Subgroups — C₅₃×#36594 #27244106

Order 13568.

2330 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#556	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2858	C₂	1	2	yes	maximal
4	C₅₃×#2547	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2545	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2538	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2512	C₂	1	8	yes	maximal, complemented
8	C₅₃×#2354	C₂	1	4	yes	maximal, complemented
9	C₄×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
10	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
11	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#556	—	2	16	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
16	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#126×C₅₃	C₂²	1	8	yes	—
18	C₂×#124×C₅₃	C₂²	1	4	yes	—
19	C₂×#124×C₅₃	C₂²	1	4	yes	—
20	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×Q₈×C₅₃	C₂²	1	4	yes	—
22	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
26	C₂×#117×C₅₃	C₂²	1	4	yes	—
27	C₅₃×#397	C₂²	1	8	yes	—
28	C₅₃×#396	C₂²	1	16	yes	—
29	C₅₃×#391	C₂²	1	4	yes	—
30	C₅₃×#390	C₂²	1	4	yes	—
31	C₅₃×#390	C₂²	1	8	yes	—
32	C₅₃×#390	C₂²	1	8	yes	—
33	C₅₃×#390	C₂²	1	2	yes	—
34	C₅₃×#389	C₂²	1	8	yes	—
35	C₅₃×#388	C₂²	1	16	yes	—
36	C₅₃×#384	C₂²	1	4	yes	—
37	C₅₃×#384	C₂²	1	4	yes	—
38	C₅₃×#384	C₂²	1	1	yes	characteristic
39	C₅₃×#380	C₂²	1	8	yes	—
40	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
41	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
42	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
43	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
44	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
45	C₄³×C₅₃	C₂²	1	1	yes	characteristic
46	C₄³×C₅₃	C₂²	1	2	yes	—
47	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
48	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
50	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
52	C₄⋊Q₈×C₅₃	—	2	8	no	—
53	C₄⋊Q₈×C₅₃	—	2	8	no	—
54	C₄⋊Q₈×C₅₃	—	4	4	no	—
55	#127×C₅₃	—	2	4	no	—
56	#126×C₅₃	—	2	32	no	—
57	#124×C₅₃	—	2	16	no	—
58	#124×C₅₃	—	4	8	no	—
59	C₂²⋊Q₈×C₅₃	—	2	16	no	—
60	C₄×Q₈×C₅₃	—	2	16	no	—
61	C₄×Q₈×C₅₃	—	2	8	no	—
62	C₄×Q₈×C₅₃	D₄	1	8	yes	complemented
63	C₄×D₄×C₅₃	—	2	8	no	—
64	C₄×D₄×C₅₃	—	2	16	no	—
65	#117×C₅₃	—	2	16	no	—
66	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
77	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
78	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
80	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
82	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
83	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
84	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
85	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
87	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
88	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
90	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
91	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
92	#95×C₅₃	C₂³	1	4	yes	—
93	#95×C₅₃	C₂³	1	2	yes	—
94	#95×C₅₃	C₂³	1	16	yes	—
95	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
96	C₂×Q₈×C₅₃	—	2	8	no	—
97	C₂×Q₈×C₅₃	—	2	4	no	—
98	C₂×Q₈×C₅₃	—	2	8	no	—
99	C₂×Q₈×C₅₃	—	2	8	no	—
100	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
101	C₂×D₄×C₅₃	—	2	4	no	—
102	C₂×D₄×C₅₃	—	2	8	no	—
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	—	2	8	no	—
107	C₂²×C₄×C₅₃	—	2	8	no	—
108	C₂²×C₄×C₅₃	—	2	16	no	—
109	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
110	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
111	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
112	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
113	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
115	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
116	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
119	C₄⋊C₄×C₅₃	—	2	8	no	—
