Subgroups — C₅₃×#34981 #27242493

Order 13568.

5126 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#534	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#120×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂³×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2847	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2617	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2412	C₂	1	8	yes	maximal, complemented
7	C₅₃×#2410	C₂	1	8	yes	maximal, complemented
8	C₅₃×#2315	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2308	C₂	1	2	yes	maximal, complemented
10	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
12	C₂³×D₄×C₅₃	C₂²	1	4	yes	complemented
13	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	complemented
14	C₅₃×#534	—	2	8	no	—
15	C₂×#123×C₅₃	C₂²	1	4	yes	complemented
16	C₂×#121×C₅₃	C₂²	1	4	yes	complemented
17	C₂×#121×C₅₃	C₂²	1	8	yes	complemented
18	C₂×#120×C₅₃	—	2	16	no	—
19	C₂×#120×C₅₃	C₂²	1	8	yes	—
20	C₂×#120×C₅₃	C₂²	1	4	yes	complemented
21	C₂×#120×C₅₃	C₂²	1	4	yes	complemented
22	C₂×#120×C₅₃	C₂²	1	4	yes	complemented
23	C₂²×C₂²⋊C₄×C₅₃	—	2	8	no	—
24	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
28	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	complemented
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
30	C₅₃×#397	C₂²	1	8	yes	complemented
31	C₅₃×#394	C₂²	1	16	yes	complemented
32	C₅₃×#392	C₂²	1	8	yes	complemented
33	C₅₃×#386	C₂²	1	8	yes	complemented
34	C₅₃×#386	C₂²	1	8	yes	complemented
35	C₅₃×#386	C₂²	1	16	yes	complemented
36	C₅₃×#385	C₂²	1	8	yes	complemented
37	C₅₃×#381	C₂²	1	4	yes	complemented
38	C₅₃×#380	C₂²	1	4	yes	complemented
39	C₅₃×#380	C₂²	1	8	yes	complemented
40	C₅₃×#379	C₂²	1	4	yes	complemented
41	C₂⁵×C₅₃	—	2	4	no	—
42	C₂⁵×C₅₃	—	2	4	no	—
43	C₂⁵×C₅₃	—	4	8	no	—
44	C₂⁵×C₅₃	D₄	1	8	yes	complemented
45	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
46	C₂⁵×C₅₃	C₂³	1	4	yes	—
47	C₂⁵×C₅₃	C₂³	1	2	yes	—
48	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
49	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₅₃	—	2	8	no	—
51	C₂²×D₄×C₅₃	—	2	16	no	—
52	C₂²×D₄×C₅₃	—	2	8	no	—
53	C₂²×D₄×C₅₃	—	2	8	no	—
54	C₂²×D₄×C₅₃	—	4	16	no	—
55	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
56	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
57	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
58	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
59	C₂²×D₄×C₅₃	C₂³	1	4	yes	complemented
60	C₂³×C₄×C₅₃	—	2	8	no	—
61	C₂³×C₄×C₅₃	—	4	8	no	—
62	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
66	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
67	#123×C₅₃	—	2	16	no	—
68	#121×C₅₃	—	2	16	no	—
69	#121×C₅₃	—	4	16	no	—
70	#120×C₅₃	—	2	16	no	—
71	#120×C₅₃	—	2	32	no	—
72	#120×C₅₃	—	2	16	no	—
73	#120×C₅₃	—	2	16	no	—
74	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	complemented
75	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	complemented
76	C₂×C₂²⋊C₄×C₅₃	—	2	32	no	—
