Subgroups — C₅₃×#17681 #27225193

Order 13568.

940 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3514	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2971	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2923	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1535	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1533	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1530	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1529	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1529	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1527	C₂	1	2	yes	maximal, complemented
10	C₅₃×#1525	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#1524	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#559	C₄	1	1	yes	characteristic
13	C₅₃×#556	C₄	1	1	yes	characteristic
14	C₅₃×#555	—	2	1	no	—
15	C₅₃×#551	—	2	2	no	—
16	D₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
17	C₅₃×#548	C₄	1	2	yes	—
18	C₅₃×#546	—	2	1	no	—
19	C₅₃×#546	C₄	1	2	yes	—
20	D₄²×C₅₃	C₄	1	1	yes	characteristic
21	C₅₃×#542	—	2	1	no	—
22	C₅₃×#538	—	2	2	no	—
23	C₅₃×#532	C₂²	1	1	yes	characteristic
24	C₅₃×#529	C₂²	1	1	yes	characteristic
25	C₅₃×#529	C₂²	1	2	yes	—
26	C₂×#124×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#518	—	2	1	no	—
28	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
29	C₈×Q₈×C₅₃	—	2	1	no	—
30	C₅₃×#436	—	2	1	no	—
31	C₅₃×#436	—	2	2	no	—
32	C₅₃×#433	—	2	2	no	—
33	C₅₃×#432	C₂²	1	1	yes	characteristic, complemented
34	C₅₃×#432	C₂²	1	2	yes	complemented
35	C₅₃×#432	C₂²	1	2	yes	complemented
36	C₅₃×#432	C₂²	1	2	yes	complemented
37	C₅₃×#424	—	2	1	no	—
38	C₅₃×#408	—	2	1	no	—
39	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
40	C₄×M₁₆×C₅₃	C₂²	1	2	yes	complemented
41	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
42	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic, complemented
43	C₅₃×#330	C₂²	1	1	yes	characteristic, complemented
44	C₅₃×#330	C₂²	1	1	yes	characteristic, complemented
45	C₅₃×#330	C₂²	1	2	yes	complemented
46	C₅₃×#329	C₂²	1	1	yes	characteristic
47	C₅₃×#329	C₂²	1	1	yes	characteristic
48	C₅₃×#329	C₂²	1	2	yes	—
49	C₅₃×#329	C₂²	1	2	yes	complemented
50	C₅₃×#329	C₂²	1	2	yes	complemented
51	C₅₃×#329	C₂²	1	2	yes	complemented
52	C₅₃×#329	C₂²	1	2	yes	complemented
53	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
56	C₂²×D₄×C₅₃	—	2	1	no	—
57	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×M₁₆×C₅₃	—	2	1	no	—
59	C₂×M₁₆×C₅₃	—	2	2	no	—
60	C₂×M₁₆×C₅₃	—	2	1	no	—
61	C₂²×C₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	—	2	1	no	—
63	C₄⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	#127×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#126×C₅₃	—	2	2	no	—
66	#126×C₅₃	—	2	2	no	—
67	#126×C₅₃	—	2	1	no	—
68	#125×C₅₃	C₂³	1	1	yes	characteristic
69	#124×C₅₃	—	2	2	no	—
70	#124×C₅₃	—	2	2	no	—
71	#124×C₅₃	—	2	2	no	—
72	#124×C₅₃	—	2	1	no	—
73	#124×C₅₃	C₂×C₄	1	2	yes	—
74	#124×C₅₃	C₂×C₄	1	2	yes	—
