Subgroups — C₂²×#123×D₅₃ #28290737

Order 13568.

15150 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133202	C₂	1	2	yes	maximal, complemented
4	C₂²×#13133201	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#13133200	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13133199	C₂	1	2	yes	maximal, complemented
7	C₂²×#13133198	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#123×D₅₃	C₂	1	48	yes	maximal, direct product factor
9	C₂²×C₄⋊C₄×D₅₃	C₂	1	2	yes	maximal, complemented
10	C₂²×C₂²⋊C₄×D₅₃	C₂	1	2	yes	maximal, complemented
11	C₂²×C₂²⋊C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#123×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
14	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
15	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
16	C₂³×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
17	C₂³×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
18	C₂²×D₄×D₅₃	—	2	16	no	—
19	C₂²×D₄×D₅₃	C₂²	1	24	yes	—
20	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic, complemented
21	C₂³×C₄×D₅₃	—	2	16	no	—
22	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
23	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
24	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
25	C₂³×C₄×D₅₃	C₂²	1	2	yes	complemented
26	C₂³×C₄×D₅₃	C₂²	1	24	yes	complemented
27	C₂×#13133202	C₂²	1	48	yes	complemented
28	C₂×#13133201	C₂²	1	24	yes	complemented
29	C₂×#13133200	C₂²	1	24	yes	complemented
30	C₂×#13133199	C₂²	1	48	yes	complemented
31	C₂×#13133198	C₂²	1	24	yes	complemented
32	#123×D₅₃	C₂²	1	256	yes	direct product factor
33	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	complemented
34	C₂²×#11719919	C₂²	1	2	yes	complemented
35	C₂²×#11719919	C₂²	1	2	yes	complemented
36	C₂×C₄⋊C₄×D₅₃	C₂²	1	48	yes	complemented
37	C₂²×C₂²⋊Dic₅₃	C₂²	1	2	yes	complemented
38	C₂²×C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
39	C₂²×#11719921	C₂²	1	2	yes	complemented
40	C₂²×#11719921	C₂²	1	2	yes	complemented
41	C₂²×#11719921	C₂²	1	2	yes	complemented
42	C₂×C₂²⋊C₄×D₅₃	C₂²	1	48	yes	complemented
43	C₂×C₂²⋊C₄×D₅₃	C₂²	1	24	yes	complemented
44	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
45	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
46	C₂×#123×C₅₃	C₂²	1	24	yes	complemented
47	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
48	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	complemented
49	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
50	C₂⁴×D₅₃	—	2	16	no	—
51	C₂⁴×D₅₃	—	2	8	no	—
52	C₂⁴×D₅₃	—	2	16	no	—
53	C₂⁴×D₅₃	D₄	1	16	yes	—
54	C₂⁴×D₅₃	C₂³	1	12	yes	—
55	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
56	C₂⁴×D₅₃	C₂³	1	12	yes	—
