Subgroups — AGL(1,ℤ/8)×#88322 #27353508

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	#41×#88322	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	#20273539	C₂	1	2	yes	maximal, complemented
4	SD₁₆⋊#88322	C₂	1	2	yes	maximal, complemented
5	SD₁₆×#88322	C₂	1	2	yes	maximal, complemented
6	D₈×#88322	C₂	1	2	yes	maximal, complemented
7	#20273467	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273106	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273102	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×AGL(1,ℤ/8)×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×#88322	—	2	1	no	—
13	Q₈×#88322	C₂²	1	1	yes	characteristic, complemented
14	Q₈×#88322	C₂²	1	1	yes	characteristic, complemented
15	D₄×#88322	—	2	1	no	—
16	D₄×#88322	—	2	1	no	—
17	D₄×#88322	—	2	1	no	—
18	D₄×#88322	—	2	1	no	—
19	D₄×#88322	C₂²	1	2	yes	—
20	D₄×#88322	C₂²	1	2	yes	—
21	D₄×#88322	C₂²	1	1	yes	characteristic, complemented
22	D₄×#88322	C₂²	1	1	yes	characteristic, complemented
23	#16810676	—	2	1	no	—
24	#16810676	C₂²	1	1	yes	characteristic
25	#16810676	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊#88322	—	2	1	no	—
28	C₂×C₄×#88322	—	2	1	no	—
29	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
30	#16810376	C₂²	1	2	yes	complemented
31	#16810374	C₂²	1	2	yes	complemented
32	Q₈⋊#88322	C₂²	1	2	yes	complemented
33	D₄⋊#88322	C₂²	1	2	yes	—
34	D₄⋊#88322	C₂²	1	2	yes	—
35	D₄⋊#88322	C₂²	1	2	yes	complemented
36	#16810344	C₂²	1	2	yes	complemented
37	C₈×#88322	C₂²	1	2	yes	complemented
38	C₂×AGL(1,ℤ/8)×C₅₃	C₄	1	1	yes	characteristic
39	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
41	C₅₃×#442	C₄	1	2	yes	—
42	C₅₃×#440	C₄	1	2	yes	—
43	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
44	C₄×D₈×C₅₃	C₂²	1	2	yes	—
45	C₅₃×#428	C₄	1	1	yes	characteristic
46	C₅₃×#418	C₄	1	1	yes	characteristic
47	C₅₃×#417	C₄	1	1	yes	characteristic
48	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
49	C₂²×#88322	—	2	1	no	—
50	C₂²×#88322	—	2	1	no	—
51	C₂²×#88322	—	2	2	no	—
52	C₂²×#88322	—	2	2	no	—
53	C₂²×#88322	—	2	1	no	—
54	C₂²×#88322	—	4	4	no	—
55	C₂²×#88322	D₄	1	1	yes	characteristic
56	C₂²×#88322	D₄	1	1	yes	characteristic
57	C₄⋊#88322	—	2	1	no	—
58	C₄⋊#88322	—	2	1	no	—
59	C₄⋊#88322	—	2	1	no	—
60	C₄⋊#88322	—	2	1	no	—
61	C₄⋊#88322	C₂³	1	1	yes	characteristic
62	C₄⋊#88322	C₂³	1	1	yes	characteristic
63	C₄⋊#88322	C₂³	1	2	yes	—
64	C₂²⋊#88322	—	2	2	no	—
65	C₂²⋊#88322	—	2	2	no	—
66	C₂²⋊#88322	—	2	1	no	—
67	C₂²⋊#88322	—	2	1	no	—
68	C₂²⋊#88322	—	2	1	no	—
69	C₂²⋊#88322	—	2	1	no	—
70	C₄×#88322	—	2	1	no	—
71	C₄×#88322	—	2	1	no	—
72	C₄×#88322	—	2	1	no	—
73	C₄×#88322	D₄	1	1	yes	characteristic
74	C₄×#88322	D₄	1	1	yes	characteristic
75	C₄×#88322	C₂³	1	2	yes	—
76	C₄×#88322	C₂³	1	1	yes	characteristic
77	C₄×#88322	C₂³	1	1	yes	characteristic
78	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
80	C₂³×C₄×C₅₃	—	2	1	no	—
81	AGL(1,ℤ/8)×C₅₃	C₈	1	8	yes	complemented
82	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
83	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
84	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
