Subgroups — D₅₃×#3549 #28291890

Order 13568.

8802 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	#20289120	C₂	1	3	yes	maximal, complemented
2	#20289119	C₂	1	4	yes	maximal, complemented
3	#20289118	C₂	1	12	yes	maximal, complemented
4	#20289117	C₂	1	6	yes	maximal, complemented
5	#20289116	C₂	1	6	yes	maximal, complemented
6	H(3,𝔽₄)×D₅₃	C₂	1	4	yes	maximal, complemented
7	#555×D₅₃	C₂	1	6	yes	maximal, complemented
8	D₄²×D₅₃	C₂	1	6	yes	maximal, complemented
9	#535×D₅₃	C₂	1	12	yes	maximal, complemented
10	#532×D₅₃	C₂	1	3	yes	maximal, complemented
11	C₅₃×#3549	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×D₅₃	C₂²	1	6	yes	complemented
13	C₂²×D₄×D₅₃	C₂²	1	12	yes	complemented
14	#16811366	C₂²	1	24	yes	complemented
15	#16811365	C₂²	1	24	yes	complemented
16	#16811365	C₂²	1	12	yes	complemented
17	#16811339	C₂²	1	6	yes	complemented
18	#16811338	C₂²	1	12	yes	complemented
19	#16811337	C₂²	1	24	yes	complemented
20	#16811336	C₂²	1	24	yes	complemented
21	#16811335	C₂²	1	24	yes	complemented
22	D₄×D₂₁₂	C₂²	1	12	yes	complemented
23	#16811258	C₂²	1	6	yes	complemented
24	#16811257	C₂²	1	24	yes	complemented
25	D₄×C₂²⋊D₅₃	C₂²	1	24	yes	complemented
26	#16811255	C₂²	1	24	yes	complemented
27	#16811192	C₂²	1	12	yes	complemented
28	#16811191	C₂²	1	24	yes	complemented
29	#16811190	C₂²	1	12	yes	complemented
30	#16811189	C₂²	1	24	yes	complemented
31	#16811189	C₂²	1	24	yes	complemented
32	#16811188	C₂²	1	24	yes	complemented
33	#16811188	C₂²	1	24	yes	complemented
34	#16811187	C₂²	1	12	yes	complemented
35	#16811186	C₂²	1	24	yes	complemented
36	#16811174	C₂²	1	3	yes	complemented
37	#16811173	C₂²	1	3	yes	complemented
38	#16811172	C₂²	1	6	yes	complemented
39	#16811171	C₂²	1	3	yes	complemented
40	#16811170	C₂²	1	12	yes	complemented
41	#16811169	C₂²	1	6	yes	complemented
42	#16811168	C₂²	1	12	yes	complemented
43	C₄⋊Q₈×D₅₃	C₂²	1	3	yes	complemented
44	#127×D₅₃	C₂²	1	3	yes	—
45	#127×D₅₃	C₂²	1	8	yes	complemented
46	#124×D₅₃	C₂²	1	24	yes	complemented
47	#124×D₅₃	C₂²	1	6	yes	complemented
48	#121×D₅₃	C₂²	1	12	yes	complemented
49	#121×D₅₃	C₂²	1	24	yes	complemented
50	#120×D₅₃	C₂²	1	8	yes	complemented
51	#120×D₅₃	C₂²	1	24	yes	complemented
52	C₄×D₄×D₅₃	C₂²	1	12	yes	complemented
53	#117×D₅₃	C₂²	1	12	yes	complemented
54	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic, complemented
55	C₅₃×H(3,𝔽₄)	C₂²	1	4	yes	complemented
56	C₅₃×#555	C₂²	1	6	yes	complemented
57	D₄²×C₅₃	C₂²	1	6	yes	complemented
58	C₅₃×#535	C₂²	1	12	yes	complemented
59	C₅₃×#532	C₂²	1	3	yes	complemented
60	C₂⁴×D₅₃	C₂³	1	8	yes	—
61	C₂×#11719949	C₂³	1	6	yes	complemented
62	C₂×#11719945	C₂³	1	6	yes	complemented
63	C₂×#11719945	C₂³	1	12	yes	complemented
64	C₂×#11719947	C₂³	1	12	yes	complemented
65	C₂×#11719947	C₂³	1	6	yes	complemented
66	#41×D₅₃	—	4	24	no	—