75	#124×C₅₃	C₂³	1	1	yes	characteristic
76	#124×C₅₃	C₂³	1	2	yes	—
77	#123×C₅₃	—	2	2	no	—
78	#123×C₅₃	—	2	2	no	—
79	#123×C₅₃	—	2	1	no	—
80	#123×C₅₃	—	2	2	no	—
81	C₂²⋊Q₈×C₅₃	—	2	2	no	—
82	C₂²⋊Q₈×C₅₃	—	2	2	no	—
83	C₂²⋊Q₈×C₅₃	—	2	1	no	—
84	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
85	C₂²⋊Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
86	#121×C₅₃	—	2	1	no	—
87	#121×C₅₃	—	2	2	no	—
88	#121×C₅₃	—	2	2	no	—
89	#121×C₅₃	—	2	1	no	—
90	#121×C₅₃	—	2	1	no	—
91	#121×C₅₃	C₂×C₄	1	2	yes	—
92	#121×C₅₃	C₂×C₄	1	1	yes	characteristic
93	#120×C₅₃	—	2	2	no	—
94	#120×C₅₃	—	2	2	no	—
95	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
96	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
97	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
99	C₄×D₄×C₅₃	—	2	1	no	—
100	C₄×D₄×C₅₃	—	2	2	no	—
101	C₄×D₄×C₅₃	—	2	1	no	—
102	C₄×D₄×C₅₃	—	2	1	no	—
103	C₄×D₄×C₅₃	—	2	1	no	—
104	C₄×D₄×C₅₃	—	2	2	no	—
105	C₄×D₄×C₅₃	—	2	1	no	—
106	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
107	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
108	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
109	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
110	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
111	C₄×D₄×C₅₃	D₄	1	2	yes	—
112	#117×C₅₃	—	2	1	no	—
113	#117×C₅₃	—	2	1	no	—
114	#117×C₅₃	—	2	2	no	—
115	#117×C₅₃	D₄	1	2	yes	—
116	#117×C₅₃	D₄	1	1	yes	characteristic
117	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
120	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
121	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
122	C₄⋊C₈×C₅₃	—	2	2	no	—
123	C₄⋊C₈×C₅₃	—	2	2	no	—
124	C₄⋊C₈×C₅₃	—	2	1	no	—
125	C₄⋊C₈×C₅₃	—	2	2	no	—
126	C₂²⋊C₈×C₅₃	—	2	2	no	—
127	C₂²⋊C₈×C₅₃	—	2	2	no	—
128	C₂²⋊C₈×C₅₃	—	2	2	no	—
129	C₂²⋊C₈×C₅₃	—	2	1	no	—
130	#97×C₅₃	—	2	1	no	—
131	#97×C₅₃	C₂³	1	1	yes	characteristic
132	#97×C₅₃	C₂³	1	1	yes	characteristic
133	#97×C₅₃	C₂³	1	1	yes	characteristic
134	#97×C₅₃	C₂³	1	1	yes	characteristic
135	#97×C₅₃	C₂³	1	2	yes	—
136	#97×C₅₃	C₂³	1	2	yes	—
137	C₄×C₈×C₅₃	—	2	1	no	—
138	C₄×C₈×C₅₃	—	2	1	no	—
139	C₄×C₈×C₅₃	—	2	2	no	—
140	C₂⁴×C₅₃	—	2	2	no	—
141	#41×C₅₃	—	4	2	no	—
142	#41×C₅₃	—	8	2	no	—
143	C₂×Q₈×C₅₃	—	2	2	no	—
144	C₂×Q₈×C₅₃	—	8	2	no	—
145	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
146	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
148	C₂×D₄×C₅₃	—	2	2	no	—
149	C₂×D₄×C₅₃	—	2	1	no	—
150	C₂×D₄×C₅₃	—	2	2	no	—
151	C₂×D₄×C₅₃	—	2	1	no	—
152	C₂×D₄×C₅₃	—	2	1	no	—
153	C₂×D₄×C₅₃	—	2	2	no	—
154	C₂×D₄×C₅₃	—	4	4	no	—
155	C₂×D₄×C₅₃	—	8	2	no	—
156	C₂×D₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
157	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
158	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
159	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
160	C₂²×C₄×C₅₃	—	2	2	no	—
161	C₂²×C₄×C₅₃	—	2	1	no	—
162	C₂²×C₄×C₅₃	—	2	2	no	—
163	C₂²×C₄×C₅₃	—	2	2	no	—
164	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