85	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
88	C₄×D₄×C₅₃	—	2	1	no	—
89	C₄×D₄×C₅₃	—	2	1	no	—
90	C₄×D₄×C₅₃	—	2	1	no	—
91	C₄×D₄×C₅₃	—	2	1	no	—
92	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
93	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₅₃	C₂³	1	2	yes	—
95	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
96	#117×C₅₃	—	2	1	no	—
97	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
100	C₂×C₄²×C₅₃	—	2	1	no	—
101	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
102	#107×C₅₃	C₂×C₄	1	2	yes	—
103	#106×C₅₃	C₂×C₄	1	2	yes	—
104	#103×C₅₃	C₂×C₄	1	2	yes	—
105	#102×C₅₃	C₂×C₄	1	2	yes	—
106	#102×C₅₃	C₂×C₄	1	2	yes	—
107	#102×C₅₃	C₂×C₄	1	2	yes	—
108	#97×C₅₃	C₂×C₄	1	2	yes	—
109	C₄×C₈×C₅₃	C₂³	1	2	yes	—
110	C₂⁴×C₅₃	—	2	1	no	—
111	#41×C₅₃	—	2	2	no	—
112	#41×C₅₃	C₂×C₈	1	4	yes	complemented
113	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×D₄×C₅₃	—	2	1	no	—
115	C₂×D₄×C₅₃	—	2	1	no	—
116	C₂×D₄×C₅₃	—	2	2	no	—
117	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	complemented
118	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
119	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₅₃	—	2	1	no	—
121	C₂²×C₄×C₅₃	—	2	1	no	—
122	C₂²×C₄×C₅₃	—	2	1	no	—
123	C₂²×C₄×C₅₃	—	2	2	no	—
124	C₂²×C₄×C₅₃	—	2	1	no	—
125	C₂²×C₄×C₅₃	—	2	2	no	—
126	C₂²×C₄×C₅₃	—	4	2	no	—
127	C₂²×C₄×C₅₃	—	4	2	no	—
128	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
131	SD₁₆×C₅₃	C₂×C₈	1	8	yes	complemented
132	D₈×C₅₃	C₂×C₈	1	8	yes	complemented
133	M₁₆×C₅₃	—	2	2	no	—
134	M₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
135	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
136	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
137	C₄⋊C₄×C₅₃	—	2	1	no	—
138	C₄⋊C₄×C₅₃	—	2	1	no	—
139	C₄⋊C₄×C₅₃	—	2	1	no	—
140	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
143	C₂²⋊C₄×C₅₃	—	2	1	no	—
144	C₂²⋊C₄×C₅₃	—	2	2	no	—
145	C₂²⋊C₄×C₅₃	—	2	1	no	—
146	C₂²⋊C₄×C₅₃	—	2	2	no	—
147	C₂²⋊C₄×C₅₃	—	2	1	no	—
148	C₄²×C₅₃	—	2	1	no	—
149	C₄²×C₅₃	—	2	1	no	—
150	C₄²×C₅₃	—	2	1	no	—
151	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₅₃	#41	1	1	yes	characteristic
154	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
155	C₂×#88322	—	2	2	no	—
156	C₂×#88322	—	2	1	no	—
157	C₂×#88322	—	2	1	no	—
158	C₂×#88322	—	2	2	no	—
159	C₂×#88322	—	2	2	no	—
160	C₂×#88322	—	4	1	no	—
161	C₂×#88322	—	4	2	no	—
162	C₂×#88322	—	4	4	no	—
163	C₂×#88322	—	4	2	no	—
164	C₂×#88322	—	4	2	no	—
165	C₂×#88322	—	4	4	no	—
166	C₂×#88322	C₂×D₄	1	1	yes	characteristic
167	C₂×#88322	C₂×D₄	1	1	yes	characteristic
168	C₂×#88322	C₂×D₄	1	1	yes	characteristic
169	C₂×#88322	C₂×D₄	1	1	yes	characteristic
170	C₂³×C₅₃	—	2	1	no	—
171	C₂³×C₅₃	—	2	2	no	—
172	C₂³×C₅₃	—	2	4	no	—
173	C₂³×C₅₃	—	4	2	no	—
174	C₂³×C₅₃	—	4	2	no	—
175	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
176	Q₈×C₅₃	—	2	1	no	—
177	Q₈×C₅₃	C₂²×C₈	1	2	yes	complemented
178	D₄×C₅₃	—	2	4	no	—
179	D₄×C₅₃	—	2	2	no	—
180	D₄×C₅₃	—	2	1	no	—
181	D₄×C₅₃	—	2	4	no	—
182	D₄×C₅₃	C₂²×C₈	1	4	yes	—
183	D₄×C₅₃	C₂²×C₈	1	2	yes	complemented
184	C₂×C₄×C₅₃	—	2	1	no	—
185	C₂×C₄×C₅₃	—	2	1	no	—
