Subgroups — #27401478

Order 13568.

864 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275274	C₂	1	2	yes	maximal, complemented
2	#20275274	C₂	1	2	yes	maximal, complemented
3	#20274895	C₂	1	2	yes	maximal
4	#20274858	C₂	1	1	yes	maximal, characteristic, complemented
5	#20274832	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274783	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²⋊C₄×#88322	C₂	1	2	yes	maximal
9	#20274461	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16810378	C₂	1	1	yes	maximal, characteristic
11	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×#88322	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	C₂×C₄⋊#88322	C₂²	1	2	yes	—
15	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
16	C₂×C₄×#88322	C₂²	1	2	yes	—
17	C₂×C₄×#88322	C₂²	1	2	yes	—
18	#16810378	C₂²	1	2	yes	—
19	#16810378	C₂²	1	1	yes	characteristic
20	#16810378	C₂²	1	1	yes	characteristic
21	#16810378	C₂²	1	2	yes	—
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	1	yes	characteristic
27	#16810378	C₂²	1	1	yes	characteristic
28	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
29	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
30	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
39	C₅₃×#376	C₄	1	1	yes	characteristic
40	C₅₃×#376	C₄	1	1	yes	characteristic
41	C₄³×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	—	2	4	no	—
51	C₄⋊#88322	—	4	2	no	—
52	C₂²⋊#88322	—	2	4	no	—
53	C₄×#88322	—	2	4	no	—
54	C₄×#88322	—	4	2	no	—
55	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
58	#117×C₅₃	—	2	4	no	—
59	#117×C₅₃	C₈	1	8	yes	complemented
60	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	—	2	4	no	—
69	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
85	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
86	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
87	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
88	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
89	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
90	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
91	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
92	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	—	2	2	no	—
95	C₂²×C₄×C₅₃	—	2	2	no	—
96	C₂²×C₄×C₅₃	—	2	2	no	—
97	C₂²×C₄×C₅₃	—	2	1	no	—
98	C₂²×C₄×C₅₃	—	2	1	no	—
99	C₂²×C₄×C₅₃	—	2	2	no	—
100	C₂²×C₄×C₅₃	—	2	2	no	—
101	C₂²×C₄×C₅₃	—	4	1	no	—
102	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
103	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	#41	1	4	yes	—
117	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₄⋊C₄×C₅₃	—	2	2	no	—
119	C₄⋊C₄×C₅₃	—	2	2	no	—
120	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
121	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
122	C₂²⋊C₄×C₅₃	—	2	2	no	—
123	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
124	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
125	C₂²⋊C₄×C₅₃	M₁₆	1	4	yes	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	2	2	no	—
128	C₄²×C₅₃	—	2	2	no	—
129	C₄²×C₅₃	—	2	2	no	—
130	C₄²×C₅₃	—	2	2	no	—
131	C₄²×C₅₃	—	2	2	no	—
132	C₄²×C₅₃	—	2	2	no	—
133	C₄²×C₅₃	—	2	2	no	—
134	C₄²×C₅₃	—	2	2	no	—
135	C₄²×C₅₃	—	2	2	no	—
136	C₄²×C₅₃	—	2	1	no	—
137	C₄²×C₅₃	—	2	1	no	—
138	C₄²×C₅₃	—	2	1	no	—
139	C₄²×C₅₃	—	2	1	no	—
140	C₄²×C₅₃	—	4	1	no	—
141	C₄²×C₅₃	—	4	1	no	—
142	C₄²×C₅₃	C₂×C₈	1	4	yes	complemented
143	C₄²×C₅₃	C₂×C₈	1	4	yes	complemented
144	C₂×#88322	—	2	2	no	—
145	C₂×#88322	—	2	2	no	—
146	C₂×#88322	—	2	2	no	—
147	C₂×#88322	—	2	2	no	—
148	C₂×#88322	—	2	2	no	—
149	C₂×#88322	—	2	2	no	—
150	C₂×#88322	—	2	2	no	—
151	C₂×#88322	—	2	2	no	—
152	C₂×#88322	—	2	2	no	—
153	C₂×#88322	—	2	2	no	—
154	C₂×#88322	—	2	2	no	—
155	C₂×#88322	—	4	8	no	—
