Subgroups — #27403113

Order 13568.

1052 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275323	C₂	1	2	yes	maximal, complemented
4	#20275274	C₂	1	2	yes	maximal, complemented
5	#20275008	C₂	1	2	yes	maximal
6	#20274832	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274695	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274695	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₄×#88322	C₂	1	2	yes	maximal, complemented
11	C₄²×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×#88322	C₂²	1	2	yes	—
13	#16810676	—	2	4	no	—
14	C₂×C₄⋊#88322	—	2	4	no	—
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#88322	C₂²	1	2	yes	—
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
21	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
22	C₂×C₂²⋊#88322	C₂²	1	2	yes	—
23	C₂×C₄×#88322	C₂²	1	2	yes	—
24	C₂×C₄×#88322	C₂²	1	2	yes	—
25	#16810378	C₂²	1	2	yes	—
26	#16810378	C₂²	1	2	yes	—
27	#16810378	C₂²	1	2	yes	—
28	#16810378	C₂²	1	2	yes	—
29	#16810378	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
31	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
32	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₅₃	C₂²	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	C₄	1	1	yes	characteristic
41	C₄×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
44	C₄³×C₅₃	C₂²	1	1	yes	characteristic
45	C₂²×#88322	—	2	4	no	—
46	C₂²×#88322	—	2	4	no	—
47	C₂²×#88322	—	4	2	no	—
48	C₂²×#88322	C₂³	1	2	yes	—
49	C₂²×#88322	C₂³	1	2	yes	—
50	C₂²×#88322	C₂³	1	2	yes	—
51	C₂²×#88322	C₂³	1	2	yes	—
52	C₄⋊#88322	—	2	2	no	—
53	C₄⋊#88322	—	2	2	no	—
54	C₄⋊#88322	—	2	2	no	—
55	C₄⋊#88322	—	2	4	no	—
56	C₄⋊#88322	D₄	1	4	yes	complemented
57	C₂²⋊#88322	—	2	4	no	—
58	C₂²⋊#88322	—	2	4	no	—
59	C₂²⋊#88322	—	4	2	no	—
60	C₄×#88322	—	2	4	no	—
61	C₄×#88322	—	4	2	no	—
62	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₅₃	—	2	4	no	—
70	C₄×D₄×C₅₃	—	2	2	no	—
71	C₄×D₄×C₅₃	—	2	2	no	—
72	C₄×D₄×C₅₃	C₈	1	8	yes	complemented
73	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₅₃	—	2	2	no	—
83	C₂×C₄²×C₅₃	—	2	2	no	—
84	C₂×C₄²×C₅₃	—	2	2	no	—
85	C₂×C₄²×C₅₃	—	2	2	no	—
86	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
100	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
101	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
102	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
103	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
104	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
105	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×D₄×C₅₃	—	2	2	no	—
108	C₂×D₄×C₅₃	—	2	2	no	—
109	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
110	C₂²×C₄×C₅₃	—	2	2	no	—
111	C₂²×C₄×C₅₃	—	2	2	no	—
112	C₂²×C₄×C₅₃	—	2	2	no	—
113	C₂²×C₄×C₅₃	—	2	2	no	—
114	C₂²×C₄×C₅₃	—	2	1	no	—
115	C₂²×C₄×C₅₃	—	2	1	no	—
116	C₂²×C₄×C₅₃	—	2	1	no	—
117	C₂²×C₄×C₅₃	—	2	1	no	—
118	C₂²×C₄×C₅₃	—	2	2	no	—
119	C₂²×C₄×C₅₃	—	2	2	no	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	—	2	2	no	—
122	C₂²×C₄×C₅₃	—	2	1	no	—
123	C₂²×C₄×C₅₃	—	2	1	no	—
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	—	2	1	no	—
126	C₂²×C₄×C₅₃	—	2	1	no	—
127	C₂²×C₄×C₅₃	—	2	1	no	—
128	C₂²×C₄×C₅₃	—	4	1	no	—
129	C₂²×C₄×C₅₃	—	4	1	no	—
130	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
131	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	complemented
132	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	#41	1	2	yes	—
145	C₂²×C₄×C₅₃	#41	1	2	yes	—
146	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
147	C₄⋊C₄×C₅₃	—	2	2	no	—
148	C₄⋊C₄×C₅₃	—	2	2	no	—
