Subgroups — C₅₃×#36597 #27244109

Order 13568.

2410 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2858	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2858	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2554	C₂	1	8	yes	maximal, complemented
6	C₅₃×#2546	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2545	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2544	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2541	C₂	1	1	yes	maximal, characteristic
10	C₅₃×#2426	C₂	1	4	yes	maximal, complemented
11	C₅₃×#2363	C₂	1	4	yes	maximal, complemented
12	C₄×#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#124×C₅₃	C₂	1	2	yes	maximal, complemented
14	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#559	—	2	8	no	—
16	C₅₃×#556	—	2	8	no	—
17	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#127×C₅₃	C₂²	1	2	yes	—
21	C₂×#126×C₅₃	C₂²	1	4	yes	—
22	C₂×#124×C₅₃	C₂²	1	2	yes	—
23	C₂×#124×C₅₃	C₂²	1	2	yes	—
24	C₂×#124×C₅₃	C₂²	1	2	yes	—
25	C₂×#124×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#124×C₅₃	C₂²	1	2	yes	—
27	C₂×#124×C₅₃	C₂²	1	2	yes	—
28	C₂×#123×C₅₃	C₂²	1	4	yes	—
29	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
30	C₂×#121×C₅₃	C₂²	1	4	yes	—
31	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
33	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
37	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
38	C₂×#117×C₅₃	C₂²	1	2	yes	—
39	C₅₃×#398	C₂²	1	4	yes	—
40	C₅₃×#396	C₂²	1	8	yes	—
41	C₅₃×#394	C₂²	1	8	yes	—
42	C₅₃×#392	C₂²	1	4	yes	—
43	C₅₃×#391	C₂²	1	2	yes	—
44	C₅₃×#391	C₂²	1	1	yes	characteristic
45	C₅₃×#390	C₂²	1	8	yes	—
46	C₅₃×#390	C₂²	1	4	yes	—
47	C₅₃×#390	C₂²	1	2	yes	—
48	C₅₃×#390	C₂²	1	1	yes	characteristic
49	C₅₃×#390	C₂²	1	4	yes	—
50	C₅₃×#390	C₂²	1	8	yes	—
51	C₅₃×#389	C₂²	1	4	yes	—
52	C₅₃×#388	C₂²	1	8	yes	—
53	C₅₃×#388	C₂²	1	8	yes	—
54	C₅₃×#388	C₂²	1	4	yes	—
55	C₅₃×#386	C₂²	1	8	yes	—
56	C₅₃×#382	C₂²	1	1	yes	characteristic
57	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
58	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
59	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
60	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
61	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
62	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
63	C₄³×C₅₃	C₂²	1	1	yes	characteristic
64	C₄³×C₅₃	C₂²	1	2	yes	—
65	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
67	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
69	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
71	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
72	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
73	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
74	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
75	C₄⋊Q₈×C₅₃	—	2	4	no	—
76	C₄⋊Q₈×C₅₃	—	4	4	no	—
77	#127×C₅₃	—	2	4	no	—
78	#127×C₅₃	—	4	4	no	—
79	#126×C₅₃	—	2	16	no	—
80	#124×C₅₃	—	2	8	no	—
81	#124×C₅₃	—	2	8	no	—
82	#124×C₅₃	—	2	4	no	—
83	#124×C₅₃	—	2	8	no	—
84	#124×C₅₃	—	2	8	no	—
85	#124×C₅₃	—	4	4	no	—
86	#123×C₅₃	—	2	16	no	—
87	C₂²⋊Q₈×C₅₃	—	2	8	no	—