156	C₂×#88322	#41	1	4	yes	—
157	C₂³×C₅₃	—	2	1	no	—
158	C₂³×C₅₃	—	2	1	no	—
159	C₂³×C₅₃	—	2	1	no	—
160	C₂³×C₅₃	—	2	1	no	—
161	C₂³×C₅₃	#117	1	2	yes	—
162	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
163	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
164	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₄×C₅₃	—	2	1	no	—
166	C₂×C₄×C₅₃	—	2	1	no	—
167	C₂×C₄×C₅₃	—	2	1	no	—
168	C₂×C₄×C₅₃	—	2	1	no	—
169	C₂×C₄×C₅₃	—	2	1	no	—
170	C₂×C₄×C₅₃	—	2	1	no	—
171	C₂×C₄×C₅₃	—	2	1	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	1	no	—
175	C₂×C₄×C₅₃	—	2	1	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	2	1	no	—
179	C₂×C₄×C₅₃	—	2	1	no	—
180	C₂×C₄×C₅₃	—	2	1	no	—
181	C₂×C₄×C₅₃	—	2	1	no	—
182	C₂×C₄×C₅₃	—	2	1	no	—
183	C₂×C₄×C₅₃	—	2	1	no	—
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	8	no	—
186	C₂×C₄×C₅₃	—	2	4	no	—
187	C₂×C₄×C₅₃	—	2	4	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	—	4	4	no	—
198	C₂×C₄×C₅₃	#117	1	2	yes	—
199	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
200	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
201	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
202	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
203	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
204	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
205	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
206	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
207	C₂×C₄×C₅₃	#131	1	2	yes	—
208	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
209	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
210	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
211	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
212	#88322	—	2	4	no	—
213	#88322	—	4	2	no	—
214	#88322	—	4	2	no	—
215	#88322	—	4	2	no	—
216	#22589	—	53	1	no	maximal, Sylow
217	C₂²×C₅₃	—	2	2	no	—
218	C₂²×C₅₃	—	2	2	no	—
219	C₂²×C₅₃	—	2	2	no	—
220	C₂²×C₅₃	—	2	2	no	—
221	C₂²×C₅₃	—	2	2	no	—
222	C₂²×C₅₃	—	2	2	no	—
223	C₂²×C₅₃	#431	1	4	yes	—
224	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
225	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
226	C₂²×C₅₃	#520	1	1	yes	characteristic
227	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
228	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
229	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
230	C₂²×C₅₃	#568	1	1	yes	characteristic
231	C₄×C₅₃	—	2	2	no	—
232	C₄×C₅₃	—	2	2	no	—
233	C₄×C₅₃	—	2	2	no	—
234	C₄×C₅₃	—	2	2	no	—
235	C₄×C₅₃	—	2	2	no	—
236	C₄×C₅₃	—	2	2	no	—
237	C₄×C₅₃	—	2	2	no	—
238	C₄×C₅₃	—	2	2	no	—
239	C₄×C₅₃	—	2	2	no	—
240	C₄×C₅₃	—	2	2	no	—
241	C₄×C₅₃	—	4	1	no	—
242	C₄×C₅₃	—	4	1	no	—
243	C₄×C₅₃	—	4	1	no	—
244	C₄×C₅₃	#551	1	2	yes	—
245	C₄×C₅₃	#552	1	2	yes	—
246	C₄×#117	D₅₃	1	1	yes	characteristic
247	#1916	—	53	2	no	—
248	#1916	—	53	2	no	—
249	#1850	—	53	2	no	—
250	#1839	—	53	1	no	—
251	#1831	—	53	1	no	—
252	#1817	—	53	1	no	—
253	C₄×C₄⋊C₈	—	53	1	no	—
254	C₈×C₂²⋊C₄	—	53	2	no	—
255	#1744	—	53	1	no	—
256	C₂×#336	—	53	1	no	—
257	C₂×C₅₃	—	2	4	no	—
258	C₂×C₅₃	#2343	1	1	yes	characteristic
259	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
260	C₂×C₅₃	#2924	1	1	yes	characteristic
261	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
262	C₂×C₅₃	#2970	1	1	yes	characteristic
263	C₂×C₅₃	#2988	1	1	yes	characteristic
264	C₂×C₅₃	#2990	1	1	yes	characteristic
265	C₂³×C₈	—	53	2	no	—
266	C₂×#117	Dic₅₃	1	1	yes	characteristic
267	C₂×#117	Dic₅₃	1	1	yes	characteristic
268	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
269	C₂×C₄⋊C₈	—	53	2	no	—
270	C₂×C₄⋊C₈	—	53	2	no	—
271	C₂×C₂²⋊C₈	—	53	2	no	—
272	C₂×C₄×C₈	—	53	2	no	—
273	C₂×C₄×C₈	—	53	2	no	—
274	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
