Subgroups — C₂×C₅₃×#2353 #27241628

Order 13568.

3374 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2353	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅₃×#388	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#387	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#386	C₂	1	2	yes	maximal
6	C₂×C₅₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×C₂²⋊C₄×C₅₃	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#95×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₅₃	C₄	1	16	yes	complemented
18	C₂×C₄×D₄×C₅₃	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
27	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
28	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#388	C₂²	1	16	yes	complemented
30	C₅₃×#387	C₂²	1	16	yes	complemented
31	C₅₃×#386	C₂²	1	16	yes	—
32	C₅₃×#383	C₂²	1	8	yes	complemented
33	C₅₃×#381	C₂²	1	8	yes	complemented
34	C₅₃×#380	C₂²	1	8	yes	complemented
35	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	complemented
36	C₄×C₂²⋊C₄×C₅₃	C₂²	1	16	yes	complemented
37	C₅₃×#376	C₂²	1	8	yes	complemented
38	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
39	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×#95×C₅₃	C₂²	1	2	yes	—
41	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#95×C₅₃	C₂²	1	2	yes	—
43	C₂×#95×C₅₃	C₂²	1	1	yes	characteristic
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	8	yes	complemented
48	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
49	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₅₃	—	2	4	no	—
51	C₂²×D₄×C₅₃	—	2	4	no	—
52	C₂²×D₄×C₅₃	C₂×C₄	1	8	yes	—
53	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
54	C₂³×C₄×C₅₃	—	2	8	no	—
55	C₂³×C₄×C₅₃	—	2	8	no	—
56	C₂³×C₄×C₅₃	—	4	4	no	—
57	C₂³×C₄×C₅₃	—	4	4	no	—
58	C₂³×C₄×C₅₃	—	4	4	no	—
59	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
60	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	complemented
61	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
62	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
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	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
71	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
72	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
73	C₄×D₄×C₅₃	C₂×C₄	1	64	yes	complemented
74	C₂×C₄⋊C₄×C₅₃	—	4	4	no	—
75	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	complemented
76	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
79	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
80	C₂×C₂²⋊C₄×C₅₃	—	4	4	no	—
81	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
82	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	complemented
83	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
84	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
88	C₂×C₄²×C₅₃	—	2	8	no	—
89	C₂×C₄²×C₅₃	—	4	4	no	—
90	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	complemented
91	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
92	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
93	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
94	#95×C₅₃	C₂³	1	4	yes	—
95	#95×C₅₃	C₂³	1	4	yes	—
96	#95×C₅₃	C₂³	1	8	yes	—
97	#95×C₅₃	C₂³	1	4	yes	—
98	#95×C₅₃	C₂³	1	8	yes	—
99	#95×C₅₃	C₂³	1	4	yes	—
100	#95×C₅₃	C₂³	1	8	yes	—
101	#95×C₅₃	C₂³	1	8	yes	—
102	#95×C₅₃	C₂³	1	8	yes	—
103	C₂⁴×C₅₃	—	2	4	no	—
104	C₂⁴×C₅₃	—	2	4	no	—
105	C₂⁴×C₅₃	—	2	4	no	—
106	C₂⁴×C₅₃	—	2	2	no	—
107	C₂⁴×C₅₃	—	2	2	no	—
