Subgroups — C₅₃×#60525 #27268037

Order 13568.

2234 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3488	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3099	C₂	1	6	yes	maximal, complemented
4	C₅₃×#3096	C₂	1	6	yes	maximal, complemented
5	C₅₃×#3094	C₂	1	3	yes	maximal, complemented
6	C₅₃×#3092	C₂	1	3	yes	maximal, complemented
7	C₅₃×#3091	C₂	1	3	yes	maximal, complemented
8	C₂×C₅₃×#481	C₂	1	3	yes	maximal, complemented
9	C₅₃×#2912	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2897	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2880	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#131×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#550	—	2	1	no	—
16	C₅₃×#548	—	2	1	no	—
17	C₅₃×#546	—	2	3	no	—
18	D₄²×C₅₃	—	2	3	no	—
19	C₅₃×#537	C₂²	1	3	yes	—
20	C₅₃×#535	C₂²	1	3	yes	—
21	C₅₃×#525	C₂²	1	3	yes	—
22	C₂×#121×C₅₃	C₂²	1	3	yes	—
23	C₅₃×#520	—	2	3	no	—
24	C₅₃×#520	—	2	1	no	—
25	C₅₃×#520	C₂²	1	1	yes	characteristic
26	C₅₃×#520	C₂²	1	2	yes	—
27	C₅₃×#520	C₂²	1	6	yes	complemented
28	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
29	C₂×C₄×D₄×C₅₃	—	2	3	no	—
30	C₂×#117×C₅₃	C₂²	1	3	yes	complemented
31	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	complemented
32	C₅₃×#485	C₂²	1	6	yes	complemented
33	C₅₃×#484	C₂²	1	3	yes	complemented
34	C₅₃×#483	C₂²	1	1	yes	characteristic, complemented
35	C₅₃×#483	C₂²	1	6	yes	complemented
36	C₅₃×#482	C₂²	1	6	yes	complemented
37	C₅₃×#482	C₂²	1	12	yes	complemented
38	C₅₃×#481	C₂²	1	3	yes	complemented
39	C₅₃×#481	C₂²	1	6	yes	complemented
40	C₅₃×#481	C₂²	1	12	yes	complemented
41	C₅₃×#480	C₂²	1	3	yes	complemented
42	C₅₃×#480	C₂²	1	6	yes	complemented
43	C₅₃×#428	C₂²	1	6	yes	complemented
44	C₅₃×#428	C₂²	1	6	yes	complemented
45	C₅₃×#427	C₂²	1	6	yes	complemented
46	C₂×#106×C₅₃	C₂²	1	6	yes	complemented
47	C₅₃×#418	C₂²	1	6	yes	complemented
48	C₅₃×#417	C₂²	1	6	yes	complemented
49	C₅₃×#416	C₂²	1	6	yes	complemented
50	C₂×#102×C₅₃	C₂²	1	6	yes	complemented
51	C₅₃×#408	C₂²	1	6	yes	complemented
52	C₅₃×#407	C₂²	1	3	yes	complemented
53	C₂×C₂²⋊C₈×C₅₃	C₂²	1	3	yes	complemented
54	H(5,𝔽₂)×C₅₃	—	2	8	no	—
55	C₂×#41×C₅₃	—	2	8	no	—
56	C₂×#41×C₅₃	—	2	1	no	—
57	C₂×#41×C₅₃	D₄	1	3	yes	—
58	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
59	C₂×#41×C₅₃	C₂³	1	3	yes	—
60	C₂×#41×C₅₃	C₂³	1	3	yes	—
61	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
63	C₂²×D₄×C₅₃	—	2	3	no	—
64	C₂²×D₄×C₅₃	—	2	3	no	—
65	C₂²×D₄×C₅₃	D₄	1	3	yes	—
