Subgroups — #28260333

Order 13568.

2620 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289489	C₂	1	2	yes	maximal, complemented
2	#20289484	C₂	1	2	yes	maximal, complemented
3	#20286520	C₂	1	2	yes	maximal, complemented
4	#20286515	C₂	1	2	yes	maximal, complemented
5	#20286509	C₂	1	4	yes	maximal, complemented
6	#20286503	C₂	1	4	yes	maximal, complemented
7	#20286502	C₂	1	4	yes	maximal, complemented
8	#20285143	C₂	1	1	yes	maximal, characteristic, complemented
9	#20284436	C₂	1	2	yes	maximal, complemented
10	#20284435	C₂	1	2	yes	maximal, complemented
11	#20284066	C₂	1	4	yes	maximal, complemented
12	C₂×#16810741	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
14	#16811459	C₂²	1	4	yes	complemented
15	#16811440	—	2	2	no	—
16	#16811440	C₂²	1	4	yes	complemented
17	#16811437	C₂²	1	4	yes	complemented
18	#16811436	—	2	2	no	—
19	#16811435	—	2	4	no	—
20	#16811435	C₂²	1	8	yes	complemented
21	#16811430	C₂²	1	8	yes	complemented
22	#16811429	—	2	4	no	—
23	#16811428	—	2	2	no	—
24	#16811428	C₂²	1	4	yes	complemented
25	#16811425	C₂²	1	4	yes	complemented
26	#16811424	—	2	2	no	—
27	#16811423	—	2	4	no	—
28	#16811423	C₂²	1	4	yes	complemented
29	#16811421	C₂²	1	4	yes	complemented
30	#16811420	—	2	4	no	—
31	C₂×#13133239	—	2	4	no	—
32	C₂×#13133239	C₂²	1	1	yes	characteristic
33	C₂×#13133239	C₂²	1	2	yes	complemented
34	C₂×#13133239	C₂²	1	2	yes	complemented
35	#135×D₅₃	—	2	4	no	—
36	C₂×#13133223	C₂²	1	2	yes	complemented
37	#16811173	C₂²	1	2	yes	complemented
38	#16811007	C₂²	1	2	yes	complemented
39	#16811007	C₂²	1	2	yes	complemented
40	#16811003	C₂²	1	1	yes	characteristic, complemented
41	#16811000	C₂²	1	1	yes	characteristic, complemented
42	#16810996	C₂²	1	2	yes	complemented
43	#16810899	C₂²	1	4	yes	complemented
44	#16810898	C₂²	1	4	yes	complemented
45	#16810894	C₂²	1	4	yes	complemented
46	#16810893	C₂²	1	4	yes	complemented
47	#16810887	C₂²	1	8	yes	complemented
48	#16810886	C₂²	1	8	yes	complemented
49	#16810807	C₂²	1	8	yes	complemented
50	#16810802	C₂²	1	8	yes	complemented
51	#16810797	C₂²	1	8	yes	complemented
52	#16810792	C₂²	1	8	yes	complemented
53	#16810741	C₂²	1	2	yes	complemented
54	#16810741	C₂²	1	2	yes	complemented
55	#16810741	C₂²	1	2	yes	complemented
56	#16810741	C₂²	1	2	yes	complemented
57	C₂×#13133153	C₂²	1	1	yes	characteristic, complemented
58	C₂×#13133145	C₂²	1	2	yes	complemented
59	#16810605	C₂²	1	4	yes	complemented
60	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
61	C₂×#131×C₅₃	C₂²	1	2	yes	complemented
62	C₅₃×#443	C₂²	1	2	yes	complemented
63	C₅₃×#443	C₂²	1	2	yes	complemented
64	C₅₃×#443	C₂²	1	2	yes	complemented
65	C₅₃×#443	C₂²	1	2	yes	complemented
66	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic, complemented
67	C₂×#104×C₅₃	C₂²	1	2	yes	complemented
68	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
69	#13133265	—	2	4	no	—
70	#13133264	—	2	4	no	—
71	#13133264	C₂³	1	8	yes	—
72	#13133263	—	2	4	no	—
73	#13133263	C₂³	1	8	yes	—
74	#13133262	—	2	4	no	—
75	#13133262	—	4	4	no	—
76	C₂×#11719945	—	2	8	no	—
77	C₂×#11719945	C₂³	1	2	yes	—
78	C₂×#11719945	C₂³	1	4	yes	—
79	#41×D₅₃	—	2	8	no	—
80	#41×D₅₃	—	4	4	no	—
81	#13133250	—	2	8	no	—
