Subgroups — #27403557

Order 13568.

752 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133239	C₂	1	1	yes	maximal, characteristic
2	#20283380	C₂	1	1	yes	maximal, characteristic, complemented
3	#20283280	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275419	C₂	1	2	yes	maximal, complemented
5	#20275315	C₂	1	2	yes	maximal, complemented
6	#20275041	C₂	1	2	yes	maximal
7	#20274853	C₂	1	1	yes	maximal, characteristic, complemented
8	#20274736	C₂	1	1	yes	maximal, characteristic, complemented
9	#20274736	C₂	1	1	yes	maximal, characteristic, complemented
10	#20274552	C₂	1	2	yes	maximal, complemented
11	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133239	C₂²	1	1	yes	characteristic
13	C₂×#13133223	C₂²	1	2	yes	—
14	C₄×#11719945	C₂²	1	1	yes	characteristic
15	C₄×#11719945	C₂²	1	1	yes	characteristic
16	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic, complemented
17	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic, complemented
18	C₄×#11719913	C₄	1	1	yes	characteristic, complemented
19	C₄×#11719913	C₄	1	1	yes	characteristic, complemented
20	C₄×#11719913	C₄	1	1	yes	characteristic, complemented
21	C₄×#11719913	C₄	1	1	yes	characteristic, complemented
22	C₄×D₄₂₄	C₄	1	1	yes	characteristic, complemented
23	C₄×D₄₂₄	C₄	1	1	yes	characteristic, complemented
24	#16810683	—	2	2	no	—
25	#16810679	—	2	1	no	—
26	#16810675	—	2	1	no	—
27	#16810661	C₂²	1	1	yes	characteristic
28	#16810649	C₂²	1	2	yes	—
29	#16810648	—	2	2	no	—
30	#16810647	C₂²	1	1	yes	characteristic
31	#16810647	C₂²	1	1	yes	characteristic
32	#16810645	—	2	1	no	—
33	C₄⋊C₈×D₅₃	—	2	1	no	—
34	C₂×#13133145	C₂²	1	2	yes	—
35	C₂×#13133145	C₂²	1	2	yes	—
36	#16810620	C₂²	1	1	yes	characteristic
37	#16810620	C₂²	1	1	yes	characteristic
38	#16810589	C₂²	1	2	yes	—
39	#16810578	C₂²	1	2	yes	—
40	#16810388	C₂²	1	2	yes	—
41	#16810388	C₂²	1	2	yes	—
42	#16810383	C₂²	1	2	yes	—
43	#16810383	C₂²	1	2	yes	—
44	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
45	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
48	C₅₃×#336	C₂²	1	2	yes	—
49	C₂×#11719945	C₂³	1	1	yes	characteristic
50	C₂×#11719945	C₂³	1	1	yes	characteristic
51	#13133239	—	2	2	no	—
52	#13133239	—	2	2	no	—
53	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719913	C₂×C₄	1	1	yes	characteristic
55	C₂×#11719913	C₂×C₄	1	1	yes	characteristic
56	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
57	#13133223	—	2	4	no	—
58	#13133223	—	4	2	no	—
59	C₂×#11719917	C₂³	1	2	yes	—
60	C₂×#11719917	C₂³	1	2	yes	—
61	C₂×#11719912	—	2	2	no	—
62	C₂×C₈×D₅₃	—	2	2	no	—
63	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
67	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
68	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
69	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
70	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
71	C₄×C₂²⋊D₅₃	—	2	1	no	—
72	C₄×C₂²⋊D₅₃	—	2	1	no	—
73	C₄×C₂²⋊D₅₃	—	2	1	no	—
74	C₄×C₂²⋊D₅₃	—	2	1	no	—
