Subgroups — C₂×C₅₃×#2423 #27242199

Order 13568.

2790 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2423	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#399	C₂	1	2	yes	maximal
5	C₂×C₅₃×#398	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#388	C₂	1	2	yes	maximal
9	C₂×C₅₃×#387	C₂	1	2	yes	maximal, complemented
10	C₂×C₅₃×#384	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#380	C₂	1	1	yes	maximal, characteristic
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×C₂²⋊Q₈×C₅₃	—	2	8	no	—
16	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	8	yes	—
17	C₂×C₄×Q₈×C₅₃	—	2	8	no	—
18	C₂×C₄×Q₈×C₅₃	C₂²	1	8	yes	complemented
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	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
32	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
34	C₅₃×#399	C₂²	1	16	yes	—
35	C₅₃×#398	C₂²	1	16	yes	complemented
36	C₅₃×#391	C₂²	1	8	yes	complemented
37	C₅₃×#389	C₂²	1	8	yes	complemented
38	C₅₃×#388	C₂²	1	16	yes	—
39	C₅₃×#387	C₂²	1	16	yes	complemented
40	C₅₃×#384	C₂²	1	8	yes	complemented
41	C₅₃×#380	C₂²	1	8	yes	—
42	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
43	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
44	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
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	D₄	1	8	yes	—
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	2	yes	—
60	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
61	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
62	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²⋊Q₈×C₅₃	—	2	32	no	—
69	C₄×Q₈×C₅₃	—	2	32	no	—
70	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
71	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
74	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
75	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
76	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
77	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	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	8	yes	—
82	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
84	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
87	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
88	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
89	C₂×C₂²⋊C₄×C₅₃	Q₈	1	8	yes	complemented
90	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
94	C₂×C₄²×C₅₃	—	2	8	no	—
95	C₂×C₄²×C₅₃	—	2	4	no	—
96	C₂×C₄²×C₅₃	—	2	4	no	—
97	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
98	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
99	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
100	#95×C₅₃	C₂³	1	4	yes	—
101	#95×C₅₃	C₂³	1	4	yes	—
102	#95×C₅₃	C₂³	1	8	yes	—
103	#95×C₅₃	C₂³	1	8	yes	—
104	#95×C₅₃	C₂³	1	8	yes	—
105	#95×C₅₃	C₂³	1	8	yes	—
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	—	2	2	no	—
111	C₂⁴×C₅₃	—	2	2	no	—
112	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
113	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
114	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
115	C₂×Q₈×C₅₃	—	2	32	no	—
116	C₂×Q₈×C₅₃	—	2	8	no	—
117	C₂×Q₈×C₅₃	—	2	8	no	—
118	C₂×Q₈×C₅₃	C₂×D₄	1	16	yes	—
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	4	no	—
123	C₂²×C₄×C₅₃	—	2	4	no	—
124	C₂²×C₄×C₅₃	—	2	2	no	—
125	C₂²×C₄×C₅₃	—	2	2	no	—
126	C₂²×C₄×C₅₃	—	2	2	no	—
127	C₂²×C₄×C₅₃	—	2	2	no	—
128	C₂²×C₄×C₅₃	—	2	4	no	—
129	C₂²×C₄×C₅₃	—	2	4	no	—
130	C₂²×C₄×C₅₃	—	2	2	no	—
131	C₂²×C₄×C₅₃	—	2	2	no	—
132	C₂²×C₄×C₅₃	—	2	4	no	—
133	C₂²×C₄×C₅₃	—	2	4	no	—
134	C₂²×C₄×C₅₃	—	2	4	no	—
135	C₂²×C₄×C₅₃	—	2	2	no	—
136	C₂²×C₄×C₅₃	—	2	2	no	—
137	C₂²×C₄×C₅₃	—	2	4	no	—
