Finite Group Databasebeta

← #27283237

Subgroups — #27283237

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274695C211yesmaximal, characteristic, complemented
2C2×#16810360C212yesmaximal, complemented
3C2×#16810357C212yesmaximal
4#20272942C211yesmaximal, characteristic, complemented
5C53×#2850C211yesmaximal, characteristic
6C2×C4⋊#88322—21no—
7C2×C22⋊#88322C2212yes—
8C2×#13133124—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—
27C4⋊#88322—42no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30#13133124—22no—
31#13133124—42no—
32#13133124—42no—
33#13133124D414yes—
34C22×Q8×C53C2×C411yescharacteristic
35C22×D4×C53—21no—
36C22×D4×C53C2×C411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40#124×C53—42no—
41C22⋊Q8×C53—22no—
42C22⋊Q8×C53—22no—
43#121×C53—22no—
44#121×C53—22no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C42×C53—42no—
55C2×C42×C53C2×C411yescharacteristic
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62#95×C53—21no—
63C24×C53—21no—
64C24×C53C2×D411yescharacteristic
65C2×Q8×C53—22no—
66C2×Q8×C53—41no—
67C2×Q8×C53—41no—
68C2×D4×C53—22no—
69C2×D4×C53—41no—
70C2×D4×C53—41no—
71C2×D4×C53—42no—
72C2×D4×C53—42no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C22×C4×C53—22no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—22no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22⋊C411yescharacteristic
91C22×C4×C53C22×C411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D412yes—
94C22×C4×C53#4112yes—
95C22×Dic53—24no—
96C22×Dic53—42no—
97C22×Dic53—42no—
98C22×Dic53—42no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C22×Dic53—42no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C22⋊C4×C53—42no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—42no—
109C22⋊C4×C53—42no—
110C22⋊C4×C53—42no—
111C22⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C42×C53—22no—
114C42×C53—22no—
115C42×C53—41no—
116C42×C53—41no—
117C2×#88322—42no—
118C2×#88322—42no—
119C2×#88322—42no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53C2×C22⋊C411yescharacteristic
132C23×C53#12112yes—
133C23×C53#13512yes—
134C23×C53AGL(1,ℤ/8)11yescharacteristic
135C23×C53#13711yescharacteristic
136Q8×C53—82no—
137D4×C53—82no—
138D4×C53—82no—
139D4×C53—82no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—44no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53#9912yes—
170C2×C4×C53C4×D412yes—
171C2×C4×C53#12411yescharacteristic
172C2×C4×C53#12711yescharacteristic
173C2×Dic53—44no—
174C2×Dic53—44no—
175C2×Dic53—44no—
176C2×Dic53—82no—
177C2×Dic53—82no—
178C2×Dic53—82no—
179C2×Dic53—82no—
180#88322—82no—
181#9716—531nomaximal, Sylow
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—22no—
189C22×C53—22no—
190C22×C53—81no—
191C22×C53—81no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53#39011yescharacteristic
198C22×C53C2×#9911yescharacteristic
199C22×C53#41011yescharacteristic
200C22×C53#41611yescharacteristic
201C22×C53#41711yescharacteristic
202C22×C53#42111yescharacteristic
203C22×C53#42111yescharacteristic
204C22×C53#46312yes—
205C22×C53#46412yes—
206C4×C53—21no—
207C4×C53—22no—
208C4×C53—21no—
209C4×C53—22no—
210C4×C53—41no—
211C4×C53—41no—
212C4×C53—41no—
213C4×C53—81no—
214C4×C53—81no—
215C4×C53—81no—
216Dic53—84no—
217#2850D5311yescharacteristic
218#1797—531no—
219C2×#328—532no—
220C2×#327—532no—
221#1434—531no—
222C2×C53—22no—
223C2×C53—22no—
224C2×C53—161no—
225C2×C53#151211yescharacteristic
226C2×C53#151311yescharacteristic
227C2×C53#151611yescharacteristic
228C2×C53#152111yescharacteristic
229C2×C53#196311yescharacteristic
230C2×C53#196411yescharacteristic
231C2×C53#198311yescharacteristic
232C2×#124Dic5311yescharacteristic, complemented
233C2×C22⋊Q8C2×D5311yescharacteristic
234C2×#121C2×D5311yescharacteristic
