Subgroups — #27406607

Order 13568.

880 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20284170	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810742	C₂	1	2	yes	maximal, complemented
3	#20283461	C₂	1	1	yes	maximal, characteristic, complemented
4	#20283283	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275543	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275490	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275427	C₂	1	2	yes	maximal, complemented
8	#20275040	C₂	1	2	yes	maximal, complemented
9	#20274991	C₂	1	2	yes	maximal, complemented
10	#20274568	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#104×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133237	C₂²	1	2	yes	—
13	C₂×#13133225	C₂²	1	2	yes	—
14	C₂×#13133224	C₂²	1	2	yes	—
15	#16811173	C₂²	1	1	yes	characteristic
16	#16810845	C₄	1	2	yes	complemented
17	#16810838	C₄	1	2	yes	complemented
18	#16810833	C₄	1	1	yes	characteristic, complemented
19	#16810828	C₄	1	1	yes	characteristic, complemented
20	#16810823	C₄	1	1	yes	characteristic, complemented
21	#16810818	C₄	1	1	yes	characteristic, complemented
22	#16810742	—	2	8	no	—
23	#16810702	—	2	1	no	—
24	#16810696	—	2	2	no	—
25	C₈×D₂₁₂	—	2	1	no	—
26	C₂×#13133152	C₂²	1	2	yes	—
27	#16810650	—	2	2	no	—
28	#16810650	C₂²	1	2	yes	—
29	#16810647	—	2	1	no	—
30	#16810647	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊#88322	—	2	1	no	—
32	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
34	C₂×#13133145	C₂²	1	2	yes	—
35	C₂×#13133145	C₂²	1	2	yes	—
36	#16810617	C₂²	1	2	yes	—
37	#16810591	C₂²	1	2	yes	—
38	C₂×#13133128	C₂²	1	2	yes	—
39	#16810387	C₂²	1	2	yes	—
40	#16810378	C₂²	1	2	yes	—
41	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
42	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
43	C₂×#104×C₅₃	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
45	C₄³×C₅₃	C₂²	1	1	yes	characteristic
46	C₅₃×#339	C₂²	1	1	yes	characteristic
47	C₅₃×#339	C₂²	1	1	yes	characteristic
48	C₂×#11719945	C₂³	1	2	yes	—
49	#13133237	—	2	4	no	—
50	#13133237	—	2	4	no	—
51	C₂×#11719925	C₂×C₄	1	2	yes	—
52	C₂×#11719923	C₂×C₄	1	2	yes	—
53	C₂×#11719924	C₂×C₄	1	2	yes	—
54	C₂×#11719922	C₂×C₄	1	2	yes	—
55	#13133225	—	2	4	no	—
56	#13133225	—	4	2	no	—
57	#13133224	—	2	4	no	—
58	#13133224	—	4	2	no	—
59	C₂×#11719917	—	2	2	no	—
60	C₂×#11719917	C₂³	1	2	yes	—
61	C₂×#11719912	—	2	2	no	—
62	C₂²×#88322	—	2	2	no	—
63	C₂²×#88322	C₂³	1	2	yes	—
64	C₂²×#88322	C₂³	1	2	yes	—
65	C₂×C₈×D₅₃	—	2	2	no	—
66	#13133217	C₂×C₄	1	1	yes	characteristic
67	#13133214	C₂×C₄	1	1	yes	characteristic
68	#13133205	C₂×C₄	1	2	yes	—
69	#13133191	—	2	1	no	—
70	#13133191	—	2	1	no	—
71	C₄×D₂₁₂	—	2	1	no	—
72	C₄×D₂₁₂	—	2	1	no	—
73	#13133152	—	2	4	no	—
74	C₄⋊#88322	—	2	1	no	—
75	C₄⋊#88322	—	2	1	no	—
76	C₄⋊#88322	—	2	2	no	—
