Subgroups — C₅₃×#36609 #27244121

Order 13568.

1938 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#2858	C₂	1	2	yes	maximal
3	C₅₃×#2592	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2580	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2513	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2448	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2447	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2367	C₂	1	4	yes	maximal, complemented
9	C₅₃×#2357	C₂	1	2	yes	maximal, complemented
10	C₄×C₄⋊Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
11	C₄×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
14	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#124×C₅₃	C₂²	1	4	yes	—
16	C₂×#124×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₅₃	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
22	C₅₃×#399	C₂²	1	4	yes	—
23	C₅₃×#397	C₂²	1	8	yes	—
24	C₅₃×#397	C₂²	1	8	yes	—
25	C₅₃×#393	C₂²	1	4	yes	—
26	C₅₃×#391	C₂²	1	4	yes	—
27	C₅₃×#391	C₂²	1	4	yes	—
28	C₅₃×#391	C₂²	1	4	yes	—
29	C₅₃×#390	C₂²	1	4	yes	—
30	C₅₃×#390	C₂²	1	4	yes	—
31	C₅₃×#390	C₂²	1	4	yes	—
32	C₅₃×#390	C₂²	1	2	yes	—
33	C₅₃×#390	C₂²	1	4	yes	—
34	C₅₃×#390	C₂²	1	4	yes	—
35	C₅₃×#389	C₂²	1	4	yes	—
36	C₅₃×#389	C₂²	1	4	yes	—
37	C₅₃×#388	C₂²	1	8	yes	—
38	C₅₃×#388	C₂²	1	8	yes	—
39	C₅₃×#388	C₂²	1	8	yes	—
40	C₅₃×#385	C₂²	1	8	yes	—
41	C₅₃×#384	C₂²	1	1	yes	characteristic
42	C₅₃×#382	C₂²	1	2	yes	—
43	C₅₃×#382	C₂²	1	2	yes	—
44	C₅₃×#381	C₂²	1	4	yes	—
45	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
46	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
48	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
49	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
50	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
52	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
53	C₄³×C₅₃	C₂²	1	1	yes	characteristic
54	C₄³×C₅₃	C₂²	1	2	yes	—
55	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
57	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
58	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
59	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
60	C₄⋊Q₈×C₅₃	—	2	8	no	—
61	#127×C₅₃	—	2	4	no	—
62	#124×C₅₃	—	2	16	no	—
63	#124×C₅₃	—	4	8	no	—
64	C₄×Q₈×C₅₃	—	2	4	no	—
65	C₄×Q₈×C₅₃	—	2	8	no	—
66	C₄×Q₈×C₅₃	—	4	8	no	—
67	C₄×D₄×C₅₃	—	2	8	no	—
68	C₄×D₄×C₅₃	—	4	8	no	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
84	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
85	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
86	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
87	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
88	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
89	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
90	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
91	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
92	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
94	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	#95×C₅₃	C₂³	1	1	yes	characteristic
98	#95×C₅₃	C₂³	1	4	yes	—
99	#95×C₅₃	C₂³	1	1	yes	characteristic
100	#95×C₅₃	C₂³	1	2	yes	—
101	#95×C₅₃	C₂³	1	2	yes	—
102	#95×C₅₃	C₂³	1	8	yes	—
103	#95×C₅₃	C₂³	1	8	yes	—
104	#95×C₅₃	C₂³	1	4	yes	—
105	#95×C₅₃	C₂³	1	8	yes	—
106	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
107	C₂×Q₈×C₅₃	—	2	2	no	—
108	C₂×Q₈×C₅₃	—	2	4	no	—
109	C₂×Q₈×C₅₃	—	2	8	no	—
110	C₂×Q₈×C₅₃	—	2	8	no	—
111	C₂×Q₈×C₅₃	—	2	8	no	—
112	C₂×D₄×C₅₃	—	2	4	no	—
113	C₂×D₄×C₅₃	—	2	8	no	—
114	C₂×D₄×C₅₃	—	2	4	no	—
115	C₂×D₄×C₅₃	—	2	8	no	—
116	C₂×D₄×C₅₃	—	2	8	no	—
117	C₂×D₄×C₅₃	—	2	4	no	—
118	C₂²×C₄×C₅₃	—	2	8	no	—
119	C₂²×C₄×C₅₃	—	4	8	no	—
120	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
121	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
123	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
124	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