235C22×C42C2×D5311yescharacteristic
236C2×C4⋊C8—1061no—
237C2×C22⋊C8—532no—
238#391Dic5311yescharacteristic, complemented
239#390Dic5311yescharacteristic, complemented
240#388—21no—
241#388—21no—
242#386—21no—
243#386—21no—
244#382Dic5311yescharacteristic, complemented
245C2×#95—532no—
246C2×#95—1061no—
247C2×#95—1061no—
248#336—1061no—
249#328—1064no—
250#327—1064no—
251C53#971611yescharacteristic, complemented, Sylow
252C22×Q8C2×Dic5311yescharacteristic
253C22×D4—21no—
254C22×D4C2×Dic5311yescharacteristic
255C23×C4—1062no—
256C23×C4C22⋊D5311yescharacteristic
257C23×C4C22⋊D5311yescharacteristic
258C23×C4C22×D5311yescharacteristic
259C22×C8—1062no—
260#124—42no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263#121—22no—
264#121—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C2×Dic5311yescharacteristic
267C2×C4⋊C4C2×Dic5311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—42no—
275C2×C42C2×Dic5311yescharacteristic
276C2×C42C2×Dic5311yescharacteristic
277C2×C42C22⋊D5311yescharacteristic
278C2×C42C22⋊D5311yescharacteristic
279C4⋊C8—2122no—
280C22⋊C8—1062no—
281C22⋊C8—1062no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—534no—
287#95—1062no—
288#95—2122no—
289#95—2122no—
290C24—21no—
291C24C2×C22⋊D5311yescharacteristic
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×D4—22no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—1064no—
315C22×C4—2122no—
316C22×C4—2122no—
317C22×C4—2122no—
318C22×C4—2122no—
319C22×C4—2122no—
320C22×C4—2122no—
321C22×C4C22⋊Dic5311yescharacteristic
322C22×C4C22⋊Dic5311yescharacteristic
323C22×C4C22⋊Dic5311yescharacteristic
324C22×C4C22⋊Dic5311yescharacteristic
325C22×C4D4×D5312yes—
326C22×C4#1171994712yes—
327C22×C4C22×Dic5311yescharacteristic
328C22×C4C2×C22⋊D5311yescharacteristic
329C2×C8—2122no—
330C2×C8—2122no—
331C2×C8—2122no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C42—22no—
344C42—22no—
345C42—41no—
346C42—41no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23C2×C22⋊Dic5311yescharacteristic
359C23#1313318812yes—
360C23#1313323712yes—
361C23#1313324211yescharacteristic
362C23#1313324711yescharacteristic
363Q8—82no—
364D4—82no—
365D4—82no—
366D4—82no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—4242no—
397C2×C4—4242no—
398C2×C4—4242no—
399C2×C4—4242no—
400C2×C4—2124no—
401C2×C4—2124no—
402C2×C4—2124no—
403C2×C4#1313313212yes—
404C2×C4D4×Dic5312yes—
405C2×C4#1313320411yescharacteristic
406C2×C4#1313321311yescharacteristic
407C8—4242no—
408C22—22no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22#1681052611yescharacteristic
424C22C2×#1313313211yescharacteristic
425C22#1681059711yescharacteristic
426C22#1681061911yescharacteristic
427C22#1681062411yescharacteristic
428C22#1681064011yescharacteristic
429C22#1681064011yescharacteristic
430C22#1681083512yes—
431C22#1681084212yes—
432C4—21no—
433C4—22no—
434C4—21no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—4244no—
443C2—22no—
444C2—22no—
445C2—161no—
446C2#2027325311yescharacteristic
447C2#2027325811yescharacteristic
448C2#2027327111yescharacteristic
449C2#2027329211yescharacteristic
450C2#2027554811yescharacteristic
451C2#2027555211yescharacteristic
452C2#2027564811yescharacteristic
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#9716
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. #20274695C2semidirect
2. C2×#16810360C2semidirect
4. #20272942C2semidirect
14. C2×#124×C53C4semidirect
18. C53×#391C4semidirect
19. C53×#390C4semidirect
24. C53×#382C4semidirect
232. C2×#124Dic53semidirect
238. #391Dic53semidirect
239. #390Dic53semidirect
244. #382Dic53semidirect
251. C53#9716semidirect

Maximal subgroups

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

subgroupindex
#202746952
C2×#168103602
C2×#168103572
#202729422
C53×#28502
#971653