Subgroups — #27540895

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287864	C₂	1	1	yes	maximal, characteristic, complemented
2	#20287852	C₂	1	1	yes	maximal, characteristic, complemented
3	#20286500	C₂	1	2	yes	maximal, complemented
4	#20276837	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276832	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276781	C₂	1	2	yes	maximal, complemented
7	#20276685	C₂	1	2	yes	maximal, complemented
8	#20276678	C₂	1	2	yes	maximal, complemented
9	#20276597	C₂	1	1	yes	maximal, characteristic, complemented
10	#20276553	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2249	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811436	C₂²	1	2	yes	complemented
13	#16811429	—	2	2	no	—
14	#16811424	C₂²	1	2	yes	—
15	#16811065	—	2	2	no	—
16	#16811064	C₂²	1	2	yes	complemented
17	#16811059	C₂²	1	2	yes	—
18	#16811057	—	2	2	no	—
19	#16811056	—	2	1	no	—
20	#16811055	C₂²	1	1	yes	characteristic
21	#16811052	—	2	1	no	—
22	Q₃₂×D₅₃	—	2	1	no	—
23	D₁₆×D₅₃	—	2	1	no	—
24	#16811039	C₂²	1	1	yes	characteristic
25	#16811039	C₂²	1	1	yes	characteristic, complemented
26	#16810895	C₂²	1	2	yes	—
27	#16810890	C₂²	1	2	yes	complemented
28	#16810883	—	2	2	no	—
29	#16810800	—	2	2	no	—
30	#16810795	—	2	2	no	—
31	#16810741	C₂²	1	1	yes	characteristic
32	#16810741	C₂²	1	2	yes	complemented
33	#16810453	C₂²	1	1	yes	characteristic, complemented
34	#16810444	C₂²	1	1	yes	characteristic, complemented
35	#16810439	C₂²	1	2	yes	complemented
36	#16810436	C₂²	1	2	yes	complemented
37	#16810431	C₂²	1	2	yes	complemented
38	#16810406	C₂²	1	1	yes	characteristic
39	#16810406	C₂²	1	1	yes	characteristic, complemented
40	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
41	C₂×SD₃₂×C₅₃	C₂²	1	2	yes	complemented
42	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic
43	C₅₃×#495	C₂²	1	2	yes	complemented
44	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
45	#13133265	—	2	2	no	—
46	#13133262	—	2	2	no	—
47	#13133249	—	4	2	no	—
48	#13133248	—	2	2	no	—
49	#13133248	—	2	2	no	—
50	#13133246	—	4	2	no	—
51	#13133244	—	4	2	no	—
52	#13133243	—	2	2	no	—
53	#13133243	—	2	2	no	—
54	#13133241	—	4	2	no	—
55	#13133240	—	2	2	no	—
56	#13133239	D₄	1	1	yes	characteristic
57	#13133239	C₂³	1	2	yes	—
58	#13133238	—	4	2	no	—
59	#13133236	—	2	2	no	—
60	Q₁₆×D₅₃	—	2	2	no	—
61	SD₁₆×D₅₃	—	4	2	no	—
62	D₈×D₅₃	—	2	2	no	—
63	#13133223	—	2	2	no	—
64	#13133223	C₂³	1	1	yes	characteristic
65	#13133223	C₂³	1	1	yes	characteristic
66	Dic₄₂₄	D₄	1	1	yes	characteristic
67	#13133163	—	2	2	no	—
68	#13133162	D₄	1	2	yes	—
69	#13133161	—	2	2	no	—
70	#13133160	—	2	2	no	—
71	D₈₄₈	D₄	1	1	yes	characteristic
