Finite Group Databasebeta

← #27280653

Subgroups — #27280653

Order 13568.

606 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
1#20277767C211yesmaximal, characteristic
2#20275701C211yesmaximal, characteristic, complemented
3#20273283C211yesmaximal, characteristic
4C2×#16810360C211yesmaximal, characteristic, complemented
5#20272595C211yesmaximal, characteristic, complemented
6#20272561C211yesmaximal, characteristic, complemented
7C53×#3187C211yesmaximal, characteristic
8C2×D4×Dic53—21no—
9C2×C22⋊#88322C2211yescharacteristic
10#16810516—21no—
11#16810509—21no—
12#16810503—21no—
13#16810503—21no—
14#16810483C2211yescharacteristic
15C22⋊C4×Dic53—21no—
16C4×C22⋊Dic53C2211yescharacteristic
17C2×#13133124C2211yescharacteristic
18#16810390—21no—
19#16810390—21no—
20#16810360—21no—
21#16810360—21no—
22#16810357—22no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×#121×C53C2211yescharacteristic
25C53×#483—21no—
26C53×#481C412yescomplemented
27C53×#465—22no—
28C53×#450—21no—
29C53×#448C412yescomplemented
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—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124D411yescharacteristic
54#13133124D411yescharacteristic
55C22×Q8×C53C2×C411yescharacteristic, complemented
56C22×D4×C53—21no—
57C22×D4×C53D411yescharacteristic
58C23×C4×C53C2311yescharacteristic
59C2×SD16×C53—22no—
60C22×C8×C53—21no—
61C22⋊Q8×C53—21no—
62C22⋊Q8×C53—21no—
63C22⋊Q8×C53—21no—
64C22⋊Q8×C53—21no—
65#121×C53—21no—
66#121×C53—21no—
67#121×C53C2×C412yes—
68#121×C53D412yes—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
71C2×C4⋊C4×C53D411yescharacteristic
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—21no—
74#106×C53—22no—
75#103×C53—22no—
76#102×C53—22no—
77C22⋊C8×C53—21no—
78C22⋊C8×C53C2×C412yescomplemented
79C24×C53—21no—
80C24×C53C2×D411yescharacteristic
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—81no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—21no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—81no—
92C2×D4×C53C22⋊C412yes—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—81no—
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×Dic53—21no—
104C22×Dic53—21no—
105C22×Dic53—21no—
106C22×Dic53—21no—
107C22×Dic53—42no—
108C22×Dic53—42no—
109C22×Dic53—42no—
110C22×Dic53—41no—
111C22×Dic53—41no—
112C22×Dic53—41no—
113C22×Dic53—81no—
114C22×Dic53—81no—
115SD16×C53—82no—
116C2×C8×C53—22no—
117C2×C8×C53—81no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53C22⋊C412yes—
124C22⋊C4×C53—22no—
125C22⋊C4×C53—22no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C22⋊Dic53—21no—
129C22⋊Dic53—21no—
130C22⋊Dic53—21no—
131C22⋊Dic53—42no—
132C22⋊Dic53—42no—
133C22⋊Dic53—41no—
134C22⋊Dic53—41no—
135C22⋊Dic53—81no—
136C22⋊Dic53SD1612yescomplemented
137C4⋊Dic53—42no—
138C4⋊Dic53—81no—
139C4×Dic53—42no—
140C4×Dic53—81no—
141C2×#88322—41no—
142C2×#88322—81no—
143C23×C53—22no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53C4×D411yescharacteristic
153C23×C53#12011yescharacteristic
154C23×C53#12111yescharacteristic
155C23×C53C2×SD1611yescharacteristic
156C23×C53#13511yescharacteristic
157C23×C53AGL(1,ℤ/8)11yescharacteristic
158C23×C53#13711yescharacteristic
159Q8×C53—42no—
160Q8×C53—81no—
161D4×C53—42no—
162D4×C53—42no—
163D4×C53—42no—
164D4×C53—42no—
165D4×C53—42no—
166D4×C53—81no—
167C2×C4×C53—22no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—81no—
179C2×C4×C53—81no—
180C2×C4×C53C2×C22⋊C411yescharacteristic
181C2×C4×C53C4×D411yescharacteristic
182C2×C4×C53#12111yescharacteristic
183C2×C4×C53#12311yescharacteristic
184C2×Dic53—22no—
185C2×Dic53—22no—
186C2×Dic53—41no—
187C2×Dic53—41no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—84no—
197C2×Dic53—82no—
198C2×Dic53—82no—
199C2×Dic53—81no—
200C2×Dic53—81no—
201C2×Dic53—81no—
202C8×C53—42no—
203#88322—81no—
204#9275—531nomaximal, Sylow
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—42no—
216C22×C53—42no—
217C22×C53—42no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53—81no—
221C22×C53—81no—
222C22×C53#38611yescharacteristic
223C22×C53C4×SD1611yescharacteristic
224C22×C53#44011yescharacteristic
225C22×C53#45011yescharacteristic
226C22×C53#45211yescharacteristic
227C22×C53#46111yescharacteristic
228C22×C53#46311yescharacteristic
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—42no—
234C4×C53—81no—
235C4×C53—81no—
236Dic53—42no—
237Dic53—42no—
238Dic53—81no—
239Dic53—81no—
240Dic53—161no—
241#3187D5311yescharacteristic
242#2364—531no—
243#1996—531no—
244#1519—531no—
245C2×#328—531no—
246#1294—531no—
247#1282—531no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53—21no—
253C2×C53—21no—
254C2×C53—82no—
255C2×C53#187711yescharacteristic
256C2×C53#190611yescharacteristic
257C2×C53#191311yescharacteristic
258C2×C22⋊Q8C2×D5311yescharacteristic