149	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
150	C₂²⋊C₄×C₅₃	—	2	2	no	—
151	C₂²⋊C₄×C₅₃	—	2	2	no	—
152	C₂²⋊C₄×C₅₃	—	2	2	no	—
153	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
154	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	complemented
155	C₂²⋊C₄×C₅₃	M₁₆	1	4	yes	—
156	C₄²×C₅₃	—	2	2	no	—
157	C₄²×C₅₃	—	2	2	no	—
158	C₄²×C₅₃	—	2	2	no	—
159	C₄²×C₅₃	—	2	2	no	—
160	C₄²×C₅₃	—	2	2	no	—
161	C₄²×C₅₃	—	2	2	no	—
162	C₄²×C₅₃	—	2	1	no	—
163	C₄²×C₅₃	—	2	1	no	—
164	C₄²×C₅₃	—	2	1	no	—
165	C₄²×C₅₃	—	2	1	no	—
166	C₄²×C₅₃	—	2	1	no	—
167	C₄²×C₅₃	—	2	1	no	—
168	C₄²×C₅₃	—	2	1	no	—
169	C₄²×C₅₃	—	2	1	no	—
170	C₄²×C₅₃	—	4	1	no	—
171	C₄²×C₅₃	C₂×C₈	1	4	yes	complemented
172	C₄²×C₅₃	C₂×D₄	1	2	yes	—
173	C₄²×C₅₃	#41	1	2	yes	—
174	C₂×#88322	—	2	2	no	—
175	C₂×#88322	—	2	2	no	—
176	C₂×#88322	—	2	2	no	—
177	C₂×#88322	—	2	2	no	—
178	C₂×#88322	—	2	2	no	—
179	C₂×#88322	—	2	2	no	—
180	C₂×#88322	—	2	2	no	—
181	C₂×#88322	—	2	2	no	—
182	C₂×#88322	—	2	2	no	—
183	C₂×#88322	—	2	2	no	—
184	C₂×#88322	—	2	2	no	—
185	C₂×#88322	—	4	8	no	—
186	C₂×#88322	C₂×D₄	1	4	yes	—
187	C₂³×C₅₃	—	2	1	no	—
188	C₂³×C₅₃	—	2	1	no	—
189	C₂³×C₅₃	—	2	1	no	—
190	C₂³×C₅₃	—	2	1	no	—
191	C₂³×C₅₃	—	2	1	no	—
192	C₂³×C₅₃	—	2	1	no	—
193	C₂³×C₅₃	—	2	1	no	—
194	C₂³×C₅₃	—	2	1	no	—
195	C₂³×C₅₃	—	2	1	no	—
196	C₂³×C₅₃	—	2	1	no	—
197	C₂³×C₅₃	#117	1	2	yes	—
198	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
199	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
200	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
201	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
202	D₄×C₅₃	—	2	8	no	—
203	C₂×C₄×C₅₃	—	2	1	no	—
204	C₂×C₄×C₅₃	—	2	1	no	—
205	C₂×C₄×C₅₃	—	2	1	no	—
206	C₂×C₄×C₅₃	—	2	1	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	—	2	1	no	—
217	C₂×C₄×C₅₃	—	2	1	no	—
218	C₂×C₄×C₅₃	—	2	1	no	—
219	C₂×C₄×C₅₃	—	2	2	no	—
220	C₂×C₄×C₅₃	—	2	2	no	—
221	C₂×C₄×C₅₃	—	2	2	no	—
222	C₂×C₄×C₅₃	—	2	2	no	—
223	C₂×C₄×C₅₃	—	2	8	no	—
224	C₂×C₄×C₅₃	—	2	8	no	—
225	C₂×C₄×C₅₃	—	2	1	no	—
226	C₂×C₄×C₅₃	—	2	1	no	—
227	C₂×C₄×C₅₃	—	2	1	no	—
228	C₂×C₄×C₅₃	—	2	1	no	—
229	C₂×C₄×C₅₃	—	2	1	no	—
230	C₂×C₄×C₅₃	—	2	1	no	—
231	C₂×C₄×C₅₃	—	2	1	no	—
232	C₂×C₄×C₅₃	—	2	1	no	—
233	C₂×C₄×C₅₃	—	2	1	no	—
234	C₂×C₄×C₅₃	—	4	4	no	—
235	C₂×C₄×C₅₃	—	4	4	no	—
236	C₂×C₄×C₅₃	—	4	1	no	—
237	C₂×C₄×C₅₃	—	4	1	no	—
238	C₂×C₄×C₅₃	—	4	1	no	—
239	C₂×C₄×C₅₃	—	4	1	no	—
240	C₂×C₄×C₅₃	#117	1	2	yes	—
241	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
242	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
243	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
244	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
245	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
246	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
247	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
248	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
249	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
250	C₂×C₄×C₅₃	#131	1	2	yes	—
251	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
255	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
257	#88322	—	2	4	no	—
258	#88322	—	4	2	no	—
259	#88322	—	4	2	no	—
260	#88322	—	4	2	no	—
261	#22748	—	53	1	no	maximal, Sylow
262	C₂²×C₅₃	—	2	2	no	—
263	C₂²×C₅₃	—	2	2	no	—
264	C₂²×C₅₃	—	2	2	no	—
265	C₂²×C₅₃	—	2	2	no	—
266	C₂²×C₅₃	—	2	2	no	—
267	C₂²×C₅₃	—	2	2	no	—
268	C₂²×C₅₃	—	2	2	no	—
269	C₂²×C₅₃	—	2	2	no	—
270	C₂²×C₅₃	—	2	2	no	—
271	C₂²×C₅₃	—	4	1	no	—
272	C₂²×C₅₃	—	4	1	no	—
273	C₂²×C₅₃	—	4	1	no	—
274	C₂²×C₅₃	—	4	1	no	—
275	C₂²×C₅₃	#431	1	4	yes	complemented
276	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