77	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
78	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
79	C₂×C₂²⋊C₄×C₅₃	—	2	16	no	—
80	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
81	C₂×C₂²⋊C₄×C₅₃	D₄	1	8	yes	complemented
82	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
87	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	complemented
90	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	complemented
91	#95×C₅₃	C₂³	1	16	yes	complemented
92	#95×C₅₃	C₂³	1	8	yes	complemented
93	C₂⁴×C₅₃	—	2	2	no	—
94	C₂⁴×C₅₃	—	2	2	no	—
95	C₂⁴×C₅₃	—	2	8	no	—
96	C₂⁴×C₅₃	—	2	32	no	—
97	C₂⁴×C₅₃	—	2	8	no	—
98	C₂⁴×C₅₃	—	2	2	no	—
99	C₂⁴×C₅₃	—	2	8	no	—
100	C₂⁴×C₅₃	—	2	2	no	—
101	C₂⁴×C₅₃	—	2	16	no	—
102	C₂⁴×C₅₃	—	2	4	no	—
103	C₂⁴×C₅₃	—	2	4	no	—
104	C₂⁴×C₅₃	—	2	4	no	—
105	C₂⁴×C₅₃	—	4	32	no	—
106	C₂⁴×C₅₃	—	4	4	no	—
107	C₂⁴×C₅₃	—	4	8	no	—
108	C₂⁴×C₅₃	—	4	4	no	—
109	C₂⁴×C₅₃	—	4	8	no	—
110	C₂⁴×C₅₃	—	4	32	no	—
111	C₂⁴×C₅₃	—	4	32	no	—
112	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
113	C₂⁴×C₅₃	C₂×D₄	1	16	yes	—
114	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
115	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
116	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
117	C₂⁴×C₅₃	#41	1	4	yes	—
118	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
119	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
120	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
121	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
122	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
123	C₂×D₄×C₅₃	—	2	16	no	—
124	C₂×D₄×C₅₃	—	2	8	no	—
125	C₂×D₄×C₅₃	—	2	8	no	—
126	C₂×D₄×C₅₃	—	2	16	no	—
127	C₂×D₄×C₅₃	—	2	8	no	—
128	C₂×D₄×C₅₃	—	2	8	no	—
129	C₂×D₄×C₅₃	—	2	8	no	—
130	C₂×D₄×C₅₃	—	2	8	no	—
131	C₂×D₄×C₅₃	—	2	32	no	—
132	C₂×D₄×C₅₃	—	2	8	no	—
133	C₂×D₄×C₅₃	—	4	16	no	—
134	C₂×D₄×C₅₃	—	4	8	no	—
135	C₂×D₄×C₅₃	—	4	64	no	—
136	C₂×D₄×C₅₃	—	4	32	no	—
137	C₂²×C₄×C₅₃	—	2	4	no	—
138	C₂²×C₄×C₅₃	—	2	8	no	—
139	C₂²×C₄×C₅₃	—	2	4	no	—
140	C₂²×C₄×C₅₃	—	2	4	no	—
141	C₂²×C₄×C₅₃	—	2	4	no	—
142	C₂²×C₄×C₅₃	—	2	16	no	—
143	C₂²×C₄×C₅₃	—	2	16	no	—
144	C₂²×C₄×C₅₃	—	4	4	no	—
145	C₂²×C₄×C₅₃	—	4	8	no	—
146	C₂²×C₄×C₅₃	—	4	4	no	—
147	C₂²×C₄×C₅₃	—	4	8	no	—
148	C₂²×C₄×C₅₃	—	4	32	no	—
149	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	complemented
150	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
154	C₄⋊C₄×C₅₃	—	2	8	no	—
155	C₄⋊C₄×C₅₃	—	4	8	no	—
156	C₂²⋊C₄×C₅₃	—	2	8	no	—
157	C₂²⋊C₄×C₅₃	—	2	16	no	—
158	C₂²⋊C₄×C₅₃	—	2	8	no	—
159	C₂²⋊C₄×C₅₃	—	2	8	no	—
160	C₂²⋊C₄×C₅₃	—	2	16	no	—
161	C₂²⋊C₄×C₅₃	—	2	16	no	—
162	C₂²⋊C₄×C₅₃	—	2	16	no	—
163	C₂²⋊C₄×C₅₃	—	4	16	no	—
164	C₂²⋊C₄×C₅₃	C₂×D₄	1	16	yes	complemented
165	C₂³×C₅₃	—	2	16	no	—
166	C₂³×C₅₃	—	2	16	no	—
167	C₂³×C₅₃	—	2	8	no	—
168	C₂³×C₅₃	—	2	16	no	—
169	C₂³×C₅₃	—	2	2	no	—
170	C₂³×C₅₃	—	2	2	no	—
171	C₂³×C₅₃	—	2	4	no	—
172	C₂³×C₅₃	—	2	4	no	—
173	C₂³×C₅₃	—	2	8	no	—
174	C₂³×C₅₃	—	2	32	no	—
175	C₂³×C₅₃	—	2	8	no	—
