Subgroups — #27403580

Order 13568.

960 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275323	C₂	1	2	yes	maximal, complemented
2	#20275274	C₂	1	2	yes	maximal, complemented
3	#20275008	C₂	1	2	yes	maximal
4	#20274858	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274783	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274695	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₄×#88322	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274461	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16810378	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×#88322	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
15	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
16	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
17	C₂×C₄×#88322	C₂²	1	2	yes	—
18	C₂×C₄×#88322	C₂²	1	2	yes	—
19	#16810378	C₂²	1	1	yes	characteristic
20	#16810378	C₂²	1	1	yes	characteristic
21	#16810378	C₂²	1	1	yes	characteristic
22	#16810378	C₂²	1	2	yes	—
23	#16810378	C₂²	1	2	yes	—
24	#16810378	C₂²	1	2	yes	—
25	#16810378	C₂²	1	2	yes	—
26	#16810378	C₂²	1	2	yes	—
27	#16810378	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
29	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
30	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
41	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
42	C₄³×C₅₃	C₂²	1	1	yes	characteristic
43	C₂²×#88322	—	2	4	no	—
44	C₂²×#88322	—	2	4	no	—
45	C₂²×#88322	—	4	2	no	—
46	C₂²×#88322	C₂³	1	2	yes	—
47	C₂²×#88322	C₂³	1	2	yes	—
48	C₂²×#88322	C₂³	1	2	yes	—
49	C₂²×#88322	C₂³	1	2	yes	—
50	C₄⋊#88322	—	4	2	no	—
51	C₂²⋊#88322	—	2	4	no	—
52	C₂²⋊#88322	—	2	4	no	—
53	C₂²⋊#88322	—	4	2	no	—
54	C₄×#88322	—	2	4	no	—
55	C₄×#88322	—	4	2	no	—
56	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₄×D₄×C₅₃	—	2	4	no	—
64	C₄×D₄×C₅₃	—	2	2	no	—
65	C₄×D₄×C₅₃	—	2	2	no	—
66	C₄×D₄×C₅₃	C₈	1	8	yes	complemented
67	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	—	4	2	no	—
77	C₂×C₄²×C₅₃	—	4	2	no	—
78	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
83	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
87	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
91	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
92	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
93	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
94	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
95	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
96	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×D₄×C₅₃	—	2	2	no	—
100	C₂×D₄×C₅₃	—	2	2	no	—
101	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
102	C₂²×C₄×C₅₃	—	2	2	no	—
103	C₂²×C₄×C₅₃	—	2	2	no	—
104	C₂²×C₄×C₅₃	—	2	2	no	—
105	C₂²×C₄×C₅₃	—	2	2	no	—
106	C₂²×C₄×C₅₃	—	2	2	no	—
107	C₂²×C₄×C₅₃	—	2	2	no	—
108	C₂²×C₄×C₅₃	—	2	1	no	—
109	C₂²×C₄×C₅₃	—	2	1	no	—
110	C₂²×C₄×C₅₃	—	2	2	no	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	4	1	no	—
113	C₂²×C₄×C₅₃	—	4	1	no	—
114	C₂²×C₄×C₅₃	—	4	1	no	—
115	C₂²×C₄×C₅₃	—	4	1	no	—
116	C₂²×C₄×C₅₃	—	4	1	no	—
117	C₂²×C₄×C₅₃	—	4	1	no	—
118	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
119	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	complemented
120	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₅₃	#41	1	2	yes	—
133	C₂²×C₄×C₅₃	#41	1	2	yes	—
134	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
135	C₄⋊C₄×C₅₃	—	2	2	no	—
136	C₄⋊C₄×C₅₃	—	2	2	no	—
137	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
138	C₂²⋊C₄×C₅₃	—	2	2	no	—