66	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
67	C₂³×C₄×C₅₃	—	2	3	no	—
68	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
69	#131×C₅₃	—	2	8	no	—
70	C₂×M₁₆×C₅₃	C₂³	1	6	yes	—
71	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
72	#127×C₅₃	—	2	3	no	—
73	#124×C₅₃	—	2	3	no	—
74	#123×C₅₃	—	2	3	no	—
75	C₂²⋊Q₈×C₅₃	—	2	3	no	—
76	C₂²⋊Q₈×C₅₃	C₂³	1	3	yes	—
77	C₂²⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
78	#121×C₅₃	—	2	6	no	—
79	#121×C₅₃	—	2	3	no	—
80	#121×C₅₃	—	2	3	no	—
81	#121×C₅₃	—	2	3	no	—
82	#121×C₅₃	C₂³	1	3	yes	—
83	#121×C₅₃	C₂³	1	6	yes	—
84	#121×C₅₃	C₂³	1	3	yes	—
85	#120×C₅₃	—	2	3	no	—
86	#120×C₅₃	—	2	6	no	—
87	C₄×Q₈×C₅₃	—	2	2	no	—
88	C₄×Q₈×C₅₃	—	2	6	no	—
89	C₄×Q₈×C₅₃	—	2	1	no	—
90	C₄×D₄×C₅₃	—	2	6	no	—
91	C₄×D₄×C₅₃	—	2	12	no	—
92	C₄×D₄×C₅₃	—	2	6	no	—
93	C₄×D₄×C₅₃	—	2	3	no	—
94	C₄×D₄×C₅₃	—	2	3	no	—
95	C₄×D₄×C₅₃	—	2	3	no	—
96	#117×C₅₃	—	2	3	no	—
97	#117×C₅₃	—	2	3	no	—
98	#117×C₅₃	C₂³	1	6	yes	—
99	#117×C₅₃	C₂³	1	6	yes	—
100	#117×C₅₃	C₂³	1	3	yes	—
101	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
102	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
103	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
104	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
105	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
106	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
107	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
109	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
110	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
111	C₂×C₄²×C₅₃	—	2	3	no	—
112	#107×C₅₃	C₂³	1	6	yes	—
113	#107×C₅₃	C₂³	1	6	yes	—
114	#106×C₅₃	C₂³	1	6	yes	—
115	#106×C₅₃	C₂³	1	12	yes	—
116	#106×C₅₃	C₂³	1	2	yes	—
117	#103×C₅₃	C₂³	1	2	yes	—
118	#103×C₅₃	C₂³	1	6	yes	—
119	#102×C₅₃	C₂³	1	6	yes	—
120	#102×C₅₃	C₂³	1	12	yes	—
121	#102×C₅₃	C₂³	1	6	yes	—
122	C₂²⋊C₈×C₅₃	C₂³	1	6	yes	—
123	C₂²⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
124	C₂²⋊C₈×C₅₃	C₂³	1	3	yes	—
125	C₂²⋊C₈×C₅₃	C₂³	1	6	yes	—
126	C₂⁴×C₅₃	—	2	3	no	—
127	C₂⁴×C₅₃	C₂×D₄	1	3	yes	—
128	#41×C₅₃	—	2	4	no	—
129	#41×C₅₃	—	2	12	no	—
130	#41×C₅₃	—	2	12	no	—
131	#41×C₅₃	—	2	12	no	—
132	#41×C₅₃	—	4	4	no	—
133	#41×C₅₃	#41	1	8	yes	—
134	C₂×Q₈×C₅₃	—	2	4	no	—
135	C₂×Q₈×C₅₃	C₂×D₄	1	3	yes	—
136	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
137	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₂×D₄×C₅₃	—	2	12	no	—
139	C₂×D₄×C₅₃	—	2	3	no	—
140	C₂×D₄×C₅₃	—	2	3	no	—
141	C₂×D₄×C₅₃	—	2	3	no	—