138	C₂²×C₄×C₅₃	—	2	4	no	—
139	C₂²×C₄×C₅₃	—	2	2	no	—
140	C₂²×C₄×C₅₃	—	2	4	no	—
141	C₂²×C₄×C₅₃	—	2	4	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	C₂×Q₈	1	4	yes	—
149	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
150	C₂²×C₄×C₅₃	#41	1	4	yes	—
151	C₂²×C₄×C₅₃	#41	1	4	yes	—
152	C₂²×C₄×C₅₃	#41	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	4	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	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
163	C₄⋊C₄×C₅₃	—	2	16	no	—
164	C₄⋊C₄×C₅₃	—	2	8	no	—
165	C₄⋊C₄×C₅₃	—	2	16	no	—
166	C₄⋊C₄×C₅₃	—	2	8	no	—
167	C₄⋊C₄×C₅₃	—	2	16	no	—
168	C₄⋊C₄×C₅₃	—	2	8	no	—
169	C₄⋊C₄×C₅₃	—	4	8	no	—
170	C₄⋊C₄×C₅₃	—	4	8	no	—
171	C₂²⋊C₄×C₅₃	—	2	16	no	—
172	C₂²⋊C₄×C₅₃	—	2	8	no	—
173	C₂²⋊C₄×C₅₃	—	4	8	no	—
174	C₂²⋊C₄×C₅₃	C₂×Q₈	1	16	yes	complemented
175	C₄²×C₅₃	—	2	16	no	—
176	C₄²×C₅₃	—	2	8	no	—
177	C₄²×C₅₃	—	2	8	no	—
178	C₂³×C₅₃	—	2	8	no	—
179	C₂³×C₅₃	—	2	8	no	—
180	C₂³×C₅₃	—	2	8	no	—
181	C₂³×C₅₃	—	2	2	no	—
182	C₂³×C₅₃	—	2	2	no	—
183	C₂³×C₅₃	—	2	2	no	—
184	C₂³×C₅₃	—	2	2	no	—
185	C₂³×C₅₃	—	2	2	no	—
186	C₂³×C₅₃	—	2	2	no	—
187	C₂³×C₅₃	—	4	4	no	—
188	C₂³×C₅₃	—	4	4	no	—
189	C₂³×C₅₃	—	4	4	no	—
190	C₂³×C₅₃	—	4	4	no	—
191	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
192	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
193	C₂³×C₅₃	C₂²×Q₈	1	2	yes	—
194	C₂³×C₅₃	C₂×#41	1	2	yes	—
195	C₂³×C₅₃	C₂×#41	1	2	yes	—
196	C₂³×C₅₃	C₂×#41	1	2	yes	—
197	C₂³×C₅₃	H(5,𝔽₂)	1	2	yes	—
198	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
199	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
200	Q₈×C₅₃	—	2	32	no	—
201	C₂×C₄×C₅₃	—	2	16	no	—
202	C₂×C₄×C₅₃	—	2	8	no	—
203	C₂×C₄×C₅₃	—	2	8	no	—
204	C₂×C₄×C₅₃	—	2	8	no	—
205	C₂×C₄×C₅₃	—	2	2	no	—
206	C₂×C₄×C₅₃	—	2	4	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	2	no	—
211	C₂×C₄×C₅₃	—	2	2	no	—
212	C₂×C₄×C₅₃	—	2	2	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	2	no	—
220	C₂×C₄×C₅₃	—	2	4	no	—
221	C₂×C₄×C₅₃	—	2	4	no	—
222	C₂×C₄×C₅₃	—	2	4	no	—
223	C₂×C₄×C₅₃	—	2	4	no	—
224	C₂×C₄×C₅₃	—	2	4	no	—
225	C₂×C₄×C₅₃	—	2	2	no	—
226	C₂×C₄×C₅₃	—	2	4	no	—
227	C₂×C₄×C₅₃	—	2	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	8	no	—
232	C₂×C₄×C₅₃	—	4	8	no	—
233	C₂×C₄×C₅₃	—	4	16	no	—
234	C₂×C₄×C₅₃	#123	1	16	yes	—
235	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
236	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
237	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
238	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
239	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
240	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
241	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
242	C₂×#2423	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
243	C₂²×C₅₃	—	2	4	no	—
244	C₂²×C₅₃	—	2	4	no	—
245	C₂²×C₅₃	—	2	4	no	—
246	C₂²×C₅₃	—	4	2	no	—
247	C₂²×C₅₃	—	4	16	no	—
248	C₂²×C₅₃	—	4	2	no	—
249	C₂²×C₅₃	—	4	2	no	—
250	C₂²×C₅₃	—	4	2	no	—
251	C₂²×C₅₃	C₂×#123	1	4	yes	—
252	C₂²×C₅₃	#533	1	4	yes	—
253	C₂²×C₅₃	#542	1	4	yes	—
254	C₂²×C₅₃	D₄×Q₈	1	4	yes	—
255	C₂²×C₅₃	#550	1	4	yes	—
256	C₂²×C₅₃	#552	1	4	yes	—
257	C₂²×C₅₃	#554	1	4	yes	—
258	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
259	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
260	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
261	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
262	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
263	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
264	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
265	C₄×C₅₃	—	2	8	no	—
266	C₄×C₅₃	—	2	4	no	—
267	C₄×C₅₃	—	2	4	no	—