67	C₂×Q₈×D₅₃	C₂³	1	6	yes	—
68	C₂²×D₂₁₂	C₂³	1	12	yes	complemented
69	C₂²×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
70	C₂×D₄×D₅₃	—	4	48	no	—
71	C₂×D₄×D₅₃	C₂³	1	12	yes	—
72	C₂×D₄×D₅₃	C₂³	1	6	yes	—
73	C₂×D₄×D₅₃	C₂³	1	24	yes	—
74	C₂×D₄×D₅₃	C₂³	1	12	yes	—
75	C₂×D₄×D₅₃	C₂³	1	24	yes	complemented
76	C₂×D₄×D₅₃	C₂³	1	24	yes	complemented
77	C₂²×C₄×D₅₃	C₂³	1	3	yes	—
78	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
79	#13133217	C₂³	1	3	yes	complemented
80	#13133216	C₂³	1	12	yes	complemented
81	#13133215	C₂³	1	6	yes	complemented
82	#13133214	C₂³	1	3	yes	—
83	#13133214	C₂³	1	8	yes	complemented
84	#13133213	C₂³	1	12	yes	—
85	#13133213	C₂³	1	6	yes	—
86	#13133213	C₂³	1	24	yes	complemented
87	#13133213	C₂³	1	24	yes	complemented
88	#13133206	C₂³	1	24	yes	complemented
89	#13133206	C₂³	1	6	yes	complemented
90	#13133205	C₂³	1	24	yes	complemented
91	#13133205	C₂³	1	6	yes	complemented
92	#13133204	C₂³	1	24	yes	complemented
93	#13133204	C₂³	1	6	yes	complemented
94	#13133203	C₂³	1	48	yes	complemented
95	#13133203	C₂³	1	48	yes	complemented
96	#13133203	C₂³	1	24	yes	complemented
97	#13133192	C₂³	1	24	yes	complemented
98	#13133192	C₂³	1	12	yes	complemented
99	#13133191	C₂³	1	24	yes	complemented
100	#13133191	C₂³	1	12	yes	complemented
101	#13133190	C₂³	1	48	yes	complemented
102	#13133190	C₂³	1	24	yes	complemented
103	#13133189	C₂³	1	48	yes	complemented
104	#13133189	C₂³	1	24	yes	complemented
105	#13133188	C₂³	1	24	yes	complemented
106	#13133188	C₂³	1	12	yes	complemented
107	#13133187	C₂³	1	24	yes	complemented
108	#13133187	C₂³	1	48	yes	complemented
109	#13133187	C₂³	1	24	yes	complemented
110	#13133186	C₂³	1	24	yes	complemented
111	#13133186	C₂³	1	8	yes	complemented
112	#13133185	C₂³	1	48	yes	complemented
113	#13133185	C₂³	1	24	yes	complemented
114	#13133185	C₂³	1	24	yes	complemented
115	C₄×D₂₁₂	C₂³	1	12	yes	complemented
116	D₄×Dic₅₃	C₂³	1	12	yes	complemented
117	#13133179	C₂³	1	12	yes	complemented
118	C₄×C₂²⋊D₅₃	C₂³	1	24	yes	complemented
119	#13133177	C₂³	1	24	yes	complemented
120	#13133176	C₂³	1	12	yes	complemented
121	#13133175	C₂³	1	24	yes	complemented
122	#13133174	C₂³	1	24	yes	complemented
123	#13133173	C₂³	1	24	yes	complemented
124	C₄⋊C₄×D₅₃	C₂³	1	12	yes	—
125	C₂²⋊C₄×D₅₃	C₂³	1	24	yes	—
126	C₂²⋊C₄×D₅₃	C₂³	1	24	yes	—
127	C₂×C₄×Dic₅₃	C₂³	1	3	yes	complemented
128	C₄²×D₅₃	C₂³	1	3	yes	—
129	C₄²×D₅₃	C₂³	1	1	yes	characteristic
130	C₄²×D₅₃	C₂³	1	8	yes	—
131	C₄²×D₅₃	C₂³	1	3	yes	complemented
132	C₄²×D₅₃	C₂³	1	1	yes	characteristic, complemented
133	C₂×#41×C₅₃	C₂³	1	6	yes	complemented
134	C₂²×D₄×C₅₃	C₂³	1	12	yes	complemented
135	C₄⋊Q₈×C₅₃	C₂³	1	3	yes	complemented
136	#127×C₅₃	C₂³	1	3	yes	—
137	#127×C₅₃	C₂³	1	8	yes	complemented
138	#124×C₅₃	C₂³	1	6	yes	complemented
139	#124×C₅₃	C₂³	1	24	yes	complemented
