Subgroups — C₅₃×#17725 #27225237

Order 13568.

884 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3509	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2988	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2925	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1554	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1552	C₂	1	2	yes	maximal, complemented
6	C₅₃×#1550	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1544	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#1542	C₂	1	2	yes	maximal, complemented
11	C₂×C₅₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#557	—	2	1	no	—
13	C₅₃×#557	C₄	1	2	yes	—
14	C₅₃×#556	—	2	1	no	—
15	C₅₃×#556	C₄	1	2	yes	—
16	C₅₃×#552	—	2	2	no	—
17	C₅₃×#552	C₄	1	4	yes	—
18	C₅₃×#541	—	2	2	no	—
19	C₅₃×#536	—	2	1	no	—
20	C₅₃×#533	C₂²	1	1	yes	characteristic
21	C₅₃×#533	C₂²	1	2	yes	—
22	C₅₃×#533	C₂²	1	2	yes	—
23	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#519	—	2	1	no	—
25	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₅₃	—	2	1	no	—
27	C₈×D₄×C₅₃	—	2	2	no	—
28	C₈×D₄×C₅₃	—	2	1	no	—
29	C₅₃×#433	—	2	2	no	—
30	C₅₃×#432	C₂²	1	2	yes	—
31	C₅₃×#431	C₂²	1	1	yes	characteristic
32	C₅₃×#431	C₂²	1	2	yes	—
33	C₅₃×#431	C₂²	1	2	yes	—
34	C₅₃×#423	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
37	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₅₃×#408	—	2	1	no	—
39	C₅₃×#405	—	2	1	no	—
40	C₅₃×#333	C₂²	1	1	yes	characteristic
41	C₅₃×#333	C₂²	1	2	yes	—
42	C₅₃×#333	C₂²	1	1	yes	characteristic
43	C₅₃×#332	C₂²	1	2	yes	—
44	C₅₃×#332	C₂²	1	2	yes	—
45	C₅₃×#332	C₂²	1	1	yes	characteristic
46	C₅₃×#332	C₂²	1	2	yes	—
47	C₅₃×#332	C₂²	1	1	yes	characteristic
48	C₅₃×#332	C₂²	1	2	yes	—
49	C₅₃×#332	C₂²	1	2	yes	—
50	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
51	C₂²×Q₈×C₅₃	—	2	1	no	—
52	C₂×M₁₆×C₅₃	—	2	1	no	—
53	C₂²×C₈×C₅₃	—	2	1	no	—
54	C₂²×C₈×C₅₃	—	2	1	no	—
55	C₂²×C₈×C₅₃	—	2	2	no	—
56	C₄⋊Q₈×C₅₃	—	2	1	no	—
57	C₄⋊Q₈×C₅₃	—	2	1	no	—
58	C₄⋊Q₈×C₅₃	—	2	2	no	—
59	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
63	#126×C₅₃	—	2	4	no	—
64	#126×C₅₃	—	2	4	no	—
65	#125×C₅₃	—	2	4	no	—
66	#125×C₅₃	—	2	4	no	—
67	#125×C₅₃	—	2	1	no	—
68	#125×C₅₃	—	2	2	no	—
69	#125×C₅₃	C₂×C₄	1	4	yes	—
70	#124×C₅₃	—	2	2	no	—
71	#123×C₅₃	—	2	4	no	—
72	C₂²⋊Q₈×C₅₃	—	2	2	no	—
73	C₂²⋊Q₈×C₅₃	—	2	2	no	—
74	C₂²⋊Q₈×C₅₃	—	2	1	no	—
75	C₂²⋊Q₈×C₅₃	—	2	2	no	—
76	C₂²⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
77	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
78	C₄×Q₈×C₅₃	—	2	2	no	—
79	C₄×Q₈×C₅₃	—	2	2	no	—
80	C₄×Q₈×C₅₃	—	2	2	no	—
81	C₄×Q₈×C₅₃	—	2	2	no	—
82	C₄×Q₈×C₅₃	—	2	2	no	—
83	C₄×Q₈×C₅₃	—	2	1	no	—
84	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
86	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
87	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
88	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
89	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₄×D₄×C₅₃	D₄	1	2	yes	—
91	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
92	#117×C₅₃	—	2	4	no	—
93	#117×C₅₃	—	2	1	no	—
94	#117×C₅₃	—	2	2	no	—
95	#117×C₅₃	C₂×C₄	1	4	yes	—
96	#117×C₅₃	D₄	1	1	yes	characteristic
97	#117×C₅₃	D₄	1	2	yes	—
98	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