268	C₄×C₅₃	—	2	4	no	—
269	C₄×C₅₃	—	4	2	no	—
270	C₄×C₅₃	—	4	4	no	—
271	C₄×C₅₃	—	4	2	no	—
272	C₄×C₅₃	—	4	4	no	—
273	C₄×C₅₃	—	4	4	no	—
274	C₄×C₅₃	C₂×#123	1	8	yes	—
275	C₂²×C₂²⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
276	C₂²×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic, complemented
277	#2423	C₂×C₅₃	1	16	yes	direct product factor
278	C₂×#399	C₂×C₅₃	1	2	yes	—
279	C₂×#398	C₂×C₅₃	1	2	yes	complemented
280	C₂×#391	C₂×C₅₃	1	1	yes	characteristic, complemented
281	C₂×#389	C₂×C₅₃	1	1	yes	characteristic, complemented
282	C₂×#388	C₂×C₅₃	1	2	yes	—
283	C₂×#387	C₂×C₅₃	1	2	yes	complemented
284	C₂×#384	C₂×C₅₃	1	1	yes	characteristic, complemented
285	C₂×#380	C₂×C₅₃	1	1	yes	characteristic
286	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic
287	C₂×C₅₃	—	4	4	no	—
288	C₂×C₅₃	#2423	1	8	yes	direct product factor
289	C₂×C₅₃	C₂²×#123	1	1	yes	characteristic
290	C₂×C₅₃	C₂×#533	1	1	yes	characteristic
291	C₂×C₅₃	C₂×#542	1	1	yes	characteristic
292	C₂×C₅₃	C₂×D₄×Q₈	1	1	yes	characteristic
293	C₂×C₅₃	C₂×#550	1	1	yes	characteristic
294	C₂×C₅₃	C₂×#552	1	1	yes	characteristic
295	C₂×C₅₃	C₂×#554	1	1	yes	characteristic
296	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
297	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
298	C₂×C₂²⋊Q₈	—	2	8	no	—
299	C₂×C₂²⋊Q₈	C₂²×C₅₃	1	8	yes	—
300	C₂×C₄×Q₈	—	2	8	no	—
301	C₂×C₄×Q₈	C₂²×C₅₃	1	8	yes	complemented
302	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
303	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
304	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
305	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
306	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
307	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
308	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
309	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
310	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
311	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
312	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
313	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
314	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
315	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
316	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
317	#399	C₂²×C₅₃	1	16	yes	—
318	#398	C₂²×C₅₃	1	16	yes	complemented
319	#391	C₂²×C₅₃	1	8	yes	complemented
320	#389	C₂²×C₅₃	1	8	yes	complemented
321	#388	C₂²×C₅₃	1	16	yes	—
322	#387	C₂²×C₅₃	1	16	yes	complemented
323	#384	C₂²×C₅₃	1	8	yes	complemented
324	#380	C₂²×C₅₃	1	8	yes	—
325	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
326	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
327	C₂×#95	C₂²×C₅₃	1	2	yes	—
328	C₂×#95	C₂²×C₅₃	1	2	yes	—
329	C₂×#95	C₂²×C₅₃	1	2	yes	—
330	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
331	C₂×#95	C₂²×C₅₃	1	2	yes	—
332	C₅₃	C₂×#2423	1	1	yes	characteristic, direct product factor, Sylow
333	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
334	C₂²×Q₈	—	2	4	no	—
335	C₂²×Q₈	—	2	4	no	—
336	C₂²×Q₈	D₄×C₅₃	1	8	yes	—
337	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
338	C₂³×C₄	—	2	4	no	—
339	C₂³×C₄	—	2	4	no	—
340	C₂³×C₄	—	2	4	no	—
341	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
342	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
344	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
345	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
346	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
347	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
348	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
