Subgroups — C₅₃×#60676 #27268188

Order 13568.

1998 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3444	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3172	C₂	1	8	yes	maximal, complemented
3	C₅₃×#3171	C₂	1	8	yes	maximal, complemented
4	C₅₃×#3162	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#3150	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#3135	C₂	1	2	yes	maximal, complemented
7	C₅₃×#3054	C₂	1	4	yes	maximal, complemented
8	C₄×#135×C₅₃	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₅₃	C₂²	1	4	yes	—
11	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#533	C₂²	1	4	yes	—
13	C₅₃×#533	C₂²	1	2	yes	—
14	C₅₃×#532	C₂²	1	2	yes	—
15	C₅₃×#532	C₂²	1	4	yes	—
16	C₅₃×#529	—	2	4	no	—
17	C₅₃×#529	—	2	4	no	—
18	C₅₃×#529	C₂²	1	8	yes	—
19	C₅₃×#525	—	2	4	no	—
20	C₅₃×#525	C₂²	1	4	yes	—
21	C₄×#41×C₅₃	—	2	4	no	—
22	C₄×#41×C₅₃	C₂²	1	4	yes	—
23	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#501	C₂²	1	8	yes	complemented
25	C₅₃×#497	C₂²	1	4	yes	complemented
26	C₅₃×#496	C₂²	1	4	yes	complemented
27	C₅₃×#491	C₂²	1	4	yes	complemented
28	C₅₃×#490	C₂²	1	4	yes	complemented
29	C₅₃×#489	C₂²	1	8	yes	complemented
30	C₅₃×#470	C₂²	1	8	yes	—
31	C₅₃×#469	C₂²	1	8	yes	—
32	C₅₃×#468	C₂²	1	16	yes	—
33	C₄×Q₁₆×C₅₃	C₂²	1	8	yes	complemented
34	C₄×SD₁₆×C₅₃	C₂²	1	16	yes	complemented
35	C₄×D₈×C₅₃	C₂²	1	8	yes	complemented
36	C₅₃×#427	C₂²	1	4	yes	—
37	C₅₃×#416	C₂²	1	8	yes	—
38	C₄×M₁₆×C₅₃	C₂²	1	8	yes	—
39	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
42	C₂×#41×C₅₃	—	2	4	no	—
43	C₂×#41×C₅₃	C₂³	1	4	yes	—
44	C₂³×C₄×C₅₃	D₄	1	2	yes	—
45	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	#135×C₅₃	—	2	16	no	—
47	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
48	C₂×SD₁₆×C₅₃	C₂³	1	8	yes	—
49	C₂×D₈×C₅₃	C₂³	1	4	yes	—
50	C₂×M₁₆×C₅₃	—	2	8	no	—
51	C₂×M₁₆×C₅₃	C₂³	1	8	yes	—
52	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
54	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
55	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
56	#127×C₅₃	C₂³	1	2	yes	—
57	#126×C₅₃	—	2	8	no	—
58	#125×C₅₃	—	2	4	no	—
59	#125×C₅₃	C₂³	1	4	yes	—
60	#124×C₅₃	—	2	4	no	—
61	#124×C₅₃	C₂³	1	4	yes	—
62	#123×C₅₃	—	2	8	no	—
63	#123×C₅₃	—	2	4	no	—
64	C₂²⋊Q₈×C₅₃	—	2	4	no	—
65	C₂²⋊Q₈×C₅₃	—	2	4	no	—
66	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
67	#121×C₅₃	—	2	4	no	—
68	#121×C₅₃	—	2	4	no	—
69	#121×C₅₃	C₂³	1	8	yes	—
70	#120×C₅₃	—	2	4	no	—
71	C₄×Q₈×C₅₃	—	2	4	no	—
72	C₄×Q₈×C₅₃	—	2	4	no	—
73	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
74	C₄×D₄×C₅₃	—	2	4	no	—
75	C₄×D₄×C₅₃	—	2	4	no	—
76	C₄×D₄×C₅₃	—	2	8	no	—
77	C₄×D₄×C₅₃	—	2	8	no	—
78	C₄×D₄×C₅₃	—	2	8	no	—
79	C₄×D₄×C₅₃	C₂³	1	8	yes	—
80	#117×C₅₃	—	2	8	no	—
81	#117×C₅₃	—	2	4	no	—
82	#117×C₅₃	—	2	4	no	—
83	#117×C₅₃	C₂³	1	4	yes	—
84	C₂×C₄²×C₅₃	—	2	4	no	—
85	C₂×C₄²×C₅₃	D₄	1	2	yes	—
