Subgroups — C₅₃×#35032 #27242544

Order 13568.

2778 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#525	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2857	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2856	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2850	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2617	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2602	C₂	1	8	yes	maximal, complemented
7	C₅₃×#2500	C₂	1	2	yes	maximal, complemented
8	C₅₃×#2499	C₂	1	2	yes	maximal, complemented
9	C₅₃×#2432	C₂	1	2	yes	maximal, complemented
10	C₅₃×#2431	C₂	1	2	yes	maximal, complemented
11	C₅₃×#2325	C₂	1	4	yes	maximal, complemented
12	C₂×C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₅₃	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
16	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic, complemented
17	C₂×#124×C₅₃	C₂²	1	2	yes	complemented
18	C₅₃×#525	—	2	16	no	—
19	C₂×#123×C₅₃	C₂²	1	4	yes	—
20	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
22	C₂×#121×C₅₃	C₂²	1	2	yes	—
23	C₂×#121×C₅₃	C₂²	1	4	yes	—
24	C₂×#121×C₅₃	C₂²	1	2	yes	—
25	C₂×#120×C₅₃	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
27	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
28	C₂×#117×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#397	C₂²	1	8	yes	—
33	C₅₃×#394	C₂²	1	8	yes	—
34	C₅₃×#394	C₂²	1	8	yes	—
35	C₅₃×#392	C₂²	1	8	yes	—
36	C₅₃×#391	C₂²	1	2	yes	complemented
37	C₅₃×#391	C₂²	1	2	yes	complemented
38	C₅₃×#390	C₂²	1	2	yes	complemented
39	C₅₃×#390	C₂²	1	2	yes	complemented
40	C₅₃×#390	C₂²	1	4	yes	complemented
41	C₅₃×#389	C₂²	1	4	yes	complemented
42	C₅₃×#388	C₂²	1	8	yes	complemented
43	C₅₃×#388	C₂²	1	8	yes	complemented
44	C₅₃×#387	C₂²	1	8	yes	complemented
45	C₅₃×#386	C₂²	1	8	yes	—
46	C₅₃×#386	C₂²	1	8	yes	complemented
47	C₅₃×#385	C₂²	1	8	yes	complemented
48	C₅₃×#384	C₂²	1	1	yes	characteristic, complemented
49	C₅₃×#384	C₂²	1	1	yes	characteristic, complemented
50	C₅₃×#382	C₂²	1	2	yes	complemented
51	C₅₃×#380	C₂²	1	4	yes	—
52	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
53	C₄×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
54	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
55	C₂×#95×C₅₃	C₂²	1	2	yes	—
56	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×#41×C₅₃	—	2	8	no	—
58	C₂×#41×C₅₃	—	4	4	no	—
59	C₂×#41×C₅₃	—	4	4	no	—
60	C₂²×Q₈×C₅₃	C₂³	1	2	yes	—
61	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
62	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
63	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
64	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
65	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
66	C₂³×C₄×C₅₃	—	4	2	no	—
67	C₂³×C₄×C₅₃	—	4	2	no	—
68	C₂³×C₄×C₅₃	D₄	1	8	yes	complemented
69	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
71	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
72	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
73	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
75	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
76	C₄⋊Q₈×C₅₃	—	4	2	no	—
77	#127×C₅₃	—	4	2	no	—
78	#124×C₅₃	—	4	4	no	—
79	#123×C₅₃	—	2	16	no	—
80	C₂²⋊Q₈×C₅₃	—	2	8	no	—
81	C₂²⋊Q₈×C₅₃	—	2	8	no	—
82	#121×C₅₃	—	2	8	no	—
83	#121×C₅₃	—	2	8	no	—
84	#121×C₅₃	—	2	16	no	—
85	#120×C₅₃	—	2	16	no	—
86	C₄×D₄×C₅₃	—	2	16	no	—
87	C₄×D₄×C₅₃	—	2	16	no	—
88	#117×C₅₃	—	2	8	no	—
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