72	#13133158	—	2	2	no	—
73	#13133157	C₂³	1	1	yes	characteristic
74	#13133157	C₂³	1	1	yes	characteristic
75	#13133156	—	2	1	no	—
76	#13133156	—	2	1	no	—
77	C₁₆×D₅₃	—	2	1	no	—
78	C₁₆×D₅₃	—	2	1	no	—
79	#13133153	D₄	1	1	yes	characteristic
80	#13133153	C₂³	1	2	yes	—
81	#13133145	—	2	2	no	—
82	#13133145	D₄	1	1	yes	characteristic
83	#13133145	D₄	1	1	yes	characteristic
84	#13133136	—	2	2	no	—
85	#13133134	—	2	2	no	—
86	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
87	C₂×SD₁₆×C₅₃	—	2	2	no	—
88	C₂×D₈×C₅₃	C₂³	1	2	yes	—
89	#124×C₅₃	—	2	2	no	—
90	Q₃₂×C₅₃	—	2	1	no	—
91	Q₃₂×C₅₃	—	2	1	no	—
92	SD₃₂×C₅₃	—	4	2	no	—
93	D₁₆×C₅₃	—	2	1	no	—
94	D₁₆×C₅₃	—	2	1	no	—
95	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
96	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₅₃	—	2	2	no	—
99	C₂×D₄×C₅₃	—	2	2	no	—
100	C₂×C₂²⋊D₅₃	—	8	2	no	—
101	#11719949	—	4	2	no	—
102	#11719949	—	4	2	no	—
103	Q₈×D₅₃	—	4	2	no	—
104	Q₈×D₅₃	—	4	2	no	—
105	#11719947	—	4	2	no	—
106	#11719947	—	4	2	no	—
107	D₄×D₅₃	—	4	2	no	—
108	D₄×D₅₃	—	4	2	no	—
109	#11719945	—	2	2	no	—
110	#11719945	—	8	2	no	—
111	#11719945	C₂×D₄	1	1	yes	characteristic
112	#11719945	C₂×D₄	1	1	yes	characteristic
113	Q₁₆×C₅₃	—	2	2	no	—
114	Q₁₆×C₅₃	—	2	2	no	—
115	SD₁₆×C₅₃	—	4	2	no	—
116	SD₁₆×C₅₃	—	4	2	no	—
117	D₈×C₅₃	—	2	2	no	—
118	D₈×C₅₃	—	2	2	no	—
119	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊C₄×C₅₃	—	8	2	no	—
122	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
123	#11719925	—	4	2	no	—
124	#11719925	—	4	2	no	—
125	#11719924	—	4	2	no	—
126	#11719924	—	4	2	no	—
127	#11719923	—	4	2	no	—
128	#11719923	—	4	2	no	—
129	#11719922	—	4	2	no	—
130	#11719922	—	4	2	no	—
131	#11719917	—	2	2	no	—
132	#11719917	C₂×D₄	1	1	yes	characteristic
133	#11719917	C₂×D₄	1	1	yes	characteristic
134	Dic₂₁₂	D₈	1	1	yes	characteristic
135	Dic₂₁₂	C₂×D₄	1	2	yes	—
136	D₄₂₄	D₈	1	1	yes	characteristic
137	D₄₂₄	C₂×D₄	1	2	yes	—
138	#11719913	—	2	2	no	—
139	#11719913	D₈	1	1	yes	characteristic
140	#11719913	D₈	1	1	yes	characteristic
141	#11719912	—	2	1	no	—
142	#11719912	—	2	1	no	—
143	#11719912	—	4	2	no	—
144	C₈×D₅₃	—	2	1	no	—
145	C₈×D₅₃	—	2	1	no	—
146	C₈×D₅₃	—	4	2	no	—
147	C₁₆×C₅₃	—	2	1	no	—
148	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
149	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
150	#11719908	—	2	1	no	—
151	#11719908	—	2	1	no	—
152	C₂³×C₅₃	—	4	2	no	—
153	C₂²×D₅₃	—	8	2	no	—
154	Q₈×C₅₃	—	4	2	no	—
155	Q₈×C₅₃	—	4	2	no	—
156	D₄×C₅₃	—	4	2	no	—
157	D₄×C₅₃	—	4	2	no	—
158	C₂×C₄×C₅₃	—	2	1	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
161	C₂²⋊D₅₃	—	2	1	no	—
162	C₂²⋊D₅₃	—	2	1	no	—
163	C₂²⋊D₅₃	—	4	2	no	—
164	C₂²⋊D₅₃	—	8	2	no	—
165	C₂²⋊D₅₃	—	8	2	no	—