77	C₄⋊#88322	—	2	2	no	—
78	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
79	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
80	C₄⋊#88322	C₂×C₄	1	2	yes	—
81	C₄⋊#88322	D₄	1	1	yes	characteristic
82	C₄⋊#88322	D₄	1	1	yes	characteristic
83	C₄⋊#88322	D₄	1	2	yes	—
84	#13133145	—	2	4	no	—
85	#13133145	—	4	2	no	—
86	#13133140	C₂×C₄	1	2	yes	—
87	#13133140	C₂×C₄	1	2	yes	—
88	#13133137	C₂×C₄	1	2	yes	—
89	#13133137	C₂×C₄	1	2	yes	—
90	#13133129	—	2	2	no	—
91	#13133129	—	2	2	no	—
92	#13133128	—	2	4	no	—
93	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
94	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
96	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₄×D₄×C₅₃	—	2	1	no	—
99	C₄×D₄×C₅₃	—	2	1	no	—
100	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
101	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
102	#117×C₅₃	—	2	1	no	—
103	#117×C₅₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₅₃	—	2	1	no	—
105	C₂×C₄²×C₅₃	—	2	1	no	—
106	C₂×C₄²×C₅₃	—	2	1	no	—
107	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
110	#104×C₅₃	—	2	2	no	—
111	#104×C₅₃	D₄	1	4	yes	complemented
112	#97×C₅₃	—	2	1	no	—
113	C₄×C₈×C₅₃	—	2	1	no	—
114	#41×C₅₃	—	2	1	no	—
115	#41×C₅₃	—	2	1	no	—
116	#41×C₅₃	C₂×D₄	1	2	yes	—
117	#41×C₅₃	#41	1	2	yes	—
118	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×D₄×C₅₃	—	2	1	no	—
120	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	—	2	1	no	—
123	C₂²×C₄×C₅₃	—	2	1	no	—
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	—	2	1	no	—
126	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
127	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
128	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
129	C₂×C₂²⋊D₅₃	—	2	2	no	—
130	#11719945	—	2	2	no	—
131	#11719945	—	2	2	no	—
132	#11719945	—	2	2	no	—
133	#11719945	—	2	2	no	—
134	C₂×D₂₁₂	—	2	1	no	—
135	C₂×D₂₁₂	—	2	1	no	—
136	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
137	C₂×C₄×D₅₃	—	2	2	no	—
138	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
139	M₁₆×C₅₃	—	2	1	no	—
140	M₁₆×C₅₃	—	2	1	no	—
141	M₁₆×C₅₃	—	2	1	no	—
142	M₁₆×C₅₃	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₅₃	—	2	1	no	—
144	C₂×C₈×C₅₃	—	2	1	no	—
145	C₄⋊C₄×C₅₃	—	2	1	no	—
146	C₄⋊C₄×C₅₃	—	2	1	no	—
147	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²⋊C₄×C₅₃	—	2	1	no	—
150	C₂²⋊C₄×C₅₃	—	2	1	no	—
151	C₂²⋊C₄×C₅₃	—	2	1	no	—
152	C₄²×C₅₃	—	2	1	no	—
153	C₄²×C₅₃	—	2	1	no	—
154	C₄²×C₅₃	—	2	1	no	—
155	C₄²×C₅₃	—	2	1	no	—
156	C₄²×C₅₃	—	2	1	no	—
157	C₄²×C₅₃	—	2	1	no	—
158	C₄²×C₅₃	—	4	2	no	—
159	C₄²×C₅₃	—	4	2	no	—
160	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
162	C₄²×C₅₃	C₂×D₄	1	2	yes	—
163	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
165	#11719925	—	2	4	no	—
166	#11719924	—	2	4	no	—