259C2×#121C2×D5311yescharacteristic
260C2×C4×D4—1061no—
261#483—21no—
262#481Dic5312yescomplemented
263#465—22no—
264#450—21no—
265#448Dic5312yescomplemented
266C2×C22⋊C8—531no—
267C2×C22⋊C8C2×D5311yescharacteristic
268#388—1061no—
269#387—1061no—
270#386—1061no—
271#386—1061no—
272#380—531no—
273C4×C22⋊C4—531no—
274C4×C22⋊C4—1061no—
275C2×#95—531no—
276#342—1061no—
277#342—1061no—
278#328—1061no—
279#328—1061no—
280#327—1062no—
281C53#927511yescharacteristic, complemented, Sylow
282C22×Q8C2×Dic5311yescharacteristic, complemented
283C22×D4—21no—
284C22×D4C22⋊D5311yescharacteristic
285C23×C4—1061no—
286C23×C4—1061no—
287C23×C4C22×D5311yescharacteristic
288C2×SD16—22no—
289C22×C8—21no—
290C22×C8—1061no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295#121—21no—
296#121—21no—
297#121C2×Dic5312yes—
298#121C22⋊D5312yes—
299C4×D4—2122no—
300C4×D4—2122no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—1061no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4C2×Dic5311yescharacteristic, complemented
305C2×C4⋊C4C22⋊D5311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—531no—
309C2×C22⋊C4—531no—
310C2×C22⋊C4—1061no—
311C2×C22⋊C4—1061no—
312C2×C22⋊C4—1061no—
313C2×C22⋊C4—1061no—
314C2×C42—1061no—
315C2×C42—1061no—
316#106—22no—
317#103—22no—
318#102—22no—
319C22⋊C8—21no—
320C22⋊C8—1061no—
321C22⋊C8—1061no—
322C22⋊C8C2×Dic5312yescomplemented
323#95—531no—
324#95—531no—
325#95—1061no—
326#95—1061no—
327#95—1061no—
328#95—1061no—
329#95—1061no—
330C24—21no—
331C24D4×D5311yescharacteristic
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—81no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—81no—
343C2×D4C22⋊Dic5312yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—81no—
348C22×C4—1061no—
349C22×C4—1061no—
350C22×C4—1061no—
351C22×C4—1061no—
352C22×C4—4241no—
353C22×C4—4241no—
354C22×C4—2122no—
355C22×C4—2122no—
356C22×C4—2122no—
357C22×C4—2121no—
358C22×C4—2121no—
359C22×C4—2121no—
360C22×C4D4×D5311yescharacteristic
361C22×C4#1171994711yescharacteristic
362C22×C4#1171994711yescharacteristic
363C22×C4C22×Dic5311yescharacteristic
364C22×C4C2×C22⋊D5311yescharacteristic
365C22×C4C2×C22⋊D5311yescharacteristic
366SD16—82no—
367C2×C8—22no—
368C2×C8—81no—
369C2×C8—4241no—
370C2×C8—2121no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—4241no—
377C4⋊C4—2122no—
378C4⋊C4C22⋊Dic5312yes—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—532no—
384C22⋊C4—1061no—
385C22⋊C4—1061no—
386C22⋊C4—1061no—
387C22⋊C4—4241no—
388C22⋊C4—2122no—
389C22⋊C4—2122no—
390C22⋊C4—2121no—
391C22⋊C4—2121no—
392C42—4241no—
393C42—2122no—
394C23—22no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23D4×Dic5311yescharacteristic
404C23#1313318511yescharacteristic
405C23#1313318911yescharacteristic
406C23SD16×D5311yescharacteristic
407C23#1313323811yescharacteristic
408C23#1313324411yescharacteristic
409C23#1313324611yescharacteristic
410Q8—42no—
411Q8—81no—
412D4—42no—
413D4—42no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—81no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—1062no—
432C2×C4—1062no—
433C2×C4—4241no—
434C2×C4—4241no—
435C2×C4—4241no—
436C2×C4—4242no—
437C2×C4—4242no—
438C2×C4—4244no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4—2121no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4—2121no—
448C2×C4—2121no—
449C2×C4C2×C22⋊Dic5311yescharacteristic
450C2×C4D4×Dic5311yescharacteristic
451C2×C4#1313318911yescharacteristic
452C2×C4#1313319811yescharacteristic
453C8—42no—
454C8—4241no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22#1681050311yescharacteristic
473C22SD16×Dic5311yescharacteristic
474C22#1681072411yescharacteristic
475C22#1681077211yescharacteristic
476C22#1681078211yescharacteristic
477C22#1681082611yescharacteristic
478C22#1681084011yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—81no—
485C4—81no—
486C4—8481no—
487C4—4241no—
488C4—4241no—
489C4—2122no—
490C4—2122no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—82no—
498C2#2027505111yescharacteristic
499C2#2027521811yescharacteristic
500C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#3187C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#3187D53
hypocenterC53#9275
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
2. #20275701C2semidirect
4. C2×#16810360C2semidirect
5. #20272595C2semidirect
6. #20272561C2semidirect
26. C53×#481C4semidirect
29. C53×#448C4semidirect
55. C22×Q8×C53C2×C4semidirect
70. C2×C4⋊C4×C53C2×C4semidirect
78. C22⋊C8×C53C2×C4semidirect
136. C22⋊Dic53SD16semidirect
262. #481Dic53semidirect
265. #448Dic53semidirect
281. C53#9275semidirect
282. C22×Q8C2×Dic53semidirect
304. C2×C4⋊C4C2×Dic53semidirect
322. C22⋊C8C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202777672
#202757012
#202732832
C2×#168103602
#202725952
#202725612
C53×#31872
#927553