Subgroups — C₅₃×#62096 #27269608

Order 13568.

2030 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3503	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
3	Q₈×SD₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3362	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3335	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3295	C₂	1	6	yes	maximal, complemented
7	C₅₃×#3246	C₂	1	3	yes	maximal, complemented
8	C₅₃×#3227	C₂	1	3	yes	maximal, complemented
9	C₅₃×#3101	C₂	1	3	yes	maximal, complemented
10	C₂×C₅₃×#476	C₂	1	3	yes	maximal, complemented
11	C₅₃×#2988	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₅₃	—	2	1	no	—
13	Q₈²×C₅₃	C₂²	1	2	yes	complemented
14	C₅₃×#557	—	2	1	no	—
15	C₅₃×#554	C₂²	1	6	yes	complemented
16	C₅₃×#550	C₂²	1	1	yes	characteristic
17	D₄×Q₈×C₅₃	—	2	3	no	—
18	D₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#548	—	2	1	no	—
20	C₅₃×#547	C₂²	1	3	yes	—
21	C₅₃×#546	—	2	3	no	—
22	D₄²×C₅₃	C₂²	1	3	yes	—
23	C₅₃×#544	—	2	3	no	—
24	C₅₃×#533	C₂²	1	3	yes	complemented
25	C₅₃×#532	C₂²	1	3	yes	—
26	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	complemented
27	C₅₃×#525	—	2	3	no	—
28	C₅₃×#520	—	2	1	no	—
29	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
31	C₅₃×#498	C₂²	1	6	yes	complemented
32	C₅₃×#486	C₂²	1	6	yes	complemented
33	C₅₃×#483	C₂²	1	6	yes	complemented
34	C₅₃×#481	C₂²	1	12	yes	complemented
35	C₅₃×#476	C₂²	1	6	yes	—
36	C₅₃×#476	C₂²	1	3	yes	—
37	C₅₃×#476	C₂²	1	3	yes	complemented
38	C₅₃×#476	C₂²	1	6	yes	complemented
39	C₅₃×#476	C₂²	1	6	yes	complemented
40	C₅₃×#475	C₂²	1	6	yes	complemented
41	C₅₃×#465	C₂²	1	12	yes	complemented
42	C₅₃×#461	C₂²	1	1	yes	characteristic
43	C₅₃×#461	C₂²	1	3	yes	complemented
44	C₅₃×#460	C₂²	1	6	yes	complemented
45	C₅₃×#450	C₂²	1	6	yes	—
46	C₈×Q₈×C₅₃	C₂²	1	2	yes	complemented
47	C₄×SD₁₆×C₅₃	C₂²	1	6	yes	—
48	C₄×SD₁₆×C₅₃	C₂²	1	6	yes	complemented
49	C₈×D₄×C₅₃	C₂²	1	6	yes	complemented
50	C₅₃×#431	C₂²	1	3	yes	complemented
51	C₂×#106×C₅₃	C₂²	1	6	yes	complemented
52	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	complemented
53	C₂×#102×C₅₃	C₂²	1	6	yes	—
54	C₂×#41×C₅₃	—	2	1	no	—
55	C₂×#41×C₅₃	—	2	3	no	—
56	C₂×#41×C₅₃	—	2	3	no	—
57	C₂×#41×C₅₃	—	4	8	no	—
58	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
59	C₂²×Q₈×C₅₃	—	2	1	no	—
60	C₂²×Q₈×C₅₃	—	2	3	no	—
61	C₂²×D₄×C₅₃	—	2	3	no	—
62	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
63	C₂³×C₄×C₅₃	—	2	3	no	—
64	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
65	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
66	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
67	C₄⋊Q₈×C₅₃	—	2	6	no	—
68	C₄⋊Q₈×C₅₃	—	2	3	no	—
69	C₄⋊Q₈×C₅₃	—	2	3	no	—
70	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