277	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
278	C₂²×C₅₃	#520	1	1	yes	characteristic
279	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
280	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
281	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
282	C₂²×C₅₃	#568	1	1	yes	characteristic
283	C₄×C₅₃	—	2	2	no	—
284	C₄×C₅₃	—	2	2	no	—
285	C₄×C₅₃	—	2	2	no	—
286	C₄×C₅₃	—	2	2	no	—
287	C₄×C₅₃	—	2	2	no	—
288	C₄×C₅₃	—	2	2	no	—
289	C₄×C₅₃	—	2	2	no	—
290	C₄×C₅₃	—	2	2	no	—
291	C₄×C₅₃	—	4	1	no	—
292	C₄×C₅₃	—	4	1	no	—
293	C₄×C₅₃	—	4	1	no	—
294	C₄×C₅₃	C₈×D₄	1	4	yes	—
295	C₄×C₅₃	#546	1	1	yes	characteristic
296	C₄×C₅₃	#548	1	1	yes	characteristic
297	C₄×C₅₃	#552	1	1	yes	characteristic
298	C₄×C₅₃	#553	1	1	yes	characteristic
299	C₂×#431	—	53	1	no	—
300	C₂²×C₄⋊C₈	—	53	1	no	—
301	C₄²×D₄	D₅₃	1	1	yes	characteristic
302	#1923	—	53	2	no	—
303	#1916	—	53	2	no	—
304	#1869	—	53	2	no	—
305	#1831	—	53	1	no	—
306	#1797	—	53	1	no	—
307	#1797	—	53	1	no	—
308	C₄×C₄⋊C₈	—	53	1	no	—
309	C₈×C₂²⋊C₄	—	53	2	no	—
310	C₂×C₅₃	—	2	4	no	—
311	C₂×C₅₃	—	4	2	no	—
312	C₂×C₅₃	#2352	1	1	yes	characteristic
313	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
314	C₂×C₅₃	#2925	1	1	yes	characteristic
315	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
316	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
317	C₂×C₅₃	#2988	1	1	yes	characteristic
318	C₂×C₅₃	#2990	1	1	yes	characteristic
319	C₂³×C₈	—	53	2	no	—
320	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
321	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
322	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
323	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
324	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
325	#431	—	106	4	no	—
326	C₂×C₄⋊C₈	—	53	1	no	—
327	C₂×C₄⋊C₈	—	53	1	no	—
328	C₂×C₄⋊C₈	—	53	1	no	—
329	C₂×C₄⋊C₈	—	53	1	no	—
330	C₂×C₄⋊C₈	—	53	2	no	—
331	C₂×C₄⋊C₈	—	106	4	no	—
332	C₂×C₂²⋊C₈	—	53	2	no	—
333	C₂×C₂²⋊C₈	—	53	2	no	—
334	C₂×C₂²⋊C₈	—	53	2	no	—
335	C₂×C₄×C₈	—	53	2	no	—
336	C₂×C₄×C₈	—	53	2	no	—
337	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
338	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
339	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
340	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
341	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
342	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
343	C₄×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
344	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
345	C₄×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
346	C₄³	C₂×D₅₃	1	1	yes	characteristic
347	#336	—	53	2	no	—
348	#336	—	53	2	no	—
349	#336	—	53	2	no	—
350	#336	—	53	2	no	—
351	#336	—	53	2	no	—
352	C₅₃	#22748	1	1	yes	characteristic, complemented, Sylow
353	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
354	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
355	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
356	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
357	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
358	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
359	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
360	C₂²×C₈	—	53	2	no	—
361	C₂²×C₈	—	53	2	no	—
362	C₂²×C₈	—	53	2	no	—
363	C₂²×C₈	—	53	2	no	—
364	C₂²×C₈	—	106	4	no	—
365	C₂²×C₈	—	106	4	no	—
366	C₂²×C₈	—	212	2	no	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	#88322	1	8	yes	complemented
371	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