166	C₂×Dic₅₃	—	8	2	no	—
167	D₂₁₂	—	2	2	no	—
168	D₂₁₂	—	8	2	no	—
169	D₂₁₂	C₂×D₈	1	1	yes	characteristic
170	D₂₁₂	C₂×D₈	1	1	yes	characteristic
171	C₄×D₅₃	—	2	1	no	—
172	C₄×D₅₃	—	2	1	no	—
173	C₄×D₅₃	—	4	2	no	—
174	C₄×D₅₃	—	8	2	no	—
175	C₄×D₅₃	—	8	2	no	—
176	Dic₁₀₆	—	2	2	no	—
177	Dic₁₀₆	—	8	2	no	—
178	Dic₁₀₆	C₂×D₈	1	1	yes	characteristic
179	Dic₁₀₆	C₂×D₈	1	1	yes	characteristic
180	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
181	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
182	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
183	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
184	#88322	—	2	1	no	—
185	#88322	—	2	1	no	—
186	#88322	—	4	2	no	—
187	#33859	—	53	1	no	maximal, Sylow
188	C₂²×C₅₃	—	8	2	no	—
189	C₂²×C₅₃	—	16	2	no	—
190	C₂²×C₅₃	#576	1	1	yes	characteristic
191	C₂×D₅₃	—	2	1	no	—
192	C₂×D₅₃	—	2	1	no	—
193	C₂×D₅₃	—	4	2	no	—
194	C₂×D₅₃	—	32	2	no	—
195	C₄×C₅₃	—	4	1	no	—
196	C₄×C₅₃	—	8	2	no	—
197	C₄×C₅₃	D₄²	1	1	yes	characteristic
198	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
199	Dic₅₃	—	2	1	no	—
200	Dic₅₃	—	2	1	no	—
201	Dic₅₃	—	4	2	no	—
202	#3417	—	53	1	no	—
203	#3415	—	53	1	no	—
204	#3290	—	53	2	no	—
205	#2249	D₅₃	1	1	yes	characteristic, complemented
206	#2222	—	53	1	no	—
207	#2221	—	53	1	no	—
208	#2212	—	53	2	no	—
209	#2194	—	53	2	no	—
210	#2193	—	53	2	no	—
211	#2178	—	53	1	no	—
212	#2170	—	53	1	no	—
213	C₂×C₅₃	—	2	1	no	—
214	C₂×C₅₃	—	16	2	no	—
215	C₂×C₅₃	D₄×D₈	1	1	yes	characteristic
216	D₅₃	—	4	1	no	—
217	D₅₃	—	4	1	no	—
218	D₅₃	—	8	2	no	—
219	#578	—	53	2	no	—
220	#577	—	106	2	no	—
221	#576	—	53	2	no	—
222	#510	—	53	2	no	—
223	#510	—	106	2	no	—
224	#509	—	53	2	no	—
225	#509	—	106	2	no	—
226	#508	—	53	1	no	—
227	#508	—	106	1	no	—
228	#508	—	106	1	no	—
229	C₂×Q₃₂	—	106	1	no	—
230	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
231	C₂×SD₃₂	C₂×D₅₃	1	2	yes	complemented
232	C₂×D₁₆	—	106	1	no	—
233	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic
234	#504	—	53	1	no	—
235	#504	—	53	1	no	—
236	#495	C₂×D₅₃	1	2	yes	complemented
237	#473	—	53	2	no	—
238	#472	—	53	2	no	—
239	#471	—	106	2	no	—
240	#455	—	106	2	no	—
241	#454	—	106	2	no	—
242	#443	—	53	1	no	—
243	#443	—	53	2	no	—
244	#368	—	53	1	no	—
245	#364	—	53	1	no	—
246	#362	—	53	2	no	—
247	#361	—	53	2	no	—
248	#359	—	53	2	no	—
249	#350	—	53	1	no	—
250	#350	—	53	1	no	—
251	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
252	C₅₃	#33859	1	1	yes	characteristic, complemented, Sylow
253	2₋¹⁺⁴	—	106	2	no	—
254	H(5,𝔽₂)	—	106	2	no	—
255	#137	—	106	2	no	—
256	#137	—	106	2	no	—
257	#137	—	212	2	no	—
258	#137	—	212	2	no	—
259	AGL(1,ℤ/8)	—	106	2	no	—
260	AGL(1,ℤ/8)	—	106	2	no	—