97	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
105	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
107	C₂×C₄²×C₅₃	—	4	2	no	—
108	C₂×C₄²×C₅₃	—	4	2	no	—
109	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
110	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
113	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
114	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
115	#95×C₅₃	C₂³	1	4	yes	—
116	#95×C₅₃	C₂³	1	4	yes	—
117	#95×C₅₃	C₂³	1	8	yes	—
118	#95×C₅₃	C₂³	1	8	yes	—
119	#95×C₅₃	C₂³	1	8	yes	—
120	C₂⁴×C₅₃	—	2	4	no	—
121	C₂⁴×C₅₃	—	4	1	no	—
122	C₂⁴×C₅₃	—	4	1	no	—
123	C₂⁴×C₅₃	—	4	2	no	—
124	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
125	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
126	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
127	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
128	#41×C₅₃	—	4	32	no	—
129	C₂×Q₈×C₅₃	—	2	4	no	—
130	C₂×Q₈×C₅₃	—	2	4	no	—
131	C₂×Q₈×C₅₃	—	2	4	no	—
132	C₂×D₄×C₅₃	—	2	8	no	—
133	C₂×D₄×C₅₃	—	2	8	no	—
134	C₂×D₄×C₅₃	—	2	8	no	—
135	C₂×D₄×C₅₃	—	2	16	no	—
136	C₂×D₄×C₅₃	—	2	4	no	—
137	C₂×D₄×C₅₃	—	2	8	no	—
138	C₂×D₄×C₅₃	—	2	4	no	—
139	C₂×D₄×C₅₃	—	2	4	no	—
140	C₂×D₄×C₅₃	—	4	8	no	—
141	C₂×D₄×C₅₃	—	4	4	no	—
142	C₂×D₄×C₅₃	—	4	4	no	—
143	C₂×D₄×C₅₃	—	4	4	no	—
144	C₂×D₄×C₅₃	—	4	4	no	—
145	C₂²×C₄×C₅₃	—	2	2	no	—
146	C₂²×C₄×C₅₃	—	2	2	no	—
147	C₂²×C₄×C₅₃	—	2	4	no	—
148	C₂²×C₄×C₅₃	—	2	4	no	—
149	C₂²×C₄×C₅₃	—	2	4	no	—
150	C₂²×C₄×C₅₃	—	2	8	no	—
151	C₂²×C₄×C₅₃	—	4	16	no	—
152	C₂²×C₄×C₅₃	—	4	2	no	—
153	C₂²×C₄×C₅₃	—	4	2	no	—
154	C₂²×C₄×C₅₃	—	4	1	no	—
155	C₂²×C₄×C₅₃	—	4	1	no	—
156	C₂²×C₄×C₅₃	—	4	2	no	—
157	C₂²×C₄×C₅₃	—	4	2	no	—
158	C₂²×C₄×C₅₃	—	4	2	no	—
159	C₂²×C₄×C₅₃	—	4	2	no	—
160	C₂²×C₄×C₅₃	—	4	4	no	—
161	C₂²×C₄×C₅₃	—	4	4	no	—
162	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
163	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
164	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
165	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
166	C₂²×C₄×C₅₃	#41	1	8	yes	complemented
167	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₅₃	C₂⁴	1	4	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	2	yes	—
174	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
176	C₄⋊C₄×C₅₃	—	2	4	no	—
177	C₄⋊C₄×C₅₃	—	2	4	no	—
178	C₄⋊C₄×C₅₃	—	2	8	no	—
179	C₄⋊C₄×C₅₃	—	2	8	no	—
180	C₄⋊C₄×C₅₃	—	2	8	no	—
181	C₄⋊C₄×C₅₃	—	4	4	no	—
182	C₄⋊C₄×C₅₃	—	4	4	no	—
183	C₂²⋊C₄×C₅₃	—	2	8	no	—
184	C₂²⋊C₄×C₅₃	—	2	8	no	—
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	8	no	—
189	C₂²⋊C₄×C₅₃	—	2	8	no	—
190	C₂²⋊C₄×C₅₃	—	2	16	no	—
191	C₂²⋊C₄×C₅₃	—	4	8	no	—
192	C₂²⋊C₄×C₅₃	—	4	8	no	—
193	C₄²×C₅₃	—	2	8	no	—
194	C₄²×C₅₃	—	2	4	no	—
195	C₄²×C₅₃	—	2	2	no	—
196	C₄²×C₅₃	—	2	2	no	—
197	C₄²×C₅₃	—	4	2	no	—
198	C₄²×C₅₃	—	4	2	no	—
199	C₂³×C₅₃	—	2	1	no	—
200	C₂³×C₅₃	—	2	1	no	—
201	C₂³×C₅₃	—	2	2	no	—
202	C₂³×C₅₃	—	2	8	no	—
203	C₂³×C₅₃	—	2	4	no	—
204	C₂³×C₅₃	—	2	2	no	—
205	C₂³×C₅₃	—	2	2	no	—
206	C₂³×C₅₃	—	2	4	no	—
207	C₂³×C₅₃	—	2	4	no	—
208	C₂³×C₅₃	—	2	8	no	—
209	C₂³×C₅₃	—	2	4	no	—
210	C₂³×C₅₃	—	2	4	no	—
211	C₂³×C₅₃	—	2	8	no	—
212	C₂³×C₅₃	—	4	4	no	—
213	C₂³×C₅₃	—	4	4	no	—
214	C₂³×C₅₃	—	4	8	no	—
215	C₂³×C₅₃	—	4	8	no	—
216	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
217	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
218	C₂³×C₅₃	C₂×#41	1	4	yes	—
219	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
220	Q₈×C₅₃	—	4	8	no	—
