Subgroups — C₄×C₅₃×#534 #27241599

Order 13568.

3770 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#534	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2325	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2322	C₂	1	4	yes	maximal, complemented
5	C₅₃×#2321	C₂	1	4	yes	maximal, complemented
6	C₅₃×#2315	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#2308	C₂	1	2	yes	maximal, complemented
8	C₄×#123×C₅₃	C₂	1	4	yes	maximal, complemented
9	C₄×#121×C₅₃	C₂	1	4	yes	maximal, complemented
10	C₄×#120×C₅₃	C₂	1	4	yes	maximal, complemented
11	C₂×C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	complemented
15	C₅₃×#534	C₄	1	16	yes	direct product factor
16	C₂×#123×C₅₃	C₂²	1	4	yes	—
17	C₂×#121×C₅₃	C₂²	1	4	yes	—
18	C₂×#120×C₅₃	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₅₃	—	2	16	no	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
23	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
24	C₂×C₄×D₄×C₅₃	C₂²	1	4	yes	complemented
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
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	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
30	C₂²×C₄²×C₅₃	C₂²	1	2	yes	complemented
31	C₂²×C₄²×C₅₃	C₂²	1	2	yes	complemented
32	C₅₃×#390	C₂²	1	4	yes	complemented
33	C₅₃×#389	C₂²	1	4	yes	complemented
34	C₅₃×#388	C₂²	1	8	yes	complemented
35	C₅₃×#388	C₂²	1	8	yes	complemented
36	C₅₃×#387	C₂²	1	8	yes	complemented
37	C₅₃×#386	C₂²	1	4	yes	—
38	C₅₃×#386	C₂²	1	4	yes	—
39	C₅₃×#386	C₂²	1	8	yes	complemented
40	C₅₃×#386	C₂²	1	8	yes	complemented
41	C₅₃×#385	C₂²	1	4	yes	complemented
42	C₅₃×#385	C₂²	1	8	yes	complemented
43	C₅₃×#385	C₂²	1	4	yes	complemented
44	C₅₃×#381	C₂²	1	4	yes	complemented
45	C₅₃×#380	C₂²	1	4	yes	complemented
46	C₅₃×#379	C₂²	1	4	yes	complemented
47	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	complemented
48	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	complemented
50	C₄×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	complemented
51	C₂×#95×C₅₃	C₂²	1	2	yes	complemented
52	C₂⁵×C₅₃	C₂³	1	2	yes	—
53	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₅₃	—	2	8	no	—
55	C₂²×D₄×C₅₃	—	2	8	no	—
56	C₂²×D₄×C₅₃	C₂×C₄	1	16	yes	complemented
57	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
58	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
59	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
60	C₂³×C₄×C₅₃	—	2	8	no	—
61	C₂³×C₄×C₅₃	—	4	4	no	—
62	C₂³×C₄×C₅₃	—	4	8	no	—
63	C₂³×C₄×C₅₃	D₄	1	8	yes	complemented
64	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
66	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
68	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
69	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
70	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
71	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
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	#123×C₅₃	C₂×C₄	1	32	yes	complemented
76	#121×C₅₃	C₂×C₄	1	32	yes	complemented
77	#120×C₅₃	C₂×C₄	1	32	yes	complemented
78	C₄×D₄×C₅₃	—	2	16	no	—
79	C₄×D₄×C₅₃	—	2	16	no	—
80	C₄×D₄×C₅₃	—	2	16	no	—
81	C₄×D₄×C₅₃	—	2	16	no	—
82	C₄×D₄×C₅₃	—	4	8	no	—
83	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
84	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
