Finite Group Databasebeta

← #27287447

Subgroups — #27287447

Order 13568.

708 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275522C211yesmaximal, characteristic, complemented
2#20275473C211yesmaximal, characteristic, complemented
3#20274715C211yesmaximal, characteristic, complemented
4C2×#16810367C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2541C211yesmaximal, characteristic
8C2×#13133150—21no—
9C2×C4⋊#88322C2211yescharacteristic
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133138—21no—
12C2×#13133138—22no—
13C2×C4×#88322C2211yescharacteristic
14#16810493C2211yescharacteristic
15#16810385—21no—
16#16810385—21no—
17#16810367—22no—
18D4⋊#88322—22no—
19C2×#127×C53—21no—
20C2×#127×C53C2211yescharacteristic
21C2×#121×C53—21no—
22C2×#121×C53—21no—
23C2×C4×D4×C53C2211yescharacteristic
24C53×#392—22no—
25C53×#390—21no—
26C4×C4⋊C4×C53C2211yescharacteristic
27C22×#88322—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33#13133150—42no—
34C4⋊#88322—21no—
35C4⋊#88322—21no—
36C4⋊#88322—21no—
37C4⋊#88322D412yescomplemented
38#13133138—44no—
39#13133138—42no—
40C4×#88322—21no—
41C4×#88322—21no—
42C22×D4×C53—22no—
43C22×D4×C53—22no—
44C22×D4×C53—21no—
45C22×D4×C53—21no—
46C22×D4×C53D411yescharacteristic
47C22×D4×C53D412yes—
48C23×C4×C53—21no—
49#127×C53—21no—
50#127×C53—21no—
51#127×C53—21no—
52#127×C53—21no—
53#127×C53—42no—
54#127×C53—42no—
55#121×C53—42no—
56#121×C53—42no—
57#121×C53—42no—
58#121×C53—42no—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C22⋊C4×C53—22no—
67C2×C22⋊C4×C53—22no—
68C2×C22⋊C4×C53—21no—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71#95×C53—21no—
72C24×C53—21no—
73C24×C53—22no—
74C2×D4×C53—21no—
75C2×D4×C53—22no—
76C2×D4×C53—21no—
77C2×D4×C53—22no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—42no—
82C2×D4×C53—42no—
83C2×D4×C53—44no—
84C2×D4×C53—44no—
85C2×D4×C53—44no—
86C2×D4×C53—44no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—82no—
100C2×D4×C53—82no—
101C2×D4×C53D812yes—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—41no—
106C22×C4×C53—41no—
107C22×C4×C53—41no—
108C22×C4×C53—41no—
109C22×C4×C53—41no—
110C22×C4×C53—41no—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×Dic53—41no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53D812yes—
120C4⋊C4×C53SD1612yes—
121C22⋊C4×C53—44no—
122C22⋊C4×C53—44no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C42×C53—41no—
126C42×C53—41no—
127C42×C53C2×D411yescharacteristic
128C42×C53#4111yescharacteristic
129C42×C53#4111yescharacteristic
130C42×C53#4111yescharacteristic
131C4⋊Dic53—42no—
132C4⋊Dic53—41no—
133C4⋊Dic53—41no—
134C4⋊Dic53—41no—
135C4⋊Dic53—41no—
136C2×#88322—22no—
137C2×#88322—22no—
138C2×#88322—41no—
139C2×#88322—41no—
140C2×#88322—41no—
141C2×#88322—41no—
142C2×#88322—41no—
143C2×#88322—41no—
144C2×#88322—41no—
145C23×C53—22no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—42no—
153C23×C53—42no—
154C23×C53—42no—
155C23×C53—42no—
156C23×C53—42no—
157C23×C53—42no—
158C23×C53—42no—
159C23×C53#12011yescharacteristic
160D4×C53—22no—
161D4×C53—22no—
162D4×C53—44no—
163D4×C53—41no—
164D4×C53—41no—
165D4×C53—82no—
166D4×C53—88no—
167D4×C53—88no—
168D4×C53—81no—
169D4×C53—81no—
170D4×C53—81no—
171D4×C53—81no—
172D4×C53—81no—
173D4×C53—81no—
174D4×C53—81no—