57	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
58	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
59	C₂²×D₂₁₂	—	2	8	no	—
60	C₂²×D₂₁₂	C₂³	1	12	yes	—
61	C₂²×C₂²⋊D₅₃	—	2	8	no	—
62	C₂²×C₂²⋊D₅₃	—	2	8	no	—
63	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
64	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
65	C₂×D₄×D₅₃	—	2	192	no	—
66	C₂×D₄×D₅₃	C₂³	1	64	yes	—
67	C₂³×Dic₅₃	—	2	8	no	—
68	C₂³×Dic₅₃	C₂³	1	2	yes	—
69	C₂³×Dic₅₃	C₂³	1	2	yes	—
70	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₂³×Dic₅₃	C₂³	1	12	yes	complemented
72	C₂²×C₄×D₅₃	—	2	16	no	—
73	C₂²×C₄×D₅₃	—	2	192	no	—
74	C₂²×C₄×D₅₃	—	2	16	no	—
75	C₂²×C₄×D₅₃	—	2	8	no	—
76	C₂²×C₄×D₅₃	—	2	16	no	—
77	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
78	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
79	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
80	C₂²×C₄×D₅₃	C₂³	1	64	yes	complemented
81	C₂²×C₄×D₅₃	C₂³	1	24	yes	complemented
82	#13133202	C₂³	1	128	yes	complemented
83	#13133201	C₂³	1	64	yes	complemented
84	#13133200	C₂³	1	64	yes	complemented
85	#13133199	C₂³	1	128	yes	complemented
86	#13133198	C₂³	1	64	yes	complemented
87	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	complemented
88	C₂×#11719919	C₂³	1	24	yes	complemented
89	C₂×#11719919	C₂³	1	24	yes	complemented
90	C₄⋊C₄×D₅₃	C₂³	1	128	yes	complemented
91	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	complemented
92	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	complemented
93	C₂×#11719921	C₂³	1	24	yes	complemented
94	C₂×#11719921	C₂³	1	24	yes	complemented
95	C₂×#11719921	C₂³	1	24	yes	complemented
96	C₂²⋊C₄×D₅₃	C₂³	1	64	yes	complemented
97	C₂²⋊C₄×D₅₃	C₂³	1	128	yes	complemented
98	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
99	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
100	C₂²×D₄×C₅₃	—	2	8	no	—
101	C₂²×D₄×C₅₃	C₂³	1	12	yes	—
102	C₂³×C₄×C₅₃	—	2	8	no	—
103	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
104	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
105	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
106	C₂³×C₄×C₅₃	C₂³	1	12	yes	complemented
107	#123×C₅₃	C₂³	1	64	yes	complemented
108	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	complemented
109	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	complemented
110	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	complemented
111	C₂⁴×C₅₃	—	2	8	no	—
112	C₂⁴×C₅₃	—	2	4	no	—
113	C₂⁴×C₅₃	—	2	8	no	—
114	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
115	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
116	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
117	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
