Subgroups — C₈×#13133245 #27353510

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719945	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#13133245	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273540	C₂	1	2	yes	maximal, complemented
5	#20273529	C₂	1	2	yes	maximal, complemented
6	C₈×#11719913	C₂	1	2	yes	maximal, complemented
7	C₈×D₄₂₄	C₂	1	2	yes	maximal, complemented
8	#20273467	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273107	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273103	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133245	C₄	1	1	yes	characteristic
13	C₂²×C₈×D₅₃	—	2	1	no	—
14	C₄×#11719945	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₁₂	C₂²	1	1	yes	characteristic
16	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
17	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
18	#16810736	C₄	1	2	yes	—
19	#16810725	C₄	1	2	yes	—
20	C₄×#11719913	C₂²	1	2	yes	—
21	C₄×D₄₂₄	C₂²	1	2	yes	—
22	C₈×D₂₁₂	—	2	1	no	—
23	C₈×D₂₁₂	—	2	1	no	—
24	C₈×D₂₁₂	C₂²	1	2	yes	—
25	C₈×D₂₁₂	C₂²	1	2	yes	—
26	C₈×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
27	C₈×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
28	C₈×C₂²⋊D₅₃	—	2	1	no	—
29	C₈×C₂²⋊D₅₃	—	2	1	no	—
30	#16810676	C₂²	1	1	yes	characteristic
31	#16810674	—	2	1	no	—
32	#16810665	C₄	1	1	yes	characteristic
33	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
34	#16810629	C₄	1	1	yes	characteristic
35	#16810625	C₄	1	1	yes	characteristic
36	C₂×#13133129	—	2	1	no	—
37	C₄×C₈×D₅₃	—	2	1	no	—
38	#16810376	C₂²	1	2	yes	complemented
39	#16810374	C₂²	1	2	yes	complemented
40	#16810355	C₂²	1	2	yes	complemented
41	#16810352	C₂²	1	2	yes	—
42	#16810352	C₂²	1	2	yes	—
43	#16810352	C₂²	1	2	yes	complemented
44	C₅₃×#431	C₂²	1	1	yes	characteristic
45	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₅₃×#322	C₂²	1	2	yes	complemented
48	C₈²×C₅₃	C₂²	1	2	yes	complemented
49	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₂₁₂	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×D₅₃	—	2	1	no	—
52	#13133245	C₈	1	8	yes	direct product factor
53	C₂×#11719913	C₂×C₄	1	2	yes	—
54	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
55	C₂²×#88322	—	2	1	no	—
56	C₂×C₈×D₅₃	—	2	1	no	—
57	C₂×C₈×D₅₃	—	2	1	no	—
58	C₂×C₈×D₅₃	—	2	2	no	—
59	C₂×C₈×D₅₃	—	2	2	no	—
60	C₂×C₈×D₅₃	—	4	4	no	—
61	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
63	C₄×D₂₁₂	—	2	1	no	—
64	C₄×D₂₁₂	—	2	1	no	—
65	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
66	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
67	C₄×D₂₁₂	C₂³	1	2	yes	—
68	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
69	C₄×C₂²⋊D₅₃	—	2	1	no	—