108	C₂⁴×C₅₃	—	2	4	no	—
109	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
110	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
111	C₂⁴×C₅₃	#41	1	4	yes	—
112	C₂⁴×C₅₃	#41	1	4	yes	—
113	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
114	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
115	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
116	C₂×D₄×C₅₃	—	2	32	no	—
117	C₂×D₄×C₅₃	—	2	8	no	—
118	C₂×D₄×C₅₃	—	2	8	no	—
119	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
120	C₂²×C₄×C₅₃	—	2	4	no	—
121	C₂²×C₄×C₅₃	—	2	4	no	—
122	C₂²×C₄×C₅₃	—	2	2	no	—
123	C₂²×C₄×C₅₃	—	2	2	no	—
124	C₂²×C₄×C₅₃	—	2	4	no	—
125	C₂²×C₄×C₅₃	—	2	4	no	—
126	C₂²×C₄×C₅₃	—	2	4	no	—
127	C₂²×C₄×C₅₃	—	2	4	no	—
128	C₂²×C₄×C₅₃	—	2	4	no	—
129	C₂²×C₄×C₅₃	—	2	4	no	—
130	C₂²×C₄×C₅₃	—	2	4	no	—
131	C₂²×C₄×C₅₃	—	2	4	no	—
132	C₂²×C₄×C₅₃	—	2	16	no	—
133	C₂²×C₄×C₅₃	—	2	4	no	—
134	C₂²×C₄×C₅₃	—	2	4	no	—
135	C₂²×C₄×C₅₃	—	2	4	no	—
136	C₂²×C₄×C₅₃	—	2	4	no	—
137	C₂²×C₄×C₅₃	—	2	4	no	—
138	C₂²×C₄×C₅₃	—	2	16	no	—
139	C₂²×C₄×C₅₃	—	4	16	no	—
140	C₂²×C₄×C₅₃	—	4	16	no	—
141	C₂²×C₄×C₅₃	—	4	8	no	—
142	C₂²×C₄×C₅₃	—	4	8	no	—
143	C₂²×C₄×C₅₃	—	4	8	no	—
144	C₂²×C₄×C₅₃	—	4	32	no	—
145	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
146	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
147	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
150	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
151	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
152	C₂²×C₄×C₅₃	#41	1	8	yes	—
153	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
158	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
162	C₄⋊C₄×C₅₃	—	4	8	no	—
163	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
164	C₂²⋊C₄×C₅₃	—	2	16	no	—
165	C₂²⋊C₄×C₅₃	—	2	16	no	—
166	C₂²⋊C₄×C₅₃	—	4	8	no	—
167	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	—
168	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
169	C₄²×C₅₃	—	2	16	no	—
170	C₄²×C₅₃	—	4	8	no	—
171	C₄²×C₅₃	C₂²×C₄	1	16	yes	complemented
172	C₂³×C₅₃	—	2	8	no	—
173	C₂³×C₅₃	—	2	8	no	—
174	C₂³×C₅₃	—	2	16	no	—
175	C₂³×C₅₃	—	2	16	no	—
176	C₂³×C₅₃	—	2	4	no	—
177	C₂³×C₅₃	—	2	4	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	4	no	—
184	C₂³×C₅₃	—	2	4	no	—
185	C₂³×C₅₃	—	4	8	no	—
186	C₂³×C₅₃	—	4	8	no	—
187	C₂³×C₅₃	#117	1	16	yes	—
188	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
189	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
190	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
191	C₂³×C₅₃	C₂×#41	1	4	yes	—
192	C₂³×C₅₃	C₂×#41	1	4	yes	—
193	C₂³×C₅₃	C₂×#41	1	2	yes	—
194	C₂³×C₅₃	C₂×#41	1	2	yes	—
195	C₂³×C₅₃	C₂×#41	1	4	yes	—
196	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
197	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
198	D₄×C₅₃	—	2	32	no	—
199	C₂×C₄×C₅₃	—	2	8	no	—
200	C₂×C₄×C₅₃	—	2	8	no	—
201	C₂×C₄×C₅₃	—	2	8	no	—
202	C₂×C₄×C₅₃	—	2	8	no	—
203	C₂×C₄×C₅₃	—	2	8	no	—
204	C₂×C₄×C₅₃	—	2	16	no	—
205	C₂×C₄×C₅₃	—	2	4	no	—
206	C₂×C₄×C₅₃	—	2	2	no	—
207	C₂×C₄×C₅₃	—	2	2	no	—
208	C₂×C₄×C₅₃	—	2	4	no	—
209	C₂×C₄×C₅₃	—	2	4	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	4	no	—
214	C₂×C₄×C₅₃	—	2	4	no	—
215	C₂×C₄×C₅₃	—	2	4	no	—
216	C₂×C₄×C₅₃	—	2	4	no	—
217	C₂×C₄×C₅₃	—	2	4	no	—
218	C₂×C₄×C₅₃	—	2	4	no	—
219	C₂×C₄×C₅₃	—	2	4	no	—
220	C₂×C₄×C₅₃	—	2	4	no	—
221	C₂×C₄×C₅₃	—	2	4	no	—
222	C₂×C₄×C₅₃	—	4	16	no	—
223	C₂×C₄×C₅₃	—	4	16	no	—
224	C₂×C₄×C₅₃	—	4	32	no	—
225	C₂×C₄×C₅₃	—	4	8	no	—
226	C₂×C₄×C₅₃	—	4	8	no	—
