Subgroups — C₂×C₅₃×#2489 #27242288

Order 13568.

2638 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2489	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#400	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#397	C₂	1	2	yes	maximal
6	C₂×C₅₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#388	C₂	1	2	yes	maximal
9	C₂×C₅₃×#382	C₂	1	2	yes	maximal, complemented
10	C₂×C₅₃×#380	C₂	1	1	yes	maximal, characteristic
11	C₂×C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
13	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#125×C₅₃	—	2	8	no	—
16	C₂×#125×C₅₃	C₂²	1	8	yes	complemented
17	C₂×C₂²⋊Q₈×C₅₃	—	2	8	no	—
18	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
32	C₅₃×#400	C₂²	1	16	yes	complemented
33	C₅₃×#397	C₂²	1	16	yes	—
34	C₅₃×#391	C₂²	1	8	yes	complemented
35	C₅₃×#391	C₂²	1	8	yes	complemented
36	C₅₃×#388	C₂²	1	16	yes	—
37	C₅₃×#382	C₂²	1	16	yes	complemented
38	C₅₃×#380	C₂²	1	8	yes	—
39	C₄×C₄⋊C₄×C₅₃	C₂²	1	8	yes	complemented
40	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
41	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×#95×C₅₃	C₂²	1	2	yes	—
44	C₂×#95×C₅₃	C₂²	1	2	yes	—
45	C₂×#95×C₅₃	C₂²	1	2	yes	—
46	C₂×#95×C₅₃	C₂²	1	2	yes	—
47	C₂×#95×C₅₃	C₂²	1	2	yes	—
48	C₂×#95×C₅₃	C₂²	1	2	yes	—
49	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₅₃	—	2	4	no	—
51	C₂²×Q₈×C₅₃	—	2	4	no	—
52	C₂²×Q₈×C₅₃	—	2	4	no	—
53	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
54	C₂³×C₄×C₅₃	—	2	4	no	—
55	C₂³×C₄×C₅₃	—	2	4	no	—
56	C₂³×C₄×C₅₃	—	2	4	no	—
57	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
61	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
68	#125×C₅₃	—	2	32	no	—
69	C₂²⋊Q₈×C₅₃	—	2	32	no	—
70	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
71	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
72	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
73	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
75	C₂×C₄⋊C₄×C₅₃	D₄	1	8	yes	—
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
79	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
82	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
83	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
84	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
85	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
87	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
90	C₂×C₄²×C₅₃	—	2	4	no	—
91	C₂×C₄²×C₅₃	—	2	8	no	—
92	C₂×C₄²×C₅₃	—	2	4	no	—
93	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
94	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
95	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
96	#95×C₅₃	C₂³	1	4	yes	—
97	#95×C₅₃	C₂³	1	4	yes	—
98	#95×C₅₃	C₂³	1	8	yes	—
99	#95×C₅₃	C₂³	1	8	yes	—
100	#95×C₅₃	C₂³	1	8	yes	—
101	#95×C₅₃	C₂³	1	8	yes	—
102	#95×C₅₃	C₂³	1	8	yes	—
103	#95×C₅₃	C₂³	1	8	yes	—
104	C₂⁴×C₅₃	—	2	2	no	—
105	C₂⁴×C₅₃	—	2	2	no	—
106	C₂⁴×C₅₃	—	2	2	no	—
107	C₂⁴×C₅₃	—	2	2	no	—
108	C₂⁴×C₅₃	—	2	2	no	—
109	C₂⁴×C₅₃	—	2	2	no	—
110	C₂⁴×C₅₃	#41	1	4	yes	—