86	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
88	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
90	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
91	C₂×C₂²⋊C₄×C₅₃	—	2	8	no	—
92	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	complemented
93	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
99	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
102	C₂×C₄²×C₅₃	—	2	8	no	—
103	C₂×C₄²×C₅₃	—	4	4	no	—
104	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
105	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
106	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
107	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
108	#95×C₅₃	C₂³	1	8	yes	—
109	#95×C₅₃	C₂³	1	8	yes	—
110	#95×C₅₃	C₂³	1	8	yes	—
111	#95×C₅₃	C₂³	1	8	yes	—
112	C₂⁴×C₅₃	—	2	4	no	—
113	C₂⁴×C₅₃	—	2	4	no	—
114	C₂⁴×C₅₃	—	4	2	no	—
115	C₂⁴×C₅₃	—	4	4	no	—
116	C₂⁴×C₅₃	—	4	2	no	—
117	C₂⁴×C₅₃	—	4	4	no	—
118	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
119	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	complemented
120	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
121	C₂⁴×C₅₃	#41	1	4	yes	—
122	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
123	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
124	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
125	C₂×D₄×C₅₃	—	2	64	no	—
126	C₂×D₄×C₅₃	—	2	8	no	—
127	C₂×D₄×C₅₃	—	2	8	no	—
128	C₂×D₄×C₅₃	—	2	8	no	—
129	C₂×D₄×C₅₃	—	2	8	no	—
130	C₂×D₄×C₅₃	—	4	4	no	—
131	C₂×D₄×C₅₃	—	4	4	no	—
132	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	—
133	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	complemented
134	C₂×D₄×C₅₃	C₂²×C₄	1	16	yes	complemented
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	8	no	—
139	C₂²×C₄×C₅₃	—	2	4	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	16	no	—
145	C₂²×C₄×C₅₃	—	4	16	no	—
146	C₂²×C₄×C₅₃	—	4	2	no	—
147	C₂²×C₄×C₅₃	—	4	2	no	—
148	C₂²×C₄×C₅₃	—	4	2	no	—
149	C₂²×C₄×C₅₃	—	4	4	no	—
150	C₂²×C₄×C₅₃	—	4	8	no	—
151	C₂²×C₄×C₅₃	—	4	4	no	—
152	C₂²×C₄×C₅₃	—	4	2	no	—
153	C₂²×C₄×C₅₃	—	4	2	no	—
154	C₂²×C₄×C₅₃	—	4	4	no	—
155	C₂²×C₄×C₅₃	—	4	32	no	—
156	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	complemented
157	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	complemented
158	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
159	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
161	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
162	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
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	8	no	—
172	C₄⋊C₄×C₅₃	—	4	4	no	—
173	C₄⋊C₄×C₅₃	—	4	4	no	—
174	C₄⋊C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
175	C₂²⋊C₄×C₅₃	—	2	8	no	—
176	C₂²⋊C₄×C₅₃	—	2	8	no	—
177	C₂²⋊C₄×C₅₃	—	2	8	no	—
178	C₂²⋊C₄×C₅₃	—	2	8	no	—
179	C₂²⋊C₄×C₅₃	—	2	8	no	—
180	C₂²⋊C₄×C₅₃	—	4	8	no	—
181	C₂²⋊C₄×C₅₃	—	4	8	no	—
182	C₂²⋊C₄×C₅₃	C₂²×C₄	1	16	yes	complemented
183	C₂²⋊C₄×C₅₃	C₂²×C₄	1	32	yes	complemented
184	C₄²×C₅₃	—	2	8	no	—
185	C₄²×C₅₃	—	2	8	no	—
186	C₄²×C₅₃	—	2	8	no	—
187	C₄²×C₅₃	—	4	4	no	—
188	C₂³×C₅₃	—	2	2	no	—
189	C₂³×C₅₃	—	2	4	no	—
190	C₂³×C₅₃	—	2	2	no	—
191	C₂³×C₅₃	—	2	4	no	—
192	C₂³×C₅₃	—	2	16	no	—
193	C₂³×C₅₃	—	2	4	no	—
194	C₂³×C₅₃	—	2	4	no	—
195	C₂³×C₅₃	—	2	8	no	—
196	C₂³×C₅₃	—	2	8	no	—
197	C₂³×C₅₃	—	4	4	no	—
198	C₂³×C₅₃	—	4	8	no	—