88	#121×C₅₃	—	2	16	no	—
89	C₄×Q₈×C₅₃	—	2	8	no	—
90	C₄×Q₈×C₅₃	—	2	8	no	—
91	C₄×Q₈×C₅₃	—	2	4	no	—
92	C₄×Q₈×C₅₃	D₄	1	8	yes	complemented
93	C₄×D₄×C₅₃	—	2	8	no	—
94	C₄×D₄×C₅₃	—	2	4	no	—
95	C₄×D₄×C₅₃	—	2	8	no	—
96	#117×C₅₃	—	2	8	no	—
97	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
105	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
113	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
117	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
118	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
119	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
120	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
121	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
122	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
123	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
125	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
126	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
127	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
128	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
130	#95×C₅₃	C₂³	1	4	yes	—
131	#95×C₅₃	C₂³	1	8	yes	—
132	#95×C₅₃	C₂³	1	4	yes	—
133	#95×C₅₃	C₂³	1	4	yes	—
134	#95×C₅₃	C₂³	1	2	yes	—
135	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
136	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
137	C₂×Q₈×C₅₃	—	2	2	no	—
138	C₂×Q₈×C₅₃	—	2	4	no	—
139	C₂×Q₈×C₅₃	—	2	4	no	—
140	C₂×Q₈×C₅₃	—	2	2	no	—
141	C₂×Q₈×C₅₃	—	2	4	no	—
142	C₂×Q₈×C₅₃	—	2	4	no	—
143	C₂×Q₈×C₅₃	—	2	2	no	—
144	C₂×Q₈×C₅₃	C₂×D₄	1	4	yes	—
145	C₂×D₄×C₅₃	—	2	2	no	—
146	C₂×D₄×C₅₃	—	2	4	no	—
147	C₂×D₄×C₅₃	—	2	4	no	—
148	C₂×D₄×C₅₃	—	2	4	no	—
149	C₂×D₄×C₅₃	—	2	8	no	—
150	C₂×D₄×C₅₃	—	2	2	no	—
151	C₂×D₄×C₅₃	—	2	4	no	—
152	C₂×D₄×C₅₃	—	2	16	no	—
153	C₂×D₄×C₅₃	—	2	4	no	—
154	C₂×D₄×C₅₃	—	2	2	no	—
155	C₂×D₄×C₅₃	—	4	8	no	—
156	C₂×D₄×C₅₃	—	4	4	no	—
157	C₂×D₄×C₅₃	—	4	8	no	—
158	C₂²×C₄×C₅₃	—	2	4	no	—
159	C₂²×C₄×C₅₃	—	2	8	no	—
160	C₂²×C₄×C₅₃	—	2	8	no	—
161	C₂²×C₄×C₅₃	—	4	8	no	—
162	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
174	C₄⋊C₄×C₅₃	—	2	4	no	—
175	C₄⋊C₄×C₅₃	—	2	2	no	—
176	C₄⋊C₄×C₅₃	—	2	2	no	—
177	C₄⋊C₄×C₅₃	—	2	8	no	—
178	C₄⋊C₄×C₅₃	—	2	4	no	—
179	C₄⋊C₄×C₅₃	—	2	4	no	—
180	C₄⋊C₄×C₅₃	—	2	8	no	—
181	C₄⋊C₄×C₅₃	—	4	8	no	—
182	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
183	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
184	C₄⋊C₄×C₅₃	#41	1	8	yes	—
185	C₂²⋊C₄×C₅₃	—	2	8	no	—
186	C₂²⋊C₄×C₅₃	—	2	8	no	—
187	C₂²⋊C₄×C₅₃	—	2	8	no	—
188	C₂²⋊C₄×C₅₃	—	2	4	no	—
189	C₂²⋊C₄×C₅₃	—	2	8	no	—
190	C₂²⋊C₄×C₅₃	—	2	8	no	—
191	C₂²⋊C₄×C₅₃	—	2	16	no	—
192	C₂²⋊C₄×C₅₃	—	2	8	no	—
193	C₂²⋊C₄×C₅₃	—	2	16	no	—
194	C₂²⋊C₄×C₅₃	—	4	4	no	—
195	C₂²⋊C₄×C₅₃	—	4	8	no	—
196	C₄²×C₅₃	—	2	8	no	—
197	C₄²×C₅₃	—	2	4	no	—
198	C₄²×C₅₃	—	2	8	no	—
199	C₄²×C₅₃	—	2	4	no	—
200	C₄²×C₅₃	—	2	2	no	—
201	C₄²×C₅₃	—	2	4	no	—
202	C₄²×C₅₃	—	4	4	no	—
203	C₄²×C₅₃	—	4	2	no	—
204	C₄²×C₅₃	C₂×D₄	1	8	yes	—
205	C₄²×C₅₃	C₂×D₄	1	4	yes	—
206	C₄²×C₅₃	#41	1	4	yes	—
207	C₂³×C₅₃	—	2	4	no	—
208	C₂³×C₅₃	—	2	2	no	—
209	C₂³×C₅₃	—	2	4	no	—
210	C₂³×C₅₃	—	2	8	no	—
211	C₂³×C₅₃	—	2	4	no	—
212	C₂³×C₅₃	—	2	2	no	—
213	C₂³×C₅₃	—	2	2	no	—