175D4×C53—81no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—22no—
179C2×C4×C53—22no—
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—41no—
187C2×C4×C53—81no—
188C2×C4×C53—81no—
189C2×C4×C53—81no—
190C2×C4×C53—81no—
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53#12311yescharacteristic
195C2×C4×C53#12311yescharacteristic
196C2×C4×C53#12411yescharacteristic
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53C2×SD1611yescharacteristic
200C2×C4×C53#13511yescharacteristic
201C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
202C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
203C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
204C2×C4×C53#13711yescharacteristic
205C2×Dic53—81no—
206C2×Dic53—81no—
207C2×Dic53—81no—
208#88322—42no—
209#88322—42no—
210#88322—81no—
211#10408—531nomaximal, Sylow
212C22×C53—42no—
213C22×C53—42no—
214C22×C53—42no—
215C22×C53—81no—
216C22×C53—81no—
217C22×C53—81no—
218C22×C53—81no—
219C22×C53—82no—
220C22×C53—82no—
221C22×C53—82no—
222C22×C53—82no—
223C22×C53—82no—
224C22×C53—82no—
225C22×C53—82no—
226C22×C53#39411yescharacteristic
227C22×C53#44711yescharacteristic
228C22×C53#44711yescharacteristic
229C22×C53#45011yescharacteristic
230C22×C53#45211yescharacteristic
231C22×C53#45311yescharacteristic
232C22×C53#45511yescharacteristic
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—42no—
236C4×C53—42no—
237C4×C53—81no—
238C4×C53#45911yescharacteristic
239C4×C53#46011yescharacteristic
240C4×C53#46611yescharacteristic
241C4×C53#46811yescharacteristic
242C4×C53#48011yescharacteristic
243C4×C53#48211yescharacteristic
244C4×C53#48611yescharacteristic
245C4×C53#49011yescharacteristic
246Dic53—161no—
247#2541D5311yescharacteristic
248#1957—531no—
249#1947—531no—
250#1802—531no—
251C2×#331—531no—
252C2×D4⋊C8—531no—
253#1279—531no—
254C2×C53—82no—
255C2×C53—162no—
256C2×C53#161911yescharacteristic
257C2×C53#163111yescharacteristic
258C2×C53#163911yescharacteristic
259C2×C53#164111yescharacteristic
260C2×C53#203311yescharacteristic
261C2×C53#203711yescharacteristic
262C2×C53#204111yescharacteristic
263C2×#127—21no—
264C2×#127C2×D5311yescharacteristic
265C2×#121—21no—
266C2×#121—21no—
267C2×C4×D4C2×D5311yescharacteristic
268C2×#107—1061no—
269C2×C4⋊C8—531no—
270C2×C4⋊C8—531no—
271C2×#102—1061no—
272C2×#102—1062no—
273C2×C4×C8—531no—
274#392—22no—
275#390—21no—
276#384—531no—
277C4×C4⋊C4C2×D5311yescharacteristic
278#340—1061no—
279#340—1061no—
280#331—1062no—
281D4⋊C8—1062no—
282C53#1040811yescharacteristic, complemented, Sylow
283C22×D4—22no—
284C22×D4—22no—
285C22×D4—21no—
286C22×D4—21no—
287C22×D4C22⋊D5311yescharacteristic
288C22×D4C22⋊D5312yes—
289C23×C4—21no—
290C22×C8—1061no—
291C22×C8—1061no—
292C22×C8—1061no—
293#127—21no—
294#127—21no—
295#127—21no—
296#127—21no—
297#127—42no—
298#127—42no—
299#121—42no—
300#121—42no—
301#121—42no—
302#121—42no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—21no—
307C2×C4⋊C4—1061no—
308C2×C4⋊C4—1061no—
309C2×C4⋊C4—1061no—
310C2×C4⋊C4C22⋊D5311yescharacteristic
311C2×C4⋊C4C22⋊D5311yescharacteristic
312C2×C4⋊C4C22⋊D5311yescharacteristic
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C42—21no—
317C2×C42C22×D5311yescharacteristic
318#107—2122no—
319C4⋊C8—532no—
320C4⋊C8—1061no—
321C4⋊C8—1061no—
322C4⋊C8—1061no—
323#102—2124no—