70	C₄×C₂²⋊D₅₃	—	2	1	no	—
71	#13133176	C₂×C₄	1	1	yes	characteristic
72	#13133174	—	2	1	no	—
73	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×#11719921	—	2	1	no	—
75	C₄²×D₅₃	—	2	1	no	—
76	#13133151	C₂×C₄	1	2	yes	—
77	#13133149	C₂×C₄	1	2	yes	—
78	C₄⋊#88322	—	2	1	no	—
79	C₄⋊#88322	C₂³	1	1	yes	characteristic
80	C₄⋊#88322	C₂³	1	1	yes	characteristic
81	C₄⋊#88322	C₂³	1	2	yes	—
82	#13133146	—	2	1	no	—
83	#13133146	—	2	1	no	—
84	#13133142	C₂×C₄	1	2	yes	—
85	#13133139	C₂×C₄	1	2	yes	—
86	#13133139	C₂×C₄	1	2	yes	—
87	#13133139	C₂×C₄	1	2	yes	—
88	C₂²⋊#88322	—	2	1	no	—
89	#13133129	—	2	2	no	—
90	#13133129	—	2	2	no	—
91	#13133129	—	2	1	no	—
92	#13133129	—	2	1	no	—
93	C₄×#88322	—	2	1	no	—
94	C₈×Dic₅₃	—	2	1	no	—
95	C₈×Dic₅₃	—	2	1	no	—
96	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
99	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₈×C₅₃	—	2	1	no	—
102	C₂²⋊C₈×C₅₃	—	2	1	no	—
103	#97×C₅₃	C₂×C₄	1	2	yes	—
104	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
105	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
106	C₄×C₈×C₅₃	C₂³	1	2	yes	—
107	C₄×C₈×C₅₃	C₂³	1	2	yes	—
108	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
110	C₂³×D₅₃	—	2	1	no	—
111	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
114	C₂×C₂²⋊D₅₃	—	2	1	no	—
115	C₂²×Dic₅₃	—	2	1	no	—
116	#11719945	—	2	2	no	—
117	#11719945	C₂×C₈	1	4	yes	complemented
118	C₂×D₂₁₂	—	2	1	no	—
119	C₂×D₂₁₂	—	2	2	no	—
120	C₂×D₂₁₂	C₂×C₈	1	4	yes	complemented
121	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
122	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₄×D₅₃	—	2	1	no	—
124	C₂×C₄×D₅₃	—	2	1	no	—
125	C₂×C₄×D₅₃	—	2	2	no	—
126	C₂×C₄×D₅₃	—	2	1	no	—
127	C₂×C₄×D₅₃	—	2	2	no	—
128	C₂×C₄×D₅₃	—	4	2	no	—
129	C₂×C₄×D₅₃	—	4	2	no	—
130	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
131	M₁₆×C₅₃	—	2	2	no	—
132	M₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
133	C₂×C₈×C₅₃	—	2	2	no	—
134	C₂×C₈×C₅₃	—	2	2	no	—
135	C₂×C₈×C₅₃	—	4	1	no	—
136	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
137	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₅₃	#41	1	1	yes	characteristic
145	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
146	C₂²⋊Dic₅₃	—	2	1	no	—
147	#11719921	—	2	2	no	—
148	#11719921	—	2	1	no	—
149	#11719921	—	2	2	no	—
150	#11719921	—	2	1	no	—
151	C₄⋊Dic₅₃	—	2	1	no	—
152	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
155	#11719919	—	2	1	no	—
156	#11719919	—	2	1	no	—
157	C₄×Dic₅₃	—	2	1	no	—
158	C₄×Dic₅₃	—	2	1	no	—
159	C₄×Dic₅₃	—	2	1	no	—
160	C₂×#88322	—	2	1	no	—
161	C₂×#88322	—	2	1	no	—
162	C₂×#88322	—	2	2	no	—
163	C₂×#88322	—	4	2	no	—
164	D₄₂₄	C₂×C₈	1	8	yes	complemented