82	#13133250	—	2	8	no	—
83	#13133250	—	2	4	no	—
84	#13133250	C₂³	1	8	yes	—
85	#13133249	—	2	8	no	—
86	#13133248	—	2	4	no	—
87	#13133247	C₂³	1	8	yes	—
88	#13133246	—	2	8	no	—
89	#13133245	—	2	8	no	—
90	#13133245	—	2	8	no	—
91	#13133245	—	2	4	no	—
92	#13133245	C₂³	1	8	yes	—
93	#13133244	—	2	8	no	—
94	#13133243	—	2	4	no	—
95	#13133242	C₂³	1	8	yes	—
96	#13133241	—	2	8	no	—
97	#13133240	—	2	4	no	—
98	#13133239	—	2	2	no	—
99	#13133239	—	2	2	no	—
100	#13133239	—	2	2	no	—
101	#13133239	—	2	2	no	—
102	#13133239	—	2	2	no	—
103	#13133239	—	2	2	no	—
104	#13133239	—	2	2	no	—
105	#13133239	—	2	2	no	—
106	#13133239	—	2	4	no	—
107	#13133239	—	2	4	no	—
108	#13133239	—	2	8	no	—
109	#13133239	—	2	8	no	—
110	#13133239	D₄	1	2	yes	complemented
111	#13133239	D₄	1	2	yes	complemented
112	#13133239	C₂³	1	2	yes	—
113	#13133239	C₂³	1	2	yes	—
114	#13133238	—	2	8	no	—
115	#13133237	—	2	8	no	—
116	#13133236	—	2	4	no	—
117	C₂×Dic₂₁₂	—	2	4	no	—
118	C₂×Dic₂₁₂	D₄	1	1	yes	characteristic, complemented
119	C₂×Dic₂₁₂	C₂³	1	2	yes	—
120	C₂×Dic₂₁₂	C₂³	1	2	yes	—
121	Q₁₆×D₅₃	—	2	4	no	—
122	C₂×#11719913	—	2	8	no	—
123	C₂×#11719913	D₄	1	2	yes	complemented
124	C₂×#11719913	C₂³	1	4	yes	—
125	C₂×#11719913	C₂³	1	4	yes	—
126	SD₁₆×D₅₃	—	2	8	no	—
127	C₂×D₄₂₄	—	2	4	no	—
128	C₂×D₄₂₄	D₄	1	1	yes	characteristic, complemented
129	C₂×D₄₂₄	C₂³	1	2	yes	—
130	C₂×D₄₂₄	C₂³	1	2	yes	—
131	D₈×D₅₃	—	2	4	no	—
132	#13133223	—	2	2	no	—
133	#13133223	—	2	2	no	—
134	#13133223	C₂³	1	4	yes	—
135	#13133223	C₂³	1	4	yes	—
136	C₂×#11719917	C₂³	1	2	yes	—
137	C₂×#11719912	—	2	2	no	—
138	C₂×C₈×D₅₃	—	2	2	no	—
139	#13133217	C₂³	1	2	yes	—
140	#13133214	C₂³	1	2	yes	—
141	#13133205	C₂³	1	4	yes	—
142	#13133191	—	4	2	no	—
143	C₄×D₂₁₂	—	4	2	no	—
144	#13133153	C₂³	1	2	yes	—
145	#13133153	C₂³	1	2	yes	—
146	#13133145	C₂³	1	4	yes	—
147	#13133145	C₂³	1	4	yes	—
148	#13133135	C₂³	1	8	yes	—
149	#13133133	C₂³	1	8	yes	—
150	C₂×#41×C₅₃	C₂³	1	2	yes	—
151	#135×C₅₃	D₄	1	2	yes	complemented
152	#135×C₅₃	D₄	1	2	yes	complemented
153	#135×C₅₃	C₂³	1	2	yes	—
154	#135×C₅₃	C₂³	1	2	yes	—
155	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic, complemented
156	C₂×SD₁₆×C₅₃	D₄	1	2	yes	complemented
157	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic, complemented
158	#131×C₅₃	—	2	2	no	—
159	#131×C₅₃	—	2	2	no	—
160	#131×C₅₃	C₂³	1	4	yes	—
161	#131×C₅₃	C₂³	1	4	yes	—
162	C₂×M₁₆×C₅₃	—	2	2	no	—
163	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
164	C₂²×C₈×C₅₃	—	2	2	no	—
165	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
166	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
167	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
168	#108×C₅₃	C₂³	1	2	yes	—
169	#108×C₅₃	C₂³	1	2	yes	—
170	#104×C₅₃	C₂³	1	4	yes	—
171	#104×C₅₃	C₂³	1	4	yes	—
172	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
173	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
174	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
175	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
176	#41×C₅₃	—	2	2	no	—
177	#41×C₅₃	—	2	2	no	—
178	#41×C₅₃	C₂×D₄	1	4	yes	—
179	#41×C₅₃	C₂⁴	1	4	yes	—
180	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
181	C₂×D₄×C₅₃	—	2	2	no	—
