Finite Group Databasebeta

← #27283085

Subgroups — #27283085

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274477C211yesmaximal, characteristic, complemented
2C2×#16810360C212yesmaximal, complemented
3C2×#16810357C212yesmaximal
4#20272945C211yesmaximal, characteristic, complemented
5C53×#2850C211yesmaximal, characteristic
6C22×C4⋊Dic53—21no—
7C2×C22⋊#88322C2212yes—
8C2×#13133128—21no—
9C2×#13133124—21no—
10C2×#13133124C2212yes—
11#16810378—21no—
12#16810360—24no—
13#16810357—24no—
14C2×#124×C53C411yescharacteristic, complemented
15C2×C22⋊Q8×C53C2211yescharacteristic
16C2×#121×C53C2211yescharacteristic
17C22×C42×C53C2211yescharacteristic
18C53×#391C411yescharacteristic, complemented
19C53×#390C411yescharacteristic, complemented
20C53×#388—21no—
21C53×#388—21no—
22C53×#386—21no—
23C53×#386—21no—
24C53×#382C411yescharacteristic, complemented
25C23×Dic53—22no—
26C22×#88322—22no—
27C2×C4⋊Dic53—24no—
28C2×C4⋊Dic53—42no—
29C2×C4⋊Dic53—42no—
30C22⋊#88322—22no—
31C22⋊#88322D414yes—
32#13133128—42no—
33#13133124—22no—
34#13133124—22no—
35#13133124—42no—
36C22×Q8×C53C2×C411yescharacteristic
37C22×D4×C53—21no—
38C22×D4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C23×C4×C53C2311yescharacteristic
42#124×C53—42no—
43C22⋊Q8×C53—22no—
44C22⋊Q8×C53—22no—
45#121×C53—22no—
46#121×C53—22no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C42×C53—42no—
57C2×C42×C53C2×C411yescharacteristic
58C2×C42×C53C2×C411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C2×C42×C53D411yescharacteristic
61#95×C53—21no—
62#95×C53—21no—
63#95×C53—21no—
64#95×C53—21no—
65C24×C53—21no—
66C24×C53C2×D411yescharacteristic
67C2×Q8×C53—22no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×D4×C53—22no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—41no—
78C2×D4×C53—41no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22⋊C411yescharacteristic
94C22×C4×C53C22⋊C411yescharacteristic
95C22×C4×C53C22⋊C411yescharacteristic
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D412yes—
99C22×C4×C53#4112yes—
100C22×Dic53—24no—
101C22×Dic53—42no—
102C22×Dic53—42no—
103C22×Dic53—42no—
104C22×Dic53—42no—
105C22×Dic53—42no—
106C22×Dic53—42no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—42no—
110C22⋊C4×C53—42no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—42no—
115C22⋊C4×C53—42no—
116C22⋊C4×C53—42no—
117C42×C53—22no—
118C42×C53—22no—
119C42×C53—41no—
120C42×C53—41no—
121C4⋊Dic53—48no—
122C2×#88322—24no—
123C2×#88322—42no—
124C2×#88322—42no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53C2×C22⋊C411yescharacteristic
137C23×C53#12411yescharacteristic
138C23×C53#12711yescharacteristic
139C23×C53AGL(1,ℤ/8)11yescharacteristic
140C23×C53AGL(1,ℤ/8)11yescharacteristic
141C23×C53#13711yescharacteristic
142C23×C53#13711yescharacteristic
143Q8×C53—82no—
144D4×C53—82no—
145D4×C53—81no—
146D4×C53—81no—
147D4×C53—81no—
148D4×C53—81no—
149C2×C4×C53—24no—
150C2×C4×C53—24no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
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#9912yes—
179C2×C4×C53#10412yes—
180C2×C4×C53C4×D412yes—
181C2×C4×C53#12112yes—
182C2×Dic53—44no—
183C2×Dic53—44no—
184C2×Dic53—44no—
185C2×Dic53—82no—
186C2×Dic53—82no—
187C2×Dic53—82no—
188C2×Dic53—82no—
189#88322—44no—
190#9682—531nomaximal, Sylow
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53—81no—
206C22×C53—81no—
207C22×C53—81no—
208C22×C53—81no—
209C22×C53—81no—
210C22×C53#39011yescharacteristic
211C22×C53C2×#9911yescharacteristic
212C22×C53#41011yescharacteristic
213C22×C53#41711yescharacteristic
214C22×C53#41711yescharacteristic
215C22×C53C2×#10411yescharacteristic
216C22×C53#42111yescharacteristic
217C22×C53#48911yescharacteristic
218C22×C53#48911yescharacteristic
219C22×C53#49611yescharacteristic
220C22×C53#49711yescharacteristic
221C4×C53—22no—
222C4×C53—22no—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—41no—
226C4×C53—41no—
227C4×C53—81no—
228C4×C53—81no—
229C4×C53—81no—
230Dic53—84no—
231#2850D5311yescharacteristic
232#1749—531no—
233C2×#328—532no—
234C2×#327—532no—
235C4⋊#95—531no—
236C2×C53—21no—
237C2×C53—21no—
238C2×C53—21no—
239C2×C53—21no—
240C2×C53—161no—
241C2×C53#150411yescharacteristic
242C2×C53#150911yescharacteristic
243C2×C53#151211yescharacteristic
244C2×C53#151311yescharacteristic
245C2×C53#195911yescharacteristic