199	C₂³×C₅₃	—	4	16	no	—
200	C₂³×C₅₃	—	4	4	no	—
201	C₂³×C₅₃	—	4	8	no	—
202	C₂³×C₅₃	—	4	8	no	—
203	C₂³×C₅₃	—	4	8	no	—
204	C₂³×C₅₃	—	4	16	no	—
205	C₂³×C₅₃	C₄×D₄	1	16	yes	complemented
206	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
207	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
208	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
209	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
210	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
211	C₂³×C₅₃	C₂×#41	1	2	yes	—
212	C₂³×C₅₃	C₂×#41	1	4	yes	—
213	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
214	D₄×C₅₃	—	2	32	no	—
215	D₄×C₅₃	—	2	32	no	—
216	D₄×C₅₃	—	4	16	no	—
217	C₂×C₄×C₅₃	—	2	16	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	2	no	—
222	C₂×C₄×C₅₃	—	2	4	no	—
223	C₂×C₄×C₅₃	—	2	4	no	—
224	C₂×C₄×C₅₃	—	2	16	no	—
225	C₂×C₄×C₅₃	—	2	8	no	—
226	C₂×C₄×C₅₃	—	2	4	no	—
227	C₂×C₄×C₅₃	—	2	4	no	—
228	C₂×C₄×C₅₃	—	2	4	no	—
229	C₂×C₄×C₅₃	—	2	8	no	—
230	C₂×C₄×C₅₃	—	2	4	no	—
231	C₂×C₄×C₅₃	—	2	4	no	—
232	C₂×C₄×C₅₃	—	2	8	no	—
233	C₂×C₄×C₅₃	—	4	4	no	—
234	C₂×C₄×C₅₃	—	4	8	no	—
235	C₂×C₄×C₅₃	—	4	8	no	—
236	C₂×C₄×C₅₃	—	4	16	no	—
237	C₂×C₄×C₅₃	—	4	8	no	—
238	C₂×C₄×C₅₃	—	4	8	no	—
239	C₂×C₄×C₅₃	—	4	8	no	—
240	C₂×C₄×C₅₃	—	4	16	no	—
241	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
242	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
243	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
244	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
246	C₂×C₄×C₅₃	H(5,𝔽₂)	1	4	yes	—
247	C₄×#534	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₂²×C₅₃	—	2	4	no	—
249	C₂²×C₅₃	—	2	8	no	—
250	C₂²×C₅₃	—	2	4	no	—
251	C₂²×C₅₃	—	2	2	no	—
252	C₂²×C₅₃	—	2	4	no	—
253	C₂²×C₅₃	—	2	2	no	—
254	C₂²×C₅₃	—	2	4	no	—
255	C₂²×C₅₃	—	2	8	no	—
256	C₂²×C₅₃	—	2	8	no	—
257	C₂²×C₅₃	—	2	16	no	—
258	C₂²×C₅₃	—	4	16	no	—
259	C₂²×C₅₃	—	4	32	no	—
260	C₂²×C₅₃	—	4	4	no	—
261	C₂²×C₅₃	—	4	4	no	—
262	C₂²×C₅₃	—	4	8	no	—
263	C₂²×C₅₃	C₂×C₄×D₄	1	8	yes	—
264	C₂²×C₅₃	C₂×C₄×D₄	1	4	yes	—
265	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
266	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
267	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
268	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	2	yes	—
269	C₂²×C₅₃	#585	1	2	yes	—
270	C₄×C₅₃	—	2	8	no	—
271	C₄×C₅₃	—	2	8	no	—
272	C₄×C₅₃	—	2	4	no	—
273	C₄×C₅₃	—	2	2	no	—
274	C₄×C₅₃	—	4	4	no	—
275	C₄×C₅₃	—	4	4	no	—
276	C₄×C₅₃	—	4	4	no	—
277	C₄×C₅₃	#534	1	4	yes	direct product factor
278	C₂×#534	C₂×C₅₃	1	1	yes	characteristic
279	C₂²×C₄×D₄	C₂×C₅₃	1	2	yes	complemented
280	#2325	C₂×C₅₃	1	4	yes	complemented
281	#2322	C₂×C₅₃	1	4	yes	complemented
282	#2321	C₂×C₅₃	1	4	yes	complemented
283	#2315	C₂×C₅₃	1	1	yes	characteristic, complemented
284	#2308	C₂×C₅₃	1	2	yes	complemented
285	C₄×#123	C₂×C₅₃	1	4	yes	complemented
286	C₄×#121	C₂×C₅₃	1	4	yes	complemented
287	C₄×#120	C₂×C₅₃	1	4	yes	complemented
288	C₂×C₄×C₂²⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
289	C₂×C₅₃	—	2	8	no	—
290	C₂×C₅₃	—	2	4	no	—
291	C₂×C₅₃	—	4	8	no	—
292	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
293	C₂×C₅₃	C₄×H(5,𝔽₂)	1	2	yes	—
294	C₂×C₅₃	C₂×#534	1	1	yes	characteristic
295	C₂×C₅₃	#3449	1	1	yes	characteristic
296	C₂×C₅₃	#3459	1	2	yes	—
297	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