142	C₂×D₄×C₅₃	—	2	6	no	—
143	C₂×D₄×C₅₃	—	2	3	no	—
144	C₂×D₄×C₅₃	—	2	12	no	—
145	C₂×D₄×C₅₃	—	4	12	no	—
146	C₂×D₄×C₅₃	—	8	6	no	—
147	C₂×D₄×C₅₃	C₂×D₄	1	6	yes	—
148	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
149	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
150	C₂×D₄×C₅₃	C₂⁴	1	3	yes	—
151	C₂²×C₄×C₅₃	—	2	12	no	—
152	C₂²×C₄×C₅₃	—	2	1	no	—
153	C₂²×C₄×C₅₃	—	2	3	no	—
154	C₂²×C₄×C₅₃	—	2	3	no	—
155	C₂²×C₄×C₅₃	—	2	6	no	—
156	C₂²×C₄×C₅₃	—	2	6	no	—
157	C₂²×C₄×C₅₃	—	2	3	no	—
158	C₂²×C₄×C₅₃	—	8	6	no	—
159	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
160	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
161	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
162	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
164	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
165	M₁₆×C₅₃	—	2	12	no	—
166	C₂×C₈×C₅₃	—	2	12	no	—
167	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
168	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
169	C₄⋊C₄×C₅₃	—	2	3	no	—
170	C₄⋊C₄×C₅₃	—	2	6	no	—
171	C₄⋊C₄×C₅₃	—	2	6	no	—
172	C₄⋊C₄×C₅₃	—	2	1	no	—
173	C₄⋊C₄×C₅₃	—	2	3	no	—
174	C₄⋊C₄×C₅₃	—	2	1	no	—
175	C₄⋊C₄×C₅₃	—	2	3	no	—
176	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
177	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
178	C₄⋊C₄×C₅₃	C₂⁴	1	3	yes	—
179	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
180	C₂²⋊C₄×C₅₃	—	2	6	no	—
181	C₂²⋊C₄×C₅₃	—	2	6	no	—
182	C₂²⋊C₄×C₅₃	—	2	3	no	—
183	C₂²⋊C₄×C₅₃	—	2	1	no	—
184	C₂²⋊C₄×C₅₃	—	2	3	no	—
185	C₂²⋊C₄×C₅₃	—	2	3	no	—
186	C₂²⋊C₄×C₅₃	—	2	6	no	—
187	C₂²⋊C₄×C₅₃	—	2	3	no	—
188	C₄²×C₅₃	—	2	6	no	—
189	C₄²×C₅₃	—	2	6	no	—
190	C₄²×C₅₃	—	2	3	no	—
191	C₂³×C₅₃	—	2	6	no	—
192	C₂³×C₅₃	—	2	6	no	—
193	C₂³×C₅₃	—	2	3	no	—
194	C₂³×C₅₃	—	4	3	no	—
195	C₂³×C₅₃	—	4	6	no	—
196	C₂³×C₅₃	—	8	3	no	—
197	C₂³×C₅₃	—	8	3	no	—
198	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
199	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
200	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
201	Q₈×C₅₃	—	2	6	no	—
202	Q₈×C₅₃	—	2	2	no	—
203	Q₈×C₅₃	#123	1	4	yes	—
204	D₄×C₅₃	—	2	12	no	—
205	D₄×C₅₃	—	2	6	no	—
206	D₄×C₅₃	—	2	6	no	—
207	D₄×C₅₃	—	4	6	no	—
208	D₄×C₅₃	—	4	6	no	—
209	D₄×C₅₃	—	4	6	no	—
210	D₄×C₅₃	—	8	3	no	—
211	D₄×C₅₃	—	8	6	no	—
212	D₄×C₅₃	#123	1	12	yes	—
213	C₂×C₄×C₅₃	—	2	6	no	—
214	C₂×C₄×C₅₃	—	2	6	no	—
215	C₂×C₄×C₅₃	—	2	6	no	—
216	C₂×C₄×C₅₃	—	2	2	no	—
217	C₂×C₄×C₅₃	—	2	6	no	—
218	C₂×C₄×C₅₃	—	2	6	no	—
219	C₂×C₄×C₅₃	—	2	1	no	—
