Finite Group Databasebeta

← #27438757

Subgroups — #27438757

Order 13568.

1080 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×#16811029C211yesmaximal, characteristic, complemented
2C2×#16810923C211yesmaximal, characteristic, complemented
3#20279881C211yesmaximal, characteristic, complemented
4#20275901C212yesmaximal, complemented
5#20275894C212yesmaximal, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20274914C212yesmaximal, complemented
9#20274914C212yesmaximal, complemented
10#20274540C211yesmaximal, characteristic, complemented
11C2×C53×#555C211yesmaximal, characteristic
12C2×#13133209C2211yescharacteristic
13#16811029—24no—
14#16810923—24no—
15C2×#13133150C2212yes—
16C2×#13133150C2211yescharacteristic
17C2×#13133148C2212yes—
18C2×#13133148C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133138C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2212yes—
23C2×C22⋊#88322C2212yes—
24C2×C22⋊#88322C2212yes—
25C2×#13133128C2211yescharacteristic
26#16810561—21no—
27#16810537—21no—
28#16810517—22no—
29#16810483C2212yes—
30#16810483C2212yes—
31C4×C4⋊Dic53C2211yescharacteristic
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C53×#555—24no—
36C53×#555—24no—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C2×#124×C53—22no—
39C2×#121×C53—21no—
40C2×#121×C53C2212yes—
41C2×C4×D4×C53C2211yescharacteristic
42C2×C4×D4×C53C2211yescharacteristic
43C2×#117×C53—21no—
44C2×#117×C53C2212yes—
45C22×#88322C2311yescharacteristic
46C22×#88322C2311yescharacteristic
47C22×#88322C2312yes—
48#13133209—22no—
49#13133209—22no—
50C2×C4⋊Dic53C2311yescharacteristic
51C2×C4⋊Dic53C2311yescharacteristic
52C2×C4⋊Dic53C2312yes—
53C2×#11719919—21no—
54C2×#11719919—21no—
55C2×C22⋊Dic53—22no—
56C2×C22⋊Dic53—22no—
57C2×C4×Dic53—21no—
58#13133150—24no—
59#13133150—22no—
60#13133148—24no—
61#13133148—22no—
62C4⋊#88322—22no—
63#13133138—24no—
64#13133138—22no—
65#13133138—22no—
66C22⋊#88322—24no—
67C22⋊#88322—24no—
68#13133128—22no—
69#13133124—22no—
70#13133124—22no—
71#13133124—21no—
72C22×Q8×C53—21no—
73C22×D4×C53—22no—
74C22×D4×C53D411yescharacteristic
75C22×D4×C53C2311yescharacteristic
76C23×C4×C53—21no—
77C23×C4×C53C2312yes—
78C4⋊Q8×C53—22no—
79C4⋊Q8×C53D414yes—
80#124×C53—216no—
81#121×C53—28no—
82#121×C53—24no—
83#121×C53—24no—
84C4×D4×C53—22no—
85C4×D4×C53—22no—
86C4×D4×C53—22no—
87C4×D4×C53D414yes—
88#117×C53—28no—
89#117×C53—24no—
90#117×C53—24no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53D411yescharacteristic
93C2×C4⋊C4×C53C2311yescharacteristic
94C2×C4⋊C4×C53C2312yes—
95C2×C22⋊C4×C53—22no—
96C2×C22⋊C4×C53—22no—
97C2×C22⋊C4×C53—21no—
98C2×C22⋊C4×C53D412yes—
99C2×C42×C53—22no—
100C2×C42×C53C2311yescharacteristic
101C24×C53—21no—
102C24×C53C2×D412yes—
103C2×Q8×C53—24no—
104C2×Q8×C53—81no—
105C2×Q8×C53—81no—
106C2×D4×C53—28no—
107C2×D4×C53—22no—
108C2×D4×C53—21no—
109C2×D4×C53—22no—
110C2×D4×C53—22no—
111C2×D4×C53—42no—
112C2×D4×C53—82no—
113C2×D4×C53—82no—
114C2×D4×C53C2×D412yes—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—24no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—44no—
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2411yescharacteristic
129C22×Dic53—22no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C4⋊C4×C53—24no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53C2×D412yes—
137C4⋊C4×C53C2×D414yes—
138C22⋊C4×C53—28no—
139C22⋊C4×C53—28no—
140C22⋊C4×C53—24no—
141C22⋊C4×C53—24no—
142C42×C53—28no—
143C42×C53C2×D412yes—