349	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
351	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
352	C₂²⋊Q₈	—	2	32	no	—
353	C₄×Q₈	—	2	32	no	—
354	C₂×C₄⋊C₄	—	2	8	no	—
355	C₂×C₄⋊C₄	—	2	4	no	—
356	C₂×C₄⋊C₄	—	2	4	no	—
357	C₂×C₄⋊C₄	—	2	8	no	—
358	C₂×C₄⋊C₄	—	2	8	no	—
359	C₂×C₄⋊C₄	—	2	4	no	—
360	C₂×C₄⋊C₄	—	4	4	no	—
361	C₂×C₄⋊C₄	—	4	4	no	—
362	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
363	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
364	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
365	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
366	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
367	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
368	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
369	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
370	C₂×C₂²⋊C₄	—	2	8	no	—
371	C₂×C₂²⋊C₄	—	2	4	no	—
372	C₂×C₂²⋊C₄	—	4	4	no	—
373	C₂×C₂²⋊C₄	Q₈×C₅₃	1	8	yes	complemented
374	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
378	C₂×C₄²	—	2	8	no	—
379	C₂×C₄²	—	2	4	no	—
380	C₂×C₄²	—	2	4	no	—
381	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
382	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
383	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
384	#95	C₂³×C₅₃	1	4	yes	—
385	#95	C₂³×C₅₃	1	4	yes	—
386	#95	C₂³×C₅₃	1	8	yes	—
387	#95	C₂³×C₅₃	1	8	yes	—
388	#95	C₂³×C₅₃	1	8	yes	—
389	#95	C₂³×C₅₃	1	8	yes	—
390	C₂⁴	—	2	2	no	—
391	C₂⁴	—	2	2	no	—
392	C₂⁴	—	2	2	no	—
393	C₂⁴	—	2	2	no	—
394	C₂⁴	—	2	2	no	—
395	C₂⁴	—	2	2	no	—
396	C₂⁴	C₂×Q₈×C₅₃	1	4	yes	—
397	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
398	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
399	C₂×Q₈	—	2	32	no	—
400	C₂×Q₈	—	2	8	no	—
401	C₂×Q₈	—	2	8	no	—
402	C₂×Q₈	C₂×D₄×C₅₃	1	16	yes	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	—	2	4	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	4	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	8	no	—
427	C₂²×C₄	—	2	8	no	—
428	C₂²×C₄	—	2	8	no	—
429	C₂²×C₄	—	4	16	no	—
430	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
431	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
432	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
433	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
434	C₂²×C₄	#41×C₅₃	1	4	yes	—
435	C₂²×C₄	#41×C₅₃	1	4	yes	—
436	C₂²×C₄	#41×C₅₃	1	4	yes	—
437	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
438	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
440	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
442	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
445	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
447	C₄⋊C₄	—	2	16	no	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	2	8	no	—
450	C₄⋊C₄	—	2	16	no	—
451	C₄⋊C₄	—	2	16	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	4	8	no	—
454	C₄⋊C₄	—	4	8	no	—
455	C₂²⋊C₄	—	2	16	no	—
456	C₂²⋊C₄	—	2	8	no	—
457	C₂²⋊C₄	—	4	8	no	—
458	C₂²⋊C₄	C₂×Q₈×C₅₃	1	16	yes	complemented
459	C₄²	—	2	16	no	—
460	C₄²	—	2	8	no	—
461	C₄²	—	2	8	no	—
462	C₂³	—	2	8	no	—
463	C₂³	—	2	8	no	—
464	C₂³	—	2	8	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	2	no	—
471	C₂³	—	4	4	no	—
472	C₂³	—	4	4	no	—
473	C₂³	—	4	4	no	—
474	C₂³	—	4	4	no	—
475	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
476	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
477	C₂³	C₂²×Q₈×C₅₃	1	2	yes	—
478	C₂³	C₂×#41×C₅₃	1	2	yes	—
479	C₂³	C₂×#41×C₅₃	1	2	yes	—
480	C₂³	C₂×#41×C₅₃	1	2	yes	—
481	C₂³	H(5,𝔽₂)×C₅₃	1	2	yes	—
482	C₂³	2₋¹⁺⁴×C₅₃	1	2	yes	—
483	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
484	Q₈	—	2	32	no	—
485	C₂×C₄	—	2	16	no	—
486	C₂×C₄	—	2	8	no	—