71	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
72	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
73	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
74	#127×C₅₃	C₂³	1	3	yes	—
75	#125×C₅₃	—	2	6	no	—
76	#125×C₅₃	—	2	3	no	—
77	#124×C₅₃	—	2	3	no	—
78	#123×C₅₃	—	2	6	no	—
79	C₂²⋊Q₈×C₅₃	—	2	3	no	—
80	C₂²⋊Q₈×C₅₃	—	2	3	no	—
81	C₂²⋊Q₈×C₅₃	—	2	12	no	—
82	C₂²⋊Q₈×C₅₃	—	2	3	no	—
83	C₂²⋊Q₈×C₅₃	C₂³	1	6	yes	—
84	#121×C₅₃	—	2	3	no	—
85	#121×C₅₃	—	2	3	no	—
86	#121×C₅₃	—	2	3	no	—
87	#121×C₅₃	C₂³	1	6	yes	—
88	#120×C₅₃	—	2	6	no	—
89	C₄×Q₈×C₅₃	—	2	3	no	—
90	C₄×Q₈×C₅₃	—	2	2	no	—
91	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
92	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
93	C₄×Q₈×C₅₃	C₂³	1	6	yes	—
94	C₄×D₄×C₅₃	—	2	3	no	—
95	C₄×D₄×C₅₃	—	2	3	no	—
96	C₄×D₄×C₅₃	—	2	3	no	—
97	C₄×D₄×C₅₃	—	2	6	no	—
98	C₄×D₄×C₅₃	D₄	1	3	yes	—
99	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₄×D₄×C₅₃	C₂³	1	3	yes	—
101	C₄×D₄×C₅₃	C₂³	1	3	yes	—
102	C₄×D₄×C₅₃	C₂³	1	3	yes	—
103	C₄×D₄×C₅₃	C₂³	1	3	yes	—
104	#117×C₅₃	—	2	3	no	—
105	#117×C₅₃	D₄	1	3	yes	—
106	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
107	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
109	C₂×C₂²⋊C₄×C₅₃	—	2	3	no	—
110	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
111	#106×C₅₃	C₂³	1	6	yes	—
112	#106×C₅₃	C₂³	1	12	yes	—
113	#106×C₅₃	C₂³	1	6	yes	—
114	C₄⋊C₈×C₅₃	C₂³	1	3	yes	—
115	C₄⋊C₈×C₅₃	C₂³	1	6	yes	—
116	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
117	#103×C₅₃	C₂³	1	6	yes	—
118	#103×C₅₃	C₂³	1	2	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	C₄×C₈×C₅₃	C₂³	1	6	yes	—
124	C₂⁴×C₅₃	—	2	3	no	—
125	#41×C₅₃	—	4	4	no	—
126	#41×C₅₃	—	4	12	no	—
127	#41×C₅₃	—	4	12	no	—
128	#41×C₅₃	SD₁₆	1	8	yes	complemented
129	C₂×Q₈×C₅₃	—	2	3	no	—
130	C₂×Q₈×C₅₃	—	2	3	no	—
131	C₂×Q₈×C₅₃	—	2	6	no	—
132	C₂×Q₈×C₅₃	—	4	4	no	—
133	C₂×Q₈×C₅₃	—	8	6	no	—
134	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
136	C₂×D₄×C₅₃	—	2	6	no	—
137	C₂×D₄×C₅₃	—	2	6	no	—
138	C₂×D₄×C₅₃	—	2	3	no	—
139	C₂×D₄×C₅₃	—	2	3	no	—
140	C₂×D₄×C₅₃	—	2	3	no	—
141	C₂×D₄×C₅₃	—	4	3	no	—
142	C₂×D₄×C₅₃	—	4	12	no	—
143	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
144	C₂×D₄×C₅₃	C₂⁴	1	3	yes	—
145	C₂×D₄×C₅₃	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	—	2	3	no	—
147	C₂²×C₄×C₅₃	—	2	6	no	—
148	C₂²×C₄×C₅₃	—	2	1	no	—
149	C₂²×C₄×C₅₃	—	2	3	no	—
150	C₂²×C₄×C₅₃	—	2	3	no	—
151	C₂²×C₄×C₅₃	—	2	3	no	—
152	C₂²×C₄×C₅₃	—	4	12	no	—
153	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
156	SD₁₆×C₅₃	—	2	4	no	—
157	SD₁₆×C₅₃	—	4	6	no	—
158	C₂×C₈×C₅₃	—	4	6	no	—
159	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
160	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
161	C₄⋊C₄×C₅₃	—	2	3	no	—
162	C₄⋊C₄×C₅₃	—	2	6	no	—