298	C₂⁴×C₄	C₂²×C₅₃	1	1	yes	characteristic
299	C₂⁴×C₄	C₂²×C₅₃	1	2	yes	complemented
300	#534	C₄×C₅₃	1	16	yes	direct product factor
301	C₂×#123	C₂²×C₅₃	1	4	yes	—
302	C₂×#121	C₂²×C₅₃	1	4	yes	—
303	C₂×#120	C₂²×C₅₃	1	4	yes	—
304	C₂×C₄×D₄	—	2	16	no	—
305	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
306	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	—
307	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
308	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
309	C₂×C₄×D₄	C₂²×C₅₃	1	4	yes	complemented
310	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	complemented
311	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
312	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
313	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
314	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	complemented
315	C₂²×C₄²	C₂²×C₅₃	1	2	yes	complemented
316	C₂²×C₄²	C₂²×C₅₃	1	2	yes	complemented
317	#390	C₂²×C₅₃	1	4	yes	complemented
318	#389	C₂²×C₅₃	1	4	yes	complemented
319	#388	C₂²×C₅₃	1	8	yes	complemented
320	#388	C₂²×C₅₃	1	8	yes	complemented
321	#387	C₂²×C₅₃	1	8	yes	complemented
322	#386	C₂²×C₅₃	1	4	yes	—
323	#386	C₂²×C₅₃	1	4	yes	—
324	#386	C₂²×C₅₃	1	8	yes	complemented
325	#386	C₂²×C₅₃	1	8	yes	complemented
326	#385	C₂²×C₅₃	1	4	yes	complemented
327	#385	C₂²×C₅₃	1	4	yes	complemented
328	#385	C₂²×C₅₃	1	8	yes	complemented
329	#381	C₂²×C₅₃	1	4	yes	complemented
330	#380	C₂²×C₅₃	1	4	yes	complemented
331	#379	C₂²×C₅₃	1	4	yes	complemented
332	C₄×C₄⋊C₄	C₂²×C₅₃	1	4	yes	complemented
333	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
334	C₄×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	complemented
335	C₄×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	complemented
336	C₂×#95	C₂²×C₅₃	1	2	yes	complemented
337	C₅₃	C₄×#534	1	1	yes	characteristic, direct product factor, Sylow
338	C₂⁵	C₂³×C₅₃	1	2	yes	—
339	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
340	C₂²×D₄	—	2	8	no	—
341	C₂²×D₄	—	2	8	no	—
342	C₂²×D₄	C₂×C₄×C₅₃	1	16	yes	complemented
343	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
344	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
345	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
346	C₂³×C₄	—	2	8	no	—
347	C₂³×C₄	—	4	4	no	—
348	C₂³×C₄	—	4	8	no	—
349	C₂³×C₄	D₄×C₅₃	1	8	yes	complemented
350	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
351	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
352	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
353	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
355	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
356	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
357	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
358	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
359	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
360	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
361	#123	C₂×C₄×C₅₃	1	32	yes	complemented
362	#121	C₂×C₄×C₅₃	1	32	yes	complemented
363	#120	C₂×C₄×C₅₃	1	32	yes	complemented
364	C₄×D₄	—	2	16	no	—
365	C₄×D₄	—	2	16	no	—
366	C₄×D₄	—	2	16	no	—
367	C₄×D₄	—	2	16	no	—
368	C₄×D₄	—	4	8	no	—
369	C₂×C₄⋊C₄	—	2	8	no	—
370	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
372	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