186	C₂×C₄×C₅₃	—	2	1	no	—
187	C₂×C₄×C₅₃	—	2	4	no	—
188	C₂×C₄×C₅₃	—	2	2	no	—
189	C₂×C₄×C₅₃	—	2	1	no	—
190	C₂×C₄×C₅₃	—	2	1	no	—
191	C₂×C₄×C₅₃	—	4	1	no	—
192	C₂×C₄×C₅₃	—	4	2	no	—
193	C₂×C₄×C₅₃	—	4	4	no	—
194	C₂×C₄×C₅₃	—	4	2	no	—
195	C₂×C₄×C₅₃	—	4	2	no	—
196	C₂×C₄×C₅₃	—	4	1	no	—
197	C₂×C₄×C₅₃	—	4	2	no	—
198	C₂×C₄×C₅₃	—	4	1	no	—
199	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
203	C₂×C₄×C₅₃	#131	1	2	yes	—
204	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
205	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
207	C₈×C₅₃	—	2	2	no	—
208	C₈×C₅₃	C₂²×C₈	1	4	yes	complemented
209	#88322	—	2	1	no	—
210	#88322	—	2	1	no	—
211	#88322	—	2	1	no	—
212	#88322	—	4	2	no	—
213	#88322	—	4	1	no	—
214	#88322	—	4	2	no	—
215	#88322	—	4	1	no	—
216	#88322	AGL(1,ℤ/8)	1	2	yes	direct product factor
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	—	2	2	no	—
221	C₂²×C₅₃	—	2	4	no	—
222	C₂²×C₅₃	—	4	8	no	—
223	C₂²×C₅₃	—	4	2	no	—
224	C₂²×C₅₃	—	4	1	no	—
225	C₂²×C₅₃	—	4	1	no	—
226	C₂²×C₅₃	—	4	2	no	—
227	C₂²×C₅₃	C₈×D₄	1	2	yes	—
228	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
229	C₄×C₅₃	—	2	2	no	—
230	C₄×C₅₃	—	2	1	no	—
231	C₄×C₅₃	—	2	1	no	—
232	C₄×C₅₃	—	2	1	no	—
233	C₄×C₅₃	—	4	1	no	—
234	C₄×C₅₃	—	4	2	no	—
235	C₄×C₅₃	—	4	1	no	—
236	C₄×C₅₃	C₈×D₄	1	2	yes	—
237	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
238	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
239	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₄×C₅₃	#575	1	1	yes	characteristic
241	C₈×#41	—	53	1	no	—
242	C₄×AGL(1,ℤ/8)	D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
252	C₂×C₅₃	—	2	2	no	—
253	C₂×C₅₃	—	4	2	no	—
254	C₂×C₅₃	—	4	4	no	—
255	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
256	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₂×C₅₃	#2955	1	1	yes	characteristic
258	C₂×AGL(1,ℤ/8)	Dic₅₃	1	1	yes	characteristic
259	C₂³×C₈	—	106	1	no	—
260	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
261	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
262	C₈×Q₈	—	53	1	no	—
263	C₈×Q₈	—	53	1	no	—
264	#442	Dic₅₃	1	2	yes	—
265	#440	Dic₅₃	1	2	yes	—
266	C₄×SD₁₆	C₂×D₅₃	1	2	yes	—
267	C₄×D₈	C₂×D₅₃	1	2	yes	—
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	—	53	1	no	—
278	#431	—	106	1	no	—
279	#428	Dic₅₃	1	1	yes	characteristic
280	C₂×C₄⋊C₈	—	53	1	no	—
281	#418	Dic₅₃	1	1	yes	characteristic
282	#417	Dic₅₃	1	1	yes	characteristic
283	C₂×C₂²⋊C₈	—	106	1	no	—
284	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
285	C₂×C₄×C₈	—	53	1	no	—
286	C₂×C₄×C₈	—	106	1	no	—
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	—	53	2	no	—
294	C₈²	—	53	2	no	—
295	C₅₃	C₈×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
296	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
297	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
298	C₂³×C₄	—	2	1	no	—
299	AGL(1,ℤ/8)	#88322	1	8	yes	direct product factor
300	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
301	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
302	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