182	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₅₃	—	2	2	no	—
184	C₂²×C₄×C₅₃	—	2	1	no	—
185	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
186	C₂×C₂²⋊D₅₃	—	2	2	no	—
187	C₂×C₂²⋊D₅₃	—	2	4	no	—
188	C₂×C₂²⋊D₅₃	—	4	8	no	—
189	#11719949	—	2	8	no	—
190	#11719949	—	4	4	no	—
191	Q₈×D₅₃	—	2	8	no	—
192	Q₈×D₅₃	—	4	4	no	—
193	#11719947	—	2	8	no	—
194	#11719947	—	4	4	no	—
195	#11719947	—	4	8	no	—
196	#11719947	—	4	8	no	—
197	D₄×D₅₃	—	2	8	no	—
198	D₄×D₅₃	—	4	4	no	—
199	D₄×D₅₃	—	4	8	no	—
200	D₄×D₅₃	—	4	8	no	—
201	#11719945	—	2	2	no	—
202	#11719945	—	2	2	no	—
203	#11719945	—	2	4	no	—
204	#11719945	—	2	4	no	—
205	#11719945	—	2	4	no	—
206	#11719945	—	2	4	no	—
207	#11719945	—	2	8	no	—
208	#11719945	—	2	8	no	—
209	#11719945	—	4	8	no	—
210	#11719945	—	4	8	no	—
211	#11719945	C₂×D₄	1	4	yes	—
212	#11719945	C₂⁴	1	4	yes	—
213	C₂×D₂₁₂	—	2	8	no	—
214	C₂×D₂₁₂	—	4	2	no	—
215	C₂×D₂₁₂	C₂×D₄	1	2	yes	—
216	C₂×D₂₁₂	C₂⁴	1	4	yes	—
217	C₂×C₄×D₅₃	—	2	2	no	—
218	C₂×C₄×D₅₃	—	2	4	no	—
219	C₂×C₄×D₅₃	—	4	8	no	—
220	C₂×Dic₁₀₆	—	2	8	no	—
221	C₂×Dic₁₀₆	C₂×D₄	1	2	yes	—
222	C₂×Dic₁₀₆	C₂⁴	1	4	yes	—
223	Q₁₆×C₅₃	C₂×D₄	1	2	yes	—
224	Q₁₆×C₅₃	C₂×D₄	1	2	yes	—
225	SD₁₆×C₅₃	C₂×D₄	1	4	yes	—
226	SD₁₆×C₅₃	C₂×D₄	1	4	yes	—
227	D₈×C₅₃	C₂×D₄	1	2	yes	—
228	D₈×C₅₃	C₂×D₄	1	2	yes	—
229	M₁₆×C₅₃	—	2	2	no	—
230	M₁₆×C₅₃	—	2	4	no	—
231	M₁₆×C₅₃	—	2	4	no	—
232	M₁₆×C₅₃	C₂⁴	1	4	yes	—
233	C₂×C₈×C₅₃	—	2	2	no	—
234	C₂×C₈×C₅₃	—	2	4	no	—
235	C₂×C₈×C₅₃	—	2	4	no	—
236	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
237	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
238	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
239	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
240	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
241	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
242	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
243	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
244	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
245	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
246	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
247	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
248	C₄²×C₅₃	—	2	1	no	—
249	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
250	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
251	#11719925	—	2	8	no	—
252	#11719924	—	2	8	no	—
253	#11719923	—	2	8	no	—
254	#11719922	—	2	8	no	—
255	#11719921	—	4	4	no	—
256	C₄⋊Dic₅₃	—	4	2	no	—
257	#11719917	—	2	2	no	—
258	#11719917	C₂⁴	1	4	yes	—
259	C₂×#88322	—	2	2	no	—
260	Dic₂₁₂	—	2	8	no	—
261	Dic₂₁₂	—	2	2	no	—
262	Dic₂₁₂	—	2	2	no	—
263	Dic₂₁₂	—	2	2	no	—
264	Dic₂₁₂	—	2	2	no	—
265	Dic₂₁₂	C₂×D₄	1	2	yes	—
266	Dic₂₁₂	C₂×D₄	1	2	yes	—
267	D₄₂₄	—	2	8	no	—
268	D₄₂₄	—	2	2	no	—
269	D₄₂₄	—	2	2	no	—
270	D₄₂₄	—	2	2	no	—
271	D₄₂₄	—	2	2	no	—
272	D₄₂₄	C₂×D₄	1	2	yes	—
273	D₄₂₄	C₂×D₄	1	2	yes	—
274	#11719913	—	2	8	no	—
275	#11719913	—	2	8	no	—
276	#11719913	—	2	4	no	—
277	#11719913	—	2	4	no	—
278	#11719913	—	2	4	no	—
279	#11719913	—	2	4	no	—
280	#11719913	C₂×D₄	1	4	yes	—
281	#11719913	C₂×D₄	1	4	yes	—
282	#11719912	—	2	4	no	—
283	#11719912	—	2	4	no	—
284	C₈×D₅₃	—	2	4	no	—