373	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
375	C₂×C₂²⋊C₄	—	2	8	no	—
376	C₂×C₂²⋊C₄	—	2	4	no	—
377	C₂×C₂²⋊C₄	—	2	8	no	—
378	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	8	yes	complemented
379	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₅₃	1	4	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	4	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
388	C₂×C₄²	—	2	8	no	—
389	C₂×C₄²	—	4	4	no	—
390	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
391	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
392	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
393	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
394	#95	C₂³×C₅₃	1	8	yes	—
395	#95	C₂³×C₅₃	1	8	yes	—
396	#95	C₂³×C₅₃	1	8	yes	—
397	#95	C₂³×C₅₃	1	8	yes	—
398	C₂⁴	—	2	4	no	—
399	C₂⁴	—	2	4	no	—
400	C₂⁴	—	4	2	no	—
401	C₂⁴	—	4	4	no	—
402	C₂⁴	—	4	2	no	—
403	C₂⁴	—	4	4	no	—
404	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
405	C₂⁴	C₂²×C₄×C₅₃	1	8	yes	complemented
406	C₂⁴	C₂×D₄×C₅₃	1	4	yes	—
407	C₂⁴	#41×C₅₃	1	4	yes	—
408	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
409	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
410	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
411	C₂×D₄	—	2	64	no	—
412	C₂×D₄	—	2	8	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	4	4	no	—
417	C₂×D₄	—	4	4	no	—
418	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	—
419	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	complemented
420	C₂×D₄	C₂²×C₄×C₅₃	1	16	yes	complemented
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	8	no	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	8	no	—
429	C₂²×C₄	—	2	8	no	—
430	C₂²×C₄	—	2	16	no	—
431	C₂²×C₄	—	4	16	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	2	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	—	4	2	no	—
436	C₂²×C₄	—	4	8	no	—
437	C₂²×C₄	—	4	4	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	4	2	no	—
440	C₂²×C₄	—	4	4	no	—
441	C₂²×C₄	—	4	32	no	—
442	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	complemented
443	C₂²×C₄	C₂²×C₄×C₅₃	1	8	yes	complemented
444	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
445	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
446	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
447	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
448	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
450	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
451	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
452	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
453	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
455	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	4	4	no	—
459	C₄⋊C₄	—	4	4	no	—
460	C₄⋊C₄	C₂²×C₄×C₅₃	1	16	yes	complemented
461	C₂²⋊C₄	—	2	8	no	—
462	C₂²⋊C₄	—	2	8	no	—
463	C₂²⋊C₄	—	2	8	no	—
464	C₂²⋊C₄	—	2	8	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	4	8	no	—
467	C₂²⋊C₄	—	4	8	no	—
468	C₂²⋊C₄	C₂²×C₄×C₅₃	1	16	yes	complemented
469	C₂²⋊C₄	C₂²×C₄×C₅₃	1	32	yes	complemented
470	C₄²	—	2	8	no	—
471	C₄²	—	2	8	no	—
472	C₄²	—	2	8	no	—
473	C₄²	—	4	4	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	4	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	4	no	—
