Subgroups — C₅₃×#34065 #27241577

Order 13568.

3134 conjugacy classes of subgroups.

The subgroup lattice has 606 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	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#2324	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2323	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2322	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2321	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2297	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2296	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅₃×#2293	C₂	1	4	yes	maximal, complemented
11	C₅₃×#2290	C₂	1	4	yes	maximal, complemented
12	C₅₃×#2280	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#525	C₄	1	16	yes	complemented
17	C₂×#123×C₅₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	2	yes	—
19	C₂×#121×C₅₃	C₂²	1	2	yes	—
20	C₂×#120×C₅₃	C₂²	1	2	yes	—
21	C₄×#41×C₅₃	—	2	8	no	—
22	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	complemented
23	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	complemented
26	C₂×#117×C₅₃	—	2	8	no	—
27	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#117×C₅₃	C₂²	1	2	yes	—
30	C₂×#117×C₅₃	C₂²	1	2	yes	—
31	C₂×#117×C₅₃	C₂²	1	4	yes	—
32	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#117×C₅₃	C₂²	1	2	yes	complemented
34	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
35	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
36	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
37	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
38	C₂²×C₄²×C₅₃	C₂²	1	2	yes	complemented
39	C₅₃×#391	C₂²	1	2	yes	complemented
40	C₅₃×#390	C₂²	1	8	yes	—
41	C₅₃×#390	C₂²	1	2	yes	complemented
42	C₅₃×#389	C₂²	1	8	yes	complemented
43	C₅₃×#389	C₂²	1	4	yes	complemented
44	C₅₃×#388	C₂²	1	8	yes	—
45	C₅₃×#388	C₂²	1	8	yes	—
46	C₅₃×#388	C₂²	1	8	yes	complemented
47	C₅₃×#387	C₂²	1	8	yes	complemented
48	C₅₃×#386	C₂²	1	8	yes	—
49	C₅₃×#386	C₂²	1	4	yes	—
50	C₅₃×#385	C₂²	1	4	yes	—
51	C₅₃×#385	C₂²	1	8	yes	complemented
52	C₅₃×#381	C₂²	1	2	yes	—
53	C₅₃×#380	C₂²	1	4	yes	—
54	C₅₃×#379	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
56	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
57	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
58	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
60	C₅₃×#376	C₂²	1	2	yes	complemented
61	C₄³×C₅₃	C₂²	1	2	yes	complemented
62	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×#41×C₅₃	—	2	4	no	—
64	C₂×#41×C₅₃	—	2	4	no	—
65	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
66	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
68	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
70	C₂³×C₄×C₅₃	—	2	4	no	—
71	C₂³×C₄×C₅₃	—	2	4	no	—
72	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	complemented
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	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
77	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
78	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
79	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
80	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
81	#123×C₅₃	C₂×C₄	1	16	yes	complemented
82	C₂²⋊Q₈×C₅₃	C₂×C₄	1	16	yes	complemented
83	#121×C₅₃	C₂×C₄	1	16	yes	complemented
84	#120×C₅₃	C₂×C₄	1	16	yes	complemented
85	C₄×Q₈×C₅₃	—	2	8	no	—
86	C₄×D₄×C₅₃	—	2	8	no	—
87	C₄×D₄×C₅₃	—	2	16	no	—
88	C₄×D₄×C₅₃	C₂×C₄	1	32	yes	complemented
89	#117×C₅₃	—	2	8	no	—
90	#117×C₅₃	—	2	8	no	—
91	#117×C₅₃	—	2	16	no	—
92	#117×C₅₃	—	2	8	no	—
93	#117×C₅₃	—	2	4	no	—
94	#117×C₅₃	—	2	4	no	—
95	#117×C₅₃	C₂×C₄	1	8	yes	complemented
96	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
97	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
98	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
103	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
106	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
