Subgroups — C₅₃×#62099 #27269611

Order 13568.

1948 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3499	C₂	1	1	yes	maximal, characteristic
2	Q₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×SD₁₆×C₅₃	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3359	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3335	C₂	1	6	yes	maximal, complemented
6	C₅₃×#3298	C₂	1	6	yes	maximal, complemented
7	C₅₃×#3246	C₂	1	3	yes	maximal, complemented
8	C₅₃×#3229	C₂	1	3	yes	maximal, complemented
9	C₅₃×#3103	C₂	1	3	yes	maximal, complemented
10	C₂×C₅₃×#475	C₂	1	3	yes	maximal, complemented
11	C₅₃×#2988	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×2₋¹⁺⁴×C₅₃	—	2	1	no	—
13	Q₈²×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#557	—	2	1	no	—
15	C₅₃×#557	C₂²	1	1	yes	characteristic
16	C₅₃×#554	—	2	3	no	—
17	C₅₃×#554	C₂²	1	3	yes	—
18	D₄×Q₈×C₅₃	—	2	1	no	—
19	D₄×Q₈×C₅₃	C₂²	1	3	yes	—
20	D₄×Q₈×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#547	—	2	3	no	—
22	C₅₃×#547	C₂²	1	6	yes	—
23	C₅₃×#536	—	2	3	no	—
24	C₅₃×#533	—	2	3	no	—
25	C₅₃×#533	C₂²	1	3	yes	—
26	C₅₃×#532	C₂²	1	3	yes	—
27	C₂×C₄⋊Q₈×C₅₃	C₂²	1	3	yes	—
28	C₄×#41×C₅₃	—	2	1	no	—
29	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₅₃	C₂²	1	3	yes	—
31	C₅₃×#498	C₂²	1	6	yes	—
32	C₅₃×#487	C₂²	1	6	yes	—
33	C₅₃×#483	C₂²	1	12	yes	—
34	C₅₃×#481	C₂²	1	6	yes	—
35	C₅₃×#476	C₂²	1	6	yes	—
36	C₅₃×#475	C₂²	1	3	yes	—
37	C₅₃×#475	C₂²	1	6	yes	—
38	C₅₃×#475	C₂²	1	6	yes	—
39	C₅₃×#475	C₂²	1	3	yes	—
40	C₅₃×#475	C₂²	1	6	yes	—
41	C₅₃×#465	C₂²	1	12	yes	—
42	C₅₃×#461	C₂²	1	6	yes	—
43	C₅₃×#460	C₂²	1	3	yes	—
44	C₅₃×#460	C₂²	1	1	yes	characteristic
45	C₅₃×#448	C₂²	1	6	yes	—
46	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
47	C₄×SD₁₆×C₅₃	C₂²	1	6	yes	—
48	C₄×SD₁₆×C₅₃	C₂²	1	6	yes	—
49	C₈×D₄×C₅₃	C₂²	1	6	yes	—
50	C₅₃×#431	C₂²	1	3	yes	—
51	C₂×#106×C₅₃	C₂²	1	6	yes	—
52	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
53	C₂×#103×C₅₃	C₂²	1	6	yes	—
54	2₋¹⁺⁴×C₅₃	—	4	8	no	—
55	C₂×#41×C₅₃	—	2	6	no	—
56	C₂×#41×C₅₃	—	2	3	no	—
57	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₅₃	—	2	2	no	—
59	C₂²×Q₈×C₅₃	—	2	3	no	—
60	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
61	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₅₃	C₂³	1	6	yes	—
63	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
64	C₄⋊Q₈×C₅₃	—	2	3	no	—
65	C₄⋊Q₈×C₅₃	—	2	3	no	—
66	C₄⋊Q₈×C₅₃	—	2	3	no	—
67	C₄⋊Q₈×C₅₃	—	2	3	no	—
68	C₄⋊Q₈×C₅₃	C₂³	1	6	yes	—
69	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
70	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
71	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	—
72	#127×C₅₃	C₂³	1	1	yes	characteristic