111	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
112	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
113	C₂×Q₈×C₅₃	—	2	8	no	—
114	C₂×Q₈×C₅₃	—	2	8	no	—
115	C₂×Q₈×C₅₃	—	2	8	no	—
116	C₂×Q₈×C₅₃	—	4	16	no	—
117	C₂²×C₄×C₅₃	—	2	4	no	—
118	C₂²×C₄×C₅₃	—	2	2	no	—
119	C₂²×C₄×C₅₃	—	2	4	no	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	—	2	2	no	—
122	C₂²×C₄×C₅₃	—	2	2	no	—
123	C₂²×C₄×C₅₃	—	2	4	no	—
124	C₂²×C₄×C₅₃	—	2	4	no	—
125	C₂²×C₄×C₅₃	—	2	2	no	—
126	C₂²×C₄×C₅₃	—	2	4	no	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	2	4	no	—
129	C₂²×C₄×C₅₃	—	2	2	no	—
130	C₂²×C₄×C₅₃	—	2	2	no	—
131	C₂²×C₄×C₅₃	—	2	4	no	—
132	C₂²×C₄×C₅₃	—	2	2	no	—
133	C₂²×C₄×C₅₃	—	2	4	no	—
134	C₂²×C₄×C₅₃	—	2	2	no	—
135	C₂²×C₄×C₅₃	—	2	2	no	—
136	C₂²×C₄×C₅₃	—	2	4	no	—
137	C₂²×C₄×C₅₃	—	2	4	no	—
138	C₂²×C₄×C₅₃	—	2	2	no	—
139	C₂²×C₄×C₅₃	—	2	2	no	—
140	C₂²×C₄×C₅₃	—	2	4	no	—
141	C₂²×C₄×C₅₃	—	2	2	no	—
142	C₂²×C₄×C₅₃	—	2	8	no	—
143	C₂²×C₄×C₅₃	—	2	8	no	—
144	C₂²×C₄×C₅₃	—	2	8	no	—
145	C₂²×C₄×C₅₃	—	4	16	no	—
146	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
147	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
148	C₂²×C₄×C₅₃	#41	1	4	yes	—
149	C₂²×C₄×C₅₃	#41	1	4	yes	—
150	C₂²×C₄×C₅₃	#41	1	4	yes	—
151	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
152	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
161	C₄⋊C₄×C₅₃	—	2	16	no	—
162	C₄⋊C₄×C₅₃	—	2	16	no	—
163	C₄⋊C₄×C₅₃	—	2	16	no	—
164	C₄⋊C₄×C₅₃	—	2	8	no	—
165	C₄⋊C₄×C₅₃	—	2	8	no	—
166	C₄⋊C₄×C₅₃	C₂×D₄	1	16	yes	—
167	C₂²⋊C₄×C₅₃	—	2	8	no	—
168	C₂²⋊C₄×C₅₃	—	2	16	no	—
169	C₂²⋊C₄×C₅₃	—	2	16	no	—
170	C₂²⋊C₄×C₅₃	—	2	8	no	—
171	C₄²×C₅₃	—	2	8	no	—
172	C₄²×C₅₃	—	2	16	no	—
173	C₄²×C₅₃	—	2	8	no	—
174	C₂³×C₅₃	—	2	8	no	—
175	C₂³×C₅₃	—	2	8	no	—
176	C₂³×C₅₃	—	2	8	no	—
177	C₂³×C₅₃	—	2	2	no	—
178	C₂³×C₅₃	—	2	2	no	—
179	C₂³×C₅₃	—	2	2	no	—
180	C₂³×C₅₃	—	2	2	no	—
181	C₂³×C₅₃	—	2	2	no	—
182	C₂³×C₅₃	—	2	2	no	—
183	C₂³×C₅₃	—	4	4	no	—
184	C₂³×C₅₃	—	4	4	no	—
185	C₂³×C₅₃	—	4	4	no	—
186	C₂³×C₅₃	—	4	4	no	—
187	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
188	C₂³×C₅₃	C₂×#41	1	4	yes	—
189	C₂³×C₅₃	C₂×#41	1	2	yes	—
190	C₂³×C₅₃	C₂×#41	1	2	yes	—
191	C₂³×C₅₃	C₂×#41	1	2	yes	—
192	C₂³×C₅₃	C₂×#41	1	2	yes	—
193	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
194	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
195	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
196	Q₈×C₅₃	—	4	16	no	—
197	C₂×C₄×C₅₃	—	2	16	no	—
198	C₂×C₄×C₅₃	—	2	8	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	4	no	—
202	C₂×C₄×C₅₃	—	2	2	no	—
203	C₂×C₄×C₅₃	—	2	2	no	—
204	C₂×C₄×C₅₃	—	2	4	no	—
205	C₂×C₄×C₅₃	—	2	2	no	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	4	no	—
208	C₂×C₄×C₅₃	—	2	2	no	—
209	C₂×C₄×C₅₃	—	2	2	no	—
210	C₂×C₄×C₅₃	—	2	4	no	—
211	C₂×C₄×C₅₃	—	2	4	no	—
212	C₂×C₄×C₅₃	—	2	4	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	4	no	—
215	C₂×C₄×C₅₃	—	2	2	no	—
216	C₂×C₄×C₅₃	—	2	2	no	—
217	C₂×C₄×C₅₃	—	2	4	no	—
218	C₂×C₄×C₅₃	—	2	2	no	—
219	C₂×C₄×C₅₃	—	2	4	no	—
220	C₂×C₄×C₅₃	—	2	2	no	—