110	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
112	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
113	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
116	C₂×C₄²×C₅₃	—	2	8	no	—
117	C₂×C₄²×C₅₃	—	2	4	no	—
118	C₂×C₄²×C₅₃	D₄	1	8	yes	complemented
119	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
120	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
121	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
122	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
123	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
124	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
125	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
126	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
127	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
129	#95×C₅₃	C₂³	1	8	yes	—
130	#95×C₅₃	C₂³	1	8	yes	—
131	#95×C₅₃	C₂³	1	8	yes	—
132	#95×C₅₃	C₂³	1	8	yes	—
133	C₂⁴×C₅₃	—	2	2	no	—
134	C₂⁴×C₅₃	—	2	2	no	—
135	C₂⁴×C₅₃	—	4	4	no	—
136	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
137	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
138	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
139	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
140	#41×C₅₃	—	2	32	no	—
141	C₂×Q₈×C₅₃	—	2	2	no	—
142	C₂×Q₈×C₅₃	—	2	2	no	—
143	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
144	C₂×D₄×C₅₃	—	2	4	no	—
145	C₂×D₄×C₅₃	—	2	4	no	—
146	C₂×D₄×C₅₃	—	2	2	no	—
147	C₂×D₄×C₅₃	—	2	2	no	—
148	C₂×D₄×C₅₃	—	4	8	no	—
149	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
150	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
151	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
152	C₂²×C₄×C₅₃	—	2	2	no	—
153	C₂²×C₄×C₅₃	—	2	2	no	—
154	C₂²×C₄×C₅₃	—	2	32	no	—
155	C₂²×C₄×C₅₃	—	2	4	no	—
156	C₂²×C₄×C₅₃	—	2	4	no	—
157	C₂²×C₄×C₅₃	—	2	4	no	—
158	C₂²×C₄×C₅₃	—	2	4	no	—
159	C₂²×C₄×C₅₃	—	2	4	no	—
160	C₂²×C₄×C₅₃	—	2	4	no	—
161	C₂²×C₄×C₅₃	—	2	4	no	—
162	C₂²×C₄×C₅₃	—	2	8	no	—
163	C₂²×C₄×C₅₃	—	4	4	no	—
164	C₂²×C₄×C₅₃	—	4	8	no	—
165	C₂²×C₄×C₅₃	—	4	4	no	—
166	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
167	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
168	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
169	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
172	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₅₃	#41	1	4	yes	—
174	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
184	C₄⋊C₄×C₅₃	—	2	8	no	—
185	C₄⋊C₄×C₅₃	—	2	8	no	—
186	C₄⋊C₄×C₅₃	—	2	8	no	—
187	C₄⋊C₄×C₅₃	—	2	4	no	—
188	C₄⋊C₄×C₅₃	—	2	4	no	—
189	C₄⋊C₄×C₅₃	—	4	8	no	—
190	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
191	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
192	C₂²⋊C₄×C₅₃	—	2	4	no	—
193	C₂²⋊C₄×C₅₃	—	2	4	no	—
194	C₂²⋊C₄×C₅₃	—	2	8	no	—
195	C₂²⋊C₄×C₅₃	—	2	8	no	—
196	C₂²⋊C₄×C₅₃	—	2	8	no	—
197	C₂²⋊C₄×C₅₃	—	4	8	no	—
198	C₂²⋊C₄×C₅₃	—	4	8	no	—
199	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
200	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
201	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
202	C₄²×C₅₃	—	2	4	no	—
203	C₄²×C₅₃	—	2	4	no	—
204	C₄²×C₅₃	—	2	4	no	—
205	C₄²×C₅₃	—	2	8	no	—
206	C₄²×C₅₃	—	2	4	no	—
207	C₄²×C₅₃	—	4	8	no	—
208	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
209	C₄²×C₅₃	C₂×D₄	1	4	yes	—
210	C₄²×C₅₃	C₂×D₄	1	4	yes	—
211	C₄²×C₅₃	C₂×D₄	1	8	yes	—
212	C₂³×C₅₃	—	2	4	no	—
213	C₂³×C₅₃	—	2	8	no	—
214	C₂³×C₅₃	—	2	2	no	—
215	C₂³×C₅₃	—	2	2	no	—
216	C₂³×C₅₃	—	2	2	no	—
217	C₂³×C₅₃	—	2	2	no	—
218	C₂³×C₅₃	—	2	8	no	—
219	C₂³×C₅₃	—	2	4	no	—
220	C₂³×C₅₃	—	2	4	no	—
221	C₂³×C₅₃	—	4	8	no	—
222	C₂³×C₅₃	—	4	8	no	—
223	C₂³×C₅₃	—	4	8	no	—
224	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
225	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
226	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
227	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
228	C₂³×C₅₃	C₂×#41	1	2	yes	—
