Finite Group Databasebeta

← #27280651

Subgroups — #27280651

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285472C211yesmaximal, characteristic
2#20275700C211yesmaximal, characteristic, complemented
3#20273283C211yesmaximal, characteristic
4C2×#16810359C211yesmaximal, characteristic, complemented
5#20272594C211yesmaximal, characteristic, complemented
6#20272563C211yesmaximal, characteristic, complemented
7C53×#2364C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810953—21no—
10#16810939C412yescomplemented
11#16810850—22no—
12#16810774—21no—
13#16810761C412yescomplemented
14C2×C22⋊#88322C2211yescharacteristic
15C2×C22⋊#88322C2211yescharacteristic
16#16810483C2211yescharacteristic
17#16810389—21no—
18#16810389—21no—
19#16810359—21no—
20#16810359—21no—
21#16810357—22no—
22C2×#121×C53C2211yescharacteristic
23C2×C4×D4×C53—21no—
24C53×#388—21no—
25C53×#387—21no—
26C53×#386—21no—
27C53×#386—21no—
28C4×C22⋊C4×C53—21no—
29C4×C22⋊C4×C53C2211yescharacteristic
30C2×#95×C53C2211yescharacteristic
31C22×Dic106C2×C411yescharacteristic, complemented
32C2×#11719923—22no—
33C22×#88322—21no—
34C22×#88322—21no—
35#13133196—21no—
36#13133196—21no—
37#13133196—21no—
38#13133196—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic, complemented
41C2×#11719919—21no—
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—21no—
44#13133148—22no—
45#13133140—22no—
46#13133138—22no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322C2×C412yescomplemented
51#13133124—21no—
52C22×D4×C53—21no—
53C22×D4×C53D411yescharacteristic
54C23×C4×C53—21no—
55C23×C4×C53—21no—
56C23×C4×C53C2311yescharacteristic
57#121×C53—21no—
58#121×C53—21no—
59#121×C53C2×C412yes—
60#121×C53D412yes—
61C4×D4×C53—42no—
62C4×D4×C53—42no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53D411yescharacteristic
71C2×C42×C53—21no—
72C2×C42×C53—21no—
73#95×C53—21no—
74#95×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77#95×C53D411yescharacteristic
78#95×C53D411yescharacteristic
79C24×C53—21no—
80C24×C53C2×D411yescharacteristic
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—81no—
89C2×D4×C53C22⋊C412yes—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—42no—
96C22×C4×C53—42no—
97C22×C4×C53—42no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—81no—
101C22×C4×C53—81no—
102C22×C4×C53—81no—
103C22×C4×C53C22×C411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C22×Dic53—21no—
110C22×Dic53—21no—
111C22×Dic53—41no—
112C2×Dic106—21no—
113C2×Dic106—21no—
114C2×Dic106—81no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—42no—
117C4⋊C4×C53C22⋊C412yes—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—22no—
120C22⋊C4×C53—22no—
121C22⋊C4×C53—21no—
122C22⋊C4×C53—21no—
123C22⋊C4×C53—42no—
124C22⋊C4×C53—42no—
125C22⋊C4×C53—81no—
126C22⋊C4×C53SD1612yescomplemented
127C42×C53—42no—
128C42×C53—81no—
129C22⋊Dic53—41no—
130C22⋊Dic53—41no—
131C22⋊Dic53—41no—
132C22⋊Dic53—41no—
133#11719923—82no—
134C4⋊Dic53—21no—
135C4⋊Dic53—21no—
136C4⋊Dic53—81no—
137#11719919—41no—
138#11719919—41no—
139C2×#88322—22no—
140C2×#88322—41no—
141C2×#88322—81no—
142C2×#88322—81no—
143C23×C53—22no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53C4×D411yescharacteristic
153C23×C53#12011yescharacteristic
154C23×C53#12111yescharacteristic
155C23×C53C2×SD1611yescharacteristic
156C23×C53#13511yescharacteristic
157C23×C53AGL(1,ℤ/8)11yescharacteristic
158C23×C53#13711yescharacteristic
159D4×C53—42no—
160D4×C53—42no—
161D4×C53—42no—
162D4×C53—42no—
163D4×C53—42no—
164D4×C53—81no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—22no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—84no—
187C2×C4×C53—82no—
188C2×C4×C53—82no—
189C2×C4×C53C2×C22⋊C411yescharacteristic
190C2×C4×C53C4×D411yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12311yescharacteristic
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—81no—
196C2×Dic53—81no—
197C2×Dic53—81no—
198C2×Dic53—81no—
199C2×Dic53—81no—
200Dic106—42no—
201Dic106—81no—
202#88322—42no—
203#88322—81no—
204#9275—531nomaximal, Sylow
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—42no—
216C22×C53—42no—
217C22×C53—42no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53—81no—
221C22×C53—81no—
222C22×C53#38611yescharacteristic
223C22×C53C4×SD1611yescharacteristic
224C22×C53#44011yescharacteristic
225C22×C53#45011yescharacteristic
226C22×C53#45211yescharacteristic
227C22×C53#46111yescharacteristic
228C22×C53#46311yescharacteristic
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—42no—
234C4×C53—42no—
235C4×C53—42no—
236C4×C53—81no—
237C4×C53—81no—
238Dic53—81no—
239Dic53—81no—
240Dic53—161no—
241#3187—531no—
242#2364D5311yescharacteristic
243#1996—531no—
244#1519—531no—
245C2×#328—531no—