75	#13133176	C₂³	1	1	yes	characteristic
76	#13133176	C₂³	1	1	yes	characteristic
77	#13133174	—	2	1	no	—
78	#13133174	—	2	1	no	—
79	C₄²×D₅₃	—	2	1	no	—
80	C₄²×D₅₃	—	2	1	no	—
81	#13133151	C₂×C₄	1	1	yes	characteristic
82	#13133151	C₂×C₄	1	1	yes	characteristic
83	#13133149	C₂×C₄	1	1	yes	characteristic
84	#13133149	C₂×C₄	1	1	yes	characteristic
85	C₄⋊#88322	—	2	1	no	—
86	C₄⋊#88322	—	2	1	no	—
87	C₄⋊#88322	—	2	1	no	—
88	C₄⋊#88322	—	2	1	no	—
89	#13133146	—	2	2	no	—
90	#13133146	—	2	2	no	—
91	#13133145	—	2	4	no	—
92	#13133145	—	4	2	no	—
93	#13133142	C₂×C₄	1	1	yes	characteristic
94	#13133142	C₂×C₄	1	1	yes	characteristic
95	#13133142	C₂×C₄	1	1	yes	characteristic
96	#13133142	C₂×C₄	1	1	yes	characteristic
97	#13133139	C₂×C₄	1	1	yes	characteristic
98	#13133139	C₂×C₄	1	1	yes	characteristic
99	#13133139	C₂×C₄	1	1	yes	characteristic
100	#13133139	C₂×C₄	1	1	yes	characteristic
101	#13133129	—	2	2	no	—
102	#13133129	—	2	2	no	—
103	#13133127	—	2	2	no	—
104	C₈×Dic₅₃	—	2	2	no	—
105	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
106	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
107	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
109	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
110	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
111	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
112	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
113	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
114	C₄×C₈×C₅₃	—	4	2	no	—
115	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
116	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
117	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
118	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
119	#95×C₅₃	—	2	1	no	—
120	C₂²×C₄×C₅₃	—	2	2	no	—
121	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
122	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
123	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
124	C₂×C₂²⋊D₅₃	—	2	1	no	—
125	C₂×C₂²⋊D₅₃	—	2	1	no	—
126	#11719945	—	2	1	no	—
127	#11719945	—	2	1	no	—
128	#11719945	—	2	1	no	—
129	#11719945	—	2	1	no	—
130	#11719945	—	2	1	no	—
131	#11719945	—	2	1	no	—
132	#11719945	#41	1	2	yes	—
133	#11719945	#41	1	2	yes	—
134	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
135	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₄×D₅₃	—	2	1	no	—
137	C₂×C₄×D₅₃	—	2	1	no	—
138	C₂×C₄×D₅₃	—	2	1	no	—
139	C₂×C₄×D₅₃	—	2	1	no	—
140	C₂×C₄×D₅₃	—	2	1	no	—
141	C₂×C₄×D₅₃	—	2	1	no	—
142	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
143	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₈×C₅₃	—	2	2	no	—
145	C₂×C₈×C₅₃	—	2	1	no	—
146	C₂×C₈×C₅₃	—	2	1	no	—
147	C₂×C₈×C₅₃	—	2	1	no	—
148	C₂×C₈×C₅₃	—	2	2	no	—
149	C₂×C₈×C₅₃	—	2	1	no	—
150	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₅₃	—	2	4	no	—
157	C₄²×C₅₃	—	2	2	no	—
158	C₄²×C₅₃	—	2	2	no	—
159	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
160	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
