Subgroups — C₅₃×#34121 #27241633

Order 13568.

2554 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2357	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2356	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2355	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2354	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2353	C₂	1	4	yes	maximal, complemented
7	C₅₃×#2352	C₂	1	4	yes	maximal, complemented
8	C₅₃×#2298	C₂	1	4	yes	maximal, complemented
9	C₅₃×#2297	C₂	1	2	yes	maximal
10	C₅₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2283	C₂	1	2	yes	maximal, complemented
12	C₄×#117×C₅₃	C₂	1	2	yes	maximal, complemented
13	C₄×#117×C₅₃	C₂	1	2	yes	maximal, complemented
14	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×#41×C₅₃	C₄	1	16	yes	complemented
17	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#117×C₅₃	C₂²	1	2	yes	—
24	C₂×#117×C₅₃	C₂²	1	4	yes	—
25	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#117×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
31	C₅₃×#391	C₂²	1	4	yes	—
32	C₅₃×#390	C₂²	1	4	yes	—
33	C₅₃×#388	C₂²	1	8	yes	—
34	C₅₃×#388	C₂²	1	8	yes	—
35	C₅₃×#387	C₂²	1	8	yes	—
36	C₅₃×#387	C₂²	1	8	yes	—
37	C₅₃×#386	C₂²	1	8	yes	—
38	C₅₃×#384	C₂²	1	2	yes	—
39	C₅₃×#383	C₂²	1	4	yes	—
40	C₅₃×#382	C₂²	1	4	yes	—
41	C₅₃×#382	C₂²	1	2	yes	—
42	C₅₃×#381	C₂²	1	4	yes	—
43	C₅₃×#380	C₂²	1	4	yes	—
44	C₅₃×#380	C₂²	1	2	yes	—
45	C₅₃×#380	C₂²	1	2	yes	—
46	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
49	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
50	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
51	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
53	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
54	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
55	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
56	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
57	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
58	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
59	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
60	C₅₃×#376	C₂²	1	4	yes	—
61	C₅₃×#376	C₂²	1	4	yes	—
62	C₂×#95×C₅₃	C₂²	1	2	yes	—
63	C₄³×C₅₃	C₂²	1	4	yes	—
64	C₂×#41×C₅₃	—	2	4	no	—
65	C₂×#41×C₅₃	—	2	4	no	—
66	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
67	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
69	C₂²×D₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
71	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
73	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
74	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
76	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
77	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
78	C₄×Q₈×C₅₃	C₂×C₄	1	8	yes	complemented
79	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	—
80	C₄×D₄×C₅₃	C₂×C₄	1	16	yes	—
81	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	complemented
82	C₄×D₄×C₅₃	C₂×C₄	1	8	yes	complemented
83	#117×C₅₃	—	2	16	no	—
84	#117×C₅₃	—	2	8	no	—
85	#117×C₅₃	C₂×C₄	1	8	yes	—
86	#117×C₅₃	C₂×C₄	1	16	yes	complemented
87	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
88	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₅₃	C₂³	1	2	yes	—
104	C₂×C₄²×C₅₃	—	2	8	no	—
105	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
106	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	complemented
107	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
108	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
109	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
110	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
111	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
113	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
115	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
116	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
117	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
119	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
120	#95×C₅₃	C₂³	1	2	yes	—
121	#95×C₅₃	C₂³	1	2	yes	—
122	#95×C₅₃	C₂³	1	4	yes	—
123	#95×C₅₃	C₂³	1	4	yes	—
124	#95×C₅₃	C₂³	1	8	yes	—
125	#95×C₅₃	C₂³	1	8	yes	—
126	#95×C₅₃	C₂³	1	8	yes	—
127	#95×C₅₃	C₂³	1	8	yes	—
128	#95×C₅₃	C₂³	1	4	yes	—
129	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
130	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
131	#41×C₅₃	—	2	32	no	—