140	#121×C₅₃	C₂³	1	12	yes	complemented
141	#121×C₅₃	C₂³	1	24	yes	complemented
142	#120×C₅₃	C₂³	1	8	yes	complemented
143	#120×C₅₃	C₂³	1	24	yes	complemented
144	C₄×D₄×C₅₃	C₂³	1	12	yes	complemented
145	#117×C₅₃	C₂³	1	12	yes	complemented
146	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic, complemented
147	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
148	C₂³×D₅₃	—	4	48	no	—
149	C₂³×D₅₃	C₂⁴	1	12	yes	—
150	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
151	C₂³×D₅₃	C₂⁴	1	24	yes	—
152	#41×C₅₃	—	4	12	no	—
153	C₂×Q₈×C₅₃	C₂⁴	1	6	yes	—
154	C₂×D₄×C₅₃	—	4	24	no	—
155	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
156	C₂×D₄×C₅₃	C₂⁴	1	6	yes	—
157	C₂×D₄×C₅₃	C₂⁴	1	24	yes	—
158	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
159	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
160	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
161	C₂×C₂²⋊D₅₃	—	4	48	no	—
162	C₂×C₂²⋊D₅₃	C₂⁴	1	24	yes	—
163	C₂×C₂²⋊D₅₃	C₂⁴	1	12	yes	—
164	C₂×C₂²⋊D₅₃	C₂⁴	1	12	yes	—
165	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
166	C₂×C₂²⋊D₅₃	C₂⁴	1	12	yes	—
167	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
168	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
169	#11719949	—	4	12	no	—
170	Q₈×D₅₃	—	4	12	no	—
171	#11719947	—	4	24	no	—
172	#11719947	—	4	12	no	—
173	D₄×D₅₃	—	2	48	no	—
174	D₄×D₅₃	—	4	48	no	—
175	D₄×D₅₃	—	4	48	no	—
176	D₄×D₅₃	—	4	48	no	—
177	D₄×D₅₃	—	4	24	no	—
178	D₄×D₅₃	—	4	12	no	—
179	#11719945	—	4	24	no	—
180	#11719945	—	4	12	no	—
181	C₂×D₂₁₂	—	4	24	no	—
182	C₂×D₂₁₂	C₂⁴	1	12	yes	—
183	C₂×D₂₁₂	C₂⁴	1	6	yes	—
184	C₂×D₂₁₂	C₂⁴	1	24	yes	—
185	C₂×D₂₁₂	C₂⁴	1	12	yes	—
186	C₂×C₄×D₅₃	—	2	48	no	—
187	C₂×C₄×D₅₃	—	2	12	no	—
188	C₂×C₄×D₅₃	C₂⁴	1	3	yes	—
189	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
190	C₂×C₄×D₅₃	C₂⁴	1	3	yes	—
191	C₂×C₄×D₅₃	C₂⁴	1	3	yes	—
192	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
193	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
194	C₂×C₄×D₅₃	C₂⁴	1	3	yes	—
195	C₂×Dic₁₀₆	C₂⁴	1	6	yes	—
196	C₂×Dic₁₀₆	C₂⁴	1	12	yes	—
197	C₄⋊C₄×C₅₃	C₂⁴	1	12	yes	—
198	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
199	C₂²⋊C₄×C₅₃	C₂⁴	1	24	yes	—
200	C₄²×C₅₃	C₂⁴	1	3	yes	—
201	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
202	C₄²×C₅₃	C₂⁴	1	8	yes	—
203	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
204	C₂²⋊Dic₅₃	C₂⁴	1	24	yes	—
205	#11719921	C₂⁴	1	48	yes	—
206	#11719921	C₂⁴	1	24	yes	—
207	#11719921	C₂⁴	1	24	yes	—
208	C₄⋊Dic₅₃	C₂⁴	1	12	yes	—
209	#11719919	C₂⁴	1	24	yes	—
210	C₄×Dic₅₃	C₂⁴	1	3	yes	—
211	C₄×Dic₅₃	C₂⁴	1	6	yes	—
212	C₄×Dic₅₃	C₂⁴	1	3	yes	—
213	C₄×Dic₅₃	C₂⁴	1	24	yes	—
214	C₂³×C₅₃	—	4	24	no	—
215	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
216	C₂³×C₅₃	C₂⁵	1	12	yes	—
217	C₂²×D₅₃	—	2	6	no	—
218	C₂²×D₅₃	—	2	24	no	—