229	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
230	Q₈×C₅₃	—	2	8	no	—
231	D₄×C₅₃	—	2	16	no	—
232	D₄×C₅₃	—	2	8	no	—
233	D₄×C₅₃	—	4	16	no	—
234	C₂×C₄×C₅₃	—	2	8	no	—
235	C₂×C₄×C₅₃	—	2	2	no	—
236	C₂×C₄×C₅₃	—	2	2	no	—
237	C₂×C₄×C₅₃	—	2	2	no	—
238	C₂×C₄×C₅₃	—	2	2	no	—
239	C₂×C₄×C₅₃	—	2	4	no	—
240	C₂×C₄×C₅₃	—	2	4	no	—
241	C₂×C₄×C₅₃	—	2	8	no	—
242	C₂×C₄×C₅₃	—	2	16	no	—
243	C₂×C₄×C₅₃	—	2	16	no	—
244	C₂×C₄×C₅₃	—	2	16	no	—
245	C₂×C₄×C₅₃	—	2	4	no	—
246	C₂×C₄×C₅₃	—	2	4	no	—
247	C₂×C₄×C₅₃	—	2	4	no	—
248	C₂×C₄×C₅₃	—	2	4	no	—
249	C₂×C₄×C₅₃	—	2	8	no	—
250	C₂×C₄×C₅₃	—	2	4	no	—
251	C₂×C₄×C₅₃	—	2	8	no	—
252	C₂×C₄×C₅₃	—	4	8	no	—
253	C₂×C₄×C₅₃	—	4	16	no	—
254	C₂×C₄×C₅₃	—	4	8	no	—
255	C₂×C₄×C₅₃	C₄×D₄	1	16	yes	—
256	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
257	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
258	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
259	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
260	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
261	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
262	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
263	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
266	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
267	#34065	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₂²×C₅₃	—	2	4	no	—
269	C₂²×C₅₃	—	2	4	no	—
270	C₂²×C₅₃	—	2	4	no	—
271	C₂²×C₅₃	—	2	4	no	—
272	C₂²×C₅₃	—	2	4	no	—
273	C₂²×C₅₃	—	2	4	no	—
274	C₂²×C₅₃	—	2	8	no	—
275	C₂²×C₅₃	—	2	4	no	—
276	C₂²×C₅₃	—	4	16	no	—
277	C₂²×C₅₃	—	4	4	no	—
278	C₂²×C₅₃	—	4	4	no	—
279	C₂²×C₅₃	—	4	4	no	—
280	C₂²×C₅₃	—	4	4	no	—
281	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
282	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
283	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
284	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
285	C₂²×C₅₃	#585	1	4	yes	—
286	C₄×C₅₃	—	2	4	no	—
287	C₄×C₅₃	—	2	4	no	—
288	C₄×C₅₃	—	2	8	no	—
289	C₄×C₅₃	—	2	4	no	—
290	C₄×C₅₃	—	2	4	no	—
291	C₄×C₅₃	—	4	4	no	—
292	C₄×C₅₃	—	4	4	no	—
293	C₄×C₅₃	—	4	4	no	—
294	C₄×C₅₃	C₂×C₄×D₄	1	4	yes	—
295	C₄×C₅₃	C₂×C₄×D₄	1	4	yes	—
296	C₂×#525	C₂×C₅₃	1	1	yes	characteristic
297	C₂×C₄×#41	C₂×C₅₃	1	1	yes	characteristic, complemented
298	C₂²×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
299	#2324	C₂×C₅₃	1	4	yes	complemented
300	#2323	C₂×C₅₃	1	4	yes	complemented
301	#2322	C₂×C₅₃	1	4	yes	complemented
302	#2321	C₂×C₅₃	1	4	yes	complemented
303	#2297	C₂×C₅₃	1	1	yes	characteristic, complemented
304	#2296	C₂×C₅₃	1	1	yes	characteristic, complemented
305	#2293	C₂×C₅₃	1	4	yes	complemented
306	#2290	C₂×C₅₃	1	4	yes	complemented
307	#2280	C₂×C₅₃	1	1	yes	characteristic, complemented
308	C₄×#117	C₂×C₅₃	1	1	yes	characteristic, complemented
309	C₂×C₅₃	—	2	4	no	—
310	C₂×C₅₃	—	4	4	no	—
311	C₂×C₅₃	—	4	4	no	—
312	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
313	C₂×C₅₃	#3428	1	2	yes	—
314	C₂×C₅₃	#3449	1	2	yes	—
315	C₂×C₅₃	#3455	1	2	yes	—
316	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
317	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
318	#525	C₄×C₅₃	1	16	yes	complemented
319	C₂×#123	C₂²×C₅₃	1	2	yes	—
320	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	2	yes	—
321	C₂×#121	C₂²×C₅₃	1	2	yes	—
322	C₂×#120	C₂²×C₅₃	1	2	yes	—
323	C₄×#41	—	2	8	no	—
324	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	complemented
325	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
326	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
327	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	complemented
328	C₂×#117	—	2	8	no	—
329	C₂×#117	C₂²×C₅₃	1	4	yes	—
330	C₂×#117	C₂²×C₅₃	1	2	yes	—
331	C₂×#117	C₂²×C₅₃	1	2	yes	—
332	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
333	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
334	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
335	C₂×#117	C₂²×C₅₃	1	2	yes	complemented