165	#11719913	C₂×C₈	1	8	yes	complemented
166	C₈×D₅₃	—	4	2	no	—
167	C₈×D₅₃	—	4	4	no	—
168	C₈×D₅₃	—	4	2	no	—
169	C₈×D₅₃	—	4	4	no	—
170	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
171	C₂²×D₅₃	—	2	1	no	—
172	C₂²×D₅₃	—	2	2	no	—
173	C₂²×D₅₃	—	2	4	no	—
174	C₂²×D₅₃	—	4	2	no	—
175	C₂²×D₅₃	—	4	2	no	—
176	C₂×C₄×C₅₃	—	2	1	no	—
177	C₂×C₄×C₅₃	—	2	1	no	—
178	C₂×C₄×C₅₃	—	4	1	no	—
179	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
183	C₂×C₄×C₅₃	#131	1	2	yes	—
184	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
187	C₂²⋊D₅₃	—	2	4	no	—
188	C₂×Dic₅₃	—	2	1	no	—
189	C₂×Dic₅₃	—	2	2	no	—
190	C₂×Dic₅₃	—	2	1	no	—
191	C₂×Dic₅₃	—	2	1	no	—
192	C₂×Dic₅₃	—	4	2	no	—
193	D₂₁₂	—	2	2	no	—
194	D₂₁₂	—	2	1	no	—
195	D₂₁₂	—	2	4	no	—
196	D₂₁₂	C₂²×C₈	1	4	yes	—
197	D₂₁₂	C₂²×C₈	1	2	yes	complemented
198	C₄×D₅₃	—	2	4	no	—
199	C₄×D₅₃	—	4	2	no	—
200	C₄×D₅₃	—	4	4	no	—
201	C₄×D₅₃	—	4	2	no	—
202	C₄×D₅₃	—	4	1	no	—
203	C₄×D₅₃	—	4	2	no	—
204	C₄×D₅₃	—	4	1	no	—
205	Dic₁₀₆	—	2	1	no	—
206	Dic₁₀₆	C₂²×C₈	1	2	yes	complemented
207	C₈×C₅₃	—	2	2	no	—
208	C₈×C₅₃	—	2	1	no	—
209	C₈×C₅₃	—	2	1	no	—
210	C₈×C₅₃	—	2	1	no	—
211	C₈×C₅₃	—	4	2	no	—
212	C₈×C₅₃	C₂²×C₈	1	4	yes	complemented
213	C₈×C₅₃	AGL(1,ℤ/8)	1	2	yes	complemented
214	#88322	—	4	2	no	—
215	#88322	—	4	1	no	—
216	#88322	—	4	1	no	—
217	C₈×AGL(1,ℤ/8)	—	53	1	no	maximal, Sylow
218	C₂²×C₅₃	—	2	1	no	—
219	C₂²×C₅₃	—	2	1	no	—
220	C₂²×C₅₃	C₈×D₄	1	2	yes	—
221	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂×D₅₃	—	2	2	no	—
223	C₂×D₅₃	—	2	4	no	—
224	C₂×D₅₃	—	4	8	no	—
225	C₂×D₅₃	—	4	2	no	—
226	C₂×D₅₃	—	4	1	no	—
227	C₂×D₅₃	—	4	1	no	—
228	C₂×D₅₃	—	4	2	no	—
229	C₄×C₅₃	—	2	1	no	—
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	C₈×D₄	1	2	yes	—
233	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
234	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
235	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
236	C₄×C₅₃	#575	1	1	yes	characteristic
237	Dic₅₃	—	2	2	no	—
238	Dic₅₃	—	4	2	no	—
239	Dic₅₃	—	4	1	no	—
240	Dic₅₃	—	4	1	no	—
241	C₈×#41	—	53	1	no	—
242	C₄×AGL(1,ℤ/8)	—	53	1	no	—
243	C₂×C₈×D₄	—	53	1	no	—
244	#1580	—	53	2	no	—
245	SD₁₆⋊C₈	—	53	2	no	—
246	C₈×SD₁₆	—	53	2	no	—
247	C₈×D₈	—	53	2	no	—
248	#1565	—	53	1	no	—
249	#1478	—	53	1	no	—
250	#1477	—	53	1	no	—
251	C₈×M₁₆	D₅₃	1	1	yes	characteristic, complemented
252	C₂×C₅₃	—	2	2	no	—
253	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
254	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
255	C₂×C₅₃	#2955	1	1	yes	characteristic
256	D₅₃	—	4	2	no	—
257	D₅₃	—	4	4	no	—
258	C₂×AGL(1,ℤ/8)	—	53	1	no	—
259	C₂³×C₈	—	106	1	no	—
260	C₄×#41	—	53	1	no	—
261	C₂×C₄×D₄	—	53	1	no	—
262	C₈×Q₈	—	53	1	no	—
263	C₈×Q₈	—	53	1	no	—
