Subgroups — #28297950

Order 13568.

10894 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289526	C₂	1	2	yes	maximal, complemented
2	#20289494	C₂	1	2	yes	maximal, complemented
3	#20289493	C₂	1	16	yes	maximal, complemented
4	#20289492	C₂	1	12	yes	maximal, complemented
5	#20289485	C₂	1	8	yes	maximal, complemented
6	#20289484	C₂	1	6	yes	maximal, complemented
7	C₂×#16811428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16811425	C₂	1	6	yes	maximal, complemented
9	#576×D₅₃	C₂	1	8	yes	maximal, complemented
10	#20289426	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3584	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811460	—	2	4	no	—
13	#16811460	C₂²	1	16	yes	complemented
14	#16811459	—	2	8	no	—
15	#16811459	C₂²	1	1	yes	characteristic
16	#16811459	C₂²	1	12	yes	complemented
17	C₂×#13133263	—	2	8	no	—
18	C₂×#13133263	C₂²	1	2	yes	complemented
19	C₂×#13133263	C₂²	1	12	yes	complemented
20	H(5,𝔽₂)×D₅₃	—	2	12	no	—
21	H(5,𝔽₂)×D₅₃	C₂²	1	16	yes	complemented
22	#16811437	C₂²	1	4	yes	complemented
23	#16811436	C₂²	1	12	yes	complemented
24	#16811432	C₂²	1	48	yes	complemented
25	#16811431	C₂²	1	16	yes	complemented
26	#16811430	C₂²	1	24	yes	complemented
27	#16811429	C₂²	1	24	yes	complemented
28	#16811428	C₂²	1	8	yes	complemented
29	#16811428	C₂²	1	8	yes	complemented
30	#16811427	C₂²	1	48	yes	complemented
31	#16811426	C₂²	1	16	yes	complemented
32	#16811425	C₂²	1	24	yes	complemented
33	#16811425	C₂²	1	12	yes	complemented
34	#16811425	C₂²	1	24	yes	complemented
35	#16811425	C₂²	1	24	yes	complemented
36	#16811424	C₂²	1	12	yes	complemented
37	#16811422	C₂²	1	48	yes	complemented
38	#16811421	C₂²	1	12	yes	complemented
39	#16811420	C₂²	1	24	yes	complemented
40	C₂×#13133245	C₂²	1	6	yes	complemented
41	C₂×#13133244	C₂²	1	24	yes	complemented
42	C₂×#13133242	C₂²	1	12	yes	complemented
43	AGL(1,ℤ/8)×D₅₃	C₂²	1	48	yes	complemented
44	C₂×#13133240	C₂²	1	12	yes	complemented
45	C₂×#13133239	C₂²	1	3	yes	complemented
46	#135×D₅₃	C₂²	1	24	yes	complemented
47	C₂²×D₄₂₄	C₂²	1	3	yes	complemented
48	C₂×D₈×D₅₃	C₂²	1	12	yes	complemented
49	#16811390	C₂²	1	1	yes	characteristic, complemented
50	#16811389	C₂²	1	6	yes	complemented
51	#16811388	C₂²	1	8	yes	complemented
52	C₂×#13133223	C₂²	1	6	yes	complemented
53	#131×D₅₃	C₂²	1	8	yes	complemented
54	C₅₃×#585	C₂²	1	2	yes	complemented
55	C₅₃×#578	C₂²	1	4	yes	complemented
56	C₅₃×#577	C₂²	1	8	yes	complemented
57	C₅₃×#576	C₂²	1	4	yes	complemented
58	C₅₃×#575	C₂²	1	6	yes	complemented
59	C₂×#135×C₅₃	C₂²	1	6	yes	complemented
60	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic, complemented
61	#13133265	—	2	16	no	—
62	#13133265	D₄	1	2	yes	—
63	#13133265	C₂³	1	24	yes	complemented
64	#13133264	—	2	12	no	—
65	#13133264	D₄	1	4	yes	—
66	#13133264	C₂³	1	8	yes	complemented
67	#13133263	—	2	8	no	—
68	#13133263	—	2	16	no	—
69	#13133263	—	2	48	no	—
70	#13133263	—	2	24	no	—
71	#13133263	—	2	24	no	—
72	#13133263	—	2	12	no	—
73	#13133263	D₄	1	4	yes	—
74	#13133263	C₂³	1	24	yes	complemented
75	#13133263	C₂³	1	48	yes	complemented
76	#13133263	C₂³	1	16	yes	complemented
