Subgroups — C₅₃×#36578 #27244090

Order 13568.

2480 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#548	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2854	C₂	1	3	yes	maximal, complemented
4	C₅₃×#2504	C₂	1	3	yes	maximal, complemented
5	C₅₃×#2487	C₂	1	3	yes	maximal, complemented
6	C₅₃×#2448	C₂	1	6	yes	maximal, complemented
7	C₅₃×#2421	C₂	1	6	yes	maximal, complemented
8	C₅₃×#2351	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2341	C₂	1	2	yes	maximal, complemented
10	Q₈×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#123×C₅₃	C₂	1	3	yes	maximal, complemented
12	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#548	—	2	8	no	—
15	C₂×#124×C₅₃	C₂²	1	3	yes	—
16	C₂×#123×C₅₃	C₂²	1	3	yes	—
17	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	3	yes	—
18	C₂×#121×C₅₃	C₂²	1	3	yes	—
19	C₂×C₄×Q₈×C₅₃	—	2	8	no	—
20	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₅₃	C₂²	1	6	yes	—
22	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
26	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
27	C₅₃×#400	C₂²	1	6	yes	—
28	C₅₃×#399	C₂²	1	6	yes	—
29	C₅₃×#397	C₂²	1	12	yes	—
30	C₅₃×#391	C₂²	1	6	yes	—
31	C₅₃×#390	C₂²	1	6	yes	—
32	C₅₃×#390	C₂²	1	6	yes	—
33	C₅₃×#389	C₂²	1	6	yes	—
34	C₅₃×#389	C₂²	1	6	yes	—
35	C₅₃×#388	C₂²	1	6	yes	—
36	C₅₃×#388	C₂²	1	12	yes	—
37	C₅₃×#387	C₂²	1	6	yes	—
38	C₅₃×#386	C₂²	1	6	yes	—
39	C₅₃×#386	C₂²	1	3	yes	—
40	C₅₃×#385	C₂²	1	3	yes	—
41	C₅₃×#382	C₂²	1	6	yes	—
42	C₅₃×#381	C₂²	1	1	yes	characteristic
43	C₅₃×#380	C₂²	1	3	yes	—
44	C₅₃×#380	C₂²	1	6	yes	—
45	C₄×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
46	C₄×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
47	C₄×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	—
48	C₂×#41×C₅₃	—	2	4	no	—
49	C₂×#41×C₅₃	—	2	8	no	—
50	C₂²×Q₈×C₅₃	—	2	8	no	—
51	C₂²×Q₈×C₅₃	D₄	1	8	yes	—
52	C₂²×Q₈×C₅₃	C₂³	1	3	yes	—
53	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
55	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
56	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
57	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
60	#124×C₅₃	—	2	12	no	—
61	#123×C₅₃	—	2	12	no	—
62	C₂²⋊Q₈×C₅₃	—	2	12	no	—
63	#121×C₅₃	—	2	12	no	—
64	C₄×Q₈×C₅₃	—	2	24	no	—
65	C₄×Q₈×C₅₃	—	2	8	no	—
66	C₄×Q₈×C₅₃	—	2	4	no	—
67	C₄×Q₈×C₅₃	—	2	8	no	—
68	C₄×D₄×C₅₃	—	2	12	no	—
69	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	3	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
77	C₂×C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₅₃	C₂³	1	3	yes	—
79	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
80	C₂×C₂²⋊C₄×C₅₃	C₂³	1	3	yes	—
81	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
84	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
85	C₂×C₄²×C₅₃	—	2	12	no	—
86	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
87	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
88	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
89	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
90	#95×C₅₃	C₂³	1	2	yes	—
91	#95×C₅₃	C₂³	1	6	yes	—
92	#95×C₅₃	C₂³	1	6	yes	—
93	#95×C₅₃	C₂³	1	2	yes	—
94	#95×C₅₃	C₂³	1	6	yes	—