336	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
337	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
338	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
339	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
340	C₂²×C₄²	C₂²×C₅₃	1	2	yes	complemented
341	#391	C₂²×C₅₃	1	2	yes	complemented
342	#390	C₂²×C₅₃	1	8	yes	—
343	#390	C₂²×C₅₃	1	2	yes	complemented
344	#389	C₂²×C₅₃	1	8	yes	complemented
345	#389	C₂²×C₅₃	1	4	yes	complemented
346	#388	C₂²×C₅₃	1	8	yes	—
347	#388	C₂²×C₅₃	1	8	yes	—
348	#388	C₂²×C₅₃	1	8	yes	complemented
349	#387	C₂²×C₅₃	1	8	yes	complemented
350	#386	C₂²×C₅₃	1	4	yes	—
351	#386	C₂²×C₅₃	1	8	yes	—
352	#385	C₂²×C₅₃	1	4	yes	—
353	#385	C₂²×C₅₃	1	8	yes	complemented
354	#381	C₂²×C₅₃	1	2	yes	—
355	#380	C₂²×C₅₃	1	4	yes	—
356	#379	C₂²×C₅₃	1	2	yes	—
357	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
358	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
359	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
360	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
361	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
362	#376	C₂²×C₅₃	1	2	yes	complemented
363	C₄³	C₂²×C₅₃	1	2	yes	complemented
364	C₅₃	#34065	1	1	yes	characteristic, direct product factor, Sylow
365	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
366	C₂×#41	—	2	4	no	—
367	C₂×#41	—	2	4	no	—
368	C₂×#41	C₂×C₄×C₅₃	1	8	yes	—
369	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
370	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
371	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
372	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
373	C₂³×C₄	—	2	4	no	—
374	C₂³×C₄	—	2	4	no	—
375	C₂³×C₄	C₂×C₄×C₅₃	1	8	yes	complemented
376	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
377	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
378	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
379	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
380	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
381	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
382	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
383	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
384	#123	C₂×C₄×C₅₃	1	16	yes	complemented
385	C₂²⋊Q₈	C₂×C₄×C₅₃	1	16	yes	complemented
386	#121	C₂×C₄×C₅₃	1	16	yes	complemented
387	#120	C₂×C₄×C₅₃	1	16	yes	complemented
388	C₄×Q₈	—	2	8	no	—
389	C₄×D₄	—	2	16	no	—
390	C₄×D₄	—	2	8	no	—
391	C₄×D₄	C₂×C₄×C₅₃	1	32	yes	complemented
392	#117	—	2	4	no	—
393	#117	—	2	4	no	—
394	#117	—	2	8	no	—
395	#117	—	2	8	no	—
396	#117	—	2	16	no	—
397	#117	—	2	8	no	—
398	#117	C₂×C₄×C₅₃	1	8	yes	complemented
399	C₂×C₄⋊C₄	—	2	8	no	—
400	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
401	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
402	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
403	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
405	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
406	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
408	C₂×C₂²⋊C₄	—	2	8	no	—
409	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
419	C₂×C₄²	—	2	4	no	—
420	C₂×C₄²	—	2	8	no	—
421	C₂×C₄²	D₄×C₅₃	1	8	yes	complemented
422	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
423	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
424	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
425	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
426	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
427	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
428	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
429	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
430	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
431	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
432	#95	C₂³×C₅₃	1	8	yes	—
433	#95	C₂³×C₅₃	1	8	yes	—
434	#95	C₂³×C₅₃	1	8	yes	—
435	#95	C₂³×C₅₃	1	8	yes	—
436	C₂⁴	—	2	2	no	—