100	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
102	C₄⋊C₈×C₅₃	—	2	1	no	—
103	C₄⋊C₈×C₅₃	—	2	2	no	—
104	C₄⋊C₈×C₅₃	—	2	1	no	—
105	C₄⋊C₈×C₅₃	—	2	1	no	—
106	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
107	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
109	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
110	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
111	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
112	C₂²⋊C₈×C₅₃	—	2	2	no	—
113	C₂²⋊C₈×C₅₃	—	2	2	no	—
114	C₂²⋊C₈×C₅₃	—	2	1	no	—
115	C₂²⋊C₈×C₅₃	—	2	2	no	—
116	#97×C₅₃	—	2	2	no	—
117	C₄×C₈×C₅₃	—	2	2	no	—
118	C₄×C₈×C₅₃	—	2	2	no	—
119	C₄×C₈×C₅₃	—	2	1	no	—
120	#41×C₅₃	—	2	2	no	—
121	#41×C₅₃	—	2	2	no	—
122	C₂×Q₈×C₅₃	—	2	2	no	—
123	C₂×Q₈×C₅₃	—	2	1	no	—
124	C₂×Q₈×C₅₃	—	2	2	no	—
125	C₂×Q₈×C₅₃	—	4	2	no	—
126	C₂×Q₈×C₅₃	—	4	2	no	—
127	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
128	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
129	C₂×D₄×C₅₃	C₂²⋊C₄	1	2	yes	—
130	C₂×D₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	—	2	2	no	—
132	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
138	M₁₆×C₅₃	—	4	2	no	—
139	C₂×C₈×C₅₃	—	2	1	no	—
140	C₂×C₈×C₅₃	—	2	1	no	—
141	C₂×C₈×C₅₃	—	2	2	no	—
142	C₂×C₈×C₅₃	—	2	2	no	—
143	C₂×C₈×C₅₃	—	2	1	no	—
144	C₂×C₈×C₅₃	—	2	1	no	—
145	C₂×C₈×C₅₃	—	4	2	no	—
146	C₂×C₈×C₅₃	—	8	1	no	—
147	C₂×C₈×C₅₃	—	8	2	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	—	2	2	no	—
150	C₄⋊C₄×C₅₃	—	2	4	no	—
151	C₄⋊C₄×C₅₃	—	2	4	no	—
152	C₄⋊C₄×C₅₃	—	2	4	no	—
153	C₄⋊C₄×C₅₃	—	2	2	no	—
154	C₄⋊C₄×C₅₃	—	2	4	no	—
155	C₄⋊C₄×C₅₃	—	2	2	no	—
156	C₄⋊C₄×C₅₃	—	2	2	no	—
157	C₄⋊C₄×C₅₃	—	2	1	no	—
158	C₄⋊C₄×C₅₃	—	2	1	no	—
159	C₄⋊C₄×C₅₃	—	2	1	no	—
160	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
161	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
162	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
163	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
164	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
165	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
166	C₂²⋊C₄×C₅₃	—	2	2	no	—
167	C₂²⋊C₄×C₅₃	—	2	4	no	—
168	C₂²⋊C₄×C₅₃	—	2	4	no	—
169	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
170	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
171	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
172	C₄²×C₅₃	—	2	4	no	—
173	C₄²×C₅₃	—	2	4	no	—
174	C₄²×C₅₃	—	2	2	no	—
175	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
176	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
177	C₄²×C₅₃	C₂×D₄	1	2	yes	—
178	C₄²×C₅₃	C₂×D₄	1	2	yes	—
179	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
182	C₂³×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
184	Q₈×C₅₃	—	2	2	no	—
185	Q₈×C₅₃	—	4	2	no	—
186	Q₈×C₅₃	—	4	2	no	—
187	Q₈×C₅₃	—	4	2	no	—
188	Q₈×C₅₃	—	4	1	no	—
189	Q₈×C₅₃	—	4	1	no	—
190	Q₈×C₅₃	—	4	4	no	—
191	D₄×C₅₃	—	2	4	no	—
192	D₄×C₅₃	—	2	2	no	—
193	C₂×C₄×C₅₃	—	2	2	no	—
194	C₂×C₄×C₅₃	—	2	2	no	—
195	C₂×C₄×C₅₃	—	2	1	no	—
196	C₂×C₄×C₅₃	—	2	2	no	—
197	C₂×C₄×C₅₃	—	2	1	no	—
198	C₂×C₄×C₅₃	—	2	2	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	2	no	—
202	C₂×C₄×C₅₃	—	4	1	no	—
203	C₂×C₄×C₅₃	—	4	2	no	—
204	C₂×C₄×C₅₃	—	4	4	no	—
