Finite Group Databasebeta

← #27383890

Subgroups — #27383890

Order 13568.

1774 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C2×#16810585C211yesmaximal, characteristic
3#20277097C211yesmaximal, characteristic
4#20274806C211yesmaximal, characteristic, complemented
5#20274805C211yesmaximal, characteristic, complemented
6#20274803C211yesmaximal, characteristic, complemented
7#20274800C214yesmaximal, complemented
8#20274433C214yesmaximal, complemented
9C53×#1824C211yesmaximal, characteristic, complemented
10C23×C4×D53—21no—
11C23×C4×D53C2211yescharacteristic
12C22×#11719912C2212yes—
13C2×#13133175C2212yes—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919—21no—
16C2×C4⋊C4×D53—21no—
17C2×C4⋊C4×D53—21no—
18C2×C4⋊C4×D53—24no—
19C2×C4⋊C4×D53C418yescomplemented
20C2×C4⋊C4×D53C2212yes—
21C2×C4⋊C4×D53C2212yes—
22#16810585—24no—
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133129C2212yes—
25#16810485—21no—
26#16810483C2211yescharacteristic
27#16810480C2212yes—
28#16810478—21no—
29C22⋊C4×Dic53—21no—
30#16810470—21no—
31#16810386C2214yescomplemented
32#16810385C2214yescomplemented
33#16810385C2214yescomplemented
34C22×C4⋊C4×C53C2211yescharacteristic
35C2×C22⋊C8×C53C2211yescharacteristic
36C53×#380C2211yescharacteristic
37C53×#340C2214yescomplemented
38C24×D53D411yescharacteristic
39C23×Dic53—21no—
40C23×Dic53D411yescharacteristic
41C22×C4×D53—28no—
42C22×C4×D53—28no—
43C22×C4×D53—22no—
44C22×C4×D53—22no—
45C22×C4×D53—48no—
46C22×C4×D53C2×C414yes—
47C22×C4×D53D411yescharacteristic
48C22×C4×D53D411yescharacteristic
49C22×C4×D53D414yes—
50C22×C4×D53C2311yescharacteristic
51C22×C4×D53C2311yescharacteristic
52C2×#11719912—24no—
53C2×#11719912—28no—
54C22×#88322C2312yes—
55#13133175—28no—
56C2×C4⋊Dic53—22no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53C2×C414yes—
59C2×C4⋊Dic53C2312yes—
60C2×C4⋊Dic53C2312yes—
61C2×#11719919—28no—
62C2×#11719919—22no—
63C2×#11719919—22no—
64C4⋊C4×D53—28no—
65C4⋊C4×D53—28no—
66C4⋊C4×D53—24no—
67C4⋊C4×D53—24no—
68C4⋊C4×D53C2×C418yes—
69C4⋊C4×D53C2×C418yes—
70C2×C22⋊Dic53—21no—
71C2×#11719921—22no—
72C2×#11719921—22no—
73C2×C4×Dic53—22no—
74C22⋊#88322—22no—
75#13133129—24no—
76#13133124—21no—
77#13133124—22no—
78C23×C4×C53—21no—
79C23×C4×C53C2311yescharacteristic
80C22×C8×C53C2312yes—
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53C2×C414yes—
84C2×C4⋊C4×C53C2312yes—
85C2×C4⋊C4×C53C2312yes—
86C2×C22⋊C4×C53—21no—
87C22⋊C8×C53—22no—
88#95×C53—21no—
89C24×C53C2×D411yescharacteristic
90C23×D53—22no—
91C23×D53—22no—
92C23×D53—24no—
93C23×D53—24no—
94C23×D53C22⋊C414yes—
95C23×D53C2×D411yescharacteristic
96C23×D53C2×D411yescharacteristic
97C22×C4×C53—24no—
98C22×C4×C53—24no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—44no—