165	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
170	M₁₆×C₅₃	—	4	4	no	—
171	M₁₆×C₅₃	—	4	2	no	—
172	M₁₆×C₅₃	—	4	2	no	—
173	C₂×C₈×C₅₃	—	2	1	no	—
174	C₂×C₈×C₅₃	—	2	2	no	—
175	C₂×C₈×C₅₃	—	2	1	no	—
176	C₂×C₈×C₅₃	—	2	2	no	—
177	C₂×C₈×C₅₃	—	2	1	no	—
178	C₂×C₈×C₅₃	—	2	1	no	—
179	C₂×C₈×C₅₃	—	4	2	no	—
180	C₄⋊C₄×C₅₃	—	2	1	no	—
181	C₄⋊C₄×C₅₃	—	2	1	no	—
182	C₄⋊C₄×C₅₃	—	2	2	no	—
183	C₄⋊C₄×C₅₃	—	2	1	no	—
184	C₄⋊C₄×C₅₃	—	2	2	no	—
185	C₄⋊C₄×C₅₃	—	2	1	no	—
186	C₄⋊C₄×C₅₃	—	2	1	no	—
187	C₄⋊C₄×C₅₃	—	2	1	no	—
188	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
189	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
190	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
191	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
192	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
193	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
194	C₂²⋊C₄×C₅₃	—	2	2	no	—
195	C₂²⋊C₄×C₅₃	—	2	1	no	—
196	C₂²⋊C₄×C₅₃	—	2	2	no	—
197	C₂²⋊C₄×C₅₃	—	2	1	no	—
198	C₂²⋊C₄×C₅₃	—	2	2	no	—
199	C₂²⋊C₄×C₅₃	—	2	2	no	—
200	C₂²⋊C₄×C₅₃	—	2	2	no	—
201	C₂²⋊C₄×C₅₃	—	2	2	no	—
202	C₂²⋊C₄×C₅₃	—	2	1	no	—
203	C₂²⋊C₄×C₅₃	—	2	2	no	—
204	C₂²⋊C₄×C₅₃	—	2	1	no	—
205	C₂²⋊C₄×C₅₃	—	2	1	no	—
206	C₂²⋊C₄×C₅₃	—	2	2	no	—
207	C₂²⋊C₄×C₅₃	—	2	2	no	—
208	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
209	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
210	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
211	C₄²×C₅₃	—	2	2	no	—
212	C₄²×C₅₃	—	2	1	no	—
213	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
214	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
215	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₅₃	C₂×D₄	1	2	yes	—
218	C₄²×C₅₃	C₂×D₄	1	2	yes	—
219	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
220	C₂³×C₅₃	—	2	2	no	—
221	C₂³×C₅₃	—	2	1	no	—
222	C₂³×C₅₃	—	4	4	no	—
223	C₂³×C₅₃	—	8	2	no	—
224	C₂³×C₅₃	—	8	2	no	—
225	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
226	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
227	Q₈×C₅₃	—	2	2	no	—
228	Q₈×C₅₃	—	4	1	no	—
229	Q₈×C₅₃	—	4	2	no	—
230	Q₈×C₅₃	—	8	2	no	—
231	D₄×C₅₃	—	2	2	no	—
232	D₄×C₅₃	—	4	2	no	—
233	D₄×C₅₃	—	4	1	no	—
234	D₄×C₅₃	—	4	2	no	—
235	D₄×C₅₃	—	4	4	no	—
236	D₄×C₅₃	—	4	2	no	—
237	D₄×C₅₃	—	8	1	no	—
238	D₄×C₅₃	—	8	2	no	—
239	D₄×C₅₃	—	8	1	no	—
240	D₄×C₅₃	—	8	2	no	—
241	C₂×C₄×C₅₃	—	2	2	no	—
242	C₂×C₄×C₅₃	—	2	2	no	—
243	C₂×C₄×C₅₃	—	2	1	no	—
244	C₂×C₄×C₅₃	—	2	1	no	—
245	C₂×C₄×C₅₃	—	2	1	no	—
246	C₂×C₄×C₅₃	—	4	2	no	—
247	C₂×C₄×C₅₃	—	4	1	no	—
248	C₂×C₄×C₅₃	—	4	2	no	—
249	C₂×C₄×C₅₃	—	4	2	no	—
250	C₂×C₄×C₅₃	—	4	1	no	—
251	C₂×C₄×C₅₃	—	4	2	no	—
252	C₂×C₄×C₅₃	—	8	2	no	—
253	C₂×C₄×C₅₃	—	8	2	no	—
254	C₂×C₄×C₅₃	—	8	2	no	—
255	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
257	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
258	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
260	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
262	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
264	C₈×C₅₃	—	2	2	no	—