219	C₂²×D₅₃	—	2	48	no	—
220	C₂²×D₅₃	—	4	24	no	—
221	C₂²×D₅₃	—	4	48	no	—
222	C₂²×D₅₃	—	8	64	no	—
223	C₂²×D₅₃	C₂⁵	1	1	yes	characteristic
224	C₂²×D₅₃	C₂⁵	1	12	yes	—
225	C₂²×D₅₃	C₂⁵	1	1	yes	characteristic
226	Q₈×C₅₃	—	4	6	no	—
227	D₄×C₅₃	—	2	24	no	—
228	D₄×C₅₃	—	4	24	no	—
229	D₄×C₅₃	—	4	6	no	—
230	D₄×C₅₃	—	4	12	no	—
231	C₂×C₄×C₅₃	—	2	24	no	—
232	C₂×C₄×C₅₃	—	2	6	no	—
233	C₂×C₄×C₅₃	C₂⁵	1	3	yes	—
234	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
235	C₂×C₄×C₅₃	C₂⁵	1	3	yes	—
236	C₂²⋊D₅₃	—	2	48	no	—
237	C₂²⋊D₅₃	—	4	12	no	—
238	C₂²⋊D₅₃	—	4	12	no	—
239	C₂²⋊D₅₃	—	4	12	no	—
240	C₂²⋊D₅₃	—	4	48	no	—
241	C₂×Dic₅₃	—	2	6	no	—
242	C₂×Dic₅₃	—	2	24	no	—
243	C₂×Dic₅₃	C₂⁵	1	3	yes	—
244	C₂×Dic₅₃	C₂⁵	1	12	yes	—
245	C₂×Dic₅₃	C₂⁵	1	3	yes	—
246	D₂₁₂	—	2	24	no	—
247	D₂₁₂	—	4	24	no	—
248	D₂₁₂	—	4	6	no	—
249	D₂₁₂	—	4	12	no	—
250	C₄×D₅₃	—	2	24	no	—
251	C₄×D₅₃	—	2	6	no	—
252	C₄×D₅₃	—	2	6	no	—
253	C₄×D₅₃	—	2	24	no	—
254	C₄×D₅₃	—	2	24	no	—
255	C₄×D₅₃	—	2	6	no	—
256	C₄×D₅₃	H(5,𝔽₂)	1	12	yes	—
257	Dic₁₀₆	—	4	12	no	—
258	Dic₁₀₆	—	4	6	no	—
259	C₂×#3549	—	53	1	no	maximal, Sylow
260	C₂²×C₅₃	—	2	3	no	—
261	C₂²×C₅₃	—	2	12	no	—
262	C₂²×C₅₃	—	2	24	no	—
263	C₂²×C₅₃	—	8	16	no	—
264	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
265	C₂×D₅₃	—	2	3	no	—
266	C₂×D₅₃	—	2	12	no	—
267	C₂×D₅₃	—	2	24	no	—
268	C₂×D₅₃	—	4	4	no	—
269	C₂×D₅₃	—	4	48	no	—
270	C₂×D₅₃	—	8	48	no	—
271	C₂×D₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
272	C₄×C₅₃	—	2	3	no	—
273	C₄×C₅₃	—	2	12	no	—
274	C₄×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
275	Dic₅₃	—	2	3	no	—
276	Dic₅₃	—	2	12	no	—
277	Dic₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
278	#3549	—	53	4	no	—
279	#3549	—	53	6	no	—
280	#3549	—	53	6	no	—
281	#3549	—	53	12	no	—
282	#3549	—	53	3	no	—
283	#3549	D₅₃	1	1	yes	characteristic, direct product factor
284	C₂×H(3,𝔽₄)	—	53	4	no	—
285	C₂×#555	—	53	6	no	—
286	C₂×D₄²	—	53	6	no	—
287	C₂×#535	—	53	12	no	—
288	C₂×#532	—	53	3	no	—
289	C₂×C₅₃	—	4	12	no	—
290	C₂×C₅₃	—	4	1	no	—
291	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	3	yes	—
292	D₅₃	—	4	1	no	—
293	D₅₃	—	4	12	no	—
294	D₅₃	#3549	1	4	yes	direct product factor
295	C₂²×#41	—	53	6	no	—
296	C₂³×D₄	—	53	12	no	—
297	H(3,𝔽₄)	—	53	24	no	—
298	H(3,𝔽₄)	—	53	12	no	—
299	H(3,𝔽₄)	—	53	24	no	—
300	H(3,𝔽₄)	C₂×D₅₃	1	4	yes	complemented
301	#555	—	53	6	no	—
302	#555	—	53	24	no	—
303	#555	—	53	24	no	—
304	#555	—	53	12	no	—
305	#555	—	53	24	no	—
306	#555	C₂×D₅₃	1	6	yes	complemented
307	D₄²	—	53	24	no	—
308	D₄²	—	53	6	no	—
309	D₄²	—	53	24	no	—
310	D₄²	—	53	24	no	—
311	D₄²	—	53	12	no	—
312	D₄²	C₂×D₅₃	1	6	yes	complemented
313	#535	—	53	12	no	—
314	#535	—	53	12	no	—
315	#535	—	53	24	no	—
316	#535	—	53	24	no	—
317	#535	—	53	24	no	—
318	#535	—	53	24	no	—
319	#535	—	53	24	no	—
320	#535	—	53	24	no	—