86	C₂×C₄²×C₅₃	D₄	1	2	yes	—
87	C₂×C₄²×C₅₃	D₄	1	2	yes	—
88	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
89	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
91	#107×C₅₃	C₂³	1	8	yes	—
92	#106×C₅₃	C₂³	1	8	yes	—
93	#103×C₅₃	C₂³	1	16	yes	—
94	#102×C₅₃	C₂³	1	16	yes	—
95	#97×C₅₃	C₂³	1	4	yes	—
96	#97×C₅₃	C₂³	1	4	yes	—
97	C₄×C₈×C₅₃	C₂³	1	8	yes	—
98	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
99	#41×C₅₃	—	2	16	no	—
100	#41×C₅₃	—	4	16	no	—
101	C₂×Q₈×C₅₃	—	2	4	no	—
102	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
103	C₂×D₄×C₅₃	—	2	4	no	—
104	C₂×D₄×C₅₃	—	2	8	no	—
105	C₂×D₄×C₅₃	—	2	4	no	—
106	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
107	C₂²×C₄×C₅₃	—	2	4	no	—
108	C₂²×C₄×C₅₃	—	2	4	no	—
109	C₂²×C₄×C₅₃	—	2	8	no	—
110	C₂²×C₄×C₅₃	—	4	4	no	—
111	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
117	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
118	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
119	Q₁₆×C₅₃	—	2	8	no	—
120	SD₁₆×C₅₃	—	2	16	no	—
121	D₈×C₅₃	—	2	8	no	—
122	M₁₆×C₅₃	—	2	16	no	—
123	C₂×C₈×C₅₃	—	2	4	no	—
124	C₂×C₈×C₅₃	#41	1	8	yes	complemented
125	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
126	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
127	C₄⋊C₄×C₅₃	—	2	4	no	—
128	C₄⋊C₄×C₅₃	—	2	4	no	—
129	C₄⋊C₄×C₅₃	—	2	8	no	—
130	C₄⋊C₄×C₅₃	—	2	8	no	—
131	C₄⋊C₄×C₅₃	—	2	8	no	—
132	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
133	C₂²⋊C₄×C₅₃	—	2	8	no	—
134	C₂²⋊C₄×C₅₃	—	2	4	no	—
135	C₂²⋊C₄×C₅₃	—	2	8	no	—
136	C₂²⋊C₄×C₅₃	—	2	8	no	—
137	C₂²⋊C₄×C₅₃	—	2	8	no	—
138	C₄²×C₅₃	—	2	8	no	—
139	C₄²×C₅₃	—	2	4	no	—
140	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₄²×C₅₃	C₂×D₄	1	4	yes	—
142	C₄²×C₅₃	C₂×D₄	1	2	yes	—
143	C₄²×C₅₃	C₂×D₄	1	4	yes	—
144	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
145	C₄²×C₅₃	C₂⁴	1	2	yes	—
146	C₄²×C₅₃	C₂⁴	1	2	yes	—
147	C₂³×C₅₃	—	2	2	no	—
148	C₂³×C₅₃	—	2	4	no	—
149	C₂³×C₅₃	—	4	2	no	—
150	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
151	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₅₃	—	2	8	no	—
153	Q₈×C₅₃	—	4	8	no	—
154	D₄×C₅₃	—	2	8	no	—
155	D₄×C₅₃	—	4	8	no	—
156	D₄×C₅₃	—	4	16	no	—
157	D₄×C₅₃	—	4	8	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	4	no	—
160	C₂×C₄×C₅₃	—	2	4	no	—
161	C₂×C₄×C₅₃	—	2	8	no	—
162	C₂×C₄×C₅₃	—	2	4	no	—
163	C₂×C₄×C₅₃	—	2	8	no	—
164	C₂×C₄×C₅₃	—	2	4	no	—
165	C₂×C₄×C₅₃	—	4	4	no	—
166	C₂×C₄×C₅₃	—	4	2	no	—
167	C₂×C₄×C₅₃	—	4	8	no	—
168	C₂×C₄×C₅₃	—	4	16	no	—
169	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
170	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
171	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
175	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
176	C₈×C₅₃	—	2	4	no	—
177	C₈×C₅₃	C₂×#41	1	8	yes	—
178	#60676	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