265	C₈×C₅₃	—	4	1	no	—
266	C₈×C₅₃	—	4	1	no	—
267	C₈×C₅₃	—	4	2	no	—
268	C₈×C₅₃	—	4	1	no	—
269	C₈×C₅₃	—	4	2	no	—
270	#17681	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
271	C₂²×C₅₃	—	2	2	no	—
272	C₂²×C₅₃	—	2	1	no	—
273	C₂²×C₅₃	—	2	1	no	—
274	C₂²×C₅₃	—	2	2	no	—
275	C₂²×C₅₃	—	2	1	no	—
276	C₂²×C₅₃	—	2	2	no	—
277	C₂²×C₅₃	—	4	2	no	—
278	C₂²×C₅₃	—	4	1	no	—
279	C₂²×C₅₃	—	4	1	no	—
280	C₂²×C₅₃	—	4	2	no	—
281	C₂²×C₅₃	—	4	2	no	—
282	C₂²×C₅₃	—	4	1	no	—
283	C₂²×C₅₃	—	16	4	no	—
284	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
285	C₄×C₅₃	—	2	1	no	—
286	C₄×C₅₃	—	2	1	no	—
287	C₄×C₅₃	—	2	2	no	—
288	C₄×C₅₃	—	2	1	no	—
289	C₄×C₅₃	—	2	2	no	—
290	C₄×C₅₃	—	2	1	no	—
291	C₄×C₅₃	—	2	2	no	—
292	C₄×C₅₃	—	2	1	no	—
293	C₄×C₅₃	—	4	2	no	—
294	C₄×C₅₃	—	4	1	no	—
295	C₄×C₅₃	—	4	1	no	—
296	C₄×C₅₃	—	4	1	no	—
297	C₄×C₅₃	#421	1	2	yes	—
298	#3514	C₂×C₅₃	1	1	yes	characteristic
299	#2971	C₂×C₅₃	1	1	yes	characteristic, complemented
300	#2923	C₂×C₅₃	1	1	yes	characteristic, complemented
301	#1535	C₂×C₅₃	1	2	yes	complemented
302	#1533	C₂×C₅₃	1	2	yes	complemented
303	#1530	C₂×C₅₃	1	1	yes	characteristic, complemented
304	#1529	C₂×C₅₃	1	1	yes	characteristic, complemented
305	#1529	C₂×C₅₃	1	2	yes	complemented
306	#1527	C₂×C₅₃	1	2	yes	complemented
307	#1525	C₂×C₅₃	1	1	yes	characteristic, complemented
308	#1524	C₂×C₅₃	1	1	yes	characteristic, complemented
309	C₂×C₅₃	—	4	1	no	—
310	C₂×C₅₃	—	4	2	no	—
311	C₂×C₅₃	—	8	2	no	—
312	C₂×C₅₃	—	8	1	no	—
313	C₂×C₅₃	#2888	1	1	yes	characteristic
314	C₂×C₅₃	C₂×#421	1	1	yes	characteristic
315	C₂×C₅₃	#2901	1	1	yes	characteristic
316	#559	C₄×C₅₃	1	1	yes	characteristic
317	#556	C₄×C₅₃	1	1	yes	characteristic
318	#555	—	2	1	no	—
319	#551	—	2	2	no	—
320	D₄×Q₈	C₄×C₅₃	1	1	yes	characteristic
321	#548	C₄×C₅₃	1	2	yes	—
322	#546	—	2	1	no	—
323	#546	C₄×C₅₃	1	2	yes	—
324	D₄²	C₄×C₅₃	1	1	yes	characteristic
325	#542	—	2	1	no	—
326	#538	—	2	2	no	—
327	#532	C₂²×C₅₃	1	1	yes	characteristic
328	#529	C₂²×C₅₃	1	1	yes	characteristic
329	#529	C₂²×C₅₃	1	2	yes	—
330	C₂×#124	C₂²×C₅₃	1	2	yes	—
331	#518	—	2	1	no	—
332	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
333	C₈×Q₈	—	2	1	no	—
334	#436	—	2	1	no	—
335	#436	—	2	2	no	—
336	#433	—	2	2	no	—
337	#432	C₂²×C₅₃	1	2	yes	complemented
338	#432	C₂²×C₅₃	1	2	yes	complemented
339	#432	C₂²×C₅₃	1	1	yes	characteristic, complemented
340	#432	C₂²×C₅₃	1	2	yes	complemented
341	#424	—	2	1	no	—
342	#408	—	2	1	no	—
343	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
344	C₄×M₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
345	C₄×M₁₆	C₂²×C₅₃	1	2	yes	complemented
346	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic, complemented
347	#330	C₂²×C₅₃	1	1	yes	characteristic, complemented
348	#330	C₂²×C₅₃	1	1	yes	characteristic, complemented
349	#330	C₂²×C₅₃	1	2	yes	complemented
350	#329	C₂²×C₅₃	1	1	yes	characteristic
351	#329	C₂²×C₅₃	1	1	yes	characteristic
352	#329	C₂²×C₅₃	1	2	yes	—
353	#329	C₂²×C₅₃	1	2	yes	complemented
