Finite Group Databasebeta

← #27280667

Subgroups — #27280667

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277767C211yesmaximal, characteristic, complemented
2#20275552C211yesmaximal, characteristic, complemented
3#20273283C211yesmaximal, characteristic, complemented
4C2×#16810357C211yesmaximal, characteristic, complemented
5#20272595C211yesmaximal, characteristic, complemented
6#20272561C211yesmaximal, characteristic, complemented
7C53×#3186C211yesmaximal, characteristic
8C2×D4×Dic53—21no—
9C2×#13133132—21no—
10C2×#13133132—21no—
11C2×C22⋊#88322C2211yescharacteristic
12#16810516—21no—
13#16810509—21no—
14#16810503—21no—
15#16810503—21no—
16#16810483C2211yescharacteristic
17C22⋊C4×Dic53—21no—
18C4×C22⋊Dic53C2211yescharacteristic
19C2×#13133124C2211yescharacteristic
20#16810357—21no—
21#16810357—21no—
22#16810357—22no—
23C2×#121×C53C2211yescharacteristic
24C2×#121×C53C2211yescharacteristic
25C53×#480—21no—
26C53×#480C412yescomplemented
27C53×#466—22no—
28C53×#447—21no—
29C53×#447C412yescomplemented
30C2×C22⋊C8×C53C2211yescharacteristic
31C23×Dic53—21no—
32C23×Dic53—21no—
33C22×#88322—21no—
34D4×Dic53—42no—
35D4×Dic53—42no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53D411yescharacteristic
43C2×C22⋊Dic53D411yescharacteristic
44C2×C4×Dic53—21no—
45C2×C4×Dic53—21no—
46#13133132—42no—
47#13133132—42no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124D411yescharacteristic
56#13133124D411yescharacteristic
57C22×D4×C53—21no—
58C22×D4×C53—21no—
59C22×D4×C53C2×C411yescharacteristic, complemented
60C22×D4×C53D411yescharacteristic
61C23×C4×C53C2311yescharacteristic
62C2×D8×C53—22no—
63C22×C8×C53—21no—
64#121×C53—21no—
65#121×C53—21no—
66#121×C53—21no—
67#121×C53—21no—
68#121×C53—21no—
69#121×C53—21no—
70#121×C53C2×C412yes—
71#121×C53D412yes—
72C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
73C2×C4⋊C4×C53D411yescharacteristic
74C2×C22⋊C4×C53—21no—
75C2×C22⋊C4×C53—21no—
76#107×C53—22no—
77#102×C53—22no—
78#102×C53—22no—
79C22⋊C8×C53—21no—
80C22⋊C8×C53C2×C412yescomplemented
81C24×C53—21no—
82C24×C53—21no—
83C24×C53C2×D411yescharacteristic
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—21no—
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—81no—
100C2×D4×C53—81no—
101C2×D4×C53C22⋊C412yes—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—81no—
105C22×C4×C53C22×C411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×Dic53—21no—
112C22×Dic53—21no—
113C22×Dic53—21no—
114C22×Dic53—21no—
115C22×Dic53—42no—
116C22×Dic53—42no—
117C22×Dic53—42no—
118C22×Dic53—41no—
119C22×Dic53—41no—
120C22×Dic53—41no—
121C22×Dic53—81no—
122C22×Dic53—81no—
123D8×C53—82no—
124C2×C8×C53—22no—
125C2×C8×C53—81no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53C22⋊C412yes—
130C22⋊C4×C53—22no—
131C22⋊C4×C53—22no—
132C22⋊C4×C53—41no—
133C22⋊C4×C53—41no—
134C22⋊Dic53—21no—
135C22⋊Dic53—21no—
136C22⋊Dic53—21no—
137C22⋊Dic53—42no—
138C22⋊Dic53—42no—
139C22⋊Dic53—41no—
140C22⋊Dic53—41no—
141C22⋊Dic53—81no—
142C22⋊Dic53D812yescomplemented
143C4⋊Dic53—42no—
144C4⋊Dic53—81no—
145C4×Dic53—42no—
146C4×Dic53—81no—
147C2×#88322—41no—
148C2×#88322—81no—
149C23×C53—22no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53—41no—
154C23×C53—41no—
155C23×C53—41no—
156C23×C53—41no—
157C23×C53—41no—
158C23×C53—41no—
159C23×C53—41no—
160C23×C53—41no—
161C23×C53—81no—
162C23×C53—81no—
163C23×C53C4×D411yescharacteristic
164C23×C53#12011yescharacteristic
165C23×C53#12111yescharacteristic
166C23×C53C2×D811yescharacteristic