221	C₂×C₄×C₅₃	—	2	2	no	—
222	C₂×C₄×C₅₃	—	2	4	no	—
223	C₂×C₄×C₅₃	—	2	2	no	—
224	C₂×C₄×C₅₃	—	4	4	no	—
225	C₂×C₄×C₅₃	—	4	8	no	—
226	C₂×C₄×C₅₃	—	4	8	no	—
227	C₂×C₄×C₅₃	—	4	4	no	—
228	C₂×C₄×C₅₃	—	4	4	no	—
229	C₂×C₄×C₅₃	—	4	8	no	—
230	C₂×C₄×C₅₃	—	4	4	no	—
231	C₂×C₄×C₅₃	—	4	16	no	—
232	C₂×C₄×C₅₃	#124	1	16	yes	—
233	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
234	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
235	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
236	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
237	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
238	C₂×#2489	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂²×C₅₃	—	2	4	no	—
240	C₂²×C₅₃	—	2	4	no	—
241	C₂²×C₅₃	—	2	4	no	—
242	C₂²×C₅₃	—	4	2	no	—
243	C₂²×C₅₃	—	4	2	no	—
244	C₂²×C₅₃	—	4	2	no	—
245	C₂²×C₅₃	—	4	2	no	—
246	C₂²×C₅₃	—	4	16	no	—
247	C₂²×C₅₃	C₂×#124	1	4	yes	—
248	C₂²×C₅₃	#529	1	4	yes	—
249	C₂²×C₅₃	#541	1	4	yes	—
250	C₂²×C₅₃	#547	1	4	yes	—
251	C₂²×C₅₃	#548	1	4	yes	—
252	C₂²×C₅₃	#552	1	4	yes	—
253	C₂²×C₅₃	#552	1	4	yes	—
254	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
255	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
256	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
257	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
258	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
259	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
260	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
261	C₄×C₅₃	—	2	8	no	—
262	C₄×C₅₃	—	2	4	no	—
263	C₄×C₅₃	—	4	2	no	—
264	C₄×C₅₃	—	4	4	no	—
265	C₄×C₅₃	—	4	4	no	—
266	C₄×C₅₃	—	4	2	no	—
267	C₄×C₅₃	—	4	2	no	—
268	C₄×C₅₃	—	4	4	no	—
269	C₄×C₅₃	—	4	2	no	—
270	C₄×C₅₃	C₂×#124	1	8	yes	—
271	C₂²×#125	C₂×C₅₃	1	1	yes	characteristic, complemented
272	C₂²×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
273	#2489	C₂×C₅₃	1	16	yes	direct product factor
274	C₂×#400	C₂×C₅₃	1	2	yes	complemented
275	C₂×#397	C₂×C₅₃	1	2	yes	—
276	C₂×#391	C₂×C₅₃	1	1	yes	characteristic, complemented
277	C₂×#391	C₂×C₅₃	1	1	yes	characteristic, complemented
278	C₂×#388	C₂×C₅₃	1	2	yes	—
279	C₂×#382	C₂×C₅₃	1	2	yes	complemented
280	C₂×#380	C₂×C₅₃	1	1	yes	characteristic
281	C₂×C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
282	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
283	C₂×C₅₃	—	4	4	no	—
284	C₂×C₅₃	#2489	1	8	yes	direct product factor
285	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
286	C₂×C₅₃	C₂×#529	1	1	yes	characteristic
287	C₂×C₅₃	C₂×#541	1	1	yes	characteristic
288	C₂×C₅₃	C₂×#547	1	1	yes	characteristic
289	C₂×C₅₃	C₂×#548	1	1	yes	characteristic
290	C₂×C₅₃	C₂×#552	1	1	yes	characteristic
291	C₂×C₅₃	C₂×#552	1	1	yes	characteristic
292	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
293	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
294	C₂×#125	—	2	8	no	—
295	C₂×#125	C₂²×C₅₃	1	8	yes	complemented
296	C₂×C₂²⋊Q₈	—	2	8	no	—
297	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	8	yes	—
298	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