214	C₂³×C₅₃	—	2	4	no	—
215	C₂³×C₅₃	—	2	8	no	—
216	C₂³×C₅₃	—	2	4	no	—
217	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
218	Q₈×C₅₃	—	2	8	no	—
219	Q₈×C₅₃	—	4	4	no	—
220	Q₈×C₅₃	—	4	8	no	—
221	D₄×C₅₃	—	4	8	no	—
222	D₄×C₅₃	—	4	4	no	—
223	D₄×C₅₃	—	4	8	no	—
224	D₄×C₅₃	—	8	4	no	—
225	D₄×C₅₃	—	8	16	no	—
226	C₂×C₄×C₅₃	—	2	4	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	4	no	—
229	C₂×C₄×C₅₃	—	2	4	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	4	no	—
232	C₂×C₄×C₅₃	—	2	2	no	—
233	C₂×C₄×C₅₃	—	2	2	no	—
234	C₂×C₄×C₅₃	—	2	2	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	4	no	—
237	C₂×C₄×C₅₃	—	2	4	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	4	no	—
240	C₂×C₄×C₅₃	—	2	2	no	—
241	C₂×C₄×C₅₃	—	2	4	no	—
242	C₂×C₄×C₅₃	—	2	2	no	—
243	C₂×C₄×C₅₃	—	2	2	no	—
244	C₂×C₄×C₅₃	—	2	4	no	—
245	C₂×C₄×C₅₃	—	2	2	no	—
246	C₂×C₄×C₅₃	—	4	8	no	—
247	C₂×C₄×C₅₃	—	8	8	no	—
248	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
249	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
251	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
254	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
255	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
256	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
257	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
258	#36597	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂²×C₅₃	—	4	2	no	—
260	C₂²×C₅₃	—	4	4	no	—
261	C₂²×C₅₃	—	4	4	no	—
262	C₂²×C₅₃	—	4	2	no	—
263	C₂²×C₅₃	—	4	4	no	—
264	C₂²×C₅₃	—	4	8	no	—
265	C₂²×C₅₃	—	4	4	no	—
266	C₂²×C₅₃	—	4	2	no	—
267	C₂²×C₅₃	—	4	4	no	—
268	C₂²×C₅₃	—	4	8	no	—
269	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
270	C₂²×C₅₃	C₂²×#41	1	2	yes	—
271	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
272	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
273	C₂²×C₅₃	#585	1	2	yes	—
274	C₄×C₅₃	—	2	4	no	—
275	C₄×C₅₃	—	2	4	no	—
276	C₄×C₅₃	—	2	4	no	—
277	C₄×C₅₃	—	2	2	no	—
278	C₄×C₅₃	—	2	8	no	—
279	C₄×C₅₃	—	4	2	no	—
280	C₄×C₅₃	—	4	2	no	—
281	C₄×C₅₃	—	4	4	no	—
282	C₄×C₅₃	—	4	2	no	—
283	C₄×C₅₃	—	4	2	no	—
284	C₄×C₅₃	#532	1	4	yes	—
285	C₄×C₅₃	#550	1	4	yes	—
286	C₂×#559	C₂×C₅₃	1	1	yes	characteristic, complemented
287	C₂×#556	C₂×C₅₃	1	1	yes	characteristic, complemented
288	#2858	C₂×C₅₃	1	2	yes	complemented
289	#2858	C₂×C₅₃	1	1	yes	characteristic, complemented
290	#2554	C₂×C₅₃	1	8	yes	complemented
291	#2546	C₂×C₅₃	1	1	yes	characteristic, complemented
292	#2545	C₂×C₅₃	1	2	yes	complemented
293	#2544	C₂×C₅₃	1	2	yes	complemented
294	#2541	C₂×C₅₃	1	1	yes	characteristic
295	#2426	C₂×C₅₃	1	4	yes	complemented
296	#2363	C₂×C₅₃	1	4	yes	complemented
297	C₄×#124	C₂×C₅₃	1	1	yes	characteristic, complemented
298	C₄×#124	C₂×C₅₃	1	2	yes	complemented
299	C₄²×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
300	C₂×C₅₃	—	8	2	no	—
301	C₂×C₅₃	—	8	4	no	—
302	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
303	C₂×C₅₃	C₂×#550	1	1	yes	characteristic
304	C₂×C₅₃	#3480	1	2	yes	—
305	C₂×C₅₃	#3500	1	1	yes	characteristic
306	C₂×C₅₃	#3514	1	2	yes	—
307	#559	—	2	8	no	—
308	#556	—	2	8	no	—