77	#13133262	—	2	24	no	—
78	#13133262	—	2	48	no	—
79	#13133262	D₄	1	6	yes	—
80	#13133262	C₂³	1	24	yes	complemented
81	C₂×#11719949	—	2	2	no	—
82	C₂×#11719949	—	2	16	no	—
83	C₂×#11719949	C₂³	1	24	yes	complemented
84	C₂×#11719945	—	2	24	no	—
85	C₂×#11719945	—	2	12	no	—
86	C₂×#11719945	C₂³	1	6	yes	—
87	C₂×#11719945	C₂³	1	6	yes	complemented
88	#41×D₅₃	—	2	16	no	—
89	#41×D₅₃	—	2	48	no	—
90	#41×D₅₃	—	2	24	no	—
91	#41×D₅₃	C₂³	1	8	yes	—
92	#41×D₅₃	C₂³	1	48	yes	complemented
93	C₂²×D₂₁₂	—	2	12	no	—
94	C₂²×D₂₁₂	C₂³	1	6	yes	complemented
95	C₂×D₄×D₅₃	—	2	6	no	—
96	C₂×D₄×D₅₃	—	2	24	no	—
97	C₂×D₄×D₅₃	—	2	48	no	—
98	C₂×D₄×D₅₃	C₂³	1	24	yes	complemented
99	#13133249	C₂³	1	48	yes	complemented
100	#13133248	C₂³	1	24	yes	complemented
101	#13133247	C₂³	1	8	yes	complemented
102	#13133246	C₂³	1	16	yes	complemented
103	#13133245	C₂³	1	48	yes	complemented
104	#13133244	C₂³	1	96	yes	complemented
105	#13133244	C₂³	1	48	yes	complemented
106	#13133243	C₂³	1	24	yes	complemented
107	#13133242	C₂³	1	48	yes	complemented
108	#13133242	C₂³	1	24	yes	complemented
109	#13133241	C₂³	1	48	yes	complemented
110	#13133240	C₂³	1	24	yes	complemented
111	#13133240	C₂³	1	48	yes	complemented
112	#13133239	C₂³	1	12	yes	complemented
113	#13133239	C₂³	1	12	yes	complemented
114	#13133238	C₂³	1	48	yes	complemented
115	#13133237	C₂³	1	48	yes	complemented
116	#13133236	C₂³	1	8	yes	complemented
117	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic, complemented
118	Q₁₆×D₅₃	C₂³	1	8	yes	complemented
119	C₂×#11719913	C₂³	1	6	yes	complemented
120	C₂×#11719924	C₂³	1	24	yes	complemented
121	SD₁₆×D₅₃	C₂³	1	48	yes	complemented
122	C₂×D₄₂₄	C₂³	1	12	yes	complemented
123	C₂×D₄₂₄	C₂³	1	3	yes	complemented
124	C₂×D₄₂₄	C₂³	1	12	yes	complemented
125	C₂×D₄₂₄	C₂³	1	6	yes	complemented
126	C₂×#11719922	C₂³	1	24	yes	complemented
127	D₈×D₅₃	C₂³	1	24	yes	complemented
128	D₈×D₅₃	C₂³	1	48	yes	complemented
129	#13133225	C₂³	1	24	yes	complemented
130	#13133224	C₂³	1	8	yes	complemented
131	#13133223	C₂³	1	12	yes	complemented
132	#13133223	C₂³	1	4	yes	complemented
133	#13133223	C₂³	1	12	yes	complemented
134	#13133223	C₂³	1	12	yes	complemented
135	C₂×#11719917	C₂³	1	6	yes	complemented
136	C₂×#11719912	C₂³	1	12	yes	complemented
137	M₁₆×D₅₃	C₂³	1	24	yes	complemented
138	C₂×C₈×D₅₃	C₂³	1	12	yes	complemented
139	2₋¹⁺⁴×C₅₃	—	2	2	no	—
140	2₋¹⁺⁴×C₅₃	C₂³	1	8	yes	complemented
141	H(5,𝔽₂)×C₅₃	—	2	6	no	—
142	H(5,𝔽₂)×C₅₃	C₂³	1	8	yes	complemented
143	C₂×#41×C₅₃	—	2	8	no	—
144	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
145	C₂×#41×C₅₃	C₂³	1	12	yes	complemented
146	#137×C₅₃	C₂³	1	24	yes	complemented
147	AGL(1,ℤ/8)×C₅₃	C₂³	1	24	yes	complemented
148	#135×C₅₃	C₂³	1	4	yes	complemented
149	#135×C₅₃	C₂³	1	12	yes	complemented
150	#135×C₅₃	C₂³	1	12	yes	complemented
151	#135×C₅₃	C₂³	1	12	yes	complemented
152	C₂×Q₁₆×C₅₃	C₂³	1	6	yes	complemented
153	C₂×SD₁₆×C₅₃	C₂³	1	12	yes	complemented
154	C₂×D₈×C₅₃	C₂³	1	6	yes	complemented
155	#131×C₅₃	C₂³	1	4	yes	complemented
156	#131×C₅₃	C₂³	1	4	yes	complemented
157	C₂×M₁₆×C₅₃	C₂³	1	3	yes	complemented
158	C₂²×C₈×C₅₃	C₂³	1	3	yes	complemented