264	#442	—	53	2	no	—
265	#440	—	53	2	no	—
266	C₄×SD₁₆	—	53	2	no	—
267	C₄×D₈	—	53	2	no	—
268	C₈×D₄	—	53	1	no	—
269	C₈×D₄	—	53	2	no	—
270	C₈×D₄	—	53	2	no	—
271	C₈×D₄	—	53	1	no	—
272	C₈×D₄	—	106	1	no	—
273	C₈×D₄	—	106	1	no	—
274	C₈×D₄	—	106	1	no	—
275	C₈×D₄	—	106	1	no	—
276	#431	—	53	1	no	—
277	#431	—	106	1	no	—
278	#431	C₂×D₅₃	1	1	yes	characteristic
279	#428	—	53	1	no	—
280	C₂×C₄⋊C₈	—	53	1	no	—
281	#418	—	53	1	no	—
282	#417	—	53	1	no	—
283	C₂×C₂²⋊C₈	—	106	1	no	—
284	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
285	C₂×C₄×C₈	—	106	1	no	—
286	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
287	#335	—	53	2	no	—
288	#334	—	53	2	no	—
289	Q₈⋊C₈	—	53	2	no	—
290	D₄⋊C₈	—	53	2	no	—
291	D₄⋊C₈	—	53	2	no	—
292	D₄⋊C₈	—	53	2	no	—
293	#322	C₂×D₅₃	1	2	yes	complemented
294	C₈²	C₂×D₅₃	1	2	yes	complemented
295	C₅₃	C₈×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
296	C₂×#41	—	53	1	no	—
297	C₂²×D₄	—	53	1	no	—
298	C₂³×C₄	—	106	1	no	—
299	AGL(1,ℤ/8)	—	53	8	no	—
300	C₂×SD₁₆	—	53	2	no	—
301	C₂×D₈	—	53	2	no	—
302	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
303	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
304	C₂²×C₈	—	106	1	no	—
305	C₂²×C₈	—	106	2	no	—
306	C₂²×C₈	—	106	1	no	—
307	C₂²×C₈	—	106	2	no	—
308	C₂²×C₈	—	106	1	no	—
309	C₂²×C₈	—	212	4	no	—
310	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
311	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
312	C₄×Q₈	—	53	1	no	—
313	C₄×Q₈	—	53	1	no	—
314	C₄×D₄	—	53	2	no	—
315	C₄×D₄	—	53	1	no	—
316	C₄×D₄	—	53	2	no	—
317	C₄×D₄	—	53	1	no	—
318	C₄×D₄	—	106	1	no	—
319	C₄×D₄	—	106	1	no	—
320	C₄×D₄	—	106	1	no	—
321	C₄×D₄	—	106	1	no	—
322	#117	—	53	1	no	—
323	#117	—	106	1	no	—
324	C₂×C₄⋊C₄	—	53	1	no	—
325	C₂×C₂²⋊C₄	—	106	1	no	—
326	C₂×C₄²	—	106	1	no	—
327	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
328	#107	—	53	2	no	—
329	#106	—	53	2	no	—
330	C₄⋊C₈	—	2	1	no	—
331	C₄⋊C₈	—	53	1	no	—
332	C₄⋊C₈	—	53	1	no	—
333	C₄⋊C₈	—	53	2	no	—
334	C₄⋊C₈	—	106	1	no	—
335	C₄⋊C₈	—	106	1	no	—
336	C₄⋊C₈	—	106	1	no	—
337	#103	—	53	2	no	—
338	#102	—	53	2	no	—
339	#102	—	53	2	no	—
340	#102	—	53	2	no	—
341	C₂²⋊C₈	—	2	1	no	—
342	C₂²⋊C₈	—	106	1	no	—
343	C₂²⋊C₈	—	106	2	no	—
344	C₂²⋊C₈	—	106	2	no	—
345	C₂²⋊C₈	—	106	1	no	—
346	C₂²⋊C₈	—	106	1	no	—
347	#97	C₄×D₅₃	1	2	yes	—
348	C₄×C₈	—	106	1	no	—
349	C₄×C₈	—	106	1	no	—
350	C₄×C₈	—	106	1	no	—
351	C₄×C₈	D₂₁₂	1	1	yes	characteristic
352	C₄×C₈	D₂₁₂	1	1	yes	characteristic
353	C₄×C₈	C₂²×D₅₃	1	2	yes	—
354	C₄×C₈	C₂²×D₅₃	1	2	yes	—
355	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
356	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
357	C₂⁴	—	106	1	no	—
358	#41	—	53	4	no	—