309	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
310	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
311	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
312	C₂×#127	C₂²×C₅₃	1	2	yes	—
313	C₂×#126	C₂²×C₅₃	1	4	yes	—
314	C₂×#124	C₂²×C₅₃	1	1	yes	characteristic
315	C₂×#124	C₂²×C₅₃	1	2	yes	—
316	C₂×#124	C₂²×C₅₃	1	2	yes	—
317	C₂×#124	C₂²×C₅₃	1	2	yes	—
318	C₂×#124	C₂²×C₅₃	1	2	yes	—
319	C₂×#124	C₂²×C₅₃	1	2	yes	—
320	C₂×#123	C₂²×C₅₃	1	4	yes	—
321	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
322	C₂×#121	C₂²×C₅₃	1	4	yes	—
323	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
324	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
325	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
326	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
327	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
328	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
329	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
330	C₂×#117	C₂²×C₅₃	1	2	yes	—
331	#398	C₂²×C₅₃	1	4	yes	—
332	#396	C₂²×C₅₃	1	8	yes	—
333	#394	C₂²×C₅₃	1	8	yes	—
334	#392	C₂²×C₅₃	1	4	yes	—
335	#391	C₂²×C₅₃	1	2	yes	—
336	#391	C₂²×C₅₃	1	1	yes	characteristic
337	#390	C₂²×C₅₃	1	4	yes	—
338	#390	C₂²×C₅₃	1	8	yes	—
339	#390	C₂²×C₅₃	1	8	yes	—
340	#390	C₂²×C₅₃	1	4	yes	—
341	#390	C₂²×C₅₃	1	2	yes	—
342	#390	C₂²×C₅₃	1	1	yes	characteristic
343	#389	C₂²×C₅₃	1	4	yes	—
344	#388	C₂²×C₅₃	1	8	yes	—
345	#388	C₂²×C₅₃	1	8	yes	—
346	#388	C₂²×C₅₃	1	4	yes	—
347	#386	C₂²×C₅₃	1	8	yes	—
348	#382	C₂²×C₅₃	1	1	yes	characteristic
349	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
350	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
351	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
352	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
353	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
354	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
355	C₄³	C₂²×C₅₃	1	1	yes	characteristic
356	C₄³	C₂²×C₅₃	1	2	yes	—
357	C₅₃	#36597	1	1	yes	characteristic, direct product factor, Sylow
358	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
359	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
360	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
361	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
362	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
363	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
364	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
365	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
366	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
367	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
368	C₄⋊Q₈	—	2	4	no	—
369	C₄⋊Q₈	—	4	4	no	—
370	#127	—	2	4	no	—
371	#127	—	4	4	no	—
372	#126	—	2	16	no	—
373	#124	—	2	8	no	—
374	#124	—	2	8	no	—
375	#124	—	2	8	no	—
376	#124	—	2	4	no	—
377	#124	—	2	8	no	—
378	#124	—	4	4	no	—
379	#123	—	2	16	no	—
380	C₂²⋊Q₈	—	2	8	no	—
381	#121	—	2	16	no	—
382	C₄×Q₈	—	2	8	no	—
383	C₄×Q₈	—	2	8	no	—
384	C₄×Q₈	—	2	4	no	—
385	C₄×Q₈	D₄×C₅₃	1	8	yes	complemented
386	C₄×D₄	—	2	8	no	—
387	C₄×D₄	—	2	4	no	—
388	C₄×D₄	—	2	8	no	—
389	#117	—	2	8	no	—