321	#535	—	53	12	no	—
322	#535	C₂×D₅₃	1	12	yes	complemented
323	#532	—	53	6	no	—
324	#532	—	53	3	no	—
325	#532	—	53	3	no	—
326	#532	—	53	6	no	—
327	#532	—	53	12	no	—
328	#532	—	53	12	no	—
329	#532	—	53	3	no	—
330	#532	C₂×D₅₃	1	3	yes	complemented
331	C₂×C₄⋊Q₈	—	53	3	no	—
332	C₂×#127	—	53	3	no	—
333	C₂×#127	—	53	8	no	—
334	C₂×#124	—	53	6	no	—
335	C₂×#124	—	53	24	no	—
336	C₂×#121	—	53	12	no	—
337	C₂×#121	—	53	24	no	—
338	C₂×#120	—	53	8	no	—
339	C₂×#120	—	53	24	no	—
340	C₂×C₄×D₄	—	53	12	no	—
341	C₂×#117	—	53	12	no	—
342	C₂²×C₄²	—	53	1	no	—
343	C₅₃	C₂×#3549	1	1	yes	characteristic, complemented, Sylow
344	C₂⁵	—	53	8	no	—
345	C₂×#41	—	53	6	no	—
346	C₂×#41	—	53	12	no	—
347	C₂×#41	—	53	6	no	—
348	C₂×#41	—	53	12	no	—
349	C₂×#41	—	53	6	no	—
350	C₂×#41	—	212	24	no	—
351	C₂×#41	C₂²×D₅₃	1	6	yes	complemented
352	C₂²×Q₈	—	53	6	no	—
353	C₂²×D₄	—	53	24	no	—
354	C₂²×D₄	—	53	12	no	—
355	C₂²×D₄	—	53	6	no	—
356	C₂²×D₄	—	53	24	no	—
357	C₂²×D₄	—	53	12	no	—
358	C₂²×D₄	—	53	24	no	—
359	C₂²×D₄	—	53	24	no	—
360	C₂²×D₄	—	53	12	no	—
361	C₂²×D₄	—	212	48	no	—
362	C₂²×D₄	C₂²×D₅₃	1	12	yes	complemented
363	C₂³×C₄	—	53	12	no	—
364	C₂³×C₄	—	53	3	no	—
365	C₄⋊Q₈	—	53	6	no	—
366	C₄⋊Q₈	—	53	12	no	—
367	C₄⋊Q₈	—	53	3	no	—
368	C₄⋊Q₈	C₂²×D₅₃	1	3	yes	complemented
369	#127	—	53	6	no	—
370	#127	—	53	12	no	—
371	#127	—	53	24	no	—
372	#127	—	53	24	no	—
373	#127	—	53	3	no	—
374	#127	—	53	8	no	—
375	#127	C₂²×D₅₃	1	3	yes	—
376	#127	C₂²×D₅₃	1	8	yes	complemented
377	#124	—	53	6	no	—
378	#124	—	53	24	no	—
379	#124	—	53	6	no	—
380	#124	—	53	24	no	—
381	#124	—	53	24	no	—
382	#124	—	53	48	no	—
383	#124	—	53	48	no	—
384	#124	—	53	6	no	—
385	#124	—	53	24	no	—
386	#124	C₂²×D₅₃	1	6	yes	complemented
387	#124	C₂²×D₅₃	1	24	yes	complemented
388	#121	—	53	12	no	—
389	#121	—	53	24	no	—
390	#121	—	53	24	no	—
391	#121	—	53	48	no	—
392	#121	—	53	12	no	—
393	#121	—	53	24	no	—
394	#121	—	53	12	no	—
395	#121	—	53	24	no	—
396	#121	—	53	24	no	—
397	#121	—	53	48	no	—
398	#121	C₂²×D₅₃	1	12	yes	complemented
399	#121	C₂²×D₅₃	1	24	yes	complemented
400	#120	—	53	8	no	—
401	#120	—	53	24	no	—
402	#120	—	53	24	no	—
403	#120	—	53	24	no	—
404	#120	—	53	48	no	—
405	#120	—	53	24	no	—
406	#120	—	53	48	no	—
407	#120	—	53	24	no	—
408	#120	C₂²×D₅₃	1	8	yes	complemented
409	#120	C₂²×D₅₃	1	24	yes	complemented
410	C₄×D₄	—	53	12	no	—
411	C₄×D₄	—	53	24	no	—
412	C₄×D₄	—	53	12	no	—
413	C₄×D₄	—	53	24	no	—
414	C₄×D₄	—	53	12	no	—
415	C₄×D₄	C₂²×D₅₃	1	12	yes	complemented
416	#117	—	53	12	no	—
417	#117	—	53	24	no	—
418	#117	—	53	24	no	—
419	#117	—	53	24	no	—
420	#117	C₂²×D₅₃	1	12	yes	complemented
421	C₂×C₄⋊C₄	—	53	12	no	—
422	C₂×C₂²⋊C₄	—	53	24	no	—
423	C₂×C₂²⋊C₄	—	53	24	no	—
424	C₂×C₄²	—	53	3	no	—
425	C₂×C₄²	—	53	8	no	—
426	C₂×C₄²	—	53	3	no	—
427	C₂×C₄²	—	53	1	no	—
428	C₂×C₄²	—	53	1	no	—