285	C₈×D₅₃	—	2	4	no	—
286	C₂³×C₅₃	—	2	2	no	—
287	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
288	C₂²×D₅₃	—	2	2	no	—
289	C₂²×D₅₃	—	2	4	no	—
290	C₂²×D₅₃	—	4	8	no	—
291	Q₈×C₅₃	—	2	2	no	—
292	Q₈×C₅₃	C₂²×D₄	1	4	yes	—
293	D₄×C₅₃	—	2	4	no	—
294	D₄×C₅₃	—	2	4	no	—
295	D₄×C₅₃	—	2	2	no	—
296	D₄×C₅₃	C₂²×D₄	1	4	yes	—
297	C₂×C₄×C₅₃	—	2	2	no	—
298	C₂×C₄×C₅₃	—	2	1	no	—
299	C₂×C₄×C₅₃	—	2	4	no	—
300	C₂×C₄×C₅₃	—	2	4	no	—
301	C₂×C₄×C₅₃	—	2	1	no	—
302	C₂×C₄×C₅₃	—	4	2	no	—
303	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
304	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
305	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
306	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
308	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
309	C₂²⋊D₅₃	—	2	4	no	—
310	C₂²⋊D₅₃	—	2	4	no	—
311	C₂²⋊D₅₃	—	4	8	no	—
312	C₂²⋊D₅₃	—	4	8	no	—
313	C₂²⋊D₅₃	—	4	4	no	—
314	C₂²⋊D₅₃	—	4	4	no	—
315	C₂×Dic₅₃	—	2	2	no	—
316	C₂×Dic₅₃	—	2	4	no	—
317	C₂×Dic₅₃	—	4	8	no	—
318	D₂₁₂	—	2	2	no	—
319	D₂₁₂	—	2	4	no	—
320	D₂₁₂	—	2	4	no	—
321	D₂₁₂	—	4	8	no	—
322	D₂₁₂	—	8	4	no	—
323	D₂₁₂	C₂²×D₄	1	4	yes	—
324	C₄×D₅₃	—	2	4	no	—
325	C₄×D₅₃	—	2	4	no	—
326	C₄×D₅₃	—	4	8	no	—
327	C₄×D₅₃	—	4	8	no	—
328	C₄×D₅₃	—	4	4	no	—
329	C₄×D₅₃	—	4	4	no	—
330	Dic₁₀₆	—	2	2	no	—
331	Dic₁₀₆	—	2	4	no	—
332	Dic₁₀₆	—	2	4	no	—
333	Dic₁₀₆	—	4	8	no	—
334	Dic₁₀₆	C₂²×D₄	1	4	yes	—
335	C₈×C₅₃	—	2	4	no	—
336	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
337	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
338	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
339	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
340	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
341	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
342	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
343	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
344	#88322	—	2	4	no	—
345	#61105	—	53	1	no	maximal, Sylow
346	C₂²×C₅₃	—	2	4	no	—
347	C₂²×C₅₃	—	4	1	no	—
348	C₂²×C₅₃	—	8	2	no	—
349	C₂²×C₅₃	D₄²	1	1	yes	characteristic
350	C₂²×C₅₃	D₄²	1	1	yes	characteristic
351	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
352	C₂×D₅₃	—	2	4	no	—
353	C₂×D₅₃	—	4	4	no	—
354	C₂×D₅₃	—	8	2	no	—
355	C₂×D₅₃	—	8	4	no	—
356	C₂×D₅₃	—	16	8	no	—
357	C₄×C₅₃	—	2	2	no	—
358	C₄×C₅₃	—	2	4	no	—
359	C₄×C₅₃	—	4	1	no	—
360	C₄×C₅₃	D₄²	1	1	yes	characteristic
361	C₄×C₅₃	D₄²	1	1	yes	characteristic
362	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
363	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
364	Dic₅₃	—	2	4	no	—
365	Dic₅₃	—	4	4	no	—
366	#3584	—	53	2	no	—
367	#3584	—	53	2	no	—
368	#3293	—	53	2	no	—
369	#3292	—	53	2	no	—
370	#3291	—	53	4	no	—
371	#3290	—	53	4	no	—
372	#3290	—	53	4	no	—
373	#3147	—	53	1	no	—
374	#3067	—	53	2	no	—
375	#3067	—	53	2	no	—
376	#3018	—	53	4	no	—
377	C₂×#443	—	53	1	no	—
378	C₂×#443	D₅₃	1	1	yes	characteristic, complemented
379	C₂×C₅₃	—	2	1	no	—
380	C₂×C₅₃	—	2	1	no	—
381	C₂×C₅₃	—	2	1	no	—
382	C₂×C₅₃	—	4	4	no	—
383	C₂×C₅₃	C₂×D₄²	1	1	yes	characteristic
384	D₅₃	—	4	4	no	—
385	D₅₃	—	8	4	no	—
386	#585	—	53	4	no	—
387	#578	—	53	4	no	—
388	#578	—	53	4	no	—
389	#578	—	106	2	no	—
390	#578	—	106	2	no	—
391	#577	—	53	8	no	—