132	C₂×Q₈×C₅₃	—	2	2	no	—
133	C₂×Q₈×C₅₃	—	2	2	no	—
134	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
135	C₂×D₄×C₅₃	—	2	4	no	—
136	C₂×D₄×C₅₃	—	2	4	no	—
137	C₂×D₄×C₅₃	—	2	2	no	—
138	C₂×D₄×C₅₃	—	2	2	no	—
139	C₂×D₄×C₅₃	C₂²×C₄	1	8	yes	—
140	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
141	C₂²×C₄×C₅₃	—	2	4	no	—
142	C₂²×C₄×C₅₃	—	2	8	no	—
143	C₂²×C₄×C₅₃	—	2	4	no	—
144	C₂²×C₄×C₅₃	—	2	4	no	—
145	C₂²×C₄×C₅₃	—	2	8	no	—
146	C₂²×C₄×C₅₃	—	2	8	no	—
147	C₂²×C₄×C₅₃	—	4	4	no	—
148	C₂²×C₄×C₅₃	—	4	2	no	—
149	C₂²×C₄×C₅₃	—	4	4	no	—
150	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
152	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
153	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
154	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
155	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₅₃	#41	1	4	yes	—
157	C₂²×C₄×C₅₃	#41	1	4	yes	—
158	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
161	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
170	C₄⋊C₄×C₅₃	—	2	8	no	—
171	C₄⋊C₄×C₅₃	—	2	4	no	—
172	C₄⋊C₄×C₅₃	—	2	8	no	—
173	C₄⋊C₄×C₅₃	—	2	8	no	—
174	C₄⋊C₄×C₅₃	—	4	4	no	—
175	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
176	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
177	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
178	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
179	C₄⋊C₄×C₅₃	#41	1	8	yes	complemented
180	C₂²⋊C₄×C₅₃	—	2	8	no	—
181	C₂²⋊C₄×C₅₃	—	2	16	no	—
182	C₂²⋊C₄×C₅₃	—	4	8	no	—
183	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
184	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
185	C₂²⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
186	C₄²×C₅₃	—	2	4	no	—
187	C₄²×C₅₃	—	2	8	no	—
188	C₄²×C₅₃	—	2	4	no	—
189	C₄²×C₅₃	—	2	16	no	—
190	C₄²×C₅₃	—	2	8	no	—
191	C₄²×C₅₃	—	4	8	no	—
192	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
193	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
194	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
195	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
196	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
197	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
198	C₄²×C₅₃	C₂²×C₄	1	8	yes	—
199	C₂³×C₅₃	—	2	2	no	—
200	C₂³×C₅₃	—	2	2	no	—
201	C₂³×C₅₃	—	2	2	no	—
202	C₂³×C₅₃	—	2	2	no	—
203	C₂³×C₅₃	—	2	4	no	—
204	C₂³×C₅₃	—	2	4	no	—
205	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
206	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
207	C₂³×C₅₃	C₂×#41	1	2	yes	—
208	C₂³×C₅₃	C₂×#41	1	2	yes	—
209	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
210	Q₈×C₅₃	—	2	8	no	—
211	D₄×C₅₃	—	2	16	no	—
212	D₄×C₅₃	—	2	8	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	2	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	2	4	no	—
217	C₂×C₄×C₅₃	—	2	2	no	—
218	C₂×C₄×C₅₃	—	2	2	no	—
219	C₂×C₄×C₅₃	—	2	4	no	—
220	C₂×C₄×C₅₃	—	2	4	no	—
221	C₂×C₄×C₅₃	—	2	2	no	—
222	C₂×C₄×C₅₃	—	2	16	no	—
223	C₂×C₄×C₅₃	—	2	8	no	—
224	C₂×C₄×C₅₃	—	2	16	no	—
225	C₂×C₄×C₅₃	—	2	4	no	—
226	C₂×C₄×C₅₃	—	2	4	no	—
227	C₂×C₄×C₅₃	—	2	8	no	—
228	C₂×C₄×C₅₃	—	2	8	no	—
229	C₂×C₄×C₅₃	—	2	4	no	—
230	C₂×C₄×C₅₃	—	2	8	no	—
231	C₂×C₄×C₅₃	—	4	4	no	—
232	C₂×C₄×C₅₃	—	4	8	no	—
233	C₂×C₄×C₅₃	—	4	16	no	—
234	C₂×C₄×C₅₃	#117	1	16	yes	complemented
235	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
236	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
237	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
238	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
239	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
240	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
241	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
242	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
243	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
244	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
245	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
246	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
247	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
248	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
249	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
250	#34121	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂²×C₅₃	—	2	2	no	—
252	C₂²×C₅₃	—	2	2	no	—
253	C₂²×C₅₃	—	2	4	no	—
254	C₂²×C₅₃	—	2	4	no	—
255	C₂²×C₅₃	—	2	4	no	—