125	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
127	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
129	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
130	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
131	C₄⋊C₄×C₅₃	—	2	4	no	—
132	C₄⋊C₄×C₅₃	—	2	4	no	—
133	C₄⋊C₄×C₅₃	—	2	4	no	—
134	C₄⋊C₄×C₅₃	—	2	4	no	—
135	C₄⋊C₄×C₅₃	—	2	4	no	—
136	C₄⋊C₄×C₅₃	—	2	2	no	—
137	C₄⋊C₄×C₅₃	—	2	8	no	—
138	C₄⋊C₄×C₅₃	—	2	8	no	—
139	C₄⋊C₄×C₅₃	—	2	8	no	—
140	C₄⋊C₄×C₅₃	—	4	8	no	—
141	C₂²⋊C₄×C₅₃	—	2	8	no	—
142	C₂²⋊C₄×C₅₃	—	2	16	no	—
143	C₂²⋊C₄×C₅₃	—	2	16	no	—
144	C₂²⋊C₄×C₅₃	—	4	8	no	—
145	C₂²⋊C₄×C₅₃	—	4	8	no	—
146	C₂²⋊C₄×C₅₃	—	4	8	no	—
147	C₄²×C₅₃	—	2	4	no	—
148	C₄²×C₅₃	—	2	4	no	—
149	C₄²×C₅₃	—	2	4	no	—
150	C₄²×C₅₃	—	2	4	no	—
151	C₄²×C₅₃	—	2	4	no	—
152	C₄²×C₅₃	—	2	4	no	—
153	C₄²×C₅₃	—	2	4	no	—
154	C₄²×C₅₃	—	2	2	no	—
155	C₄²×C₅₃	—	4	8	no	—
156	C₄²×C₅₃	#41	1	4	yes	—
157	C₄²×C₅₃	#41	1	8	yes	—
158	C₄²×C₅₃	#41	1	4	yes	—
159	C₂³×C₅₃	—	2	4	no	—
160	C₂³×C₅₃	—	2	4	no	—
161	C₂³×C₅₃	—	2	8	no	—
162	C₂³×C₅₃	—	2	8	no	—
163	C₂³×C₅₃	—	2	4	no	—
164	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
165	Q₈×C₅₃	—	2	8	no	—
166	Q₈×C₅₃	—	4	16	no	—
167	D₄×C₅₃	—	4	8	no	—
168	D₄×C₅₃	—	4	16	no	—
169	C₂×C₄×C₅₃	—	2	2	no	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	2	no	—
172	C₂×C₄×C₅₃	—	2	2	no	—
173	C₂×C₄×C₅₃	—	2	2	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	2	4	no	—
176	C₂×C₄×C₅₃	—	2	8	no	—
177	C₂×C₄×C₅₃	—	2	8	no	—
178	C₂×C₄×C₅₃	—	2	2	no	—
179	C₂×C₄×C₅₃	—	2	2	no	—
180	C₂×C₄×C₅₃	—	2	2	no	—
181	C₂×C₄×C₅₃	—	2	8	no	—
182	C₂×C₄×C₅₃	—	2	8	no	—
183	C₂×C₄×C₅₃	—	2	2	no	—
184	C₂×C₄×C₅₃	—	2	2	no	—
185	C₂×C₄×C₅₃	—	2	2	no	—
186	C₂×C₄×C₅₃	—	2	2	no	—
187	C₂×C₄×C₅₃	—	2	2	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	8	8	no	—
190	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
191	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
192	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
193	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
194	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
195	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
196	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
197	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
198	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
199	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
200	#36609	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂²×C₅₃	—	4	4	no	—
202	C₂²×C₅₃	—	4	4	no	—
203	C₂²×C₅₃	—	4	8	no	—
204	C₂²×C₅₃	—	4	4	no	—
205	C₂²×C₅₃	—	4	8	no	—
206	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
207	C₂²×C₅₃	C₂²×#41	1	2	yes	—
208	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
209	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
210	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
211	C₄×C₅₃	—	2	4	no	—
212	C₄×C₅₃	—	2	2	no	—
213	C₄×C₅₃	—	2	2	no	—
214	C₄×C₅₃	—	2	4	no	—
215	C₄×C₅₃	—	4	8	no	—
216	C₄×C₅₃	—	4	2	no	—
217	C₄×C₅₃	—	4	4	no	—
218	C₄×C₅₃	—	4	2	no	—
219	C₄×C₅₃	—	4	2	no	—
220	C₄×C₅₃	#556	1	4	yes	—
221	C₄×C₅₃	#559	1	8	yes	—
222	#2862	C₂×C₅₃	1	1	yes	characteristic, complemented
223	#2858	C₂×C₅₃	1	2	yes	—
224	#2592	C₂×C₅₃	1	4	yes	complemented
225	#2580	C₂×C₅₃	1	2	yes	complemented
226	#2513	C₂×C₅₃	1	4	yes	complemented
227	#2448	C₂×C₅₃	1	4	yes	complemented
228	#2447	C₂×C₅₃	1	4	yes	complemented
229	#2367	C₂×C₅₃	1	4	yes	complemented
230	#2357	C₂×C₅₃	1	2	yes	complemented
231	C₄×C₄⋊Q₈	C₂×C₅₃	1	2	yes	complemented
232	C₄×#127	C₂×C₅₃	1	1	yes	characteristic
233	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
234	C₂×C₅₃	—	8	4	no	—
235	C₂×C₅₃	#3444	1	1	yes	characteristic
236	C₂×C₅₃	C₂×#556	1	1	yes	characteristic
237	C₂×C₅₃	C₂×#559	1	2	yes	—
238	C₂×C₅₃	#3508	1	1	yes	characteristic
239	C₂×C₅₃	#3511	1	2	yes	—
240	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
241	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
242	C₂×#124	C₂²×C₅₃	1	4	yes	—