221	D₄×C₅₃	—	4	16	no	—
222	D₄×C₅₃	—	4	16	no	—
223	D₄×C₅₃	—	4	16	no	—
224	D₄×C₅₃	—	4	8	no	—
225	D₄×C₅₃	—	8	8	no	—
226	C₂×C₄×C₅₃	—	2	2	no	—
227	C₂×C₄×C₅₃	—	2	2	no	—
228	C₂×C₄×C₅₃	—	2	1	no	—
229	C₂×C₄×C₅₃	—	2	1	no	—
230	C₂×C₄×C₅₃	—	2	2	no	—
231	C₂×C₄×C₅₃	—	2	2	no	—
232	C₂×C₄×C₅₃	—	2	2	no	—
233	C₂×C₄×C₅₃	—	2	2	no	—
234	C₂×C₄×C₅₃	—	2	8	no	—
235	C₂×C₄×C₅₃	—	2	16	no	—
236	C₂×C₄×C₅₃	—	2	4	no	—
237	C₂×C₄×C₅₃	—	2	4	no	—
238	C₂×C₄×C₅₃	—	2	4	no	—
239	C₂×C₄×C₅₃	—	2	4	no	—
240	C₂×C₄×C₅₃	—	2	8	no	—
241	C₂×C₄×C₅₃	—	2	4	no	—
242	C₂×C₄×C₅₃	—	2	4	no	—
243	C₂×C₄×C₅₃	—	2	4	no	—
244	C₂×C₄×C₅₃	—	2	4	no	—
245	C₂×C₄×C₅₃	—	2	8	no	—
246	C₂×C₄×C₅₃	—	4	4	no	—
247	C₂×C₄×C₅₃	—	4	8	no	—
248	C₂×C₄×C₅₃	—	4	8	no	—
249	C₂×C₄×C₅₃	—	4	16	no	—
250	C₂×C₄×C₅₃	—	4	8	no	—
251	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
252	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
255	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
256	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
257	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
258	#35032	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
259	C₂²×C₅₃	—	2	2	no	—
260	C₂²×C₅₃	—	2	2	no	—
261	C₂²×C₅₃	—	2	4	no	—
262	C₂²×C₅₃	—	2	4	no	—
263	C₂²×C₅₃	—	2	4	no	—
264	C₂²×C₅₃	—	2	4	no	—
265	C₂²×C₅₃	—	2	4	no	—
266	C₂²×C₅₃	—	4	16	no	—
267	C₂²×C₅₃	—	4	4	no	—
268	C₂²×C₅₃	—	4	4	no	—
269	C₂²×C₅₃	—	4	4	no	—
270	C₂²×C₅₃	—	4	4	no	—
271	C₂²×C₅₃	—	4	4	no	—
272	C₂²×C₅₃	—	4	8	no	—
273	C₂²×C₅₃	—	4	8	no	—
274	C₂²×C₅₃	#532	1	4	yes	—
275	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
276	C₂²×C₅₃	C₂²×#41	1	2	yes	—
277	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
278	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
279	C₂²×C₅₃	#585	1	2	yes	—
280	C₄×C₅₃	—	2	4	no	—
281	C₄×C₅₃	—	2	2	no	—
282	C₄×C₅₃	—	2	4	no	—
283	C₄×C₅₃	—	2	4	no	—
284	C₄×C₅₃	—	4	4	no	—
285	C₄×C₅₃	—	4	4	no	—
286	C₄×C₅₃	—	4	4	no	—
287	C₄×C₅₃	—	4	4	no	—
288	C₄×C₅₃	#534	1	4	yes	—
289	C₂×#525	C₂×C₅₃	1	2	yes	complemented
290	#2857	C₂×C₅₃	1	1	yes	characteristic, complemented
291	#2856	C₂×C₅₃	1	1	yes	characteristic, complemented
292	#2850	C₂×C₅₃	1	2	yes	complemented
293	#2617	C₂×C₅₃	1	4	yes	complemented
294	#2602	C₂×C₅₃	1	8	yes	complemented
295	#2500	C₂×C₅₃	1	2	yes	complemented
296	#2499	C₂×C₅₃	1	2	yes	complemented
297	#2432	C₂×C₅₃	1	2	yes	complemented
298	#2431	C₂×C₅₃	1	2	yes	complemented
299	#2325	C₂×C₅₃	1	4	yes	complemented
300	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
301	C₂×C₅₃	—	2	4	no	—
302	C₂×C₅₃	—	4	4	no	—
303	C₂×C₅₃	—	8	4	no	—
304	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
305	C₂×C₅₃	C₂×#534	1	1	yes	characteristic
306	C₂×C₅₃	#3449	1	1	yes	characteristic
307	C₂×C₅₃	#3505	1	2	yes	—
308	C₂×C₅₃	#3506	1	2	yes	—
309	C₂²×#41	C₂²×C₅₃	1	2	yes	—
310	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
311	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
312	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic, complemented
313	C₂×#124	C₂²×C₅₃	1	2	yes	complemented
314	#525	—	2	16	no	—
315	C₂×#123	C₂²×C₅₃	1	4	yes	—
316	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
317	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
318	C₂×#121	C₂²×C₅₃	1	4	yes	—
319	C₂×#121	C₂²×C₅₃	1	2	yes	—
320	C₂×#121	C₂²×C₅₃	1	2	yes	—
321	C₂×#120	C₂²×C₅₃	1	4	yes	—
322	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
323	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
324	C₂×#117	C₂²×C₅₃	1	2	yes	—