163	C₄⋊C₄×C₅₃	—	2	3	no	—
164	C₄⋊C₄×C₅₃	—	2	6	no	—
165	C₄⋊C₄×C₅₃	—	2	1	no	—
166	C₄⋊C₄×C₅₃	—	2	3	no	—
167	C₄⋊C₄×C₅₃	—	2	1	no	—
168	C₄⋊C₄×C₅₃	—	2	6	no	—
169	C₄⋊C₄×C₅₃	—	2	6	no	—
170	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
171	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
172	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
173	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
174	C₄⋊C₄×C₅₃	C₂⁴	1	3	yes	—
175	C₂²⋊C₄×C₅₃	—	2	6	no	—
176	C₂²⋊C₄×C₅₃	—	2	1	no	—
177	C₂²⋊C₄×C₅₃	—	2	3	no	—
178	C₂²⋊C₄×C₅₃	—	2	3	no	—
179	C₂²⋊C₄×C₅₃	—	2	3	no	—
180	C₂²⋊C₄×C₅₃	—	2	6	no	—
181	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
182	C₄²×C₅₃	—	2	6	no	—
183	C₄²×C₅₃	C₂×D₄	1	3	yes	—
184	C₄²×C₅₃	C₂×D₄	1	3	yes	—
185	C₄²×C₅₃	C₂⁴	1	3	yes	—
186	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
187	C₂³×C₅₃	—	2	1	no	—
188	C₂³×C₅₃	—	2	3	no	—
189	C₂³×C₅₃	—	4	6	no	—
190	C₂³×C₅₃	—	4	6	no	—
191	C₂³×C₅₃	—	8	3	no	—
192	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
193	Q₈×C₅₃	—	2	2	no	—
194	Q₈×C₅₃	—	2	2	no	—
195	Q₈×C₅₃	—	4	6	no	—
196	Q₈×C₅₃	—	4	6	no	—
197	Q₈×C₅₃	—	8	6	no	—
198	Q₈×C₅₃	C₂×SD₁₆	1	4	yes	—
199	D₄×C₅₃	—	2	3	no	—
200	D₄×C₅₃	—	2	1	no	—
201	D₄×C₅₃	—	2	3	no	—
202	D₄×C₅₃	—	4	6	no	—
203	D₄×C₅₃	—	4	12	no	—
204	D₄×C₅₃	—	4	6	no	—
205	D₄×C₅₃	—	8	3	no	—
206	D₄×C₅₃	C₂×SD₁₆	1	12	yes	—
207	D₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
208	C₂×C₄×C₅₃	—	2	3	no	—
209	C₂×C₄×C₅₃	—	2	3	no	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	—	2	3	no	—
212	C₂×C₄×C₅₃	—	2	6	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	4	3	no	—
215	C₂×C₄×C₅₃	—	4	6	no	—
216	C₂×C₄×C₅₃	—	4	2	no	—
217	C₂×C₄×C₅₃	—	4	6	no	—
218	C₂×C₄×C₅₃	—	4	6	no	—
219	C₂×C₄×C₅₃	—	4	6	no	—
220	C₂×C₄×C₅₃	—	8	6	no	—
221	C₂×C₄×C₅₃	C₂×SD₁₆	1	12	yes	—
222	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
224	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
225	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
227	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
228	C₈×C₅₃	—	2	2	no	—
229	C₈×C₅₃	—	2	6	no	—
230	#62096	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂²×C₅₃	—	2	3	no	—
232	C₂²×C₅₃	—	2	3	no	—
233	C₂²×C₅₃	—	2	2	no	—
234	C₂²×C₅₃	—	4	1	no	—
235	C₂²×C₅₃	—	4	1	no	—
236	C₂²×C₅₃	—	4	3	no	—
237	C₂²×C₅₃	—	4	3	no	—
238	C₂²×C₅₃	—	4	3	no	—
239	C₂²×C₅₃	—	8	12	no	—
240	C₂²×C₅₃	C₂²×SD₁₆	1	6	yes	—
241	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
242	C₄×C₅₃	—	2	1	no	—
243	C₄×C₅₃	—	2	3	no	—
244	C₄×C₅₃	—	2	3	no	—
245	C₄×C₅₃	—	4	6	no	—
246	C₄×C₅₃	—	4	2	no	—
247	C₄×C₅₃	—	4	1	no	—
248	C₄×C₅₃	—	4	3	no	—
249	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
250	C₄×C₅₃	C₂²×SD₁₆	1	6	yes	—
251	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