243	C₂×#124	C₂²×C₅₃	1	4	yes	—
244	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
245	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
246	C₂×C₄×Q₈	C₂²×C₅₃	1	4	yes	—
247	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
248	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
249	#399	C₂²×C₅₃	1	4	yes	—
250	#397	C₂²×C₅₃	1	8	yes	—
251	#397	C₂²×C₅₃	1	8	yes	—
252	#393	C₂²×C₅₃	1	4	yes	—
253	#391	C₂²×C₅₃	1	4	yes	—
254	#391	C₂²×C₅₃	1	4	yes	—
255	#391	C₂²×C₅₃	1	4	yes	—
256	#390	C₂²×C₅₃	1	4	yes	—
257	#390	C₂²×C₅₃	1	2	yes	—
258	#390	C₂²×C₅₃	1	4	yes	—
259	#390	C₂²×C₅₃	1	4	yes	—
260	#390	C₂²×C₅₃	1	4	yes	—
261	#390	C₂²×C₅₃	1	4	yes	—
262	#389	C₂²×C₅₃	1	4	yes	—
263	#389	C₂²×C₅₃	1	4	yes	—
264	#388	C₂²×C₅₃	1	8	yes	—
265	#388	C₂²×C₅₃	1	8	yes	—
266	#388	C₂²×C₅₃	1	8	yes	—
267	#385	C₂²×C₅₃	1	8	yes	—
268	#384	C₂²×C₅₃	1	1	yes	characteristic
269	#382	C₂²×C₅₃	1	2	yes	—
270	#382	C₂²×C₅₃	1	2	yes	—
271	#381	C₂²×C₅₃	1	4	yes	—
272	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
273	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
274	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
275	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
276	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
277	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
278	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
279	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
280	C₄³	C₂²×C₅₃	1	1	yes	characteristic
281	C₄³	C₂²×C₅₃	1	2	yes	—
282	C₅₃	#36609	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
284	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
285	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
286	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
287	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
288	C₄⋊Q₈	—	2	8	no	—
289	#127	—	2	4	no	—
290	#124	—	2	16	no	—
291	#124	—	4	8	no	—
292	C₄×Q₈	—	2	4	no	—
293	C₄×Q₈	—	2	8	no	—
294	C₄×Q₈	—	4	8	no	—
295	C₄×D₄	—	2	8	no	—
296	C₄×D₄	—	4	8	no	—
297	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
298	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
302	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
303	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
305	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
306	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
307	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
308	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
309	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
313	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
314	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
315	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
316	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
317	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
318	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
319	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
320	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
321	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
322	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
324	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
325	#95	C₂³×C₅₃	1	1	yes	characteristic
326	#95	C₂³×C₅₃	1	4	yes	—
327	#95	C₂³×C₅₃	1	1	yes	characteristic
328	#95	C₂³×C₅₃	1	2	yes	—
329	#95	C₂³×C₅₃	1	2	yes	—
330	#95	C₂³×C₅₃	1	8	yes	—
331	#95	C₂³×C₅₃	1	8	yes	—
332	#95	C₂³×C₅₃	1	4	yes	—
333	#95	C₂³×C₅₃	1	8	yes	—
334	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	4	no	—
337	C₂×Q₈	—	2	8	no	—
338	C₂×Q₈	—	2	8	no	—
339	C₂×Q₈	—	2	8	no	—
340	C₂×D₄	—	2	4	no	—
341	C₂×D₄	—	2	8	no	—
342	C₂×D₄	—	2	4	no	—
343	C₂×D₄	—	2	8	no	—
344	C₂×D₄	—	2	8	no	—
345	C₂×D₄	—	2	4	no	—
346	C₂²×C₄	—	2	8	no	—
347	C₂²×C₄	—	4	8	no	—
348	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
349	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
351	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