205	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
206	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
208	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
209	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₄×C₅₃	#135	1	2	yes	—
212	C₂×C₄×C₅₃	#135	1	2	yes	—
213	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
216	C₈×C₅₃	—	4	1	no	—
217	C₈×C₅₃	—	4	1	no	—
218	C₈×C₅₃	—	4	2	no	—
219	C₈×C₅₃	—	4	2	no	—
220	C₈×C₅₃	—	4	1	no	—
221	C₈×C₅₃	—	4	1	no	—
222	#17725	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂²×C₅₃	—	2	1	no	—
224	C₂²×C₅₃	—	2	1	no	—
225	C₂²×C₅₃	—	2	2	no	—
226	C₂²×C₅₃	—	2	2	no	—
227	C₂²×C₅₃	—	2	2	no	—
228	C₂²×C₅₃	#416	1	2	yes	—
229	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	2	2	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	—	2	2	no	—
234	C₄×C₅₃	—	2	2	no	—
235	C₄×C₅₃	—	2	1	no	—
236	C₄×C₅₃	—	4	2	no	—
237	C₄×C₅₃	—	4	4	no	—
238	C₄×C₅₃	—	4	2	no	—
239	C₄×C₅₃	#416	1	2	yes	—
240	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
241	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
242	C₄×C₅₃	#577	1	1	yes	characteristic
243	C₄×C₅₃	#578	1	1	yes	characteristic
244	#3509	C₂×C₅₃	1	1	yes	characteristic
245	#2988	C₂×C₅₃	1	1	yes	characteristic
246	#2925	C₂×C₅₃	1	1	yes	characteristic
247	#1554	C₂×C₅₃	1	2	yes	complemented
248	#1552	C₂×C₅₃	1	2	yes	complemented
249	#1550	C₂×C₅₃	1	1	yes	characteristic, complemented
250	#1546	C₂×C₅₃	1	1	yes	characteristic, complemented
251	#1544	C₂×C₅₃	1	2	yes	complemented
252	#1542	C₂×C₅₃	1	1	yes	characteristic, complemented
253	#1542	C₂×C₅₃	1	2	yes	complemented
254	C₂×#332	C₂×C₅₃	1	1	yes	characteristic, complemented
255	C₂×C₅₃	—	2	2	no	—
256	C₂×C₅₃	—	4	2	no	—
257	C₂×C₅₃	#2889	1	1	yes	characteristic
258	C₂×C₅₃	C₂×#416	1	1	yes	characteristic
259	C₂×C₅₃	#2898	1	1	yes	characteristic
260	#557	—	2	1	no	—
261	#557	C₄×C₅₃	1	2	yes	—
262	#556	—	2	1	no	—
263	#556	C₄×C₅₃	1	2	yes	—
264	#552	—	2	2	no	—
265	#552	C₄×C₅₃	1	4	yes	—
266	#541	—	2	2	no	—
267	#536	—	2	1	no	—
268	#533	C₂²×C₅₃	1	1	yes	characteristic
269	#533	C₂²×C₅₃	1	2	yes	—
270	#533	C₂²×C₅₃	1	2	yes	—
271	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
272	#519	—	2	1	no	—
273	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
274	C₈×Q₈	—	2	1	no	—
275	C₈×D₄	—	2	2	no	—
276	C₈×D₄	—	2	1	no	—
277	#433	—	2	2	no	—
278	#432	C₂²×C₅₃	1	2	yes	—
279	#431	C₂²×C₅₃	1	2	yes	—
280	#431	C₂²×C₅₃	1	1	yes	characteristic
281	#431	C₂²×C₅₃	1	2	yes	—
282	#423	C₂²×C₅₃	1	1	yes	characteristic
283	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
284	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
285	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
286	#408	—	2	1	no	—
287	#405	—	2	1	no	—
288	#333	C₂²×C₅₃	1	1	yes	characteristic
289	#333	C₂²×C₅₃	1	2	yes	—
290	#333	C₂²×C₅₃	1	1	yes	characteristic
291	#332	C₂²×C₅₃	1	2	yes	—
292	#332	C₂²×C₅₃	1	2	yes	—
293	#332	C₂²×C₅₃	1	1	yes	characteristic
294	#332	C₂²×C₅₃	1	2	yes	—
295	#332	C₂²×C₅₃	1	1	yes	characteristic
296	#332	C₂²×C₅₃	1	2	yes	—
297	#332	C₂²×C₅₃	1	2	yes	—
298	C₅₃	#17725	1	1	yes	characteristic, direct product factor, Sylow
299	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
300	C₂²×Q₈	—	2	1	no	—