102C22×C4×C53C22×C412yes—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D412yes—
105C22×C4×C53C2411yescharacteristic
106C22×Dic53—24no—
107C22×Dic53—24no—
108C22×Dic53—22no—
109C22×Dic53—22no—
110C22×Dic53—44no—
111C22×Dic53C22⋊C414yes—
112C22×Dic53C2×D411yescharacteristic
113C22×Dic53C2×D411yescharacteristic
114C2×C4×D53—24no—
115C2×C4×D53—24no—
116C2×C4×D53—24no—
117C2×C4×D53—216no—
118C2×C4×D53—216no—
119C2×C4×D53—28no—
120C2×C4×D53—28no—
121C2×C4×D53—48no—
122C2×C4×D53—432no—
123C2×C4×D53—48no—
124C2×C4×D53C22⋊C414yes—
125C2×C4×D53C22⋊C414yes—
126C2×C4×D53C22×C412yes—
127C2×C4×D53C22×C412yes—
128C2×C4×D53C2×D412yes—
129C2×C4×D53C2×D412yes—
130C2×C4×D53#4118yes—
131C2×C8×C53—22no—
132C2×C8×C53—24no—
133C4⋊C4×C53—24no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—24no—
136C4⋊C4×C53C22×C414yes—
137C22⋊C4×C53—42no—
138C22⋊Dic53—42no—
139#11719921—44no—
140#11719921—44no—
141C4⋊Dic53—24no—
142C4⋊Dic53—22no—
143C4⋊Dic53—24no—
144C4⋊Dic53C22×C414yes—
145#11719919—28no—
146#11719919—28no—
147C4×Dic53—44no—
148C2×#88322—22no—
149C2×#88322—24no—
150#11719912—48no—
151C23×C53—22no—
152C23×C53—22no—
153C23×C53C2×C22⋊C412yes—
154C23×C53#13712yes—
155C23×C53#13712yes—
156C23×C53C22×D411yescharacteristic
157C22×D53—216no—
158C22×D53—216no—
159C22×D53—22no—
160C22×D53—22no—
161C22×D53—21no—
162C22×D53—21no—
163C22×D53—21no—
164C22×D53—21no—
165C22×D53—28no—
166C22×D53—28no—
167C22×D53—28no—
168C22×D53C2×C22⋊C412yes—
169C22×D53C2×C22⋊C412yes—
170C22×D53#12314yes—
171C22×D53AGL(1,ℤ/8)14yes—
172C2×C4×C53—22no—
173C2×C4×C53—24no—
174C2×C4×C53—24no—
175C2×C4×C53—24no—
176C2×C4×C53—48no—
177C2×C4×C53—44no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53C2×C22⋊C412yes—
182C2×C4×C53C23×C411yescharacteristic
183C2×C4×C53C22×D411yescharacteristic
184C2×C4×C53C2×#4114yes—
185C2×Dic53—22no—
186C2×Dic53—28no—
187C2×Dic53—24no—
188C2×Dic53—48no—
189C2×Dic53—44no—
190C2×Dic53—42no—
191C2×Dic53—42no—
192C2×Dic53—42no—
193C2×Dic53C2×C22⋊C412yes—
194C2×Dic53C2×C22⋊C412yes—
195C2×Dic53#12314yes—
196C4×D53—28no—
197C4×D53—24no—
198C4×D53—24no—
199C4×D53—28no—
200C4×D53—416no—
201C4×D53—416no—
202C4×D53#11718yes—
203C4×D53#12318yes—
204C8×C53—42no—
205#88322—42no—
206#20700—531nomaximal, Sylow
207C22×C53—24no—
208C22×C53—24no—
209C22×C53—22no—
210C22×C53—22no—
211C22×C53#41914yes—
212C22×C53C22×C22⋊C411yescharacteristic
213C22×C53C2×#12312yes—
214C22×C53C2×AGL(1,ℤ/8)12yes—
215C22×C53C2×#13711yescharacteristic
216C22×C53C2×#13711yescharacteristic
217C2×D53—232no—
218C2×D53—24no—
219C2×D53—24no—
220C2×D53—24no—
221C2×D53—24no—
222C2×D53—22no—
223C2×D53—22no—
224C2×D53—22no—
225C2×D53—22no—
226C2×D53#38114yes—
227C2×D53#41814yes—
228C4×C53—22no—
229C4×C53—44no—
230C4×C53—82no—
231C4×C53C2×#11712yes—