392	#577	—	53	8	no	—
393	#577	—	106	4	no	—
394	#577	—	106	4	no	—
395	#576	—	53	4	no	—
396	#576	—	53	4	no	—
397	#576	—	106	2	no	—
398	#576	—	106	2	no	—
399	#575	—	53	4	no	—
400	#575	—	53	4	no	—
401	#575	—	106	4	no	—
402	#575	—	106	4	no	—
403	C₂×#135	—	53	1	no	—
404	C₂×#135	—	53	2	no	—
405	C₂×#135	—	53	2	no	—
406	C₂×#135	—	106	4	no	—
407	C₂×#135	—	106	4	no	—
408	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
409	C₂×#131	—	53	2	no	—
410	C₂×#131	C₂×D₅₃	1	2	yes	complemented
411	#532	—	53	2	no	—
412	#495	—	53	2	no	—
413	#495	—	53	2	no	—
414	#494	—	53	1	no	—
415	#493	—	53	1	no	—
416	#492	—	53	2	no	—
417	#473	—	53	4	no	—
418	#473	—	53	4	no	—
419	#472	—	53	4	no	—
420	#472	—	53	4	no	—
421	#471	—	53	8	no	—
422	#471	—	53	8	no	—
423	#456	—	53	8	no	—
424	#455	—	53	8	no	—
425	#454	—	53	8	no	—
426	#453	—	53	8	no	—
427	#443	—	53	2	no	—
428	#443	—	53	2	no	—
429	#443	—	53	2	no	—
430	#443	—	53	2	no	—
431	#443	C₂×D₅₃	1	2	yes	complemented
432	#443	C₂×D₅₃	1	2	yes	complemented
433	#443	C₂×D₅₃	1	2	yes	complemented
434	#443	C₂×D₅₃	1	2	yes	complemented
435	C₂×#108	—	53	1	no	—
436	C₂×#108	C₂×D₅₃	1	1	yes	characteristic, complemented
437	C₂×#104	—	53	2	no	—
438	C₂×#104	C₂×D₅₃	1	2	yes	complemented
439	#413	—	53	4	no	—
440	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
441	C₅₃	#61105	1	1	yes	characteristic, complemented, Sylow
442	2₋¹⁺⁴	—	53	8	no	—
443	2₋¹⁺⁴	—	106	4	no	—
444	2₋¹⁺⁴	—	106	4	no	—
445	H(5,𝔽₂)	—	53	8	no	—
446	H(5,𝔽₂)	—	106	4	no	—
447	H(5,𝔽₂)	—	106	4	no	—
448	H(5,𝔽₂)	—	212	4	no	—
449	C₂×#41	—	53	2	no	—
450	C₂×#41	—	53	4	no	—
451	C₂×#41	—	106	8	no	—
452	C₂×#41	—	106	8	no	—
453	C₂×#41	—	212	4	no	—
454	C₂×#41	C₂²×D₅₃	1	2	yes	—
455	#137	—	53	8	no	—
456	#137	—	53	8	no	—
457	#137	—	106	8	no	—
458	#137	—	106	8	no	—
459	#137	—	106	4	no	—
460	#137	—	106	4	no	—
461	#137	—	106	8	no	—
462	#137	—	106	8	no	—
463	AGL(1,ℤ/8)	—	53	8	no	—
464	AGL(1,ℤ/8)	—	53	8	no	—
465	AGL(1,ℤ/8)	—	106	8	no	—
466	AGL(1,ℤ/8)	—	106	8	no	—
467	AGL(1,ℤ/8)	—	106	4	no	—
468	AGL(1,ℤ/8)	—	106	4	no	—
469	AGL(1,ℤ/8)	—	106	8	no	—
470	AGL(1,ℤ/8)	—	106	8	no	—
471	#135	—	53	2	no	—
472	#135	—	53	2	no	—
473	#135	—	53	2	no	—
474	#135	—	53	2	no	—
475	#135	—	106	4	no	—
476	#135	—	106	8	no	—
477	#135	—	106	4	no	—
478	#135	—	106	8	no	—
479	#135	—	106	8	no	—
480	#135	—	106	4	no	—
481	#135	—	106	8	no	—
482	#135	—	106	4	no	—
483	#135	—	106	2	no	—
484	#135	—	106	2	no	—
485	#135	—	106	2	no	—
486	#135	—	106	2	no	—
487	#135	—	106	2	no	—
488	#135	—	106	2	no	—
489	#135	—	106	2	no	—
490	#135	—	106	2	no	—
491	#135	D₂₁₂	1	2	yes	complemented
492	#135	D₂₁₂	1	2	yes	complemented
493	#135	C₂²×D₅₃	1	2	yes	—
494	#135	C₂²×D₅₃	1	2	yes	—
495	C₂×Q₁₆	—	53	1	no	—
496	C₂×Q₁₆	—	53	2	no	—
497	C₂×Q₁₆	—	53	2	no	—
498	C₂×Q₁₆	—	106	4	no	—
499	C₂×Q₁₆	—	106	4	no	—
500	C₂×Q₁₆	D₂₁₂	1	1	yes	characteristic, complemented
501	C₂×SD₁₆	—	53	2	no	—
502	C₂×SD₁₆	—	53	4	no	—
503	C₂×SD₁₆	—	53	4	no	—
504	C₂×SD₁₆	—	106	8	no	—
505	C₂×SD₁₆	—	106	8	no	—
506	C₂×SD₁₆	D₂₁₂	1	2	yes	complemented
507	C₂×D₈	—	53	1	no	—
508	C₂×D₈	—	53	2	no	—
509	C₂×D₈	—	53	2	no	—
510	C₂×D₈	—	106	4	no	—
511	C₂×D₈	—	106	4	no	—
512	C₂×D₈	D₂₁₂	1	1	yes	characteristic, complemented