325	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
326	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
327	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
328	#397	C₂²×C₅₃	1	8	yes	—
329	#394	C₂²×C₅₃	1	8	yes	—
330	#394	C₂²×C₅₃	1	8	yes	—
331	#392	C₂²×C₅₃	1	8	yes	—
332	#391	C₂²×C₅₃	1	2	yes	complemented
333	#391	C₂²×C₅₃	1	2	yes	complemented
334	#390	C₂²×C₅₃	1	2	yes	complemented
335	#390	C₂²×C₅₃	1	4	yes	complemented
336	#390	C₂²×C₅₃	1	2	yes	complemented
337	#389	C₂²×C₅₃	1	4	yes	complemented
338	#388	C₂²×C₅₃	1	8	yes	complemented
339	#388	C₂²×C₅₃	1	8	yes	complemented
340	#387	C₂²×C₅₃	1	8	yes	complemented
341	#386	C₂²×C₅₃	1	8	yes	—
342	#386	C₂²×C₅₃	1	8	yes	complemented
343	#385	C₂²×C₅₃	1	8	yes	complemented
344	#384	C₂²×C₅₃	1	1	yes	characteristic, complemented
345	#384	C₂²×C₅₃	1	1	yes	characteristic, complemented
346	#382	C₂²×C₅₃	1	2	yes	complemented
347	#380	C₂²×C₅₃	1	4	yes	—
348	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
349	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
350	C₄×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
351	C₂×#95	C₂²×C₅₃	1	2	yes	—
352	C₅₃	#35032	1	1	yes	characteristic, direct product factor, Sylow
353	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
354	C₂×#41	—	2	8	no	—
355	C₂×#41	—	4	4	no	—
356	C₂×#41	—	4	4	no	—
357	C₂²×Q₈	C₂³×C₅₃	1	2	yes	—
358	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
359	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
360	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
361	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
362	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
363	C₂³×C₄	—	4	2	no	—
364	C₂³×C₄	—	4	2	no	—
365	C₂³×C₄	D₄×C₅₃	1	8	yes	complemented
366	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
367	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
368	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
369	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
370	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
371	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
372	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
373	C₄⋊Q₈	—	4	2	no	—
374	#127	—	4	2	no	—
375	#124	—	4	4	no	—
376	#123	—	2	16	no	—
377	C₂²⋊Q₈	—	2	8	no	—
378	C₂²⋊Q₈	—	2	8	no	—
379	#121	—	2	8	no	—
380	#121	—	2	8	no	—
381	#121	—	2	16	no	—
382	#120	—	2	16	no	—
383	C₄×D₄	—	2	16	no	—
384	C₄×D₄	—	2	16	no	—
385	#117	—	2	8	no	—
386	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
387	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
388	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
389	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
392	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
393	C₂×C₂²⋊C₄	—	2	8	no	—
394	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
402	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
403	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
404	C₂×C₄²	—	4	2	no	—
405	C₂×C₄²	—	4	2	no	—
406	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
407	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
408	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
409	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
411	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
412	#95	C₂³×C₅₃	1	4	yes	—
413	#95	C₂³×C₅₃	1	4	yes	—
414	#95	C₂³×C₅₃	1	8	yes	—
415	#95	C₂³×C₅₃	1	8	yes	—
416	#95	C₂³×C₅₃	1	8	yes	—