252	C₄×C₅₃	#585	1	1	yes	characteristic
253	#3503	C₂×C₅₃	1	1	yes	characteristic, complemented
254	D₄×#41	C₂×C₅₃	1	1	yes	characteristic
255	Q₈×SD₁₆	C₂×C₅₃	1	2	yes	complemented
256	#3362	C₂×C₅₃	1	2	yes	complemented
257	#3335	C₂×C₅₃	1	6	yes	complemented
258	#3295	C₂×C₅₃	1	6	yes	complemented
259	#3246	C₂×C₅₃	1	3	yes	complemented
260	#3227	C₂×C₅₃	1	3	yes	complemented
261	#3101	C₂×C₅₃	1	3	yes	complemented
262	C₂×#476	C₂×C₅₃	1	3	yes	complemented
263	#2988	C₂×C₅₃	1	1	yes	characteristic, complemented
264	C₂×C₅₃	—	2	6	no	—
265	C₂×C₅₃	—	4	2	no	—
266	C₂×C₅₃	—	8	3	no	—
267	C₂×C₅₃	#3489	1	1	yes	characteristic
268	C₂×C₅₃	C₂³×SD₁₆	1	1	yes	characteristic
269	C₂×C₅₃	#3585	1	1	yes	characteristic
270	C₂²×#41	—	2	1	no	—
271	Q₈²	C₂²×C₅₃	1	2	yes	complemented
272	#557	—	2	1	no	—
273	#554	C₂²×C₅₃	1	6	yes	complemented
274	#550	C₂²×C₅₃	1	1	yes	characteristic
275	D₄×Q₈	—	2	3	no	—
276	D₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
277	#548	—	2	1	no	—
278	#547	C₂²×C₅₃	1	3	yes	—
279	#546	—	2	3	no	—
280	D₄²	C₂²×C₅₃	1	3	yes	—
281	#544	—	2	3	no	—
282	#533	C₂²×C₅₃	1	3	yes	complemented
283	#532	C₂²×C₅₃	1	3	yes	—
284	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	complemented
285	#525	—	2	3	no	—
286	#520	—	2	1	no	—
287	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
288	C₂×C₄×D₄	C₂²×C₅₃	1	3	yes	—
289	#498	C₂²×C₅₃	1	6	yes	complemented
290	#486	C₂²×C₅₃	1	6	yes	complemented
291	#483	C₂²×C₅₃	1	6	yes	complemented
292	#481	C₂²×C₅₃	1	12	yes	complemented
293	#476	C₂²×C₅₃	1	3	yes	—
294	#476	C₂²×C₅₃	1	6	yes	—
295	#476	C₂²×C₅₃	1	6	yes	complemented
296	#476	C₂²×C₅₃	1	3	yes	complemented
297	#476	C₂²×C₅₃	1	6	yes	complemented
298	#475	C₂²×C₅₃	1	6	yes	complemented
299	#465	C₂²×C₅₃	1	12	yes	complemented
300	#461	C₂²×C₅₃	1	1	yes	characteristic
301	#461	C₂²×C₅₃	1	3	yes	complemented
302	#460	C₂²×C₅₃	1	6	yes	complemented
303	#450	C₂²×C₅₃	1	6	yes	—
304	C₈×Q₈	C₂²×C₅₃	1	2	yes	complemented
305	C₄×SD₁₆	C₂²×C₅₃	1	6	yes	—
306	C₄×SD₁₆	C₂²×C₅₃	1	6	yes	complemented
307	C₈×D₄	C₂²×C₅₃	1	6	yes	complemented
308	#431	C₂²×C₅₃	1	3	yes	complemented
309	C₂×#106	C₂²×C₅₃	1	6	yes	complemented
310	C₂×C₄⋊C₈	C₂²×C₅₃	1	3	yes	complemented
311	C₂×#102	C₂²×C₅₃	1	6	yes	—
312	C₅₃	#62096	1	1	yes	characteristic, direct product factor, Sylow
313	C₂×#41	—	2	1	no	—
314	C₂×#41	—	2	3	no	—
315	C₂×#41	—	2	3	no	—
316	C₂×#41	—	4	8	no	—
317	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
318	C₂²×Q₈	—	2	1	no	—
319	C₂²×Q₈	—	2	3	no	—
320	C₂²×D₄	—	2	3	no	—
321	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
322	C₂³×C₄	—	2	3	no	—
323	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
324	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
325	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