220	C₂×C₄×C₅₃	—	2	3	no	—
221	C₂×C₄×C₅₃	—	2	2	no	—
222	C₂×C₄×C₅₃	—	2	1	no	—
223	C₂×C₄×C₅₃	—	4	6	no	—
224	C₂×C₄×C₅₃	—	4	2	no	—
225	C₂×C₄×C₅₃	—	4	6	no	—
226	C₂×C₄×C₅₃	—	8	6	no	—
227	C₂×C₄×C₅₃	—	8	6	no	—
228	C₂×C₄×C₅₃	—	8	3	no	—
229	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
231	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
232	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
233	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
234	C₈×C₅₃	—	2	2	no	—
235	C₈×C₅₃	—	2	6	no	—
236	#60525	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
237	C₂²×C₅₃	—	2	1	no	—
238	C₂²×C₅₃	—	2	3	no	—
239	C₂²×C₅₃	—	2	1	no	—
240	C₂²×C₅₃	—	2	3	no	—
241	C₂²×C₅₃	—	2	3	no	—
242	C₂²×C₅₃	—	2	3	no	—
243	C₂²×C₅₃	—	2	3	no	—
244	C₂²×C₅₃	—	4	12	no	—
245	C₂²×C₅₃	—	4	3	no	—
246	C₂²×C₅₃	—	4	3	no	—
247	C₂²×C₅₃	—	4	3	no	—
248	C₂²×C₅₃	—	16	6	no	—
249	C₂²×C₅₃	C₂×#123	1	6	yes	—
250	C₂²×C₅₃	#577	1	2	yes	—
251	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
252	C₄×C₅₃	—	2	1	no	—
253	C₄×C₅₃	—	2	3	no	—
254	C₄×C₅₃	—	4	6	no	—
255	C₄×C₅₃	—	4	1	no	—
256	C₄×C₅₃	—	4	1	no	—
257	C₄×C₅₃	—	4	2	no	—
258	C₄×C₅₃	—	4	3	no	—
259	C₄×C₅₃	C₂×#123	1	6	yes	—
260	C₄×C₅₃	C₂²×#41	1	2	yes	—
261	#3488	C₂×C₅₃	1	1	yes	characteristic
262	C₂×#520	C₂×C₅₃	1	1	yes	characteristic, complemented
263	#3099	C₂×C₅₃	1	6	yes	complemented
264	#3096	C₂×C₅₃	1	6	yes	complemented
265	#3094	C₂×C₅₃	1	3	yes	complemented
266	#3092	C₂×C₅₃	1	3	yes	complemented
267	#3091	C₂×C₅₃	1	3	yes	complemented
268	C₂×#481	C₂×C₅₃	1	3	yes	complemented
269	#2912	C₂×C₅₃	1	2	yes	complemented
270	#2897	C₂×C₅₃	1	2	yes	complemented
271	#2880	C₂×C₅₃	1	1	yes	characteristic, complemented
272	C₂×C₅₃	—	2	2	no	—
273	C₂×C₅₃	—	2	6	no	—
274	C₂×C₅₃	—	4	3	no	—
275	C₂×C₅₃	—	8	3	no	—
276	C₂×C₅₃	C₂²×#123	1	1	yes	characteristic
277	C₂×C₅₃	C₂×#577	1	1	yes	characteristic
278	C₂×C₅₃	#3585	1	1	yes	characteristic
279	C₂×H(5,𝔽₂)	C₂²×C₅₃	1	1	yes	characteristic
280	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
281	C₂×#131	C₂²×C₅₃	1	2	yes	—
282	#550	—	2	1	no	—
283	#548	—	2	1	no	—
284	#546	—	2	3	no	—
285	D₄²	—	2	3	no	—
286	#537	C₂²×C₅₃	1	3	yes	—
287	#535	C₂²×C₅₃	1	3	yes	—
288	#525	C₂²×C₅₃	1	3	yes	—
289	C₂×#121	C₂²×C₅₃	1	3	yes	—
290	#520	—	2	3	no	—
291	#520	—	2	1	no	—
292	#520	C₂²×C₅₃	1	2	yes	—
293	#520	C₂²×C₅₃	1	1	yes	characteristic
294	#520	C₂²×C₅₃	1	6	yes	complemented
295	C₂×C₄×Q₈	—	2	1	no	—
296	C₂×C₄×D₄	—	2	3	no	—
297	C₂×#117	C₂²×C₅₃	1	3	yes	complemented