139	C₂²⋊C₄×C₅₃	—	2	2	no	—
140	C₂²⋊C₄×C₅₃	—	2	2	no	—
141	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
142	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
143	C₂²⋊C₄×C₅₃	M₁₆	1	4	yes	—
144	C₄²×C₅₃	—	2	2	no	—
145	C₄²×C₅₃	—	2	2	no	—
146	C₄²×C₅₃	—	2	1	no	—
147	C₄²×C₅₃	—	2	1	no	—
148	C₄²×C₅₃	—	2	2	no	—
149	C₄²×C₅₃	—	2	2	no	—
150	C₄²×C₅₃	—	2	2	no	—
151	C₄²×C₅₃	—	2	2	no	—
152	C₄²×C₅₃	—	4	1	no	—
153	C₄²×C₅₃	—	4	1	no	—
154	C₄²×C₅₃	—	4	1	no	—
155	C₄²×C₅₃	—	4	1	no	—
156	C₄²×C₅₃	—	4	1	no	—
157	C₄²×C₅₃	C₂×C₈	1	4	yes	complemented
158	C₂×#88322	—	2	2	no	—
159	C₂×#88322	—	2	2	no	—
160	C₂×#88322	—	2	2	no	—
161	C₂×#88322	—	2	2	no	—
162	C₂×#88322	—	2	2	no	—
163	C₂×#88322	—	2	2	no	—
164	C₂×#88322	—	2	2	no	—
165	C₂×#88322	—	2	2	no	—
166	C₂×#88322	—	2	2	no	—
167	C₂×#88322	—	2	2	no	—
168	C₂×#88322	—	2	2	no	—
169	C₂×#88322	—	4	8	no	—
170	C₂×#88322	#41	1	4	yes	—
171	C₂³×C₅₃	—	2	1	no	—
172	C₂³×C₅₃	—	2	1	no	—
173	C₂³×C₅₃	—	2	1	no	—
174	C₂³×C₅₃	—	2	1	no	—
175	C₂³×C₅₃	—	2	1	no	—
176	C₂³×C₅₃	—	2	1	no	—
177	C₂³×C₅₃	—	2	1	no	—
178	C₂³×C₅₃	—	2	1	no	—
179	C₂³×C₅₃	—	2	1	no	—
180	C₂³×C₅₃	—	2	1	no	—
181	C₂³×C₅₃	#117	1	2	yes	—
182	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
183	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
184	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
185	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
186	D₄×C₅₃	—	2	8	no	—
187	C₂×C₄×C₅₃	—	2	1	no	—
188	C₂×C₄×C₅₃	—	2	1	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	1	no	—
191	C₂×C₄×C₅₃	—	2	1	no	—
192	C₂×C₄×C₅₃	—	2	1	no	—
193	C₂×C₄×C₅₃	—	2	1	no	—
194	C₂×C₄×C₅₃	—	2	1	no	—
195	C₂×C₄×C₅₃	—	2	1	no	—
196	C₂×C₄×C₅₃	—	2	1	no	—
197	C₂×C₄×C₅₃	—	2	1	no	—
198	C₂×C₄×C₅₃	—	2	1	no	—
199	C₂×C₄×C₅₃	—	2	1	no	—
200	C₂×C₄×C₅₃	—	2	1	no	—
201	C₂×C₄×C₅₃	—	2	1	no	—
202	C₂×C₄×C₅₃	—	2	1	no	—
203	C₂×C₄×C₅₃	—	2	1	no	—
204	C₂×C₄×C₅₃	—	2	1	no	—
205	C₂×C₄×C₅₃	—	2	4	no	—
206	C₂×C₄×C₅₃	—	2	4	no	—
207	C₂×C₄×C₅₃	—	2	1	no	—
208	C₂×C₄×C₅₃	—	2	1	no	—
209	C₂×C₄×C₅₃	—	2	1	no	—
210	C₂×C₄×C₅₃	—	2	1	no	—
211	C₂×C₄×C₅₃	—	2	1	no	—
212	C₂×C₄×C₅₃	—	2	1	no	—
213	C₂×C₄×C₅₃	—	2	1	no	—
214	C₂×C₄×C₅₃	—	2	1	no	—
215	C₂×C₄×C₅₃	—	2	1	no	—
216	C₂×C₄×C₅₃	—	4	4	no	—
217	C₂×C₄×C₅₃	—	4	4	no	—
218	C₂×C₄×C₅₃	—	4	4	no	—
219	C₂×C₄×C₅₃	—	4	4	no	—
220	C₂×C₄×C₅₃	—	4	2	no	—
221	C₂×C₄×C₅₃	—	4	2	no	—
222	C₂×C₄×C₅₃	#117	1	2	yes	—
223	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
224	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
225	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
226	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
227	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
228	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
229	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
230	C₂×C₄×C₅₃	#131	1	2	yes	—
231	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
232	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
233	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
234	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
235	#88322	—	2	4	no	—
236	#88322	—	4	2	no	—
237	#88322	—	4	2	no	—
238	#88322	—	4	2	no	—
239	#22785	—	53	1	no	maximal, Sylow
240	C₂²×C₅₃	—	2	2	no	—
241	C₂²×C₅₃	—	2	2	no	—
242	C₂²×C₅₃	—	2	2	no	—
243	C₂²×C₅₃	—	2	2	no	—
244	C₂²×C₅₃	—	2	2	no	—
245	C₂²×C₅₃	—	2	2	no	—
246	C₂²×C₅₃	—	2	2	no	—
247	C₂²×C₅₃	—	2	2	no	—
248	C₂²×C₅₃	—	2	2	no	—
249	C₂²×C₅₃	—	4	1	no	—
250	C₂²×C₅₃	—	4	1	no	—
251	C₂²×C₅₃	—	4	1	no	—
252	C₂²×C₅₃	—	4	1	no	—
253	C₂²×C₅₃	#431	1	4	yes	—
