Finite Group Databasebeta

← #27284372

Subgroups — #27284372

Order 13568.

734 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
1C2×#16810953C211yesmaximal, characteristic, complemented
2#20280520C211yesmaximal, characteristic
3#20274801C211yesmaximal, characteristic, complemented
4C2×#16810359C211yesmaximal, characteristic, complemented
5C2×#16810347C211yesmaximal, characteristic
6#20272594C211yesmaximal, characteristic, complemented
7C2×C53×#397C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810953—22no—
10#16810953—22no—
11C2×#13133148—21no—
12C2×#13133140—21no—
13C2×C22⋊#88322C2211yescharacteristic
14C2×C22⋊#88322C2211yescharacteristic
15#16810559—21no—
16#16810546—22no—
17#16810535—21no—
18#16810496—21no—
19#16810483C2211yescharacteristic
20#16810479—21no—
21#16810385—22no—
22#16810359—22no—
23#16810347—22no—
24C22×C4⋊C4×C53C2211yescharacteristic
25C22×C22⋊C4×C53—21no—
26C22×C22⋊C4×C53C2211yescharacteristic
27C53×#397—22no—
28C53×#397—22no—
29C2×#95×C53—21no—
30C2×#95×C53C2211yescharacteristic
31C22×Dic106D411yescharacteristic
32C22×#88322—21no—
33C22×#88322—21no—
34#13133196—21no—
35#13133196—21no—
36#13133196—21no—
37#13133196—21no—
38C2×C4⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53D412yes—
40C2×#11719919—22no—
41C2×#11719919—21no—
42C2×C22⋊Dic53—22no—
43C2×C22⋊Dic53—22no—
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53—21no—
46#13133148—24no—
47#13133140—24no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C22⋊#88322—21no—
51C22⋊#88322D412yes—
52#13133124—22no—
53#13133124—21no—
54C25×C53D411yescharacteristic
55C23×C4×C53—21no—
56C23×C4×C53—21no—
57C23×C4×C53D411yescharacteristic
58C23×C4×C53C2311yescharacteristic
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D412yes—
65C2×C22⋊C4×C53—24no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53—42no—
73C2×C22⋊C4×C53—42no—
74#95×C53—24no—
75#95×C53—21no—
76#95×C53D412yes—
77C24×C53—21no—
78C24×C53—21no—
79C24×C53—21no—
80C24×C53—21no—
81C24×C53—21no—
82C24×C53—21no—
83C24×C53—42no—
84C24×C53—42no—
85C24×C53SD1612yes—
86C24×C53C2×D411yescharacteristic
87C22×C4×C53—21no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—42no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—42no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C22×Dic53—22no—
111C22×Dic53—21no—
112C22×Dic53—21no—
113C2×Dic106—21no—
114C2×Dic106—21no—
115C2×Dic106—42no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53—41no—
120C22⋊C4×C53—48no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C22⋊Dic53—41no—
126C22⋊Dic53—41no—
127C22⋊Dic53—41no—
128C22⋊Dic53—41no—
129C22⋊Dic53—82no—
130C22⋊Dic53—82no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133C4⋊Dic53—42no—
134#11719919—41no—
135#11719919—41no—
136#11719919—82no—
137C2×#88322—22no—
138C2×#88322—42no—
139C2×#88322—42no—
140C2×#88322—41no—
141C23×C53—22no—
142C23×C53—22no—
143C23×C53—22no—
144C23×C53—22no—
145C23×C53—22no—
146C23×C53—44no—
147C23×C53—44no—
148C23×C53—42no—
149C23×C53—42no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53—42no—
154C23×C53—42no—
155C23×C53—41no—
156C23×C53—82no—
157C23×C53—82no—
158C23×C53—81no—
159C23×C53—81no—
160C23×C53#12011yescharacteristic
161C23×C53#12011yescharacteristic
162C23×C53#12111yescharacteristic
163C23×C53C2×SD1611yescharacteristic
164C23×C53#13511yescharacteristic
165C23×C53AGL(1,ℤ/8)11yescharacteristic
166C23×C53#13711yescharacteristic
167C2×C4×C53—22no—
168C2×C4×C53—22no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—42no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53—82no—
183C2×C4×C53—82no—
184C2×C4×C53—81no—
185C2×C4×C53—81no—
186C2×C4×C53—81no—
187C2×C4×C53—81no—
188C2×C4×C53—81no—
189C2×C4×C53#12011yescharacteristic
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12711yescharacteristic
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—42no—
196C2×Dic53—42no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—42no—
200Dic106—42no—
201Dic106—81no—
202#88322—42no—
203#88322—81no—
204#9926—531nomaximal, Sylow
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—22no—
208C22×C53—22no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—44no—
214C22×C53—44no—
215C22×C53—42no—
216C22×C53—82no—
217C22×C53—82no—
218C22×C53—82no—
219C22×C53—82no—
220C22×C53—81no—
221C22×C53—81no—
222C22×C53#39211yescharacteristic
223C22×C53#44811yescharacteristic
224C22×C53#44911yescharacteristic
225C22×C53#45011yescharacteristic
226C22×C53#45211yescharacteristic
227C22×C53#45512yes—
228C22×C53#45812yes—
229C22×C53#46511yescharacteristic
230C22×C53#46811yescharacteristic
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—82no—