261	AGL(1,ℤ/8)	—	212	2	no	—
262	AGL(1,ℤ/8)	—	212	2	no	—
263	#135	—	53	1	no	—
264	#135	—	53	2	no	—
265	#135	—	106	2	no	—
266	#135	—	106	2	no	—
267	#135	—	212	2	no	—
268	C₂×Q₁₆	—	106	2	no	—
269	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
270	C₂×SD₁₆	—	2	2	no	—
271	C₂×SD₁₆	—	212	2	no	—
272	C₂×D₈	—	106	2	no	—
273	C₂×D₈	C₂²×D₅₃	1	2	yes	—
274	#131	—	53	1	no	—
275	#131	—	53	1	no	—
276	#131	—	106	2	no	—
277	#124	—	2	2	no	—
278	Q₃₂	—	2	1	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	—	53	1	no	—
281	Q₃₂	—	106	2	no	—
282	SD₃₂	—	4	2	no	—
283	SD₃₂	—	53	2	no	—
284	SD₃₂	—	106	2	no	—
285	SD₃₂	—	106	2	no	—
286	D₁₆	—	2	1	no	—
287	D₁₆	—	2	1	no	—
288	D₁₆	—	53	1	no	—
289	D₁₆	—	106	2	no	—
290	M₃₂	—	53	1	no	—
291	M₃₂	—	53	1	no	—
292	M₃₂	—	106	1	no	—
293	M₃₂	—	106	1	no	—
294	C₂×C₁₆	—	106	1	no	—
295	C₂×C₁₆	—	106	1	no	—
296	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
297	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
298	#108	—	53	1	no	—
299	#108	—	53	2	no	—
300	#104	—	53	1	no	—
301	#104	—	53	1	no	—
302	#104	—	106	2	no	—
303	#101	—	106	2	no	—
304	#100	—	106	2	no	—
305	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
306	#41	—	53	1	no	—
307	#41	—	53	1	no	—
308	#41	—	106	2	no	—
309	#41	—	424	2	no	—
310	#41	—	212	2	no	—
311	#41	—	212	2	no	—
312	#41	—	212	2	no	—
313	#41	—	212	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	212	2	no	—
316	C₂×Q₈	—	212	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	424	2	no	—
319	C₂×D₄	—	212	2	no	—
320	C₂×D₄	—	212	2	no	—
321	Q₁₆	—	2	2	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	53	1	no	—
324	Q₁₆	—	53	2	no	—
325	Q₁₆	—	212	2	no	—
326	Q₁₆	—	212	2	no	—
327	SD₁₆	—	4	2	no	—
328	SD₁₆	—	4	2	no	—
329	SD₁₆	—	53	1	no	—
330	SD₁₆	—	53	1	no	—
331	SD₁₆	—	106	2	no	—
332	SD₁₆	—	212	2	no	—
333	SD₁₆	—	212	2	no	—
334	SD₁₆	—	212	2	no	—
335	SD₁₆	—	212	2	no	—
336	D₈	—	2	2	no	—
337	D₈	—	2	2	no	—
338	D₈	—	53	1	no	—
339	D₈	—	53	2	no	—
340	D₈	—	212	2	no	—
341	D₈	—	212	2	no	—
342	M₁₆	—	53	1	no	—
343	M₁₆	—	53	1	no	—
344	M₁₆	—	106	1	no	—
345	M₁₆	—	106	1	no	—
346	M₁₆	—	106	2	no	—
347	M₁₆	—	212	2	no	—
348	C₂×C₈	—	106	1	no	—
349	C₂×C₈	—	106	1	no	—
350	C₂×C₈	—	212	2	no	—
351	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
352	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
353	C₂²⋊C₄	—	8	2	no	—
354	C₄²	D₄×D₅₃	1	1	yes	characteristic
355	C₁₆	—	2	1	no	—
356	C₁₆	—	106	1	no	—
357	C₁₆	—	106	1	no	—
358	C₁₆	D₄×D₅₃	1	1	yes	characteristic
359	C₁₆	D₄×D₅₃	1	1	yes	characteristic
360	C₂³	—	4	2	no	—
361	C₂³	—	424	2	no	—
362	Q₈	—	4	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	53	1	no	—
365	Q₈	—	53	1	no	—
366	Q₈	—	106	2	no	—