354	#329	C₂²×C₅₃	1	2	yes	complemented
355	#329	C₂²×C₅₃	1	2	yes	complemented
356	#329	C₂²×C₅₃	1	2	yes	complemented
357	C₅₃	#17681	1	1	yes	characteristic, direct product factor, Sylow
358	C₂×#41	C₂×C₄×C₅₃	1	2	yes	—
359	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
360	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
361	C₂²×D₄	—	2	1	no	—
362	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
363	C₂×M₁₆	—	2	1	no	—
364	C₂×M₁₆	—	2	2	no	—
365	C₂×M₁₆	—	2	1	no	—
366	C₂²×C₈	—	2	1	no	—
367	C₄⋊Q₈	—	2	1	no	—
368	C₄⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
369	#127	C₂×C₄×C₅₃	1	1	yes	characteristic
370	#126	—	2	2	no	—
371	#126	—	2	2	no	—
372	#126	—	2	1	no	—
373	#125	C₂³×C₅₃	1	1	yes	characteristic
374	#124	—	2	1	no	—
375	#124	—	2	2	no	—
376	#124	—	2	2	no	—
377	#124	—	2	2	no	—
378	#124	C₂×C₄×C₅₃	1	2	yes	—
379	#124	C₂×C₄×C₅₃	1	2	yes	—
380	#124	C₂³×C₅₃	1	1	yes	characteristic
381	#124	C₂³×C₅₃	1	2	yes	—
382	#123	—	2	1	no	—
383	#123	—	2	2	no	—
384	#123	—	2	2	no	—
385	#123	—	2	2	no	—
386	C₂²⋊Q₈	—	2	1	no	—
387	C₂²⋊Q₈	—	2	2	no	—
388	C₂²⋊Q₈	—	2	2	no	—
389	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
390	C₂²⋊Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
391	#121	—	2	2	no	—
392	#121	—	2	1	no	—
393	#121	—	2	1	no	—
394	#121	—	2	1	no	—
395	#121	—	2	2	no	—
396	#121	C₂×C₄×C₅₃	1	2	yes	—
397	#121	C₂×C₄×C₅₃	1	1	yes	characteristic
398	#120	—	2	2	no	—
399	#120	—	2	2	no	—
400	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
401	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
402	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
403	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	—	2	1	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	2	1	no	—
411	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
412	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
413	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
414	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
415	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
416	C₄×D₄	D₄×C₅₃	1	2	yes	—
417	#117	—	2	1	no	—
418	#117	—	2	1	no	—
419	#117	—	2	2	no	—
420	#117	D₄×C₅₃	1	2	yes	—
421	#117	D₄×C₅₃	1	1	yes	characteristic
422	C₂×C₂²⋊C₄	—	2	2	no	—
423	C₂×C₂²⋊C₄	—	2	2	no	—
424	C₂×C₂²⋊C₄	—	2	2	no	—
425	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
426	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
427	C₄⋊C₈	—	2	2	no	—
428	C₄⋊C₈	—	2	2	no	—
429	C₄⋊C₈	—	2	1	no	—
430	C₄⋊C₈	—	2	2	no	—
431	C₂²⋊C₈	—	2	2	no	—
432	C₂²⋊C₈	—	2	2	no	—
433	C₂²⋊C₈	—	2	2	no	—
434	C₂²⋊C₈	—	2	1	no	—
435	#97	—	2	1	no	—
436	#97	C₂³×C₅₃	1	1	yes	characteristic
437	#97	C₂³×C₅₃	1	1	yes	characteristic
438	#97	C₂³×C₅₃	1	1	yes	characteristic
439	#97	C₂³×C₅₃	1	1	yes	characteristic
440	#97	C₂³×C₅₃	1	2	yes	—
441	#97	C₂³×C₅₃	1	2	yes	—
442	C₄×C₈	—	2	1	no	—
443	C₄×C₈	—	2	1	no	—
444	C₄×C₈	—	2	2	no	—
445	C₂⁴	—	2	2	no	—
446	#41	—	4	2	no	—
447	#41	—	8	2	no	—
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	—	8	2	no	—