236C4×C53—82no—
237C4×C53—81no—
238Dic53—82no—
239Dic53—81no—
240Dic53—81no—
241C2×#483—531no—
242#2626—531no—
243C2×#397D5311yescharacteristic
244#1823—531no—
245C2×#328—531no—
246C2×#323—531no—
247#1294—531no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53—82no—
253C2×C53#160812yes—
254C2×C53#161212yes—
255C2×C53#200011yescharacteristic
256C2×C53#200411yescharacteristic
257C2×C53#202211yescharacteristic
258C2×C22⋊Q8—531no—
259C22×C4⋊C4C2×D5311yescharacteristic
260C22×C22⋊C4—21no—
261C22×C22⋊C4C2×D5311yescharacteristic
262#483—1062no—
263#483—1062no—
264C2×#106—1061no—
265C2×#103—1061no—
266C2×C22⋊C8—531no—
267C2×C22⋊C8—531no—
268#399—1061no—
269#397—22no—
270#397—22no—
271#396—1062no—
272#393—1061no—
273#385—1061no—
274#380—531no—
275#379—1061no—
276C2×#95—21no—
277C2×#95C2×D5311yescharacteristic
278#340—1062no—
279#328—1062no—
280#323—1062no—
281C53#992611yescharacteristic, complemented, Sylow
282C25C22⋊D5311yescharacteristic
283C22×Q8—531no—
284C23×C4—21no—
285C23×C4—21no—
286C23×C4C22⋊D5311yescharacteristic
287C23×C4C22×D5311yescharacteristic
288C22×C8—1061no—
289C22×C8—1061no—
290C22⋊Q8—1061no—
291C22⋊Q8—1061no—
292C22⋊Q8—1061no—
293C22⋊Q8—1061no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—531no—
300C2×C4⋊C4—532no—
301C2×C4⋊C4—1062no—
302C2×C4⋊C4—1061no—
303C2×C4⋊C4C22⋊D5312yes—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—1062no—
314C2×C22⋊C4—1062no—
315C2×C22⋊C4—1061no—
316C2×C22⋊C4—1061no—
317#106—1064no—
318#103—1064no—
319C22⋊C8—532no—
320C22⋊C8—1061no—
321C22⋊C8—1061no—
322C22⋊C8—1061no—
323#95—24no—
324#95—21no—
325#95—1062no—
326#95—1061no—
327#95C22⋊D5312yes—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—42no—
335C24—42no—
336C24#1171992312yes—
337C24C2×C22⋊D5311yescharacteristic
338C2×Q8—1061no—
339C2×Q8—1061no—
340C2×Q8—2122no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—1062no—
359C22×C4—1061no—
360C22×C4—1061no—
361C22×C4D4×D5311yescharacteristic
362C22×C4D4×D5311yescharacteristic
363C22×C4D4×D5311yescharacteristic
364C22×C4#1171994711yescharacteristic
365C22×C4C2×C22⋊D5311yescharacteristic
366C22×C4C2×C22⋊D5311yescharacteristic
367C2×C8—1062no—
368C2×C8—2122no—
369C2×C8—2121no—
370C2×C8—2122no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—1061no—
376C4⋊C4—1061no—
377C4⋊C4—4242no—
378C4⋊C4—2122no—
379C4⋊C4—2121no—
380C4⋊C4—2121no—
381C22⋊C4—48no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—4242no—
387C22⋊C4—4242no—
388C22⋊C4—2121no—
389C22⋊C4—2121no—
390C22⋊C4—2121no—
391C22⋊C4—2121no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—44no—
398C23—42no—
399C23—44no—
400C23—42no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—42no—
405C23—42no—
406C23—41no—
407C23—82no—
408C23—81no—
409C23—82no—
410C23—81no—
411C23#1313318511yescharacteristic
412C23#1313318611yescharacteristic
413C23#1313318911yescharacteristic
414C23C2×#1171992311yescharacteristic
415C23#1313323711yescharacteristic
416C23#1313324311yescharacteristic
417C23#1313324711yescharacteristic
418Q8—4241no—
419Q8—2122no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—2122no—
443C2×C4—2121no—
444C2×C4—2121no—
445C2×C4—2122no—
446C2×C4—2122no—
447C2×C4—2122no—
448C2×C4—2122no—
449C2×C4#1313318511yescharacteristic
450C2×C4#1313318811yescharacteristic
451C2×C4#1313318911yescharacteristic
452C2×C4#1313321311yescharacteristic
453C8—4241no—
454C8—2122no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—42no—
465C22—44no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—81no—
470C22—82no—
471C22—81no—
472C22#1681053111yescharacteristic
473C22#1681076111yescharacteristic
474C22#1681076611yescharacteristic
475C22#1681077411yescharacteristic
476C22#1681078511yescharacteristic
477C22#1681080012yes—
478C22#1681081212yes—
479C22#1681085011yescharacteristic
480C22#1681086711yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—82no—
486C4—82no—
487C4—81no—
488C4—4241no—
489C4—4242no—
490C4—4241no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—82no—
496C2#2027370512yes—
497C2#2027372712yes—
498C2#2027571811yescharacteristic
499C2#2027574411yescharacteristic
500C2#2027584311yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC2×C53×#397C2
socleC23×C53#120
prime radicalC25×C53D4
hypercenterC2×#397D53
hypocenterC53#9926
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
1. C2×#16810953C2semidirect
3. #20274801C2semidirect
4. C2×#16810359C2semidirect
6. #20272594C2semidirect
281. C53#9926semidirect

Maximal subgroups

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

subgroupindex
C2×#168109532
#202805202
#202748012
C2×#168103592
C2×#168103472
#202725942
C2×C53×#3972
#992653