179	C₂²×C₅₃	—	2	2	no	—
180	C₂²×C₅₃	—	2	2	no	—
181	C₂²×C₅₃	—	2	4	no	—
182	C₂²×C₅₃	—	4	4	no	—
183	C₂²×C₅₃	—	4	4	no	—
184	C₂²×C₅₃	—	4	8	no	—
185	C₂²×C₅₃	#532	1	2	yes	—
186	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
187	C₄×C₅₃	—	2	2	no	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	—	2	4	no	—
190	C₄×C₅₃	—	4	4	no	—
191	C₄×C₅₃	—	4	8	no	—
192	C₄×C₅₃	#532	1	2	yes	—
193	C₄×C₅₃	#575	1	4	yes	—
194	C₄×C₅₃	C₂²×#41	1	2	yes	—
195	#3444	C₂×C₅₃	1	2	yes	complemented
196	#3172	C₂×C₅₃	1	8	yes	complemented
197	#3171	C₂×C₅₃	1	8	yes	complemented
198	#3162	C₂×C₅₃	1	1	yes	characteristic, complemented
199	#3150	C₂×C₅₃	1	1	yes	characteristic, complemented
200	#3135	C₂×C₅₃	1	2	yes	complemented
201	#3054	C₂×C₅₃	1	4	yes	complemented
202	C₄×#135	C₂×C₅₃	1	4	yes	complemented
203	C₂×C₄×M₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
204	C₂×C₅₃	—	2	2	no	—
205	C₂×C₅₃	—	4	2	no	—
206	C₂×C₅₃	—	8	4	no	—
207	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
208	C₂×C₅₃	C₂×#575	1	2	yes	—
209	C₂×#135	C₂²×C₅₃	1	4	yes	—
210	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
211	#533	C₂²×C₅₃	1	4	yes	—
212	#533	C₂²×C₅₃	1	2	yes	—
213	#532	C₂²×C₅₃	1	2	yes	—
214	#532	C₂²×C₅₃	1	4	yes	—
215	#529	—	2	4	no	—
216	#529	—	2	4	no	—
217	#529	C₂²×C₅₃	1	8	yes	—
218	#525	—	2	4	no	—
219	#525	C₂²×C₅₃	1	4	yes	—
220	C₄×#41	—	2	4	no	—
221	C₄×#41	C₂²×C₅₃	1	4	yes	—
222	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
223	#501	C₂²×C₅₃	1	8	yes	complemented
224	#497	C₂²×C₅₃	1	4	yes	complemented
225	#496	C₂²×C₅₃	1	4	yes	complemented
226	#491	C₂²×C₅₃	1	4	yes	complemented
227	#490	C₂²×C₅₃	1	4	yes	complemented
228	#489	C₂²×C₅₃	1	8	yes	complemented
229	#470	C₂²×C₅₃	1	8	yes	—
230	#469	C₂²×C₅₃	1	8	yes	—
231	#468	C₂²×C₅₃	1	16	yes	—
232	C₄×Q₁₆	C₂²×C₅₃	1	8	yes	complemented
233	C₄×SD₁₆	C₂²×C₅₃	1	16	yes	complemented
234	C₄×D₈	C₂²×C₅₃	1	8	yes	complemented
235	#427	C₂²×C₅₃	1	4	yes	—
236	#416	C₂²×C₅₃	1	8	yes	—
237	C₄×M₁₆	C₂²×C₅₃	1	8	yes	—
238	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
239	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
240	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
241	C₅₃	#60676	1	1	yes	characteristic, direct product factor, Sylow
242	C₂×#41	—	2	4	no	—
243	C₂×#41	C₂³×C₅₃	1	4	yes	—
244	C₂³×C₄	D₄×C₅₃	1	2	yes	—
245	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
246	#135	—	2	16	no	—
247	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
248	C₂×SD₁₆	C₂³×C₅₃	1	8	yes	—
249	C₂×D₈	C₂³×C₅₃	1	4	yes	—
250	C₂×M₁₆	—	2	8	no	—
251	C₂×M₁₆	C₂³×C₅₃	1	8	yes	—
252	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
253	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
254	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
255	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