298	C₂²×C₄⋊C₄	C₂²×C₅₃	1	3	yes	complemented
299	#485	C₂²×C₅₃	1	6	yes	complemented
300	#484	C₂²×C₅₃	1	3	yes	complemented
301	#483	C₂²×C₅₃	1	1	yes	characteristic, complemented
302	#483	C₂²×C₅₃	1	6	yes	complemented
303	#482	C₂²×C₅₃	1	6	yes	complemented
304	#482	C₂²×C₅₃	1	12	yes	complemented
305	#481	C₂²×C₅₃	1	3	yes	complemented
306	#481	C₂²×C₅₃	1	6	yes	complemented
307	#481	C₂²×C₅₃	1	12	yes	complemented
308	#480	C₂²×C₅₃	1	3	yes	complemented
309	#480	C₂²×C₅₃	1	6	yes	complemented
310	#428	C₂²×C₅₃	1	6	yes	complemented
311	#428	C₂²×C₅₃	1	6	yes	complemented
312	#427	C₂²×C₅₃	1	6	yes	complemented
313	C₂×#106	C₂²×C₅₃	1	6	yes	complemented
314	#418	C₂²×C₅₃	1	6	yes	complemented
315	#417	C₂²×C₅₃	1	6	yes	complemented
316	#416	C₂²×C₅₃	1	6	yes	complemented
317	C₂×#102	C₂²×C₅₃	1	6	yes	complemented
318	#408	C₂²×C₅₃	1	6	yes	complemented
319	#407	C₂²×C₅₃	1	3	yes	complemented
320	C₂×C₂²⋊C₈	C₂²×C₅₃	1	3	yes	complemented
321	C₅₃	#60525	1	1	yes	characteristic, direct product factor, Sylow
322	H(5,𝔽₂)	—	2	8	no	—
323	C₂×#41	—	2	8	no	—
324	C₂×#41	—	2	1	no	—
325	C₂×#41	D₄×C₅₃	1	3	yes	—
326	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
327	C₂×#41	C₂³×C₅₃	1	3	yes	—
328	C₂×#41	C₂³×C₅₃	1	3	yes	—
329	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
330	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
331	C₂²×D₄	—	2	3	no	—
332	C₂²×D₄	—	2	3	no	—
333	C₂²×D₄	D₄×C₅₃	1	3	yes	—
334	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
335	C₂³×C₄	—	2	3	no	—
336	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
337	#131	—	2	8	no	—
338	C₂×M₁₆	C₂³×C₅₃	1	6	yes	—
339	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
340	#127	—	2	3	no	—
341	#124	—	2	3	no	—
342	#123	—	2	3	no	—
343	C₂²⋊Q₈	—	2	3	no	—
344	C₂²⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
345	C₂²⋊Q₈	C₂³×C₅₃	1	3	yes	—
346	#121	—	2	6	no	—
347	#121	—	2	3	no	—
348	#121	—	2	3	no	—
349	#121	—	2	3	no	—
350	#121	C₂³×C₅₃	1	3	yes	—
351	#121	C₂³×C₅₃	1	6	yes	—
352	#121	C₂³×C₅₃	1	3	yes	—
353	#120	—	2	3	no	—
354	#120	—	2	6	no	—
355	C₄×Q₈	—	2	2	no	—
356	C₄×Q₈	—	2	6	no	—
357	C₄×Q₈	—	2	1	no	—
358	C₄×D₄	—	2	6	no	—
359	C₄×D₄	—	2	12	no	—
360	C₄×D₄	—	2	6	no	—
361	C₄×D₄	—	2	3	no	—
362	C₄×D₄	—	2	3	no	—
363	C₄×D₄	—	2	3	no	—
364	#117	—	2	3	no	—
365	#117	—	2	3	no	—
366	#117	C₂³×C₅₃	1	6	yes	—
367	#117	C₂³×C₅₃	1	6	yes	—
368	#117	C₂³×C₅₃	1	3	yes	—
369	C₂×C₄⋊C₄	—	2	3	no	—
370	C₂×C₄⋊C₄	—	2	3	no	—
371	C₂×C₄⋊C₄	—	2	3	no	—