227	C₂×C₄×C₅₃	—	4	8	no	—
228	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
229	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
230	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
231	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
232	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
233	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
234	C₂×#2353	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂²×C₅₃	—	2	32	no	—
236	C₂²×C₅₃	—	2	4	no	—
237	C₂²×C₅₃	—	2	4	no	—
238	C₂²×C₅₃	—	2	8	no	—
239	C₂²×C₅₃	—	2	8	no	—
240	C₂²×C₅₃	—	2	4	no	—
241	C₂²×C₅₃	—	2	4	no	—
242	C₂²×C₅₃	—	2	4	no	—
243	C₂²×C₅₃	—	4	16	no	—
244	C₂²×C₅₃	—	4	4	no	—
245	C₂²×C₅₃	—	4	4	no	—
246	C₂²×C₅₃	C₂×#117	1	8	yes	—
247	C₂²×C₅₃	C₂×#117	1	4	yes	—
248	C₂²×C₅₃	C₄×#41	1	4	yes	—
249	C₂²×C₅₃	#518	1	4	yes	—
250	C₂²×C₅₃	#551	1	8	yes	—
251	C₂²×C₅₃	#553	1	8	yes	—
252	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
253	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
254	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
255	C₂²×C₅₃	C₂²×#41	1	2	yes	—
256	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
257	C₄×C₅₃	—	2	4	no	—
258	C₄×C₅₃	—	2	4	no	—
259	C₄×C₅₃	—	2	4	no	—
260	C₄×C₅₃	—	2	4	no	—
261	C₄×C₅₃	—	2	4	no	—
262	C₄×C₅₃	—	2	8	no	—
263	C₄×C₅₃	—	4	4	no	—
264	C₄×C₅₃	—	4	4	no	—
265	C₄×C₅₃	—	4	4	no	—
266	C₂²×C₄×D₄	C₂×C₅₃	1	1	yes	characteristic
267	#2353	C₂×C₅₃	1	16	yes	direct product factor
268	C₂×#388	C₂×C₅₃	1	2	yes	complemented
269	C₂×#387	C₂×C₅₃	1	2	yes	complemented
270	C₂×#386	C₂×C₅₃	1	2	yes	—
271	C₂×#383	C₂×C₅₃	1	1	yes	characteristic, complemented
272	C₂×#381	C₂×C₅₃	1	1	yes	characteristic, complemented
273	C₂×#380	C₂×C₅₃	1	1	yes	characteristic, complemented
274	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	2	yes	complemented
275	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
276	C₂×#376	C₂×C₅₃	1	1	yes	characteristic, complemented
277	C₂²×#95	C₂×C₅₃	1	1	yes	characteristic, complemented
278	C₂×C₅₃	—	2	8	no	—
279	C₂×C₅₃	—	4	4	no	—
280	C₂×C₅₃	#2353	1	8	yes	direct product factor
281	C₂×C₅₃	C₂²×#117	1	1	yes	characteristic
282	C₂×C₅₃	C₂×C₄×#41	1	1	yes	characteristic
283	C₂×C₅₃	C₂×#518	1	1	yes	characteristic
284	C₂×C₅₃	C₂×#551	1	2	yes	—
285	C₂×C₅₃	C₂×#553	1	2	yes	—
286	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
287	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	—
288	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
289	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
290	C₂×C₄×D₄	C₄×C₅₃	1	16	yes	complemented
291	C₂×C₄×D₄	C₂²×C₅₃	1	8	yes	—
292	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
293	C₂²×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
294	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
295	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
296	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
297	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
298	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
299	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
300	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
301	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
302	#388	C₂²×C₅₃	1	16	yes	complemented
303	#387	C₂²×C₅₃	1	16	yes	complemented
304	#386	C₂²×C₅₃	1	16	yes	—
305	#383	C₂²×C₅₃	1	8	yes	complemented
306	#381	C₂²×C₅₃	1	8	yes	complemented
307	#380	C₂²×C₅₃	1	8	yes	complemented
308	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	complemented