144C42×C53C2×D412yes—
145C22⋊Dic53—44no—
146C22⋊Dic53—82no—
147C4⋊Dic53—21no—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—22no—
152C4⋊Dic53—22no—
153#11719919—42no—
154#11719919—42no—
155C4×Dic53—81no—
156C2×#88322—22no—
157C2×#88322—22no—
158C2×#88322—21no—
159C2×#88322—21no—
160C2×#88322—22no—
161C2×#88322—22no—
162C2×#88322#4114yes—
163C23×C53—22no—
164C23×C53—22no—
165C23×C53—22no—
166C23×C53—22no—
167C23×C53—22no—
168C23×C53—22no—
169C23×C53—22no—
170C23×C53—22no—
171C23×C53—42no—
172C23×C53—42no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53C22×D411yescharacteristic
176Q8×C53—84no—
177D4×C53—22no—
178D4×C53—22no—
179D4×C53—44no—
180D4×C53—42no—
181D4×C53—88no—
182C2×C4×C53—22no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—22no—
186C2×C4×C53—22no—
187C2×C4×C53—24no—
188C2×C4×C53—22no—
189C2×C4×C53—22no—
190C2×C4×C53—48no—
191C2×C4×C53—42no—
192C2×C4×C53—42no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53#12012yes—
199C2×C4×C53#12012yes—
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C2×#4112yes—
203C2×C4×C532−1+412yes—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—42no—
207C2×Dic53—42no—
208C2×Dic53—42no—
209C2×Dic53—42no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212#88322—24no—
213#88322—41no—
214#88322—41no—
215#25710—531nomaximal, Sylow
216C22×C53—42no—
217C22×C53—42no—
218C22×C53—42no—
219C22×C53—42no—
220C22×C53—42no—
221C22×C53—42no—
222C22×C53—42no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—82no—
227C22×C53—82no—
228C22×C53C2×#12011yescharacteristic
229C22×C53#54712yes—
230C22×C53#57512yes—
231C22×C53#57611yescharacteristic
232C22×C53#57711yescharacteristic
233C4×C53—22no—
234C4×C53—22no—
235C4×C53—42no—
236C4×C53—42no—
237C4×C53—44no—
238C4×C53#53211yescharacteristic
239C4×C53#53611yescharacteristic
240C4×C53#54812yes—
241Dic53—82no—
242Dic53—81no—
243Dic53—81no—
244C2×#555D5311yescharacteristic
245C2×#501—531no—
246C2×#478—531no—
247#2562—531no—
248#2034—532no—
249#2033—532no—
250#1988—531no—
251#1955—531no—
252#1854—532no—
253#1854—532no—
254#1762—531no—
255C2×C53—82no—
256C2×C53—82no—
257C2×C53#243211yescharacteristic
258C2×C53#300511yescharacteristic
259C2×C53#301211yescharacteristic
260C2×C53#334712yes—
261C2×C53#334812yes—
262#555—24no—
263#555—24no—
264C2×C4⋊Q8C2×D5311yescharacteristic
265C2×#125—531no—
266C2×#124—22no—
267C2×#121—21no—
268C2×#121C2×D5312yes—
269C2×C4×D4C2×D5311yescharacteristic
270C2×C4×D4C2×D5311yescharacteristic
271C2×#117—21no—
272C2×#117C2×D5312yes—
273#501—1064no—
274#478—1064no—
275C2×#107—531no—
276C2×#107—532no—
277C2×#106—531no—
278C2×#106—532no—
279C2×C4⋊C8—531no—
280C2×#102—531no—
281C2×#102—531no—
282C2×#102—532no—
283C2×C22⋊C8—532no—
284C2×C22⋊C8—532no—
285C2×#97—531no—
286#400—1061no—
287#394—1061no—
288#388—1062no—
289#380—532no—
290#380—532no—
291C4×C4⋊C4—531no—
292#340—532no—
293#340—532no—
294#340—532no—
295C53#2571011yescharacteristic, complemented, Sylow
296C22×Q8—21no—
297C22×D4—22no—
298C22×D4C22⋊D5311yescharacteristic
299C22×D4C22×D5311yescharacteristic
300C23×C4—21no—
301C23×C4C22×D5312yes—
302C22×C8—532no—
303C22×C8—531no—
304C22×C8—531no—
305C4⋊Q8—22no—
306C4⋊Q8C22⋊D5314yes—
307#125—1062no—
308#125—1062no—
309#124—216no—
310#121—28no—
311#121—24no—
312#121—24no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4C22⋊D5314yes—
317#117—28no—
318#117—24no—
319#117—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—531no—