118	C₂³×D₅₃	—	2	8	no	—
119	C₂³×D₅₃	—	2	128	no	—
120	C₂³×D₅₃	—	2	96	no	—
121	C₂³×D₅₃	—	2	4	no	—
122	C₂³×D₅₃	—	2	8	no	—
123	C₂³×D₅₃	—	2	48	no	—
124	C₂³×D₅₃	—	2	96	no	—
125	C₂³×D₅₃	—	4	64	no	—
126	C₂³×D₅₃	C₂×D₄	1	96	yes	—
127	C₂³×D₅₃	C₂×D₄	1	8	yes	—
128	C₂³×D₅₃	C₂×D₄	1	8	yes	—
129	C₂³×D₅₃	#41	1	16	yes	—
130	C₂³×D₅₃	C₂⁴	1	16	yes	—
131	C₂³×D₅₃	C₂⁴	1	6	yes	—
132	C₂³×D₅₃	C₂⁴	1	16	yes	—
133	C₂³×D₅₃	C₂⁴	1	6	yes	—
134	C₂³×D₅₃	C₂⁴	1	6	yes	—
135	C₂×D₄×C₅₃	—	2	48	no	—
136	C₂×D₄×C₅₃	C₂⁴	1	16	yes	—
137	C₂²×C₄×C₅₃	—	2	8	no	—
138	C₂²×C₄×C₅₃	—	2	48	no	—
139	C₂²×C₄×C₅₃	—	2	4	no	—
140	C₂²×C₄×C₅₃	—	2	8	no	—
141	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
142	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
143	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
144	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	complemented
145	C₂×C₂²⋊D₅₃	—	2	48	no	—
146	C₂×C₂²⋊D₅₃	—	2	48	no	—
147	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
148	C₂×C₂²⋊D₅₃	C₂⁴	1	16	yes	—
149	C₂²×Dic₅₃	—	2	8	no	—
150	C₂²×Dic₅₃	—	2	48	no	—
151	C₂²×Dic₅₃	—	2	4	no	—
152	C₂²×Dic₅₃	—	2	8	no	—
153	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
154	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
155	C₂²×Dic₅₃	C₂⁴	1	6	yes	—
156	C₂²×Dic₅₃	C₂⁴	1	16	yes	complemented
157	D₄×D₅₃	—	2	256	no	—
158	C₂×D₂₁₂	—	2	48	no	—
159	C₂×D₂₁₂	C₂⁴	1	16	yes	—
160	C₂×C₄×D₅₃	—	2	96	no	—
161	C₂×C₄×D₅₃	—	2	256	no	—
162	C₂×C₄×D₅₃	—	2	96	no	—
163	C₂×C₄×D₅₃	—	2	48	no	—
164	C₂×C₄×D₅₃	—	2	96	no	—
165	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
166	C₂×C₄×D₅₃	C₂⁴	1	32	yes	—
167	C₂×C₄×D₅₃	C₂⁴	1	16	yes	—
168	C₂×C₄×D₅₃	C₂⁴	1	32	yes	complemented
169	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
170	C₂²⋊C₄×C₅₃	C₂⁴	1	32	yes	complemented
171	C₂²⋊C₄×C₅₃	C₂⁴	1	16	yes	complemented
172	C₂²⋊Dic₅₃	C₂⁴	1	32	yes	complemented
173	C₂²⋊Dic₅₃	C₂⁴	1	16	yes	complemented
174	#11719921	C₂⁴	1	32	yes	complemented
175	#11719921	C₂⁴	1	32	yes	complemented
176	#11719921	C₂⁴	1	32	yes	complemented
177	C₄⋊Dic₅₃	C₂⁴	1	32	yes	complemented
178	#11719919	C₂⁴	1	32	yes	complemented
179	#11719919	C₂⁴	1	32	yes	complemented
180	C₂³×C₅₃	—	2	32	no	—
181	C₂³×C₅₃	—	2	24	no	—
182	C₂³×C₅₃	—	2	12	no	—
183	C₂³×C₅₃	—	2	24	no	—
184	C₂³×C₅₃	—	4	16	no	—
185	C₂³×C₅₃	C₂²×D₄	1	24	yes	—
186	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
187	C₂³×C₅₃	C₂×#41	1	8	yes	—
188	C₂³×C₅₃	C₂⁵	1	4	yes	—
189	C₂³×C₅₃	C₂⁵	1	4	yes	—
190	C₂³×C₅₃	C₂⁵	1	3	yes	—
191	C₂²×D₅₃	—	2	32	no	—
192	C₂²×D₅₃	—	2	24	no	—
193	C₂²×D₅₃	—	2	384	no	—
194	C₂²×D₅₃	—	2	64	no	—
195	C₂²×D₅₃	—	2	12	no	—
196	C₂²×D₅₃	—	2	24	no	—
197	C₂²×D₅₃	—	2	32	no	—
198	C₂²×D₅₃	—	2	64	no	—
199	C₂²×D₅₃	—	4	16	no	—