429	C₂×C₄²	—	53	3	no	—
430	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic, complemented
431	C₂⁴	—	53	12	no	—
432	C₂⁴	—	53	1	no	—
433	C₂⁴	—	53	24	no	—
434	C₂⁴	—	212	48	no	—
435	C₂⁴	C₂³×D₅₃	1	8	yes	—
436	#41	—	4	12	no	—
437	#41	—	212	24	no	—
438	#41	—	212	12	no	—
439	#41	—	212	24	no	—
440	#41	—	212	12	no	—
441	#41	—	212	12	no	—
442	C₂×Q₈	—	53	12	no	—
443	C₂×Q₈	—	53	6	no	—
444	C₂×Q₈	—	212	12	no	—
445	C₂×Q₈	C₂³×D₅₃	1	6	yes	—
446	C₂×D₄	—	4	24	no	—
447	C₂×D₄	—	53	24	no	—
448	C₂×D₄	—	53	12	no	—
449	C₂×D₄	—	53	48	no	—
450	C₂×D₄	—	53	12	no	—
451	C₂×D₄	—	53	12	no	—
452	C₂×D₄	—	53	12	no	—
453	C₂×D₄	—	53	6	no	—
454	C₂×D₄	—	53	24	no	—
455	C₂×D₄	—	53	12	no	—
456	C₂×D₄	—	106	48	no	—
457	C₂×D₄	—	212	24	no	—
458	C₂×D₄	—	212	48	no	—
459	C₂×D₄	—	212	48	no	—
460	C₂×D₄	—	212	48	no	—
461	C₂×D₄	—	212	24	no	—
462	C₂×D₄	—	212	12	no	—
463	C₂×D₄	—	212	48	no	—
464	C₂×D₄	C₂³×D₅₃	1	12	yes	—
465	C₂×D₄	C₂³×D₅₃	1	6	yes	—
466	C₂×D₄	C₂³×D₅₃	1	24	yes	—
467	C₂×D₄	C₂³×D₅₃	1	12	yes	—
468	C₂²×C₄	—	53	3	no	—
469	C₂²×C₄	—	53	3	no	—
470	C₂²×C₄	—	53	12	no	—
471	C₂²×C₄	—	53	3	no	—
472	C₂²×C₄	—	53	12	no	—
473	C₂²×C₄	—	53	3	no	—
474	C₂²×C₄	—	53	3	no	—
475	C₂²×C₄	—	53	12	no	—
476	C₂²×C₄	—	53	12	no	—
477	C₂²×C₄	—	106	48	no	—
478	C₂²×C₄	—	106	12	no	—
479	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
480	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
481	C₄⋊C₄	—	53	24	no	—
482	C₄⋊C₄	—	53	12	no	—
483	C₄⋊C₄	C₂³×D₅₃	1	12	yes	—
484	C₂²⋊C₄	—	53	24	no	—
485	C₂²⋊C₄	—	53	24	no	—
486	C₂²⋊C₄	—	53	48	no	—
487	C₂²⋊C₄	—	53	24	no	—
488	C₂²⋊C₄	—	53	24	no	—
489	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
490	C₂²⋊C₄	C₂³×D₅₃	1	24	yes	—
491	C₄²	—	53	3	no	—
492	C₄²	—	53	6	no	—
493	C₄²	—	53	3	no	—
494	C₄²	—	53	24	no	—
495	C₄²	C₂³×D₅₃	1	3	yes	—
496	C₄²	C₂³×D₅₃	1	1	yes	characteristic
497	C₄²	C₂³×D₅₃	1	8	yes	—
498	C₂³	—	4	24	no	—
499	C₂³	—	53	1	no	—
500	C₂³	—	53	12	no	—
501	C₂³	—	53	1	no	—
502	C₂³	—	106	6	no	—
503	C₂³	—	106	24	no	—
504	C₂³	—	106	48	no	—
505	C₂³	—	424	64	no	—
506	C₂³	—	212	24	no	—
507	C₂³	—	212	48	no	—
508	C₂³	C₂⁴×D₅₃	1	12	yes	—
509	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
510	Q₈	—	4	6	no	—
511	Q₈	—	212	12	no	—
512	Q₈	—	212	6	no	—
513	D₄	—	2	24	no	—
514	D₄	—	4	24	no	—
515	D₄	—	4	12	no	—
516	D₄	—	4	6	no	—
517	D₄	—	106	48	no	—
518	D₄	—	106	24	no	—
519	D₄	—	212	24	no	—
520	D₄	—	212	12	no	—
521	D₄	—	212	6	no	—
522	D₄	—	212	48	no	—
523	D₄	—	212	12	no	—
524	D₄	—	212	12	no	—
525	D₄	—	212	12	no	—
526	C₂×C₄	—	2	24	no	—
527	C₂×C₄	—	2	6	no	—
528	C₂×C₄	—	53	3	no	—
529	C₂×C₄	—	53	12	no	—
530	C₂×C₄	—	53	3	no	—
531	C₂×C₄	—	53	12	no	—
532	C₂×C₄	—	106	24	no	—
533	C₂×C₄	—	106	6	no	—
534	C₂×C₄	—	106	24	no	—
535	C₂×C₄	—	106	6	no	—
536	C₂×C₄	—	106	24	no	—
537	C₂×C₄	—	106	6	no	—
538	C₂×C₄	—	106	24	no	—