176	C₂³×C₅₃	—	2	32	no	—
177	C₂³×C₅₃	—	2	2	no	—
178	C₂³×C₅₃	—	2	2	no	—
179	C₂³×C₅₃	—	2	8	no	—
180	C₂³×C₅₃	—	2	4	no	—
181	C₂³×C₅₃	—	2	4	no	—
182	C₂³×C₅₃	—	2	4	no	—
183	C₂³×C₅₃	—	2	16	no	—
184	C₂³×C₅₃	—	4	8	no	—
185	C₂³×C₅₃	—	4	16	no	—
186	C₂³×C₅₃	—	4	8	no	—
187	C₂³×C₅₃	—	4	8	no	—
188	C₂³×C₅₃	—	4	8	no	—
189	C₂³×C₅₃	—	4	32	no	—
190	C₂³×C₅₃	—	4	32	no	—
191	C₂³×C₅₃	—	4	128	no	—
192	C₂³×C₅₃	—	4	8	no	—
193	C₂³×C₅₃	—	4	8	no	—
194	C₂³×C₅₃	—	4	16	no	—
195	C₂³×C₅₃	—	4	8	no	—
196	C₂³×C₅₃	—	8	8	no	—
197	C₂³×C₅₃	—	8	8	no	—
198	C₂³×C₅₃	#121	1	16	yes	complemented
199	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
200	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
201	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
202	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
203	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
204	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
205	C₂³×C₅₃	C₂×#41	1	4	yes	—
206	C₂³×C₅₃	C₂×#41	1	2	yes	—
207	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
208	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
209	D₄×C₅₃	—	4	32	no	—
210	D₄×C₅₃	—	4	16	no	—
211	D₄×C₅₃	—	4	16	no	—
212	D₄×C₅₃	—	4	32	no	—
213	D₄×C₅₃	—	8	8	no	—
214	C₂×C₄×C₅₃	—	2	8	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	2	4	no	—
217	C₂×C₄×C₅₃	—	2	8	no	—
218	C₂×C₄×C₅₃	—	2	4	no	—
219	C₂×C₄×C₅₃	—	2	4	no	—
220	C₂×C₄×C₅₃	—	2	8	no	—
221	C₂×C₄×C₅₃	—	2	16	no	—
222	C₂×C₄×C₅₃	—	4	8	no	—
223	C₂×C₄×C₅₃	—	4	32	no	—
224	C₂×C₄×C₅₃	—	4	8	no	—
225	C₂×C₄×C₅₃	—	4	16	no	—
226	C₂×C₄×C₅₃	—	4	8	no	—
227	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
228	#34981	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂²×C₅₃	—	2	32	no	—
230	C₂²×C₅₃	—	2	4	no	—
231	C₂²×C₅₃	—	2	8	no	—
232	C₂²×C₅₃	—	2	4	no	—
233	C₂²×C₅₃	—	2	4	no	—
234	C₂²×C₅₃	—	2	4	no	—
235	C₂²×C₅₃	—	2	8	no	—
236	C₂²×C₅₃	—	2	8	no	—
237	C₂²×C₅₃	—	2	4	no	—
238	C₂²×C₅₃	—	2	8	no	—
239	C₂²×C₅₃	—	2	8	no	—
240	C₂²×C₅₃	—	2	4	no	—
241	C₂²×C₅₃	—	2	8	no	—
242	C₂²×C₅₃	—	4	16	no	—
243	C₂²×C₅₃	—	4	64	no	—
244	C₂²×C₅₃	—	4	16	no	—
245	C₂²×C₅₃	—	4	4	no	—
246	C₂²×C₅₃	—	4	4	no	—
247	C₂²×C₅₃	—	4	4	no	—
248	C₂²×C₅₃	—	4	8	no	—
249	C₂²×C₅₃	—	4	8	no	—
250	C₂²×C₅₃	—	4	8	no	—
251	C₂²×C₅₃	—	8	16	no	—
252	C₂²×C₅₃	C₂×#121	1	8	yes	—
253	C₂²×C₅₃	C₂×#121	1	4	yes	—
254	C₂²×C₅₃	#534	1	8	yes	complemented
255	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
256	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
257	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
258	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
259	C₄×C₅₃	—	2	8	no	—
260	C₄×C₅₃	—	4	4	no	—
261	C₄×C₅₃	—	4	8	no	—
262	C₄×C₅₃	—	4	4	no	—
263	C₂×#534	C₂×C₅₃	1	1	yes	characteristic, complemented
264	C₂²×#120	C₂×C₅₃	1	2	yes	complemented
265	C₂³×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
266	#2847	C₂×C₅₃	1	4	yes	complemented
267	#2617	C₂×C₅₃	1	4	yes	complemented
268	#2412	C₂×C₅₃	1	8	yes	complemented
269	#2410	C₂×C₅₃	1	8	yes	complemented