437	C₂⁴	—	2	2	no	—
438	C₂⁴	—	4	4	no	—
439	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
440	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
441	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
442	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
443	#41	—	2	32	no	—
444	C₂×Q₈	—	2	2	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	4	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	4	8	no	—
452	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
453	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
454	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	32	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	8	no	—
466	C₂²×C₄	—	4	4	no	—
467	C₂²×C₄	—	4	8	no	—
468	C₂²×C₄	—	4	4	no	—
469	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
470	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
471	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
472	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
473	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
474	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
475	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
476	C₂²×C₄	#41×C₅₃	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
478	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
480	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
482	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
485	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
487	C₄⋊C₄	—	2	8	no	—
488	C₄⋊C₄	—	2	8	no	—
489	C₄⋊C₄	—	2	8	no	—
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	4	8	no	—
493	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
494	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	—	2	8	no	—
499	C₂²⋊C₄	—	2	8	no	—
500	C₂²⋊C₄	—	4	8	no	—
501	C₂²⋊C₄	—	4	8	no	—
502	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
503	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
504	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
505	C₄²	—	2	4	no	—
506	C₄²	—	2	4	no	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	8	no	—
509	C₄²	—	2	4	no	—
510	C₄²	—	4	8	no	—
511	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
512	C₄²	C₂×D₄×C₅₃	1	4	yes	—
513	C₄²	C₂×D₄×C₅₃	1	4	yes	—
514	C₄²	C₂×D₄×C₅₃	1	8	yes	—
515	C₂³	—	2	4	no	—
516	C₂³	—	2	8	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	2	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	4	no	—
522	C₂³	—	2	8	no	—
523	C₂³	—	2	4	no	—
524	C₂³	—	4	8	no	—
525	C₂³	—	4	8	no	—
526	C₂³	—	4	8	no	—
527	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
528	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
529	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
530	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
531	C₂³	C₂×#41×C₅₃	1	2	yes	—
532	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
533	Q₈	—	2	8	no	—
534	D₄	—	2	16	no	—
535	D₄	—	2	8	no	—
536	D₄	—	4	16	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	8	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	16	no	—
546	C₂×C₄	—	2	16	no	—
547	C₂×C₄	—	2	16	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	8	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	8	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	4	8	no	—
556	C₂×C₄	—	4	16	no	—
557	C₂×C₄	—	4	8	no	—
558	C₂×C₄	C₄×D₄×C₅₃	1	16	yes	—
559	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
560	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
561	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
562	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
563	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
564	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
565	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
566	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
567	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
568	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