254	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
255	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
256	C₂²×C₅₃	#518	1	1	yes	characteristic
257	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
258	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
259	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
260	C₂²×C₅₃	#568	1	1	yes	characteristic
261	C₄×C₅₃	—	2	2	no	—
262	C₄×C₅₃	—	2	2	no	—
263	C₄×C₅₃	—	2	2	no	—
264	C₄×C₅₃	—	2	2	no	—
265	C₄×C₅₃	—	2	2	no	—
266	C₄×C₅₃	—	2	2	no	—
267	C₄×C₅₃	—	2	2	no	—
268	C₄×C₅₃	—	2	2	no	—
269	C₄×C₅₃	—	2	2	no	—
270	C₄×C₅₃	—	4	1	no	—
271	C₄×C₅₃	—	4	1	no	—
272	C₄×C₅₃	—	4	1	no	—
273	C₄×C₅₃	#551	1	2	yes	—
274	C₄×C₅₃	#553	1	2	yes	—
275	C₄²×D₄	D₅₃	1	1	yes	characteristic
276	#1923	—	53	2	no	—
277	#1916	—	53	2	no	—
278	#1869	—	53	2	no	—
279	#1839	—	53	1	no	—
280	#1817	—	53	1	no	—
281	#1797	—	53	1	no	—
282	C₈×C₂²⋊C₄	—	53	2	no	—
283	C₄×C₂²⋊C₈	—	53	1	no	—
284	#1744	—	53	1	no	—
285	C₂×#336	—	53	1	no	—
286	C₂×C₅₃	—	2	4	no	—
287	C₂×C₅₃	—	4	2	no	—
288	C₂×C₅₃	#2353	1	1	yes	characteristic
289	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
290	C₂×C₅₃	#2925	1	1	yes	characteristic
291	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
292	C₂×C₅₃	#2970	1	1	yes	characteristic
293	C₂×C₅₃	#2972	1	1	yes	characteristic
294	C₂×C₅₃	#2974	1	1	yes	characteristic
295	C₂³×C₈	—	53	2	no	—
296	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
297	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
298	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
299	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
300	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
301	C₂×C₄⋊C₈	—	53	2	no	—
302	C₂×C₂²⋊C₈	—	53	2	no	—
303	C₂×C₂²⋊C₈	—	53	2	no	—
304	C₂×C₂²⋊C₈	—	53	2	no	—
305	C₂×C₄×C₈	—	53	2	no	—
306	C₂×C₄×C₈	—	53	2	no	—
307	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
308	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
309	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
310	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
311	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
312	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
313	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
314	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
315	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
316	C₄³	C₂×D₅₃	1	1	yes	characteristic
317	#336	—	53	1	no	—
318	#336	—	53	1	no	—
319	#336	—	53	1	no	—
320	#336	—	53	2	no	—
321	#336	—	53	2	no	—
322	#336	—	53	2	no	—
323	#336	—	53	2	no	—
324	#336	—	53	2	no	—
325	#336	—	53	1	no	—
326	C₅₃	#22785	1	1	yes	characteristic, complemented, Sylow
327	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
328	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
329	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
330	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
331	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
332	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
333	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
334	C₂²×C₈	—	53	2	no	—
335	C₂²×C₈	—	53	2	no	—
336	C₂²×C₈	—	53	2	no	—
337	C₂²×C₈	—	53	2	no	—
338	C₂²×C₈	—	106	4	no	—
339	C₂²×C₈	—	106	4	no	—
340	C₂²×C₈	—	212	2	no	—
341	C₄×D₄	—	2	2	no	—
342	C₄×D₄	—	2	2	no	—
343	C₄×D₄	—	2	4	no	—
344	C₄×D₄	#88322	1	8	yes	complemented
345	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
346	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
347	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