161	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
162	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
163	C₂²⋊Dic₅₃	—	2	1	no	—
164	C₂²⋊Dic₅₃	—	2	1	no	—
165	#11719921	—	2	1	no	—
166	#11719921	—	2	1	no	—
167	#11719921	—	2	1	no	—
168	#11719921	—	2	1	no	—
169	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
171	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
172	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
173	#11719919	—	2	1	no	—
174	#11719919	—	2	1	no	—
175	#11719919	—	2	1	no	—
176	#11719919	—	2	1	no	—
177	C₄×Dic₅₃	—	2	1	no	—
178	C₄×Dic₅₃	—	2	1	no	—
179	C₄×Dic₅₃	—	2	1	no	—
180	C₄×Dic₅₃	—	2	1	no	—
181	C₄×Dic₅₃	—	2	1	no	—
182	C₄×Dic₅₃	—	2	1	no	—
183	#11719917	—	2	2	no	—
184	#11719917	—	2	2	no	—
185	#11719917	—	2	2	no	—
186	#11719917	—	2	2	no	—
187	C₂×#88322	—	2	2	no	—
188	C₂×#88322	—	2	2	no	—
189	Dic₂₁₂	—	2	2	no	—
190	D₄₂₄	—	2	2	no	—
191	#11719913	—	2	2	no	—
192	#11719913	—	2	2	no	—
193	#11719912	—	4	4	no	—
194	C₈×D₅₃	—	4	4	no	—
195	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
196	C₂²×D₅₃	—	2	1	no	—
197	C₂²×D₅₃	—	2	1	no	—
198	C₂×C₄×C₅₃	—	2	2	no	—
199	C₂×C₄×C₅₃	—	2	4	no	—
200	C₂×C₄×C₅₃	—	4	1	no	—
201	C₂×C₄×C₅₃	—	4	2	no	—
202	C₂×C₄×C₅₃	—	4	2	no	—
203	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
206	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
208	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
210	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
211	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
212	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
213	C₂²⋊D₅₃	—	2	2	no	—
214	C₂²⋊D₅₃	—	2	2	no	—
215	C₂²⋊D₅₃	—	4	1	no	—
216	C₂²⋊D₅₃	—	4	1	no	—
217	C₂×Dic₅₃	—	2	1	no	—
218	C₂×Dic₅₃	—	2	1	no	—
219	C₂×Dic₅₃	—	2	1	no	—
220	C₂×Dic₅₃	—	2	1	no	—
221	C₂×Dic₅₃	—	2	1	no	—
222	C₂×Dic₅₃	—	2	1	no	—
223	D₂₁₂	—	2	1	no	—
224	D₂₁₂	—	2	1	no	—
225	D₂₁₂	—	2	1	no	—
226	D₂₁₂	#117	1	2	yes	—
227	C₄×D₅₃	—	2	2	no	—
228	C₄×D₅₃	—	2	2	no	—
229	C₄×D₅₃	—	4	1	no	—
230	C₄×D₅₃	—	4	1	no	—
231	C₄×D₅₃	—	4	2	no	—
232	C₄×D₅₃	—	4	2	no	—
233	C₄×D₅₃	—	8	1	no	—
234	C₄×D₅₃	—	8	1	no	—
235	Dic₁₀₆	—	2	1	no	—
236	Dic₁₀₆	—	2	1	no	—
237	Dic₁₀₆	—	2	1	no	—
238	Dic₁₀₆	#117	1	2	yes	—
239	C₈×C₅₃	—	2	1	no	—
240	C₈×C₅₃	—	2	1	no	—
241	C₈×C₅₃	—	2	1	no	—
242	C₈×C₅₃	—	2	1	no	—
243	C₈×C₅₃	—	2	2	no	—
244	C₈×C₅₃	—	2	2	no	—
245	#88322	—	4	2	no	—
246	#88322	—	4	2	no	—
247	#22783	—	53	1	no	maximal, Sylow
248	C₂²×C₅₃	—	2	1	no	—
249	C₂²×C₅₃	—	2	1	no	—
250	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
251	C₂²×C₅₃	#552	1	1	yes	characteristic
252	C₂²×C₅₃	#553	1	1	yes	characteristic
253	C₂×D₅₃	—	2	2	no	—
254	C₂×D₅₃	—	4	1	no	—
255	C₂×D₅₃	—	4	1	no	—
256	C₂×D₅₃	—	4	1	no	—
257	C₂×D₅₃	—	4	1	no	—
258	C₂×D₅₃	—	4	1	no	—
259	C₄×C₅₃	—	2	1	no	—
260	C₄×C₅₃	—	2	1	no	—
261	C₄×C₅₃	—	2	1	no	—
262	C₄×C₅₃	—	4	4	no	—
263	C₄×C₅₃	#443	1	2	yes	—
264	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
265	C₄×C₅₃	#520	1	1	yes	characteristic
266	C₄×C₅₃	#546	1	1	yes	characteristic