326	C₄⋊Q₈	—	2	6	no	—
327	C₄⋊Q₈	—	2	3	no	—
328	C₄⋊Q₈	—	2	3	no	—
329	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
330	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
331	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
332	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
333	#127	C₂³×C₅₃	1	3	yes	—
334	#125	—	2	6	no	—
335	#125	—	2	3	no	—
336	#124	—	2	3	no	—
337	#123	—	2	6	no	—
338	C₂²⋊Q₈	—	2	3	no	—
339	C₂²⋊Q₈	—	2	3	no	—
340	C₂²⋊Q₈	—	2	3	no	—
341	C₂²⋊Q₈	—	2	12	no	—
342	C₂²⋊Q₈	C₂³×C₅₃	1	6	yes	—
343	#121	—	2	3	no	—
344	#121	—	2	3	no	—
345	#121	—	2	3	no	—
346	#121	C₂³×C₅₃	1	6	yes	—
347	#120	—	2	6	no	—
348	C₄×Q₈	—	2	2	no	—
349	C₄×Q₈	—	2	3	no	—
350	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
351	C₄×Q₈	C₂³×C₅₃	1	6	yes	—
352	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
353	C₄×D₄	—	2	6	no	—
354	C₄×D₄	—	2	3	no	—
355	C₄×D₄	—	2	3	no	—
356	C₄×D₄	—	2	3	no	—
357	C₄×D₄	D₄×C₅₃	1	3	yes	—
358	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
359	C₄×D₄	C₂³×C₅₃	1	3	yes	—
360	C₄×D₄	C₂³×C₅₃	1	3	yes	—
361	C₄×D₄	C₂³×C₅₃	1	3	yes	—
362	C₄×D₄	C₂³×C₅₃	1	3	yes	—
363	#117	—	2	3	no	—
364	#117	D₄×C₅₃	1	3	yes	—
365	C₂×C₄⋊C₄	—	2	3	no	—
366	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
367	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
368	C₂×C₂²⋊C₄	—	2	3	no	—
369	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
370	#106	C₂³×C₅₃	1	6	yes	—
371	#106	C₂³×C₅₃	1	12	yes	—
372	#106	C₂³×C₅₃	1	6	yes	—
373	C₄⋊C₈	C₂³×C₅₃	1	3	yes	—
374	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
375	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
376	#103	C₂³×C₅₃	1	6	yes	—
377	#103	C₂³×C₅₃	1	2	yes	—
378	#102	C₂³×C₅₃	1	6	yes	—
379	#102	C₂³×C₅₃	1	12	yes	—
380	#102	C₂³×C₅₃	1	6	yes	—
381	C₂²⋊C₈	C₂³×C₅₃	1	6	yes	—
382	C₄×C₈	C₂³×C₅₃	1	6	yes	—
383	C₂⁴	—	2	3	no	—
384	#41	—	4	4	no	—
385	#41	—	4	12	no	—
386	#41	—	4	12	no	—
387	#41	SD₁₆×C₅₃	1	8	yes	complemented
388	C₂×Q₈	—	2	3	no	—
389	C₂×Q₈	—	2	3	no	—
390	C₂×Q₈	—	2	6	no	—
391	C₂×Q₈	—	4	4	no	—
392	C₂×Q₈	—	8	6	no	—
393	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
394	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
395	C₂×D₄	—	2	6	no	—
396	C₂×D₄	—	2	6	no	—
397	C₂×D₄	—	2	3	no	—
398	C₂×D₄	—	2	3	no	—
399	C₂×D₄	—	2	3	no	—
400	C₂×D₄	—	4	3	no	—
401	C₂×D₄	—	4	12	no	—
402	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
403	C₂×D₄	C₂⁴×C₅₃	1	3	yes	—
404	C₂×D₄	C₂⁴×C₅₃	1	1	yes	characteristic
405	C₂²×C₄	—	2	3	no	—
406	C₂²×C₄	—	2	6	no	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	3	no	—
409	C₂²×C₄	—	2	3	no	—
410	C₂²×C₄	—	2	3	no	—
411	C₂²×C₄	—	4	12	no	—
412	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
413	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