95	#95×C₅₃	C₂³	1	6	yes	—
96	#95×C₅₃	C₂³	1	12	yes	—
97	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
98	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
99	#41×C₅₃	—	4	16	no	—
100	C₂×Q₈×C₅₃	—	2	32	no	—
101	C₂×Q₈×C₅₃	—	2	12	no	—
102	C₂×Q₈×C₅₃	C₂×D₄	1	12	yes	—
103	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
104	C₂×Q₈×C₅₃	#41	1	8	yes	—
105	C₂×D₄×C₅₃	—	2	6	no	—
106	C₂×D₄×C₅₃	—	2	6	no	—
107	C₂×D₄×C₅₃	—	2	6	no	—
108	C₂×D₄×C₅₃	—	2	6	no	—
109	C₂×D₄×C₅₃	—	2	12	no	—
110	C₂²×C₄×C₅₃	—	2	2	no	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	2	12	no	—
113	C₂²×C₄×C₅₃	—	2	6	no	—
114	C₂²×C₄×C₅₃	—	2	6	no	—
115	C₂²×C₄×C₅₃	—	2	12	no	—
116	C₂²×C₄×C₅₃	—	4	6	no	—
117	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
118	C₂²×C₄×C₅₃	#41	1	4	yes	complemented
119	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
120	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
122	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
123	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
124	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
126	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
127	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
128	C₄⋊C₄×C₅₃	—	2	12	no	—
129	C₄⋊C₄×C₅₃	—	2	12	no	—
130	C₄⋊C₄×C₅₃	—	2	6	no	—
131	C₄⋊C₄×C₅₃	—	2	6	no	—
132	C₄⋊C₄×C₅₃	—	2	12	no	—
133	C₄⋊C₄×C₅₃	—	2	12	no	—
134	C₄⋊C₄×C₅₃	—	4	6	no	—
135	C₂²⋊C₄×C₅₃	—	2	2	no	—
136	C₂²⋊C₄×C₅₃	—	2	6	no	—
137	C₂²⋊C₄×C₅₃	—	2	2	no	—
138	C₂²⋊C₄×C₅₃	—	2	6	no	—
139	C₂²⋊C₄×C₅₃	—	2	12	no	—
140	C₂²⋊C₄×C₅₃	—	2	12	no	—
141	C₂²⋊C₄×C₅₃	—	4	12	no	—
142	C₂²⋊C₄×C₅₃	—	4	6	no	—
143	C₄²×C₅₃	—	2	12	no	—
144	C₄²×C₅₃	—	2	12	no	—
145	C₄²×C₅₃	—	2	6	no	—
146	C₄²×C₅₃	—	2	12	no	—
147	C₂³×C₅₃	—	2	2	no	—
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	2	3	no	—
150	C₂³×C₅₃	—	2	6	no	—
151	C₂³×C₅₃	—	2	3	no	—
152	C₂³×C₅₃	—	2	3	no	—
153	C₂³×C₅₃	—	2	6	no	—
154	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
155	C₂³×C₅₃	C₂×#41	1	2	yes	—
156	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
157	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
158	Q₈×C₅₃	—	2	24	no	—
159	Q₈×C₅₃	#123	1	16	yes	complemented
160	D₄×C₅₃	—	4	12	no	—
161	D₄×C₅₃	—	4	12	no	—
162	C₂×C₄×C₅₃	—	2	8	no	—
163	C₂×C₄×C₅₃	—	2	2	no	—
164	C₂×C₄×C₅₃	—	2	6	no	—
165	C₂×C₄×C₅₃	—	2	2	no	—
166	C₂×C₄×C₅₃	—	2	24	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	6	no	—
169	C₂×C₄×C₅₃	—	2	3	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	3	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	6	no	—
175	C₂×C₄×C₅₃	—	2	6	no	—
176	C₂×C₄×C₅₃	—	2	3	no	—
177	C₂×C₄×C₅₃	—	2	2	no	—
178	C₂×C₄×C₅₃	—	2	6	no	—
179	C₂×C₄×C₅₃	—	2	6	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	6	no	—
182	C₂×C₄×C₅₃	—	2	2	no	—
183	C₂×C₄×C₅₃	—	4	12	no	—
184	C₂×C₄×C₅₃	—	4	12	no	—
185	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
186	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
187	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
188	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
189	#36578	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂²×C₅₃	—	2	2	no	—
191	C₂²×C₅₃	—	2	4	no	—
