Finite Group Databasebeta

← #27466126

Subgroups — #27466126

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22⋊Q8×Dic53C211yesmaximal, characteristic
2C2×#16810492C211yesmaximal, characteristic
3#20276276C212yesmaximal, complemented
4#20276256C212yesmaximal, complemented
5#20275477C212yesmaximal, complemented
6#20275023C211yesmaximal, characteristic
7#20274910C211yesmaximal, characteristic
8#20274908C211yesmaximal, characteristic
9#20274718C212yesmaximal, complemented
10#20274693C211yesmaximal, characteristic
11C53×#1872C211yesmaximal, characteristic
12C2×Q8×Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919—21no—
15C22×#11719919—21no—
16C22×C4×Dic53C2211yescharacteristic
17C2×#13133149C2212yes—
18C2×#13133148C2212yes—
19C2×#13133146C2212yes—
20C2×#13133141C2212yes—
21C2×C22⋊#88322C2211yescharacteristic
22#16810530C2211yescharacteristic
23#16810522C2211yescharacteristic
24#16810509—21no—
25#16810498—21no—
26#16810492—21no—
27#16810492—21no—
28#16810492C2212yes—
29#16810492C2212yes—
30#16810483C2211yescharacteristic
31C4⋊C4×Dic53—21no—
32C4⋊C4×Dic53C2211yescharacteristic
33C22⋊C4×Dic53—21no—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C2×C22⋊Q8×C53C2211yescharacteristic
39C22×C4⋊C4×C53C2211yescharacteristic
40C2×#103×C53C2212yes—
41C2×C22⋊C8×C53C2211yescharacteristic
42C53×#340C2212yes—
43C23×Dic53—21no—
44C23×Dic53D411yescharacteristic
45C23×Dic53D411yescharacteristic
46C22×#88322C2312yes—
47Q8×Dic53—24no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—42no—
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2312yes—
55C2×#11719919—22no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919—22no—
59C2×#11719919—44no—
60C2×#11719919—44no—
61C2×C22⋊Dic53—21no—
62C2×C4×Dic53—24no—
63C2×C4×Dic53—21no—
64C2×C4×Dic53—21no—
65C2×C4×Dic53D411yescharacteristic
66C2×C4×Dic53D411yescharacteristic
67C2×C4×Dic53C2311yescharacteristic
68C2×C4×Dic53C2311yescharacteristic
69#13133149—42no—
70#13133148—42no—
71#13133146—24no—
72#13133141—42no—
73C22⋊#88322—22no—
74#13133124—21no—
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78#13133124—21no—
79C22×Q8×C53C2311yescharacteristic
80C23×C4×C53—21no—
81C23×C4×C53C2311yescharacteristic
82C22×C8×C53C2312yes—
83C22⋊Q8×C53—22no—
84C22⋊Q8×C53Q814yes—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53—42no—
88C2×C4⋊C4×C53C2312yes—
89C2×C4⋊C4×C53C2311yescharacteristic
90C2×C22⋊C4×C53—21no—
91#103×C53—42no—
92C22⋊C8×C53—22no—
93C24×C53C2×D411yescharacteristic
94C2×Q8×C53—21no—
95C2×Q8×C53—22no—
96C2×Q8×C53—22no—
97C2×Q8×C53C2×Q812yes—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—22no—
103C22×C4×C53—42no—
104C22×C4×C53—42no—
105C22×C4×C53—42no—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×Q812yes—
108C22×C4×C53#4112yes—
109C22×C4×C53C2411yescharacteristic
110C22×Dic53—22no—
111C22×Dic53—22no—
112C22×Dic53—22no—
113C22×Dic53—22no—
114C22×Dic53—22no—
115C22×Dic53—21no—
116C22×Dic53—21no—
117C22×Dic53—22no—
118C22×Dic53—42no—
119C22×Dic53—42no—
120C22×Dic53—42no—
121C22×Dic53SD1614yes—
122C22×Dic53C2×D411yescharacteristic
123C22×Dic53C2×D411yescharacteristic
124C22×Dic53C2×D411yescharacteristic
125C22×Dic53C2×D411yescharacteristic
126C2×C8×C53—22no—
127C2×C8×C53—22no—
128C2×C8×C53—22no—
129C4⋊C4×C53—24no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—42no—
134C4⋊C4×C53C2×Q812yes—
135C22⋊C4×C53—24no—
136C22⋊Dic53—42no—
137C4⋊Dic53—22no—
138C4⋊Dic53—22no—
139C4⋊Dic53—22no—
140C4⋊Dic53—22no—
141C4⋊Dic53—42no—
142C4⋊Dic53—42no—
143C4⋊Dic53—42no—