348	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
349	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
350	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
351	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
352	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
353	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
354	C₂×C₄²	—	4	2	no	—
355	C₂×C₄²	—	4	2	no	—
356	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
357	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
358	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
359	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
360	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
361	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
362	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
363	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
364	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
365	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
366	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
367	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
368	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
369	C₄⋊C₈	—	212	2	no	—
370	C₂²⋊C₈	—	106	4	no	—
371	C₂²⋊C₈	—	106	4	no	—
372	C₂²⋊C₈	—	212	2	no	—
373	C₄×C₈	—	106	4	no	—
374	C₄×C₈	—	212	2	no	—
375	#95	C₂×Dic₅₃	1	1	yes	characteristic
376	#95	C₂×Dic₅₃	1	1	yes	characteristic
377	#95	C₂×Dic₅₃	1	1	yes	characteristic
378	#95	C₂×Dic₅₃	1	1	yes	characteristic
379	#95	C₂×Dic₅₃	1	1	yes	characteristic
380	#95	C₂×Dic₅₃	1	1	yes	characteristic
381	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
382	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	C₂×#88322	1	4	yes	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	4	1	no	—
397	C₂²×C₄	—	4	1	no	—
398	C₂²×C₄	—	4	1	no	—
399	C₂²×C₄	—	4	1	no	—
400	C₂²×C₄	—	4	1	no	—
401	C₂²×C₄	—	4	1	no	—
402	C₂²×C₄	C₂×#88322	1	4	yes	—
403	C₂²×C₄	C₂×#88322	1	4	yes	complemented
404	C₂²×C₄	#11719945	1	2	yes	—
405	C₂²×C₄	#11719945	1	2	yes	—
406	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
407	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
408	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
409	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
410	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
411	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
412	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
413	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
414	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
415	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
416	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
417	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
418	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
419	C₂×C₈	—	53	4	no	—
420	C₂×C₈	—	106	2	no	—
421	C₂×C₈	—	106	2	no	—
422	C₂×C₈	—	106	2	no	—
423	C₂×C₈	—	106	2	no	—
424	C₂×C₈	—	106	2	no	—
425	C₂×C₈	—	106	2	no	—
426	C₂×C₈	—	106	2	no	—
427	C₂×C₈	—	106	2	no	—
428	C₂×C₈	—	106	2	no	—
429	C₂×C₈	—	106	2	no	—
430	C₂×C₈	—	106	2	no	—
431	C₂×C₈	—	212	8	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	C₂×#88322	1	4	yes	complemented
435	C₂²⋊C₄	—	2	2	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	C₂×#88322	1	4	yes	—
439	C₂²⋊C₄	C₂×#88322	1	4	yes	complemented
440	C₂²⋊C₄	#11719917	1	4	yes	—
441	C₄²	—	2	2	no	—
442	C₄²	—	2	2	no	—
443	C₄²	—	2	2	no	—
444	C₄²	—	2	1	no	—
445	C₄²	—	2	1	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	2	no	—