275	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
276	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
277	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
278	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
279	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
280	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
281	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
282	#376	Dic₅₃	1	1	yes	characteristic
283	#376	Dic₅₃	1	1	yes	characteristic
284	C₄³	C₂×D₅₃	1	1	yes	characteristic
285	C₄³	C₂×D₅₃	1	1	yes	characteristic
286	#336	—	53	1	no	—
287	#336	—	53	1	no	—
288	#336	—	53	2	no	—
289	#336	—	53	1	no	—
290	#336	—	53	2	no	—
291	#336	—	53	2	no	—
292	#336	—	53	2	no	—
293	#336	—	53	2	no	—
294	#336	—	53	2	no	—
295	#336	—	53	1	no	—
296	C₅₃	#22589	1	1	yes	characteristic, complemented, Sylow
297	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
298	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
299	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
300	C₂²×C₈	—	53	2	no	—
301	C₂²×C₈	—	53	2	no	—
302	C₂²×C₈	—	53	2	no	—
303	C₂²×C₈	—	53	2	no	—
304	C₂²×C₈	—	106	4	no	—
305	C₂²×C₈	—	106	4	no	—
306	C₂²×C₈	—	212	2	no	—
307	#117	—	2	4	no	—
308	#117	#88322	1	8	yes	complemented
309	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
314	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
315	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
316	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
317	C₂×C₄²	—	2	4	no	—
318	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
319	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
320	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
321	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
322	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
323	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
324	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
325	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
326	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
327	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
328	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
329	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
330	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
331	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
332	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
333	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
334	C₄⋊C₈	—	106	4	no	—
335	C₄⋊C₈	—	212	2	no	—
336	C₂²⋊C₈	—	106	4	no	—
337	C₄×C₈	—	106	4	no	—
338	C₄×C₈	—	212	2	no	—
339	#95	C₂×Dic₅₃	1	1	yes	characteristic
340	#95	C₂×Dic₅₃	1	1	yes	characteristic
341	#95	C₂×Dic₅₃	1	1	yes	characteristic
342	#95	C₂×Dic₅₃	1	1	yes	characteristic
343	#95	C₂×Dic₅₃	1	1	yes	characteristic
344	#95	C₂×Dic₅₃	1	1	yes	characteristic
345	#95	C₂×Dic₅₃	1	1	yes	characteristic
346	#95	C₂×Dic₅₃	1	1	yes	characteristic
347	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	4	1	no	—
356	C₂²×C₄	C₂×#88322	1	4	yes	—
357	C₂²×C₄	#11719945	1	4	yes	—
358	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
359	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
360	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
361	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
362	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
363	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
364	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
365	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
366	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
367	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
368	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