414	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
415	SD₁₆	—	2	4	no	—
416	SD₁₆	—	4	6	no	—
417	C₂×C₈	—	4	6	no	—
418	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
419	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
420	C₄⋊C₄	—	2	3	no	—
421	C₄⋊C₄	—	2	6	no	—
422	C₄⋊C₄	—	2	3	no	—
423	C₄⋊C₄	—	2	6	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	3	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	6	no	—
428	C₄⋊C₄	—	2	6	no	—
429	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
430	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
431	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
432	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
433	C₄⋊C₄	C₂⁴×C₅₃	1	3	yes	—
434	C₂²⋊C₄	—	2	6	no	—
435	C₂²⋊C₄	—	2	1	no	—
436	C₂²⋊C₄	—	2	3	no	—
437	C₂²⋊C₄	—	2	3	no	—
438	C₂²⋊C₄	—	2	3	no	—
439	C₂²⋊C₄	—	2	6	no	—
440	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
441	C₄²	—	2	6	no	—
442	C₄²	C₂×D₄×C₅₃	1	3	yes	—
443	C₄²	C₂×D₄×C₅₃	1	3	yes	—
444	C₄²	C₂⁴×C₅₃	1	3	yes	—
445	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
446	C₂³	—	2	1	no	—
447	C₂³	—	2	3	no	—
448	C₂³	—	4	6	no	—
449	C₂³	—	4	6	no	—
450	C₂³	—	8	3	no	—
451	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
452	Q₈	—	2	2	no	—
453	Q₈	—	2	2	no	—
454	Q₈	—	4	6	no	—
455	Q₈	—	4	6	no	—
456	Q₈	—	8	6	no	—
457	Q₈	C₂×SD₁₆×C₅₃	1	4	yes	—
458	D₄	—	2	3	no	—
459	D₄	—	2	1	no	—
460	D₄	—	2	3	no	—
461	D₄	—	4	6	no	—
462	D₄	—	4	12	no	—
463	D₄	—	4	6	no	—
464	D₄	—	8	3	no	—
465	D₄	C₂×SD₁₆×C₅₃	1	12	yes	—
466	D₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
467	C₂×C₄	—	2	3	no	—
468	C₂×C₄	—	2	3	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	6	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	3	no	—
473	C₂×C₄	—	4	3	no	—
474	C₂×C₄	—	4	6	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	6	no	—
477	C₂×C₄	—	4	6	no	—
478	C₂×C₄	—	4	6	no	—
479	C₂×C₄	—	8	6	no	—
480	C₂×C₄	C₂×SD₁₆×C₅₃	1	12	yes	—
481	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
483	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
484	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
485	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
486	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
487	C₈	—	2	2	no	—
488	C₈	—	2	6	no	—
489	C₂²	—	2	3	no	—
490	C₂²	—	2	3	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	4	1	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	3	no	—
495	C₂²	—	4	3	no	—
496	C₂²	—	4	3	no	—
497	C₂²	—	8	12	no	—
498	C₂²	C₂²×SD₁₆×C₅₃	1	6	yes	—
499	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
500	C₄	—	2	3	no	—
501	C₄	—	2	1	no	—
502	C₄	—	2	3	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	1	no	—
505	C₄	—	4	3	no	—
506	C₄	—	4	6	no	—