256	#127	C₂³×C₅₃	1	2	yes	—
257	#126	—	2	8	no	—
258	#125	—	2	4	no	—
259	#125	C₂³×C₅₃	1	4	yes	—
260	#124	—	2	4	no	—
261	#124	C₂³×C₅₃	1	4	yes	—
262	#123	—	2	8	no	—
263	#123	—	2	4	no	—
264	C₂²⋊Q₈	—	2	4	no	—
265	C₂²⋊Q₈	—	2	4	no	—
266	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
267	#121	—	2	4	no	—
268	#121	—	2	4	no	—
269	#121	C₂³×C₅₃	1	8	yes	—
270	#120	—	2	4	no	—
271	C₄×Q₈	—	2	4	no	—
272	C₄×Q₈	—	2	4	no	—
273	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
274	C₄×D₄	—	2	8	no	—
275	C₄×D₄	—	2	8	no	—
276	C₄×D₄	—	2	8	no	—
277	C₄×D₄	—	2	4	no	—
278	C₄×D₄	—	2	4	no	—
279	C₄×D₄	C₂³×C₅₃	1	8	yes	—
280	#117	—	2	4	no	—
281	#117	—	2	8	no	—
282	#117	—	2	4	no	—
283	#117	C₂³×C₅₃	1	4	yes	—
284	C₂×C₄²	—	2	4	no	—
285	C₂×C₄²	D₄×C₅₃	1	2	yes	—
286	C₂×C₄²	D₄×C₅₃	1	2	yes	—
287	C₂×C₄²	D₄×C₅₃	1	2	yes	—
288	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
290	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
291	#107	C₂³×C₅₃	1	8	yes	—
292	#106	C₂³×C₅₃	1	8	yes	—
293	#103	C₂³×C₅₃	1	16	yes	—
294	#102	C₂³×C₅₃	1	16	yes	—
295	#97	C₂³×C₅₃	1	4	yes	—
296	#97	C₂³×C₅₃	1	4	yes	—
297	C₄×C₈	C₂³×C₅₃	1	8	yes	—
298	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
299	#41	—	2	16	no	—
300	#41	—	4	16	no	—
301	C₂×Q₈	—	2	4	no	—
302	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	8	no	—
305	C₂×D₄	—	2	4	no	—
306	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
307	C₂²×C₄	—	2	4	no	—
308	C₂²×C₄	—	2	4	no	—
309	C₂²×C₄	—	2	8	no	—
310	C₂²×C₄	—	4	4	no	—
311	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
313	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
314	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
317	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
318	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
319	Q₁₆	—	2	8	no	—
320	SD₁₆	—	2	16	no	—
321	D₈	—	2	8	no	—
322	M₁₆	—	2	16	no	—
323	C₂×C₈	—	2	4	no	—
324	C₂×C₈	#41×C₅₃	1	8	yes	complemented
325	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
326	C₂×C₈	C₂⁴×C₅₃	1	4	yes	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	8	no	—
330	C₄⋊C₄	—	2	8	no	—
331	C₄⋊C₄	—	2	8	no	—
332	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
333	C₂²⋊C₄	—	2	8	no	—
334	C₂²⋊C₄	—	2	4	no	—
335	C₂²⋊C₄	—	2	8	no	—
336	C₂²⋊C₄	—	2	8	no	—
337	C₂²⋊C₄	—	2	8	no	—
338	C₄²	—	2	8	no	—
339	C₄²	—	2	4	no	—
340	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
341	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
342	C₄²	C₂×D₄×C₅₃	1	4	yes	—
343	C₄²	C₂×D₄×C₅₃	1	2	yes	—
344	C₄²	C₂×D₄×C₅₃	1	4	yes	—
345	C₄²	C₂⁴×C₅₃	1	2	yes	—
346	C₄²	C₂⁴×C₅₃	1	2	yes	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	4	no	—
349	C₂³	—	4	2	no	—
350	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
351	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
352	Q₈	—	2	8	no	—
