Finite Group Databasebeta

← #27284386

Subgroups — #27284386

Order 13568.

778 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
1C2×#16810958C211yesmaximal, characteristic, complemented
2#20280520C211yesmaximal, characteristic
3#20274805C211yesmaximal, characteristic, complemented
4C2×#16810359C211yesmaximal, characteristic, complemented
5C2×#16810347C211yesmaximal, characteristic
6#20272594C211yesmaximal, characteristic, complemented
7C2×C53×#399C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810958—22no—
10#16810958—22no—
11C2×#13133150—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×C53—21no—
25C22×C4⋊C4×C53C2211yescharacteristic
26C22×C22⋊C4×C53—21no—
27C22×C22⋊C4×C53C2211yescharacteristic
28C53×#399—22no—
29C53×#399—22no—
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#13133150—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—24no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—44no—
66C2×C4⋊C4×C53D412yes—
67C2×C22⋊C4×C53—24no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—21no—
74C2×C22⋊C4×C53—44no—
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×C53Q1612yes—
86C24×C53C2×D411yescharacteristic
87C22×C4×C53—24no—
88C22×C4×C53—22no—
89C22×C4×C53—21no—
90C22×C4×C53—22no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—42no—
98C22×C4×C53—42no—
99C22×C4×C53—42no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×Dic53—22no—
109C22×Dic53—21no—
110C22×Dic53—21no—
111C2×Dic106—21no—
112C2×Dic106—21no—
113C2×Dic106—42no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—48no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C22⋊C4×C53—48no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C22⋊Dic53—41no—
125C22⋊Dic53—41no—
126C22⋊Dic53—41no—
127C22⋊Dic53—41no—
128C22⋊Dic53—82no—
129C22⋊Dic53—82no—
130C4⋊Dic53—21no—
131C4⋊Dic53—21no—
132C4⋊Dic53—42no—
133#11719919—41no—
134#11719919—41no—
135#11719919—82no—
136C2×#88322—22no—
137C2×#88322—42no—
138C2×#88322—42no—
139C2×#88322—41no—
140C23×C53—22no—
141C23×C53—22no—
142C23×C53—22no—
143C23×C53—22no—
144C23×C53—22no—
145C23×C53—44no—
146C23×C53—44no—
147C23×C53—42no—
148C23×C53—42no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—42no—
153C23×C53—42no—
154C23×C53—41no—
155C23×C53—82no—
156C23×C53—82no—
157C23×C53—81no—
158C23×C53—81no—
159C23×C53#12011yescharacteristic
160C23×C53#12011yescharacteristic
161C23×C53#12111yescharacteristic
162C23×C53C2×Q1611yescharacteristic
163C23×C53#13511yescharacteristic
164C23×C53#13711yescharacteristic
165C23×C53#13711yescharacteristic
166C2×C4×C53—22no—
167C2×C4×C53—22no—
168C2×C4×C53—24no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—44no—
174C2×C4×C53—44no—
175C2×C4×C53—42no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—41no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—82no—
184C2×C4×C53—82no—
185C2×C4×C53—81no—
186C2×C4×C53#12011yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12711yescharacteristic
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—42no—
193C2×Dic53—42no—
194C2×Dic53—42no—
195C2×Dic53—42no—
196C2×Dic53—42no—
197Dic106—42no—
198Dic106—81no—
199#88322—42no—
200#88322—81no—
201#9928—531nomaximal, Sylow
202C22×C53—22no—
203C22×C53—22no—
204C22×C53—22no—
205C22×C53—22no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—44no—
211C22×C53—44no—
212C22×C53—42no—
213C22×C53—82no—
214C22×C53—82no—
215C22×C53—82no—
216C22×C53—82no—
217C22×C53—81no—
218C22×C53—81no—
219C22×C53#39211yescharacteristic
220C22×C53#45111yescharacteristic
221C22×C53#45111yescharacteristic
222C22×C53#45211yescharacteristic
223C22×C53#45211yescharacteristic
224C22×C53#45312yes—
225C22×C53#45812yes—
226C22×C53#46711yescharacteristic
227C22×C53#47011yescharacteristic
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—44no—
233C4×C53—81no—
234C4×C53—82no—
235Dic53—82no—
236Dic53—81no—
237Dic53—81no—
238C2×#484—531no—
239#2626—531no—
240C2×#399D5311yescharacteristic
241#1824—531no—
242C2×#328—531no—
243C2×#323—531no—
244#1294—531no—
245C2×C53—21no—
246C2×C53—21no—
247C2×C53—21no—
248C2×C53—21no—
249C2×C53—82no—
250C2×C53#161412yes—
251C2×C53#161812yes—
252C2×C53#200111yescharacteristic
253C2×C53#200211yescharacteristic
254C2×C53#202211yescharacteristic
255C2×C22⋊Q8—531no—