120	C₄⋊C₄×C₅₃	—	2	4	no	—
121	C₄⋊C₄×C₅₃	—	2	16	no	—
122	C₄⋊C₄×C₅₃	—	2	8	no	—
123	C₄⋊C₄×C₅₃	—	2	8	no	—
124	C₄⋊C₄×C₅₃	—	2	8	no	—
125	C₄⋊C₄×C₅₃	—	2	4	no	—
126	C₄⋊C₄×C₅₃	—	2	16	no	—
127	C₄⋊C₄×C₅₃	—	4	8	no	—
128	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
129	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
130	C₂²⋊C₄×C₅₃	—	2	16	no	—
131	C₂²⋊C₄×C₅₃	—	2	8	no	—
132	C₂²⋊C₄×C₅₃	—	2	16	no	—
133	C₂²⋊C₄×C₅₃	—	2	16	no	—
134	C₂²⋊C₄×C₅₃	—	4	16	no	—
135	C₄²×C₅₃	—	2	16	no	—
136	C₄²×C₅₃	—	2	8	no	—
137	C₄²×C₅₃	—	2	4	no	—
138	C₄²×C₅₃	—	2	2	no	—
139	C₄²×C₅₃	—	2	4	no	—
140	C₄²×C₅₃	—	2	8	no	—
141	C₄²×C₅₃	C₂×D₄	1	4	yes	—
142	C₄²×C₅₃	C₂×D₄	1	8	yes	—
143	C₄²×C₅₃	#41	1	8	yes	—
144	C₄²×C₅₃	#41	1	4	yes	—
145	C₂³×C₅₃	—	2	4	no	—
146	C₂³×C₅₃	—	2	8	no	—
147	C₂³×C₅₃	—	2	4	no	—
148	C₂³×C₅₃	—	2	4	no	—
149	C₂³×C₅₃	—	2	8	no	—
150	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
151	Q₈×C₅₃	—	2	8	no	—
152	Q₈×C₅₃	—	4	16	no	—
153	D₄×C₅₃	—	4	8	no	—
154	D₄×C₅₃	—	4	16	no	—
155	C₂×C₄×C₅₃	—	2	4	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	4	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	2	2	no	—
160	C₂×C₄×C₅₃	—	2	4	no	—
161	C₂×C₄×C₅₃	—	2	8	no	—
162	C₂×C₄×C₅₃	—	2	8	no	—
163	C₂×C₄×C₅₃	—	2	8	no	—
164	C₂×C₄×C₅₃	—	2	4	no	—
165	C₂×C₄×C₅₃	—	2	8	no	—
166	C₂×C₄×C₅₃	—	2	4	no	—
167	C₂×C₄×C₅₃	—	2	4	no	—
168	C₂×C₄×C₅₃	—	4	16	no	—
169	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
170	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
171	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
173	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
175	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
176	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
177	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
178	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
179	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
180	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
181	#36594	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂²×C₅₃	—	4	4	no	—
183	C₂²×C₅₃	—	4	4	no	—
184	C₂²×C₅₃	—	4	8	no	—
185	C₂²×C₅₃	—	4	4	no	—
186	C₂²×C₅₃	—	4	8	no	—
187	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
188	C₂²×C₅₃	C₂²×#41	1	2	yes	—
189	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
190	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
191	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	2	yes	—
192	C₄×C₅₃	—	2	4	no	—
193	C₄×C₅₃	—	2	4	no	—
194	C₄×C₅₃	—	2	2	no	—
195	C₄×C₅₃	—	2	4	no	—
196	C₄×C₅₃	—	4	4	no	—
197	C₄×C₅₃	—	4	8	no	—
198	C₄×C₅₃	—	4	4	no	—
199	C₄×C₅₃	#532	1	4	yes	—
200	C₄×C₅₃	#550	1	4	yes	—
201	C₄×C₅₃	#556	1	8	yes	complemented
202	C₂×#556	C₂×C₅₃	1	2	yes	complemented
203	#2868	C₂×C₅₃	1	1	yes	characteristic, complemented
204	#2858	C₂×C₅₃	1	2	yes	—
205	#2547	C₂×C₅₃	1	2	yes	complemented
206	#2545	C₂×C₅₃	1	4	yes	complemented
207	#2538	C₂×C₅₃	1	4	yes	complemented
208	#2512	C₂×C₅₃	1	8	yes	complemented
209	#2354	C₂×C₅₃	1	4	yes	complemented
210	C₄×C₄⋊Q₈	C₂×C₅₃	1	2	yes	complemented
211	C₄×#127	C₂×C₅₃	1	1	yes	characteristic
212	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
213	C₂×C₅₃	—	8	4	no	—
214	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
215	C₂×C₅₃	C₂×#550	1	1	yes	characteristic
216	C₂×C₅₃	C₂×#556	1	2	yes	—
217	C₂×C₅₃	#3499	1	1	yes	characteristic
218	C₂×C₅₃	#3511	1	2	yes	—
219	#556	—	2	16	no	—
220	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
221	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
222	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
223	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
224	C₂×#126	C₂²×C₅₃	1	8	yes	—
225	C₂×#124	C₂²×C₅₃	1	4	yes	—
226	C₂×#124	C₂²×C₅₃	1	4	yes	—
227	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	4	yes	—
228	C₂×C₄×Q₈	C₂²×C₅₃	1	4	yes	—
229	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