270	#2315	C₂×C₅₃	1	1	yes	characteristic, complemented
271	#2308	C₂×C₅₃	1	2	yes	complemented
272	C₂×C₅₃	—	2	4	no	—
273	C₂×C₅₃	—	2	8	no	—
274	C₂×C₅₃	—	2	8	no	—
275	C₂×C₅₃	—	4	4	no	—
276	C₂×C₅₃	—	8	4	no	—
277	C₂×C₅₃	C₂²×#121	1	1	yes	characteristic
278	C₂×C₅₃	C₂×#534	1	2	yes	—
279	C₂×C₅₃	#3484	1	2	yes	—
280	C₂×C₅₃	#3490	1	2	yes	—
281	C₂⁶	C₂²×C₅₃	1	1	yes	characteristic
282	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
283	C₂³×D₄	C₂²×C₅₃	1	4	yes	complemented
284	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	complemented
285	#534	—	2	8	no	—
286	C₂×#123	C₂²×C₅₃	1	4	yes	complemented
287	C₂×#121	C₂²×C₅₃	1	4	yes	complemented
288	C₂×#121	C₂²×C₅₃	1	8	yes	complemented
289	C₂×#120	—	2	16	no	—
290	C₂×#120	C₂²×C₅₃	1	8	yes	—
291	C₂×#120	C₂²×C₅₃	1	4	yes	complemented
292	C₂×#120	C₂²×C₅₃	1	4	yes	complemented
293	C₂×#120	C₂²×C₅₃	1	4	yes	complemented
294	C₂²×C₂²⋊C₄	—	2	8	no	—
295	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
296	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
297	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
298	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	complemented
299	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	complemented
300	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
301	#397	C₂²×C₅₃	1	8	yes	complemented
302	#394	C₂²×C₅₃	1	16	yes	complemented
303	#392	C₂²×C₅₃	1	8	yes	complemented
304	#386	C₂²×C₅₃	1	8	yes	complemented
305	#386	C₂²×C₅₃	1	8	yes	complemented
306	#386	C₂²×C₅₃	1	16	yes	complemented
307	#385	C₂²×C₅₃	1	8	yes	complemented
308	#381	C₂²×C₅₃	1	4	yes	complemented
309	#380	C₂²×C₅₃	1	4	yes	complemented
310	#380	C₂²×C₅₃	1	8	yes	complemented
311	#379	C₂²×C₅₃	1	4	yes	complemented
312	C₅₃	#34981	1	1	yes	characteristic, direct product factor, Sylow
313	C₂⁵	—	2	4	no	—
314	C₂⁵	—	2	4	no	—
315	C₂⁵	—	4	8	no	—
316	C₂⁵	D₄×C₅₃	1	8	yes	complemented
317	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
318	C₂⁵	C₂³×C₅₃	1	4	yes	—
319	C₂⁵	C₂³×C₅₃	1	2	yes	—
320	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
321	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
322	C₂²×D₄	—	2	8	no	—
323	C₂²×D₄	—	2	16	no	—
324	C₂²×D₄	—	2	8	no	—
325	C₂²×D₄	—	2	8	no	—
326	C₂²×D₄	—	4	16	no	—
327	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
328	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
329	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
330	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
331	C₂²×D₄	C₂³×C₅₃	1	4	yes	complemented
332	C₂³×C₄	—	2	8	no	—
333	C₂³×C₄	—	4	8	no	—
334	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
335	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
336	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
337	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
338	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
339	#123	—	2	16	no	—
340	#121	—	2	16	no	—
341	#121	—	4	16	no	—
342	#120	—	2	32	no	—
