Subgroups — C₄×#16810893 #27390972

Order 13568.

946 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810893	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133242	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#20275625	C₂	1	1	yes	maximal, characteristic, complemented
5	#20275500	C₂	1	1	yes	maximal, characteristic, complemented
6	#20275377	C₂	1	2	yes	maximal, complemented
7	#20275223	C₂	1	2	yes	maximal, complemented
8	#20274779	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13133133	C₂	1	2	yes	maximal, complemented
11	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133242	C₂²	1	2	yes	—
13	C₂²×D₄₂₄	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₂₁₂	C₂²	1	2	yes	—
15	#16810893	C₄	1	8	yes	direct product factor
16	#16810733	—	2	2	no	—
17	#16810721	—	2	2	no	—
18	C₄×#11719924	—	2	2	no	—
19	C₄×D₄₂₄	—	2	1	no	—
20	C₄×D₄₂₄	—	2	1	no	—
21	C₄×D₄₂₄	—	2	1	no	—
22	C₄×D₄₂₄	—	2	1	no	—
23	C₄×#11719922	—	2	2	no	—
24	C₂×#13133153	C₂²	1	1	yes	characteristic
25	C₂×#13133153	C₂²	1	1	yes	characteristic
26	#16810666	C₂²	1	2	yes	—
27	C₂×#13133151	C₂²	1	1	yes	characteristic
28	#16810627	C₂²	1	2	yes	—
29	#16810624	C₂²	1	2	yes	—
30	C₂×#13133139	C₂²	1	2	yes	—
31	C₂×#13133133	C₂²	1	2	yes	—
32	C₂×#13133133	C₂²	1	2	yes	—
33	C₄×#11719917	C₂²	1	2	yes	—
34	#16810391	C₂²	1	2	yes	—
35	#16810391	C₂²	1	2	yes	—
36	#16810388	C₂²	1	2	yes	—
37	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#405	—	2	1	no	—
41	C₅₃×#405	—	2	1	no	—
42	C₅₃×#405	C₂²	1	1	yes	characteristic
43	C₅₃×#405	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₅₃	—	2	1	no	—
45	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	—	2	1	no	—
47	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
48	C₂²×D₂₁₂	C₂³	1	2	yes	—
49	C₂²×C₄×D₅₃	—	2	2	no	—
50	#13133242	—	2	4	no	—
51	#13133242	C₂×C₄	1	8	yes	complemented
52	C₂×#11719924	—	2	2	no	—
53	C₂×D₄₂₄	—	2	1	no	—
54	C₂×D₄₂₄	—	2	1	no	—
55	C₂×D₄₂₄	—	2	2	no	—
56	C₂×D₄₂₄	C₂×C₄	1	4	yes	complemented
57	C₂×#11719922	—	2	2	no	—
58	C₂×#11719917	C₂³	1	2	yes	—
59	C₂×#11719917	C₂³	1	2	yes	—
60	C₄×D₂₁₂	—	2	2	no	—
61	C₄×D₂₁₂	—	2	2	no	—
62	C₄×D₂₁₂	—	2	2	no	—
63	C₄×D₂₁₂	—	2	2	no	—
64	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
65	C₂×#11719921	—	2	2	no	—
66	#13133153	—	2	2	no	—
67	#13133153	C₂×C₄	1	4	yes	complemented
68	#13133151	—	2	1	no	—
69	#13133151	—	2	1	no	—
70	#13133150	—	2	2	no	—
71	#13133148	—	2	2	no	—
72	#13133141	—	2	2	no	—
73	#13133139	—	2	2	no	—
74	#13133139	—	2	2	no	—
75	#13133138	—	2	2	no	—
76	#13133137	—	2	2	no	—
77	#13133137	—	2	2	no	—
78	#13133133	—	2	4	no	—
79	#13133133	C₂×C₄	1	8	yes	complemented
80	#13133128	—	2	2	no	—
81	C₄×#88322	—	2	2	no	—
82	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
83	#131×C₅₃	—	2	2	no	—
84	#131×C₅₃	—	2	2	no	—
85	#131×C₅₃	—	2	2	no	—