372	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
373	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
374	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
375	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
377	C₂×C₂²⋊C₄	—	2	3	no	—
378	C₂×C₂²⋊C₄	—	2	3	no	—
379	C₂×C₄²	—	2	3	no	—
380	#107	C₂³×C₅₃	1	6	yes	—
381	#107	C₂³×C₅₃	1	6	yes	—
382	#106	C₂³×C₅₃	1	6	yes	—
383	#106	C₂³×C₅₃	1	12	yes	—
384	#106	C₂³×C₅₃	1	2	yes	—
385	#103	C₂³×C₅₃	1	2	yes	—
386	#103	C₂³×C₅₃	1	6	yes	—
387	#102	C₂³×C₅₃	1	6	yes	—
388	#102	C₂³×C₅₃	1	12	yes	—
389	#102	C₂³×C₅₃	1	6	yes	—
390	C₂²⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
391	C₂²⋊C₈	C₂³×C₅₃	1	3	yes	—
392	C₂²⋊C₈	C₂³×C₅₃	1	6	yes	—
393	C₂²⋊C₈	C₂³×C₅₃	1	6	yes	—
394	C₂⁴	—	2	3	no	—
395	C₂⁴	C₂×D₄×C₅₃	1	3	yes	—
396	#41	—	2	4	no	—
397	#41	—	2	12	no	—
398	#41	—	2	12	no	—
399	#41	—	2	12	no	—
400	#41	—	4	4	no	—
401	#41	#41×C₅₃	1	8	yes	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	C₂×D₄×C₅₃	1	3	yes	—
404	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
405	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
406	C₂×D₄	—	2	12	no	—
407	C₂×D₄	—	2	3	no	—
408	C₂×D₄	—	2	3	no	—
409	C₂×D₄	—	2	3	no	—
410	C₂×D₄	—	2	6	no	—
411	C₂×D₄	—	2	3	no	—
412	C₂×D₄	—	2	12	no	—
413	C₂×D₄	—	4	12	no	—
414	C₂×D₄	—	8	6	no	—
415	C₂×D₄	C₂×D₄×C₅₃	1	6	yes	—
416	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
417	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
418	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
419	C₂²×C₄	—	2	12	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	3	no	—
422	C₂²×C₄	—	2	3	no	—
423	C₂²×C₄	—	2	6	no	—
424	C₂²×C₄	—	2	6	no	—
425	C₂²×C₄	—	2	3	no	—
426	C₂²×C₄	—	8	6	no	—
427	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
428	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
429	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
430	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
431	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
432	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
433	M₁₆	—	2	12	no	—
434	C₂×C₈	—	2	12	no	—
435	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
436	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
437	C₄⋊C₄	—	2	3	no	—
438	C₄⋊C₄	—	2	6	no	—
439	C₄⋊C₄	—	2	6	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	3	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	3	no	—
444	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