232C4×C53C2×#12312yes—
233Dic53—22no—
234Dic53—44no—
235Dic53—82no—
236Dic53#38114yes—
237C23×C4⋊C4—531no—
238C2×#407—531no—
239#2280—531no—
240#1824—531no—
241#1824—531no—
242#1824—531no—
243#1824D5311yescharacteristic, complemented
244#1823—534no—
245#1736—534no—
246C2×C53—24no—
247C2×C53C2×#38111yescharacteristic
248C2×C53C2×#41811yescharacteristic
249C2×C53C2×#41911yescharacteristic
250C2×C53#309212yes—
251C2×C53#309212yes—
252D53—24no—
253D53—24no—
254C24×C4—531no—
255C24×C4—1061no—
256C22×M16—532no—
257C2×#117—532no—
258C22×C4⋊C4—531no—
259C22×C4⋊C4—538no—
260C22×C4⋊C4—532no—
261C22×C4⋊C4—532no—
262C22×C4⋊C4—1061no—
263C22×C4⋊C4—1061no—
264C22×C4⋊C4—1064no—
265C22×C4⋊C4—1061no—
266C22×C4⋊C4C2×D5311yescharacteristic
267#407—1064no—
268C2×C22⋊C8—531no—
269C2×C22⋊C8—532no—
270C2×C22⋊C8C2×D5311yescharacteristic
271#381—1061no—
272#380—531no—
273#380—532no—
274#380C2×D5311yescharacteristic
275#379—1061no—
276C4×C22⋊C4—1061no—
277C4×C22⋊C4—1061no—
278#340—534no—
279#340—534no—
280#340—534no—
281#340C2×D5314yescomplemented
282C53#2070011yescharacteristic, complemented, Sylow
283C25—531no—
284C23×C4—21no—
285C23×C4—531no—
286C23×C4—531no—
287C23×C4—534no—
288C23×C4—531no—
289C23×C4—531no—
290C23×C4—534no—
291C23×C4—531no—
292C23×C4—1068no—
293C23×C4—1068no—
294C23×C4—1062no—
295C23×C4—1062no—
296C23×C4—1061no—
297C23×C4—2128no—
298C23×C4C22×D5311yescharacteristic
299C2×M16—1064no—
300C2×M16—1068no—
301C22×C8—532no—
302C22×C8C22×D5312yes—
303#117—1068no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—534no—
307C2×C4⋊C4—532no—
308C2×C4⋊C4—538no—
309C2×C4⋊C4—538no—
310C2×C4⋊C4—532no—
311C2×C4⋊C4—1068no—
312C2×C4⋊C4—1068no—
313C2×C4⋊C4—1064no—
314C2×C4⋊C4—1064no—
315C2×C4⋊C4—1068no—
316C2×C4⋊C4—1062no—
317C2×C4⋊C4—1062no—
318C2×C4⋊C4—1061no—
319C2×C4⋊C4—1062no—
320C2×C4⋊C4C4×D5314yes—
321C2×C4⋊C4C22×D5312yes—
322C2×C4⋊C4C22×D5312yes—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—1062no—
325C2×C22⋊C4—1062no—
326C2×C22⋊C4—1061no—
327C2×C42—1062no—
328C22⋊C8—22no—
329C22⋊C8—1064no—
330C22⋊C8—1062no—
331#95—21no—
332#95—1062no—
333#95—1061no—
334C24—531no—
335C24—531no—
336C24—534no—
337C24—1064no—
338C24—1064no—
339C24—1062no—
340C24—1062no—
341C24D4×D5311yescharacteristic
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—44no—
347C22×C4—531no—
348C22×C4—532no—
349C22×C4—532no—
350C22×C4—538no—
351C22×C4—534no—
352C22×C4—531no—
353C22×C4—534no—
354C22×C4—534no—
355C22×C4—532no—
356C22×C4—532no—
357C22×C4—1064no—
358C22×C4—10616no—
359C22×C4—10616no—
360C22×C4—1064no—
361C22×C4—1068no—
362C22×C4—1068no—
363C22×C4—1064no—
364C22×C4—1064no—
365C22×C4—1064no—
366C22×C4—1062no—
367C22×C4—1062no—
368C22×C4—2128no—
369C22×C4—21232no—
370C22×C4—2128no—
371C22×C4—2124no—
372C22×C4C2×C4×D5312yes—