299	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
300	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
301	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
302	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
303	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
304	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
305	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
306	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
307	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
308	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
309	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
310	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
311	#400	C₂²×C₅₃	1	16	yes	complemented
312	#397	C₂²×C₅₃	1	16	yes	—
313	#391	C₂²×C₅₃	1	8	yes	complemented
314	#391	C₂²×C₅₃	1	8	yes	complemented
315	#388	C₂²×C₅₃	1	16	yes	—
316	#382	C₂²×C₅₃	1	16	yes	complemented
317	#380	C₂²×C₅₃	1	8	yes	—
318	C₄×C₄⋊C₄	C₂²×C₅₃	1	8	yes	complemented
319	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
320	C₂×#95	C₂²×C₅₃	1	2	yes	—
321	C₂×#95	C₂²×C₅₃	1	2	yes	—
322	C₂×#95	C₂²×C₅₃	1	2	yes	—
323	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
324	C₂×#95	C₂²×C₅₃	1	2	yes	—
325	C₂×#95	C₂²×C₅₃	1	2	yes	—
326	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
327	C₂×#95	C₂²×C₅₃	1	2	yes	—
328	C₅₃	C₂×#2489	1	1	yes	characteristic, direct product factor, Sylow
329	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
330	C₂²×Q₈	—	2	4	no	—
331	C₂²×Q₈	—	2	4	no	—
332	C₂²×Q₈	—	2	4	no	—
333	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
334	C₂³×C₄	—	2	4	no	—
335	C₂³×C₄	—	2	4	no	—
336	C₂³×C₄	—	2	4	no	—
337	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
338	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
341	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
342	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
344	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
345	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
346	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
347	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
348	#125	—	2	32	no	—
349	C₂²⋊Q₈	—	2	32	no	—
350	C₂×C₄⋊C₄	—	2	8	no	—
351	C₂×C₄⋊C₄	—	2	8	no	—
352	C₂×C₄⋊C₄	—	2	8	no	—
353	C₂×C₄⋊C₄	—	2	4	no	—
354	C₂×C₄⋊C₄	—	2	4	no	—
355	C₂×C₄⋊C₄	D₄×C₅₃	1	8	yes	—
356	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
357	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
358	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
359	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
360	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
361	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
362	C₂×C₂²⋊C₄	—	2	4	no	—
363	C₂×C₂²⋊C₄	—	2	8	no	—
364	C₂×C₂²⋊C₄	—	2	8	no	—
365	C₂×C₂²⋊C₄	—	2	4	no	—
366	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
367	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
368	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