167	#11719923	—	2	4	no	—
168	#11719922	—	2	4	no	—
169	#11719921	—	2	2	no	—
170	#11719921	—	2	2	no	—
171	C₄⋊Dic₅₃	—	2	1	no	—
172	C₄⋊Dic₅₃	—	2	1	no	—
173	#11719917	—	2	2	no	—
174	#11719917	—	2	2	no	—
175	#11719917	—	4	4	no	—
176	C₂×#88322	—	2	2	no	—
177	C₂×#88322	—	2	2	no	—
178	C₂×#88322	—	2	2	no	—
179	C₂×#88322	—	2	2	no	—
180	C₂×#88322	—	2	2	no	—
181	C₂×#88322	—	4	4	no	—
182	C₂×#88322	C₂²×C₄	1	2	yes	—
183	C₂×#88322	C₂²×C₄	1	2	yes	—
184	C₂×#88322	C₂×D₄	1	2	yes	—
185	C₂×#88322	C₂×D₄	1	2	yes	—
186	#11719912	—	4	4	no	—
187	C₈×D₅₃	—	4	4	no	—
188	C₂³×C₅₃	—	2	1	no	—
189	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
190	C₂²×D₅₃	—	2	2	no	—
191	Q₈×C₅₃	—	2	1	no	—
192	Q₈×C₅₃	C₄×D₄	1	2	yes	—
193	D₄×C₅₃	—	2	2	no	—
194	D₄×C₅₃	—	2	2	no	—
195	D₄×C₅₃	—	2	1	no	—
196	D₄×C₅₃	C₄×D₄	1	2	yes	—
197	C₂×C₄×C₅₃	—	2	1	no	—
198	C₂×C₄×C₅₃	—	2	2	no	—
199	C₂×C₄×C₅₃	—	2	2	no	—
200	C₂×C₄×C₅₃	—	2	2	no	—
201	C₂×C₄×C₅₃	—	2	2	no	—
202	C₂×C₄×C₅₃	—	2	1	no	—
203	C₂×C₄×C₅₃	—	2	1	no	—
204	C₂×C₄×C₅₃	—	2	2	no	—
205	C₂×C₄×C₅₃	—	4	1	no	—
206	C₂×C₄×C₅₃	—	4	1	no	—
207	C₂×C₄×C₅₃	—	4	1	no	—
208	C₂×C₄×C₅₃	—	4	1	no	—
209	C₂×C₄×C₅₃	—	4	1	no	—
210	C₂×C₄×C₅₃	—	4	2	no	—
211	C₂×C₄×C₅₃	—	8	1	no	—
212	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
213	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
216	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
217	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
221	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
222	C₂²⋊D₅₃	—	4	2	no	—
223	C₂²⋊D₅₃	—	4	2	no	—
224	C₂×Dic₅₃	—	2	2	no	—
225	D₂₁₂	—	2	2	no	—
226	D₂₁₂	—	2	2	no	—
227	D₂₁₂	—	4	2	no	—
228	D₂₁₂	—	8	1	no	—
229	C₄×D₅₃	—	4	2	no	—
230	C₄×D₅₃	—	4	2	no	—
231	Dic₁₀₆	—	2	2	no	—
232	Dic₁₀₆	—	2	2	no	—
233	C₈×C₅₃	—	2	2	no	—
234	C₈×C₅₃	—	4	1	no	—
235	#88322	—	2	2	no	—
236	#88322	—	2	2	no	—
237	#88322	—	2	2	no	—
238	#88322	—	2	2	no	—
239	#88322	—	4	2	no	—
240	#23016	—	53	1	no	maximal, Sylow
241	C₂²×C₅₃	—	2	1	no	—
242	C₂²×C₅₃	—	2	1	no	—
243	C₂²×C₅₃	—	2	2	no	—
244	C₂²×C₅₃	—	4	1	no	—
245	C₂²×C₅₃	—	4	1	no	—
246	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
247	C₂²×C₅₃	#547	1	1	yes	characteristic
248	C₂²×C₅₃	#553	1	1	yes	characteristic
249	C₂×D₅₃	—	4	2	no	—
250	C₂×D₅₃	—	4	2	no	—
251	C₂×D₅₃	—	4	2	no	—
252	C₄×C₅₃	—	2	2	no	—
253	C₄×C₅₃	—	2	1	no	—
254	C₄×C₅₃	—	2	1	no	—
255	C₄×C₅₃	—	2	1	no	—
256	C₄×C₅₃	—	4	2	no	—
257	C₄×C₅₃	—	4	2	no	—
258	C₄×C₅₃	—	4	1	no	—
259	C₄×C₅₃	—	4	1	no	—
260	C₄×C₅₃	#444	1	2	yes	complemented
261	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
262	C₄×C₅₃	#520	1	1	yes	characteristic
263	C₄×C₅₃	D₄²	1	1	yes	characteristic
264	C₄×C₅₃	#548	1	1	yes	characteristic
265	Dic₅₃	—	4	2	no	—
266	#3035	—	53	1	no	—
267	C₂×#444	—	53	2	no	—
268	#2935	—	53	1	no	—
269	#2906	—	53	1	no	—
270	#1962	—	53	1	no	—
271	#1950	—	53	1	no	—