445	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
446	C₄⋊C₄	C₂⁴×C₅₃	1	3	yes	—
447	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
448	C₂²⋊C₄	—	2	6	no	—
449	C₂²⋊C₄	—	2	6	no	—
450	C₂²⋊C₄	—	2	3	no	—
451	C₂²⋊C₄	—	2	1	no	—
452	C₂²⋊C₄	—	2	3	no	—
453	C₂²⋊C₄	—	2	3	no	—
454	C₂²⋊C₄	—	2	6	no	—
455	C₂²⋊C₄	—	2	3	no	—
456	C₄²	—	2	6	no	—
457	C₄²	—	2	6	no	—
458	C₄²	—	2	3	no	—
459	C₂³	—	2	6	no	—
460	C₂³	—	2	6	no	—
461	C₂³	—	2	3	no	—
462	C₂³	—	4	3	no	—
463	C₂³	—	4	6	no	—
464	C₂³	—	8	3	no	—
465	C₂³	—	8	3	no	—
466	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
467	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
468	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
469	Q₈	—	2	6	no	—
470	Q₈	—	2	2	no	—
471	Q₈	#123×C₅₃	1	4	yes	—
472	D₄	—	2	12	no	—
473	D₄	—	2	6	no	—
474	D₄	—	2	6	no	—
475	D₄	—	4	6	no	—
476	D₄	—	4	6	no	—
477	D₄	—	4	6	no	—
478	D₄	—	8	3	no	—
479	D₄	—	8	6	no	—
480	D₄	#123×C₅₃	1	12	yes	—
481	C₂×C₄	—	2	6	no	—
482	C₂×C₄	—	2	6	no	—
483	C₂×C₄	—	2	6	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	6	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	6	no	—
489	C₂×C₄	—	2	3	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	4	6	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	6	no	—
494	C₂×C₄	—	8	6	no	—
495	C₂×C₄	—	8	6	no	—
496	C₂×C₄	—	8	3	no	—
497	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
498	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
499	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
500	C₂×C₄	C₂×#41×C₅₃	1	12	yes	—
501	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
502	C₈	—	2	2	no	—
503	C₈	—	2	6	no	—
504	C₂²	—	2	3	no	—
505	C₂²	—	2	1	no	—
506	C₂²	—	2	3	no	—
507	C₂²	—	2	1	no	—
508	C₂²	—	2	3	no	—
509	C₂²	—	2	3	no	—
510	C₂²	—	2	3	no	—
511	C₂²	—	4	12	no	—
512	C₂²	—	4	3	no	—
513	C₂²	—	4	3	no	—
514	C₂²	—	4	3	no	—
515	C₂²	—	16	6	no	—
516	C₂²	C₂×#123×C₅₃	1	6	yes	—
517	C₂²	C₅₃×#577	1	2	yes	—
518	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
519	C₄	—	2	1	no	—
520	C₄	—	2	3	no	—
521	C₄	—	4	2	no	—
522	C₄	—	4	3	no	—
523	C₄	—	4	6	no	—
524	C₄	—	4	1	no	—
525	C₄	—	4	1	no	—
526	C₄	C₂×#123×C₅₃	1	6	yes	—
527	C₄	C₂²×#41×C₅₃	1	2	yes	—
528	C₂	—	2	2	no	—
529	C₂	—	2	6	no	—
530	C₂	—	4	3	no	—
531	C₂	—	8	3	no	—
532	C₂	C₂²×#123×C₅₃	1	1	yes	characteristic
533	C₂	C₂×C₅₃×#577	1	1	yes	characteristic
534	C₂	C₅₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3488	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#520	C₂	semidirect