539	C₂×C₄	—	106	6	no	—
540	C₂×C₄	C₂⁴×D₅₃	1	3	yes	—
541	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
542	C₂×C₄	C₂⁴×D₅₃	1	3	yes	—
543	C₂²	—	2	3	no	—
544	C₂²	—	2	12	no	—
545	C₂²	—	2	24	no	—
546	C₂²	—	8	16	no	—
547	C₂²	—	53	6	no	—
548	C₂²	—	106	3	no	—
549	C₂²	—	106	12	no	—
550	C₂²	—	106	24	no	—
551	C₂²	—	424	48	no	—
552	C₂²	—	212	48	no	—
553	C₂²	—	212	4	no	—
554	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
555	C₄	—	2	12	no	—
556	C₄	—	2	3	no	—
557	C₄	—	53	6	no	—
558	C₄	—	106	12	no	—
559	C₄	—	106	3	no	—
560	C₄	H(5,𝔽₂)×D₅₃	1	6	yes	—
561	C₂	—	4	12	no	—
562	C₂	—	4	1	no	—
563	C₂	—	53	4	no	—
564	C₂	—	212	12	no	—
565	C₂	—	212	1	no	—
566	C₂	C₂×H(5,𝔽₂)×D₅₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁶
Fitting	C₅₃×#3549	C₂
hypercenter	#3549	D₅₃
hypocenter	C₅₃	C₂×#3549
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #3549

Complements

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

N	H ≅ G/N	type
1. #20289120	C₂	semidirect
2. #20289119	C₂	semidirect
3. #20289118	C₂	semidirect
4. #20289117	C₂	semidirect
5. #20289116	C₂	semidirect
6. H(3,𝔽₄)×D₅₃	C₂	semidirect
7. #555×D₅₃	C₂	semidirect
8. D₄²×D₅₃	C₂	semidirect
9. #535×D₅₃	C₂	semidirect
10. #532×D₅₃	C₂	semidirect
11. C₅₃×#3549	C₂	semidirect
12. C₂×#41×D₅₃	C₂²	semidirect
13. C₂²×D₄×D₅₃	C₂²	semidirect
14. #16811366	C₂²	semidirect
15. #16811365	C₂²	semidirect
16. #16811365	C₂²	semidirect
17. #16811339	C₂²	semidirect
18. #16811338	C₂²	semidirect
19. #16811337	C₂²	semidirect
20. #16811336	C₂²	semidirect
21. #16811335	C₂²	semidirect
22. D₄×D₂₁₂	C₂²	semidirect
23. #16811258	C₂²	semidirect
24. #16811257	C₂²	semidirect
25. D₄×C₂²⋊D₅₃	C₂²	semidirect
26. #16811255	C₂²	semidirect
27. #16811192	C₂²	semidirect
28. #16811191	C₂²	semidirect
29. #16811190	C₂²	semidirect
30. #16811189	C₂²	semidirect
31. #16811189	C₂²	semidirect
32. #16811188	C₂²	semidirect
33. #16811188	C₂²	semidirect
34. #16811187	C₂²	semidirect
35. #16811186	C₂²	semidirect
36. #16811174	C₂²	semidirect
37. #16811173	C₂²	semidirect
38. #16811172	C₂²	semidirect
39. #16811171	C₂²	semidirect
40. #16811170	C₂²	semidirect
41. #16811169	C₂²	semidirect
42. #16811168	C₂²	semidirect
43. C₄⋊Q₈×D₅₃	C₂²	semidirect
45. #127×D₅₃	C₂²	semidirect
46. #124×D₅₃	C₂²	semidirect
47. #124×D₅₃	C₂²	semidirect
48. #121×D₅₃	C₂²	semidirect
49. #121×D₅₃	C₂²	semidirect
50. #120×D₅₃	C₂²	semidirect
51. #120×D₅₃	C₂²	semidirect
52. C₄×D₄×D₅₃	C₂²	semidirect
53. #117×D₅₃	C₂²	semidirect
54. C₂×C₄²×D₅₃	C₂²	semidirect
55. C₅₃×H(3,𝔽₄)	C₂²	semidirect
56. C₅₃×#555	C₂²	semidirect
57. D₄²×C₅₃	C₂²	semidirect
58. C₅₃×#535	C₂²	semidirect
59. C₅₃×#532	C₂²	semidirect
61. C₂×#11719949	C₂³	semidirect
62. C₂×#11719945	C₂³	semidirect
63. C₂×#11719945	C₂³	semidirect
64. C₂×#11719947	C₂³	semidirect
65. C₂×#11719947	C₂³	semidirect