86	#131×C₅₃	C₂×C₄	1	4	yes	complemented
87	C₂×M₁₆×C₅₃	—	2	1	no	—
88	C₂×M₁₆×C₅₃	—	2	1	no	—
89	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
91	C₂²×C₈×C₅₃	—	2	1	no	—
92	C₂²×C₈×C₅₃	—	2	1	no	—
93	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
95	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
96	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
97	C₄×D₄×C₅₃	—	2	1	no	—
98	C₄×D₄×C₅₃	—	2	1	no	—
99	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
100	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
101	#117×C₅₃	—	2	1	no	—
102	#117×C₅₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₅₃	—	2	1	no	—
104	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
105	#97×C₅₃	—	2	1	no	—
106	#97×C₅₃	—	2	1	no	—
107	#97×C₅₃	—	2	1	no	—
108	C₄×C₈×C₅₃	—	2	1	no	—
109	C₄×C₈×C₅₃	—	2	1	no	—
110	C₄×C₈×C₅₃	—	2	1	no	—
111	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
112	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
113	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
114	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
115	C₂³×D₅₃	—	2	2	no	—
116	#41×C₅₃	—	2	1	no	—
117	#41×C₅₃	—	2	1	no	—
118	#41×C₅₃	C₂²×C₄	1	2	yes	—
119	#41×C₅₃	#41	1	2	yes	—
120	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
121	C₂×D₄×C₅₃	—	2	1	no	—
122	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
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	C₂²×Dic₅₃	—	2	2	no	—
130	C₂×D₂₁₂	—	2	2	no	—
131	C₂×D₂₁₂	—	2	2	no	—
132	C₂×D₂₁₂	—	2	2	no	—
133	C₂×D₂₁₂	—	2	2	no	—
134	C₂×D₂₁₂	—	2	2	no	—
135	C₂×D₂₁₂	C₂²×C₄	1	4	yes	—
136	C₂×C₄×D₅₃	—	4	2	no	—
137	C₂×C₄×D₅₃	—	4	2	no	—
138	C₂×C₄×D₅₃	—	8	4	no	—
139	M₁₆×C₅₃	—	2	4	no	—
140	M₁₆×C₅₃	—	2	1	no	—
141	M₁₆×C₅₃	—	2	1	no	—
142	M₁₆×C₅₃	—	2	4	no	—
143	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
144	M₁₆×C₅₃	#41	1	2	yes	—
145	C₂×C₈×C₅₃	—	2	4	no	—
146	C₂×C₈×C₅₃	—	2	1	no	—
147	C₂×C₈×C₅₃	—	2	1	no	—
148	C₂×C₈×C₅₃	—	2	1	no	—
149	C₂×C₈×C₅₃	—	2	1	no	—
150	C₂×C₈×C₅₃	—	2	4	no	—
151	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
152	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
154	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
155	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
156	C₂×C₈×C₅₃	#41	1	2	yes	—
157	C₄⋊C₄×C₅₃	—	2	1	no	—
158	C₄⋊C₄×C₅₃	—	2	1	no	—
159	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
160	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
161	C₂²⋊C₄×C₅₃	—	2	1	no	—
162	C₂²⋊C₄×C₅₃	—	2	1	no	—
163	C₂²⋊C₄×C₅₃	—	2	1	no	—
164	C₄²×C₅₃	—	2	1	no	—
165	C₄²×C₅₃	—	2	1	no	—
166	C₄²×C₅₃	—	2	1	no	—
167	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
171	#11719924	—	2	8	no	—
172	#11719922	—	2	8	no	—
173	#11719921	—	4	2	no	—
174	#11719921	—	4	2	no	—
175	C₄⋊Dic₅₃	—	2	2	no	—
176	C₄⋊Dic₅₃	—	2	2	no	—