513	#131	—	2	2	no	—
514	#131	—	2	2	no	—
515	#131	—	53	4	no	—
516	#131	—	53	4	no	—
517	#131	—	106	2	no	—
518	#131	—	106	2	no	—
519	#131	C₂²×D₅₃	1	4	yes	—
520	#131	C₂²×D₅₃	1	4	yes	—
521	C₂×M₁₆	—	2	2	no	—
522	C₂×M₁₆	—	53	2	no	—
523	C₂×M₁₆	—	106	2	no	—
524	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
525	C₂²×C₈	—	2	2	no	—
526	C₂²×C₈	—	106	2	no	—
527	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
528	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
529	C₄⋊Q₈	—	53	2	no	—
530	#127	—	53	2	no	—
531	#124	—	53	4	no	—
532	#121	—	212	2	no	—
533	C₄×D₄	—	212	2	no	—
534	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
535	#108	—	53	2	no	—
536	#108	—	53	2	no	—
537	#108	C₂²×D₅₃	1	2	yes	—
538	#108	C₂²×D₅₃	1	2	yes	—
539	#104	—	53	4	no	—
540	#104	—	53	4	no	—
541	#104	C₂²×D₅₃	1	4	yes	—
542	#104	C₂²×D₅₃	1	4	yes	—
543	#101	—	53	8	no	—
544	#100	—	53	8	no	—
545	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
546	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
547	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
548	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
549	#41	—	2	2	no	—
550	#41	—	2	2	no	—
551	#41	—	53	4	no	—
552	#41	—	53	4	no	—
553	#41	—	106	8	no	—
554	#41	—	106	8	no	—
555	#41	—	106	2	no	—
556	#41	—	106	2	no	—
557	#41	—	106	8	no	—
558	#41	—	106	8	no	—
559	#41	—	106	4	no	—
560	#41	—	106	4	no	—
561	#41	—	106	4	no	—
562	#41	—	106	4	no	—
563	#41	—	212	8	no	—
564	#41	—	212	8	no	—
565	#41	—	212	8	no	—
566	#41	—	212	8	no	—
567	#41	—	212	4	no	—
568	#41	—	212	4	no	—
569	#41	C₂×D₂₁₂	1	4	yes	—
570	#41	C₂³×D₅₃	1	4	yes	—
571	C₂×Q₈	—	53	2	no	—
572	C₂×Q₈	—	53	4	no	—
573	C₂×Q₈	—	106	8	no	—
574	C₂×Q₈	—	106	8	no	—
575	C₂×Q₈	—	212	4	no	—
576	C₂×Q₈	C₂×D₂₁₂	1	2	yes	—
577	C₂×D₄	—	2	2	no	—
578	C₂×D₄	—	53	2	no	—
579	C₂×D₄	—	53	4	no	—
580	C₂×D₄	—	106	8	no	—
581	C₂×D₄	—	106	2	no	—
582	C₂×D₄	—	106	8	no	—
583	C₂×D₄	—	106	4	no	—
584	C₂×D₄	—	212	2	no	—
585	C₂×D₄	—	212	8	no	—
586	C₂×D₄	—	212	4	no	—
587	C₂×D₄	—	212	8	no	—
588	C₂×D₄	—	212	8	no	—
589	C₂×D₄	C₂×D₂₁₂	1	2	yes	—
590	C₂²×C₄	—	2	2	no	—
591	C₂²×C₄	—	2	1	no	—
592	C₂²×C₄	—	106	2	no	—
593	C₂²×C₄	—	106	4	no	—
594	C₂²×C₄	—	212	8	no	—
595	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
596	Q₁₆	—	53	2	no	—
597	Q₁₆	—	53	2	no	—
598	Q₁₆	—	106	8	no	—
599	Q₁₆	—	106	8	no	—
600	Q₁₆	—	106	2	no	—
601	Q₁₆	—	106	2	no	—
602	Q₁₆	—	106	2	no	—
603	Q₁₆	—	106	2	no	—
604	Q₁₆	C₂×D₂₁₂	1	2	yes	—
605	Q₁₆	C₂×D₂₁₂	1	2	yes	—
606	SD₁₆	—	53	4	no	—
607	SD₁₆	—	53	4	no	—
608	SD₁₆	—	106	8	no	—
609	SD₁₆	—	106	8	no	—
610	SD₁₆	—	106	8	no	—
611	SD₁₆	—	106	8	no	—
612	SD₁₆	—	106	4	no	—
613	SD₁₆	—	106	4	no	—
614	SD₁₆	—	106	4	no	—
615	SD₁₆	—	106	4	no	—
616	SD₁₆	C₂×D₂₁₂	1	4	yes	—
617	SD₁₆	C₂×D₂₁₂	1	4	yes	—
618	D₈	—	53	2	no	—
619	D₈	—	53	2	no	—
620	D₈	—	106	8	no	—
621	D₈	—	106	8	no	—
622	D₈	—	106	2	no	—
623	D₈	—	106	2	no	—
624	D₈	—	106	2	no	—
625	D₈	—	106	2	no	—
626	D₈	C₂×D₂₁₂	1	2	yes	—
627	D₈	C₂×D₂₁₂	1	2	yes	—
628	M₁₆	—	2	2	no	—
629	M₁₆	—	2	4	no	—
630	M₁₆	—	2	4	no	—
631	M₁₆	—	53	4	no	—
632	M₁₆	—	106	2	no	—
633	M₁₆	—	106	4	no	—
634	M₁₆	—	106	4	no	—
635	M₁₆	C₂³×D₅₃	1	4	yes	—
636	C₂×C₈	—	2	2	no	—
637	C₂×C₈	—	2	4	no	—