159	C₂³×D₅₃	—	2	6	no	—
160	C₂³×D₅₃	—	2	24	no	—
161	#41×C₅₃	—	2	8	no	—
162	#41×C₅₃	—	2	24	no	—
163	#41×C₅₃	—	2	12	no	—
164	#41×C₅₃	—	2	12	no	—
165	#41×C₅₃	C₂×D₄	1	4	yes	—
166	#41×C₅₃	C₂⁴	1	8	yes	—
167	#41×C₅₃	C₂⁴	1	24	yes	—
168	#41×C₅₃	C₂⁴	1	4	yes	—
169	C₂×Q₈×C₅₃	—	2	8	no	—
170	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
171	C₂×Q₈×C₅₃	C₂⁴	1	12	yes	—
172	C₂×D₄×C₅₃	—	2	12	no	—
173	C₂×D₄×C₅₃	—	2	24	no	—
174	C₂×D₄×C₅₃	C₂×D₄	1	3	yes	—
175	C₂×D₄×C₅₃	C₂⁴	1	12	yes	—
176	C₂²×C₄×C₅₃	—	2	12	no	—
177	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
178	C₂×C₂²⋊D₅₃	—	2	6	no	—
179	C₂×C₂²⋊D₅₃	—	2	48	no	—
180	C₂×C₂²⋊D₅₃	—	2	24	no	—
181	C₂×C₂²⋊D₅₃	C₂×D₄	1	12	yes	—
182	#11719949	—	2	16	no	—
183	#11719949	—	2	24	no	—
184	#11719949	—	2	48	no	—
185	#11719949	C₂×D₄	1	8	yes	—
186	#11719949	C₂⁴	1	48	yes	—
187	Q₈×D₅₃	—	2	24	no	—
188	Q₈×D₅₃	C₂×D₄	1	8	yes	—
189	Q₈×D₅₃	C₂⁴	1	16	yes	—
190	#11719947	—	2	16	no	—
191	#11719947	—	2	48	no	—
192	#11719947	—	2	24	no	—
193	#11719947	C₂×D₄	1	24	yes	—
194	#11719947	C₂⁴	1	16	yes	—
195	D₄×D₅₃	—	2	48	no	—
196	D₄×D₅₃	—	2	48	no	—
197	D₄×D₅₃	—	2	96	no	—
198	D₄×D₅₃	—	2	48	no	—
199	D₄×D₅₃	—	2	24	no	—
200	D₄×D₅₃	—	2	48	no	—
201	D₄×D₅₃	C₂×D₄	1	24	yes	—
202	D₄×D₅₃	C₂⁴	1	48	yes	—
203	#11719945	—	2	12	no	—
204	#11719945	—	2	48	no	—
205	#11719945	—	2	8	no	—
206	#11719945	—	2	48	no	—
207	#11719945	—	2	48	no	—
208	#11719945	—	2	24	no	—
209	#11719945	C₂×D₄	1	12	yes	—
210	#11719945	C₂×D₄	1	12	yes	—
211	#11719945	C₂⁴	1	4	yes	—
212	#11719945	C₂⁴	1	12	yes	—
213	#11719945	C₂⁴	1	24	yes	—
214	C₂×D₂₁₂	—	2	24	no	—
215	C₂×D₂₁₂	—	2	6	no	—
216	C₂×D₂₁₂	—	2	12	no	—
217	C₂×D₂₁₂	—	2	48	no	—
218	C₂×D₂₁₂	—	2	24	no	—
219	C₂×D₂₁₂	C₂×D₄	1	6	yes	—
220	C₂×D₂₁₂	C₂⁴	1	24	yes	—
221	C₂×D₂₁₂	C₂⁴	1	6	yes	—
222	C₂×C₄×D₅₃	—	2	6	no	—
223	C₂×C₄×D₅₃	—	2	24	no	—
224	C₂×C₄×D₅₃	—	2	48	no	—
225	C₂×C₄×D₅₃	C₂⁴	1	12	yes	—
226	C₂×Dic₁₀₆	—	2	8	no	—
227	C₂×Dic₁₀₆	C₂×D₄	1	6	yes	—
228	C₂×Dic₁₀₆	C₂⁴	1	2	yes	—
229	Q₁₆×C₅₃	C₂⁴	1	4	yes	—
230	Q₁₆×C₅₃	C₂⁴	1	12	yes	—
231	SD₁₆×C₅₃	C₂⁴	1	8	yes	—
232	SD₁₆×C₅₃	C₂⁴	1	24	yes	—
233	D₈×C₅₃	C₂⁴	1	4	yes	—
234	D₈×C₅₃	C₂⁴	1	12	yes	—
235	M₁₆×C₅₃	C₂⁴	1	12	yes	—
236	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
237	C₂×C₈×C₅₃	C₂⁴	1	3	yes	—
238	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
239	C₂×C₈×C₅₃	C₂⁴	1	6	yes	—
240	#11719925	C₂⁴	1	16	yes	—
241	#11719924	C₂⁴	1	48	yes	—
242	#11719923	C₂⁴	1	16	yes	—
243	#11719922	C₂⁴	1	48	yes	—
244	#11719917	C₂⁴	1	12	yes	—
245	#11719917	C₂⁴	1	4	yes	—
246	C₂×#88322	C₂⁴	1	12	yes	—
247	Dic₂₁₂	C₂⁴	1	2	yes	—
248	Dic₂₁₂	C₂⁴	1	2	yes	—
249	D₄₂₄	C₂⁴	1	6	yes	—
250	D₄₂₄	C₂⁴	1	24	yes	—
251	D₄₂₄	C₂⁴	1	6	yes	—
252	#11719913	C₂⁴	1	24	yes	—
253	#11719912	C₂⁴	1	24	yes	—
254	#11719912	C₂⁴	1	8	yes	—
255	C₈×D₅₃	C₂⁴	1	8	yes	—
256	C₈×D₅₃	C₂⁴	1	24	yes	—
257	C₂³×C₅₃	—	2	12	no	—
258	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
259	C₂²×D₅₃	—	2	6	no	—
260	C₂²×D₅₃	—	2	24	no	—
261	C₂²×D₅₃	—	2	48	no	—
262	C₂²×D₅₃	—	16	24	no	—
263	C₂²×D₅₃	—	16	6	no	—
264	C₂²×D₅₃	C₂²×D₄	1	12	yes	—
265	Q₈×C₅₃	—	2	12	no	—
266	Q₈×C₅₃	C₂²×D₄	1	4	yes	—
267	Q₈×C₅₃	C₂⁵	1	8	yes	—