372	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
374	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
375	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
376	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
377	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
379	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
380	C₂×C₄²	—	2	2	no	—
381	C₂×C₄²	—	2	2	no	—
382	C₂×C₄²	—	2	2	no	—
383	C₂×C₄²	—	2	2	no	—
384	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
385	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
386	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
387	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
388	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
389	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
390	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
391	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
392	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
393	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
394	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
395	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
396	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
397	C₄⋊C₈	—	53	4	no	—
398	C₄⋊C₈	—	106	4	no	—
399	C₄⋊C₈	—	106	2	no	—
400	C₄⋊C₈	—	106	2	no	—
401	C₄⋊C₈	—	106	2	no	—
402	C₂²⋊C₈	—	106	4	no	—
403	C₂²⋊C₈	—	106	4	no	—
404	C₂²⋊C₈	—	212	2	no	—
405	C₄×C₈	—	106	4	no	—
406	C₄×C₈	—	212	2	no	—
407	#95	C₂×Dic₅₃	1	1	yes	characteristic
408	#95	C₂×Dic₅₃	1	1	yes	characteristic
409	#95	C₂×Dic₅₃	1	1	yes	characteristic
410	#95	C₂×Dic₅₃	1	1	yes	characteristic
411	#95	C₂×Dic₅₃	1	1	yes	characteristic
412	#95	C₂×Dic₅₃	1	1	yes	characteristic
413	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
414	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	C₂×#88322	1	4	yes	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	4	1	no	—
437	C₂²×C₄	—	4	1	no	—
438	C₂²×C₄	C₂×#88322	1	4	yes	—
439	C₂²×C₄	C₂×#88322	1	4	yes	complemented
440	C₂²×C₄	#11719945	1	2	yes	—
441	C₂²×C₄	#11719945	1	2	yes	—
442	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
443	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
444	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
445	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
446	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
447	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
448	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
449	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
450	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
451	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
452	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
453	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
454	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
455	C₂×C₈	—	53	4	no	—
456	C₂×C₈	—	106	2	no	—
457	C₂×C₈	—	106	2	no	—
458	C₂×C₈	—	106	2	no	—
459	C₂×C₈	—	106	2	no	—
460	C₂×C₈	—	106	2	no	—
461	C₂×C₈	—	106	2	no	—
462	C₂×C₈	—	106	2	no	—
463	C₂×C₈	—	106	2	no	—
464	C₂×C₈	—	106	2	no	—
465	C₂×C₈	—	106	2	no	—
466	C₂×C₈	—	106	2	no	—
467	C₂×C₈	—	212	8	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	C₂×#88322	1	4	yes	complemented
471	C₂²⋊C₄	—	2	2	no	—
472	C₂²⋊C₄	—	2	2	no	—
473	C₂²⋊C₄	—	2	2	no	—
474	C₂²⋊C₄	C₂×#88322	1	4	yes	—
475	C₂²⋊C₄	C₂×#88322	1	4	yes	complemented
476	C₂²⋊C₄	#11719917	1	4	yes	—
477	C₄²	—	2	2	no	—
478	C₄²	—	2	2	no	—
479	C₄²	—	2	2	no	—
480	C₄²	—	2	2	no	—
481	C₄²	—	2	2	no	—
482	C₄²	—	2	2	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	4	1	no	—