390	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
391	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
394	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
396	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
397	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
398	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
412	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
413	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
414	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
415	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
416	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
418	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
419	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
420	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
421	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
423	#95	C₂³×C₅₃	1	4	yes	—
424	#95	C₂³×C₅₃	1	8	yes	—
425	#95	C₂³×C₅₃	1	4	yes	—
426	#95	C₂³×C₅₃	1	4	yes	—
427	#95	C₂³×C₅₃	1	2	yes	—
428	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
429	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
430	C₂×Q₈	—	2	2	no	—
431	C₂×Q₈	—	2	4	no	—
432	C₂×Q₈	—	2	4	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	—	2	4	no	—
435	C₂×Q₈	—	2	4	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	C₂×D₄×C₅₃	1	4	yes	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	8	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	16	no	—
446	C₂×D₄	—	2	4	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	4	8	no	—
449	C₂×D₄	—	4	8	no	—
450	C₂×D₄	—	4	4	no	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	8	no	—
453	C₂²×C₄	—	2	8	no	—
454	C₂²×C₄	—	4	8	no	—
455	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
456	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
459	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
460	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
462	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
463	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
464	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
467	C₄⋊C₄	—	2	4	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	2	8	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	8	no	—
474	C₄⋊C₄	—	4	8	no	—
475	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
476	C₄⋊C₄	C₂×D₄×C₅₃	1	8	yes	—
477	C₄⋊C₄	#41×C₅₃	1	8	yes	—
478	C₂²⋊C₄	—	2	8	no	—
479	C₂²⋊C₄	—	2	8	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	2	16	no	—
485	C₂²⋊C₄	—	2	8	no	—
486	C₂²⋊C₄	—	2	16	no	—
487	C₂²⋊C₄	—	4	8	no	—
488	C₂²⋊C₄	—	4	4	no	—
489	C₄²	—	2	8	no	—
490	C₄²	—	2	4	no	—
491	C₄²	—	2	8	no	—
492	C₄²	—	2	4	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	2	4	no	—
495	C₄²	—	4	4	no	—
496	C₄²	—	4	2	no	—
497	C₄²	C₂×D₄×C₅₃	1	8	yes	—
498	C₄²	C₂×D₄×C₅₃	1	4	yes	—
499	C₄²	#41×C₅₃	1	4	yes	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	4	no	—
502	C₂³	—	2	2	no	—
503	C₂³	—	2	4	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	2	4	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	2	4	no	—
509	C₂³	—	2	8	no	—