417	C₂⁴	—	2	4	no	—
418	C₂⁴	—	4	1	no	—
419	C₂⁴	—	4	1	no	—
420	C₂⁴	—	4	2	no	—
421	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
422	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
423	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
424	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
425	#41	—	4	32	no	—
426	C₂×Q₈	—	2	4	no	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	—	2	4	no	—
429	C₂×D₄	—	2	8	no	—
430	C₂×D₄	—	2	8	no	—
431	C₂×D₄	—	2	8	no	—
432	C₂×D₄	—	2	16	no	—
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	8	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	4	8	no	—
438	C₂×D₄	—	4	4	no	—
439	C₂×D₄	—	4	4	no	—
440	C₂×D₄	—	4	4	no	—
441	C₂×D₄	—	4	4	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	8	no	—
448	C₂²×C₄	—	4	16	no	—
449	C₂²×C₄	—	4	2	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	—	4	2	no	—
453	C₂²×C₄	—	4	1	no	—
454	C₂²×C₄	—	4	1	no	—
455	C₂²×C₄	—	4	2	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	—	4	4	no	—
458	C₂²×C₄	—	4	4	no	—
459	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
460	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
461	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
462	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
463	C₂²×C₄	#41×C₅₃	1	8	yes	complemented
464	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
465	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
466	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
467	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
468	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
471	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
472	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	8	no	—
476	C₄⋊C₄	—	2	8	no	—
477	C₄⋊C₄	—	2	8	no	—
478	C₄⋊C₄	—	4	4	no	—
479	C₄⋊C₄	—	4	4	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	—	2	8	no	—
486	C₂²⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	—	2	16	no	—
488	C₂²⋊C₄	—	4	8	no	—
489	C₂²⋊C₄	—	4	8	no	—
490	C₄²	—	2	8	no	—
491	C₄²	—	2	4	no	—
492	C₄²	—	2	2	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	4	2	no	—
495	C₄²	—	4	2	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	8	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	2	no	—
503	C₂³	—	2	4	no	—
504	C₂³	—	2	4	no	—
505	C₂³	—	2	8	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	8	no	—
509	C₂³	—	4	4	no	—
510	C₂³	—	4	4	no	—
511	C₂³	—	4	8	no	—
512	C₂³	—	4	8	no	—
513	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
514	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
515	C₂³	C₂×#41×C₅₃	1	4	yes	—
516	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
517	Q₈	—	4	8	no	—
518	D₄	—	4	16	no	—
519	D₄	—	4	16	no	—
520	D₄	—	4	16	no	—
521	D₄	—	4	8	no	—
522	D₄	—	8	8	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	8	no	—
532	C₂×C₄	—	2	16	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	8	no	—
543	C₂×C₄	—	4	4	no	—
544	C₂×C₄	—	4	8	no	—
545	C₂×C₄	—	4	8	no	—
546	C₂×C₄	—	4	16	no	—
547	C₂×C₄	—	4	8	no	—
548	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
549	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
550	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
551	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
552	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
553	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
554	C₂×C₄	H(5,𝔽₂)×C₅₃	1	2	yes	—
555	C₂²	—	2	2	no	—