492	C₄²	C₂×#88322	1	4	yes	complemented
493	C₄²	D₄×D₅₃	1	2	yes	—
494	C₄²	#11719947	1	2	yes	—
495	C₂³	—	2	1	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	2	1	no	—
500	C₂³	—	2	1	no	—
501	C₂³	—	2	1	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	2	1	no	—
505	C₂³	#13133176	1	2	yes	—
506	C₂³	C₂²×#88322	1	2	yes	—
507	C₂³	C₂²×#88322	1	2	yes	—
508	C₂³	C₂×#11719917	1	2	yes	—
509	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
510	D₄	—	2	8	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	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	8	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	4	4	no	—
543	C₂×C₄	—	4	4	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	#13133176	1	2	yes	—
549	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
550	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
551	C₂×C₄	C₂²×#88322	1	2	yes	—
552	C₂×C₄	C₂²×#88322	1	2	yes	—
553	C₂×C₄	C₂²×#88322	1	2	yes	—
554	C₂×C₄	C₂²×#88322	1	2	yes	—
555	C₂×C₄	C₂²×#88322	1	2	yes	—
556	C₂×C₄	C₂×#11719917	1	2	yes	—
557	C₂×C₄	C₂×#11719917	1	2	yes	—
558	C₂×C₄	#13133224	1	2	yes	—
559	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
560	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
561	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
562	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
563	C₂×C₄	#13133263	1	1	yes	characteristic
564	C₂×C₄	#13133264	1	1	yes	characteristic
565	C₈	—	106	4	no	—
566	C₈	—	212	2	no	—
567	C₈	—	212	2	no	—
568	C₈	—	212	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	4	1	no	—
582	C₂²	#16810676	1	4	yes	complemented
583	C₂²	C₂×#13133176	1	1	yes	characteristic
584	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
585	C₂²	#16811106	1	1	yes	characteristic
586	C₂²	C₂³×#88322	1	1	yes	characteristic
587	C₂²	C₂²×#11719917	1	1	yes	characteristic
588	C₂²	C₂×#13133224	1	1	yes	characteristic
589	C₂²	#16811390	1	1	yes	characteristic
590	C₄	—	2	2	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	1	no	—
600	C₄	—	4	1	no	—
601	C₄	D₄×#88322	1	4	yes	—
602	C₄	#16811269	1	1	yes	characteristic
603	C₄	#16811282	1	1	yes	characteristic
604	C₄	#16811313	1	1	yes	characteristic
605	C₄	#16811319	1	1	yes	characteristic
606	C₂	—	2	4	no	—
607	C₂	—	4	2	no	—
608	C₂	#20277644	1	1	yes	characteristic
609	C₂	C₂×#16810676	1	1	yes	characteristic
610	C₂	#20283381	1	1	yes	characteristic
611	C₂	C₂×D₄×#88322	1	1	yes	characteristic
612	C₂	D₄×#11719917	1	1	yes	characteristic
613	C₂	#20283826	1	1	yes	characteristic
614	C₂	#20283840	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	#22748
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. C₂×#16810676	C₂	semidirect
2. C₂²×C₄⋊#88322	C₂	semidirect
3. #20275323	C₂	semidirect
4. #20275274	C₂	semidirect
6. #20274832	C₂	semidirect
7. #20274695	C₂	semidirect
8. #20274695	C₂	semidirect
9. C₄×C₄⋊#88322	C₂	semidirect
10. C₂²⋊C₄×#88322	C₂	semidirect
56. C₄⋊#88322	D₄	semidirect
72. C₄×D₄×C₅₃	C₈	semidirect
131. C₂²×C₄×C₅₃	C₂×C₈	semidirect
149. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
154. C₂²⋊C₄×C₅₃	C₂×C₈	semidirect
171. C₄²×C₅₃	C₂×C₈	semidirect
275. C₂²×C₅₃	#431	semidirect
352. C₅₃	#22748	semidirect
370. C₄×D₄	#88322	semidirect
439. C₂²×C₄	C₂×#88322	semidirect
470. C₄⋊C₄	C₂×#88322	semidirect
475. C₂²⋊C₄	C₂×#88322	semidirect
492. C₄²	C₂×#88322	semidirect
582. C₂²	#16810676	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂²×C₄⋊#88322	2
#20275323	2
#20275274	2
#20275008	2
#20274832	2
#20274695	2
C₄×C₄⋊#88322	2
C₂²⋊C₄×#88322	2
C₄²×D₄×C₅₃	2
#22748	53