487	C₂×C₄	—	2	8	no	—
488	C₂×C₄	—	2	8	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	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	4	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	4	4	no	—
513	C₂×C₄	—	4	8	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	8	no	—
516	C₂×C₄	—	4	8	no	—
517	C₂×C₄	—	4	16	no	—
518	C₂×C₄	#123×C₅₃	1	16	yes	—
519	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
520	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
521	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
522	C₂×C₄	C₂²×Q₈×C₅₃	1	4	yes	—
523	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
524	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
525	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
526	C₂²	—	2	4	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	2	4	no	—
529	C₂²	—	4	16	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	C₂×#123×C₅₃	1	4	yes	—
535	C₂²	C₅₃×#533	1	4	yes	—
536	C₂²	C₅₃×#542	1	4	yes	—
537	C₂²	D₄×Q₈×C₅₃	1	4	yes	—
538	C₂²	C₅₃×#550	1	4	yes	—
539	C₂²	C₅₃×#552	1	4	yes	—
540	C₂²	C₅₃×#554	1	4	yes	—
541	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
542	C₂²	C₂³×Q₈×C₅₃	1	1	yes	characteristic
543	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
544	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
545	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
546	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic
547	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
548	C₄	—	2	4	no	—
549	C₄	—	2	4	no	—
550	C₄	—	2	4	no	—
551	C₄	—	2	8	no	—
552	C₄	—	4	4	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	4	no	—
555	C₄	—	4	4	no	—
556	C₄	—	4	2	no	—
557	C₄	C₂×#123×C₅₃	1	8	yes	—
558	C₂	—	4	4	no	—
559	C₂	C₅₃×#2423	1	8	yes	direct product factor
560	C₂	C₂²×#123×C₅₃	1	1	yes	characteristic
561	C₂	C₂×C₅₃×#533	1	1	yes	characteristic
562	C₂	C₂×C₅₃×#542	1	1	yes	characteristic
563	C₂	C₂×D₄×Q₈×C₅₃	1	1	yes	characteristic
564	C₂	C₂×C₅₃×#550	1	1	yes	characteristic
565	C₂	C₂×C₅₃×#552	1	1	yes	characteristic
566	C₂	C₂×C₅₃×#554	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
2. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
3. C₅₃×#2423	C₂	direct (N × H)
5. C₂×C₅₃×#398	C₂	semidirect
6. C₂×C₅₃×#391	C₂	semidirect
7. C₂×C₅₃×#389	C₂	semidirect
9. C₂×C₅₃×#387	C₂	semidirect
10. C₂×C₅₃×#384	C₂	semidirect
18. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
35. C₅₃×#398	C₂²	semidirect
36. C₅₃×#391	C₂²	semidirect
37. C₅₃×#389	C₂²	semidirect
39. C₅₃×#387	C₂²	semidirect
40. C₅₃×#384	C₂²	semidirect
89. C₂×C₂²⋊C₄×C₅₃	Q₈	semidirect
174. C₂²⋊C₄×C₅₃	C₂×Q₈	semidirect
242. C₂×#2423	C₅₃	direct (N × H)
276. C₂²×C₄×Q₈	C₂×C₅₃	semidirect
277. #2423	C₂×C₅₃	direct (N × H)
279. C₂×#398	C₂×C₅₃	semidirect
280. C₂×#391	C₂×C₅₃	semidirect
281. C₂×#389	C₂×C₅₃	semidirect
283. C₂×#387	C₂×C₅₃	semidirect
284. C₂×#384	C₂×C₅₃	semidirect
288. C₂×C₅₃	#2423	direct (N × H)
301. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
318. #398	C₂²×C₅₃	semidirect
319. #391	C₂²×C₅₃	semidirect
320. #389	C₂²×C₅₃	semidirect
322. #387	C₂²×C₅₃	semidirect
323. #384	C₂²×C₅₃	semidirect
332. C₅₃	C₂×#2423	direct (N × H)
373. C₂×C₂²⋊C₄	Q₈×C₅₃	semidirect
458. C₂²⋊C₄	C₂×Q₈×C₅₃	semidirect
559. C₂	C₅₃×#2423	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×C₂²⋊Q₈×C₅₃	2
C₂²×C₄×Q₈×C₅₃	2
C₅₃×#2423	2
C₂×C₅₃×#399	2
C₂×C₅₃×#398	2
C₂×C₅₃×#391	2
C₂×C₅₃×#389	2
C₂×C₅₃×#388	2
C₂×C₅₃×#387	2
C₂×C₅₃×#384	2
C₂×C₅₃×#380	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
C₂×#2423	53

Subgroups — C2×C53×#2423 #27242199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2423 #27242199