638	C₂×C₈	—	2	4	no	—
639	C₂×C₈	—	106	2	no	—
640	C₂×C₈	—	106	4	no	—
641	C₂×C₈	—	106	4	no	—
642	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
643	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
644	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
645	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
646	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
647	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
648	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
649	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
650	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
651	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
652	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
653	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
654	C₄⋊C₄	—	212	2	no	—
655	C₂²⋊C₄	—	212	4	no	—
656	C₄²	—	2	1	no	—
657	C₄²	C₂³×D₅₃	1	1	yes	characteristic
658	C₄²	C₂³×D₅₃	1	1	yes	characteristic
659	C₂³	—	2	2	no	—
660	C₂³	—	106	2	no	—
661	C₂³	—	106	4	no	—
662	C₂³	—	212	8	no	—
663	C₂³	#13133263	1	1	yes	characteristic
664	Q₈	—	2	2	no	—
665	Q₈	—	53	4	no	—
666	Q₈	—	106	2	no	—
667	Q₈	—	106	4	no	—
668	Q₈	—	106	4	no	—
669	Q₈	—	212	8	no	—
670	Q₈	C₂²×D₂₁₂	1	4	yes	—
671	D₄	—	2	4	no	—
672	D₄	—	2	4	no	—
673	D₄	—	2	2	no	—
674	D₄	—	53	4	no	—
675	D₄	—	106	4	no	—
676	D₄	—	106	4	no	—
677	D₄	—	106	4	no	—
678	D₄	—	106	4	no	—
679	D₄	—	106	2	no	—
680	D₄	—	424	4	no	—
681	D₄	—	212	8	no	—
682	D₄	—	212	8	no	—
683	D₄	—	212	8	no	—
684	D₄	—	212	4	no	—
685	D₄	—	212	4	no	—
686	D₄	C₂²×D₂₁₂	1	4	yes	—
687	C₂×C₄	—	2	1	no	—
688	C₂×C₄	—	2	2	no	—
689	C₂×C₄	—	2	4	no	—
690	C₂×C₄	—	2	4	no	—
691	C₂×C₄	—	2	1	no	—
692	C₂×C₄	—	4	2	no	—
693	C₂×C₄	—	106	2	no	—
694	C₂×C₄	—	106	4	no	—
695	C₂×C₄	—	106	4	no	—
696	C₂×C₄	—	106	4	no	—
697	C₂×C₄	—	212	8	no	—
698	C₂×C₄	—	212	8	no	—
699	C₂×C₄	—	212	8	no	—
700	C₂×C₄	—	212	4	no	—
701	C₂×C₄	—	212	4	no	—
702	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
703	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
704	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
705	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
706	C₂×C₄	#13133263	1	1	yes	characteristic
707	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
708	C₈	—	2	4	no	—
709	C₈	—	106	4	no	—
710	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
711	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
712	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
713	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
714	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
715	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
716	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
717	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
718	C₂²	—	2	4	no	—
719	C₂²	—	4	1	no	—
720	C₂²	—	8	2	no	—
721	C₂²	—	848	8	no	—
722	C₂²	—	106	4	no	—
723	C₂²	—	424	4	no	—
724	C₂²	—	424	2	no	—
725	C₂²	—	212	4	no	—
726	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
727	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
728	C₂²	C₂×#13133263	1	1	yes	characteristic
729	C₄	—	2	2	no	—
730	C₄	—	2	4	no	—
731	C₄	—	4	1	no	—
732	C₄	—	106	4	no	—
733	C₄	—	212	4	no	—