268	D₄×C₅₃	—	2	8	no	—
269	D₄×C₅₃	—	2	24	no	—
270	D₄×C₅₃	—	2	12	no	—
271	D₄×C₅₃	C₂²×D₄	1	12	yes	—
272	D₄×C₅₃	C₂⁵	1	8	yes	—
273	C₂×C₄×C₅₃	—	2	12	no	—
274	C₂×C₄×C₅₃	—	2	8	no	—
275	C₂×C₄×C₅₃	—	2	24	no	—
276	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
277	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
278	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
279	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
280	C₂²⋊D₅₃	—	2	24	no	—
281	C₂²⋊D₅₃	—	2	16	no	—
282	C₂²⋊D₅₃	—	2	48	no	—
283	C₂²⋊D₅₃	C₂²×D₄	1	24	yes	—
284	C₂²⋊D₅₃	C₂²×D₄	1	8	yes	—
285	C₂×Dic₅₃	—	2	2	no	—
286	C₂×Dic₅₃	—	2	16	no	—
287	C₂×Dic₅₃	C₂²×D₄	1	12	yes	—
288	D₂₁₂	—	2	48	no	—
289	D₂₁₂	—	2	6	no	—
290	D₂₁₂	—	2	24	no	—
291	D₂₁₂	C₂²×D₄	1	4	yes	—
292	D₂₁₂	C₂²×D₄	1	12	yes	—
293	D₂₁₂	C₂⁵	1	12	yes	—
294	C₄×D₅₃	—	2	16	no	—
295	C₄×D₅₃	—	2	24	no	—
296	C₄×D₅₃	—	2	48	no	—
297	C₄×D₅₃	C₂²×D₄	1	24	yes	—
298	C₄×D₅₃	C₂⁵	1	8	yes	—
299	Dic₁₀₆	—	2	2	no	—
300	Dic₁₀₆	—	2	16	no	—
301	Dic₁₀₆	C₂²×D₄	1	4	yes	—
302	Dic₁₀₆	C₂²×D₄	1	12	yes	—
303	Dic₁₀₆	C₂⁵	1	4	yes	—
304	C₈×C₅₃	C₂⁵	1	1	yes	characteristic
305	C₈×C₅₃	C₂⁵	1	6	yes	—
306	C₈×C₅₃	C₂⁵	1	1	yes	characteristic
307	#88322	C₂⁵	1	8	yes	—
308	#63091	—	53	1	no	maximal, Sylow
309	C₂²×C₅₃	—	2	8	no	—
310	C₂²×C₅₃	—	4	3	no	—
311	C₂²×C₅₃	—	8	12	no	—
312	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
313	C₂²×C₅₃	C₂³×D₄	1	6	yes	—
314	C₂×D₅₃	—	2	12	no	—
315	C₂×D₅₃	—	4	12	no	—
316	C₂×D₅₃	—	8	48	no	—
317	C₂×D₅₃	—	8	24	no	—
318	C₂×D₅₃	—	8	6	no	—
319	C₂×D₅₃	C₂³×D₄	1	8	yes	—
320	C₄×C₅₃	—	2	8	no	—
321	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
322	C₄×C₅₃	C₂³×D₄	1	6	yes	—
323	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
324	Dic₅₃	—	2	4	no	—
325	Dic₅₃	C₂³×D₄	1	8	yes	—
326	H(7,𝔽₂)	—	53	2	no	—
327	#3585	—	53	2	no	—
328	#3585	—	53	16	no	—
329	#3585	—	53	12	no	—
330	#3584	—	53	8	no	—
331	#3584	—	53	6	no	—
332	#3584	D₅₃	1	1	yes	characteristic, complemented
333	C₂×#576	—	53	8	no	—
334	C₂×#576	—	53	6	no	—
335	C₂×#576	—	53	1	no	—
336	#3573	—	53	1	no	—
337	C₂×C₅₃	—	2	6	no	—
338	C₂×C₅₃	—	2	1	no	—
339	C₂×C₅₃	—	4	8	no	—
340	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
341	D₅₃	—	2	8	no	—
342	D₅₃	—	4	12	no	—
343	#585	—	53	1	no	—
344	#585	—	53	16	no	—
345	#585	—	53	12	no	—
346	#585	—	106	8	no	—
347	#585	—	106	4	no	—
348	#585	C₂×D₅₃	1	2	yes	complemented
349	C₂×H(5,𝔽₂)	—	53	16	no	—
350	C₂×H(5,𝔽₂)	—	53	2	no	—
351	C₂×H(5,𝔽₂)	—	53	12	no	—
352	C₂×H(5,𝔽₂)	—	106	8	no	—
353	C₂×H(5,𝔽₂)	—	106	12	no	—
354	#578	—	53	4	no	—
355	#578	—	53	12	no	—
356	#578	C₂×D₅₃	1	4	yes	complemented
357	#577	—	53	16	no	—
358	#577	—	53	48	no	—
359	#577	—	53	24	no	—
360	#577	—	53	24	no	—
361	#577	C₂×D₅₃	1	8	yes	complemented
362	#576	—	53	16	no	—
363	#576	—	53	48	no	—
364	#576	—	53	24	no	—
365	#576	—	53	12	no	—
366	#576	—	53	24	no	—
367	#576	—	53	24	no	—
368	#576	—	53	12	no	—
369	#576	—	53	8	no	—
370	#576	—	53	8	no	—
371	#576	C₂×D₅₃	1	4	yes	complemented
372	#575	—	53	12	no	—
373	#575	—	53	48	no	—
374	#575	—	53	24	no	—
375	#575	C₂×D₅₃	1	6	yes	complemented
376	C₂×AGL(1,ℤ/8)	—	53	12	no	—