167C23×C53#13511yescharacteristic
168C23×C53AGL(1,ℤ/8)11yescharacteristic
169C23×C53AGL(1,ℤ/8)11yescharacteristic
170D4×C53—42no—
171D4×C53—42no—
172D4×C53—42no—
173D4×C53—42no—
174D4×C53—42no—
175D4×C53—42no—
176D4×C53—81no—
177D4×C53—81no—
178D4×C53—81no—
179D4×C53—81no—
180D4×C53—81no—
181D4×C53—81no—
182C2×C4×C53—22no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53C2×C22⋊C411yescharacteristic
195C2×C4×C53C4×D411yescharacteristic
196C2×C4×C53#12111yescharacteristic
197C2×C4×C53#12311yescharacteristic
198C2×Dic53—22no—
199C2×Dic53—22no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53—41no—
203C2×Dic53—41no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—84no—
211C2×Dic53—82no—
212C2×Dic53—82no—
213C2×Dic53—81no—
214C2×Dic53—81no—
215C2×Dic53—81no—
216C8×C53—42no—
217#88322—81no—
218#9277—531nomaximal, Sylow
219C22×C53—21no—
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—21no—
224C22×C53—21no—
225C22×C53—41no—
226C22×C53—41no—
227C22×C53—41no—
228C22×C53—41no—
229C22×C53—42no—
230C22×C53—42no—
231C22×C53—42no—
232C22×C53—81no—
233C22×C53—81no—
234C22×C53—81no—
235C22×C53—81no—
236C22×C53—81no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53—161no—
240C22×C53—161no—
241C22×C53#38611yescharacteristic
242C22×C53C4×D811yescharacteristic
243C22×C53#44211yescharacteristic
244C22×C53#44711yescharacteristic
245C22×C53#44911yescharacteristic
246C22×C53#45911yescharacteristic
247C22×C53#46311yescharacteristic
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—42no—
253C4×C53—81no—
254Dic53—42no—
255Dic53—42no—
256Dic53—81no—
257Dic53—81no—
258Dic53—161no—
259#3186D5311yescharacteristic
260#2364—531no—
261#1964—531no—
262#1519—531no—
263C2×#327—531no—
264#1294—531no—
265#1282—531no—
266C2×C53—21no—
267C2×C53—21no—
268C2×C53—21no—
269C2×C53—21no—
270C2×C53—21no—
271C2×C53—21no—
272C2×C53—82no—
273C2×C53—161no—
274C2×C53#187911yescharacteristic
275C2×C53#190611yescharacteristic
276C2×C53#191311yescharacteristic
277C2×#121C2×D5311yescharacteristic
278C2×#121C2×D5311yescharacteristic
279C2×C4×D4—1061no—
280#480—21no—
281#480Dic5312yescomplemented
282#466—22no—
283#447—21no—
284#447Dic5312yescomplemented
285C2×#99—1061no—
286C2×#99—1061no—
287C2×C22⋊C8—531no—
288C2×C22⋊C8C2×D5311yescharacteristic
289#388—1061no—
290#387—1061no—
291#386—1061no—
292#386—1061no—
293#380—531no—
294C4×C22⋊C4—531no—
295C4×C22⋊C4—1061no—
296C2×#95—531no—
297#327—1061no—
298#327—1061no—
299#327—1062no—
300C53#927711yescharacteristic, complemented, Sylow
301C22×D4—21no—
302C22×D4—21no—
303C22×D4C2×Dic5311yescharacteristic, complemented
304C22×D4C22⋊D5311yescharacteristic
305C23×C4—1061no—
306C23×C4—1061no—
307C23×C4C22×D5311yescharacteristic
308C2×D8—22no—
309C22×C8—21no—
310C22×C8—1061no—
311#121—21no—
312#121—21no—
313#121—21no—
314#121—21no—
315#121—21no—
316#121—21no—
317#121C2×Dic5312yes—
318#121C22⋊D5312yes—
319C4×D4—2122no—
320C4×D4—2122no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4—1061no—
323C2×C4⋊C4C2×Dic5311yescharacteristic, complemented
324C2×C4⋊C4C22⋊D5311yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—531no—
328C2×C22⋊C4—531no—
329C2×C22⋊C4—1061no—
330C2×C22⋊C4—1061no—
331C2×C22⋊C4—1061no—
332C2×C22⋊C4—1061no—
333C2×C42—1061no—
334C2×C42—1061no—
335#107—22no—
336#102—22no—
337#102—22no—
338#99—2122no—
339#99—2122no—
340C22⋊C8—21no—
341C22⋊C8—1061no—
342C22⋊C8—1061no—
343C22⋊C8C2×Dic5312yescomplemented
344#95—531no—
345#95—531no—
346#95—1061no—
347#95—1061no—