200	C₂²×D₅₃	—	4	192	no	—
201	C₂²×D₅₃	#123	1	64	yes	direct product factor
202	C₂²×D₅₃	C₂²×D₄	1	24	yes	—
203	C₂²×D₅₃	C₂²×D₄	1	64	yes	—
204	C₂²×D₅₃	C₂²×D₄	1	24	yes	—
205	C₂²×D₅₃	C₂×#41	1	48	yes	—
206	C₂²×D₅₃	C₂⁵	1	4	yes	—
207	C₂²×D₅₃	C₂⁵	1	4	yes	—
208	C₂²×D₅₃	C₂⁵	1	4	yes	—
209	D₄×C₅₃	—	2	32	no	—
210	C₂×C₄×C₅₃	—	2	24	no	—
211	C₂×C₄×C₅₃	—	2	32	no	—
212	C₂×C₄×C₅₃	—	2	12	no	—
213	C₂×C₄×C₅₃	—	2	24	no	—
214	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
215	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
216	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
217	C₂²⋊D₅₃	—	2	32	no	—
218	C₂²⋊D₅₃	—	2	32	no	—
219	C₂×Dic₅₃	—	2	24	no	—
220	C₂×Dic₅₃	—	2	32	no	—
221	C₂×Dic₅₃	—	2	12	no	—
222	C₂×Dic₅₃	—	2	24	no	—
223	C₂×Dic₅₃	C₂⁵	1	8	yes	—
224	C₂×Dic₅₃	C₂⁵	1	4	yes	—
225	C₂×Dic₅₃	C₂⁵	1	8	yes	—
226	D₂₁₂	—	2	32	no	—
227	C₄×D₅₃	—	2	64	no	—
228	C₄×D₅₃	—	2	64	no	—
229	C₄×D₅₃	—	2	32	no	—
230	C₄×D₅₃	—	2	64	no	—
231	C₂³×#123	—	53	1	no	maximal, Sylow
232	C₂²×C₅₃	—	2	8	no	—
233	C₂²×C₅₃	—	2	48	no	—
234	C₂²×C₅₃	—	2	4	no	—
235	C₂²×C₅₃	—	2	8	no	—
236	C₂²×C₅₃	—	4	24	no	—
237	C₂²×C₅₃	C₂×#123	1	16	yes	complemented
238	C₂²×C₅₃	C₂³×D₄	1	8	yes	—
239	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
240	C₂²×C₅₃	C₂²×#41	1	12	yes	—
241	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
242	C₂×D₅₃	—	2	8	no	—
243	C₂×D₅₃	—	2	48	no	—
244	C₂×D₅₃	—	2	128	no	—
245	C₂×D₅₃	—	2	4	no	—
246	C₂×D₅₃	—	2	8	no	—
247	C₂×D₅₃	—	4	24	no	—
248	C₂×D₅₃	—	4	64	no	—
249	C₂×D₅₃	C₂×#123	1	96	yes	direct product factor
250	C₂×D₅₃	C₂³×D₄	1	8	yes	—
251	C₂×D₅₃	C₂³×D₄	1	8	yes	—
252	C₂×D₅₃	C₂²×#41	1	16	yes	—
253	C₄×C₅₃	—	2	8	no	—
254	C₄×C₅₃	—	2	4	no	—
255	C₄×C₅₃	—	2	8	no	—
256	Dic₅₃	—	2	8	no	—
257	Dic₅₃	—	2	4	no	—
258	Dic₅₃	—	2	8	no	—
259	C₂⁴×D₄	—	53	1	no	—
260	C₂⁵×C₄	—	53	1	no	—
261	C₂²×#123	—	53	1	no	—
262	C₂²×#123	—	53	1	no	—
263	C₂²×#123	—	53	48	no	—
264	C₂²×#123	—	53	2	no	—
265	C₂²×#123	—	53	2	no	—
266	C₂²×#123	—	53	1	no	—
267	C₂²×#123	D₅₃	1	1	yes	characteristic, direct product factor
268	C₂³×C₄⋊C₄	—	53	2	no	—
269	C₂³×C₂²⋊C₄	—	53	2	no	—
270	C₂³×C₂²⋊C₄	—	53	1	no	—
271	C₂×C₅₃	—	2	8	no	—
272	C₂×C₅₃	—	4	4	no	—
273	C₂×C₅₃	C₂²×#123	1	12	yes	complemented
274	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
275	C₂×C₅₃	C₂³×#41	1	2	yes	—
276	D₅₃	—	2	8	no	—
277	D₅₃	—	4	4	no	—
278	D₅₃	C₂²×#123	1	16	yes	direct product factor
279	C₂⁶	—	53	1	no	—
280	C₂⁶	—	53	1	no	—
281	C₂³×D₄	—	53	1	no	—
282	C₂³×D₄	—	53	1	no	—
283	C₂³×D₄	—	53	24	no	—
284	C₂³×D₄	—	53	1	no	—
285	C₂³×D₄	—	106	16	no	—
286	C₂³×D₄	C₂×D₅₃	1	1	yes	characteristic
287	C₂⁴×C₄	—	53	1	no	—
288	C₂⁴×C₄	—	53	2	no	—
289	C₂⁴×C₄	—	53	2	no	—
290	C₂⁴×C₄	—	53	1	no	—
291	C₂⁴×C₄	—	53	2	no	—
292	C₂⁴×C₄	—	53	24	no	—