192	C₂²×C₅₃	—	2	2	no	—
193	C₂²×C₅₃	—	2	4	no	—
194	C₂²×C₅₃	—	4	3	no	—
195	C₂²×C₅₃	—	4	3	no	—
196	C₂²×C₅₃	—	4	6	no	—
197	C₂²×C₅₃	—	4	3	no	—
198	C₂²×C₅₃	—	4	6	no	—
199	C₂²×C₅₃	#548	1	4	yes	—
200	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
201	C₂²×C₅₃	C₂²×#41	1	2	yes	—
202	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
203	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
204	C₂²×C₅₃	#585	1	2	yes	—
205	C₄×C₅₃	—	2	6	no	—
206	C₄×C₅₃	—	2	4	no	—
207	C₄×C₅₃	—	4	1	no	—
208	C₄×C₅₃	—	4	3	no	—
209	C₄×C₅₃	—	4	6	no	—
210	C₄×C₅₃	—	4	1	no	—
211	C₄×C₅₃	—	4	6	no	—
212	C₄×C₅₃	—	4	2	no	—
213	C₄×C₅₃	C₂×#123	1	12	yes	—
214	C₂×#548	C₂×C₅₃	1	1	yes	characteristic
215	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
216	#2854	C₂×C₅₃	1	3	yes	complemented
217	#2504	C₂×C₅₃	1	3	yes	complemented
218	#2487	C₂×C₅₃	1	3	yes	complemented
219	#2448	C₂×C₅₃	1	6	yes	complemented
220	#2421	C₂×C₅₃	1	6	yes	complemented
221	#2351	C₂×C₅₃	1	2	yes	complemented
222	#2341	C₂×C₅₃	1	2	yes	complemented
223	Q₈×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
224	C₄×#123	C₂×C₅₃	1	3	yes	complemented
225	C₂×C₅₃	—	2	4	no	—
226	C₂×C₅₃	—	8	3	no	—
227	C₂×C₅₃	C₂²×#123	1	1	yes	characteristic
228	C₂×C₅₃	C₂×#548	1	1	yes	characteristic
229	C₂×C₅₃	#3480	1	2	yes	—
230	C₂×C₅₃	#3487	1	1	yes	characteristic
231	C₂×C₅₃	#3516	1	2	yes	—
232	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
233	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
234	#548	—	2	8	no	—
235	C₂×#124	C₂²×C₅₃	1	3	yes	—
236	C₂×#123	C₂²×C₅₃	1	3	yes	—
237	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	3	yes	—
238	C₂×#121	C₂²×C₅₃	1	3	yes	—
239	C₂×C₄×Q₈	—	2	8	no	—
240	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
241	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
242	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
243	C₂×C₄×Q₈	C₂²×C₅₃	1	6	yes	—
244	C₂×C₄×D₄	C₂²×C₅₃	1	3	yes	—
245	C₂²×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
246	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
247	#400	C₂²×C₅₃	1	6	yes	—
248	#399	C₂²×C₅₃	1	6	yes	—
249	#397	C₂²×C₅₃	1	12	yes	—
250	#391	C₂²×C₅₃	1	6	yes	—
251	#390	C₂²×C₅₃	1	6	yes	—
252	#390	C₂²×C₅₃	1	6	yes	—
253	#389	C₂²×C₅₃	1	6	yes	—
254	#389	C₂²×C₅₃	1	6	yes	—
255	#388	C₂²×C₅₃	1	12	yes	—
256	#388	C₂²×C₅₃	1	6	yes	—
257	#387	C₂²×C₅₃	1	6	yes	—
258	#386	C₂²×C₅₃	1	6	yes	—
259	#386	C₂²×C₅₃	1	3	yes	—
260	#385	C₂²×C₅₃	1	3	yes	—
261	#382	C₂²×C₅₃	1	6	yes	—
262	#381	C₂²×C₅₃	1	1	yes	characteristic
263	#380	C₂²×C₅₃	1	3	yes	—
264	#380	C₂²×C₅₃	1	6	yes	—
265	C₄×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
266	C₄×C₂²⋊C₄	C₂²×C₅₃	1	3	yes	—
267	C₄×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	—
268	C₅₃	#36578	1	1	yes	characteristic, direct product factor, Sylow
269	C₂×#41	—	2	4	no	—
270	C₂×#41	—	2	8	no	—
271	C₂²×Q₈	—	2	8	no	—
272	C₂²×Q₈	D₄×C₅₃	1	8	yes	—
273	C₂²×Q₈	C₂³×C₅₃	1	3	yes	—
274	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
275	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
276	C₂²×D₄	C₂³×C₅₃	1	3	yes	—
277	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