144#11719919—44no—
145#11719919—44no—
146#11719919—44no—
147#11719919—44no—
148C4×Dic53—22no—
149C4×Dic53—22no—
150C4×Dic53—22no—
151C4×Dic53—42no—
152C4×Dic53—42no—
153C4×Dic53#4114yes—
154C2×#88322—22no—
155C2×#88322—22no—
156C2×#88322—22no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53C22⋊Q812yes—
162C23×C53C2×SD1612yes—
163C23×C53#13712yes—
164C23×C53C22×D411yescharacteristic
165Q8×C53—24no—
166Q8×C53—42no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—24no—
170C2×C4×C53—24no—
171C2×C4×C53—22no—
172C2×C4×C53—22no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—84no—
181C2×C4×C53C22⋊Q812yes—
182C2×C4×C53C22×Q811yescharacteristic
183C2×C4×C53C2×#4112yes—
184C2×C4×C53C2×#4111yescharacteristic
185C2×C4×C53H(5,𝔽2)12yes—
186C2×Dic53—28no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—21no—
191C2×Dic53—21no—
192C2×Dic53—21no—
193C2×Dic53—28no—
194C2×Dic53—41no—
195C2×Dic53—42no—
196C2×Dic53—41no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53—41no—
203C2×Dic53—84no—
204C2×Dic53#12312yes—
205C2×Dic53#12312yes—
206C2×Dic53C2×SD1612yes—
207C2×Dic53C2×SD1612yes—
208C2×Dic53AGL(1,ℤ/8)12yes—
209C2×Dic53#13712yes—
210C8×C53—42no—
211#88322—42no—
212#27680—531nomaximal, Sylow
213C22×C53—22no—
214C22×C53—22no—
215C22×C53—22no—
216C22×C53—22no—
217C22×C53—22no—
218C22×C53—22no—
219C22×C53#47514yes—
220C22×C53C2×C22⋊Q811yescharacteristic
221C22×C53C2×#12311yescharacteristic
222C22×C53#53511yescharacteristic
223C22×C53C22×SD1611yescharacteristic
224C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
225C22×C53C2×#13711yescharacteristic
226C22×C53C2×#13711yescharacteristic
227C4×C53—22no—
228C4×C53—42no—
229C4×C53—42no—
230C4×C53—82no—
231C4×C53#55312yes—
232C4×C53#55412yes—
233Dic53—22no—
234Dic53—22no—
235Dic53—22no—
236Dic53—81no—
237Dic53—82no—
238Dic53—81no—
239Dic53#48314yes—
240C4×C22⋊Q8—531no—
241C2×#384—531no—
242#2099—532no—
243#2096—532no—
244#1948—532no—
245#1872—531no—
246#1872D5311yescharacteristic
247#1853—531no—
248#1853—531no—
249#1803—532no—
250#1797—531no—
251C2×C53—24no—
252C2×C53#248711yescharacteristic
253C2×C53C2×#47511yescharacteristic
254C2×C53#307911yescharacteristic
255C2×C53C2×#48311yescharacteristic
256C2×C53#309211yescharacteristic
257C2×C53#320811yescharacteristic
258C2×C53#321811yescharacteristic
259C2×C22⋊Q8C2×D5311yescharacteristic
260C2×C4×Q8—531no—
261C22×C4⋊C4—531no—
262C22×C4⋊C4—1061no—
263C22×C4⋊C4—1061no—
264C22×C4⋊C4C2×D5311yescharacteristic
265C22×C42—531no—
266C2×#106—532no—
267C2×#106—532no—
268C2×C4⋊C8—532no—
269C2×#103—532no—
270C2×#103C2×D5312yes—
271C2×C22⋊C8—531no—
272C2×C22⋊C8C2×D5311yescharacteristic
273#391—531no—
274#389—531no—
275#387—1061no—
276#385—1061no—
277#384—532no—
278#384—532no—
279#384—1061no—
280#384—1061no—
281#380—531no—
282C4×C4⋊C4—531no—
283C4×C4⋊C4—1061no—
284C4×C22⋊C4—1061no—
285#340—532no—
286#340—532no—
287#340—532no—
288#340C2×D5312yes—
289#336—532no—
290C53#2768011yescharacteristic, complemented, Sylow
291C22×Q8C22×D5311yescharacteristic
292C23×C4—21no—
293C23×C4—531no—
294C23×C4—531no—
295C23×C4—1061no—
296C23×C4C22×D5311yescharacteristic
297C22×C8—532no—
298C22×C8C22×D5312yes—
299C22⋊Q8—22no—
300C22⋊Q8Dic10614yes—
301C4×Q8—1064no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—532no—
306C2×C4⋊C4—531no—
307C2×C4⋊C4—531no—
308C2×C4⋊C4—1062no—
309C2×C4⋊C4—1062no—
310C2×C4⋊C4—1062no—
311C2×C4⋊C4—1062no—
312C2×C4⋊C4—1061no—
313C2×C4⋊C4—1061no—
314C2×C4⋊C4—1061no—
315C2×C4⋊C4—2122no—
316C2×C4⋊C4—2124no—
317C2×C4⋊C4—2124no—