293	C₂⁴×C₄	—	106	16	no	—
294	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
295	C₂×#123	—	53	24	no	—
296	C₂×#123	—	53	24	no	—
297	C₂×#123	—	53	256	no	—
298	C₂×#123	—	53	48	no	—
299	C₂×#123	—	53	48	no	—
300	C₂×#123	—	53	24	no	—
301	C₂×#123	C₂×D₅₃	1	24	yes	direct product factor
302	C₂²×C₄⋊C₄	—	53	2	no	—
303	C₂²×C₄⋊C₄	—	53	2	no	—
304	C₂²×C₄⋊C₄	—	53	2	no	—
305	C₂²×C₄⋊C₄	—	53	48	no	—
306	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
307	C₂²×C₂²⋊C₄	—	53	1	no	—
308	C₂²×C₂²⋊C₄	—	53	2	no	—
309	C₂²×C₂²⋊C₄	—	53	2	no	—
310	C₂²×C₂²⋊C₄	—	53	2	no	—
311	C₂²×C₂²⋊C₄	—	53	2	no	—
312	C₂²×C₂²⋊C₄	—	53	24	no	—
313	C₂²×C₂²⋊C₄	—	53	48	no	—
314	C₂²×C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
315	C₂²×C₂²⋊C₄	C₂×D₅₃	1	2	yes	complemented
316	C₅₃	C₂³×#123	1	1	yes	characteristic, complemented, Sylow
317	C₂⁵	—	53	1	no	—
318	C₂⁵	—	53	16	no	—
319	C₂⁵	—	53	12	no	—
320	C₂⁵	—	53	1	no	—
321	C₂⁵	—	53	12	no	—
322	C₂⁵	—	53	1	no	—
323	C₂⁵	—	106	16	no	—
324	C₂⁵	—	106	8	no	—
325	C₂⁵	—	106	16	no	—
326	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
327	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
328	C₂²×D₄	—	2	8	no	—
329	C₂²×D₄	—	53	12	no	—
330	C₂²×D₄	—	53	12	no	—
331	C₂²×D₄	—	53	64	no	—
332	C₂²×D₄	—	53	12	no	—
333	C₂²×D₄	—	106	8	no	—
334	C₂²×D₄	—	106	192	no	—
335	C₂²×D₄	—	106	8	no	—
336	C₂²×D₄	—	106	8	no	—
337	C₂²×D₄	C₂²×D₅₃	1	12	yes	—
338	C₂³×C₄	—	2	8	no	—
339	C₂³×C₄	—	53	12	no	—
340	C₂³×C₄	—	53	2	no	—
341	C₂³×C₄	—	53	2	no	—
342	C₂³×C₄	—	53	1	no	—
343	C₂³×C₄	—	53	24	no	—
344	C₂³×C₄	—	53	24	no	—
345	C₂³×C₄	—	53	12	no	—
346	C₂³×C₄	—	53	24	no	—
347	C₂³×C₄	—	53	64	no	—
348	C₂³×C₄	—	106	16	no	—
349	C₂³×C₄	—	106	16	no	—
350	C₂³×C₄	—	106	192	no	—
351	C₂³×C₄	—	106	8	no	—
352	C₂³×C₄	—	106	16	no	—
353	C₂³×C₄	—	106	8	no	—
354	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
355	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
356	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
357	C₂³×C₄	C₂²×D₅₃	1	12	yes	complemented
358	#123	—	53	64	no	—
359	#123	—	53	64	no	—
360	#123	—	53	128	no	—
361	#123	—	53	128	no	—
362	#123	—	53	64	no	—
363	#123	C₂²×D₅₃	1	64	yes	direct product factor
364	C₂×C₄⋊C₄	—	53	24	no	—
365	C₂×C₄⋊C₄	—	53	24	no	—
366	C₂×C₄⋊C₄	—	53	24	no	—
367	C₂×C₄⋊C₄	—	53	128	no	—
368	C₂×C₄⋊C₄	C₂²×D₅₃	1	24	yes	complemented
369	C₂×C₂²⋊C₄	—	53	12	no	—
370	C₂×C₂²⋊C₄	—	53	24	no	—
371	C₂×C₂²⋊C₄	—	53	24	no	—
372	C₂×C₂²⋊C₄	—	53	24	no	—
373	C₂×C₂²⋊C₄	—	53	24	no	—
374	C₂×C₂²⋊C₄	—	53	64	no	—
375	C₂×C₂²⋊C₄	—	53	128	no	—
376	C₂×C₂²⋊C₄	C₂²×D₅₃	1	12	yes	complemented
377	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	complemented
378	C₂⁴	—	2	8	no	—
379	C₂⁴	—	2	4	no	—
380	C₂⁴	—	2	8	no	—