301	C₂×M₁₆	—	2	1	no	—
302	C₂²×C₈	—	2	1	no	—
303	C₂²×C₈	—	2	1	no	—
304	C₂²×C₈	—	2	2	no	—
305	C₄⋊Q₈	—	2	2	no	—
306	C₄⋊Q₈	—	2	1	no	—
307	C₄⋊Q₈	—	2	1	no	—
308	C₄⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
309	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
310	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
311	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
312	#126	—	2	4	no	—
313	#126	—	2	4	no	—
314	#125	—	2	4	no	—
315	#125	—	2	4	no	—
316	#125	—	2	1	no	—
317	#125	—	2	2	no	—
318	#125	C₂×C₄×C₅₃	1	4	yes	—
319	#124	—	2	2	no	—
320	#123	—	2	4	no	—
321	C₂²⋊Q₈	—	2	2	no	—
322	C₂²⋊Q₈	—	2	2	no	—
323	C₂²⋊Q₈	—	2	1	no	—
324	C₂²⋊Q₈	—	2	2	no	—
325	C₂²⋊Q₈	C₂×C₄×C₅₃	1	2	yes	—
326	C₂²⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
327	C₄×Q₈	—	2	2	no	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×Q₈	—	2	2	no	—
330	C₄×Q₈	—	2	2	no	—
331	C₄×Q₈	—	2	2	no	—
332	C₄×Q₈	—	2	1	no	—
333	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
334	C₄×Q₈	C₂×C₄×C₅₃	1	2	yes	—
335	C₄×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
336	C₄×Q₈	D₄×C₅₃	1	1	yes	characteristic
337	C₄×D₄	C₂×C₄×C₅₃	1	2	yes	—
338	C₄×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
339	C₄×D₄	D₄×C₅₃	1	2	yes	—
340	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
341	#117	—	2	1	no	—
342	#117	—	2	2	no	—
343	#117	—	2	4	no	—
344	#117	C₂×C₄×C₅₃	1	4	yes	—
345	#117	D₄×C₅₃	1	1	yes	characteristic
346	#117	D₄×C₅₃	1	2	yes	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
349	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
350	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
351	C₄⋊C₈	—	2	1	no	—
352	C₄⋊C₈	—	2	2	no	—
353	C₄⋊C₈	—	2	1	no	—
354	C₄⋊C₈	—	2	1	no	—
355	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
356	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
357	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
358	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
359	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
360	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
361	C₂²⋊C₈	—	2	2	no	—
362	C₂²⋊C₈	—	2	2	no	—
363	C₂²⋊C₈	—	2	1	no	—
364	C₂²⋊C₈	—	2	2	no	—
365	#97	—	2	2	no	—
366	C₄×C₈	—	2	2	no	—
367	C₄×C₈	—	2	2	no	—
368	C₄×C₈	—	2	1	no	—
369	#41	—	2	2	no	—
370	#41	—	2	2	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	4	2	no	—
375	C₂×Q₈	—	4	2	no	—
376	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
377	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
378	C₂×D₄	C₂²⋊C₄×C₅₃	1	2	yes	—
379	C₂×D₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
382	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
383	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
384	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
387	M₁₆	—	4	2	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	4	2	no	—
395	C₂×C₈	—	8	1	no	—
396	C₂×C₈	—	8	2	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
410	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
411	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
412	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
413	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