322C2×C4⋊C4—531no—
323C2×C4⋊C4—532no—
324C2×C4⋊C4—1061no—
325C2×C4⋊C4—1061no—
326C2×C4⋊C4C22⋊D5311yescharacteristic
327C2×C4⋊C4C22×D5311yescharacteristic
328C2×C4⋊C4C22×D5312yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—1062no—
333C2×C22⋊C4—1062no—
334C2×C22⋊C4C22⋊D5312yes—
335C2×C42—22no—
336C2×C42—1061no—
337C2×C42C22×D5311yescharacteristic
338#107—1062no—
339#107—1064no—
340#106—1062no—
341#106—1064no—
342C4⋊C8—1062no—
343#102—1064no—
344#102—1062no—
345#102—1062no—
346C22⋊C8—1064no—
347C22⋊C8—1064no—
348#97—1062no—
349#95—1061no—
350#95—1062no—
351#95—1062no—
352C24—21no—
353C24C2×C22⋊D5312yes—
354C2×Q8—24no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×D4—28no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—42no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4C2×C22⋊D5312yes—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—44no—
376C22×C4—1062no—
377C22×C4—1061no—
378C22×C4—1061no—
379C22×C4C2×C22⋊D5312yes—
380C22×C4C2×C22⋊D5311yescharacteristic
381C22×C4C2×C22⋊D5311yescharacteristic
382C22×C4C23×D5311yescharacteristic
383C2×C8—534no—
384C2×C8—1062no—
385C2×C8—1062no—
386C2×C8—1061no—
387C2×C8—1061no—
388C2×C8—1062no—
389C2×C8—1062no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—1061no—
395C4⋊C4—1061no—
396C4⋊C4—1061no—
397C4⋊C4—1061no—
398C4⋊C4—1062no—
399C4⋊C4—1062no—
400C4⋊C4—2122no—
401C4⋊C4—2122no—
402C4⋊C4C2×C22⋊D5312yes—
403C4⋊C4C2×C22⋊D5314yes—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—4242no—
409C22⋊C4—2124no—
410C42—28no—
411C42—4241no—
412C42C2×C22⋊D5312yes—
413C42C2×C22⋊D5312yes—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23C22×C22⋊D5311yescharacteristic
427Q8—84no—
428D4—22no—
429D4—22no—
430D4—44no—
431D4—42no—
432D4—88no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—48no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—2122no—
450C2×C4—2121no—
451C2×C4—2121no—
452C2×C4—2122no—
453C2×C4—2122no—
454C2×C4—2122no—
455C2×C4—2121no—
456C2×C4—2121no—
457C2×C4#1313318612yes—
458C2×C4#1313318612yes—
459C2×C4C22×C22⋊D5311yescharacteristic
460C2×C4C22×C22⋊D5311yescharacteristic
461C2×C4#41×D5312yes—
462C2×C4#1313326412yes—
463C8—1064no—
464C8—2121no—
465C8—2121no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—82no—
477C22—82no—
478C22C2×#1313318611yescharacteristic
479C22#1681127212yes—
480C22#1681142112yes—
481C22#1681142611yescharacteristic
482C22#1681143411yescharacteristic
483C4—22no—
484C4—22no—
485C4—42no—
486C4—42no—
487C4—44no—
488C4—4241no—
489C4—4241no—
490C4—4242no—
491C4#1681116911yescharacteristic
492C4#1681119411yescharacteristic
493C4#1681128012yes—
494C2—82no—
495C2—82no—
496C2#2027833911yescharacteristic
497C2#2028395711yescharacteristic
498C2#2028401911yescharacteristic
499C2#2028721512yes—
500C2#2028723012yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#555C2
socleC23×C53C22×D4
hypercenterC2×#555D53
hypocenterC53#25710
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#16811029C2semidirect
2. C2×#16810923C2semidirect
3. #20279881C2semidirect
4. #20275901C2semidirect
5. #20275894C2semidirect
6. #20275668C2semidirect
7. #20275514C2semidirect
8. #20274914C2semidirect
9. #20274914C2semidirect
10. #20274540C2semidirect
295. C53#25710semidirect

Maximal subgroups

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

subgroupindex
C2×#168110292
C2×#168109232
#202798812
#202759012
#202758942
#202756682
#202755142
#202749142
#202745402
C2×C53×#5552
#2571053