267	C₄×C₅₃	#548	1	1	yes	characteristic
268	Dic₅₃	—	2	2	no	—
269	Dic₅₃	—	4	1	no	—
270	Dic₅₃	—	4	1	no	—
271	Dic₅₃	—	4	1	no	—
272	Dic₅₃	—	4	1	no	—
273	Dic₅₃	—	4	1	no	—
274	C₄×#135	—	53	1	no	—
275	#2925	—	53	1	no	—
276	#2906	—	53	1	no	—
277	#1937	—	53	2	no	—
278	#1922	—	53	2	no	—
279	#1876	—	53	2	no	—
280	#1837	—	53	1	no	—
281	#1806	—	53	1	no	—
282	#1806	—	53	1	no	—
283	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
284	#1764	—	53	2	no	—
285	C₂×C₅₃	—	2	1	no	—
286	C₂×C₅₃	—	2	1	no	—
287	C₂×C₅₃	#2352	1	1	yes	characteristic
288	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
289	C₂×C₅₃	#2957	1	1	yes	characteristic
290	D₅₃	—	4	2	no	—
291	D₅₃	—	8	1	no	—
292	C₂×#135	—	53	1	no	—
293	C₂×#131	—	53	2	no	—
294	C₄×#41	—	53	1	no	—
295	C₄×#41	—	53	1	no	—
296	C₄×Q₁₆	—	53	1	no	—
297	C₄×Q₁₆	—	53	1	no	—
298	C₄×SD₁₆	—	53	1	no	—
299	C₄×SD₁₆	—	53	1	no	—
300	C₄×SD₁₆	—	53	1	no	—
301	C₄×SD₁₆	—	53	1	no	—
302	C₄×D₈	—	53	1	no	—
303	C₄×D₈	—	53	1	no	—
304	#433	—	106	2	no	—
305	#432	—	106	1	no	—
306	#431	—	106	1	no	—
307	#427	—	53	1	no	—
308	#424	—	53	2	no	—
309	#424	—	106	2	no	—
310	#423	—	53	1	no	—
311	#423	—	53	1	no	—
312	#423	—	106	1	no	—
313	C₂×C₄⋊C₈	—	106	1	no	—
314	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
315	C₂×#104	—	53	2	no	—
316	C₂×#104	—	53	2	no	—
317	#416	—	53	1	no	—
318	#416	—	53	1	no	—
319	#408	—	53	2	no	—
320	#405	—	53	2	no	—
321	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
322	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
323	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
324	#341	—	53	2	no	—
325	#341	—	53	2	no	—
326	#339	—	53	2	no	—
327	#339	—	53	2	no	—
328	#336	C₂×D₅₃	1	2	yes	—
329	C₅₃	#22783	1	1	yes	characteristic, complemented, Sylow
330	C₂×#41	—	53	1	no	—
331	C₂×#41	—	53	1	no	—
332	#135	—	106	2	no	—
333	#135	—	106	2	no	—
334	C₂×Q₁₆	—	53	1	no	—
335	C₂×SD₁₆	—	53	1	no	—
336	C₂×SD₁₆	—	53	1	no	—
337	C₂×D₈	—	53	1	no	—
338	#131	—	106	4	no	—
339	#131	—	212	2	no	—
340	C₂×M₁₆	—	53	2	no	—
341	C₂×M₁₆	—	53	2	no	—
342	C₂×M₁₆	—	106	2	no	—
343	C₂²×C₈	—	106	2	no	—
344	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
345	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
346	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
347	C₄×Q₈	—	53	1	no	—
348	C₄×Q₈	—	53	1	no	—
349	C₄×Q₈	—	53	1	no	—
350	C₄×Q₈	—	53	1	no	—
351	C₄×D₄	—	53	1	no	—
352	C₄×D₄	—	53	1	no	—
353	C₄×D₄	—	53	1	no	—
354	C₄×D₄	—	53	1	no	—
355	C₄×D₄	—	106	1	no	—
356	C₄×D₄	—	106	1	no	—
357	C₄×D₄	—	106	1	no	—
358	C₄×D₄	—	106	1	no	—
359	#117	—	53	1	no	—
360	#117	—	53	1	no	—
361	#117	—	106	1	no	—
362	#117	—	106	1	no	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄²	—	106	1	no	—
365	C₂×C₄²	—	106	1	no	—
366	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
367	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
368	#107	—	53	1	no	—
369	#107	—	53	1	no	—
370	#106	—	53	1	no	—
371	#106	—	53	1	no	—
372	C₄⋊C₈	—	106	2	no	—
373	C₄⋊C₈	—	106	2	no	—
374	C₄⋊C₈	—	106	1	no	—