734	C₄	D₄×D₂₁₂	1	1	yes	characteristic
735	C₄	D₄×D₂₁₂	1	1	yes	characteristic
736	C₄	C₂²×D₄×D₅₃	1	1	yes	characteristic
737	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
738	C₂	—	2	1	no	—
739	C₂	—	2	1	no	—
740	C₂	—	2	1	no	—
741	C₂	—	4	4	no	—
742	C₂	—	424	4	no	—
743	C₂	—	212	4	no	—
744	C₂	C₂×D₄×D₂₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₁₂
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂×C₅₃	C₂×D₄²
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	#61105
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #20289489	C₂	semidirect
2. #20289484	C₂	semidirect
3. #20286520	C₂	semidirect
4. #20286515	C₂	semidirect
5. #20286509	C₂	semidirect
6. #20286503	C₂	semidirect
7. #20286502	C₂	semidirect
8. #20285143	C₂	semidirect
9. #20284436	C₂	semidirect
10. #20284435	C₂	semidirect
11. #20284066	C₂	semidirect
12. C₂×#16810741	C₂	semidirect
13. C₂×C₅₃×#443	C₂	semidirect
14. #16811459	C₂²	semidirect
16. #16811440	C₂²	semidirect
17. #16811437	C₂²	semidirect
20. #16811435	C₂²	semidirect
21. #16811430	C₂²	semidirect
24. #16811428	C₂²	semidirect
25. #16811425	C₂²	semidirect
28. #16811423	C₂²	semidirect
29. #16811421	C₂²	semidirect
33. C₂×#13133239	C₂²	semidirect
34. C₂×#13133239	C₂²	semidirect
36. C₂×#13133223	C₂²	semidirect
37. #16811173	C₂²	semidirect
38. #16811007	C₂²	semidirect
39. #16811007	C₂²	semidirect
40. #16811003	C₂²	semidirect
41. #16811000	C₂²	semidirect
42. #16810996	C₂²	semidirect
43. #16810899	C₂²	semidirect
44. #16810898	C₂²	semidirect
45. #16810894	C₂²	semidirect
46. #16810893	C₂²	semidirect
47. #16810887	C₂²	semidirect
48. #16810886	C₂²	semidirect
49. #16810807	C₂²	semidirect
50. #16810802	C₂²	semidirect
51. #16810797	C₂²	semidirect
52. #16810792	C₂²	semidirect
53. #16810741	C₂²	semidirect
54. #16810741	C₂²	semidirect
55. #16810741	C₂²	semidirect
56. #16810741	C₂²	semidirect
57. C₂×#13133153	C₂²	semidirect
58. C₂×#13133145	C₂²	semidirect
59. #16810605	C₂²	semidirect
61. C₂×#131×C₅₃	C₂²	semidirect
62. C₅₃×#443	C₂²	semidirect
63. C₅₃×#443	C₂²	semidirect
64. C₅₃×#443	C₂²	semidirect
65. C₅₃×#443	C₂²	semidirect
66. C₂×#108×C₅₃	C₂²	semidirect
67. C₂×#104×C₅₃	C₂²	semidirect
68. C₂×C₄×C₈×C₅₃	C₂²	semidirect
110. #13133239	D₄	semidirect
111. #13133239	D₄	semidirect
118. C₂×Dic₂₁₂	D₄	semidirect
123. C₂×#11719913	D₄	semidirect
128. C₂×D₄₂₄	D₄	semidirect
151. #135×C₅₃	D₄	semidirect
152. #135×C₅₃	D₄	semidirect
155. C₂×Q₁₆×C₅₃	D₄	semidirect
156. C₂×SD₁₆×C₅₃	D₄	semidirect
157. C₂×D₈×C₅₃	D₄	semidirect
378. C₂×#443	D₅₃	semidirect
410. C₂×#131	C₂×D₅₃	semidirect
431. #443	C₂×D₅₃	semidirect
432. #443	C₂×D₅₃	semidirect
433. #443	C₂×D₅₃	semidirect
434. #443	C₂×D₅₃	semidirect
436. C₂×#108	C₂×D₅₃	semidirect
438. C₂×#104	C₂×D₅₃	semidirect
440. C₂×C₄×C₈	C₂×D₅₃	semidirect
441. C₅₃	#61105	semidirect
491. #135	D₂₁₂	semidirect
492. #135	D₂₁₂	semidirect
500. C₂×Q₁₆	D₂₁₂	semidirect
506. C₂×SD₁₆	D₂₁₂	semidirect
512. C₂×D₈	D₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
#20289489	2
#20289484	2
#20286520	2
#20286515	2
#20286509	2
#20286503	2
#20286502	2
#20285143	2
#20284436	2
#20284435	2
#20284066	2
C₂×#16810741	2
C₂×C₅₃×#443	2
#61105	53