303	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
304	C₂²×C₈	—	53	1	no	—
305	C₂²×C₈	—	53	1	no	—
306	C₂²×C₈	—	106	2	no	—
307	C₂²×C₈	—	106	2	no	—
308	C₂²×C₈	—	106	1	no	—
309	C₂²×C₈	—	106	1	no	—
310	C₂²×C₈	—	106	1	no	—
311	C₂²×C₈	—	212	4	no	—
312	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
313	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
319	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
320	C₄×D₄	C₂²×D₅₃	1	2	yes	—
321	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
322	#117	—	2	1	no	—
323	#117	C₂×Dic₅₃	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
325	C₂×C₂²⋊C₄	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
328	#107	C₂×Dic₅₃	1	2	yes	—
329	#106	C₂×Dic₅₃	1	2	yes	—
330	C₄⋊C₈	—	53	1	no	—
331	C₄⋊C₈	—	53	1	no	—
332	C₄⋊C₈	—	53	2	no	—
333	C₄⋊C₈	—	106	1	no	—
334	C₄⋊C₈	—	106	1	no	—
335	C₄⋊C₈	—	106	1	no	—
336	C₄⋊C₈	—	106	1	no	—
337	#103	C₂×Dic₅₃	1	2	yes	—
338	#102	C₂×Dic₅₃	1	2	yes	—
339	#102	C₂×Dic₅₃	1	2	yes	—
340	#102	C₂×Dic₅₃	1	2	yes	—
341	C₂²⋊C₈	—	106	2	no	—
342	C₂²⋊C₈	—	106	2	no	—
343	C₂²⋊C₈	—	106	1	no	—
344	C₂²⋊C₈	—	106	1	no	—
345	C₂²⋊C₈	—	106	1	no	—
346	C₂²⋊C₈	—	106	1	no	—
347	#97	C₂×Dic₅₃	1	2	yes	—
348	C₄×C₈	—	53	1	no	—
349	C₄×C₈	—	53	1	no	—
350	C₄×C₈	—	53	1	no	—
351	C₄×C₈	—	53	1	no	—
352	C₄×C₈	—	53	2	no	—
353	C₄×C₈	—	106	1	no	—
354	C₄×C₈	—	106	1	no	—
355	C₄×C₈	—	106	1	no	—
356	C₄×C₈	C₂²×D₅₃	1	2	yes	—
357	C₂⁴	—	2	1	no	—
358	#41	—	2	2	no	—
359	#41	C₂×#88322	1	4	yes	complemented
360	C₂×Q₈	C₂²×Dic₅₃	1	1	yes	characteristic
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	2	1	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	C₂×#88322	1	4	yes	complemented
365	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
366	C₂×D₄	C₂²×Dic₅₃	1	1	yes	characteristic
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
376	C₂²×C₄	#11719947	1	1	yes	characteristic
377	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
378	SD₁₆	C₂×#88322	1	8	yes	complemented
379	D₈	C₂×#88322	1	8	yes	complemented
380	M₁₆	—	2	2	no	—
381	M₁₆	C₂×#88322	1	4	yes	complemented
382	C₂×C₈	—	53	1	no	—
383	C₂×C₈	—	53	1	no	—
384	C₂×C₈	—	53	1	no	—
385	C₂×C₈	—	53	1	no	—
386	C₂×C₈	—	106	2	no	—
387	C₂×C₈	—	106	1	no	—
388	C₂×C₈	—	106	1	no	—
389	C₂×C₈	—	106	2	no	—
390	C₂×C₈	—	106	2	no	—
391	C₂×C₈	—	212	4	no	—
392	C₂×C₈	—	212	2	no	—
393	C₂×C₈	—	212	1	no	—
394	C₂×C₈	—	212	4	no	—
395	C₂×C₈	—	212	2	no	—
396	C₂×C₈	—	212	2	no	—
397	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
398	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
403	C₄⋊C₄	C₂²×Dic₅₃	1	1	yes	characteristic
404	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
405	C₂²⋊C₄	—	2	1	no	—
406	C₂²⋊C₄	—	2	2	no	—
407	C₂²⋊C₄	—	2	1	no	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	1	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	1	no	—
413	C₄²	D₄×D₅₃	1	1	yes	characteristic
414	C₄²	#11719947	1	1	yes	characteristic
415	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
416	C₄²	C₂³×D₅₃	1	1	yes	characteristic
417	C₂³	—	2	1	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
423	Q₈	—	2	1	no	—