309	C₄×C₂²⋊C₄	C₂²×C₅₃	1	16	yes	complemented
310	#376	C₂²×C₅₃	1	8	yes	complemented
311	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
312	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
313	C₂×#95	C₂²×C₅₃	1	2	yes	—
314	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
315	C₂×#95	C₂²×C₅₃	1	2	yes	—
316	C₂×#95	C₂²×C₅₃	1	1	yes	characteristic
317	C₂×#95	C₂²×C₅₃	1	2	yes	—
318	C₂×#95	C₂²×C₅₃	1	2	yes	—
319	C₂×#95	C₂²×C₅₃	1	2	yes	—
320	C₂×#95	C₂²×C₅₃	1	8	yes	complemented
321	C₅₃	C₂×#2353	1	1	yes	characteristic, direct product factor, Sylow
322	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
323	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
324	C₂²×D₄	—	2	4	no	—
325	C₂²×D₄	—	2	4	no	—
326	C₂²×D₄	C₂×C₄×C₅₃	1	8	yes	—
327	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
328	C₂³×C₄	—	2	8	no	—
329	C₂³×C₄	—	2	8	no	—
330	C₂³×C₄	—	4	4	no	—
331	C₂³×C₄	—	4	4	no	—
332	C₂³×C₄	—	4	4	no	—
333	C₂³×C₄	C₂×C₄×C₅₃	1	8	yes	—
334	C₂³×C₄	C₂×C₄×C₅₃	1	8	yes	complemented
335	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
336	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
337	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
338	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
339	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
341	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
342	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
344	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
345	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
346	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
347	C₄×D₄	C₂×C₄×C₅₃	1	64	yes	complemented
348	C₂×C₄⋊C₄	—	4	4	no	—
349	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	complemented
350	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
351	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
352	C₂×C₂²⋊C₄	—	2	8	no	—
353	C₂×C₂²⋊C₄	—	2	8	no	—
354	C₂×C₂²⋊C₄	—	4	4	no	—
355	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
356	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	8	yes	complemented
357	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
360	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
361	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
362	C₂×C₄²	—	2	8	no	—
363	C₂×C₄²	—	4	4	no	—
364	C₂×C₄²	C₂×C₄×C₅₃	1	8	yes	complemented
365	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
366	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
367	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
368	#95	C₂³×C₅₃	1	4	yes	—
369	#95	C₂³×C₅₃	1	4	yes	—
370	#95	C₂³×C₅₃	1	8	yes	—
371	#95	C₂³×C₅₃	1	4	yes	—
372	#95	C₂³×C₅₃	1	8	yes	—
373	#95	C₂³×C₅₃	1	4	yes	—
374	#95	C₂³×C₅₃	1	8	yes	—
375	#95	C₂³×C₅₃	1	8	yes	—
376	#95	C₂³×C₅₃	1	8	yes	—
377	C₂⁴	—	2	4	no	—
378	C₂⁴	—	2	4	no	—
379	C₂⁴	—	2	4	no	—
380	C₂⁴	—	2	2	no	—
381	C₂⁴	—	2	2	no	—
382	C₂⁴	—	2	4	no	—
383	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
384	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
385	C₂⁴	#41×C₅₃	1	4	yes	—
386	C₂⁴	#41×C₅₃	1	4	yes	—
387	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
388	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
389	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
390	C₂×D₄	—	2	32	no	—
391	C₂×D₄	—	2	8	no	—
392	C₂×D₄	—	2	8	no	—