73	#125×C₅₃	—	2	3	no	—
74	#125×C₅₃	—	2	3	no	—
75	#125×C₅₃	—	2	3	no	—
76	#124×C₅₃	—	2	3	no	—
77	#123×C₅₃	—	2	12	no	—
78	C₂²⋊Q₈×C₅₃	—	2	6	no	—
79	C₂²⋊Q₈×C₅₃	—	2	6	no	—
80	C₂²⋊Q₈×C₅₃	—	2	6	no	—
81	C₂²⋊Q₈×C₅₃	—	2	6	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	C₂³	1	6	yes	—
86	C₄×Q₈×C₅₃	—	2	3	no	—
87	C₄×Q₈×C₅₃	—	2	3	no	—
88	C₄×Q₈×C₅₃	—	2	3	no	—
89	C₄×Q₈×C₅₃	—	2	1	no	—
90	C₄×Q₈×C₅₃	—	2	1	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	1	yes	characteristic
94	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
95	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
96	C₄×Q₈×C₅₃	C₂³	1	3	yes	—
97	C₄×D₄×C₅₃	—	2	3	no	—
98	C₄×D₄×C₅₃	—	2	3	no	—
99	C₄×D₄×C₅₃	—	2	3	no	—
100	C₄×D₄×C₅₃	D₄	1	3	yes	—
101	C₄×D₄×C₅₃	C₂³	1	6	yes	—
102	C₄×D₄×C₅₃	C₂³	1	3	yes	—
103	#117×C₅₃	—	2	3	no	—
104	#117×C₅₃	D₄	1	3	yes	—
105	C₂×C₄⋊C₄×C₅₃	—	2	3	no	—
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×C₄²×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	12	yes	—
119	#103×C₅₃	C₂³	1	6	yes	—
120	#102×C₅₃	C₂³	1	2	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	#41×C₅₃	—	4	24	no	—
125	#41×C₅₃	—	4	12	no	—
126	#41×C₅₃	SD₁₆	1	8	yes	complemented
127	C₂×Q₈×C₅₃	—	2	3	no	—
128	C₂×Q₈×C₅₃	—	2	6	no	—
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	3	no	—
133	C₂×Q₈×C₅₃	—	4	8	no	—
134	C₂×Q₈×C₅₃	—	4	12	no	—
135	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Q₈×C₅₃	C₂⁴	1	3	yes	—
137	C₂×Q₈×C₅₃	C₂⁴	1	1	yes	characteristic
138	C₂×D₄×C₅₃	—	2	6	no	—
139	C₂×D₄×C₅₃	—	2	1	no	—
140	C₂×D₄×C₅₃	—	2	1	no	—
141	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
142	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₅₃	—	2	3	no	—
144	C₂²×C₄×C₅₃	—	2	6	no	—
145	C₂²×C₄×C₅₃	—	2	3	no	—
146	C₂²×C₄×C₅₃	—	2	6	no	—
147	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
148	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
149	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
150	SD₁₆×C₅₃	—	2	4	no	—
151	SD₁₆×C₅₃	—	4	6	no	—
152	C₂×C₈×C₅₃	—	4	6	no	—
153	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
154	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
155	C₄⋊C₄×C₅₃	—	2	1	no	—
156	C₄⋊C₄×C₅₃	—	2	3	no	—
157	C₄⋊C₄×C₅₃	—	2	3	no	—
158	C₄⋊C₄×C₅₃	—	2	3	no	—
159	C₄⋊C₄×C₅₃	—	2	3	no	—
160	C₄⋊C₄×C₅₃	—	2	6	no	—
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	3	no	—
165	C₄⋊C₄×C₅₃	—	2	3	no	—
166	C₄⋊C₄×C₅₃	—	2	6	no	—
167	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
168	C₄⋊C₄×C₅₃	C₂×D₄	1	3	yes	—
169	C₄⋊C₄×C₅₃	C₂⁴	1	1	yes	characteristic
170	C₄⋊C₄×C₅₃	C₂⁴	1	3	yes	—
171	C₄⋊C₄×C₅₃	C₂⁴	1	6	yes	—
172	C₂²⋊C₄×C₅₃	—	2	6	no	—
173	C₂²⋊C₄×C₅₃	—	2	6	no	—
174	C₂²⋊C₄×C₅₃	—	2	6	no	—