377	C₂×AGL(1,ℤ/8)	—	53	48	no	—
378	C₂×AGL(1,ℤ/8)	—	53	24	no	—
379	C₂×AGL(1,ℤ/8)	—	53	6	no	—
380	C₂×#135	—	53	24	no	—
381	C₂×#135	—	53	12	no	—
382	C₂×#135	—	53	3	no	—
383	C₂×#135	C₂×D₅₃	1	6	yes	complemented
384	C₂²×D₈	—	53	12	no	—
385	C₂²×D₈	—	53	3	no	—
386	#568	—	53	1	no	—
387	#568	—	53	8	no	—
388	#568	—	53	6	no	—
389	C₂×#131	—	53	8	no	—
390	C₂×#131	—	53	6	no	—
391	C₂×#131	C₂×D₅₃	1	1	yes	characteristic, complemented
392	C₅₃	#63091	1	1	yes	characteristic, complemented, Sylow
393	2₋¹⁺⁴	—	2	2	no	—
394	2₋¹⁺⁴	—	53	4	no	—
395	2₋¹⁺⁴	—	53	2	no	—
396	2₋¹⁺⁴	—	53	8	no	—
397	2₋¹⁺⁴	—	53	24	no	—
398	2₋¹⁺⁴	—	106	16	no	—
399	2₋¹⁺⁴	—	106	12	no	—
400	2₋¹⁺⁴	C₂²×D₅₃	1	8	yes	complemented
401	H(5,𝔽₂)	—	2	6	no	—
402	H(5,𝔽₂)	—	53	4	no	—
403	H(5,𝔽₂)	—	53	6	no	—
404	H(5,𝔽₂)	—	53	48	no	—
405	H(5,𝔽₂)	—	53	24	no	—
406	H(5,𝔽₂)	—	53	16	no	—
407	H(5,𝔽₂)	—	53	24	no	—
408	H(5,𝔽₂)	—	106	24	no	—
409	H(5,𝔽₂)	—	106	48	no	—
410	H(5,𝔽₂)	—	106	48	no	—
411	H(5,𝔽₂)	—	106	12	no	—
412	H(5,𝔽₂)	—	106	16	no	—
413	H(5,𝔽₂)	—	106	24	no	—
414	H(5,𝔽₂)	—	106	24	no	—
415	H(5,𝔽₂)	—	106	8	no	—
416	H(5,𝔽₂)	C₂²×D₅₃	1	8	yes	complemented
417	C₂×#41	—	2	8	no	—
418	C₂×#41	—	53	8	no	—
419	C₂×#41	—	53	6	no	—
420	C₂×#41	—	53	48	no	—
421	C₂×#41	—	53	24	no	—
422	C₂×#41	—	53	6	no	—
423	C₂×#41	—	106	12	no	—
424	C₂×#41	—	106	24	no	—
425	C₂×#41	—	106	16	no	—
426	C₂×#41	—	106	16	no	—
427	C₂×#41	—	106	48	no	—
428	C₂×#41	—	106	24	no	—
429	C₂×#41	—	106	2	no	—
430	C₂×#41	C₂²×D₅₃	1	1	yes	characteristic
431	C₂×#41	C₂²×D₅₃	1	12	yes	complemented
432	C₂²×D₄	—	53	24	no	—
433	C₂²×D₄	—	53	6	no	—
434	C₂²×D₄	—	106	12	no	—
435	C₂²×D₄	—	106	24	no	—
436	C₂²×D₄	—	106	48	no	—
437	C₂²×D₄	—	106	6	no	—
438	#137	—	53	8	no	—
439	#137	—	53	24	no	—
440	#137	—	53	48	no	—
441	#137	—	53	16	no	—
442	#137	C₂²×D₅₃	1	24	yes	complemented
443	AGL(1,ℤ/8)	—	53	48	no	—
444	AGL(1,ℤ/8)	—	53	24	no	—
445	AGL(1,ℤ/8)	—	53	24	no	—
446	AGL(1,ℤ/8)	—	53	96	no	—
447	AGL(1,ℤ/8)	—	53	48	no	—
448	AGL(1,ℤ/8)	—	53	48	no	—
449	AGL(1,ℤ/8)	—	53	48	no	—
450	AGL(1,ℤ/8)	C₂²×D₅₃	1	24	yes	complemented
451	#135	—	53	48	no	—
452	#135	—	53	24	no	—
453	#135	—	53	48	no	—
454	#135	—	53	48	no	—
455	#135	—	53	8	no	—
456	#135	—	53	12	no	—
457	#135	—	53	12	no	—
458	#135	C₂²×D₅₃	1	12	yes	complemented
459	#135	C₂²×D₅₃	1	4	yes	complemented
460	#135	C₂²×D₅₃	1	12	yes	complemented
461	#135	C₂²×D₅₃	1	12	yes	complemented
462	C₂×Q₁₆	—	53	8	no	—
463	C₂×Q₁₆	—	53	1	no	—
464	C₂×Q₁₆	C₂²×D₅₃	1	6	yes	complemented
465	C₂×SD₁₆	—	53	24	no	—
466	C₂×SD₁₆	—	53	48	no	—
467	C₂×SD₁₆	—	53	6	no	—
468	C₂×SD₁₆	C₂²×D₅₃	1	12	yes	complemented
469	C₂×D₈	—	53	24	no	—
470	C₂×D₈	—	53	24	no	—
471	C₂×D₈	—	53	48	no	—
472	C₂×D₈	—	53	12	no	—
473	C₂×D₈	—	53	3	no	—
474	C₂×D₈	—	53	12	no	—
475	C₂×D₈	—	53	6	no	—
476	C₂×D₈	C₂²×D₅₃	1	6	yes	complemented
477	#131	—	53	8	no	—
478	#131	—	53	24	no	—
479	#131	—	53	12	no	—
480	#131	—	53	4	no	—
481	#131	—	53	12	no	—
482	#131	—	53	12	no	—
483	#131	C₂²×D₅₃	1	4	yes	complemented
484	#131	C₂²×D₅₃	1	4	yes	complemented
485	C₂×M₁₆	—	53	6	no	—
486	C₂×M₁₆	—	53	24	no	—
487	C₂×M₁₆	—	53	12	no	—
488	C₂×M₁₆	C₂²×D₅₃	1	3	yes	complemented