373C22×C4D4×D5311yescharacteristic
374C22×C4D4×D5312yes—
375C22×C4C23×D5311yescharacteristic
376M16—2128no—
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—1062no—
380C2×C8—1064no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—534no—
385C4⋊C4—1068no—
386C4⋊C4—1068no—
387C4⋊C4—1064no—
388C4⋊C4—1062no—
389C4⋊C4—1064no—
390C4⋊C4C2×C4×D5314yes—
391C22⋊C4—42no—
392C22⋊C4—2124no—
393C22⋊C4—2124no—
394C22⋊C4—2122no—
395C42—2124no—
396C23—22no—
397C23—22no—
398C23—532no—
399C23—532no—
400C23—534no—
401C23—534no—
402C23—1068no—
403C23—1068no—
404C23—10616no—
405C23—10616no—
406C23—1062no—
407C23—1068no—
408C23—1062no—
409C23—1061no—
410C23—1061no—
411C23—1061no—
412C23—1061no—
413C23C22⋊C4×D5312yes—
414C23#1313324612yes—
415C23#1313324912yes—
416C23C2×D4×D5311yescharacteristic
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—532no—
427C2×C4—534no—
428C2×C4—538no—
429C2×C4—538no—
430C2×C4—532no—
431C2×C4—1068no—
432C2×C4—1068no—
433C2×C4—1068no—
434C2×C4—1062no—
435C2×C4—1064no—
436C2×C4—1064no—
437C2×C4—1064no—
438C2×C4—21216no—
439C2×C4—21216no—
440C2×C4—2122no—
441C2×C4—2122no—
442C2×C4—2122no—
443C2×C4—2128no—
444C2×C4—2124no—
445C2×C4C22⋊C4×D5312yes—
446C2×C4C22×C4×D5311yescharacteristic
447C2×C4C2×D4×D5311yescharacteristic
448C2×C4#41×D5314yes—
449C8—42no—
450C8—2122no—
451C22—22no—
452C22—22no—
453C22—24no—
454C22—24no—
455C22—534no—
456C22—534no—
457C22—10632no—
458C22—1062no—
459C22—1062no—
460C22—1062no—
461C22—1062no—
462C22—1064no—
463C22—1064no—
464C22—1064no—
465C22—1064no—
466C22#1681063014yes—
467C22C2×C22⋊C4×D5311yescharacteristic
468C22#123×D5312yes—
469C22AGL(1,ℤ/8)×D5312yes—
470C22C2×#1313324611yescharacteristic
471C22C2×#1313324911yescharacteristic
472C4—22no—
473C4—44no—
474C4—82no—
475C4—534no—
476C4—1062no—
477C4—4242no—
478C4—2124no—
479C4#117×D5312yes—
480C4#123×D5312yes—
481C2—24no—
482C2—1064no—
483C2—1064no—
484C2#381×D5311yescharacteristic
485C2#418×D5311yescharacteristic
486C2C2×#1681063011yescharacteristic
487C2#2028463712yes—
488C2#2028464212yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1824C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1824D53
hypocenterC53#20700
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
4. #20274806C2semidirect
5. #20274805C2semidirect
6. #20274803C2semidirect
7. #20274800C2semidirect
8. #20274433C2semidirect
9. C53×#1824C2semidirect
19. C2×C4⋊C4×D53C4semidirect
31. #16810386C22semidirect
32. #16810385C22semidirect
33. #16810385C22semidirect
37. C53×#340C22semidirect
243. #1824D53semidirect
281. #340C2×D53semidirect
282. C53#20700semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D532
C2×#168105852
#202770972
#202748062
#202748052
#202748032
#202748002
#202744332
C53×#18242
#2070053