175	C₂²⋊C₄×C₅₃	C₂×D₄	1	6	yes	—
176	C₄²×C₅₃	—	2	1	no	—
177	C₄²×C₅₃	—	2	3	no	—
178	C₄²×C₅₃	—	2	3	no	—
179	C₄²×C₅₃	—	2	3	no	—
180	C₄²×C₅₃	C₂×D₄	1	3	yes	—
181	C₄²×C₅₃	C₂×D₄	1	3	yes	—
182	C₄²×C₅₃	C₂⁴	1	3	yes	—
183	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
184	C₂³×C₅₃	—	2	2	no	—
185	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
186	Q₈×C₅₃	—	2	1	no	—
187	Q₈×C₅₃	—	2	3	no	—
188	Q₈×C₅₃	—	2	3	no	—
189	Q₈×C₅₃	—	4	6	no	—
190	Q₈×C₅₃	—	4	12	no	—
191	Q₈×C₅₃	—	4	6	no	—
192	Q₈×C₅₃	—	8	3	no	—
193	Q₈×C₅₃	C₂×SD₁₆	1	4	yes	—
194	Q₈×C₅₃	2₋¹⁺⁴	1	2	yes	complemented
195	D₄×C₅₃	—	2	2	no	—
196	D₄×C₅₃	—	2	2	no	—
197	D₄×C₅₃	—	4	2	no	—
198	D₄×C₅₃	—	4	12	no	—
199	D₄×C₅₃	—	4	2	no	—
200	D₄×C₅₃	C₂×SD₁₆	1	12	yes	—
201	C₂×C₄×C₅₃	—	2	1	no	—
202	C₂×C₄×C₅₃	—	2	3	no	—
203	C₂×C₄×C₅₃	—	2	3	no	—
204	C₂×C₄×C₅₃	—	2	6	no	—
205	C₂×C₄×C₅₃	—	2	1	no	—
206	C₂×C₄×C₅₃	—	2	3	no	—
207	C₂×C₄×C₅₃	—	2	3	no	—
208	C₂×C₄×C₅₃	—	4	3	no	—
209	C₂×C₄×C₅₃	—	4	6	no	—
210	C₂×C₄×C₅₃	—	4	12	no	—
211	C₂×C₄×C₅₃	—	4	12	no	—
212	C₂×C₄×C₅₃	—	8	3	no	—
213	C₂×C₄×C₅₃	C₂×SD₁₆	1	12	yes	—
214	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
216	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
217	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
218	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
220	C₈×C₅₃	—	2	2	no	—
221	C₈×C₅₃	—	2	6	no	—
222	#62099	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂²×C₅₃	—	2	3	no	—
224	C₂²×C₅₃	—	2	3	no	—
225	C₂²×C₅₃	—	4	2	no	—
226	C₂²×C₅₃	—	4	2	no	—
227	C₂²×C₅₃	—	4	2	no	—
228	C₂²×C₅₃	C₂²×SD₁₆	1	6	yes	—
229	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	2	3	no	—
232	C₄×C₅₃	—	2	3	no	—
233	C₄×C₅₃	—	2	2	no	—
234	C₄×C₅₃	—	4	1	no	—
235	C₄×C₅₃	—	4	3	no	—
236	C₄×C₅₃	—	4	3	no	—
237	C₄×C₅₃	—	4	6	no	—
238	C₄×C₅₃	C₂²×SD₁₆	1	6	yes	—
239	C₄×C₅₃	C₂²×SD₁₆	1	2	yes	—
240	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
241	C₄×C₅₃	#585	1	1	yes	characteristic
242	#3499	C₂×C₅₃	1	1	yes	characteristic
243	Q₈×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
244	Q₈×SD₁₆	C₂×C₅₃	1	2	yes	complemented
245	#3359	C₂×C₅₃	1	2	yes	complemented
246	#3335	C₂×C₅₃	1	6	yes	complemented
247	#3298	C₂×C₅₃	1	6	yes	complemented
248	#3246	C₂×C₅₃	1	3	yes	complemented
249	#3229	C₂×C₅₃	1	3	yes	complemented
250	#3103	C₂×C₅₃	1	3	yes	complemented
251	C₂×#475	C₂×C₅₃	1	3	yes	complemented
252	#2988	C₂×C₅₃	1	1	yes	characteristic, complemented
253	C₂×C₅₃	—	2	6	no	—
254	C₂×C₅₃	—	8	2	no	—
255	C₂×C₅₃	#3489	1	1	yes	characteristic
256	C₂×C₅₃	C₂³×SD₁₆	1	1	yes	characteristic
257	C₂×C₅₃	#3586	1	1	yes	characteristic
258	C₂×2₋¹⁺⁴	—	2	1	no	—
259	Q₈²	C₂²×C₅₃	1	1	yes	characteristic
260	#557	—	2	1	no	—