507	C₄	C₂²×SD₁₆×C₅₃	1	2	yes	—
508	C₄	C₂²×SD₁₆×C₅₃	1	6	yes	—
509	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
510	C₄	C₅₃×#585	1	1	yes	characteristic
511	C₂	—	2	6	no	—
512	C₂	—	4	2	no	—
513	C₂	—	8	3	no	—
514	C₂	C₅₃×#3489	1	1	yes	characteristic
515	C₂	C₂³×SD₁₆×C₅₃	1	1	yes	characteristic
516	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	D₄×#41×C₅₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
1. C₅₃×#3503	C₂	semidirect
3. Q₈×SD₁₆×C₅₃	C₂	semidirect
4. C₅₃×#3362	C₂	semidirect
5. C₅₃×#3335	C₂	semidirect
6. C₅₃×#3295	C₂	semidirect
7. C₅₃×#3246	C₂	semidirect
8. C₅₃×#3227	C₂	semidirect
9. C₅₃×#3101	C₂	semidirect
10. C₂×C₅₃×#476	C₂	semidirect
11. C₅₃×#2988	C₂	semidirect
13. Q₈²×C₅₃	C₂²	semidirect
15. C₅₃×#554	C₂²	semidirect
24. C₅₃×#533	C₂²	semidirect
26. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
31. C₅₃×#498	C₂²	semidirect
32. C₅₃×#486	C₂²	semidirect
33. C₅₃×#483	C₂²	semidirect
34. C₅₃×#481	C₂²	semidirect
37. C₅₃×#476	C₂²	semidirect
38. C₅₃×#476	C₂²	semidirect
39. C₅₃×#476	C₂²	semidirect
40. C₅₃×#475	C₂²	semidirect
41. C₅₃×#465	C₂²	semidirect
43. C₅₃×#461	C₂²	semidirect
44. C₅₃×#460	C₂²	semidirect
46. C₈×Q₈×C₅₃	C₂²	semidirect
48. C₄×SD₁₆×C₅₃	C₂²	semidirect
49. C₈×D₄×C₅₃	C₂²	semidirect
50. C₅₃×#431	C₂²	semidirect
51. C₂×#106×C₅₃	C₂²	semidirect
52. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
128. #41×C₅₃	SD₁₆	semidirect
230. #62096	C₅₃	direct (N × H)
253. #3503	C₂×C₅₃	semidirect
255. Q₈×SD₁₆	C₂×C₅₃	semidirect
256. #3362	C₂×C₅₃	semidirect
257. #3335	C₂×C₅₃	semidirect
258. #3295	C₂×C₅₃	semidirect
259. #3246	C₂×C₅₃	semidirect
260. #3227	C₂×C₅₃	semidirect
261. #3101	C₂×C₅₃	semidirect
262. C₂×#476	C₂×C₅₃	semidirect
263. #2988	C₂×C₅₃	semidirect
271. Q₈²	C₂²×C₅₃	semidirect
273. #554	C₂²×C₅₃	semidirect
282. #533	C₂²×C₅₃	semidirect
284. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
289. #498	C₂²×C₅₃	semidirect
290. #486	C₂²×C₅₃	semidirect
291. #483	C₂²×C₅₃	semidirect
292. #481	C₂²×C₅₃	semidirect
295. #476	C₂²×C₅₃	semidirect
296. #476	C₂²×C₅₃	semidirect
297. #476	C₂²×C₅₃	semidirect
298. #475	C₂²×C₅₃	semidirect
299. #465	C₂²×C₅₃	semidirect
301. #461	C₂²×C₅₃	semidirect
302. #460	C₂²×C₅₃	semidirect
304. C₈×Q₈	C₂²×C₅₃	semidirect
306. C₄×SD₁₆	C₂²×C₅₃	semidirect
307. C₈×D₄	C₂²×C₅₃	semidirect
308. #431	C₂²×C₅₃	semidirect
309. C₂×#106	C₂²×C₅₃	semidirect
310. C₂×C₄⋊C₈	C₂²×C₅₃	semidirect
312. C₅₃	#62096	direct (N × H)
387. #41	SD₁₆×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3503	2
D₄×#41×C₅₃	2
Q₈×SD₁₆×C₅₃	2
C₅₃×#3362	2
C₅₃×#3335	2
C₅₃×#3295	2
C₅₃×#3246	2
C₅₃×#3227	2
C₅₃×#3101	2
C₂×C₅₃×#476	2
C₅₃×#2988	2
#62096	53

Subgroups — C53×#62096 #27269608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#62096 #27269608