450	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
451	C₂×Q₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
452	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	4	4	no	—
460	C₂×D₄	—	8	2	no	—
461	C₂×D₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
462	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
463	C₂×D₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
464	C₂×D₄	C₂²×C₄×C₅₃	1	2	yes	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
470	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
473	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
474	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
475	M₁₆	—	4	4	no	—
476	M₁₆	—	4	2	no	—
477	M₁₆	—	4	2	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	2	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	2	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	4	2	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
494	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
495	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
496	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
497	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
498	C₄⋊C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	—	2	1	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	1	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	1	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	1	no	—
510	C₂²⋊C₄	—	2	1	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
514	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
515	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
516	C₄²	—	2	2	no	—
517	C₄²	—	2	1	no	—
518	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
519	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
520	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
521	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
522	C₄²	C₂×D₄×C₅₃	1	2	yes	—
523	C₄²	C₂×D₄×C₅₃	1	2	yes	—
524	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
525	C₂³	—	2	1	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	4	4	no	—
528	C₂³	—	8	2	no	—
529	C₂³	—	8	2	no	—
530	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
531	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
532	Q₈	—	2	2	no	—
533	Q₈	—	4	1	no	—
534	Q₈	—	4	2	no	—
535	Q₈	—	8	2	no	—
536	D₄	—	2	2	no	—
537	D₄	—	4	2	no	—
538	D₄	—	4	1	no	—
539	D₄	—	4	2	no	—
540	D₄	—	4	4	no	—
541	D₄	—	4	2	no	—
542	D₄	—	8	1	no	—
543	D₄	—	8	2	no	—
544	D₄	—	8	1	no	—
545	D₄	—	8	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	4	2	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	8	2	no	—
558	C₂×C₄	—	8	2	no	—
559	C₂×C₄	—	8	2	no	—
560	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
561	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
562	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
563	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
564	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