256	C₂²×C₅₃	—	2	4	no	—
257	C₂²×C₅₃	—	2	4	no	—
258	C₂²×C₅₃	—	4	4	no	—
259	C₂²×C₅₃	—	4	4	no	—
260	C₂²×C₅₃	C₂×#117	1	4	yes	—
261	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
262	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
263	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
264	C₂²×C₅₃	C₂²×#41	1	2	yes	—
265	C₂²×C₅₃	#585	1	2	yes	—
266	C₄×C₅₃	—	2	4	no	—
267	C₄×C₅₃	—	2	4	no	—
268	C₄×C₅₃	—	2	2	no	—
269	C₄×C₅₃	—	2	2	no	—
270	C₄×C₅₃	—	2	4	no	—
271	C₄×C₅₃	—	2	2	no	—
272	C₄×C₅₃	—	2	8	no	—
273	C₄×C₅₃	—	4	4	no	—
274	C₄×C₅₃	—	4	8	no	—
275	C₄×C₅₃	C₂×#117	1	4	yes	—
276	C₄×C₅₃	C₂×#117	1	4	yes	—
277	C₄×C₅₃	C₄×#41	1	4	yes	—
278	C₂×C₄×#41	C₂×C₅₃	1	1	yes	characteristic
279	#2357	C₂×C₅₃	1	2	yes	complemented
280	#2356	C₂×C₅₃	1	2	yes	complemented
281	#2355	C₂×C₅₃	1	2	yes	complemented
282	#2354	C₂×C₅₃	1	2	yes	complemented
283	#2353	C₂×C₅₃	1	4	yes	complemented
284	#2352	C₂×C₅₃	1	4	yes	complemented
285	#2298	C₂×C₅₃	1	4	yes	complemented
286	#2297	C₂×C₅₃	1	2	yes	—
287	#2285	C₂×C₅₃	1	1	yes	characteristic, complemented
288	#2283	C₂×C₅₃	1	2	yes	complemented
289	C₄×#117	C₂×C₅₃	1	2	yes	complemented
290	C₄×#117	C₂×C₅₃	1	2	yes	complemented
291	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
292	C₂×C₅₃	—	2	4	no	—
293	C₂×C₅₃	—	4	4	no	—
294	C₂×C₅₃	C₂²×#117	1	1	yes	characteristic
295	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
296	C₂×C₅₃	#3428	1	1	yes	characteristic
297	C₂×C₅₃	#3475	1	2	yes	—
298	C₂×C₅₃	#3480	1	2	yes	—
299	C₂²×#41	C₂²×C₅₃	1	1	yes	characteristic
300	C₄×#41	C₄×C₅₃	1	16	yes	complemented
301	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
302	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
303	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
304	C₂×C₄×D₄	C₂²×C₅₃	1	2	yes	—
305	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
306	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
307	C₂×#117	C₂²×C₅₃	1	4	yes	—
308	C₂×#117	C₂²×C₅₃	1	2	yes	—
309	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
310	C₂×#117	C₂²×C₅₃	1	2	yes	—
311	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
312	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
313	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
314	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
315	#391	C₂²×C₅₃	1	4	yes	—
316	#390	C₂²×C₅₃	1	4	yes	—
317	#388	C₂²×C₅₃	1	8	yes	—
318	#388	C₂²×C₅₃	1	8	yes	—
319	#387	C₂²×C₅₃	1	8	yes	—
320	#387	C₂²×C₅₃	1	8	yes	—
321	#386	C₂²×C₅₃	1	8	yes	—
322	#384	C₂²×C₅₃	1	2	yes	—
323	#383	C₂²×C₅₃	1	4	yes	—
324	#382	C₂²×C₅₃	1	4	yes	—
325	#382	C₂²×C₅₃	1	2	yes	—
326	#381	C₂²×C₅₃	1	4	yes	—
327	#380	C₂²×C₅₃	1	2	yes	—
328	#380	C₂²×C₅₃	1	4	yes	—
329	#380	C₂²×C₅₃	1	2	yes	—
330	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
331	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
332	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
333	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
334	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
335	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
336	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
337	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
338	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
339	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
340	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
341	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
342	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
343	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
344	#376	C₂²×C₅₃	1	4	yes	—
345	#376	C₂²×C₅₃	1	4	yes	—
346	C₂×#95	C₂²×C₅₃	1	2	yes	—
347	C₄³	C₂²×C₅₃	1	4	yes	—
348	C₅₃	#34121	1	1	yes	characteristic, direct product factor, Sylow
349	C₂×#41	—	2	4	no	—
350	C₂×#41	—	2	4	no	—
351	C₂×#41	C₂×C₄×C₅₃	1	8	yes	—
352	C₂²×Q₈	C₂³×C₅₃	1	1	yes	characteristic
353	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
354	C₂²×D₄	C₂³×C₅₃	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
356	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
358	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
359	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
360	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
361	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
362	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