375	C₄⋊C₈	—	106	1	no	—
376	C₄⋊C₈	—	106	1	no	—
377	C₄⋊C₈	—	106	1	no	—
378	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
379	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
380	C₄⋊C₈	D₂₁₂	1	2	yes	—
381	#104	—	106	4	no	—
382	#104	—	212	2	no	—
383	#103	—	53	1	no	—
384	#103	—	53	1	no	—
385	#103	—	53	1	no	—
386	#103	—	53	1	no	—
387	#102	—	53	1	no	—
388	#102	—	53	1	no	—
389	#102	—	53	1	no	—
390	#102	—	53	1	no	—
391	C₂²⋊C₈	—	106	2	no	—
392	C₂²⋊C₈	—	106	2	no	—
393	#97	—	106	2	no	—
394	C₄×C₈	—	4	2	no	—
395	C₄×C₈	—	106	2	no	—
396	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
397	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
398	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
399	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
400	#95	—	2	1	no	—
401	#41	—	53	2	no	—
402	#41	—	53	2	no	—
403	#41	—	106	1	no	—
404	#41	—	106	1	no	—
405	#41	—	106	1	no	—
406	#41	—	106	1	no	—
407	#41	—	106	1	no	—
408	#41	—	106	1	no	—
409	C₂×Q₈	—	53	1	no	—
410	C₂×Q₈	—	53	1	no	—
411	C₂×D₄	—	53	1	no	—
412	C₂×D₄	—	53	1	no	—
413	C₂×D₄	—	106	1	no	—
414	C₂×D₄	—	106	1	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	106	1	no	—
417	C₂²×C₄	—	106	1	no	—
418	C₂²×C₄	—	106	1	no	—
419	C₂²×C₄	—	106	1	no	—
420	C₂²×C₄	—	106	1	no	—
421	C₂²×C₄	—	106	1	no	—
422	C₂²×C₄	#11719945	1	1	yes	characteristic
423	C₂²×C₄	#11719945	1	1	yes	characteristic
424	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
425	Q₁₆	—	106	2	no	—
426	SD₁₆	—	106	2	no	—
427	SD₁₆	—	106	2	no	—
428	D₈	—	106	2	no	—
429	M₁₆	—	106	2	no	—
430	M₁₆	—	106	2	no	—
431	M₁₆	—	106	2	no	—
432	M₁₆	—	106	2	no	—
433	M₁₆	—	212	4	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	106	2	no	—
441	C₂×C₈	—	106	2	no	—
442	C₂×C₈	—	212	4	no	—
443	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
444	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
445	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
446	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
447	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
448	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	53	1	no	—
451	C₄⋊C₄	—	53	1	no	—
452	C₄⋊C₄	—	53	1	no	—
453	C₄⋊C₄	—	53	1	no	—
454	C₄⋊C₄	—	106	1	no	—
455	C₄⋊C₄	—	106	1	no	—
456	C₄⋊C₄	—	106	1	no	—
457	C₄⋊C₄	—	106	1	no	—
458	C₂²⋊C₄	—	106	1	no	—
459	C₂²⋊C₄	—	106	1	no	—
460	C₂²⋊C₄	—	106	1	no	—
461	C₂²⋊C₄	—	106	1	no	—
462	C₂²⋊C₄	—	106	1	no	—
463	C₂²⋊C₄	—	106	1	no	—
464	C₄²	—	2	2	no	—
465	C₄²	—	2	2	no	—
466	C₄²	—	106	1	no	—
467	C₄²	—	106	1	no	—
468	C₄²	—	106	1	no	—
469	C₄²	—	106	1	no	—
470	C₄²	—	106	1	no	—
471	C₄²	—	106	1	no	—
472	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
473	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
474	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
475	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
476	C₂³	—	106	1	no	—
477	C₂³	—	106	1	no	—
478	C₂³	C₂×#11719945	1	1	yes	characteristic
479	Q₈	—	53	2	no	—
480	Q₈	—	106	1	no	—
481	Q₈	—	106	1	no	—