381	C₂⁴	—	53	8	no	—
382	C₂⁴	—	53	16	no	—
383	C₂⁴	—	53	6	no	—
384	C₂⁴	—	53	96	no	—
385	C₂⁴	—	53	8	no	—
386	C₂⁴	—	53	16	no	—
387	C₂⁴	—	53	6	no	—
388	C₂⁴	—	53	16	no	—
389	C₂⁴	—	53	6	no	—
390	C₂⁴	—	106	128	no	—
391	C₂⁴	—	106	96	no	—
392	C₂⁴	—	106	48	no	—
393	C₂⁴	—	106	96	no	—
394	C₂⁴	—	106	8	no	—
395	C₂⁴	—	106	4	no	—
396	C₂⁴	—	106	8	no	—
397	C₂⁴	—	212	64	no	—
398	C₂⁴	D₄×D₅₃	1	8	yes	—
399	C₂⁴	C₂³×D₅₃	1	6	yes	—
400	C₂⁴	C₂³×D₅₃	1	6	yes	—
401	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
402	C₂×D₄	—	2	48	no	—
403	C₂×D₄	—	53	16	no	—
404	C₂×D₄	—	53	16	no	—
405	C₂×D₄	—	53	16	no	—
406	C₂×D₄	—	106	48	no	—
407	C₂×D₄	—	106	256	no	—
408	C₂×D₄	—	106	48	no	—
409	C₂×D₄	—	106	48	no	—
410	C₂×D₄	C₂³×D₅₃	1	16	yes	—
411	C₂²×C₄	—	2	8	no	—
412	C₂²×C₄	—	2	48	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	8	no	—
415	C₂²×C₄	—	53	12	no	—
416	C₂²×C₄	—	53	12	no	—
417	C₂²×C₄	—	53	6	no	—
418	C₂²×C₄	—	53	16	no	—
419	C₂²×C₄	—	53	32	no	—
420	C₂²×C₄	—	53	32	no	—
421	C₂²×C₄	—	53	16	no	—
422	C₂²×C₄	—	53	32	no	—
423	C₂²×C₄	—	106	96	no	—
424	C₂²×C₄	—	106	96	no	—
425	C₂²×C₄	—	106	96	no	—
426	C₂²×C₄	—	106	256	no	—
427	C₂²×C₄	—	106	48	no	—
428	C₂²×C₄	—	106	48	no	—
429	C₂²×C₄	—	106	8	no	—
430	C₂²×C₄	—	106	4	no	—
431	C₂²×C₄	—	106	8	no	—
432	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
433	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
434	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
435	C₂²×C₄	C₂³×D₅₃	1	16	yes	complemented
436	C₄⋊C₄	—	53	32	no	—
437	C₄⋊C₄	—	53	32	no	—
438	C₄⋊C₄	—	53	32	no	—
439	C₄⋊C₄	C₂³×D₅₃	1	32	yes	complemented
440	C₂²⋊C₄	—	53	16	no	—
441	C₂²⋊C₄	—	53	32	no	—
442	C₂²⋊C₄	—	53	32	no	—
443	C₂²⋊C₄	—	53	32	no	—
444	C₂²⋊C₄	—	53	32	no	—
445	C₂²⋊C₄	C₂³×D₅₃	1	16	yes	complemented
446	C₂²⋊C₄	C₂³×D₅₃	1	32	yes	complemented
447	C₂³	—	2	32	no	—
448	C₂³	—	2	24	no	—
449	C₂³	—	2	12	no	—
450	C₂³	—	2	24	no	—
451	C₂³	—	4	16	no	—
452	C₂³	—	53	64	no	—
453	C₂³	—	53	24	no	—
454	C₂³	—	53	48	no	—
455	C₂³	—	53	4	no	—
456	C₂³	—	53	64	no	—
457	C₂³	—	53	24	no	—
458	C₂³	—	53	4	no	—
459	C₂³	—	53	4	no	—
460	C₂³	—	106	384	no	—
461	C₂³	—	106	64	no	—
462	C₂³	—	106	32	no	—
463	C₂³	—	106	64	no	—
464	C₂³	—	106	32	no	—
465	C₂³	—	106	24	no	—
466	C₂³	—	106	12	no	—
467	C₂³	—	106	24	no	—
468	C₂³	—	212	192	no	—
469	C₂³	—	212	16	no	—
470	C₂³	C₂×D₄×D₅₃	1	24	yes	—
471	C₂³	C₂×D₄×D₅₃	1	4	yes	—
472	C₂³	#41×D₅₃	1	8	yes	—
473	C₂³	C₂⁴×D₅₃	1	4	yes	—
474	C₂³	C₂⁴×D₅₃	1	4	yes	—
475	C₂³	C₂⁴×D₅₃	1	3	yes	—
476	D₄	—	2	32	no	—
477	D₄	—	106	32	no	—
478	D₄	—	106	32	no	—
479	D₄	—	106	32	no	—
480	C₂×C₄	—	2	24	no	—
481	C₂×C₄	—	2	32	no	—
482	C₂×C₄	—	2	12	no	—
483	C₂×C₄	—	2	24	no	—
484	C₂×C₄	—	53	8	no	—
485	C₂×C₄	—	53	8	no	—
486	C₂×C₄	—	53	4	no	—