177	#11719917	—	2	2	no	—
178	#11719917	—	2	2	no	—
179	#11719917	—	2	2	no	—
180	#11719917	C₂²×C₄	1	4	yes	—
181	C₂×#88322	—	2	2	no	—
182	C₂×#88322	—	2	2	no	—
183	D₄₂₄	—	2	4	no	—
184	D₄₂₄	—	2	4	no	—
185	C₂³×C₅₃	—	2	1	no	—
186	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
187	C₂²×D₅₃	—	2	4	no	—
188	C₂²×D₅₃	—	4	2	no	—
189	C₂²×D₅₃	—	4	2	no	—
190	C₂²×D₅₃	—	8	2	no	—
191	C₂²×D₅₃	—	8	2	no	—
192	Q₈×C₅₃	—	2	1	no	—
193	Q₈×C₅₃	C₄×D₄	1	2	yes	—
194	D₄×C₅₃	—	2	2	no	—
195	D₄×C₅₃	—	2	2	no	—
196	D₄×C₅₃	—	2	1	no	—
197	D₄×C₅₃	C₄×D₄	1	2	yes	—
198	C₂×C₄×C₅₃	—	2	1	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	—	4	1	no	—
205	C₂×C₄×C₅₃	—	4	2	no	—
206	C₂×C₄×C₅₃	—	4	2	no	—
207	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
208	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
209	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
210	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
211	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
212	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
217	C₂×Dic₅₃	—	4	2	no	—
218	C₂×Dic₅₃	—	8	2	no	—
219	D₂₁₂	—	2	4	no	—
220	D₂₁₂	—	2	4	no	—
221	D₂₁₂	—	4	2	no	—
222	D₂₁₂	—	4	2	no	—
223	C₄×D₅₃	—	8	4	no	—
224	C₄×D₅₃	—	8	4	no	—
225	C₈×C₅₃	—	2	2	no	—
226	C₈×C₅₃	—	2	1	no	—
227	C₈×C₅₃	—	2	2	no	—
228	C₈×C₅₃	—	2	1	no	—
229	C₈×C₅₃	C₄×D₄	1	2	yes	—
230	C₈×C₅₃	C₄×D₄	1	2	yes	—
231	#88322	—	2	4	no	—
232	#88322	—	4	2	no	—
233	C₄×#472	—	53	1	no	maximal, Sylow
234	C₂²×C₅₃	—	2	1	no	—
235	C₂²×C₅₃	—	2	1	no	—
236	C₂²×C₅₃	—	2	2	no	—
237	C₂²×C₅₃	—	4	1	no	—
238	C₂²×C₅₃	—	4	1	no	—
239	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
240	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
241	C₂²×C₅₃	#529	1	1	yes	characteristic
242	C₂×D₅₃	—	4	4	no	—
243	C₂×D₅₃	—	8	4	no	—
244	C₂×D₅₃	—	8	2	no	—
245	C₂×D₅₃	—	8	2	no	—
246	C₂×D₅₃	—	8	4	no	—
247	C₄×C₅₃	—	2	1	no	—
248	C₄×C₅₃	—	2	1	no	—
249	C₄×C₅₃	—	2	1	no	—
250	C₄×C₅₃	—	2	2	no	—
251	C₄×C₅₃	—	4	1	no	—
252	C₄×C₅₃	—	4	1	no	—
253	C₄×C₅₃	#472	1	2	yes	complemented
254	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
255	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
256	C₄×C₅₃	#525	1	1	yes	characteristic
257	C₄×C₅₃	#532	1	1	yes	characteristic
258	Dic₅₃	—	8	2	no	—
259	C₂×#472	—	53	1	no	—
260	C₄×AGL(1,ℤ/8)	—	53	2	no	—
261	C₂×C₄×D₈	—	53	1	no	—
262	C₄×#131	D₅₃	1	1	yes	characteristic, complemented
263	#1978	—	53	1	no	—
264	#1952	—	53	1	no	—
265	#1931	—	53	2	no	—
266	#1907	—	53	2	no	—
267	#1816	—	53	1	no	—
268	C₄×#108	—	53	1	no	—
269	C₄×#100	—	53	2	no	—
270	C₂×C₅₃	—	2	1	no	—
271	C₂×C₅₃	—	2	1	no	—
272	C₂×C₅₃	—	4	2	no	—
273	C₂×C₅₃	C₄×#121	1	1	yes	characteristic
274	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
275	C₂×C₅₃	#3067	1	1	yes	characteristic
276	D₅₃	—	8	4	no	—
277	C₂×AGL(1,ℤ/8)	—	53	2	no	—
278	C₂²×D₈	—	53	1	no	—