318C2×C4⋊C4C22×D5312yes—
319C2×C4⋊C4C22×D5311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—1061no—
322C2×C42—531no—
323C2×C42—531no—
324C2×C42—531no—
325C2×C42—531no—
326C2×C42—1064no—
327C2×C42—1061no—
328C2×C42—1061no—
329#106—2122no—
330#106—2122no—
331C4⋊C8—1064no—
332#103—42no—
333#103—2122no—
334C22⋊C8—22no—
335C22⋊C8—1062no—
336#95—1061no—
337#95—1061no—
338#95—1061no—
339#95—1061no—
340#95—1061no—
341C24D4×D5311yescharacteristic
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8C2×Dic10612yes—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—531no—
355C22×C4—534no—
356C22×C4—531no—
357C22×C4—531no—
358C22×C4—531no—
359C22×C4—1062no—
360C22×C4—1062no—
361C22×C4—1062no—
362C22×C4—1062no—
363C22×C4—1062no—
364C22×C4—1061no—
365C22×C4—1061no—
366C22×C4—1062no—
367C22×C4—2122no—
368C22×C4—2122no—
369C22×C4—2122no—
370C22×C4C2×Dic10612yes—
371C22×C4D4×D5311yescharacteristic
372C22×C4#1171994712yes—
373C22×C4C23×D5311yescharacteristic
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—1062no—
378C2×C8—1062no—
379C2×C8—1062no—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—1062no—
386C4⋊C4—1062no—
387C4⋊C4—1062no—
388C4⋊C4—1062no—
389C4⋊C4—2122no—
390C4⋊C4—2124no—
391C4⋊C4—2124no—
392C4⋊C4—2124no—
393C4⋊C4—2122no—
394C4⋊C4—2122no—
395C4⋊C4—2124no—
396C4⋊C4C2×Dic10612yes—
397C22⋊C4—24no—
398C22⋊C4—2122no—
399C42—534no—
400C42—1062no—
401C42—1062no—
402C42—1062no—
403C42—2122no—
404C42—2122no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23#1313319412yes—
410C23SD16×D5312yes—
411C23#1313324912yes—
412C23C2×D4×D5311yescharacteristic
413Q8—24no—
414Q8—42no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—84no—
429C2×C4—532no—
430C2×C4—532no—
431C2×C4—532no—
432C2×C4—532no—
433C2×C4—532no—
434C2×C4—532no—
435C2×C4—1068no—
436C2×C4—1068no—
437C2×C4—1061no—
438C2×C4—1061no—
439C2×C4—1061no—
440C2×C4—1061no—
441C2×C4—1061no—
442C2×C4—1061no—
443C2×C4—4244no—
444C2×C4—2121no—
445C2×C4—2122no—
446C2×C4—2122no—
447C2×C4—2121no—
448C2×C4—2121no—
449C2×C4—2121no—
450C2×C4—2121no—
451C2×C4—2122no—
452C2×C4—2121no—
453C2×C4#1313319412yes—
454C2×C4C22×Dic10611yescharacteristic
455C2×C4#41×D5312yes—
456C2×C4C2×#1171994711yescharacteristic
457C2×C4#1313326212yes—
458C8—42no—
459C8—2122no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22#1681090614yes—
467C22C2×#1313319411yescharacteristic
468C22#123×D5311yescharacteristic
469C22#1681118811yescharacteristic
470C22C2×SD16×D5311yescharacteristic
471C22AGL(1,ℤ/8)×D5311yescharacteristic
472C22#137×D5311yescharacteristic
473C22C2×#1313324911yescharacteristic
474C4—22no—
475C4—42no—
476C4—42no—
477C4—82no—
478C4—534no—
479C4—1062no—
480C4—1062no—
481C4—1062no—
482C4—4241no—
483C4—4242no—
484C4—4241no—
485C4#1681131612yes—
486C4#1681133212yes—
487C2—24no—
488C2#2027902911yescharacteristic
489C2C2×#1681090611yescharacteristic
490C2#2028452611yescharacteristic
491C2#483×D5311yescharacteristic
492C2#2028464211yescharacteristic
493C2#2028565411yescharacteristic
494C2#2028574511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1872C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1872D53
hypocenterC53#27680
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
3. #20276276C2semidirect
4. #20276256C2semidirect
5. #20275477C2semidirect
9. #20274718C2semidirect
290. C53#27680semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×Dic532
C2×#168104922
#202762762
#202762562
#202754772
#202750232
#202749102
#202749082
#202747182
#202746932
C53×#18722
#2768053