3. C₅₃×#3099	C₂	semidirect
4. C₅₃×#3096	C₂	semidirect
5. C₅₃×#3094	C₂	semidirect
6. C₅₃×#3092	C₂	semidirect
7. C₅₃×#3091	C₂	semidirect
8. C₂×C₅₃×#481	C₂	semidirect
9. C₅₃×#2912	C₂	semidirect
10. C₅₃×#2897	C₂	semidirect
11. C₅₃×#2880	C₂	semidirect
27. C₅₃×#520	C₂²	semidirect
30. C₂×#117×C₅₃	C₂²	semidirect
31. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
32. C₅₃×#485	C₂²	semidirect
33. C₅₃×#484	C₂²	semidirect
34. C₅₃×#483	C₂²	semidirect
35. C₅₃×#483	C₂²	semidirect
36. C₅₃×#482	C₂²	semidirect
37. C₅₃×#482	C₂²	semidirect
38. C₅₃×#481	C₂²	semidirect
39. C₅₃×#481	C₂²	semidirect
40. C₅₃×#481	C₂²	semidirect
41. C₅₃×#480	C₂²	semidirect
42. C₅₃×#480	C₂²	semidirect
43. C₅₃×#428	C₂²	semidirect
44. C₅₃×#428	C₂²	semidirect
45. C₅₃×#427	C₂²	semidirect
46. C₂×#106×C₅₃	C₂²	semidirect
47. C₅₃×#418	C₂²	semidirect
48. C₅₃×#417	C₂²	semidirect
49. C₅₃×#416	C₂²	semidirect
50. C₂×#102×C₅₃	C₂²	semidirect
51. C₅₃×#408	C₂²	semidirect
52. C₅₃×#407	C₂²	semidirect
53. C₂×C₂²⋊C₈×C₅₃	C₂²	semidirect
236. #60525	C₅₃	direct (N × H)
262. C₂×#520	C₂×C₅₃	semidirect
263. #3099	C₂×C₅₃	semidirect
264. #3096	C₂×C₅₃	semidirect
265. #3094	C₂×C₅₃	semidirect
266. #3092	C₂×C₅₃	semidirect
267. #3091	C₂×C₅₃	semidirect
268. C₂×#481	C₂×C₅₃	semidirect
269. #2912	C₂×C₅₃	semidirect
270. #2897	C₂×C₅₃	semidirect
271. #2880	C₂×C₅₃	semidirect
294. #520	C₂²×C₅₃	semidirect
297. C₂×#117	C₂²×C₅₃	semidirect
298. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
299. #485	C₂²×C₅₃	semidirect
300. #484	C₂²×C₅₃	semidirect
301. #483	C₂²×C₅₃	semidirect
302. #483	C₂²×C₅₃	semidirect
303. #482	C₂²×C₅₃	semidirect
304. #482	C₂²×C₅₃	semidirect
305. #481	C₂²×C₅₃	semidirect
306. #481	C₂²×C₅₃	semidirect
307. #481	C₂²×C₅₃	semidirect
308. #480	C₂²×C₅₃	semidirect
309. #480	C₂²×C₅₃	semidirect
310. #428	C₂²×C₅₃	semidirect
311. #428	C₂²×C₅₃	semidirect
312. #427	C₂²×C₅₃	semidirect
313. C₂×#106	C₂²×C₅₃	semidirect
314. #418	C₂²×C₅₃	semidirect
315. #417	C₂²×C₅₃	semidirect
316. #416	C₂²×C₅₃	semidirect
317. C₂×#102	C₂²×C₅₃	semidirect
318. #408	C₂²×C₅₃	semidirect
319. #407	C₂²×C₅₃	semidirect
320. C₂×C₂²⋊C₈	C₂²×C₅₃	semidirect
321. C₅₃	#60525	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×#3488	2
C₂×C₅₃×#520	2
C₅₃×#3099	2
C₅₃×#3096	2
C₅₃×#3094	2
C₅₃×#3092	2
C₅₃×#3091	2
C₂×C₅₃×#481	2
C₅₃×#2912	2
C₅₃×#2897	2
C₅₃×#2880	2
#60525	53

Subgroups — C53×#60525 #27268037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60525 #27268037