278	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
279	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
280	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
281	#124	—	2	12	no	—
282	#123	—	2	12	no	—
283	C₂²⋊Q₈	—	2	12	no	—
284	#121	—	2	12	no	—
285	C₄×Q₈	—	2	24	no	—
286	C₄×Q₈	—	2	8	no	—
287	C₄×Q₈	—	2	4	no	—
288	C₄×Q₈	—	2	8	no	—
289	C₄×D₄	—	2	12	no	—
290	C₂×C₄⋊C₄	—	2	12	no	—
291	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
292	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
293	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
294	C₂×C₄⋊C₄	C₂³×C₅₃	1	3	yes	—
295	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
296	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
297	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
298	C₂×C₂²⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
299	C₂×C₂²⋊C₄	C₂³×C₅₃	1	3	yes	—
300	C₂×C₂²⋊C₄	C₂³×C₅₃	1	6	yes	—
301	C₂×C₂²⋊C₄	C₂³×C₅₃	1	3	yes	—
302	C₂×C₂²⋊C₄	C₂³×C₅₃	1	6	yes	—
303	C₂×C₂²⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
304	C₂×C₂²⋊C₄	C₂³×C₅₃	1	6	yes	—
305	C₂×C₂²⋊C₄	C₂³×C₅₃	1	12	yes	—
306	C₂×C₄²	—	2	12	no	—
307	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
308	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
309	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
310	C₂×C₄²	C₂³×C₅₃	1	6	yes	—
311	#95	C₂³×C₅₃	1	2	yes	—
312	#95	C₂³×C₅₃	1	6	yes	—
313	#95	C₂³×C₅₃	1	6	yes	—
314	#95	C₂³×C₅₃	1	2	yes	—
315	#95	C₂³×C₅₃	1	6	yes	—
316	#95	C₂³×C₅₃	1	6	yes	—
317	#95	C₂³×C₅₃	1	12	yes	—
318	C₂⁴	C₂⁴×C₅₃	1	3	yes	—
319	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
320	#41	—	4	16	no	—
321	C₂×Q₈	—	2	32	no	—
322	C₂×Q₈	—	2	12	no	—
323	C₂×Q₈	C₂×D₄×C₅₃	1	12	yes	—
324	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
325	C₂×Q₈	#41×C₅₃	1	8	yes	—
326	C₂×D₄	—	2	6	no	—
327	C₂×D₄	—	2	6	no	—
328	C₂×D₄	—	2	6	no	—
329	C₂×D₄	—	2	6	no	—
330	C₂×D₄	—	2	12	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	12	no	—
334	C₂²×C₄	—	2	6	no	—
335	C₂²×C₄	—	2	6	no	—
336	C₂²×C₄	—	2	12	no	—
337	C₂²×C₄	—	4	6	no	—
338	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
339	C₂²×C₄	#41×C₅₃	1	4	yes	complemented
340	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
341	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
342	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
343	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
344	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
345	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
346	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
347	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
348	C₂²×C₄	C₂⁴×C₅₃	1	3	yes	—
349	C₄⋊C₄	—	2	12	no	—
350	C₄⋊C₄	—	2	12	no	—
351	C₄⋊C₄	—	2	6	no	—
352	C₄⋊C₄	—	2	6	no	—
353	C₄⋊C₄	—	2	12	no	—
354	C₄⋊C₄	—	2	12	no	—
355	C₄⋊C₄	—	4	6	no	—
356	C₂²⋊C₄	—	2	2	no	—
357	C₂²⋊C₄	—	2	6	no	—
358	C₂²⋊C₄	—	2	2	no	—
359	C₂²⋊C₄	—	2	6	no	—
360	C₂²⋊C₄	—	2	12	no	—
361	C₂²⋊C₄	—	2	12	no	—
362	C₂²⋊C₄	—	4	6	no	—
363	C₂²⋊C₄	—	4	12	no	—
364	C₄²	—	2	12	no	—
365	C₄²	—	2	12	no	—
366	C₄²	—	2	6	no	—
367	C₄²	—	2	12	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	2	no	—