261	#557	C₂²×C₅₃	1	1	yes	characteristic
262	#554	—	2	3	no	—
263	#554	C₂²×C₅₃	1	3	yes	—
264	D₄×Q₈	—	2	1	no	—
265	D₄×Q₈	C₂²×C₅₃	1	3	yes	—
266	D₄×Q₈	C₂²×C₅₃	1	2	yes	—
267	#547	—	2	3	no	—
268	#547	C₂²×C₅₃	1	6	yes	—
269	#536	—	2	3	no	—
270	#533	—	2	3	no	—
271	#533	C₂²×C₅₃	1	3	yes	—
272	#532	C₂²×C₅₃	1	3	yes	—
273	C₂×C₄⋊Q₈	C₂²×C₅₃	1	3	yes	—
274	C₄×#41	—	2	1	no	—
275	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
276	C₂×C₄×Q₈	C₂²×C₅₃	1	3	yes	—
277	#498	C₂²×C₅₃	1	6	yes	—
278	#487	C₂²×C₅₃	1	6	yes	—
279	#483	C₂²×C₅₃	1	12	yes	—
280	#481	C₂²×C₅₃	1	6	yes	—
281	#476	C₂²×C₅₃	1	6	yes	—
282	#475	C₂²×C₅₃	1	3	yes	—
283	#475	C₂²×C₅₃	1	6	yes	—
284	#475	C₂²×C₅₃	1	6	yes	—
285	#475	C₂²×C₅₃	1	6	yes	—
286	#475	C₂²×C₅₃	1	3	yes	—
287	#465	C₂²×C₅₃	1	12	yes	—
288	#461	C₂²×C₅₃	1	6	yes	—
289	#460	C₂²×C₅₃	1	1	yes	characteristic
290	#460	C₂²×C₅₃	1	3	yes	—
291	#448	C₂²×C₅₃	1	6	yes	—
292	C₈×Q₈	C₂²×C₅₃	1	2	yes	—
293	C₄×SD₁₆	C₂²×C₅₃	1	6	yes	—
294	C₄×SD₁₆	C₂²×C₅₃	1	6	yes	—
295	C₈×D₄	C₂²×C₅₃	1	6	yes	—
296	#431	C₂²×C₅₃	1	3	yes	—
297	C₂×#106	C₂²×C₅₃	1	6	yes	—
298	C₂×C₄⋊C₈	C₂²×C₅₃	1	3	yes	—
299	C₂×#103	C₂²×C₅₃	1	6	yes	—
300	C₅₃	#62099	1	1	yes	characteristic, direct product factor, Sylow
301	2₋¹⁺⁴	—	4	8	no	—
302	C₂×#41	—	2	6	no	—
303	C₂×#41	—	2	3	no	—
304	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
305	C₂²×Q₈	—	2	2	no	—
306	C₂²×Q₈	—	2	3	no	—
307	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
308	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
309	C₂×SD₁₆	C₂³×C₅₃	1	6	yes	—
310	C₂²×C₈	C₂³×C₅₃	1	6	yes	—
311	C₄⋊Q₈	—	2	3	no	—
312	C₄⋊Q₈	—	2	3	no	—
313	C₄⋊Q₈	—	2	3	no	—
314	C₄⋊Q₈	—	2	3	no	—
315	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
316	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
317	C₄⋊Q₈	C₂³×C₅₃	1	6	yes	—
318	C₄⋊Q₈	C₂³×C₅₃	1	3	yes	—
319	#127	C₂³×C₅₃	1	1	yes	characteristic
320	#125	—	2	3	no	—
321	#125	—	2	3	no	—
322	#125	—	2	3	no	—
323	#124	—	2	3	no	—
324	#123	—	2	12	no	—
325	C₂²⋊Q₈	—	2	3	no	—
326	C₂²⋊Q₈	—	2	6	no	—
327	C₂²⋊Q₈	—	2	6	no	—
328	C₂²⋊Q₈	—	2	6	no	—
329	C₂²⋊Q₈	—	2	6	no	—
330	C₂²⋊Q₈	C₂³×C₅₃	1	6	yes	—
331	#121	—	2	3	no	—
332	#121	C₂³×C₅₃	1	6	yes	—
333	C₄×Q₈	—	2	1	no	—
334	C₄×Q₈	—	2	1	no	—
335	C₄×Q₈	—	2	3	no	—
336	C₄×Q₈	—	2	3	no	—
337	C₄×Q₈	—	2	3	no	—
338	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
339	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
340	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
341	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
342	C₄×Q₈	C₂³×C₅₃	1	3	yes	—
343	C₄×Q₈	C₂³×C₅₃	1	1	yes	characteristic
344	C₄×D₄	—	2	3	no	—
345	C₄×D₄	—	2	3	no	—
346	C₄×D₄	—	2	3	no	—
347	C₄×D₄	D₄×C₅₃	1	3	yes	—
348	C₄×D₄	C₂³×C₅₃	1	6	yes	—