343	#120	—	2	16	no	—
344	#120	—	2	16	no	—
345	#120	—	2	16	no	—
346	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	complemented
347	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	complemented
348	C₂×C₂²⋊C₄	—	2	4	no	—
349	C₂×C₂²⋊C₄	—	2	8	no	—
350	C₂×C₂²⋊C₄	—	2	16	no	—
351	C₂×C₂²⋊C₄	—	2	8	no	—
352	C₂×C₂²⋊C₄	—	2	32	no	—
353	C₂×C₂²⋊C₄	D₄×C₅₃	1	8	yes	complemented
354	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	complemented
362	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	complemented
363	#95	C₂³×C₅₃	1	16	yes	complemented
364	#95	C₂³×C₅₃	1	8	yes	complemented
365	C₂⁴	—	2	2	no	—
366	C₂⁴	—	2	2	no	—
367	C₂⁴	—	2	8	no	—
368	C₂⁴	—	2	32	no	—
369	C₂⁴	—	2	8	no	—
370	C₂⁴	—	2	2	no	—
371	C₂⁴	—	2	8	no	—
372	C₂⁴	—	2	2	no	—
373	C₂⁴	—	2	16	no	—
374	C₂⁴	—	2	4	no	—
375	C₂⁴	—	2	4	no	—
376	C₂⁴	—	2	4	no	—
377	C₂⁴	—	4	32	no	—
378	C₂⁴	—	4	4	no	—
379	C₂⁴	—	4	8	no	—
380	C₂⁴	—	4	4	no	—
381	C₂⁴	—	4	8	no	—
382	C₂⁴	—	4	32	no	—
383	C₂⁴	—	4	32	no	—
384	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
385	C₂⁴	C₂×D₄×C₅₃	1	16	yes	—
386	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
387	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
388	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
389	C₂⁴	#41×C₅₃	1	4	yes	—
390	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
391	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
392	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
393	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
394	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
395	C₂×D₄	—	2	16	no	—
396	C₂×D₄	—	2	8	no	—
397	C₂×D₄	—	2	8	no	—
398	C₂×D₄	—	2	16	no	—
399	C₂×D₄	—	2	8	no	—
400	C₂×D₄	—	2	8	no	—
401	C₂×D₄	—	2	8	no	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	—	2	32	no	—
404	C₂×D₄	—	2	8	no	—
405	C₂×D₄	—	4	16	no	—
406	C₂×D₄	—	4	8	no	—
407	C₂×D₄	—	4	64	no	—
408	C₂×D₄	—	4	32	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	8	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	16	no	—
415	C₂²×C₄	—	2	16	no	—
416	C₂²×C₄	—	4	4	no	—
417	C₂²×C₄	—	4	8	no	—
418	C₂²×C₄	—	4	4	no	—
419	C₂²×C₄	—	4	8	no	—
420	C₂²×C₄	—	4	32	no	—
421	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	complemented
422	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
423	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
424	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
425	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
426	C₄⋊C₄	—	2	8	no	—
427	C₄⋊C₄	—	4	8	no	—
428	C₂²⋊C₄	—	2	8	no	—
429	C₂²⋊C₄	—	2	16	no	—
430	C₂²⋊C₄	—	2	8	no	—
431	C₂²⋊C₄	—	2	8	no	—
432	C₂²⋊C₄	—	2	16	no	—
433	C₂²⋊C₄	—	2	16	no	—
434	C₂²⋊C₄	—	2	16	no	—
435	C₂²⋊C₄	—	4	16	no	—
436	C₂²⋊C₄	C₂×D₄×C₅₃	1	16	yes	complemented
437	C₂³	—	2	16	no	—
438	C₂³	—	2	16	no	—
439	C₂³	—	2	8	no	—
440	C₂³	—	2	16	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	2	2	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	2	4	no	—
445	C₂³	—	2	8	no	—