246#1294—531no—
247#1282—531no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53—21no—
253C2×C53—21no—
254C2×C53—82no—
255C2×C53#187711yescharacteristic
256C2×C53#190611yescharacteristic
257C2×C53#191311yescharacteristic
258C2×C22⋊Q8—531no—
259C2×#121C2×D5311yescharacteristic
260C2×C4×D4—21no—
261#483—1061no—
262#481—532no—
263#465—1062no—
264#450—1061no—
265#448—532no—
266C2×C22⋊C8—531no—
267C2×C22⋊C8—531no—
268#388—21no—
269#387—21no—
270#386—21no—
271#386—21no—
272#380—531no—
273C4×C22⋊C4—21no—
274C4×C22⋊C4C2×D5311yescharacteristic
275C2×#95C2×D5311yescharacteristic
276#342—1061no—
277#342—1061no—
278#328—1061no—
279#328—1061no—
280#327—1062no—
281C53#927511yescharacteristic, complemented, Sylow
282C22×Q8—531no—
283C22×D4—21no—
284C22×D4C22⋊D5311yescharacteristic
285C23×C4—21no—
286C23×C4—21no—
287C23×C4C22×D5311yescharacteristic
288C2×SD16—1062no—
289C22×C8—1061no—
290C22×C8—1061no—
291C22⋊Q8—1061no—
292C22⋊Q8—1061no—
293C22⋊Q8—1061no—
294C22⋊Q8—1061no—
295#121—21no—
296#121—21no—
297#121C4×D5312yes—
298#121D21212yes—
299C4×D4—42no—
300C4×D4—42no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—531no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4C22⋊D5311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—1061no—
311C2×C22⋊C4—1061no—
312C2×C22⋊C4C22⋊D5311yescharacteristic
313C2×C22⋊C4C22⋊D5311yescharacteristic
314C2×C42—21no—
315C2×C42—21no—
316#106—1062no—
317#103—1062no—
318#102—1062no—
319C22⋊C8—532no—
320C22⋊C8—1061no—
321C22⋊C8—1061no—
322C22⋊C8—1061no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—1061no—
328#95C22⋊D5311yescharacteristic
329#95C22⋊D5311yescharacteristic
330C24—21no—
331C24C2×C22⋊D5311yescharacteristic
332C2×Q8—1061no—
333C2×Q8—1061no—
334C2×Q8—4241no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—81no—
343C2×D4#1171992112yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—1061no—
358C22×C4—1061no—
359C22×C4—2121no—
360C22×C4C2×C4×D5311yescharacteristic
361C22×C4C2×D21211yescharacteristic
362C22×C4#1171994511yescharacteristic
363C22×C4#1171994511yescharacteristic
364C22×C4C2×C22⋊D5311yescharacteristic
365C22×C4C2×C22⋊D5311yescharacteristic
366SD16—4242no—
367C2×C8—1062no—
368C2×C8—4241no—
369C2×C8—4241no—
370C2×C8—2121no—
371C4⋊C4—21no—
372C4⋊C4—42no—
373C4⋊C4—1061no—
374C4⋊C4—1061no—
375C4⋊C4—4241no—
376C4⋊C4—2121no—
377C4⋊C4—2121no—
378C4⋊C4#1171992112yes—
379C22⋊C4—21no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—81no—
387C22⋊C4—2121no—
388C22⋊C4—2121no—
389C22⋊C4—2121no—
390C22⋊C4—2121no—
391C22⋊C4#1171992312yescomplemented
392C42—42no—
393C42—81no—
394C23—22no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23C4×C22⋊D5311yescharacteristic
404C23#1313318611yescharacteristic
405C23#1313319111yescharacteristic
406C23C2×#1171992311yescharacteristic
407C23#1313323711yescharacteristic
408C23#1313324311yescharacteristic
409C23#1313324711yescharacteristic
410Q8—4241no—
411Q8—2122no—
412D4—42no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—81no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—84no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—4241no—
443C2×C4—4241no—
444C2×C4—4241no—
445C2×C4—4241no—
446C2×C4—4241no—
447C2×C4—2121no—
448C2×C4—2121no—
449C2×C4C2×#1171992111yescharacteristic
450C2×C4C4×C22⋊D5311yescharacteristic
451C2×C4#1313319111yescharacteristic
452C2×C4#1313320011yescharacteristic
453C8—4241no—
454C8—2122no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22#1681050211yescharacteristic
473C22C4×#1171992311yescharacteristic
474C22#1681072311yescharacteristic
475C22#1681077411yescharacteristic
476C22#1681078511yescharacteristic
477C22#1681082811yescharacteristic
478C22#1681083811yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—81no—
488C4—8481no—
489C4—4241no—
490C4—4241no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—82no—
498C2#2027504911yescharacteristic
499C2#2027521611yescharacteristic
500C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2364C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#2364D53
hypocenterC53#9275
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2364

Complements

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

NH ≅ G/Ntype
2. #20275700C2semidirect
4. C2×#16810359C2semidirect
5. #20272594C2semidirect
6. #20272563C2semidirect
10. #16810939C4semidirect
13. #16810761C4semidirect
31. C22×Dic106C2×C4semidirect
40. C2×C4⋊Dic53C2×C4semidirect
50. C22⋊#88322C2×C4semidirect
126. C22⋊C4×C53SD16semidirect
281. C53#9275semidirect
391. C22⋊C4#11719923semidirect

Maximal subgroups

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

subgroupindex
#202854722
#202757002
#202732832
C2×#168103592
#202725942
#202725632
C53×#23642
#927553