556	C₂²	—	2	4	no	—
557	C₂²	—	2	4	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	4	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	4	16	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	4	8	no	—
567	C₂²	—	4	8	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	4	4	no	—
570	C₂²	C₅₃×#532	1	4	yes	—
571	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
572	C₂²	C₂²×#41×C₅₃	1	2	yes	—
573	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
574	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
575	C₂²	C₅₃×#585	1	2	yes	—
576	C₄	—	2	4	no	—
577	C₄	—	2	4	no	—
578	C₄	—	2	4	no	—
579	C₄	—	2	2	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	4	no	—
583	C₄	—	4	4	no	—
584	C₄	C₅₃×#534	1	4	yes	—
585	C₂	—	2	4	no	—
586	C₂	—	4	4	no	—
587	C₂	—	8	4	no	—
588	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
589	C₂	C₂×C₅₃×#534	1	1	yes	characteristic
590	C₂	C₅₃×#3449	1	1	yes	characteristic
591	C₂	C₅₃×#3505	1	2	yes	—
592	C₂	C₅₃×#3506	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
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×#525	C₂	semidirect
2. C₅₃×#2857	C₂	semidirect
3. C₅₃×#2856	C₂	semidirect
4. C₅₃×#2850	C₂	semidirect
5. C₅₃×#2617	C₂	semidirect
6. C₅₃×#2602	C₂	semidirect
7. C₅₃×#2500	C₂	semidirect
8. C₅₃×#2499	C₂	semidirect
9. C₅₃×#2432	C₂	semidirect
10. C₅₃×#2431	C₂	semidirect
11. C₅₃×#2325	C₂	semidirect
12. C₂×C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
15. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
16. C₂×#127×C₅₃	C₂²	semidirect
17. C₂×#124×C₅₃	C₂²	semidirect
36. C₅₃×#391	C₂²	semidirect
37. C₅₃×#391	C₂²	semidirect
38. C₅₃×#390	C₂²	semidirect
39. C₅₃×#390	C₂²	semidirect
40. C₅₃×#390	C₂²	semidirect
41. C₅₃×#389	C₂²	semidirect
42. C₅₃×#388	C₂²	semidirect
43. C₅₃×#388	C₂²	semidirect
44. C₅₃×#387	C₂²	semidirect
46. C₅₃×#386	C₂²	semidirect
47. C₅₃×#385	C₂²	semidirect
48. C₅₃×#384	C₂²	semidirect
49. C₅₃×#384	C₂²	semidirect
50. C₅₃×#382	C₂²	semidirect
68. C₂³×C₄×C₅₃	D₄	semidirect
166. C₂²×C₄×C₅₃	#41	semidirect
258. #35032	C₅₃	direct (N × H)
289. C₂×#525	C₂×C₅₃	semidirect
290. #2857	C₂×C₅₃	semidirect
291. #2856	C₂×C₅₃	semidirect
292. #2850	C₂×C₅₃	semidirect
293. #2617	C₂×C₅₃	semidirect
294. #2602	C₂×C₅₃	semidirect
295. #2500	C₂×C₅₃	semidirect
296. #2499	C₂×C₅₃	semidirect
297. #2432	C₂×C₅₃	semidirect
298. #2431	C₂×C₅₃	semidirect
299. #2325	C₂×C₅₃	semidirect
300. C₂×C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
311. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
312. C₂×#127	C₂²×C₅₃	semidirect
313. C₂×#124	C₂²×C₅₃	semidirect
332. #391	C₂²×C₅₃	semidirect
333. #391	C₂²×C₅₃	semidirect
334. #390	C₂²×C₅₃	semidirect
335. #390	C₂²×C₅₃	semidirect
336. #390	C₂²×C₅₃	semidirect
337. #389	C₂²×C₅₃	semidirect
338. #388	C₂²×C₅₃	semidirect
339. #388	C₂²×C₅₃	semidirect
340. #387	C₂²×C₅₃	semidirect
342. #386	C₂²×C₅₃	semidirect
343. #385	C₂²×C₅₃	semidirect
344. #384	C₂²×C₅₃	semidirect
345. #384	C₂²×C₅₃	semidirect
346. #382	C₂²×C₅₃	semidirect
352. C₅₃	#35032	direct (N × H)
365. C₂³×C₄	D₄×C₅₃	semidirect
463. C₂²×C₄	#41×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×#525	2
C₅₃×#2857	2
C₅₃×#2856	2
C₅₃×#2850	2
C₅₃×#2617	2
C₅₃×#2602	2
C₅₃×#2500	2
C₅₃×#2499	2
C₅₃×#2432	2
C₅₃×#2431	2
C₅₃×#2325	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
#35032	53

Subgroups — C53×#35032 #27242544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#35032 #27242544