272	#1938	—	53	2	no	—
273	#1876	—	53	2	no	—
274	#1866	—	53	2	no	—
275	#1767	—	53	1	no	—
276	C₄×#104	D₅₃	1	1	yes	characteristic, complemented
277	C₂×C₅₃	—	2	1	no	—
278	C₂×C₅₃	—	2	1	no	—
279	C₂×C₅₃	—	4	2	no	—
280	C₂×C₅₃	#2361	1	1	yes	characteristic
281	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
282	C₂×C₅₃	#2956	1	1	yes	characteristic
283	D₅₃	—	8	2	no	—
284	C₂×#135	—	53	2	no	—
285	C₂×#131	—	53	2	no	—
286	C₂×#131	—	53	2	no	—
287	#532	—	53	1	no	—
288	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
289	#464	—	53	2	no	—
290	#463	—	53	2	no	—
291	#462	—	53	1	no	—
292	#461	—	53	1	no	—
293	#460	—	53	1	no	—
294	#459	—	53	1	no	—
295	#444	—	106	8	no	—
296	#436	—	106	1	no	—
297	#435	—	106	2	no	—
298	C₈×D₄	—	106	1	no	—
299	C₂×#108	—	53	2	no	—
300	#424	—	53	2	no	—
301	#424	—	106	2	no	—
302	#423	—	53	1	no	—
303	#423	—	106	1	no	—
304	C₂×C₄⋊C₈	—	53	1	no	—
305	C₂×C₄⋊C₈	—	53	1	no	—
306	C₂×C₄⋊C₈	—	106	1	no	—
307	C₂×#104	—	53	2	no	—
308	C₂×#104	—	53	2	no	—
309	C₂×#104	C₂×D₅₃	1	1	yes	characteristic
310	C₂×#104	C₂×D₅₃	1	1	yes	characteristic
311	#416	—	53	2	no	—
312	#408	—	53	2	no	—
313	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
314	C₂×#97	—	53	2	no	—
315	C₄³	C₂×D₅₃	1	1	yes	characteristic
316	#341	—	53	2	no	—
317	#339	C₂×D₅₃	1	1	yes	characteristic
318	#339	C₂×D₅₃	1	1	yes	characteristic
319	#336	—	53	2	no	—
320	C₅₃	#23016	1	1	yes	characteristic, complemented, Sylow
321	C₂×#41	—	53	2	no	—
322	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
323	#135	—	106	4	no	—
324	#135	—	106	4	no	—
325	C₂×Q₁₆	—	53	2	no	—
326	C₂×SD₁₆	—	53	2	no	—
327	C₂×SD₁₆	—	53	2	no	—
328	C₂×D₈	—	53	2	no	—
329	#131	—	106	4	no	—
330	#131	—	106	4	no	—
331	#131	—	212	2	no	—
332	#131	—	212	2	no	—
333	C₂×M₁₆	—	53	2	no	—
334	C₂×M₁₆	—	106	2	no	—
335	C₂×M₁₆	—	106	2	no	—
336	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
337	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
338	C₂²×C₈	—	53	2	no	—
339	C₂²×C₈	—	53	2	no	—
340	C₂²×C₈	—	106	2	no	—
341	C₂²×C₈	—	106	2	no	—
342	C₄⋊Q₈	—	53	1	no	—
343	#127	—	53	1	no	—
344	#124	—	53	2	no	—
345	#121	—	106	1	no	—
346	#121	—	106	1	no	—
347	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
348	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
349	C₄×D₄	—	2	1	no	—
350	C₄×D₄	—	2	1	no	—
351	C₄×D₄	—	106	1	no	—
352	C₄×D₄	—	106	1	no	—
353	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
354	C₄×D₄	C₄×D₅₃	1	1	yes	characteristic
355	#117	—	2	1	no	—
356	#117	C₂²×D₅₃	1	1	yes	characteristic
357	C₂×C₄²	—	2	1	no	—
358	C₂×C₄²	—	2	1	no	—
359	C₂×C₄²	—	2	1	no	—
360	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
361	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
362	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
363	#108	—	106	4	no	—
364	C₄⋊C₈	—	53	1	no	—
