×C₈×C₅₃	—	2	2	no	—
111	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
113	#41×C₅₃	—	4	8	no	—
114	#41×C₅₃	—	4	8	no	—
115	#41×C₅₃	—	4	8	no	—
116	#41×C₅₃	—	4	8	no	—
117	C₂×Q₈×C₅₃	—	2	2	no	—
118	C₂×Q₈×C₅₃	—	2	2	no	—
119	C₂×Q₈×C₅₃	—	4	4	no	—
120	C₂×D₄×C₅₃	—	2	4	no	—
121	C₂×D₄×C₅₃	—	2	4	no	—
122	C₂×D₄×C₅₃	—	2	2	no	—
123	C₂×D₄×C₅₃	—	2	2	no	—
124	C₂×D₄×C₅₃	—	2	2	no	—
125	C₂×D₄×C₅₃	—	2	2	no	—
126	C₂×D₄×C₅₃	—	2	2	no	—
127	C₂×D₄×C₅₃	—	2	2	no	—
128	C₂×D₄×C₅₃	—	4	4	no	—
129	C₂×D₄×C₅₃	—	4	4	no	—
130	C₂×D₄×C₅₃	—	4	4	no	—
131	C₂×D₄×C₅₃	—	4	8	no	—
132	C₂²×C₄×C₅₃	—	2	4	no	—
133	C₂²×C₄×C₅₃	—	2	4	no	—
134	C₂²×C₄×C₅₃	—	2	4	no	—
135	C₂²×C₄×C₅₃	—	4	2	no	—
136	C₂²×C₄×C₅₃	—	4	2	no	—
137	C₂²×C₄×C₅₃	—	4	8	no	—
138	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
152	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
157	M₁₆×C₅₃	—	4	4	no	—
158	C₂×C₈×C₅₃	—	2	4	no	—
159	C₂×C₈×C₅₃	—	2	4	no	—
160	C₂×C₈×C₅₃	—	4	4	no	—
161	C₄⋊C₄×C₅₃	—	2	2	no	—
162	C₄⋊C₄×C₅₃	—	2	2	no	—
163	C₄⋊C₄×C₅₃	—	2	2	no	—
164	C₄⋊C₄×C₅₃	—	2	2	no	—
165	C₂²⋊C₄×C₅₃	—	2	4	no	—
166	C₂²⋊C₄×C₅₃	—	2	4	no	—
167	C₂²⋊C₄×C₅₃	—	2	4	no	—
168	C₂²⋊C₄×C₅₃	—	2	4	no	—
169	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
170	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
171	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
172	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
173	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
174	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
175	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
176	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
177	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
185	C₂³×C₅₃	—	2	2	no	—
186	C₂³×C₅₃	—	2	2	no	—
187	C₂³×C₅₃	—	2	4	no	—
188	C₂³×C₅₃	—	2	8	no	—
189	C₂³×C₅₃	—	4	1	no	—
190	C₂³×C₅₃	—	4	4	no	—
191	C₂³×C₅₃	—	4	4	no	—
192	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
193	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
195	Q₈×C₅₃	—	4	4	no	—
196	Q₈×C₅₃	—	4	4	no	—
197	D₄×C₅₃	—	4	4	no	—
198	D₄×C₅₃	—	4	4	no	—
199	D₄×C₅₃	—	4	4	no	—
200	D₄×C₅₃	—	4	4	no	—
201	D₄×C₅₃	—	4	8	no	—
202	D₄×C₅₃	—	4	8	no	—
203	D₄×C₅₃	—	4	4	no	—
204	D₄×C₅₃	—	4	4	no	—
205	C₂×C₄×C₅₃	—	2	2	no	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	2	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	2	no	—
210	C₂×C₄×C₅₃	—	2	2	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	2	2	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	4	1	no	—
217	C₂×C₄×C₅₃	—	4	1	no	—
218	C₂×C₄×C₅₃	—	4	1	no	—
219	C₂×C₄×C₅₃	—	4	1	no	—
220	C₂×C₄×C₅₃	—	4	1	no	—
221	C₂×C₄×C₅₃	—	4	1	no	—
222	C₂×C₄×C₅₃	—	4	1	no	—
223	C₂×C₄×C₅₃	—	4	8	no	—
224	C₂×C₄×C₅₃	—	4	8	no	—
225	C₂×C₄×C₅₃	—	4	4	no	—
226	C₂×C₄×C₅₃	#100	1	4	yes	—
227	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
231	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
239	C₈×C₅₃	—	4	4	no	—
240	C₈×C₅₃	—	4	4	no	—
241	#17403	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂²×C₅₃	—	2	1	no	—
243	C₂²×C₅₃	—	2	1	no	—
244	C₂²×C₅₃	—	2	1	no	—
245	C₂²×C₅₃	—	2	1	no	—
246	C₂²×C₅₃	—	2	1	no	—
247	C₂²×C₅₃	—	2	1	no	—
248	C₂²×C₅₃	—	2	1	no	—
249	C₂²×C₅₃	—	4	4	no	—
250	C₂²×C₅₃	—	4	4	no	—
251	C₂²×C₅₃	—	4	4	no	—
252	C₂²×C₅₃	—	4	4	no	—
253	C₂²×C₅₃	—	8	8	no	—
254	C₂²×C₅₃	C₂×#100	1	2	yes	—
255	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
256	C₂²×C₅₃	#568	1	2	yes	—
257	C₄×C₅₃	—	2	1	no	—
258	C₄×C₅₃	—	2	1	no	—
259	C₄×C₅₃	—	2	1	no	—
260	C₄×C₅₃	—	2	1	no	—
261	C₄×C₅₃	—	2	1	no	—
262	C₄×C₅₃	—	2	1	no	—
263	C₄×C₅₃	—	2	1	no	—
264	C₄×C₅₃	—	2	1	no	—
265	C₄×C₅₃	—	2	1	no	—
266	C₄×C₅₃	—	4	4	no	—
267	C₄×C₅₃	#410	1	2	yes	—
268	C₄×C₅₃	C₂×#100	1	2	yes	—
269	C₄×C₅₃	C₂×#100	1	2	yes	—
270	C₂×#532	C₂×C₅₃	1	1	yes	characteristic
271	#2920	C₂×C₅₃	1	1	yes	characteristic, complemented
272	C₂×#432	C₂×C₅₃	1	1	yes	characteristic, complemented
273	#1471	C₂×C₅₃	1	4	yes	complemented
274	#1469	C₂×C₅₃	1	4	yes	complemented
275	#1466	C₂×C₅₃	1	4	yes	complemented
276	C₂×C₅₃	—	2	2	no	—
277	C₂×C₅₃	—	2	2	no	—
278	C₂×C₅₃	—	4	1	no	—
279	C₂×C₅₃	—	8	4	no	—
280	C₂×C₅₃	#2879	1	1	yes	characteristic
281	C₂×C₅₃	C₂×#410	1	1	yes	characteristic
282	C₂×C₅₃	C₂²×#100	1	1	yes	characteristic
283	C₂²×#41	C₄×C₅₃	1	2	yes	—
284	#532	—	2	1	no	—
285	#532	—	2	1	no	—
286	#532	—	2	1	no	—
287	#532	—	2	1	no	—
288	#532	—	2	1	no	—
289	#532	—	2	1	no	—
290	#532	—	2	1	no	—
291	#532	—	2	1	no	—
292	C₂×C₄⋊Q₈	C₄×C₅₃	1	1	yes	characteristic
293	C₂×#127	C₄×C₅₃	1	1	yes	characteristic