489	C₂²×C₈	—	53	12	no	—
490	C₂²×C₈	C₂²×D₅₃	1	3	yes	complemented
491	C₂⁴	—	106	24	no	—
492	C₂⁴	—	106	6	no	—
493	#41	—	2	8	no	—
494	#41	—	2	24	no	—
495	#41	—	2	12	no	—
496	#41	—	2	12	no	—
497	#41	—	53	8	no	—
498	#41	—	53	24	no	—
499	#41	—	53	12	no	—
500	#41	—	53	12	no	—
501	#41	—	53	12	no	—
502	#41	—	53	4	no	—
503	#41	—	53	48	no	—
504	#41	—	53	16	no	—
505	#41	—	53	24	no	—
506	#41	—	106	48	no	—
507	#41	—	106	48	no	—
508	#41	—	106	8	no	—
509	#41	—	106	24	no	—
510	#41	—	106	48	no	—
511	#41	—	106	48	no	—
512	#41	—	106	48	no	—
513	#41	—	106	16	no	—
514	#41	—	106	24	no	—
515	#41	—	106	24	no	—
516	#41	—	106	16	no	—
517	#41	—	106	12	no	—
518	#41	D₄×D₅₃	1	4	yes	—
519	#41	C₂³×D₅₃	1	24	yes	—
520	#41	C₂³×D₅₃	1	8	yes	—
521	#41	C₂³×D₅₃	1	4	yes	—
522	C₂×Q₈	—	2	8	no	—
523	C₂×Q₈	—	53	8	no	—
524	C₂×Q₈	—	53	6	no	—
525	C₂×Q₈	—	53	16	no	—
526	C₂×Q₈	—	53	2	no	—
527	C₂×Q₈	—	106	8	no	—
528	C₂×Q₈	—	106	24	no	—
529	C₂×Q₈	D₄×D₅₃	1	1	yes	characteristic
530	C₂×Q₈	C₂³×D₅₃	1	12	yes	—
531	C₂×D₄	—	2	12	no	—
532	C₂×D₄	—	2	24	no	—
533	C₂×D₄	—	53	12	no	—
534	C₂×D₄	—	53	24	no	—
535	C₂×D₄	—	53	48	no	—
536	C₂×D₄	—	53	6	no	—
537	C₂×D₄	—	53	24	no	—
538	C₂×D₄	—	53	6	no	—
539	C₂×D₄	—	106	24	no	—
540	C₂×D₄	—	106	48	no	—
541	C₂×D₄	—	106	24	no	—
542	C₂×D₄	—	106	48	no	—
543	C₂×D₄	—	106	24	no	—
544	C₂×D₄	—	106	48	no	—
545	C₂×D₄	—	106	96	no	—
546	C₂×D₄	—	106	24	no	—
547	C₂×D₄	—	106	48	no	—
548	C₂×D₄	—	106	48	no	—
549	C₂×D₄	—	106	6	no	—
550	C₂×D₄	—	106	48	no	—
551	C₂×D₄	—	106	12	no	—
552	C₂×D₄	—	106	6	no	—
553	C₂×D₄	D₄×D₅₃	1	3	yes	—
554	C₂×D₄	C₂³×D₅₃	1	12	yes	—
555	C₂²×C₄	—	2	12	no	—
556	C₂²×C₄	—	53	12	no	—
557	C₂²×C₄	—	106	24	no	—
558	C₂²×C₄	—	106	48	no	—
559	C₂²×C₄	—	106	6	no	—
560	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
561	Q₁₆	—	53	16	no	—
562	Q₁₆	—	53	2	no	—
563	Q₁₆	—	53	2	no	—
564	Q₁₆	C₂³×D₅₃	1	4	yes	—
565	Q₁₆	C₂³×D₅₃	1	12	yes	—
566	SD₁₆	—	53	48	no	—
567	SD₁₆	—	53	16	no	—
568	SD₁₆	—	53	24	no	—
569	SD₁₆	C₂³×D₅₃	1	24	yes	—
570	SD₁₆	C₂³×D₅₃	1	8	yes	—
571	D₈	—	53	48	no	—
572	D₈	—	53	6	no	—
573	D₈	—	53	24	no	—
574	D₈	—	53	6	no	—
575	D₈	C₂³×D₅₃	1	4	yes	—
576	D₈	C₂³×D₅₃	1	12	yes	—
577	M₁₆	—	53	12	no	—
578	M₁₆	—	53	4	no	—
579	M₁₆	—	53	24	no	—
580	M₁₆	—	53	8	no	—
581	M₁₆	C₂³×D₅₃	1	12	yes	—
582	C₂×C₈	—	53	12	no	—
583	C₂×C₈	—	53	8	no	—
584	C₂×C₈	—	53	24	no	—
585	C₂×C₈	C₂³×D₅₃	1	6	yes	—
586	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
587	C₂×C₈	C₂³×D₅₃	1	6	yes	—
588	C₂×C₈	C₂³×D₅₃	1	3	yes	—
589	C₂³	—	2	12	no	—
590	C₂³	—	53	12	no	—
591	C₂³	—	848	6	no	—
592	C₂³	—	848	24	no	—
593	C₂³	—	106	48	no	—
594	C₂³	—	106	24	no	—
595	C₂³	—	106	6	no	—
596	C₂³	C₂×D₄×D₅₃	1	3	yes	—
597	Q₈	—	2	12	no	—
598	Q₈	—	53	4	no	—
599	Q₈	—	53	12	no	—
600	Q₈	—	53	4	no	—
601	Q₈	—	106	16	no	—
602	Q₈	—	106	2	no	—
603	Q₈	C₂×D₄×D₅₃	1	4	yes	—
604	Q₈	C₂⁴×D₅₃	1	8	yes	—
605	D₄	—	2	24	no	—
606	D₄	—	2	8	no	—
607	D₄	—	2	12	no	—
608	D₄	—	53	24	no	—
609	D₄	—	53	8	no	—
610	D₄	—	53	4	no	—
611	D₄	—	53	12	no	—
612	D₄	—	53	12	no	—
613	D₄	—	106	48	no	—
614	D₄	—	106	48	no	—
615	D₄	—	106	16	no	—
616	D₄	—	106	24	no	—
617	D₄	—	106	6	no	—
618	D₄	—	106	24	no	—