256C22×C4⋊C4—21no—
257C22×C4⋊C4C2×D5311yescharacteristic
258C22×C22⋊C4—21no—
259C22×C22⋊C4C2×D5311yescharacteristic
260#484—1062no—
261#484—1062no—
262C2×#107—1061no—
263C2×#103—1061no—
264C2×C22⋊C8—531no—
265C2×C22⋊C8—531no—
266#399—22no—
267#399—22no—
268#399—1061no—
269#396—1062no—
270#393—1061no—
271#385—1061no—
272#380—531no—
273#379—1061no—
274C2×#95C2×D5311yescharacteristic
275#340—1062no—
276#328—1062no—
277#323—1062no—
278C53#992811yescharacteristic, complemented, Sylow
279C25C22⋊D5311yescharacteristic
280C22×Q8—531no—
281C23×C4—21no—
282C23×C4—21no—
283C23×C4C22⋊D5311yescharacteristic
284C23×C4C22×D5311yescharacteristic
285C22×C8—1061no—
286C22×C8—1061no—
287C22⋊Q8—1061no—
288C22⋊Q8—1061no—
289C22⋊Q8—1061no—
290C22⋊Q8—1061no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—531no—
299C2×C4⋊C4—532no—
300C2×C4⋊C4—1062no—
301C2×C4⋊C4—1061no—
302C2×C4⋊C4C22⋊D5312yes—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—1062no—
312C2×C22⋊C4—1062no—
313C2×C22⋊C4—1061no—
314C2×C22⋊C4—1061no—
315#107—1064no—
316#103—1064no—
317C22⋊C8—532no—
318C22⋊C8—1061no—
319C22⋊C8—1061no—
320C22⋊C8—1061no—
321#95—21no—
322#95—1062no—
323#95—1061no—
324#95C22⋊D5312yes—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—42no—
332C24—42no—
333C24#1171992512yes—
334C24C2×C22⋊D5311yescharacteristic
335C2×Q8—1061no—
336C2×Q8—1061no—
337C2×Q8—2122no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—1062no—
354C22×C4—1061no—
355C22×C4—1061no—
356C22×C4D4×D5311yescharacteristic
357C22×C4D4×D5311yescharacteristic
358C22×C4D4×D5311yescharacteristic
359C22×C4#1171994711yescharacteristic
360C22×C4C2×C22⋊D5311yescharacteristic
361C22×C4C2×C22⋊D5311yescharacteristic
362C2×C8—1062no—
363C2×C8—2122no—
364C2×C8—2121no—
365C2×C8—2122no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—48no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—1061no—
372C4⋊C4—1061no—
373C4⋊C4—4242no—
374C4⋊C4—2122no—
375C4⋊C4—2121no—
376C4⋊C4—2121no—
377C22⋊C4—48no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—4242no—
383C22⋊C4—4242no—
384C22⋊C4—2121no—
385C22⋊C4—2121no—
386C22⋊C4—2121no—
387C22⋊C4—2121no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—44no—
394C23—42no—
395C23—44no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—42no—
402C23—41no—
403C23—82no—
404C23—81no—
405C23—82no—
406C23—81no—
407C23#1313318511yescharacteristic
408C23#1313318611yescharacteristic
409C23#1313318911yescharacteristic
410C23C2×#1171992511yescharacteristic
411C23#1313323711yescharacteristic
412C23#1313324711yescharacteristic
413C23#1313324811yescharacteristic
414Q8—4241no—
415Q8—2122no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—2122no—
437C2×C4—2121no—
438C2×C4—2121no—
439C2×C4—2122no—
440C2×C4—2122no—
441C2×C4—2122no—
442C2×C4—2122no—
443C2×C4#1313318511yescharacteristic
444C2×C4#1313318811yescharacteristic
445C2×C4#1313318911yescharacteristic
446C2×C4#1313321311yescharacteristic
447C8—4241no—
448C8—2122no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—21no—
456C22—21no—
457C22—44no—
458C22—42no—
459C22—44no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—81no—
464C22—82no—
465C22—81no—
466C22#1681053111yescharacteristic
467C22#1681077811yescharacteristic
468C22#1681077911yescharacteristic
469C22#1681078311yescharacteristic
470C22#1681078511yescharacteristic
471C22#1681079012yes—
472C22#1681081212yes—
473C22#1681086211yescharacteristic
474C22#1681087911yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—44no—
480C4—81no—
481C4—82no—
482C4—4241no—
483C4—4242no—
484C4—4241no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—82no—
490C2#2027373912yes—
491C2#2027376112yes—
492C2#2027572311yescharacteristic
493C2#2027573011yescharacteristic
494C2#2027584311yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC2×C53×#399C2
socleC23×C53#120
prime radicalC25×C53D4
hypercenterC2×#399D53
hypocenterC53#9928
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C2×#16810958C2semidirect
3. #20274805C2semidirect
4. C2×#16810359C2semidirect
6. #20272594C2semidirect
278. C53#9928semidirect

Maximal subgroups

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

subgroupindex
C2×#168109582
#202805202
#202748052
C2×#168103592
C2×#168103472
#202725942
C2×C53×#3992
#992853