230	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
231	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
232	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
233	C₂×#117	C₂²×C₅₃	1	4	yes	—
234	#397	C₂²×C₅₃	1	8	yes	—
235	#396	C₂²×C₅₃	1	16	yes	—
236	#391	C₂²×C₅₃	1	4	yes	—
237	#390	C₂²×C₅₃	1	4	yes	—
238	#390	C₂²×C₅₃	1	2	yes	—
239	#390	C₂²×C₅₃	1	8	yes	—
240	#390	C₂²×C₅₃	1	8	yes	—
241	#389	C₂²×C₅₃	1	8	yes	—
242	#388	C₂²×C₅₃	1	16	yes	—
243	#384	C₂²×C₅₃	1	1	yes	characteristic
244	#384	C₂²×C₅₃	1	4	yes	—
245	#384	C₂²×C₅₃	1	4	yes	—
246	#380	C₂²×C₅₃	1	8	yes	—
247	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
248	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
249	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
250	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
251	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
252	C₄³	C₂²×C₅₃	1	1	yes	characteristic
253	C₄³	C₂²×C₅₃	1	2	yes	—
254	C₅₃	#36594	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
256	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
257	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
258	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
259	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
260	C₄⋊Q₈	—	2	8	no	—
261	C₄⋊Q₈	—	2	8	no	—
262	C₄⋊Q₈	—	4	4	no	—
263	#127	—	2	4	no	—
264	#126	—	2	32	no	—
265	#124	—	2	16	no	—
266	#124	—	4	8	no	—
267	C₂²⋊Q₈	—	2	16	no	—
268	C₄×Q₈	—	2	16	no	—
269	C₄×Q₈	—	2	8	no	—
270	C₄×Q₈	D₄×C₅₃	1	8	yes	complemented
271	C₄×D₄	—	2	8	no	—
272	C₄×D₄	—	2	16	no	—
273	#117	—	2	16	no	—
274	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
275	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
276	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
277	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
278	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
280	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
281	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
282	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
283	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
284	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
285	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
286	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
289	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
290	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
291	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
292	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
293	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
295	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
296	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
297	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
298	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
299	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
300	#95	C₂³×C₅₃	1	4	yes	—
301	#95	C₂³×C₅₃	1	2	yes	—
302	#95	C₂³×C₅₃	1	16	yes	—
303	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
304	C₂×Q₈	—	2	8	no	—
305	C₂×Q₈	—	2	4	no	—
306	C₂×Q₈	—	2	8	no	—
307	C₂×Q₈	—	2	8	no	—
308	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
309	C₂×D₄	—	2	4	no	—
310	C₂×D₄	—	2	8	no	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	2	8	no	—
313	C₂×D₄	—	2	4	no	—
314	C₂×D₄	—	2	8	no	—
315	C₂²×C₄	—	2	8	no	—
316	C₂²×C₄	—	2	16	no	—
317	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
318	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
319	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
320	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
321	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
322	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
323	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
324	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