482	Q₈	—	106	1	no	—
483	D₄	—	53	2	no	—
484	D₄	—	106	2	no	—
485	D₄	—	106	2	no	—
486	D₄	—	106	1	no	—
487	D₄	—	106	1	no	—
488	D₄	—	106	1	no	—
489	D₄	—	212	1	no	—
490	D₄	—	212	1	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	106	1	no	—
497	C₂×C₄	—	106	2	no	—
498	C₂×C₄	—	106	2	no	—
499	C₂×C₄	—	106	1	no	—
500	C₂×C₄	—	106	1	no	—
501	C₂×C₄	—	106	1	no	—
502	C₂×C₄	—	106	1	no	—
503	C₂×C₄	—	106	1	no	—
504	C₂×C₄	—	424	1	no	—
505	C₂×C₄	—	424	1	no	—
506	C₂×C₄	—	212	1	no	—
507	C₂×C₄	—	212	1	no	—
508	C₂×C₄	—	212	2	no	—
509	C₂×C₄	—	212	2	no	—
510	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
511	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
512	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
513	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
514	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
515	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
516	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
517	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
518	C₂×C₄	#13133262	1	1	yes	characteristic
519	C₂×C₄	#13133265	1	1	yes	characteristic
520	C₈	—	2	2	no	—
521	C₈	—	2	2	no	—
522	C₈	—	2	1	no	—
523	C₈	—	2	1	no	—
524	C₈	—	2	1	no	—
525	C₈	—	2	1	no	—
526	C₈	—	212	2	no	—
527	C₈	—	212	2	no	—
528	C₂²	—	2	1	no	—
529	C₂²	—	2	1	no	—
530	C₂²	—	106	2	no	—
531	C₂²	—	212	1	no	—
532	C₂²	—	212	1	no	—
533	C₂²	—	212	1	no	—
534	C₂²	—	212	1	no	—
535	C₂²	—	212	1	no	—
536	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
537	C₂²	#16811314	1	1	yes	characteristic
538	C₂²	#16811323	1	1	yes	characteristic
539	C₄	—	2	1	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	4	4	no	—
543	C₄	—	106	2	no	—
544	C₄	—	212	1	no	—
545	C₄	—	212	1	no	—
546	C₄	—	212	1	no	—
547	C₄	—	212	1	no	—
548	C₄	—	212	1	no	—
549	C₄	#16810741	1	2	yes	—
550	C₄	#117×D₅₃	1	1	yes	characteristic
551	C₄	#16811104	1	1	yes	characteristic
552	C₄	#16811268	1	1	yes	characteristic
553	C₄	#16811285	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	424	1	no	—
557	C₂	—	212	2	no	—
558	C₂	#20277647	1	1	yes	characteristic
559	C₂	C₂×#16810741	1	1	yes	characteristic
560	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#22783
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20283380	C₂	semidirect
3. #20283280	C₂	semidirect
4. #20275419	C₂	semidirect
5. #20275315	C₂	semidirect
7. #20274853	C₂	semidirect
8. #20274736	C₂	semidirect
9. #20274736	C₂	semidirect
10. #20274552	C₂	semidirect
11. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
16. C₄×Dic₂₁₂	C₄	semidirect
17. C₄×Dic₂₁₂	C₄	semidirect
18. C₄×#11719913	C₄	semidirect
19. C₄×#11719913	C₄	semidirect
20. C₄×#11719913	C₄	semidirect
21. C₄×#11719913	C₄	semidirect
22. C₄×D₄₂₄	C₄	semidirect
23. C₄×D₄₂₄	C₄	semidirect
283. C₈×C₄⋊C₄	D₅₃	semidirect
329. C₅₃	#22783	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133239	2
#20283380	2
#20283280	2
#20275419	2
#20275315	2
#20275041	2
#20274853	2
#20274736	2
#20274552	2
C₈×C₄⋊C₄×C₅₃	2
#22783	53