363	C₄×Q₈	C₂×C₄×C₅₃	1	8	yes	complemented
364	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	—
365	C₄×D₄	C₂×C₄×C₅₃	1	16	yes	—
366	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	complemented
367	C₄×D₄	C₂×C₄×C₅₃	1	8	yes	complemented
368	#117	—	2	8	no	—
369	#117	—	2	16	no	—
370	#117	C₂×C₄×C₅₃	1	8	yes	—
371	#117	C₂×C₄×C₅₃	1	16	yes	complemented
372	C₂×C₄⋊C₄	—	2	8	no	—
373	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
374	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
376	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
377	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
378	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
382	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₅₃	1	2	yes	—
389	C₂×C₄²	—	2	8	no	—
390	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	—
391	C₂×C₄²	C₂×C₄×C₅₃	1	16	yes	complemented
392	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
393	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
394	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
395	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
396	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
397	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
398	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
399	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
400	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
401	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
402	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
403	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
405	#95	C₂³×C₅₃	1	2	yes	—
406	#95	C₂³×C₅₃	1	2	yes	—
407	#95	C₂³×C₅₃	1	4	yes	—
408	#95	C₂³×C₅₃	1	4	yes	—
409	#95	C₂³×C₅₃	1	8	yes	—
410	#95	C₂³×C₅₃	1	8	yes	—
411	#95	C₂³×C₅₃	1	8	yes	—
412	#95	C₂³×C₅₃	1	8	yes	—
413	#95	C₂³×C₅₃	1	4	yes	—
414	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
415	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
416	#41	—	2	32	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	C₂²×C₄×C₅₃	1	8	yes	—
425	C₂×D₄	C₂²×C₄×C₅₃	1	4	yes	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	8	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	8	no	—
431	C₂²×C₄	—	2	8	no	—
432	C₂²×C₄	—	4	4	no	—
433	C₂²×C₄	—	4	4	no	—
434	C₂²×C₄	—	4	2	no	—
435	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
436	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
437	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
438	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
439	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	—
440	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
441	C₂²×C₄	#41×C₅₃	1	4	yes	—
442	C₂²×C₄	#41×C₅₃	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
444	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
445	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
447	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
448	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
449	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
450	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
452	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
453	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
454	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
455	C₄⋊C₄	—	2	8	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	2	8	no	—
459	C₄⋊C₄	—	4	4	no	—
460	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
461	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
462	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
463	C₄⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
464	C₄⋊C₄	#41×C₅₃	1	8	yes	complemented
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	16	no	—
467	C₂²⋊C₄	—	4	8	no	—
468	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
469	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
470	C₂²⋊C₄	C₂²×C₄×C₅₃	1	8	yes	—
471	C₄²	—	2	4	no	—
472	C₄²	—	2	8	no	—
473	C₄²	—	2	4	no	—
474	C₄²	—	2	16	no	—
475	C₄²	—	2	8	no	—
476	C₄²	—	4	8	no	—
477	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
478	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
479	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
480	C₄²	C₂²×C₄×C₅₃	1	4	yes	—
481	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
482	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