619	D₄	C₂×D₄×D₅₃	1	12	yes	—
620	D₄	C₂⁴×D₅₃	1	8	yes	—
621	C₂×C₄	—	2	12	no	—
622	C₂×C₄	—	2	8	no	—
623	C₂×C₄	—	2	24	no	—
624	C₂×C₄	—	53	12	no	—
625	C₂×C₄	—	53	24	no	—
626	C₂×C₄	—	53	8	no	—
627	C₂×C₄	—	106	48	no	—
628	C₂×C₄	—	106	16	no	—
629	C₂×C₄	—	106	16	no	—
630	C₂×C₄	—	106	2	no	—
631	C₂×C₄	—	106	24	no	—
632	C₂×C₄	C₂×D₄×D₅₃	1	6	yes	—
633	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
634	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
635	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
636	C₈	—	53	8	no	—
637	C₈	C₂⁴×D₅₃	1	1	yes	characteristic
638	C₈	C₂⁴×D₅₃	1	6	yes	—
639	C₈	C₂⁴×D₅₃	1	1	yes	characteristic
640	C₂²	—	2	8	no	—
641	C₂²	—	4	3	no	—
642	C₂²	—	8	12	no	—
643	C₂²	—	53	8	no	—
644	C₂²	—	106	12	no	—
645	C₂²	—	424	6	no	—
646	C₂²	—	424	24	no	—
647	C₂²	—	424	48	no	—
648	C₂²	—	212	12	no	—
649	C₂²	C₂²×D₄×D₅₃	1	6	yes	—
650	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
651	C₄	—	2	8	no	—
652	C₄	—	53	8	no	—
653	C₄	—	106	4	no	—
654	C₄	C₂²×D₄×D₅₃	1	1	yes	characteristic
655	C₄	C₂²×D₄×D₅₃	1	6	yes	—
656	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
657	C₂	—	2	1	no	—
658	C₂	—	2	6	no	—
659	C₂	—	4	8	no	—
660	C₂	—	106	8	no	—
661	C₂	—	212	12	no	—
662	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂³×D₄×D₅₃
commutator, cosocle	C₄×C₅₃	C₂⁶
Frattini	C₄	C₂⁵×D₅₃
Fitting	C₅₃×#3584	C₂
socle	C₂×C₅₃	C₂⁴×D₄
hypercenter	#3584	D₅₃
hypocenter	C₅₃	#63091
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ #3584

Complements

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

N	H ≅ G/N	type
1. #20289526	C₂	semidirect
2. #20289494	C₂	semidirect
3. #20289493	C₂	semidirect
4. #20289492	C₂	semidirect
5. #20289485	C₂	semidirect
6. #20289484	C₂	semidirect
7. C₂×#16811428	C₂	semidirect
8. C₂×#16811425	C₂	semidirect
9. #576×D₅₃	C₂	semidirect
10. #20289426	C₂	semidirect
11. C₅₃×#3584	C₂	semidirect
13. #16811460	C₂²	semidirect
16. #16811459	C₂²	semidirect
18. C₂×#13133263	C₂²	semidirect
19. C₂×#13133263	C₂²	semidirect
21. H(5,𝔽₂)×D₅₃	C₂²	semidirect
22. #16811437	C₂²	semidirect
23. #16811436	C₂²	semidirect
24. #16811432	C₂²	semidirect
25. #16811431	C₂²	semidirect
26. #16811430	C₂²	semidirect
27. #16811429	C₂²	semidirect
28. #16811428	C₂²	semidirect
29. #16811428	C₂²	semidirect
30. #16811427	C₂²	semidirect
31. #16811426	C₂²	semidirect
32. #16811425	C₂²	semidirect
33. #16811425	C₂²	semidirect
34. #16811425	C₂²	semidirect
35. #16811425	C₂²	semidirect
36. #16811424	C₂²	semidirect
37. #16811422	C₂²	semidirect
38. #16811421	C₂²	semidirect
39. #16811420	C₂²	semidirect
40. C₂×#13133245	C₂²	semidirect
41. C₂×#13133244	C₂²	semidirect
42. C₂×#13133242	C₂²	semidirect
43. AGL(1,ℤ/8)×D₅₃	C₂²	semidirect
44. C₂×#13133240	C₂²	semidirect
45. C₂×#13133239	C₂²	semidirect
46. #135×D₅₃	C₂²	semidirect
47. C₂²×D₄₂₄	C₂²	semidirect
48. C₂×D₈×D₅₃	C₂²	semidirect
49. #16811390	C₂²	semidirect
50. #16811389	C₂²	semidirect
51. #16811388	C₂²	semidirect
52. C₂×#13133223	C₂²	semidirect
53. #131×D₅₃	C₂²	semidirect
54. C₅₃×#585	C₂²	semidirect
55. C₅₃×#578	C₂²	semidirect
56. C₅₃×#577	C₂²	semidirect
57. C₅₃×#576	C₂²	semidirect