365	C₄⋊C₈	—	53	1	no	—
366	C₄⋊C₈	—	53	2	no	—
367	C₄⋊C₈	—	53	1	no	—
368	C₄⋊C₈	—	53	1	no	—
369	C₄⋊C₈	—	53	2	no	—
370	C₄⋊C₈	—	106	2	no	—
371	C₄⋊C₈	—	106	2	no	—
372	C₄⋊C₈	—	106	1	no	—
373	C₄⋊C₈	—	106	1	no	—
374	#104	—	2	2	no	—
375	#104	—	106	4	no	—
376	#104	—	212	2	no	—
377	#104	D₂₁₂	1	4	yes	complemented
378	#103	—	53	2	no	—
379	#103	—	53	2	no	—
380	#102	—	53	2	no	—
381	#102	—	53	2	no	—
382	C₂²⋊C₈	—	106	2	no	—
383	C₂²⋊C₈	—	106	2	no	—
384	#97	—	2	1	no	—
385	#97	—	106	4	no	—
386	C₄×C₈	—	2	1	no	—
387	#41	—	2	1	no	—
388	#41	—	2	1	no	—
389	#41	—	106	2	no	—
390	#41	—	106	2	no	—
391	#41	—	106	2	no	—
392	#41	—	106	2	no	—
393	#41	C₂×D₂₁₂	1	2	yes	—
394	#41	#11719945	1	2	yes	—
395	C₂×Q₈	—	53	2	no	—
396	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
397	C₂×D₄	—	2	1	no	—
398	C₂×D₄	—	53	2	no	—
399	C₂×D₄	—	106	2	no	—
400	C₂×D₄	—	106	1	no	—
401	C₂×D₄	—	106	1	no	—
402	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	106	2	no	—
409	C₂²×C₄	#11719947	1	1	yes	characteristic
410	C₂²×C₄	#11719947	1	1	yes	characteristic
411	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
412	Q₁₆	—	106	4	no	—
413	SD₁₆	—	106	4	no	—
414	SD₁₆	—	106	4	no	—
415	D₈	—	106	4	no	—
416	M₁₆	—	2	1	no	—
417	M₁₆	—	2	1	no	—
418	M₁₆	—	2	1	no	—
419	M₁₆	—	106	2	no	—
420	M₁₆	—	106	2	no	—
421	M₁₆	—	212	4	no	—
422	M₁₆	—	212	4	no	—
423	M₁₆	C₂×D₂₁₂	1	2	yes	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	1	no	—
426	C₂×C₈	—	53	2	no	—
427	C₂×C₈	—	53	2	no	—
428	C₂×C₈	—	53	2	no	—
429	C₂×C₈	—	53	2	no	—
430	C₂×C₈	—	106	2	no	—
431	C₂×C₈	—	106	2	no	—
432	C₂×C₈	—	106	2	no	—
433	C₂×C₈	—	106	2	no	—
434	C₂×C₈	—	106	2	no	—
435	C₂×C₈	—	212	4	no	—
436	C₂×C₈	—	212	4	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	106	1	no	—
440	C₄⋊C₄	—	106	1	no	—
441	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
442	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
443	C₂²⋊C₄	—	2	1	no	—
444	C₂²⋊C₄	—	2	1	no	—
445	C₂²⋊C₄	—	2	1	no	—
446	C₂²⋊C₄	—	106	2	no	—
447	C₂²⋊C₄	—	106	2	no	—
448	C₄²	—	2	1	no	—
449	C₄²	—	2	1	no	—
450	C₄²	—	2	1	no	—
451	C₄²	—	2	1	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	1	no	—
454	C₄²	—	4	2	no	—
455	C₄²	—	4	2	no	—
456	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
457	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
458	C₄²	C₂×D₂₁₂	1	2	yes	—
459	C₄²	D₄×D₅₃	1	1	yes	characteristic
460	C₄²	D₄×D₅₃	1	1	yes	characteristic
461	C₂³	—	2	1	no	—
462	C₂³	—	106	2	no	—
463	C₂³	C₂×#11719947	1	1	yes	characteristic
464	Q₈	—	2	1	no	—
465	Q₈	—	106	2	no	—
466	Q₈	—	106	2	no	—
467	Q₈	C₄×D₂₁₂	1	2	yes	—
468	D₄	—	2	2	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	1	no	—
471	D₄	—	106	2	no	—
472	D₄	—	106	2	no	—
473	D₄	—	424	1	no	—
474	D₄	—	212	2	no	—
475	D₄	—	212	2	no	—
476	D₄	—	212	2	no	—