510	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
511	Q₈	—	2	8	no	—
512	Q₈	—	4	4	no	—
513	Q₈	—	4	8	no	—
514	D₄	—	4	8	no	—
515	D₄	—	4	4	no	—
516	D₄	—	4	8	no	—
517	D₄	—	8	4	no	—
518	D₄	—	8	16	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	4	8	no	—
540	C₂×C₄	—	8	8	no	—
541	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
542	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
543	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
544	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
545	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
546	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
547	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
548	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
549	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
550	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	4	no	—
553	C₂²	—	4	8	no	—
554	C₂²	—	4	4	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	4	2	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	8	no	—
559	C₂²	—	4	2	no	—
560	C₂²	—	4	4	no	—
561	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
562	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
563	C₂²	C₂²×#41×C₅₃	1	2	yes	—
564	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
565	C₂²	C₅₃×#585	1	2	yes	—
566	C₄	—	2	4	no	—
567	C₄	—	2	4	no	—
568	C₄	—	2	2	no	—
569	C₄	—	2	8	no	—
570	C₄	—	2	4	no	—
571	C₄	—	4	2	no	—
572	C₄	—	4	4	no	—
573	C₄	—	4	2	no	—
574	C₄	—	4	2	no	—
575	C₄	—	4	2	no	—
576	C₄	C₅₃×#532	1	4	yes	—
577	C₄	C₅₃×#550	1	4	yes	—
578	C₂	—	8	4	no	—
579	C₂	—	8	2	no	—
580	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
581	C₂	C₂×C₅₃×#550	1	1	yes	characteristic
582	C₂	C₅₃×#3480	1	2	yes	—
583	C₂	C₅₃×#3500	1	1	yes	characteristic
584	C₂	C₅₃×#3514	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₅₃×#2541	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
1. C₂×C₅₃×#559	C₂	semidirect
2. C₂×C₅₃×#556	C₂	semidirect
3. C₅₃×#2858	C₂	semidirect
4. C₅₃×#2858	C₂	semidirect
5. C₅₃×#2554	C₂	semidirect
6. C₅₃×#2546	C₂	semidirect
7. C₅₃×#2545	C₂	semidirect
8. C₅₃×#2544	C₂	semidirect
10. C₅₃×#2426	C₂	semidirect
11. C₅₃×#2363	C₂	semidirect
12. C₄×#124×C₅₃	C₂	semidirect
13. C₄×#124×C₅₃	C₂	semidirect
14. C₄²×Q₈×C₅₃	C₂	semidirect
92. C₄×Q₈×C₅₃	D₄	semidirect
258. #36597	C₅₃	direct (N × H)
286. C₂×#559	C₂×C₅₃	semidirect
287. C₂×#556	C₂×C₅₃	semidirect
288. #2858	C₂×C₅₃	semidirect
289. #2858	C₂×C₅₃	semidirect
290. #2554	C₂×C₅₃	semidirect
291. #2546	C₂×C₅₃	semidirect
292. #2545	C₂×C₅₃	semidirect
293. #2544	C₂×C₅₃	semidirect
295. #2426	C₂×C₅₃	semidirect
296. #2363	C₂×C₅₃	semidirect
297. C₄×#124	C₂×C₅₃	semidirect
298. C₄×#124	C₂×C₅₃	semidirect
299. C₄²×Q₈	C₂×C₅₃	semidirect
357. C₅₃	#36597	direct (N × H)
385. C₄×Q₈	D₄×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×#559	2
C₂×C₅₃×#556	2
C₅₃×#2858	2
C₅₃×#2554	2
C₅₃×#2546	2
C₅₃×#2545	2
C₅₃×#2544	2
C₅₃×#2541	2
C₅₃×#2426	2
C₅₃×#2363	2
C₄×#124×C₅₃	2
C₄²×Q₈×C₅₃	2
#36597	53

Subgroups — C53×#36597 #27244109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#36597 #27244109