349	C₄×D₄	C₂³×C₅₃	1	3	yes	—
350	#117	—	2	3	no	—
351	#117	D₄×C₅₃	1	3	yes	—
352	C₂×C₄⋊C₄	—	2	3	no	—
353	C₂×C₄⋊C₄	—	2	3	no	—
354	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
355	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
356	C₂×C₄²	—	2	3	no	—
357	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
358	#106	C₂³×C₅₃	1	6	yes	—
359	#106	C₂³×C₅₃	1	12	yes	—
360	#106	C₂³×C₅₃	1	6	yes	—
361	C₄⋊C₈	C₂³×C₅₃	1	3	yes	—
362	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
363	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
364	#103	C₂³×C₅₃	1	6	yes	—
365	#103	C₂³×C₅₃	1	12	yes	—
366	#103	C₂³×C₅₃	1	6	yes	—
367	#102	C₂³×C₅₃	1	2	yes	—
368	#102	C₂³×C₅₃	1	6	yes	—
369	C₂²⋊C₈	C₂³×C₅₃	1	6	yes	—
370	C₄×C₈	C₂³×C₅₃	1	6	yes	—
371	#41	—	4	24	no	—
372	#41	—	4	12	no	—
373	#41	SD₁₆×C₅₃	1	8	yes	complemented
374	C₂×Q₈	—	2	3	no	—
375	C₂×Q₈	—	2	6	no	—
376	C₂×Q₈	—	2	3	no	—
377	C₂×Q₈	—	2	3	no	—
378	C₂×Q₈	—	2	6	no	—
379	C₂×Q₈	—	4	3	no	—
380	C₂×Q₈	—	4	8	no	—
381	C₂×Q₈	—	4	12	no	—
382	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
383	C₂×Q₈	C₂⁴×C₅₃	1	3	yes	—
384	C₂×Q₈	C₂⁴×C₅₃	1	1	yes	characteristic
385	C₂×D₄	—	2	6	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	C₂×D₄×C₅₃	1	3	yes	—
389	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
390	C₂²×C₄	—	2	3	no	—
391	C₂²×C₄	—	2	6	no	—
392	C₂²×C₄	—	2	3	no	—
393	C₂²×C₄	—	2	6	no	—
394	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
395	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
396	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
397	SD₁₆	—	2	4	no	—
398	SD₁₆	—	4	6	no	—
399	C₂×C₈	—	4	6	no	—
400	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
401	C₂×C₈	C₂⁴×C₅₃	1	6	yes	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	3	no	—
404	C₄⋊C₄	—	2	3	no	—
405	C₄⋊C₄	—	2	3	no	—
406	C₄⋊C₄	—	2	3	no	—
407	C₄⋊C₄	—	2	6	no	—
408	C₄⋊C₄	—	2	3	no	—
409	C₄⋊C₄	—	2	6	no	—
410	C₄⋊C₄	—	2	3	no	—
411	C₄⋊C₄	—	2	3	no	—
412	C₄⋊C₄	—	2	3	no	—
413	C₄⋊C₄	—	2	6	no	—
414	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
415	C₄⋊C₄	C₂×D₄×C₅₃	1	3	yes	—
416	C₄⋊C₄	C₂⁴×C₅₃	1	1	yes	characteristic
417	C₄⋊C₄	C₂⁴×C₅₃	1	3	yes	—
418	C₄⋊C₄	C₂⁴×C₅₃	1	6	yes	—
419	C₂²⋊C₄	—	2	6	no	—
420	C₂²⋊C₄	—	2	6	no	—
421	C₂²⋊C₄	—	2	6	no	—
422	C₂²⋊C₄	C₂×D₄×C₅₃	1	6	yes	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	3	no	—
425	C₄²	—	2	3	no	—
426	C₄²	—	2	3	no	—
427	C₄²	C₂×D₄×C₅₃	1	3	yes	—
428	C₄²	C₂×D₄×C₅₃	1	3	yes	—
429	C₄²	C₂⁴×C₅₃	1	3	yes	—
430	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
431	C₂³	—	2	2	no	—
432	C₂³	C₂²×D₄×C₅₃	1	3	yes	—
433	Q₈	—	2	1	no	—
434	Q₈	—	2	3	no	—
435	Q₈	—	2	3	no	—
436	Q₈	—	4	6	no	—
437	Q₈	—	4	12	no	—