279	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
280	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
281	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
282	C₂×C₄×D₄	—	53	2	no	—
283	#472	—	53	8	no	—
284	#442	—	106	2	no	—
285	#440	—	106	2	no	—
286	C₄×SD₁₆	—	106	2	no	—
287	C₄×D₈	—	106	2	no	—
288	C₄×D₈	—	106	1	no	—
289	C₄×D₈	—	106	1	no	—
290	C₄×D₈	—	106	1	no	—
291	C₄×D₈	—	106	1	no	—
292	C₂×#108	—	53	1	no	—
293	C₂×#108	—	53	1	no	—
294	#428	—	53	2	no	—
295	C₂×#107	—	53	1	no	—
296	#418	—	53	2	no	—
297	#417	—	53	2	no	—
298	C₂×#102	—	53	2	no	—
299	C₂×#100	—	53	2	no	—
300	C₂×#100	—	53	2	no	—
301	#405	—	2	1	no	—
302	#405	—	2	1	no	—
303	#405	C₂×D₅₃	1	1	yes	characteristic
304	#405	C₂×D₅₃	1	1	yes	characteristic
305	C₄×M₁₆	—	2	1	no	—
306	C₄×M₁₆	—	53	2	no	—
307	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
308	C₂×C₄×C₈	—	2	1	no	—
309	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
310	#343	—	53	2	no	—
311	#343	—	53	2	no	—
312	#341	—	53	2	no	—
313	C₅₃	C₄×#472	1	1	yes	characteristic, complemented, Sylow
314	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
315	C₂²×D₄	—	53	2	no	—
316	C₂³×C₄	—	106	2	no	—
317	AGL(1,ℤ/8)	—	53	8	no	—
318	AGL(1,ℤ/8)	—	106	4	no	—
319	C₂×SD₁₆	—	106	2	no	—
320	C₂×D₈	—	53	4	no	—
321	C₂×D₈	—	106	2	no	—
322	C₂×D₈	—	106	2	no	—
323	C₂×D₈	—	106	1	no	—
324	C₂×D₈	—	106	1	no	—
325	#131	—	2	2	no	—
326	#131	—	2	2	no	—
327	#131	—	2	2	no	—
328	#131	C₄×D₅₃	1	4	yes	complemented
329	C₂×M₁₆	—	2	1	no	—
330	C₂×M₁₆	—	2	1	no	—
331	C₂×M₁₆	—	53	2	no	—
332	C₂×M₁₆	—	53	2	no	—
333	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
334	C₂×M₁₆	C₂²×D₅₃	1	1	yes	characteristic
335	C₂²×C₈	—	2	1	no	—
336	C₂²×C₈	—	2	1	no	—
337	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
338	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
339	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
340	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
341	C₄×D₄	—	2	1	no	—
342	C₄×D₄	—	2	1	no	—
343	C₄×D₄	—	106	2	no	—
344	C₄×D₄	—	106	2	no	—
345	C₄×D₄	—	106	2	no	—
346	C₄×D₄	—	106	2	no	—
347	C₄×D₄	D₂₁₂	1	1	yes	characteristic
348	C₄×D₄	D₂₁₂	1	1	yes	characteristic
349	#117	—	2	1	no	—
350	#117	C₂²×D₅₃	1	1	yes	characteristic
351	C₂×C₄⋊C₄	—	53	2	no	—
352	C₂×C₂²⋊C₄	—	106	2	no	—
353	C₂×C₄²	—	2	1	no	—
354	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
355	#108	—	53	4	no	—
356	#108	—	106	2	no	—
357	#107	—	106	1	no	—
358	#107	—	106	1	no	—
359	#107	—	106	2	no	—
360	#106	—	106	2	no	—
361	#103	—	106	2	no	—
362	#102	—	106	2	no	—
363	#102	—	106	2	no	—
364	#102	—	106	2	no	—
365	#102	—	106	2	no	—
366	#102	—	106	2	no	—
367	#100	—	53	8	no	—
368	#100	—	106	4	no	—
369	#97	—	2	1	no	—
370	#97	—	2	1	no	—
371	#97	—	2	1	no	—
372	#97	—	106	2	no	—
373	C₄×C₈	—	2	1	no	—
374	C₄×C₈	—	2	1	no	—