359	#41	—	106	2	no	—
360	C₂×Q₈	—	53	1	no	—
361	C₂×D₄	—	53	4	no	—
362	C₂×D₄	—	53	1	no	—
363	C₂×D₄	—	53	2	no	—
364	C₂×D₄	—	106	1	no	—
365	C₂×D₄	—	106	1	no	—
366	C₂×D₄	—	106	2	no	—
367	C₂²×C₄	—	106	1	no	—
368	C₂²×C₄	—	106	1	no	—
369	C₂²×C₄	—	106	1	no	—
370	C₂²×C₄	—	106	2	no	—
371	C₂²×C₄	—	106	1	no	—
372	C₂²×C₄	—	106	2	no	—
373	C₂²×C₄	—	212	2	no	—
374	C₂²×C₄	—	212	2	no	—
375	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
377	C₂²×C₄	#11719945	1	1	yes	characteristic
378	SD₁₆	—	53	8	no	—
379	D₈	—	53	8	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	C₈×D₅₃	1	4	yes	complemented
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	4	1	no	—
385	C₂×C₈	—	106	1	no	—
386	C₂×C₈	—	106	1	no	—
387	C₂×C₈	—	106	2	no	—
388	C₂×C₈	—	212	4	no	—
389	C₂×C₈	—	212	2	no	—
390	C₂×C₈	—	212	2	no	—
391	C₂×C₈	—	212	4	no	—
392	C₂×C₈	—	212	2	no	—
393	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
394	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
395	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
396	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
397	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
398	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
399	C₄⋊C₄	—	53	1	no	—
400	C₄⋊C₄	—	53	1	no	—
401	C₄⋊C₄	—	53	2	no	—
402	C₄⋊C₄	—	106	1	no	—
403	C₄⋊C₄	—	106	1	no	—
404	C₄⋊C₄	—	106	1	no	—
405	C₂²⋊C₄	—	106	1	no	—
406	C₂²⋊C₄	—	106	2	no	—
407	C₂²⋊C₄	—	106	1	no	—
408	C₂²⋊C₄	—	106	2	no	—
409	C₂²⋊C₄	—	106	1	no	—
410	C₄²	—	106	1	no	—
411	C₄²	—	106	1	no	—
412	C₄²	—	106	1	no	—
413	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
414	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
415	C₄²	#11719945	1	1	yes	characteristic
416	C₄²	C₂³×D₅₃	1	1	yes	characteristic
417	C₂³	—	106	1	no	—
418	C₂³	—	106	2	no	—
419	C₂³	—	106	4	no	—
420	C₂³	—	212	2	no	—
421	C₂³	—	212	2	no	—
422	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
423	Q₈	—	53	2	no	—
424	Q₈	—	106	1	no	—
425	D₄	—	53	4	no	—
426	D₄	—	53	2	no	—
427	D₄	—	106	4	no	—
428	D₄	—	106	2	no	—
429	D₄	—	106	1	no	—
430	D₄	—	106	4	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	106	1	no	—
435	C₂×C₄	—	106	4	no	—
436	C₂×C₄	—	106	1	no	—
437	C₂×C₄	—	106	1	no	—
438	C₂×C₄	—	106	2	no	—
439	C₂×C₄	—	212	2	no	—
440	C₂×C₄	—	212	4	no	—
441	C₂×C₄	—	212	2	no	—
442	C₂×C₄	—	212	2	no	—
443	C₂×C₄	—	212	1	no	—
444	C₂×C₄	—	212	1	no	—
445	C₂×C₄	—	212	2	no	—
446	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
447	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
448	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
449	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
450	C₂×C₄	#13133223	1	2	yes	—