477	D₄	C₄×D₂₁₂	1	2	yes	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	8	1	no	—
493	C₂×C₄	—	106	2	no	—
494	C₂×C₄	—	212	2	no	—
495	C₂×C₄	—	212	2	no	—
496	C₂×C₄	#13133177	1	1	yes	characteristic
497	C₂×C₄	#13133177	1	1	yes	characteristic
498	C₂×C₄	#13133177	1	1	yes	characteristic
499	C₂×C₄	#13133177	1	1	yes	characteristic
500	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
501	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
502	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
503	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
504	C₂×C₄	#13133263	1	1	yes	characteristic
505	C₂×C₄	#13133264	1	1	yes	characteristic
506	C₈	—	2	2	no	—
507	C₈	—	4	1	no	—
508	C₈	—	106	2	no	—
509	C₈	—	106	2	no	—
510	C₈	—	106	2	no	—
511	C₈	—	106	2	no	—
512	C₈	—	212	2	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	1	no	—
518	C₂²	—	212	2	no	—
519	C₂²	—	212	2	no	—
520	C₂²	—	212	2	no	—
521	C₂²	C₂×#13133177	1	1	yes	characteristic
522	C₂²	#16811277	1	1	yes	characteristic
523	C₂²	#16811319	1	1	yes	characteristic
524	C₄	—	2	1	no	—
525	C₄	—	2	2	no	—
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	1	no	—
531	C₄	—	4	1	no	—
532	C₄	—	212	2	no	—
533	C₄	#16810742	1	2	yes	complemented
534	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
535	C₄	#16811103	1	1	yes	characteristic
536	C₄	D₄×D₂₁₂	1	1	yes	characteristic
537	C₄	#16811282	1	1	yes	characteristic
538	C₂	—	2	1	no	—
539	C₂	—	2	1	no	—
540	C₂	—	4	2	no	—
541	C₂	—	424	2	no	—
542	C₂	#20277729	1	1	yes	characteristic
543	C₂	C₂×#16810742	1	1	yes	characteristic
544	C₂	#20283621	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#104×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	#20284170	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#104	D₅₃
hypocenter	C₅₃	#23016
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#104

Complements

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

N	H ≅ G/N	type
2. C₂×#16810742	C₂	semidirect
3. #20283461	C₂	semidirect
4. #20283283	C₂	semidirect
5. #20275543	C₂	semidirect
6. #20275490	C₂	semidirect
7. #20275427	C₂	semidirect
8. #20275040	C₂	semidirect
9. #20274991	C₂	semidirect
10. #20274568	C₂	semidirect
11. C₄×#104×C₅₃	C₂	semidirect
16. #16810845	C₄	semidirect
17. #16810838	C₄	semidirect
18. #16810833	C₄	semidirect
19. #16810828	C₄	semidirect
20. #16810823	C₄	semidirect
21. #16810818	C₄	semidirect
111. #104×C₅₃	D₄	semidirect
260. C₄×C₅₃	#444	semidirect
276. C₄×#104	D₅₃	semidirect
320. C₅₃	#23016	semidirect
377. #104	D₂₁₂	semidirect
533. C₄	#16810742	semidirect

Maximal subgroups

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

subgroup	index
#20284170	2
C₂×#16810742	2
#20283461	2
#20283283	2
#20275543	2
#20275490	2
#20275427	2
#20275040	2
#20274991	2
#20274568	2
C₄×#104×C₅₃	2
#23016	53