375	C₄×C₈	—	2	1	no	—
376	C₄×C₈	—	106	2	no	—
377	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
378	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
379	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
380	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
381	C₂⁴	—	106	2	no	—
382	#41	—	2	1	no	—
383	#41	—	2	1	no	—
384	#41	C₂×C₄×D₅₃	1	2	yes	—
385	#41	#11719945	1	2	yes	—
386	C₂×Q₈	C₂×D₂₁₂	1	1	yes	characteristic
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	53	4	no	—
389	C₂×D₄	—	106	2	no	—
390	C₂×D₄	—	106	2	no	—
391	C₂×D₄	—	106	2	no	—
392	C₂×D₄	—	106	2	no	—
393	C₂×D₄	—	106	2	no	—
394	C₂×D₄	C₂×D₂₁₂	1	1	yes	characteristic
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	106	2	no	—
399	C₂²×C₄	—	424	4	no	—
400	C₂²×C₄	—	212	2	no	—
401	C₂²×C₄	—	212	2	no	—
402	C₂²×C₄	#11719945	1	1	yes	characteristic
403	C₂²×C₄	#11719945	1	1	yes	characteristic
404	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
405	SD₁₆	—	106	8	no	—
406	D₈	—	106	8	no	—
407	D₈	—	106	4	no	—
408	D₈	—	106	4	no	—
409	M₁₆	—	2	4	no	—
410	M₁₆	—	2	1	no	—
411	M₁₆	—	2	1	no	—
412	M₁₆	—	2	4	no	—
413	M₁₆	—	53	4	no	—
414	M₁₆	—	106	2	no	—
415	M₁₆	—	106	2	no	—
416	M₁₆	—	106	2	no	—
417	M₁₆	C₂×C₄×D₅₃	1	2	yes	—
418	M₁₆	#11719945	1	2	yes	—
419	C₂×C₈	—	2	4	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	4	no	—
425	C₂×C₈	—	106	2	no	—
426	C₂×C₈	—	106	2	no	—
427	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
428	C₂×C₈	#11719945	1	2	yes	—
429	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
430	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
431	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
432	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	—	2	1	no	—
435	C₄⋊C₄	—	106	2	no	—
436	C₄⋊C₄	—	106	2	no	—
437	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
438	C₄⋊C₄	C₂×D₂₁₂	1	1	yes	characteristic
439	C₂²⋊C₄	—	2	1	no	—
440	C₂²⋊C₄	—	2	1	no	—
441	C₂²⋊C₄	—	2	1	no	—
442	C₂²⋊C₄	—	212	2	no	—
443	C₂²⋊C₄	—	212	2	no	—
444	C₄²	—	2	1	no	—
445	C₄²	—	2	1	no	—
446	C₄²	—	2	1	no	—
447	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
448	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
449	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
450	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
451	C₂³	—	2	1	no	—
452	C₂³	—	106	4	no	—
453	C₂³	—	424	2	no	—
454	C₂³	—	424	2	no	—
455	C₂³	—	212	2	no	—
456	C₂³	—	212	2	no	—
457	C₂³	C₂×#11719945	1	1	yes	characteristic
458	Q₈	—	2	1	no	—
459	Q₈	C₄×D₂₁₂	1	2	yes	—
460	D₄	—	2	2	no	—
461	D₄	—	2	2	no	—
462	D₄	—	2	1	no	—
463	D₄	—	106	4	no	—
464	D₄	—	106	4	no	—
465	D₄	—	212	2	no	—
466	D₄	—	212	2	no	—
467	D₄	C₄×D₂₁₂	1	2	yes	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	424	4	no	—
478	C₂×C₄	—	424	4	no	—
479	C₂×C₄	—	424	2	no	—
480	C₂×C₄	—	212	2	no	—