483	C₄²	C₂²×C₄×C₅₃	1	8	yes	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	4	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	4	no	—
490	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
491	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
492	C₂³	C₂×#41×C₅₃	1	2	yes	—
493	C₂³	C₂×#41×C₅₃	1	2	yes	—
494	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
495	Q₈	—	2	8	no	—
496	D₄	—	2	16	no	—
497	D₄	—	2	8	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	16	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	16	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	8	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	4	4	no	—
517	C₂×C₄	—	4	8	no	—
518	C₂×C₄	—	4	16	no	—
519	C₂×C₄	#117×C₅₃	1	16	yes	complemented
520	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
521	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
522	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
523	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
524	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
525	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
526	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
527	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
528	C₂×C₄	C₂³×C₄×C₅₃	1	2	yes	—
529	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
530	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
531	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
532	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
533	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
534	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
535	C₂²	—	2	4	no	—
536	C₂²	—	2	4	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	2	4	no	—
541	C₂²	—	2	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	C₂×#117×C₅₃	1	4	yes	—
545	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
546	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
547	C₂²	C₂²×#41×C₅₃	1	2	yes	—
548	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
549	C₂²	C₅₃×#585	1	2	yes	—
550	C₄	—	2	8	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	4	no	—
553	C₄	—	2	2	no	—
554	C₄	—	2	2	no	—
555	C₄	—	2	4	no	—
556	C₄	—	2	2	no	—
557	C₄	—	4	4	no	—
558	C₄	—	4	8	no	—
559	C₄	C₂×#117×C₅₃	1	4	yes	—
560	C₄	C₂×#117×C₅₃	1	4	yes	—
561	C₄	C₄×#41×C₅₃	1	4	yes	—
562	C₂	—	2	4	no	—
563	C₂	—	4	4	no	—
564	C₂	C₂²×#117×C₅₃	1	1	yes	characteristic
565	C₂	C₂×C₄×#41×C₅₃	1	1	yes	characteristic
566	C₂	C₅₃×#3428	1	1	yes	characteristic
567	C₂	C₅₃×#3475	1	2	yes	—
568	C₂	C₅₃×#3480	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	C₂²×#41×C₅₃	C₂²
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. C₅₃×#2357	C₂	semidirect
3. C₅₃×#2356	C₂	semidirect
4. C₅₃×#2355	C₂	semidirect
5. C₅₃×#2354	C₂	semidirect
6. C₅₃×#2353	C₂	semidirect
7. C₅₃×#2352	C₂	semidirect
8. C₅₃×#2298	C₂	semidirect
10. C₅₃×#2285	C₂	semidirect
11. C₅₃×#2283	C₂	semidirect
12. C₄×#117×C₅₃	C₂	semidirect
13. C₄×#117×C₅₃	C₂	semidirect
14. C₂×C₄×C₄⋊C₄×C₅₃	C₂	semidirect
16. C₄×#41×C₅₃	C₄	semidirect
77. C₄×Q₈×C₅₃	C₂×C₄	semidirect
78. C₄×Q₈×C₅₃	C₂×C₄	semidirect
81. C₄×D₄×C₅₃	C₂×C₄	semidirect
82. C₄×D₄×C₅₃	C₂×C₄	semidirect
86. #117×C₅₃	C₂×C₄	semidirect
106. C₂×C₄²×C₅₃	C₂×C₄	semidirect
179. C₄⋊C₄×C₅₃	#41	semidirect
234. C₂×C₄×C₅₃	#117	semidirect
250. #34121	C₅₃	direct (N × H)
279. #2357	C₂×C₅₃	semidirect
280. #2356	C₂×C₅₃	semidirect
281. #2355	C₂×C₅₃	semidirect
282. #2354	C₂×C₅₃	semidirect
283. #2353	C₂×C₅₃	semidirect
284. #2352	C₂×C₅₃	semidirect
285. #2298	C₂×C₅₃	semidirect
287. #2285	C₂×C₅₃	semidirect
288. #2283	C₂×C₅₃	semidirect
289. C₄×#117	C₂×C₅₃	semidirect
290. C₄×#117	C₂×C₅₃	semidirect
291. C₂×C₄×C₄⋊C₄	C₂×C₅₃	semidirect
300. C₄×#41	C₄×C₅₃	semidirect
348. C₅₃	#34121	direct (N × H)
362. C₄×Q₈	C₂×C₄×C₅₃	semidirect
363. C₄×Q₈	C₂×C₄×C₅₃	semidirect
366. C₄×D₄	C₂×C₄×C₅₃	semidirect
367. C₄×D₄	C₂×C₄×C₅₃	semidirect
371. #117	C₂×C₄×C₅₃	semidirect
391. C₂×C₄²	C₂×C₄×C₅₃	semidirect
464. C₄⋊C₄	#41×C₅₃	semidirect
519. C₂×C₄	#117×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₅₃	2
C₅₃×#2357	2
C₅₃×#2356	2
C₅₃×#2355	2
C₅₃×#2354	2
C₅₃×#2353	2
C₅₃×#2352	2
C₅₃×#2298	2
C₅₃×#2297	2
C₅₃×#2285	2
C₅₃×#2283	2
C₄×#117×C₅₃	2
C₂×C₄×C₄⋊C₄×C₅₃	2
#34121	53

Subgroups — C53×#34121 #27241633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34121 #27241633