451	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
452	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
453	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	2	1	no	—
456	C₈	—	2	1	no	—
457	C₈	—	2	1	no	—
458	C₈	—	4	2	no	—
459	C₈	—	212	1	no	—
460	C₈	—	212	2	no	—
461	C₈	—	212	1	no	—
462	C₈	C₂×C₈×D₅₃	1	4	yes	complemented
463	C₈	#13133245	1	2	yes	direct product factor
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	106	2	no	—
467	C₂²	—	106	4	no	—
468	C₂²	—	212	8	no	—
469	C₂²	—	212	2	no	—
470	C₂²	—	212	1	no	—
471	C₂²	—	212	1	no	—
472	C₂²	—	212	2	no	—
473	C₂²	C₈×D₂₁₂	1	2	yes	—
474	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	106	2	no	—
479	C₄	—	212	2	no	—
480	C₄	—	212	1	no	—
481	C₄	—	212	1	no	—
482	C₄	C₈×D₂₁₂	1	2	yes	—
483	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
484	C₄	C₂×#13133223	1	1	yes	characteristic
485	C₄	C₂×#13133245	1	1	yes	characteristic
486	C₄	#16811423	1	1	yes	characteristic
487	C₂	—	2	2	no	—
488	C₂	—	212	4	no	—
489	C₂	—	212	2	no	—
490	C₂	C₂×C₈×D₂₁₂	1	1	yes	characteristic
491	C₂	C₄×#13133245	1	1	yes	characteristic
492	C₂	#20283619	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133245	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	C₈×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₈×#11719945	C₂	semidirect
3. C₂×C₈×D₂₁₂	C₂	semidirect
4. #20273540	C₂	semidirect
5. #20273529	C₂	semidirect
6. C₈×#11719913	C₂	semidirect
7. C₈×D₄₂₄	C₂	semidirect
8. #20273467	C₂	semidirect
9. #20273107	C₂	semidirect
10. #20273103	C₂	semidirect
11. C₈×M₁₆×C₅₃	C₂	semidirect
16. C₈×Dic₁₀₆	C₂²	semidirect
17. C₈×Dic₁₀₆	C₂²	semidirect
26. C₈×D₂₁₂	C₂²	semidirect
27. C₈×D₂₁₂	C₂²	semidirect
38. #16810376	C₂²	semidirect
39. #16810374	C₂²	semidirect
40. #16810355	C₂²	semidirect
43. #16810352	C₂²	semidirect
47. C₅₃×#322	C₂²	semidirect
48. C₈²×C₅₃	C₂²	semidirect
52. #13133245	C₈	direct (N × H)
117. #11719945	C₂×C₈	semidirect
120. C₂×D₂₁₂	C₂×C₈	semidirect
132. M₁₆×C₅₃	C₂×C₈	semidirect
164. D₄₂₄	C₂×C₈	semidirect
165. #11719913	C₂×C₈	semidirect
197. D₂₁₂	C₂²×C₈	semidirect
206. Dic₁₀₆	C₂²×C₈	semidirect
212. C₈×C₅₃	C₂²×C₈	semidirect
213. C₈×C₅₃	AGL(1,ℤ/8)	semidirect
251. C₈×M₁₆	D₅₃	semidirect
293. #322	C₂×D₅₃	semidirect
294. C₈²	C₂×D₅₃	semidirect
295. C₅₃	C₈×AGL(1,ℤ/8)	semidirect
381. M₁₆	C₈×D₅₃	semidirect
462. C₈	C₂×C₈×D₅₃	semidirect
463. C₈	#13133245	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×#11719945	2
C₄×#13133245	2
C₂×C₈×D₂₁₂	2
#20273540	2
#20273529	2
C₈×#11719913	2
C₈×D₄₂₄	2
#20273467	2
#20273107	2
#20273103	2
C₈×M₁₆×C₅₃	2
C₈×AGL(1,ℤ/8)	53

Subgroups — C8×#13133245 #27353510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#13133245 #27353510