446	C₂³	—	2	32	no	—
447	C₂³	—	2	8	no	—
448	C₂³	—	2	32	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	8	no	—
452	C₂³	—	2	4	no	—
453	C₂³	—	2	4	no	—
454	C₂³	—	2	4	no	—
455	C₂³	—	2	16	no	—
456	C₂³	—	4	8	no	—
457	C₂³	—	4	8	no	—
458	C₂³	—	4	16	no	—
459	C₂³	—	4	8	no	—
460	C₂³	—	4	8	no	—
461	C₂³	—	4	32	no	—
462	C₂³	—	4	32	no	—
463	C₂³	—	4	128	no	—
464	C₂³	—	4	8	no	—
465	C₂³	—	4	8	no	—
466	C₂³	—	4	16	no	—
467	C₂³	—	4	8	no	—
468	C₂³	—	8	8	no	—
469	C₂³	—	8	8	no	—
470	C₂³	#121×C₅₃	1	16	yes	complemented
471	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
472	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
473	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
474	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
475	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
476	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
477	C₂³	C₂×#41×C₅₃	1	4	yes	—
478	C₂³	C₂×#41×C₅₃	1	2	yes	—
479	C₂³	H(5,𝔽₂)×C₅₃	1	8	yes	—
480	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
481	D₄	—	4	32	no	—
482	D₄	—	4	16	no	—
483	D₄	—	4	16	no	—
484	D₄	—	4	32	no	—
485	D₄	—	8	8	no	—
486	C₂×C₄	—	2	8	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	8	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	8	no	—
493	C₂×C₄	—	2	16	no	—
494	C₂×C₄	—	4	8	no	—
495	C₂×C₄	—	4	32	no	—
496	C₂×C₄	—	4	8	no	—
497	C₂×C₄	—	4	16	no	—
498	C₂×C₄	—	4	8	no	—
499	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
500	C₂²	—	2	32	no	—
501	C₂²	—	2	4	no	—
502	C₂²	—	2	8	no	—
503	C₂²	—	2	8	no	—
504	C₂²	—	2	4	no	—
505	C₂²	—	2	8	no	—
506	C₂²	—	2	8	no	—
507	C₂²	—	2	4	no	—
508	C₂²	—	2	8	no	—
509	C₂²	—	2	4	no	—
510	C₂²	—	2	4	no	—
511	C₂²	—	2	4	no	—
512	C₂²	—	2	8	no	—
513	C₂²	—	4	16	no	—
514	C₂²	—	4	64	no	—
515	C₂²	—	4	16	no	—
516	C₂²	—	4	4	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	8	no	—
519	C₂²	—	4	8	no	—
520	C₂²	—	4	4	no	—
521	C₂²	—	4	8	no	—
522	C₂²	—	8	16	no	—
523	C₂²	C₂×#121×C₅₃	1	8	yes	—
524	C₂²	C₂×#121×C₅₃	1	4	yes	—
525	C₂²	C₅₃×#534	1	8	yes	complemented
526	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
527	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
528	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
529	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	4	yes	—
530	C₄	—	2	8	no	—
531	C₄	—	4	4	no	—
532	C₄	—	4	4	no	—
533	C₄	—	4	8	no	—
534	C₂	—	2	4	no	—
535	C₂	—	2	8	no	—
536	C₂	—	2	8	no	—
537	C₂	—	4	4	no	—
538	C₂	—	8	4	no	—
539	C₂	C₂²×#121×C₅₃	1	1	yes	characteristic
540	C₂	C₂×C₅₃×#534	1	2	yes	—
541	C₂	C₅₃×#3484	1	2	yes	—
542	C₂	C₅₃×#3490	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×#534	C₂	semidirect
2. C₂²×#120×C₅₃	C₂	semidirect
3. C₂³×C₂²⋊C₄×C₅₃	C₂	semidirect
4. C₅₃×#2847	C₂	semidirect
5. C₅₃×#2617	C₂	semidirect
6. C₅₃×#2412	C₂	semidirect
7. C₅₃×#2410	C₂	semidirect
8. C₅₃×#2315	C₂	semidirect
9. C₅₃×#2308	C₂	semidirect
12. C₂³×D₄×C₅₃	C₂²	semidirect
13. C₂⁴×C₄×C₅₃	C₂²	semidirect
15. C₂×#123×C₅₃	C₂²	semidirect
16. C₂×#121×C₅₃	C₂²	semidirect