369	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
370	C₂×C₄²	—	2	4	no	—
371	C₂×C₄²	—	2	8	no	—
372	C₂×C₄²	—	2	4	no	—
373	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
374	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
375	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
376	#95	C₂³×C₅₃	1	4	yes	—
377	#95	C₂³×C₅₃	1	4	yes	—
378	#95	C₂³×C₅₃	1	8	yes	—
379	#95	C₂³×C₅₃	1	8	yes	—
380	#95	C₂³×C₅₃	1	8	yes	—
381	#95	C₂³×C₅₃	1	8	yes	—
382	#95	C₂³×C₅₃	1	8	yes	—
383	#95	C₂³×C₅₃	1	8	yes	—
384	C₂⁴	—	2	2	no	—
385	C₂⁴	—	2	2	no	—
386	C₂⁴	—	2	2	no	—
387	C₂⁴	—	2	2	no	—
388	C₂⁴	—	2	2	no	—
389	C₂⁴	—	2	2	no	—
390	C₂⁴	#41×C₅₃	1	4	yes	—
391	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
392	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
393	C₂×Q₈	—	2	8	no	—
394	C₂×Q₈	—	2	8	no	—
395	C₂×Q₈	—	2	8	no	—
396	C₂×Q₈	—	4	16	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	8	no	—
424	C₂²×C₄	—	2	8	no	—
425	C₂²×C₄	—	4	16	no	—
426	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
427	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
428	C₂²×C₄	#41×C₅₃	1	4	yes	—
429	C₂²×C₄	#41×C₅₃	1	4	yes	—
430	C₂²×C₄	#41×C₅₃	1	4	yes	—
431	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
432	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
433	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
434	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
437	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
438	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
440	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
441	C₄⋊C₄	—	2	16	no	—
442	C₄⋊C₄	—	2	16	no	—
443	C₄⋊C₄	—	2	16	no	—
444	C₄⋊C₄	—	2	8	no	—
445	C₄⋊C₄	—	2	8	no	—
446	C₄⋊C₄	C₂×D₄×C₅₃	1	16	yes	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₂²⋊C₄	—	2	16	no	—
449	C₂²⋊C₄	—	2	16	no	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₄²	—	2	8	no	—
452	C₄²	—	2	16	no	—
453	C₄²	—	2	8	no	—
454	C₂³	—	2	8	no	—
455	C₂³	—	2	8	no	—
456	C₂³	—	2	8	no	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	2	2	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	2	2	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	4	4	no	—
464	C₂³	—	4	4	no	—
465	C₂³	—	4	4	no	—
466	C₂³	—	4	4	no	—
467	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
468	C₂³	C₂×#41×C₅₃	1	4	yes	—
469	C₂³	C₂×#41×C₅₃	1	2	yes	—
470	C₂³	C₂×#41×C₅₃	1	2	yes	—
471	C₂³	C₂×#41×C₅₃	1	2	yes	—
472	C₂³	C₂×#41×C₅₃	1	2	yes	—
473	C₂³	2₋¹⁺⁴×C₅₃	1	2	yes	—
474	C₂³	2₋¹⁺⁴×C₅₃	1	2	yes	—
475	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
476	Q₈	—	4	16	no	—
477	C₂×C₄	—	2	16	no	—
478	C₂×C₄	—	2	8	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	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	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	4	8	no	—
506	C₂×C₄	—	4	8	no	—
507	C₂×C₄	—	4	4	no	—
508	C₂×C₄	—	4	4	no	—
509	C₂×C₄	—	4	8	no	—
510	C₂×C₄	—	4	4	no	—
511	C₂×C₄	—	4	16	no	—
512	C₂×C₄	#124×C₅₃	1	16	yes	—