370	C₂³	—	2	3	no	—
371	C₂³	—	2	3	no	—
372	C₂³	—	2	6	no	—
373	C₂³	—	2	3	no	—
374	C₂³	—	2	6	no	—
375	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
376	C₂³	C₂×#41×C₅₃	1	2	yes	—
377	C₂³	2₋¹⁺⁴×C₅₃	1	2	yes	—
378	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
379	Q₈	—	2	24	no	—
380	Q₈	#123×C₅₃	1	16	yes	complemented
381	D₄	—	4	12	no	—
382	D₄	—	4	12	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	6	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	24	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	6	no	—
390	C₂×C₄	—	2	3	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	6	no	—
394	C₂×C₄	—	2	6	no	—
395	C₂×C₄	—	2	3	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	3	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	6	no	—
401	C₂×C₄	—	2	6	no	—
402	C₂×C₄	—	2	6	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	4	12	no	—
405	C₂×C₄	—	4	12	no	—
406	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
407	C₂×C₄	C₂²×D₄×C₅₃	1	6	yes	—
408	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
409	C₂×C₄	C₂×#41×C₅₃	1	12	yes	—
410	C₂²	—	2	4	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	4	no	—
413	C₂²	—	2	2	no	—
414	C₂²	—	4	3	no	—
415	C₂²	—	4	6	no	—
416	C₂²	—	4	3	no	—
417	C₂²	—	4	6	no	—
418	C₂²	—	4	3	no	—
419	C₂²	C₅₃×#548	1	4	yes	—
420	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
421	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
422	C₂²	C₂²×#41×C₅₃	1	2	yes	—
423	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
424	C₂²	C₅₃×#585	1	2	yes	—
425	C₄	—	2	4	no	—
426	C₄	—	2	6	no	—
427	C₄	—	4	3	no	—
428	C₄	—	4	6	no	—
429	C₄	—	4	6	no	—
430	C₄	—	4	1	no	—
431	C₄	—	4	2	no	—
432	C₄	—	4	1	no	—
433	C₄	C₂×#123×C₅₃	1	12	yes	—
434	C₂	—	2	4	no	—
435	C₂	—	8	3	no	—
436	C₂	C₂²×#123×C₅₃	1	1	yes	characteristic
437	C₂	C₂×C₅₃×#548	1	1	yes	characteristic
438	C₂	C₅₃×#3480	1	2	yes	—
439	C₂	C₅₃×#3487	1	1	yes	characteristic
440	C₂	C₅₃×#3516	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂×C₅₃×#548	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
3. C₅₃×#2854	C₂	semidirect
4. C₅₃×#2504	C₂	semidirect
5. C₅₃×#2487	C₂	semidirect
6. C₅₃×#2448	C₂	semidirect
7. C₅₃×#2421	C₂	semidirect
8. C₅₃×#2351	C₂	semidirect
9. C₅₃×#2341	C₂	semidirect
10. Q₈×C₂²⋊C₄×C₅₃	C₂	semidirect
11. C₄×#123×C₅₃	C₂	semidirect
118. C₂²×C₄×C₅₃	#41	semidirect
159. Q₈×C₅₃	#123	semidirect
189. #36578	C₅₃	direct (N × H)
215. C₂²×C₄×Q₈	C₂×C₅₃	semidirect
216. #2854	C₂×C₅₃	semidirect
217. #2504	C₂×C₅₃	semidirect
218. #2487	C₂×C₅₃	semidirect
219. #2448	C₂×C₅₃	semidirect
220. #2421	C₂×C₅₃	semidirect
221. #2351	C₂×C₅₃	semidirect
222. #2341	C₂×C₅₃	semidirect
223. Q₈×C₂²⋊C₄	C₂×C₅₃	semidirect
224. C₄×#123	C₂×C₅₃	semidirect
268. C₅₃	#36578	direct (N × H)
339. C₂²×C₄	#41×C₅₃	semidirect
380. Q₈	#123×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#548	2
C₂²×C₄×Q₈×C₅₃	2
C₅₃×#2854	2
C₅₃×#2504	2
C₅₃×#2487	2
C₅₃×#2448	2
C₅₃×#2421	2
C₅₃×#2351	2
C₅₃×#2341	2
Q₈×C₂²⋊C₄×C₅₃	2
C₄×#123×C₅₃	2
#36578	53

Subgroups — C53×#36578 #27244090

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36578 #27244090