294	C₂×#124	C₂²×C₅₃	1	2	yes	—
295	C₂×#121	C₄×C₅₃	1	2	yes	—
296	C₂×#121	C₂²×C₅₃	1	2	yes	—
297	C₂×C₄×D₄	C₄×C₅₃	1	2	yes	—
298	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
299	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
300	#432	—	2	1	no	—
301	#432	—	2	1	no	—
302	#432	—	2	1	no	—
303	#432	C₂²×C₅₃	1	2	yes	—
304	#432	C₂²×C₅₃	1	2	yes	—
305	#432	C₂²×C₅₃	1	2	yes	—
306	#431	—	2	1	no	—
307	#431	C₂²×C₅₃	1	2	yes	—
308	#423	—	2	1	no	—
309	C₂×C₄⋊C₈	—	2	1	no	—
310	#407	C₂²×C₅₃	1	2	yes	—
311	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
312	C₄×M₁₆	—	2	1	no	—
313	C₂×#97	—	2	1	no	—
314	#323	C₂²×C₅₃	1	16	yes	—
315	C₅₃	#17403	1	1	yes	characteristic, direct product factor, Sylow
316	C₂×#41	—	2	2	no	—
317	C₂×#41	—	2	2	no	—
318	C₂×#41	—	2	2	no	—
319	C₂×#41	—	2	2	no	—
320	C₂×#41	—	4	8	no	—
321	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
322	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
323	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
324	C₂²×D₄	C₂×C₄×C₅₃	1	4	yes	—
325	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
326	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
327	C₂³×C₄	C₂×C₄×C₅₃	1	4	yes	—
328	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
329	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
330	C₂×M₁₆	—	2	2	no	—
331	C₂²×C₈	—	2	2	no	—
332	C₄⋊Q₈	—	2	1	no	—
333	C₄⋊Q₈	—	2	1	no	—
334	C₄⋊Q₈	—	2	1	no	—
335	C₄⋊Q₈	—	2	1	no	—
336	#127	—	2	1	no	—
337	#127	—	2	1	no	—
338	#127	—	2	1	no	—
339	#127	—	2	1	no	—
340	#124	—	2	2	no	—
341	#124	—	2	2	no	—
342	#124	—	2	2	no	—
343	#124	—	2	2	no	—
344	#121	—	2	2	no	—
345	#121	—	2	2	no	—
346	#121	—	2	2	no	—
347	#121	—	2	2	no	—
348	#121	—	2	2	no	—
349	#121	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	C₄×D₄	—	2	2	no	—
353	C₄×D₄	—	2	2	no	—
354	C₄×D₄	—	2	2	no	—
355	C₄×D₄	—	2	2	no	—
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
361	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
362	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
363	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
364	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
365	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
366	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
367	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
368	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
369	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
370	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
371	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
372	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
373	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
374	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
375	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
376	C₄⋊C₈	—	2	2	no	—
377	C₄⋊C₈	—	2	2	no	—
378	C₄⋊C₈	—	2	2	no	—
379	C₄⋊C₈	—	2	2	no	—
380	C₂²⋊C₈	—	2	2	no	—
381	C₂²⋊C₈	—	2	2	no	—
382	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
383	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
384	#97	—	2	2	no	—
385	#97	—	2	2	no	—
386	#97	—	2	2	no	—
387	C₄×C₈	—	2	2	no	—
388	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
389	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
390	#41	—	4	8	no	—
391	#41	—	4	8	no	—
392	#41	—	4	8	no	—
393	#41	—	4	8	no	—
394	C₂×Q₈	—	2	2	no	—
395	C₂×Q₈	—	2	2	no	—
396	C₂×Q₈	—	4	4	no	—
397	C₂×D₄	—	2	4	no	—
398	C₂×D₄	—	2	4	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	4	4	no	—
406	C₂×D₄	—	4	4	no	—
407	C₂×D₄	—	4	4	no	—
408	C₂×D₄	—	4	8	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	4	2	no	—
413	C₂²×C₄	—	4	2	no	—
414	C₂²×C₄	—	4	8	no	—
415	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
416	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
417	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
418	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
419	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
420	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
421	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
422	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
423	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
424	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
425	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
426	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
427	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