353	Q₈	—	4	8	no	—
354	D₄	—	2	8	no	—
355	D₄	—	4	8	no	—
356	D₄	—	4	16	no	—
357	D₄	—	4	8	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	8	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	—	4	8	no	—
368	C₂×C₄	—	4	16	no	—
369	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
370	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
371	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
372	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
373	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
374	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
375	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
376	C₈	—	2	4	no	—
377	C₈	C₂×#41×C₅₃	1	8	yes	—
378	C₂²	—	2	2	no	—
379	C₂²	—	2	4	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	4	4	no	—
382	C₂²	—	4	8	no	—
383	C₂²	—	4	4	no	—
384	C₂²	C₅₃×#532	1	2	yes	—
385	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
386	C₄	—	2	4	no	—
387	C₄	—	2	2	no	—
388	C₄	—	2	2	no	—
389	C₄	—	4	8	no	—
390	C₄	—	4	4	no	—
391	C₄	C₅₃×#532	1	2	yes	—
392	C₄	C₅₃×#575	1	4	yes	—
393	C₄	C₂²×#41×C₅₃	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	4	2	no	—
396	C₂	—	8	4	no	—
397	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
398	C₂	C₂×C₅₃×#575	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
socle	C₂²×C₅₃	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3444	C₂	semidirect
2. C₅₃×#3172	C₂	semidirect
3. C₅₃×#3171	C₂	semidirect
4. C₅₃×#3162	C₂	semidirect
5. C₅₃×#3150	C₂	semidirect
6. C₅₃×#3135	C₂	semidirect
7. C₅₃×#3054	C₂	semidirect
8. C₄×#135×C₅₃	C₂	semidirect
9. C₂×C₄×M₁₆×C₅₃	C₂	semidirect
24. C₅₃×#501	C₂²	semidirect
25. C₅₃×#497	C₂²	semidirect
26. C₅₃×#496	C₂²	semidirect
27. C₅₃×#491	C₂²	semidirect
28. C₅₃×#490	C₂²	semidirect
29. C₅₃×#489	C₂²	semidirect
33. C₄×Q₁₆×C₅₃	C₂²	semidirect
34. C₄×SD₁₆×C₅₃	C₂²	semidirect
35. C₄×D₈×C₅₃	C₂²	semidirect
124. C₂×C₈×C₅₃	#41	semidirect
178. #60676	C₅₃	direct (N × H)
195. #3444	C₂×C₅₃	semidirect
196. #3172	C₂×C₅₃	semidirect
197. #3171	C₂×C₅₃	semidirect
198. #3162	C₂×C₅₃	semidirect
199. #3150	C₂×C₅₃	semidirect
200. #3135	C₂×C₅₃	semidirect
201. #3054	C₂×C₅₃	semidirect
202. C₄×#135	C₂×C₅₃	semidirect
203. C₂×C₄×M₁₆	C₂×C₅₃	semidirect
223. #501	C₂²×C₅₃	semidirect
224. #497	C₂²×C₅₃	semidirect
225. #496	C₂²×C₅₃	semidirect
226. #491	C₂²×C₅₃	semidirect
227. #490	C₂²×C₅₃	semidirect
228. #489	C₂²×C₅₃	semidirect
232. C₄×Q₁₆	C₂²×C₅₃	semidirect
233. C₄×SD₁₆	C₂²×C₅₃	semidirect
234. C₄×D₈	C₂²×C₅₃	semidirect
241. C₅₃	#60676	direct (N × H)
324. C₂×C₈	#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#3444	2
C₅₃×#3172	2
C₅₃×#3171	2
C₅₃×#3162	2
C₅₃×#3150	2
C₅₃×#3135	2
C₅₃×#3054	2
C₄×#135×C₅₃	2
C₂×C₄×M₁₆×C₅₃	2
#60676	53

Subgroups — C53×#60676 #27268188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60676 #27268188