414	C₄⋊C₄	C₂²×C₄×C₅₃	1	2	yes	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
419	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
420	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
421	C₄²	—	2	4	no	—
422	C₄²	—	2	4	no	—
423	C₄²	—	2	2	no	—
424	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
425	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
426	C₄²	C₂×D₄×C₅₃	1	2	yes	—
427	C₄²	C₂×D₄×C₅₃	1	2	yes	—
428	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
429	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
430	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
431	C₂³	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
432	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
433	Q₈	—	2	2	no	—
434	Q₈	—	4	2	no	—
435	Q₈	—	4	2	no	—
436	Q₈	—	4	2	no	—
437	Q₈	—	4	1	no	—
438	Q₈	—	4	1	no	—
439	Q₈	—	4	4	no	—
440	D₄	—	2	4	no	—
441	D₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	4	no	—
454	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
455	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
456	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
457	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
458	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
459	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
460	C₂×C₄	#135×C₅₃	1	2	yes	—
461	C₂×C₄	#135×C₅₃	1	2	yes	—
462	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
464	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
465	C₈	—	4	1	no	—
466	C₈	—	4	1	no	—
467	C₈	—	4	2	no	—
468	C₈	—	4	2	no	—
469	C₈	—	4	1	no	—
470	C₈	—	4	1	no	—
471	C₂²	—	2	1	no	—
472	C₂²	—	2	1	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	C₅₃×#416	1	2	yes	—
477	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
478	C₄	—	2	1	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	2	no	—
482	C₄	—	2	2	no	—
483	C₄	—	2	1	no	—
484	C₄	—	4	2	no	—
485	C₄	—	4	4	no	—
486	C₄	—	4	2	no	—
487	C₄	C₅₃×#416	1	2	yes	—
488	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
489	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
490	C₄	C₅₃×#577	1	1	yes	characteristic
491	C₄	C₅₃×#578	1	1	yes	characteristic
492	C₂	—	2	2	no	—
493	C₂	—	4	2	no	—
494	C₂	C₅₃×#2889	1	1	yes	characteristic
495	C₂	C₂×C₅₃×#416	1	1	yes	characteristic
496	C₂	C₅₃×#2898	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	C₂×#41×C₅₃	D₄
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
4. C₅₃×#1554	C₂	semidirect
5. C₅₃×#1552	C₂	semidirect
6. C₅₃×#1550	C₂	semidirect
7. C₅₃×#1546	C₂	semidirect
8. C₅₃×#1544	C₂	semidirect
9. C₅₃×#1542	C₂	semidirect
10. C₅₃×#1542	C₂	semidirect
11. C₂×C₅₃×#332	C₂	semidirect
222. #17725	C₅₃	direct (N × H)
247. #1554	C₂×C₅₃	semidirect
248. #1552	C₂×C₅₃	semidirect
249. #1550	C₂×C₅₃	semidirect
250. #1546	C₂×C₅₃	semidirect
251. #1544	C₂×C₅₃	semidirect
252. #1542	C₂×C₅₃	semidirect
253. #1542	C₂×C₅₃	semidirect
254. C₂×#332	C₂×C₅₃	semidirect
298. C₅₃	#17725	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×#3509	2
C₅₃×#2988	2
C₅₃×#2925	2
C₅₃×#1554	2
C₅₃×#1552	2
C₅₃×#1550	2
C₅₃×#1546	2
C₅₃×#1544	2
C₅₃×#1542	2
C₂×C₅₃×#332	2
#17725	53

Subgroups — C53×#17725 #27225237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17725 #27225237