487	C₂×C₄	—	106	64	no	—
488	C₂×C₄	—	106	64	no	—
489	C₂×C₄	—	106	64	no	—
490	C₂×C₄	—	106	32	no	—
491	C₂×C₄	—	106	32	no	—
492	C₂×C₄	—	106	24	no	—
493	C₂×C₄	—	106	12	no	—
494	C₂×C₄	—	106	24	no	—
495	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
496	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
497	C₂×C₄	C₂⁴×D₅₃	1	4	yes	—
498	C₂²	—	2	48	no	—
499	C₂²	—	2	8	no	—
500	C₂²	—	2	4	no	—
501	C₂²	—	2	8	no	—
502	C₂²	—	4	24	no	—
503	C₂²	—	53	96	no	—
504	C₂²	—	53	8	no	—
505	C₂²	—	53	16	no	—
506	C₂²	—	53	8	no	—
507	C₂²	—	106	128	no	—
508	C₂²	—	106	48	no	—
509	C₂²	—	106	8	no	—
510	C₂²	—	106	4	no	—
511	C₂²	—	106	8	no	—
512	C₂²	—	212	64	no	—
513	C₂²	—	212	24	no	—
514	C₂²	#123×D₅₃	1	16	yes	direct product factor
515	C₂²	C₂²×D₄×D₅₃	1	8	yes	—
516	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
517	C₂²	C₂×#41×D₅₃	1	12	yes	—
518	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
519	C₄	—	2	8	no	—
520	C₄	—	2	4	no	—
521	C₄	—	2	8	no	—
522	C₄	—	106	8	no	—
523	C₄	—	106	4	no	—
524	C₄	—	106	8	no	—
525	C₂	—	2	8	no	—
526	C₂	—	4	4	no	—
527	C₂	—	53	16	no	—
528	C₂	—	106	8	no	—
529	C₂	—	212	4	no	—
530	C₂	C₂×#123×D₅₃	1	12	yes	direct product factor
531	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
532	C₂	C₂²×#41×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#123×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×D₄×D₅₃	C₂
hypercenter	C₂²×#123	D₅₃
hypocenter	C₅₃	C₂³×#123
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#123

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₅₃	C₂	semidirect
3. C₂²×#13133202	C₂	semidirect
4. C₂²×#13133201	C₂	semidirect
5. C₂²×#13133200	C₂	semidirect
6. C₂²×#13133199	C₂	semidirect
7. C₂²×#13133198	C₂	semidirect
8. C₂×#123×D₅₃	C₂	direct (N × H)
9. C₂²×C₄⋊C₄×D₅₃	C₂	semidirect
10. C₂²×C₂²⋊C₄×D₅₃	C₂	semidirect
11. C₂²×C₂²⋊C₄×D₅₃	C₂	semidirect
12. C₂²×#123×C₅₃	C₂	semidirect
20. C₂⁴×Dic₅₃	C₂²	semidirect
25. C₂³×C₄×D₅₃	C₂²	semidirect
26. C₂³×C₄×D₅₃	C₂²	semidirect
27. C₂×#13133202	C₂²	semidirect
28. C₂×#13133201	C₂²	semidirect
29. C₂×#13133200	C₂²	semidirect
30. C₂×#13133199	C₂²	semidirect
31. C₂×#13133198	C₂²	semidirect
32. #123×D₅₃	C₂²	direct (N × H)
33. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
34. C₂²×#11719919	C₂²	semidirect
35. C₂²×#11719919	C₂²	semidirect
36. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
37. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
38. C₂²×C₂²⋊Dic₅₃	C₂²	semidirect
39. C₂²×#11719921	C₂²	semidirect
40. C₂²×#11719921	C₂²	semidirect
41. C₂²×#11719921	C₂²	semidirect
42. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
43. C₂×C₂²⋊C₄×D₅₃	C₂²	semidirect
45. C₂⁴×C₄×C₅₃	C₂²	semidirect