68. C₂²×D₂₁₂	C₂³	semidirect
69. C₂²×C₂²⋊D₅₃	C₂³	semidirect
75. C₂×D₄×D₅₃	C₂³	semidirect
76. C₂×D₄×D₅₃	C₂³	semidirect
79. #13133217	C₂³	semidirect
80. #13133216	C₂³	semidirect
81. #13133215	C₂³	semidirect
83. #13133214	C₂³	semidirect
86. #13133213	C₂³	semidirect
87. #13133213	C₂³	semidirect
88. #13133206	C₂³	semidirect
89. #13133206	C₂³	semidirect
90. #13133205	C₂³	semidirect
91. #13133205	C₂³	semidirect
92. #13133204	C₂³	semidirect
93. #13133204	C₂³	semidirect
94. #13133203	C₂³	semidirect
95. #13133203	C₂³	semidirect
96. #13133203	C₂³	semidirect
97. #13133192	C₂³	semidirect
98. #13133192	C₂³	semidirect
99. #13133191	C₂³	semidirect
100. #13133191	C₂³	semidirect
101. #13133190	C₂³	semidirect
102. #13133190	C₂³	semidirect
103. #13133189	C₂³	semidirect
104. #13133189	C₂³	semidirect
105. #13133188	C₂³	semidirect
106. #13133188	C₂³	semidirect
107. #13133187	C₂³	semidirect
108. #13133187	C₂³	semidirect
109. #13133187	C₂³	semidirect
110. #13133186	C₂³	semidirect
111. #13133186	C₂³	semidirect
112. #13133185	C₂³	semidirect
113. #13133185	C₂³	semidirect
114. #13133185	C₂³	semidirect
115. C₄×D₂₁₂	C₂³	semidirect
116. D₄×Dic₅₃	C₂³	semidirect
117. #13133179	C₂³	semidirect
118. C₄×C₂²⋊D₅₃	C₂³	semidirect
119. #13133177	C₂³	semidirect
120. #13133176	C₂³	semidirect
121. #13133175	C₂³	semidirect
122. #13133174	C₂³	semidirect
123. #13133173	C₂³	semidirect
127. C₂×C₄×Dic₅₃	C₂³	semidirect
131. C₄²×D₅₃	C₂³	semidirect
132. C₄²×D₅₃	C₂³	semidirect
133. C₂×#41×C₅₃	C₂³	semidirect
134. C₂²×D₄×C₅₃	C₂³	semidirect
135. C₄⋊Q₈×C₅₃	C₂³	semidirect
137. #127×C₅₃	C₂³	semidirect
138. #124×C₅₃	C₂³	semidirect
139. #124×C₅₃	C₂³	semidirect
140. #121×C₅₃	C₂³	semidirect
141. #121×C₅₃	C₂³	semidirect
142. #120×C₅₃	C₂³	semidirect
143. #120×C₅₃	C₂³	semidirect
144. C₄×D₄×C₅₃	C₂³	semidirect
145. #117×C₅₃	C₂³	semidirect
146. C₂×C₄²×C₅₃	C₂³	semidirect
283. #3549	D₅₃	direct (N × H)
294. D₅₃	#3549	direct (N × H)
300. H(3,𝔽₄)	C₂×D₅₃	semidirect
306. #555	C₂×D₅₃	semidirect
312. D₄²	C₂×D₅₃	semidirect
322. #535	C₂×D₅₃	semidirect
330. #532	C₂×D₅₃	semidirect
343. C₅₃	C₂×#3549	semidirect
351. C₂×#41	C₂²×D₅₃	semidirect
362. C₂²×D₄	C₂²×D₅₃	semidirect
368. C₄⋊Q₈	C₂²×D₅₃	semidirect
376. #127	C₂²×D₅₃	semidirect
386. #124	C₂²×D₅₃	semidirect
387. #124	C₂²×D₅₃	semidirect
398. #121	C₂²×D₅₃	semidirect
399. #121	C₂²×D₅₃	semidirect
408. #120	C₂²×D₅₃	semidirect
409. #120	C₂²×D₅₃	semidirect
415. C₄×D₄	C₂²×D₅₃	semidirect
420. #117	C₂²×D₅₃	semidirect
430. C₂×C₄²	C₂²×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20289120	2
#20289119	2
#20289118	2
#20289117	2
#20289116	2
H(3,𝔽₄)×D₅₃	2
#555×D₅₃	2
D₄²×D₅₃	2
#535×D₅₃	2
#532×D₅₃	2
C₅₃×#3549	2
C₂×#3549	53

Subgroups — D53×#3549 #28291890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×#3549 #28291890