438	Q₈	—	4	6	no	—
439	Q₈	—	8	3	no	—
440	Q₈	C₂×SD₁₆×C₅₃	1	4	yes	—
441	Q₈	2₋¹⁺⁴×C₅₃	1	2	yes	complemented
442	D₄	—	2	2	no	—
443	D₄	—	2	2	no	—
444	D₄	—	4	2	no	—
445	D₄	—	4	12	no	—
446	D₄	—	4	2	no	—
447	D₄	C₂×SD₁₆×C₅₃	1	12	yes	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	3	no	—
450	C₂×C₄	—	2	3	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	3	no	—
453	C₂×C₄	—	2	3	no	—
454	C₂×C₄	—	2	6	no	—
455	C₂×C₄	—	4	3	no	—
456	C₂×C₄	—	4	6	no	—
457	C₂×C₄	—	4	12	no	—
458	C₂×C₄	—	4	12	no	—
459	C₂×C₄	—	8	3	no	—
460	C₂×C₄	C₂×SD₁₆×C₅₃	1	12	yes	—
461	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
463	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
464	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
465	C₂×C₄	C₂²×D₄×C₅₃	1	3	yes	—
466	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
467	C₈	—	2	2	no	—
468	C₈	—	2	6	no	—
469	C₂²	—	2	3	no	—
470	C₂²	—	2	3	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	C₂²×SD₁₆×C₅₃	1	6	yes	—
475	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
476	C₄	—	2	3	no	—
477	C₄	—	2	2	no	—
478	C₄	—	2	1	no	—
479	C₄	—	2	3	no	—
480	C₄	—	4	3	no	—
481	C₄	—	4	6	no	—
482	C₄	—	4	1	no	—
483	C₄	—	4	3	no	—
484	C₄	C₂²×SD₁₆×C₅₃	1	6	yes	—
485	C₄	C₂²×SD₁₆×C₅₃	1	2	yes	—
486	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
487	C₄	C₅₃×#585	1	1	yes	characteristic
488	C₂	—	2	6	no	—
489	C₂	—	8	2	no	—
490	C₂	C₅₃×#3489	1	1	yes	characteristic
491	C₂	C₂³×SD₁₆×C₅₃	1	1	yes	characteristic
492	C₂	C₅₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
prime radical	C₅₃×#3499	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
2. Q₈×#41×C₅₃	C₂	semidirect
3. Q₈×SD₁₆×C₅₃	C₂	semidirect
4. C₅₃×#3359	C₂	semidirect
5. C₅₃×#3335	C₂	semidirect
6. C₅₃×#3298	C₂	semidirect
7. C₅₃×#3246	C₂	semidirect
8. C₅₃×#3229	C₂	semidirect
9. C₅₃×#3103	C₂	semidirect
10. C₂×C₅₃×#475	C₂	semidirect
11. C₅₃×#2988	C₂	semidirect
126. #41×C₅₃	SD₁₆	semidirect
194. Q₈×C₅₃	2₋¹⁺⁴	semidirect
222. #62099	C₅₃	direct (N × H)
243. Q₈×#41	C₂×C₅₃	semidirect
244. Q₈×SD₁₆	C₂×C₅₃	semidirect
245. #3359	C₂×C₅₃	semidirect
246. #3335	C₂×C₅₃	semidirect
247. #3298	C₂×C₅₃	semidirect
248. #3246	C₂×C₅₃	semidirect
249. #3229	C₂×C₅₃	semidirect
250. #3103	C₂×C₅₃	semidirect
251. C₂×#475	C₂×C₅₃	semidirect
252. #2988	C₂×C₅₃	semidirect
300. C₅₃	#62099	direct (N × H)
373. #41	SD₁₆×C₅₃	semidirect
441. Q₈	2₋¹⁺⁴×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3499	2
Q₈×#41×C₅₃	2
Q₈×SD₁₆×C₅₃	2
C₅₃×#3359	2
C₅₃×#3335	2
C₅₃×#3298	2
C₅₃×#3246	2
C₅₃×#3229	2
C₅₃×#3103	2
C₂×C₅₃×#475	2
C₅₃×#2988	2
#62099	53

Subgroups — C53×#62099 #27269611

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#62099 #27269611