325	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
326	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
327	C₄⋊C₄	—	2	8	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	16	no	—
330	C₄⋊C₄	—	2	8	no	—
331	C₄⋊C₄	—	2	8	no	—
332	C₄⋊C₄	—	2	8	no	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	16	no	—
335	C₄⋊C₄	—	4	8	no	—
336	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
337	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
338	C₂²⋊C₄	—	2	16	no	—
339	C₂²⋊C₄	—	2	8	no	—
340	C₂²⋊C₄	—	2	16	no	—
341	C₂²⋊C₄	—	2	16	no	—
342	C₂²⋊C₄	—	4	16	no	—
343	C₄²	—	2	16	no	—
344	C₄²	—	2	8	no	—
345	C₄²	—	2	4	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	4	no	—
348	C₄²	—	2	8	no	—
349	C₄²	C₂×D₄×C₅₃	1	4	yes	—
350	C₄²	C₂×D₄×C₅₃	1	8	yes	—
351	C₄²	#41×C₅₃	1	8	yes	—
352	C₄²	#41×C₅₃	1	4	yes	—
353	C₂³	—	2	4	no	—
354	C₂³	—	2	8	no	—
355	C₂³	—	2	4	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	2	8	no	—
358	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
359	Q₈	—	2	8	no	—
360	Q₈	—	4	16	no	—
361	D₄	—	4	8	no	—
362	D₄	—	4	16	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	4	no	—
369	C₂×C₄	—	2	8	no	—
370	C₂×C₄	—	2	4	no	—
371	C₂×C₄	—	2	8	no	—
372	C₂×C₄	—	2	8	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	4	16	no	—
377	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
378	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
379	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
380	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
381	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
382	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
383	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
384	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
385	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
386	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
387	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
388	C₂×C₄	2₋¹⁺⁴×C₅₃	1	4	yes	—
389	C₂²	—	4	4	no	—
390	C₂²	—	4	8	no	—
391	C₂²	—	4	4	no	—
392	C₂²	—	4	8	no	—
393	C₂²	—	4	4	no	—
394	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
395	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
396	C₂²	C₂²×#41×C₅₃	1	2	yes	—
397	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
398	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	2	yes	—
399	C₄	—	2	4	no	—
400	C₄	—	2	2	no	—
401	C₄	—	2	4	no	—
402	C₄	—	2	4	no	—
403	C₄	—	4	8	no	—
404	C₄	—	4	4	no	—
405	C₄	—	4	4	no	—
406	C₄	C₅₃×#532	1	4	yes	—
407	C₄	C₅₃×#550	1	4	yes	—
408	C₄	C₅₃×#556	1	8	yes	complemented
409	C₂	—	8	4	no	—
410	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
411	C₂	C₂×C₅₃×#550	1	1	yes	characteristic
412	C₂	C₂×C₅₃×#556	1	2	yes	—
413	C₂	C₅₃×#3499	1	1	yes	characteristic
414	C₂	C₅₃×#3511	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
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
1. C₂×C₅₃×#556	C₂	semidirect
2. C₅₃×#2868	C₂	semidirect
4. C₅₃×#2547	C₂	semidirect
5. C₅₃×#2545	C₂	semidirect
6. C₅₃×#2538	C₂	semidirect
7. C₅₃×#2512	C₂	semidirect
8. C₅₃×#2354	C₂	semidirect
9. C₄×C₄⋊Q₈×C₅₃	C₂	semidirect
11. C₄²×Q₈×C₅₃	C₂	semidirect
62. C₄×Q₈×C₅₃	D₄	semidirect
181. #36594	C₅₃	direct (N × H)
201. C₄×C₅₃	#556	semidirect
202. C₂×#556	C₂×C₅₃	semidirect
203. #2868	C₂×C₅₃	semidirect
205. #2547	C₂×C₅₃	semidirect
206. #2545	C₂×C₅₃	semidirect
207. #2538	C₂×C₅₃	semidirect
208. #2512	C₂×C₅₃	semidirect
209. #2354	C₂×C₅₃	semidirect
210. C₄×C₄⋊Q₈	C₂×C₅₃	semidirect
212. C₄²×Q₈	C₂×C₅₃	semidirect
254. C₅₃	#36594	direct (N × H)
270. C₄×Q₈	D₄×C₅₃	semidirect
408. C₄	C₅₃×#556	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#556	2
C₅₃×#2868	2
C₅₃×#2858	2
C₅₃×#2547	2
C₅₃×#2545	2
C₅₃×#2538	2
C₅₃×#2512	2
C₅₃×#2354	2
C₄×C₄⋊Q₈×C₅₃	2
C₄×#127×C₅₃	2
C₄²×Q₈×C₅₃	2
#36594	53

Subgroups — C53×#36594 #27244106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36594 #27244106