565	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
566	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
567	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
568	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
569	C₈	—	2	2	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	1	no	—
572	C₈	—	4	2	no	—
573	C₈	—	4	1	no	—
574	C₈	—	4	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	1	no	—
580	C₂²	—	2	2	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	2	no	—
585	C₂²	—	4	2	no	—
586	C₂²	—	4	1	no	—
587	C₂²	—	16	4	no	—
588	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
589	C₄	—	2	1	no	—
590	C₄	—	2	1	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	1	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	1	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	1	no	—
600	C₄	—	4	1	no	—
601	C₄	C₅₃×#421	1	2	yes	—
602	C₂	—	4	2	no	—
603	C₂	—	4	1	no	—
604	C₂	—	8	1	no	—
605	C₂	—	8	2	no	—
606	C₂	C₅₃×#2888	1	1	yes	characteristic
607	C₂	C₂×C₅₃×#421	1	1	yes	characteristic
608	C₂	C₅₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3514	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2971	C₂	semidirect
3. C₅₃×#2923	C₂	semidirect
4. C₅₃×#1535	C₂	semidirect
5. C₅₃×#1533	C₂	semidirect
6. C₅₃×#1530	C₂	semidirect
7. C₅₃×#1529	C₂	semidirect
8. C₅₃×#1529	C₂	semidirect
9. C₅₃×#1527	C₂	semidirect
10. C₅₃×#1525	C₂	semidirect
11. C₅₃×#1524	C₂	semidirect
33. C₅₃×#432	C₂²	semidirect
34. C₅₃×#432	C₂²	semidirect
35. C₅₃×#432	C₂²	semidirect
36. C₅₃×#432	C₂²	semidirect
39. C₄×M₁₆×C₅₃	C₂²	semidirect
40. C₄×M₁₆×C₅₃	C₂²	semidirect
41. C₄×M₁₆×C₅₃	C₂²	semidirect
42. C₂×#97×C₅₃	C₂²	semidirect
43. C₅₃×#330	C₂²	semidirect
44. C₅₃×#330	C₂²	semidirect
45. C₅₃×#330	C₂²	semidirect
49. C₅₃×#329	C₂²	semidirect
50. C₅₃×#329	C₂²	semidirect
51. C₅₃×#329	C₂²	semidirect
52. C₅₃×#329	C₂²	semidirect
270. #17681	C₅₃	direct (N × H)
299. #2971	C₂×C₅₃	semidirect
300. #2923	C₂×C₅₃	semidirect
301. #1535	C₂×C₅₃	semidirect
302. #1533	C₂×C₅₃	semidirect
303. #1530	C₂×C₅₃	semidirect
304. #1529	C₂×C₅₃	semidirect
305. #1529	C₂×C₅₃	semidirect
306. #1527	C₂×C₅₃	semidirect
307. #1525	C₂×C₅₃	semidirect
308. #1524	C₂×C₅₃	semidirect
337. #432	C₂²×C₅₃	semidirect
338. #432	C₂²×C₅₃	semidirect
339. #432	C₂²×C₅₃	semidirect
340. #432	C₂²×C₅₃	semidirect
343. C₄×M₁₆	C₂²×C₅₃	semidirect
344. C₄×M₁₆	C₂²×C₅₃	semidirect
345. C₄×M₁₆	C₂²×C₅₃	semidirect
346. C₂×#97	C₂²×C₅₃	semidirect
347. #330	C₂²×C₅₃	semidirect
348. #330	C₂²×C₅₃	semidirect
349. #330	C₂²×C₅₃	semidirect
353. #329	C₂²×C₅₃	semidirect
354. #329	C₂²×C₅₃	semidirect
355. #329	C₂²×C₅₃	semidirect
356. #329	C₂²×C₅₃	semidirect
357. C₅₃	#17681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3514	2
C₅₃×#2971	2
C₅₃×#2923	2
C₅₃×#1535	2
C₅₃×#1533	2
C₅₃×#1530	2
C₅₃×#1529	2
C₅₃×#1527	2
C₅₃×#1525	2
C₅₃×#1524	2
#17681	53

Subgroups — C53×#17681 #27225193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17681 #27225193