58. C₅₃×#575	C₂²	semidirect
59. C₂×#135×C₅₃	C₂²	semidirect
60. C₂×#131×C₅₃	C₂²	semidirect
63. #13133265	C₂³	semidirect
66. #13133264	C₂³	semidirect
74. #13133263	C₂³	semidirect
75. #13133263	C₂³	semidirect
76. #13133263	C₂³	semidirect
80. #13133262	C₂³	semidirect
83. C₂×#11719949	C₂³	semidirect
87. C₂×#11719945	C₂³	semidirect
92. #41×D₅₃	C₂³	semidirect
94. C₂²×D₂₁₂	C₂³	semidirect
98. C₂×D₄×D₅₃	C₂³	semidirect
99. #13133249	C₂³	semidirect
100. #13133248	C₂³	semidirect
101. #13133247	C₂³	semidirect
102. #13133246	C₂³	semidirect
103. #13133245	C₂³	semidirect
104. #13133244	C₂³	semidirect
105. #13133244	C₂³	semidirect
106. #13133243	C₂³	semidirect
107. #13133242	C₂³	semidirect
108. #13133242	C₂³	semidirect
109. #13133241	C₂³	semidirect
110. #13133240	C₂³	semidirect
111. #13133240	C₂³	semidirect
112. #13133239	C₂³	semidirect
113. #13133239	C₂³	semidirect
114. #13133238	C₂³	semidirect
115. #13133237	C₂³	semidirect
116. #13133236	C₂³	semidirect
117. C₂×Dic₂₁₂	C₂³	semidirect
118. Q₁₆×D₅₃	C₂³	semidirect
119. C₂×#11719913	C₂³	semidirect
120. C₂×#11719924	C₂³	semidirect
121. SD₁₆×D₅₃	C₂³	semidirect
122. C₂×D₄₂₄	C₂³	semidirect
123. C₂×D₄₂₄	C₂³	semidirect
124. C₂×D₄₂₄	C₂³	semidirect
125. C₂×D₄₂₄	C₂³	semidirect
126. C₂×#11719922	C₂³	semidirect
127. D₈×D₅₃	C₂³	semidirect
128. D₈×D₅₃	C₂³	semidirect
129. #13133225	C₂³	semidirect
130. #13133224	C₂³	semidirect
131. #13133223	C₂³	semidirect
132. #13133223	C₂³	semidirect
133. #13133223	C₂³	semidirect
134. #13133223	C₂³	semidirect
135. C₂×#11719917	C₂³	semidirect
136. C₂×#11719912	C₂³	semidirect
137. M₁₆×D₅₃	C₂³	semidirect
138. C₂×C₈×D₅₃	C₂³	semidirect
140. 2₋¹⁺⁴×C₅₃	C₂³	semidirect
142. H(5,𝔽₂)×C₅₃	C₂³	semidirect
145. C₂×#41×C₅₃	C₂³	semidirect
146. #137×C₅₃	C₂³	semidirect
147. AGL(1,ℤ/8)×C₅₃	C₂³	semidirect
148. #135×C₅₃	C₂³	semidirect
149. #135×C₅₃	C₂³	semidirect
150. #135×C₅₃	C₂³	semidirect
151. #135×C₅₃	C₂³	semidirect
152. C₂×Q₁₆×C₅₃	C₂³	semidirect
153. C₂×SD₁₆×C₅₃	C₂³	semidirect
154. C₂×D₈×C₅₃	C₂³	semidirect
155. #131×C₅₃	C₂³	semidirect
156. #131×C₅₃	C₂³	semidirect
157. C₂×M₁₆×C₅₃	C₂³	semidirect
158. C₂²×C₈×C₅₃	C₂³	semidirect
332. #3584	D₅₃	semidirect
348. #585	C₂×D₅₃	semidirect
356. #578	C₂×D₅₃	semidirect
361. #577	C₂×D₅₃	semidirect
371. #576	C₂×D₅₃	semidirect
375. #575	C₂×D₅₃	semidirect
383. C₂×#135	C₂×D₅₃	semidirect
391. C₂×#131	C₂×D₅₃	semidirect
392. C₅₃	#63091	semidirect
400. 2₋¹⁺⁴	C₂²×D₅₃	semidirect
416. H(5,𝔽₂)	C₂²×D₅₃	semidirect
431. C₂×#41	C₂²×D₅₃	semidirect
442. #137	C₂²×D₅₃	semidirect
450. AGL(1,ℤ/8)	C₂²×D₅₃	semidirect
458. #135	C₂²×D₅₃	semidirect
459. #135	C₂²×D₅₃	semidirect
460. #135	C₂²×D₅₃	semidirect
461. #135	C₂²×D₅₃	semidirect
464. C₂×Q₁₆	C₂²×D₅₃	semidirect
468. C₂×SD₁₆	C₂²×D₅₃	semidirect
476. C₂×D₈	C₂²×D₅₃	semidirect
483. #131	C₂²×D₅₃	semidirect
484. #131	C₂²×D₅₃	semidirect
488. C₂×M₁₆	C₂²×D₅₃	semidirect
490. C₂²×C₈	C₂²×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20289526	2
#20289494	2
#20289493	2
#20289492	2
#20289485	2
#20289484	2
C₂×#16811428	2
C₂×#16811425	2
#576×D₅₃	2
#20289426	2
C₅₃×#3584	2
#63091	53