449	C₄²	—	4	1	no	—
450	C₄²	—	4	1	no	—
451	C₄²	—	4	1	no	—
452	C₄²	—	4	1	no	—
453	C₄²	—	4	1	no	—
454	C₄²	C₂×#88322	1	4	yes	complemented
455	C₂³	—	2	1	no	—
456	C₂³	—	2	1	no	—
457	C₂³	—	2	1	no	—
458	C₂³	—	2	1	no	—
459	C₂³	—	2	1	no	—
460	C₂³	—	2	1	no	—
461	C₂³	—	2	1	no	—
462	C₂³	—	2	1	no	—
463	C₂³	—	2	1	no	—
464	C₂³	—	2	1	no	—
465	C₂³	#13133176	1	2	yes	—
466	C₂³	C₂²×#88322	1	2	yes	—
467	C₂³	C₂²×#88322	1	2	yes	—
468	C₂³	C₂×#11719917	1	2	yes	—
469	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
470	D₄	—	2	8	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	4	4	no	—
501	C₂×C₄	—	4	4	no	—
502	C₂×C₄	—	4	4	no	—
503	C₂×C₄	—	4	4	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	#13133176	1	2	yes	—
507	C₂×C₄	C₂²×#88322	1	2	yes	—
508	C₂×C₄	C₂²×#88322	1	2	yes	—
509	C₂×C₄	C₂²×#88322	1	2	yes	—
510	C₂×C₄	C₂²×#88322	1	2	yes	—
511	C₂×C₄	C₂²×#88322	1	2	yes	—
512	C₂×C₄	C₂×#11719917	1	2	yes	—
513	C₂×C₄	C₂×#11719917	1	2	yes	—
514	C₂×C₄	#13133224	1	2	yes	—
515	C₂×C₄	#41×D₅₃	1	2	yes	—
516	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
517	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
518	C₂×C₄	#13133262	1	2	yes	—
519	C₈	—	106	4	no	—
520	C₈	—	212	2	no	—
521	C₈	—	212	2	no	—
522	C₈	—	212	2	no	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	2	2	no	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	2	no	—
529	C₂²	—	2	2	no	—
530	C₂²	—	2	2	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	4	1	no	—
536	C₂²	#16810676	1	4	yes	—
537	C₂²	C₂×#13133176	1	1	yes	characteristic
538	C₂²	#41×Dic₅₃	1	1	yes	characteristic
539	C₂²	#16811098	1	1	yes	characteristic
540	C₂²	C₂³×#88322	1	1	yes	characteristic
541	C₂²	C₂²×#11719917	1	1	yes	characteristic
542	C₂²	C₂×#13133224	1	1	yes	characteristic
543	C₂²	#16811390	1	1	yes	characteristic
544	C₄	—	2	2	no	—
545	C₄	—	2	2	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	2	no	—
548	C₄	—	2	2	no	—
549	C₄	—	2	2	no	—
550	C₄	—	2	2	no	—
551	C₄	—	2	2	no	—
552	C₄	—	2	2	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	#16811302	1	2	yes	—
557	C₄	#16811322	1	2	yes	—
558	C₂	—	2	4	no	—
559	C₂	—	4	2	no	—
560	C₂	#20277655	1	1	yes	characteristic
561	C₂	C₂×#16810676	1	1	yes	characteristic
562	C₂	#20283381	1	1	yes	characteristic
563	C₂	#41×#88322	1	1	yes	characteristic
564	C₂	#20283729	1	1	yes	characteristic
565	C₂	#20283739	1	1	yes	characteristic
566	C₂	#20283750	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₄²×D₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂²×D₄×C₅₃	C₂×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×D₄	D₅₃
hypocenter	C₅₃	#22785
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20275323	C₂	semidirect
2. #20275274	C₂	semidirect
4. #20274858	C₂	semidirect
5. #20274783	C₂	semidirect
6. #20274695	C₂	semidirect
7. C₂²⋊C₄×#88322	C₂	semidirect
8. C₄×C₂²⋊#88322	C₂	semidirect
9. #20274461	C₂	semidirect
10. C₂×#16810378	C₂	semidirect
66. C₄×D₄×C₅₃	C₈	semidirect
119. C₂²×C₄×C₅₃	C₂×C₈	semidirect
137. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
142. C₂²⋊C₄×C₅₃	C₂×C₈	semidirect
157. C₄²×C₅₃	C₂×C₈	semidirect
326. C₅₃	#22785	semidirect
344. C₄×D₄	#88322	semidirect
403. C₂²×C₄	C₂×#88322	semidirect
434. C₄⋊C₄	C₂×#88322	semidirect
439. C₂²⋊C₄	C₂×#88322	semidirect
454. C₄²	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
#20275323	2
#20275274	2
#20275008	2
#20274858	2
#20274783	2
#20274695	2
C₂²⋊C₄×#88322	2
C₄×C₂²⋊#88322	2
#20274461	2
C₂×#16810378	2
C₄²×D₄×C₅₃	2
#22785	53