478	C₂³	—	2	16	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	2	8	no	—
483	C₂³	—	4	4	no	—
484	C₂³	—	4	8	no	—
485	C₂³	—	4	16	no	—
486	C₂³	—	4	4	no	—
487	C₂³	—	4	8	no	—
488	C₂³	—	4	8	no	—
489	C₂³	—	4	8	no	—
490	C₂³	—	4	16	no	—
491	C₂³	C₄×D₄×C₅₃	1	16	yes	complemented
492	C₂³	C₂³×C₄×C₅₃	1	8	yes	—
493	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
494	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
495	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
496	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
497	C₂³	C₂×#41×C₅₃	1	4	yes	—
498	C₂³	C₂×#41×C₅₃	1	2	yes	—
499	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
500	D₄	—	2	32	no	—
501	D₄	—	2	32	no	—
502	D₄	—	4	16	no	—
503	C₂×C₄	—	2	16	no	—
504	C₂×C₄	—	2	4	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	16	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	8	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	8	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	4	8	no	—
522	C₂×C₄	—	4	16	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	—	4	8	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	4	16	no	—
527	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
528	C₂×C₄	C₂³×C₄×C₅₃	1	8	yes	—
529	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
530	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
531	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
532	C₂×C₄	H(5,𝔽₂)×C₅₃	1	4	yes	—
533	C₂²	—	2	4	no	—
534	C₂²	—	2	2	no	—
535	C₂²	—	2	4	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	2	8	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	8	no	—
542	C₂²	—	2	16	no	—
543	C₂²	—	4	16	no	—
544	C₂²	—	4	32	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	8	no	—
547	C₂²	—	4	4	no	—
548	C₂²	C₂×C₄×D₄×C₅₃	1	8	yes	—
549	C₂²	C₂×C₄×D₄×C₅₃	1	4	yes	—
550	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
551	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
552	C₂²	C₂²×#41×C₅₃	1	1	yes	characteristic
553	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	2	yes	—
554	C₂²	C₅₃×#585	1	2	yes	—
555	C₄	—	2	8	no	—
556	C₄	—	2	8	no	—
557	C₄	—	2	4	no	—
558	C₄	—	2	2	no	—
559	C₄	—	4	4	no	—
560	C₄	—	4	4	no	—
561	C₄	—	4	4	no	—
562	C₄	C₅₃×#534	1	4	yes	direct product factor
563	C₂	—	2	8	no	—
564	C₂	—	2	4	no	—
565	C₂	—	4	8	no	—
566	C₂	C₂²×C₄×D₄×C₅₃	1	1	yes	characteristic
567	C₂	C₄×H(5,𝔽₂)×C₅₃	1	2	yes	—
568	C₂	C₂×C₅₃×#534	1	1	yes	characteristic
569	C₂	C₅₃×#3449	1	1	yes	characteristic
570	C₂	C₅₃×#3459	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×#534	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. C₂²×C₄×D₄×C₅₃	C₂	semidirect
3. C₅₃×#2325	C₂	semidirect
4. C₅₃×#2322	C₂	semidirect
5. C₅₃×#2321	C₂	semidirect
6. C₅₃×#2315	C₂	semidirect
7. C₅₃×#2308	C₂	semidirect
8. C₄×#123×C₅₃	C₂	semidirect
9. C₄×#121×C₅₃	C₂	semidirect
10. C₄×#120×C₅₃	C₂	semidirect
11. C₂×C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
14. C₂⁴×C₄×C₅₃	C₂²	semidirect
15. C₅₃×#534	C₄	direct (N × H)
22. C₂×C₄×D₄×C₅₃	C₂²	semidirect
23. C₂×C₄×D₄×C₅₃	C₂²	semidirect
24. C₂×C₄×D₄×C₅₃	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
29. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
30. C₂²×C₄²×C₅₃	C₂²	semidirect
31. C₂²×C₄²×C₅₃	C₂²	semidirect
32. C₅₃×#390	C₂²	semidirect
33. C₅₃×#389	C₂²	semidirect
34. C₅₃×#388	C₂²	semidirect