46. C₂×#123×C₅₃	C₂²	semidirect
47. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
48. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
49. C₂²×C₂²⋊C₄×C₅₃	C₂²	semidirect
71. C₂³×Dic₅₃	C₂³	semidirect
80. C₂²×C₄×D₅₃	C₂³	semidirect
81. C₂²×C₄×D₅₃	C₂³	semidirect
82. #13133202	C₂³	semidirect
83. #13133201	C₂³	semidirect
84. #13133200	C₂³	semidirect
85. #13133199	C₂³	semidirect
86. #13133198	C₂³	semidirect
87. C₂×C₄⋊Dic₅₃	C₂³	semidirect
88. C₂×#11719919	C₂³	semidirect
89. C₂×#11719919	C₂³	semidirect
90. C₄⋊C₄×D₅₃	C₂³	semidirect
91. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
92. C₂×C₂²⋊Dic₅₃	C₂³	semidirect
93. C₂×#11719921	C₂³	semidirect
94. C₂×#11719921	C₂³	semidirect
95. C₂×#11719921	C₂³	semidirect
96. C₂²⋊C₄×D₅₃	C₂³	semidirect
97. C₂²⋊C₄×D₅₃	C₂³	semidirect
106. C₂³×C₄×C₅₃	C₂³	semidirect
107. #123×C₅₃	C₂³	semidirect
108. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
109. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
110. C₂×C₂²⋊C₄×C₅₃	C₂³	semidirect
144. C₂²×C₄×C₅₃	C₂⁴	semidirect
156. C₂²×Dic₅₃	C₂⁴	semidirect
168. C₂×C₄×D₅₃	C₂⁴	semidirect
169. C₄⋊C₄×C₅₃	C₂⁴	semidirect
170. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
171. C₂²⋊C₄×C₅₃	C₂⁴	semidirect
172. C₂²⋊Dic₅₃	C₂⁴	semidirect
173. C₂²⋊Dic₅₃	C₂⁴	semidirect
174. #11719921	C₂⁴	semidirect
175. #11719921	C₂⁴	semidirect
176. #11719921	C₂⁴	semidirect
177. C₄⋊Dic₅₃	C₂⁴	semidirect
178. #11719919	C₂⁴	semidirect
179. #11719919	C₂⁴	semidirect
201. C₂²×D₅₃	#123	direct (N × H)
237. C₂²×C₅₃	C₂×#123	semidirect
249. C₂×D₅₃	C₂×#123	direct (N × H)
267. C₂²×#123	D₅₃	direct (N × H)
273. C₂×C₅₃	C₂²×#123	semidirect
278. D₅₃	C₂²×#123	direct (N × H)
294. C₂⁴×C₄	C₂×D₅₃	semidirect
301. C₂×#123	C₂×D₅₃	direct (N × H)
306. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
314. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
315. C₂²×C₂²⋊C₄	C₂×D₅₃	semidirect
316. C₅₃	C₂³×#123	semidirect
357. C₂³×C₄	C₂²×D₅₃	semidirect
363. #123	C₂²×D₅₃	direct (N × H)
368. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
376. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
377. C₂×C₂²⋊C₄	C₂²×D₅₃	semidirect
435. C₂²×C₄	C₂³×D₅₃	semidirect
439. C₄⋊C₄	C₂³×D₅₃	semidirect
445. C₂²⋊C₄	C₂³×D₅₃	semidirect
446. C₂²⋊C₄	C₂³×D₅₃	semidirect
514. C₂²	#123×D₅₃	direct (N × H)
530. C₂	C₂×#123×D₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×D₅₃	2
C₂⁴×C₄×D₅₃	2
C₂²×#13133202	2
C₂²×#13133201	2
C₂²×#13133200	2
C₂²×#13133199	2
C₂²×#13133198	2
C₂×#123×D₅₃	2
C₂²×C₄⋊C₄×D₅₃	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂²×#123×C₅₃	2
C₂³×#123	53

Subgroups — C22×#123×D53 #28290737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#123×D₅₃ #28290737