246C2×C53#196411yescharacteristic
247C2×C53#198011yescharacteristic
248C2×#124Dic5311yescharacteristic, complemented
249C2×C22⋊Q8C2×D5311yescharacteristic
250C2×#121C2×D5311yescharacteristic
251C22×C4⋊C4—1061no—
252C22×C42C2×D5311yescharacteristic
253C2×C22⋊C8—532no—
254C2×#97—1061no—
255#391Dic5311yescharacteristic, complemented
256#390Dic5311yescharacteristic, complemented
257#388—21no—
258#388—21no—
259#386—21no—
260#386—21no—
261#382Dic5311yescharacteristic, complemented
262C2×#95—532no—
263C2×#95—1061no—
264#336—1061no—
265#328—1064no—
266#327—1064no—
267C53#968211yescharacteristic, complemented, Sylow
268C22×Q8C2×Dic5311yescharacteristic
269C22×D4—21no—
270C22×D4C2×Dic5311yescharacteristic
271C23×C4—1062no—
272C23×C4C22⋊D5311yescharacteristic
273C23×C4C22⋊D5311yescharacteristic
274C23×C4C22×D5311yescharacteristic
275C22×C8—1062no—
276#124—42no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279#121—22no—
280#121—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—1064no—
283C2×C4⋊C4—2122no—
284C2×C4⋊C4—2122no—
285C2×C4⋊C4C2×Dic5311yescharacteristic
286C2×C4⋊C4C2×Dic5311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C42—42no—
294C2×C42C2×Dic5311yescharacteristic
295C2×C42C2×Dic5311yescharacteristic
296C2×C42C22⋊D5311yescharacteristic
297C2×C42C22⋊D5311yescharacteristic
298C22⋊C8—534no—
299C22⋊C8—1062no—
300#97—2122no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—1062no—
306#95—1062no—
307#95—2122no—
308C24—21no—
309C24C2×C22⋊D5311yescharacteristic
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—1064no—
336C22×C4—2122no—
337C22×C4—2122no—
338C22×C4—2122no—
339C22×C4—2122no—
340C22×C4—2122no—
341C22×C4—2122no—
342C22×C4C22⋊Dic5311yescharacteristic
343C22×C4C22⋊Dic5311yescharacteristic
344C22×C4C22⋊Dic5311yescharacteristic
345C22×C4C22⋊Dic5311yescharacteristic
346C22×C4D4×D5312yes—
347C22×C4#1171994712yes—
348C22×C4C22×Dic5311yescharacteristic
349C22×C4C2×C22⋊D5311yescharacteristic
350C2×C8—1064no—
351C2×C8—2122no—
352C2×C8—2122no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—42no—
356C4⋊C4—2128no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C42—22no—
365C42—22no—
366C42—41no—
367C42—41no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23C2×C22⋊Dic5311yescharacteristic
380C23#1313320411yescharacteristic
381C23#1313321311yescharacteristic
382C23#1313324211yescharacteristic
383C23#1313324211yescharacteristic
384C23#1313324711yescharacteristic
385C23#1313324711yescharacteristic
386Q8—82no—
387D4—82no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4—81no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—4242no—
422C2×C4—4242no—
423C2×C4—4242no—
424C2×C4—4242no—
425C2×C4—2124no—
426C2×C4—2124no—
427C2×C4—2124no—
428C2×C4#1313313212yes—
429C2×C4#1313314412yes—
430C2×C4D4×Dic5312yes—
431C2×C4#1313318812yes—
432C8—2124no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22#1681052611yescharacteristic
453C22C2×#1313313211yescharacteristic
454C22#1681059711yescharacteristic
455C22#1681062411yescharacteristic
456C22#1681062411yescharacteristic
457C22C2×#1313314411yescharacteristic
458C22#1681064011yescharacteristic
459C22#1681098211yescharacteristic
460C22#1681098211yescharacteristic
461C22#1681101311yescharacteristic
462C22#1681101811yescharacteristic
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—41no—
468C4—41no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—4244no—
473C2—21no—
474C2—21no—
475C2—21no—
476C2—21no—
477C2—161no—
478C2#2027321911yescharacteristic
479C2#2027324011yescharacteristic
480C2#2027325311yescharacteristic
481C2#2027325811yescharacteristic
482C2#2027553211yescharacteristic
483C2#2027555211yescharacteristic
484C2#2027563511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2850C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#2850D53
hypocenterC53#9682
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. #20274477C2semidirect
2. C2×#16810360C2semidirect
4. #20272945C2semidirect
14. C2×#124×C53C4semidirect
18. C53×#391C4semidirect
19. C53×#390C4semidirect
24. C53×#382C4semidirect
248. C2×#124Dic53semidirect
255. #391Dic53semidirect
256. #390Dic53semidirect
261. #382Dic53semidirect
267. C53#9682semidirect

Maximal subgroups

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

subgroupindex
#202744772
C2×#168103602
C2×#168103572
#202729452
C53×#28502
#968253