369	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
370	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
371	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
372	C₂×C₈	—	53	4	no	—
373	C₂×C₈	—	106	2	no	—
374	C₂×C₈	—	106	2	no	—
375	C₂×C₈	—	106	2	no	—
376	C₂×C₈	—	106	2	no	—
377	C₂×C₈	—	106	2	no	—
378	C₂×C₈	—	106	2	no	—
379	C₂×C₈	—	106	2	no	—
380	C₂×C₈	—	106	2	no	—
381	C₂×C₈	—	106	2	no	—
382	C₂×C₈	—	106	2	no	—
383	C₂×C₈	—	106	2	no	—
384	C₂×C₈	—	212	8	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	C₂×#88322	1	4	yes	complemented
388	C₄⋊C₄	C₂×#88322	1	4	yes	complemented
389	C₂²⋊C₄	—	2	2	no	—
390	C₂²⋊C₄	C₂×#88322	1	4	yes	—
391	C₂²⋊C₄	C₂×#88322	1	4	yes	—
392	C₂²⋊C₄	#11719917	1	4	yes	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	2	1	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	4	1	no	—
408	C₄²	—	4	1	no	—
409	C₄²	C₂×#88322	1	4	yes	complemented
410	C₄²	C₂×#88322	1	4	yes	complemented
411	C₂³	—	2	1	no	—
412	C₂³	—	2	1	no	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	#13133176	1	2	yes	—
416	C₂³	C₂²×#88322	1	2	yes	—
417	C₂³	C₂×#11719917	1	2	yes	—
418	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	8	no	—
440	C₂×C₄	—	2	4	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	#13133176	1	2	yes	—
453	C₂×C₄	C₂²×#88322	1	2	yes	—
454	C₂×C₄	C₂²×#88322	1	2	yes	—
455	C₂×C₄	C₂²×#88322	1	2	yes	—
456	C₂×C₄	C₂²×#88322	1	2	yes	—
457	C₂×C₄	C₂²×#88322	1	2	yes	—
458	C₂×C₄	C₂²×#88322	1	2	yes	—
459	C₂×C₄	C₂×#11719917	1	2	yes	—
460	C₂×C₄	C₂×#11719917	1	2	yes	—
461	C₂×C₄	#13133224	1	2	yes	—
462	C₂×C₄	#41×D₅₃	1	2	yes	—
463	C₂×C₄	C₂×#11719945	1	2	yes	—
464	C₂×C₄	#13133263	1	1	yes	characteristic
465	C₂×C₄	#13133264	1	1	yes	characteristic
466	C₈	—	106	4	no	—
467	C₈	—	212	2	no	—
468	C₈	—	212	2	no	—
469	C₈	—	212	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	#16810676	1	4	yes	—
477	C₂²	C₂×#13133176	1	1	yes	characteristic
478	C₂²	#41×Dic₅₃	1	1	yes	characteristic
479	C₂²	#16811106	1	1	yes	characteristic
480	C₂²	C₂³×#88322	1	1	yes	characteristic
481	C₂²	C₂²×#11719917	1	1	yes	characteristic
482	C₂²	C₂×#13133224	1	1	yes	characteristic
483	C₂²	#16811390	1	1	yes	characteristic
484	C₄	—	2	2	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	2	no	—
487	C₄	—	2	2	no	—
488	C₄	—	2	2	no	—
489	C₄	—	2	2	no	—
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	2	2	no	—
494	C₄	—	4	1	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	#16811297	1	2	yes	—
498	C₄	#16811306	1	2	yes	—
499	C₂	—	2	4	no	—
500	C₂	#20277575	1	1	yes	characteristic
501	C₂	C₂×#16810676	1	1	yes	characteristic
502	C₂	#20283373	1	1	yes	characteristic
503	C₂	#41×#88322	1	1	yes	characteristic
504	C₂	#20283729	1	1	yes	characteristic
505	C₂	#20283826	1	1	yes	characteristic
506	C₂	#20283840	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₄×#117×C₅₃	C₂
socle	C₂³×C₅₃	C₂³×C₄
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#117	D₅₃
hypocenter	C₅₃	#22589
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
1. #20275274	C₂	semidirect
2. #20275274	C₂	semidirect
4. #20274858	C₂	semidirect
5. #20274832	C₂	semidirect
7. C₄×C₄⋊#88322	C₂	semidirect
9. #20274461	C₂	semidirect
59. #117×C₅₃	C₈	semidirect
120. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
121. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
142. C₄²×C₅₃	C₂×C₈	semidirect
143. C₄²×C₅₃	C₂×C₈	semidirect
296. C₅₃	#22589	semidirect
308. #117	#88322	semidirect
387. C₄⋊C₄	C₂×#88322	semidirect
388. C₄⋊C₄	C₂×#88322	semidirect
409. C₄²	C₂×#88322	semidirect
410. 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
#20275274	2
#20274895	2
#20274858	2
#20274832	2
#20274783	2
C₄×C₄⋊#88322	2
C₂²⋊C₄×#88322	2
#20274461	2
C₂×#16810378	2
C₄×#117×C₅₃	2
#22589	53