35. C₅₃×#388	C₂²	semidirect
36. C₅₃×#387	C₂²	semidirect
39. C₅₃×#386	C₂²	semidirect
40. C₅₃×#386	C₂²	semidirect
41. C₅₃×#385	C₂²	semidirect
42. C₅₃×#385	C₂²	semidirect
43. C₅₃×#385	C₂²	semidirect
44. C₅₃×#381	C₂²	semidirect
45. C₅₃×#380	C₂²	semidirect
46. C₅₃×#379	C₂²	semidirect
47. C₄×C₄⋊C₄×C₅₃	C₂²	semidirect
48. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
49. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
50. C₄×C₂²⋊C₄×C₅₃	C₂²	semidirect
51. C₂×#95×C₅₃	C₂²	semidirect
56. C₂²×D₄×C₅₃	C₂×C₄	semidirect
63. C₂³×C₄×C₅₃	D₄	semidirect
75. #123×C₅₃	C₂×C₄	semidirect
76. #121×C₅₃	C₂×C₄	semidirect
77. #120×C₅₃	C₂×C₄	semidirect
92. C₂×C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
119. C₂⁴×C₅₃	C₂²×C₄	semidirect
133. C₂×D₄×C₅₃	C₂²×C₄	semidirect
134. C₂×D₄×C₅₃	C₂²×C₄	semidirect
156. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
157. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
174. C₄⋊C₄×C₅₃	C₂²×C₄	semidirect
182. C₂²⋊C₄×C₅₃	C₂²×C₄	semidirect
183. C₂²⋊C₄×C₅₃	C₂²×C₄	semidirect
205. C₂³×C₅₃	C₄×D₄	semidirect
247. C₄×#534	C₅₃	direct (N × H)
277. C₄×C₅₃	#534	direct (N × H)
279. C₂²×C₄×D₄	C₂×C₅₃	semidirect
280. #2325	C₂×C₅₃	semidirect
281. #2322	C₂×C₅₃	semidirect
282. #2321	C₂×C₅₃	semidirect
283. #2315	C₂×C₅₃	semidirect
284. #2308	C₂×C₅₃	semidirect
285. C₄×#123	C₂×C₅₃	semidirect
286. C₄×#121	C₂×C₅₃	semidirect
287. C₄×#120	C₂×C₅₃	semidirect
288. C₂×C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
299. C₂⁴×C₄	C₂²×C₅₃	semidirect
300. #534	C₄×C₅₃	direct (N × H)
307. C₂×C₄×D₄	C₂²×C₅₃	semidirect
308. C₂×C₄×D₄	C₂²×C₅₃	semidirect
309. C₂×C₄×D₄	C₂²×C₅₃	semidirect
310. C₂²×C₄⋊C₄	C₂²×C₅₃	semidirect
314. C₂²×C₂²⋊C₄	C₂²×C₅₃	semidirect
315. C₂²×C₄²	C₂²×C₅₃	semidirect
316. C₂²×C₄²	C₂²×C₅₃	semidirect
317. #390	C₂²×C₅₃	semidirect
318. #389	C₂²×C₅₃	semidirect
319. #388	C₂²×C₅₃	semidirect
320. #388	C₂²×C₅₃	semidirect
321. #387	C₂²×C₅₃	semidirect
324. #386	C₂²×C₅₃	semidirect
325. #386	C₂²×C₅₃	semidirect
326. #385	C₂²×C₅₃	semidirect
327. #385	C₂²×C₅₃	semidirect
328. #385	C₂²×C₅₃	semidirect
329. #381	C₂²×C₅₃	semidirect
330. #380	C₂²×C₅₃	semidirect
331. #379	C₂²×C₅₃	semidirect
332. C₄×C₄⋊C₄	C₂²×C₅₃	semidirect
333. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
334. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
335. C₄×C₂²⋊C₄	C₂²×C₅₃	semidirect
336. C₂×#95	C₂²×C₅₃	semidirect
337. C₅₃	C₄×#534	direct (N × H)
342. C₂²×D₄	C₂×C₄×C₅₃	semidirect
349. C₂³×C₄	D₄×C₅₃	semidirect
361. #123	C₂×C₄×C₅₃	semidirect
362. #121	C₂×C₄×C₅₃	semidirect
363. #120	C₂×C₄×C₅₃	semidirect
378. C₂×C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
405. C₂⁴	C₂²×C₄×C₅₃	semidirect
419. C₂×D₄	C₂²×C₄×C₅₃	semidirect
420. C₂×D₄	C₂²×C₄×C₅₃	semidirect
442. C₂²×C₄	C₂²×C₄×C₅₃	semidirect
443. C₂²×C₄	C₂²×C₄×C₅₃	semidirect
460. C₄⋊C₄	C₂²×C₄×C₅₃	semidirect
468. C₂²⋊C₄	C₂²×C₄×C₅₃	semidirect
469. C₂²⋊C₄	C₂²×C₄×C₅₃	semidirect
491. C₂³	C₄×D₄×C₅₃	semidirect
562. C₄	C₅₃×#534	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#534	2
C₂²×C₄×D₄×C₅₃	2
C₅₃×#2325	2
C₅₃×#2322	2
C₅₃×#2321	2
C₅₃×#2315	2
C₅₃×#2308	2
C₄×#123×C₅₃	2
C₄×#121×C₅₃	2
C₄×#120×C₅₃	2
C₂×C₄×C₂²⋊C₄×C₅₃	2
C₄×#534	53

Subgroups — C4×C53×#534 #27241599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#534 #27241599