17. C₂×#121×C₅₃	C₂²	semidirect
20. C₂×#120×C₅₃	C₂²	semidirect
21. C₂×#120×C₅₃	C₂²	semidirect
22. C₂×#120×C₅₃	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
28. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
29. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
30. C₅₃×#397	C₂²	semidirect
31. C₅₃×#394	C₂²	semidirect
32. C₅₃×#392	C₂²	semidirect
33. C₅₃×#386	C₂²	semidirect
34. C₅₃×#386	C₂²	semidirect
35. C₅₃×#386	C₂²	semidirect
36. C₅₃×#385	C₂²	semidirect
37. C₅₃×#381	C₂²	semidirect
38. C₅₃×#380	C₂²	semidirect
39. C₅₃×#380	C₂²	semidirect
40. C₅₃×#379	C₂²	semidirect
44. C₂⁵×C₅₃	D₄	semidirect
59. C₂²×D₄×C₅₃	C₂³	semidirect
74. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
75. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
81. C₂×C₂²⋊C₄×C₅₃	D₄	semidirect
89. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
90. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
91. #95×C₅₃	C₂³	semidirect
92. #95×C₅₃	C₂³	semidirect
149. C₂²×C₄×C₅₃	C₂×D₄	semidirect
164. C₂²⋊C₄×C₅₃	C₂×D₄	semidirect
198. C₂³×C₅₃	#121	semidirect
228. #34981	C₅₃	direct (N × H)
254. C₂²×C₅₃	#534	semidirect
263. C₂×#534	C₂×C₅₃	semidirect
264. C₂²×#120	C₂×C₅₃	semidirect
265. C₂³×C₂²⋊C₄	C₂×C₅₃	semidirect
266. #2847	C₂×C₅₃	semidirect
267. #2617	C₂×C₅₃	semidirect
268. #2412	C₂×C₅₃	semidirect
269. #2410	C₂×C₅₃	semidirect
270. #2315	C₂×C₅₃	semidirect
271. #2308	C₂×C₅₃	semidirect
283. C₂³×D₄	C₂²×C₅₃	semidirect
284. C₂⁴×C₄	C₂²×C₅₃	semidirect
286. C₂×#123	C₂²×C₅₃	semidirect
287. C₂×#121	C₂²×C₅₃	semidirect
288. C₂×#121	C₂²×C₅₃	semidirect
291. C₂×#120	C₂²×C₅₃	semidirect
292. C₂×#120	C₂²×C₅₃	semidirect
293. C₂×#120	C₂²×C₅₃	semidirect
298. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
299. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
300. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
301. #397	C₂²×C₅₃	semidirect
302. #394	C₂²×C₅₃	semidirect
303. #392	C₂²×C₅₃	semidirect
304. #386	C₂²×C₅₃	semidirect
305. #386	C₂²×C₅₃	semidirect
306. #386	C₂²×C₅₃	semidirect
307. #385	C₂²×C₅₃	semidirect
308. #381	C₂²×C₅₃	semidirect
309. #380	C₂²×C₅₃	semidirect
310. #380	C₂²×C₅₃	semidirect
311. #379	C₂²×C₅₃	semidirect
312. C₅₃	#34981	direct (N × H)
316. C₂⁵	D₄×C₅₃	semidirect
331. C₂²×D₄	C₂³×C₅₃	semidirect
346. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
347. C₂×C₄⋊C₄	C₂³×C₅₃	semidirect
353. C₂×C₂²⋊C₄	D₄×C₅₃	semidirect
361. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
362. C₂×C₂²⋊C₄	C₂³×C₅₃	semidirect
363. #95	C₂³×C₅₃	semidirect
364. #95	C₂³×C₅₃	semidirect
421. C₂²×C₄	C₂×D₄×C₅₃	semidirect
436. C₂²⋊C₄	C₂×D₄×C₅₃	semidirect
470. C₂³	#121×C₅₃	semidirect
525. C₂²	C₅₃×#534	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#534	2
C₂²×#120×C₅₃	2
C₂³×C₂²⋊C₄×C₅₃	2
C₅₃×#2847	2
C₅₃×#2617	2
C₅₃×#2412	2
C₅₃×#2410	2
C₅₃×#2315	2
C₅₃×#2308	2
#34981	53

Subgroups — C53×#34981 #27242493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34981 #27242493