393	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	4	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	4	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	4	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	16	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	16	no	—
413	C₂²×C₄	—	4	16	no	—
414	C₂²×C₄	—	4	16	no	—
415	C₂²×C₄	—	4	8	no	—
416	C₂²×C₄	—	4	8	no	—
417	C₂²×C₄	—	4	8	no	—
418	C₂²×C₄	—	4	32	no	—
419	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
420	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
421	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
422	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
423	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
424	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	—
425	C₂²×C₄	C₂²×C₄×C₅₃	1	16	yes	complemented
426	C₂²×C₄	#41×C₅₃	1	8	yes	—
427	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
428	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
429	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
430	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
432	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
433	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
434	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
435	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
436	C₄⋊C₄	—	4	8	no	—
437	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	complemented
438	C₂²⋊C₄	—	2	16	no	—
439	C₂²⋊C₄	—	2	16	no	—
440	C₂²⋊C₄	—	4	8	no	—
441	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	—
442	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	complemented
443	C₄²	—	2	16	no	—
444	C₄²	—	4	8	no	—
445	C₄²	C₂²×C₄×C₅₃	1	16	yes	complemented
446	C₂³	—	2	8	no	—
447	C₂³	—	2	8	no	—
448	C₂³	—	2	16	no	—
449	C₂³	—	2	16	no	—
450	C₂³	—	2	4	no	—
451	C₂³	—	2	4	no	—
452	C₂³	—	2	8	no	—
453	C₂³	—	2	8	no	—
454	C₂³	—	2	8	no	—
455	C₂³	—	2	4	no	—
456	C₂³	—	2	2	no	—
457	C₂³	—	2	2	no	—
458	C₂³	—	2	4	no	—
459	C₂³	—	4	8	no	—
460	C₂³	—	4	8	no	—
461	C₂³	#117×C₅₃	1	16	yes	—
462	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
463	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
464	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
465	C₂³	C₂×#41×C₅₃	1	4	yes	—
466	C₂³	C₂×#41×C₅₃	1	4	yes	—
467	C₂³	C₂×#41×C₅₃	1	2	yes	—
468	C₂³	C₂×#41×C₅₃	1	2	yes	—
469	C₂³	C₂×#41×C₅₃	1	4	yes	—
470	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
471	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
472	D₄	—	2	32	no	—
473	C₂×C₄	—	2	8	no	—
474	C₂×C₄	—	2	8	no	—
475	C₂×C₄	—	2	8	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	2	8	no	—
478	C₂×C₄	—	2	16	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	4	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	4	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	4	16	no	—
497	C₂×C₄	—	4	16	no	—
498	C₂×C₄	—	4	32	no	—
499	C₂×C₄	—	4	8	no	—
500	C₂×C₄	—	4	8	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
503	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
504	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
505	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
506	C₂×C₄	C₂³×C₄×C₅₃	1	4	yes	—
507	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
508	C₂²	—	2	8	no	—
509	C₂²	—	2	8	no	—
510	C₂²	—	2	32	no	—
511	C₂²	—	2	4	no	—
512	C₂²	—	2	4	no	—
513	C₂²	—	2	4	no	—
514	C₂²	—	2	4	no	—
515	C₂²	—	2	4	no	—
516	C₂²	—	4	4	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	16	no	—
519	C₂²	C₂×#117×C₅₃	1	8	yes	—
520	C₂²	C₂×#117×C₅₃	1	4	yes	—
521	C₂²	C₄×#41×C₅₃	1	4	yes	—