348#95—1061no—
349#95—1061no—
350#95—1061no—
351C24—21no—
352C24—21no—
353C24D4×D5311yescharacteristic
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4C22⋊Dic5312yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—81no—
375C22×C4—1061no—
376C22×C4—1061no—
377C22×C4—1061no—
378C22×C4—1061no—
379C22×C4—4241no—
380C22×C4—4241no—
381C22×C4—2122no—
382C22×C4—2122no—
383C22×C4—2122no—
384C22×C4—2121no—
385C22×C4—2121no—
386C22×C4—2121no—
387C22×C4D4×D5311yescharacteristic
388C22×C4#1171994711yescharacteristic
389C22×C4#1171994711yescharacteristic
390C22×C4C22×Dic5311yescharacteristic
391C22×C4C2×C22⋊D5311yescharacteristic
392C22×C4C2×C22⋊D5311yescharacteristic
393D8—82no—
394C2×C8—22no—
395C2×C8—81no—
396C2×C8—4241no—
397C2×C8—2121no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—4241no—
402C4⋊C4—2122no—
403C4⋊C4C22⋊Dic5312yes—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—532no—
409C22⋊C4—1061no—
410C22⋊C4—1061no—
411C22⋊C4—1061no—
412C22⋊C4—4241no—
413C22⋊C4—2122no—
414C22⋊C4—2122no—
415C22⋊C4—2121no—
416C22⋊C4—2121no—
417C42—4241no—
418C42—2122no—
419C23—22no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—81no—
432C23—81no—
433C23D4×Dic5311yescharacteristic
434C23#1313318511yescharacteristic
435C23#1313318911yescharacteristic
436C23D8×D5311yescharacteristic
437C23#1313323611yescharacteristic
438C23#1313324111yescharacteristic
439C23#1313324111yescharacteristic
440D4—42no—
441D4—42no—
442D4—42no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—81no—
447D4—81no—
448D4—81no—
449D4—81no—
450D4—81no—
451D4—81no—
452C2×C4—22no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—1062no—
465C2×C4—1062no—
466C2×C4—4241no—
467C2×C4—4241no—
468C2×C4—4241no—
469C2×C4—4242no—
470C2×C4—4242no—
471C2×C4—4244no—
472C2×C4—2121no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4—2121no—
478C2×C4—2121no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4C2×C22⋊Dic5311yescharacteristic
483C2×C4D4×Dic5311yescharacteristic
484C2×C4#1313318911yescharacteristic
485C2×C4#1313319811yescharacteristic
486C8—42no—
487C8—4241no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—161no—
509C22—161no—
510C22#1681050311yescharacteristic
511C22D8×Dic5311yescharacteristic
512C22#1681073511yescharacteristic
513C22#1681075511yescharacteristic
514C22#1681076511yescharacteristic
515C22#1681081611yescharacteristic
516C22#1681083611yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—42no—
522C4—81no—
523C4—8481no—
524C4—4241no—
525C4—4241no—
526C4—2122no—
527C4—2122no—
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—82no—
535C2—161no—
536C2#2027506511yescharacteristic
537C2#2027521811yescharacteristic
538C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#3186C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenter#3186D53
hypocenterC53#9277
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. #20277767C2semidirect
2. #20275552C2semidirect
3. #20273283C2semidirect
4. C2×#16810357C2semidirect
5. #20272595C2semidirect
6. #20272561C2semidirect
26. C53×#480C4semidirect
29. C53×#447C4semidirect
59. C22×D4×C53C2×C4semidirect
72. C2×C4⋊C4×C53C2×C4semidirect
80. C22⋊C8×C53C2×C4semidirect
142. C22⋊Dic53D8semidirect
281. #480Dic53semidirect
284. #447Dic53semidirect
300. C53#9277semidirect
303. C22×D4C2×Dic53semidirect
323. C2×C4⋊C4C2×Dic53semidirect
343. C22⋊C8C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202777672
#202755522
#202732832
C2×#168103572
#202725952
#202725612
C53×#31862
#927753