513	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
514	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
515	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
516	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
517	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
518	C₂²	—	2	4	no	—
519	C₂²	—	2	4	no	—
520	C₂²	—	2	4	no	—
521	C₂²	—	4	2	no	—
522	C₂²	—	4	2	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	4	16	no	—
525	C₂²	—	4	2	no	—
526	C₂²	C₂×#124×C₅₃	1	4	yes	—
527	C₂²	C₅₃×#529	1	4	yes	—
528	C₂²	C₅₃×#541	1	4	yes	—
529	C₂²	C₅₃×#547	1	4	yes	—
530	C₂²	C₅₃×#548	1	4	yes	—
531	C₂²	C₅₃×#552	1	4	yes	—
532	C₂²	C₅₃×#552	1	4	yes	—
533	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
534	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
535	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
536	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
537	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
538	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
539	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
540	C₄	—	2	8	no	—
541	C₄	—	2	4	no	—
542	C₄	—	4	2	no	—
543	C₄	—	4	2	no	—
544	C₄	—	4	4	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	4	no	—
548	C₄	—	4	4	no	—
549	C₄	C₂×#124×C₅₃	1	8	yes	—
550	C₂	—	4	4	no	—
551	C₂	C₅₃×#2489	1	8	yes	direct product factor
552	C₂	C₂²×#124×C₅₃	1	1	yes	characteristic
553	C₂	C₂×C₅₃×#529	1	1	yes	characteristic
554	C₂	C₂×C₅₃×#541	1	1	yes	characteristic
555	C₂	C₂×C₅₃×#547	1	1	yes	characteristic
556	C₂	C₂×C₅₃×#548	1	1	yes	characteristic
557	C₂	C₂×C₅₃×#552	1	1	yes	characteristic
558	C₂	C₂×C₅₃×#552	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×#125×C₅₃	C₂	semidirect
3. C₅₃×#2489	C₂	direct (N × H)
4. C₂×C₅₃×#400	C₂	semidirect
6. C₂×C₅₃×#391	C₂	semidirect
7. C₂×C₅₃×#391	C₂	semidirect
9. C₂×C₅₃×#382	C₂	semidirect
11. C₂×C₄×C₄⋊C₄×C₅₃	C₂	semidirect
16. C₂×#125×C₅₃	C₂²	semidirect
32. C₅₃×#400	C₂²	semidirect
34. C₅₃×#391	C₂²	semidirect
35. C₅₃×#391	C₂²	semidirect
37. C₅₃×#382	C₂²	semidirect
39. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
238. C₂×#2489	C₅₃	direct (N × H)
271. C₂²×#125	C₂×C₅₃	semidirect
273. #2489	C₂×C₅₃	direct (N × H)
274. C₂×#400	C₂×C₅₃	semidirect
276. C₂×#391	C₂×C₅₃	semidirect
277. C₂×#391	C₂×C₅₃	semidirect
279. C₂×#382	C₂×C₅₃	semidirect
281. C₂×C₄×C₄⋊C₄	C₂×C₅₃	semidirect
284. C₂×C₅₃	#2489	direct (N × H)
295. C₂×#125	C₂²×C₅₃	semidirect
311. #400	C₂²×C₅₃	semidirect
313. #391	C₂²×C₅₃	semidirect
314. #391	C₂²×C₅₃	semidirect
316. #382	C₂²×C₅₃	semidirect
318. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
328. C₅₃	C₂×#2489	direct (N × H)
551. C₂	C₅₃×#2489	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#125×C₅₃	2
C₂²×C₂²⋊Q₈×C₅₃	2
C₅₃×#2489	2
C₂×C₅₃×#400	2
C₂×C₅₃×#397	2
C₂×C₅₃×#391	2
C₂×C₅₃×#388	2
C₂×C₅₃×#382	2
C₂×C₅₃×#380	2
C₂×C₄×C₄⋊C₄×C₅₃	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
C₂×#2489	53

Subgroups — C2×C53×#2489 #27242288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2489 #27242288