428	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
429	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
430	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
433	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
434	M₁₆	—	4	4	no	—
435	C₂×C₈	—	2	4	no	—
436	C₂×C₈	—	2	4	no	—
437	C₂×C₈	—	4	4	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₂²⋊C₄	—	2	4	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
447	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
448	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
449	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
450	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
451	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
452	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
453	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
454	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
455	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
456	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
457	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
458	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
459	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
460	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
461	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
462	C₂³	—	2	2	no	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	4	no	—
465	C₂³	—	2	8	no	—
466	C₂³	—	4	1	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	4	4	no	—
469	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
470	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
471	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
472	Q₈	—	4	4	no	—
473	Q₈	—	4	4	no	—
474	D₄	—	4	4	no	—
475	D₄	—	4	4	no	—
476	D₄	—	4	4	no	—
477	D₄	—	4	4	no	—
478	D₄	—	4	8	no	—
479	D₄	—	4	8	no	—
480	D₄	—	4	4	no	—
481	D₄	—	4	4	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	1	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	4	1	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	8	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	—	4	4	no	—
503	C₂×C₄	#100×C₅₃	1	4	yes	—
504	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
505	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
506	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
507	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
508	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
509	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
510	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
511	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
512	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
513	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
514	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
515	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
516	C₈	—	4	4	no	—
517	C₈	—	4	4	no	—
518	C₂²	—	2	1	no	—
519	C₂²	—	2	1	no	—
520	C₂²	—	2	1	no	—
521	C₂²	—	2	1	no	—
522	C₂²	—	2	1	no	—
523	C₂²	—	2	1	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	4	4	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	4	no	—
529	C₂²	—	8	8	no	—
530	C₂²	C₂×#100×C₅₃	1	2	yes	—
531	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
532	C₂²	C₅₃×#568	1	2	yes	—
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	2	1	no	—
536	C₄	—	2	1	no	—
537	C₄	—	2	1	no	—
538	C₄	—	2	1	no	—
539	C₄	—	2	1	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	4	4	no	—
543	C₄	C₅₃×#410	1	2	yes	—
544	C₄	C₂×#100×C₅₃	1	2	yes	—
545	C₄	C₂×#100×C₅₃	1	2	yes	—
546	C₂	—	2	2	no	—
547	C₂	—	2	2	no	—
548	C₂	—	4	1	no	—
549	C₂	—	8	4	no	—
550	C₂	C₅₃×#2879	1	1	yes	characteristic
551	C₂	C₂×C₅₃×#410	1	1	yes	characteristic
552	C₂	C₂²×#100×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂×C₅₃×#532	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2920	C₂	semidirect
3. C₂×C₅₃×#432	C₂	semidirect
4. C₅₃×#1471	C₂	semidirect
5. C₅₃×#1469	C₂	semidirect
6. C₅₃×#1466	C₂	semidirect
241. #17403	C₅₃	direct (N × H)
271. #2920	C₂×C₅₃	semidirect
272. C₂×#432	C₂×C₅₃	semidirect
273. #1471	C₂×C₅₃	semidirect
274. #1469	C₂×C₅₃	semidirect
275. #1466	C₂×C₅₃	semidirect
315. C₅₃	#17403	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×#532	2
C₅₃×#2920	2
C₂×C₅₃×#432	2
C₅₃×#1471	2
C₅₃×#1469	2
C₅₃×#1466	2
#17403	53

Subgroups — C53×#17403 #27224915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17403 #27224915