352	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
353	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
354	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
355	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
356	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
357	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
358	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
359	C₄⋊C₄	—	2	4	no	—
360	C₄⋊C₄	—	2	4	no	—
361	C₄⋊C₄	—	2	4	no	—
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	8	no	—
366	C₄⋊C₄	—	2	8	no	—
367	C₄⋊C₄	—	2	8	no	—
368	C₄⋊C₄	—	4	8	no	—
369	C₂²⋊C₄	—	2	8	no	—
370	C₂²⋊C₄	—	2	16	no	—
371	C₂²⋊C₄	—	2	16	no	—
372	C₂²⋊C₄	—	4	8	no	—
373	C₂²⋊C₄	—	4	8	no	—
374	C₂²⋊C₄	—	4	8	no	—
375	C₄²	—	2	4	no	—
376	C₄²	—	2	4	no	—
377	C₄²	—	2	4	no	—
378	C₄²	—	2	4	no	—
379	C₄²	—	2	4	no	—
380	C₄²	—	2	4	no	—
381	C₄²	—	2	4	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	4	8	no	—
384	C₄²	#41×C₅₃	1	4	yes	—
385	C₄²	#41×C₅₃	1	8	yes	—
386	C₄²	#41×C₅₃	1	4	yes	—
387	C₂³	—	2	8	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	2	4	no	—
390	C₂³	—	2	8	no	—
391	C₂³	—	2	4	no	—
392	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
393	Q₈	—	2	8	no	—
394	Q₈	—	4	16	no	—
395	D₄	—	4	8	no	—
396	D₄	—	4	16	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	8	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	8	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	8	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	8	8	no	—
418	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
419	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
420	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
421	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
422	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
423	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
424	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
425	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
426	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
427	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
428	C₂²	—	4	4	no	—
429	C₂²	—	4	8	no	—
430	C₂²	—	4	4	no	—
431	C₂²	—	4	8	no	—
432	C₂²	—	4	4	no	—
433	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
434	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
435	C₂²	C₂²×#41×C₅₃	1	2	yes	—
436	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
437	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
438	C₄	—	2	2	no	—
439	C₄	—	2	4	no	—
440	C₄	—	2	4	no	—
441	C₄	—	2	2	no	—
442	C₄	—	4	8	no	—
443	C₄	—	4	4	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	2	no	—
446	C₄	—	4	2	no	—
447	C₄	C₅₃×#556	1	4	yes	—
448	C₄	C₅₃×#559	1	8	yes	—
449	C₂	—	8	4	no	—
450	C₂	C₅₃×#3444	1	1	yes	characteristic
451	C₂	C₂×C₅₃×#556	1	1	yes	characteristic
452	C₂	C₂×C₅₃×#559	1	2	yes	—
453	C₂	C₅₃×#3508	1	1	yes	characteristic
454	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. C₅₃×#2862	C₂	semidirect
3. C₅₃×#2592	C₂	semidirect
4. C₅₃×#2580	C₂	semidirect
5. C₅₃×#2513	C₂	semidirect
6. C₅₃×#2448	C₂	semidirect
7. C₅₃×#2447	C₂	semidirect
8. C₅₃×#2367	C₂	semidirect
9. C₅₃×#2357	C₂	semidirect
10. C₄×C₄⋊Q₈×C₅₃	C₂	semidirect
12. C₄²×Q₈×C₅₃	C₂	semidirect
200. #36609	C₅₃	direct (N × H)
222. #2862	C₂×C₅₃	semidirect
224. #2592	C₂×C₅₃	semidirect
225. #2580	C₂×C₅₃	semidirect
226. #2513	C₂×C₅₃	semidirect
227. #2448	C₂×C₅₃	semidirect
228. #2447	C₂×C₅₃	semidirect
229. #2367	C₂×C₅₃	semidirect
230. #2357	C₂×C₅₃	semidirect
231. C₄×C₄⋊Q₈	C₂×C₅₃	semidirect
233. C₄²×Q₈	C₂×C₅₃	semidirect
282. C₅₃	#36609	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×#2862	2
C₅₃×#2858	2
C₅₃×#2592	2
C₅₃×#2580	2
C₅₃×#2513	2
C₅₃×#2448	2
C₅₃×#2447	2
C₅₃×#2367	2
C₅₃×#2357	2
C₄×C₄⋊Q₈×C₅₃	2
C₄×#127×C₅₃	2
C₄²×Q₈×C₅₃	2
#36609	53

Subgroups — C53×#36609 #27244121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36609 #27244121