569	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
570	C₂²	—	2	4	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	2	4	no	—
574	C₂²	—	2	4	no	—
575	C₂²	—	2	4	no	—
576	C₂²	—	2	8	no	—
577	C₂²	—	2	4	no	—
578	C₂²	—	4	16	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	4	4	no	—
583	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
584	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
585	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
586	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
587	C₂²	C₅₃×#585	1	4	yes	—
588	C₄	—	2	4	no	—
589	C₄	—	2	4	no	—
590	C₄	—	2	4	no	—
591	C₄	—	2	4	no	—
592	C₄	—	2	8	no	—
593	C₄	—	4	4	no	—
594	C₄	—	4	4	no	—
595	C₄	—	4	4	no	—
596	C₄	C₂×C₄×D₄×C₅₃	1	4	yes	—
597	C₄	C₂×C₄×D₄×C₅₃	1	4	yes	—
598	C₂	—	2	4	no	—
599	C₂	—	4	4	no	—
600	C₂	—	4	4	no	—
601	C₂	C₂²×C₄×D₄×C₅₃	1	1	yes	characteristic
602	C₂	C₅₃×#3428	1	2	yes	—
603	C₂	C₅₃×#3449	1	2	yes	—
604	C₂	C₅₃×#3455	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂×C₅₃×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#41×C₅₃	C₂	semidirect
3. C₂²×#117×C₅₃	C₂	semidirect
4. C₅₃×#2324	C₂	semidirect
5. C₅₃×#2323	C₂	semidirect
6. C₅₃×#2322	C₂	semidirect
7. C₅₃×#2321	C₂	semidirect
8. C₅₃×#2297	C₂	semidirect
9. C₅₃×#2296	C₂	semidirect
10. C₅₃×#2293	C₂	semidirect
11. C₅₃×#2290	C₂	semidirect
12. C₅₃×#2280	C₂	semidirect
13. C₄×#117×C₅₃	C₂	semidirect
16. C₅₃×#525	C₄	semidirect
22. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
25. C₂×C₄×D₄×C₅₃	C₂²	semidirect
33. C₂×#117×C₅₃	C₂²	semidirect
38. C₂²×C₄²×C₅₃	C₂²	semidirect
39. C₅₃×#391	C₂²	semidirect
41. C₅₃×#390	C₂²	semidirect
42. C₅₃×#389	C₂²	semidirect
43. C₅₃×#389	C₂²	semidirect
46. C₅₃×#388	C₂²	semidirect
47. C₅₃×#387	C₂²	semidirect
51. C₅₃×#385	C₂²	semidirect
55. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
56. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
59. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
60. C₅₃×#376	C₂²	semidirect
61. C₄³×C₅₃	C₂²	semidirect
72. C₂³×C₄×C₅₃	C₂×C₄	semidirect
81. #123×C₅₃	C₂×C₄	semidirect
82. C₂²⋊Q₈×C₅₃	C₂×C₄	semidirect
83. #121×C₅₃	C₂×C₄	semidirect
84. #120×C₅₃	C₂×C₄	semidirect
88. C₄×D₄×C₅₃	C₂×C₄	semidirect
95. #117×C₅₃	C₂×C₄	semidirect
118. C₂×C₄²×C₅₃	D₄	semidirect
267. #34065	C₅₃	direct (N × H)
297. C₂×C₄×#41	C₂×C₅₃	semidirect
298. C₂²×#117	C₂×C₅₃	semidirect
299. #2324	C₂×C₅₃	semidirect
300. #2323	C₂×C₅₃	semidirect
301. #2322	C₂×C₅₃	semidirect
302. #2321	C₂×C₅₃	semidirect
303. #2297	C₂×C₅₃	semidirect
304. #2296	C₂×C₅₃	semidirect
305. #2293	C₂×C₅₃	semidirect
306. #2290	C₂×C₅₃	semidirect
307. #2280	C₂×C₅₃	semidirect
308. C₄×#117	C₂×C₅₃	semidirect
318. #525	C₄×C₅₃	semidirect
324. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
327. C₂×C₄×D₄	C₂²×C₅₃	semidirect
335. C₂×#117	C₂²×C₅₃	semidirect
340. C₂²×C₄²	C₂²×C₅₃	semidirect
341. #391	C₂²×C₅₃	semidirect
343. #390	C₂²×C₅₃	semidirect
344. #389	C₂²×C₅₃	semidirect
345. #389	C₂²×C₅₃	semidirect
348. #388	C₂²×C₅₃	semidirect
349. #387	C₂²×C₅₃	semidirect
353. #385	C₂²×C₅₃	semidirect
357. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
358. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
361. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
362. #376	C₂²×C₅₃	semidirect
363. C₄³	C₂²×C₅₃	semidirect
364. C₅₃	#34065	direct (N × H)
375. C₂³×C₄	C₂×C₄×C₅₃	semidirect
384. #123	C₂×C₄×C₅₃	semidirect
385. C₂²⋊Q₈	C₂×C₄×C₅₃	semidirect
386. #121	C₂×C₄×C₅₃	semidirect
387. #120	C₂×C₄×C₅₃	semidirect
391. C₄×D₄	C₂×C₄×C₅₃	semidirect
398. #117	C₂×C₄×C₅₃	semidirect
421. C₂×C₄²	D₄×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
C₂×C₄×#41×C₅₃	2
C₂²×#117×C₅₃	2
C₅₃×#2324	2
C₅₃×#2323	2
C₅₃×#2322	2
C₅₃×#2321	2
C₅₃×#2297	2
C₅₃×#2296	2
C₅₃×#2293	2
C₅₃×#2290	2
C₅₃×#2280	2
C₄×#117×C₅₃	2
#34065	53

Subgroups — C53×#34065 #27241577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34065 #27241577