481	C₂×C₄	#13133191	1	1	yes	characteristic
482	C₂×C₄	#13133191	1	1	yes	characteristic
483	C₂×C₄	#13133191	1	1	yes	characteristic
484	C₂×C₄	#13133191	1	1	yes	characteristic
485	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
486	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
487	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
488	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
489	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
490	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
491	C₈	—	2	2	no	—
492	C₈	—	2	1	no	—
493	C₈	—	2	2	no	—
494	C₈	—	2	1	no	—
495	C₈	—	106	4	no	—
496	C₈	—	212	2	no	—
497	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
498	C₈	C₄×C₂²⋊D₅₃	1	2	yes	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	1	no	—
501	C₂²	—	2	2	no	—
502	C₂²	—	4	1	no	—
503	C₂²	—	4	1	no	—
504	C₂²	—	424	4	no	—
505	C₂²	—	424	4	no	—
506	C₂²	—	424	2	no	—
507	C₂²	—	424	2	no	—
508	C₂²	—	212	4	no	—
509	C₂²	C₄×#11719945	1	1	yes	characteristic
510	C₂²	C₂×#13133191	1	1	yes	characteristic
511	C₂²	#16811157	1	1	yes	characteristic
512	C₄	—	2	1	no	—
513	C₄	—	2	1	no	—
514	C₄	—	2	1	no	—
515	C₄	—	2	2	no	—
516	C₄	—	4	1	no	—
517	C₄	—	4	1	no	—
518	C₄	—	424	2	no	—
519	C₄	#16810893	1	2	yes	direct product factor
520	C₄	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
521	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
522	C₄	#16811134	1	1	yes	characteristic
523	C₄	#16811173	1	1	yes	characteristic
524	C₂	—	2	1	no	—
525	C₂	—	2	1	no	—
526	C₂	—	4	2	no	—
527	C₂	—	424	4	no	—
528	C₂	C₄×#13133191	1	1	yes	characteristic
529	C₂	C₂×#16810893	1	1	yes	characteristic
530	C₂	#20284435	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#121
prime radical	C₂×#16810893	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	C₄×#472
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. C₄×#13133242	C₂	semidirect
3. C₂×C₄×D₄₂₄	C₂	semidirect
4. #20275625	C₂	semidirect
5. #20275500	C₂	semidirect
6. #20275377	C₂	semidirect
7. #20275223	C₂	semidirect
8. #20274779	C₂	semidirect
9. C₄×#13133153	C₂	semidirect
10. C₄×#13133133	C₂	semidirect
11. C₄×#131×C₅₃	C₂	semidirect
15. #16810893	C₄	direct (N × H)
51. #13133242	C₂×C₄	semidirect
56. C₂×D₄₂₄	C₂×C₄	semidirect
67. #13133153	C₂×C₄	semidirect
79. #13133133	C₂×C₄	semidirect
86. #131×C₅₃	C₂×C₄	semidirect
253. C₄×C₅₃	#472	semidirect
262. C₄×#131	D₅₃	semidirect
313. C₅₃	C₄×#472	semidirect
328. #131	C₄×D₅₃	semidirect
519. C₄	#16810893	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×#16810893	2
C₄×#13133242	2
C₂×C₄×D₄₂₄	2
#20275625	2
#20275500	2
#20275377	2
#20275223	2
#20274779	2
C₄×#13133153	2
C₄×#13133133	2
C₄×#131×C₅₃	2
C₄×#472	53

Subgroups — C4×#16810893 #27390972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810893 #27390972