522	C₂²	C₅₃×#518	1	4	yes	—
523	C₂²	C₅₃×#551	1	8	yes	—
524	C₂²	C₅₃×#553	1	8	yes	—
525	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
526	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
527	C₂²	C₂²×#41×C₅₃	1	2	yes	—
528	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
529	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
530	C₄	—	2	8	no	—
531	C₄	—	2	4	no	—
532	C₄	—	2	4	no	—
533	C₄	—	2	4	no	—
534	C₄	—	2	4	no	—
535	C₄	—	2	4	no	—
536	C₄	—	4	4	no	—
537	C₄	—	4	4	no	—
538	C₄	—	4	4	no	—
539	C₂	—	2	8	no	—
540	C₂	—	4	4	no	—
541	C₂	C₅₃×#2353	1	8	yes	direct product factor
542	C₂	C₂²×#117×C₅₃	1	1	yes	characteristic
543	C₂	C₂×C₄×#41×C₅₃	1	1	yes	characteristic
544	C₂	C₂×C₅₃×#518	1	1	yes	characteristic
545	C₂	C₂×C₅₃×#551	1	2	yes	—
546	C₂	C₂×C₅₃×#553	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2353	C₂	direct (N × H)
3. C₂×C₅₃×#388	C₂	semidirect
4. C₂×C₅₃×#387	C₂	semidirect
6. C₂×C₅₃×#383	C₂	semidirect
7. C₂×C₅₃×#381	C₂	semidirect
8. C₂×C₅₃×#380	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
10. C₂×C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
11. C₂×C₅₃×#376	C₂	semidirect
12. C₂²×#95×C₅₃	C₂	semidirect
17. C₂×C₄×D₄×C₅₃	C₄	semidirect
29. C₅₃×#388	C₂²	semidirect
30. C₅₃×#387	C₂²	semidirect
32. C₅₃×#383	C₂²	semidirect
33. C₅₃×#381	C₂²	semidirect
34. C₅₃×#380	C₂²	semidirect
35. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
36. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
37. C₅₃×#376	C₂²	semidirect
47. C₂×#95×C₅₃	C₂²	semidirect
60. C₂³×C₄×C₅₃	C₂×C₄	semidirect
73. C₄×D₄×C₅₃	C₂×C₄	semidirect
75. C₂×C₄⋊C₄×C₅₃	C₂×C₄	semidirect
82. C₂×C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
90. C₂×C₄²×C₅₃	C₂×C₄	semidirect
151. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
163. C₄⋊C₄×C₅₃	C₂²×C₄	semidirect
168. C₂²⋊C₄×C₅₃	C₂²×C₄	semidirect
171. C₄²×C₅₃	C₂²×C₄	semidirect
234. C₂×#2353	C₅₃	direct (N × H)
267. #2353	C₂×C₅₃	direct (N × H)
268. C₂×#388	C₂×C₅₃	semidirect
269. C₂×#387	C₂×C₅₃	semidirect
271. C₂×#383	C₂×C₅₃	semidirect
272. C₂×#381	C₂×C₅₃	semidirect
273. C₂×#380	C₂×C₅₃	semidirect
274. C₂×C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
275. C₂×C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
276. C₂×#376	C₂×C₅₃	semidirect
277. C₂²×#95	C₂×C₅₃	semidirect
280. C₂×C₅₃	#2353	direct (N × H)
290. C₂×C₄×D₄	C₄×C₅₃	semidirect
302. #388	C₂²×C₅₃	semidirect
303. #387	C₂²×C₅₃	semidirect
305. #383	C₂²×C₅₃	semidirect
306. #381	C₂²×C₅₃	semidirect
307. #380	C₂²×C₅₃	semidirect
308. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
309. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
310. #376	C₂²×C₅₃	semidirect
320. C₂×#95	C₂²×C₅₃	semidirect
321. C₅₃	C₂×#2353	direct (N × H)
334. C₂³×C₄	C₂×C₄×C₅₃	semidirect
347. C₄×D₄	C₂×C₄×C₅₃	semidirect
349. C₂×C₄⋊C₄	C₂×C₄×C₅₃	semidirect
356. C₂×C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
364. C₂×C₄²	C₂×C₄×C₅₃	semidirect
425. C₂²×C₄	C₂²×C₄×C₅₃	semidirect
437. C₄⋊C₄	C₂²×C₄×C₅₃	semidirect
442. C₂²⋊C₄	C₂²×C₄×C₅₃	semidirect
445. C₄²	C₂²×C₄×C₅₃	semidirect
541. C₂	C₅₃×#2353	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×C₄×D₄×C₅₃	2
C₅₃×#2353	2
C₂×C₅₃×#388	2
C₂×C₅₃×#387	2
C₂×C₅₃×#386	2
C₂×C₅₃×#383	2
C₂×C₅₃×#381	2
C₂×C₅₃×#380	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
C₂×C₅₃×#376	2
C₂²×#95×C₅₃	2
C₂×#2353	53

Subgroups — C2×C53×#2353 #27241628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2353 #27241628