367	Q₈	—	424	2	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	53	1	no	—
371	D₄	—	53	1	no	—
372	D₄	—	106	1	no	—
373	D₄	—	106	1	no	—
374	D₄	—	106	2	no	—
375	D₄	—	424	2	no	—
376	D₄	—	424	2	no	—
377	D₄	—	424	2	no	—
378	D₄	—	212	2	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	106	1	no	—
382	C₂×C₄	—	106	1	no	—
383	C₂×C₄	—	424	2	no	—
384	C₂×C₄	—	424	2	no	—
385	C₂×C₄	—	424	2	no	—
386	C₂×C₄	—	212	2	no	—
387	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
388	C₈	—	106	1	no	—
389	C₈	—	106	1	no	—
390	C₈	—	212	2	no	—
391	C₈	D₈×D₅₃	1	1	yes	characteristic
392	C₈	D₈×D₅₃	1	1	yes	characteristic
393	C₈	C₂×D₄×D₅₃	1	1	yes	characteristic
394	C₈	#13133262	1	1	yes	characteristic
395	C₂²	—	8	2	no	—
396	C₂²	—	16	2	no	—
397	C₂²	—	106	1	no	—
398	C₂²	—	106	1	no	—
399	C₂²	—	1696	2	no	—
400	C₂²	—	212	2	no	—
401	C₂²	#16811424	1	1	yes	characteristic
402	C₄	—	4	1	no	—
403	C₄	—	8	2	no	—
404	C₄	—	106	1	no	—
405	C₄	—	106	1	no	—
406	C₄	—	212	2	no	—
407	C₄	#16811258	1	1	yes	characteristic
408	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	16	2	no	—
411	C₂	—	424	2	no	—
412	C₂	—	212	1	no	—
413	C₂	—	212	1	no	—
414	C₂	#20286390	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20286390
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2249	C₂
socle	C₂×C₅₃	D₄×D₈
hypercenter	#2249	D₅₃
hypocenter	C₅₃	#33859
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2249

Complements

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

N	H ≅ G/N	type
1. #20287864	C₂	semidirect
2. #20287852	C₂	semidirect
3. #20286500	C₂	semidirect
4. #20276837	C₂	semidirect
5. #20276832	C₂	semidirect
6. #20276781	C₂	semidirect
7. #20276685	C₂	semidirect
8. #20276678	C₂	semidirect
9. #20276597	C₂	semidirect
10. #20276553	C₂	semidirect
11. C₅₃×#2249	C₂	semidirect
12. #16811436	C₂²	semidirect
16. #16811064	C₂²	semidirect
25. #16811039	C₂²	semidirect
27. #16810890	C₂²	semidirect
32. #16810741	C₂²	semidirect
33. #16810453	C₂²	semidirect
34. #16810444	C₂²	semidirect
35. #16810439	C₂²	semidirect
36. #16810436	C₂²	semidirect
37. #16810431	C₂²	semidirect
39. #16810406	C₂²	semidirect
40. C₂×Q₃₂×C₅₃	C₂²	semidirect
41. C₂×SD₃₂×C₅₃	C₂²	semidirect
43. C₅₃×#495	C₂²	semidirect
44. C₄×C₁₆×C₅₃	C₂²	semidirect
205. #2249	D₅₃	semidirect
230. C₂×Q₃₂	C₂×D₅₃	semidirect
231. C₂×SD₃₂	C₂×D₅₃	semidirect
236. #495	C₂×D₅₃	semidirect
251. C₄×C₁₆	C₂×D₅₃	semidirect
252. C₅₃	#33859	semidirect

Maximal subgroups

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

subgroup	index
#20287864	2
#20287852	2
#20286500	2
#20276837	2
#20276832	2
#20276781	2
#20276685	2
#20276678	2
#20276597	2
#20276553	2
C₅₃×#2249	2
#33859	53