324#102—2122no—
325C4×C8—1061no—
326C4×C8—1061no—
327#95—21no—
328C24—21no—
329C24—22no—
330C2×D4—21no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4#1171992212yes—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—2121no—
368C22×C4C2×C22⋊D5311yescharacteristic
369C22×C4C2×C22⋊D5311yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371C2×C8—1062no—
372C2×C8—1062no—
373C2×C8—2121no—
374C2×C8—2121no—
375C2×C8—2121no—
376C2×C8—2121no—
377C2×C8—2121no—
378C2×C8—2121no—
379C2×C8—2121no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—2122no—
385C4⋊C4—2121no—
386C4⋊C4—2121no—
387C4⋊C4—2121no—
388C4⋊C4—2121no—
389C4⋊C4#1171992212yes—
390C4⋊C4#1171992312yes—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C42—41no—
396C42—41no—
397C42D4×D5311yescharacteristic
398C42#1171994711yescharacteristic
399C42#1171994711yescharacteristic
400C42#1171994711yescharacteristic
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23#1313318611yescharacteristic
416D4—22no—
417D4—22no—
418D4—44no—
419D4—41no—
420D4—41no—
421D4—82no—
422D4—88no—
423D4—88no—
424D4—81no—
425D4—81no—
426D4—81no—
427D4—81no—
428D4—81no—
429D4—81no—
430D4—81no—
431D4—81no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—4241no—
448C2×C4—4241no—
449C2×C4—4241no—
450C2×C4#1313318811yescharacteristic
451C2×C4#1313318911yescharacteristic
452C2×C4#1313318911yescharacteristic
453C2×C4#1313319811yescharacteristic
454C2×C4#1313319811yescharacteristic
455C2×C4#1313320411yescharacteristic
456C2×C4C2×#1171992211yescharacteristic
457C2×C4C2×#1171992211yescharacteristic
458C2×C4C2×#1171992311yescharacteristic
459C2×C4#1313323711yescharacteristic
460C2×C4#1313324311yescharacteristic
461C2×C4#1313324311yescharacteristic
462C2×C4#1313324311yescharacteristic
463C2×C4#1313324711yescharacteristic
464C8—4241no—
465C8—2122no—
466C8—2122no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22#1681053711yescharacteristic
482C22#1681075711yescharacteristic
483C22#1681075711yescharacteristic
484C22#1681077411yescharacteristic
485C22#1681078511yescharacteristic
486C22#1681079111yescharacteristic
487C22#1681080111yescharacteristic
488C4—21no—
489C4—21no—
490C4—42no—
491C4—42no—
492C4—81no—
493C4—8481no—
494C4#1681081511yescharacteristic
495C4#1681082011yescharacteristic
496C4#1681085711yescharacteristic
497C4#1681086711yescharacteristic
498C4#1681093411yescharacteristic
499C4#1681094411yescharacteristic
500C4#1681096911yescharacteristic
501C4#1681098711yescharacteristic
502C2—82no—
503C2—162no—
504C2#2027376911yescharacteristic
505C2#2027384211yescharacteristic
506C2#2027389411yescharacteristic
507C2#2027390811yescharacteristic
508C2#2027589411yescharacteristic
509C2#2027592211yescharacteristic
510C2#2027594611yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#2541C2
socleC23×C53#120
hypercenter#2541D53
hypocenterC53#10408
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. #20275522C2semidirect
2. #20275473C2semidirect
3. #20274715C2semidirect
4. C2×#16810367C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272556C2semidirect
37. C4⋊#88322D4semidirect
282. C53#10408semidirect

Maximal subgroups

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

subgroupindex
#202755222
#202754732
#202747152
C2×#168103672
C2×D4⋊#883222
#202725562
C53×#25412
#1040853