424	Q₈	C₂²×#88322	1	2	yes	complemented
425	D₄	—	2	4	no	—
426	D₄	—	2	2	no	—
427	D₄	—	2	1	no	—
428	D₄	—	2	4	no	—
429	D₄	C₂²×#88322	1	4	yes	—
430	D₄	C₂²×#88322	1	2	yes	complemented
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	4	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
447	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
448	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
449	C₂×C₄	C₂²×#88322	1	2	yes	—
450	C₂×C₄	#13133224	1	2	yes	—
451	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
452	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
453	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	53	2	no	—
456	C₈	—	106	1	no	—
457	C₈	—	106	1	no	—
458	C₈	—	106	1	no	—
459	C₈	—	212	2	no	—
460	C₈	—	212	1	no	—
461	C₈	—	212	2	no	—
462	C₈	—	212	1	no	—
463	C₈	C₂²×#88322	1	4	yes	complemented
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	2	4	no	—
468	C₂²	—	4	8	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	1	no	—
471	C₂²	—	4	1	no	—
472	C₂²	—	4	2	no	—
473	C₂²	D₄×#88322	1	2	yes	—
474	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	2	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	1	no	—
481	C₄	—	4	1	no	—
482	C₄	D₄×#88322	1	2	yes	—
483	C₄	C₂³×#88322	1	1	yes	characteristic
484	C₄	C₂×#13133224	1	1	yes	characteristic
485	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
486	C₄	#16811419	1	1	yes	characteristic
487	C₂	—	2	2	no	—
488	C₂	—	4	4	no	—
489	C₂	—	4	2	no	—
490	C₂	C₂×D₄×#88322	1	1	yes	characteristic
491	C₂	AGL(1,ℤ/8)×Dic₅₃	1	1	yes	characteristic
492	C₂	#20283618	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #41×#88322	C₂	semidirect
2. C₂×D₄×#88322	C₂	semidirect
3. #20273539	C₂	semidirect
4. SD₁₆⋊#88322	C₂	semidirect
5. SD₁₆×#88322	C₂	semidirect
6. D₈×#88322	C₂	semidirect
7. #20273467	C₂	semidirect
8. #20273106	C₂	semidirect
9. #20273102	C₂	semidirect
10. M₁₆×#88322	C₂	semidirect
13. Q₈×#88322	C₂²	semidirect
14. Q₈×#88322	C₂²	semidirect
21. D₄×#88322	C₂²	semidirect
22. D₄×#88322	C₂²	semidirect
30. #16810376	C₂²	semidirect
31. #16810374	C₂²	semidirect
32. Q₈⋊#88322	C₂²	semidirect
35. D₄⋊#88322	C₂²	semidirect
36. #16810344	C₂²	semidirect
37. C₈×#88322	C₂²	semidirect
81. AGL(1,ℤ/8)×C₅₃	C₈	semidirect
112. #41×C₅₃	C₂×C₈	semidirect
117. C₂×D₄×C₅₃	C₂×C₈	semidirect
131. SD₁₆×C₅₃	C₂×C₈	semidirect
132. D₈×C₅₃	C₂×C₈	semidirect
134. M₁₆×C₅₃	C₂×C₈	semidirect
177. Q₈×C₅₃	C₂²×C₈	semidirect
183. D₄×C₅₃	C₂²×C₈	semidirect
208. C₈×C₅₃	C₂²×C₈	semidirect
216. #88322	AGL(1,ℤ/8)	direct (N × H)
295. C₅₃	C₈×AGL(1,ℤ/8)	semidirect
299. AGL(1,ℤ/8)	#88322	direct (N × H)
359. #41	C₂×#88322	semidirect
364. C₂×D₄	C₂×#88322	semidirect
378. SD₁₆	C₂×#88322	semidirect
379. D₈	C₂×#88322	semidirect
381. M₁₆	C₂×#88322	semidirect
424. Q₈	C₂²×#88322	semidirect
430. D₄	C₂²×#88322	semidirect
463. C₈	C₂²×#88322	semidirect

Maximal subgroups

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

subgroup	index
#41×#88322	2
C₂×D₄×#88322	2
#20273539	2
SD₁₆⋊#88322	2
SD₁₆×#88322	2
D₈×#88322	2
#20273467	2
#20273106	2
#20273102	2
M₁₆×#88322	2
C₄×AGL(1,ℤ/8)×C₅₃	2
C₈×AGL(1,ℤ/8)	53