Finite Group Databasebeta

← #27274976

Subgroups — #27274976

Order 13568.

712 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#20274477C211yesmaximal, characteristic, complemented
2C2×#16810357C212yesmaximal
3#20272942C211yesmaximal, characteristic, complemented
4#20272593C212yesmaximal, complemented
5C4×#121×C53C211yesmaximal, characteristic
6C2×C22⋊#88322C2212yes—
7C2×#13133128—21no—
8C2×#13133124—21no—
9C2×#13133124—21no—
10C2×#13133124C2212yes—
11#16810378—21no—
12#16810357C418yescomplemented
13C2×#121×C53C2211yescharacteristic
14C2×C4×D4×C53—21no—
15C2×C4×D4×C53C411yescharacteristic, complemented
16C22×C42×C53C2211yescharacteristic
17C53×#390C411yescharacteristic, complemented
18C53×#389C411yescharacteristic, complemented
19C53×#388—21no—
20C53×#386—21no—
21C53×#380C2211yescharacteristic
22C4×C4⋊C4×C53C411yescharacteristic, complemented
23C4×C22⋊C4×C53—21no—
24C23×Dic53—22no—
25C22×#88322—22no—
26C22⋊#88322—22no—
27C22⋊#88322C2×C414yescomplemented
28#13133128—42no—
29#13133124—22no—
30#13133124—42no—
31#13133124—42no—
32#13133124C2×C414yescomplemented
33C22×D4×C53—21no—
34C22×D4×C53C2×C411yescharacteristic
35C23×C4×C53—21no—
36C23×C4×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39#121×C53—22no—
40#121×C53C2×C414yes—
41C4×D4×C53—44no—
42C4×D4×C53—42no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C42×C53—21no—
53C2×C42×C53—42no—
54C2×C42×C53C2×C411yescharacteristic
55C2×C42×C53C2×C411yescharacteristic
56C2×C42×C53D411yescharacteristic
57C2×C42×C53D411yescharacteristic
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61C24×C53—21no—
62C24×C53C2×D411yescharacteristic
63C2×D4×C53—24no—
64C2×D4×C53—21no—
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C2×D4×C53—42no—
68C2×D4×C53—41no—
69C2×D4×C53—41no—
70C2×D4×C53C4212yes—
71C22×C4×C53—21no—
72C22×C4×C53—22no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—22no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—41no—
79C22×C4×C53—41no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53—42no—
84C22×C4×C53—42no—
85C22×C4×C53—42no—
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53C22⋊C411yescharacteristic
88C22×C4×C53C22⋊C411yescharacteristic
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C22×C412yes—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4112yes—
94C22×Dic53—24no—
95C22×Dic53—42no—
96C22×Dic53—42no—
97C22×Dic53—42no—
98C22×Dic53—42no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—42no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53C4212yes—
106C22⋊C4×C53—24no—
107C22⋊C4×C53—42no—
108C22⋊C4×C53—42no—
109C22⋊C4×C53—42no—
110C22⋊C4×C53—42no—
111C42×C53—22no—
112C42×C53—22no—
113C42×C53—41no—
114C42×C53—41no—
115C42×C53—42no—
116C2×#88322—24no—
117C2×#88322—42no—
118C2×#88322—42no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—22no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53C2×C22⋊C411yescharacteristic
131C23×C53C4×D412yes—
132C23×C53AGL(1,ℤ/8)11yescharacteristic
133C23×C53AGL(1,ℤ/8)12yes—
134C23×C53#13711yescharacteristic
135D4×C53—42no—
136D4×C53—42no—
137D4×C53—44no—
138D4×C53—44no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—22no—
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—84no—
169C2×C4×C53#9912yes—
170C2×C4×C53C2×C4211yescharacteristic
171C2×C4×C53#11711yescharacteristic
172C2×C4×C53C4×D412yes—
173C2×Dic53—44no—
174C2×Dic53—44no—
175C2×Dic53—44no—
176C2×Dic53—82no—
177C2×Dic53—82no—
178C2×Dic53—82no—
179C2×Dic53—82no—
180#88322—44no—
181#8216—531nomaximal, Sylow
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—22no—
190C22×C53—42no—
191C22×C53—42no—
192C22×C53—42no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53C4×C22⋊C411yescharacteristic
198C22×C53C2×#9911yescharacteristic
199C22×C53#41011yescharacteristic
200C22×C53#41711yescharacteristic
201C22×C53#41811yescharacteristic
202C22×C53#42111yescharacteristic
203C22×C53#42111yescharacteristic
204C22×C53#44012yes—
205C22×C53#44212yes—
206C4×C53—22no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—22no—
210C4×C53—42no—
211C4×C53—41no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53—81no—
215C4×C53—81no—
216Dic53—84no—
217C4×#121D5311yescharacteristic
218#1749—531no—
219C2×#327—532no—
220#1434—531no—
221#1294—532no—
222C2×C53—22no—
223C2×C53—22no—
224C2×C53—82no—
225C2×C53#151011yescharacteristic
226C2×C53#151211yescharacteristic
227C2×C53#151311yescharacteristic
228C2×C53#151411yescharacteristic
229C2×C53C4×#9911yescharacteristic
230C2×C53#175911yescharacteristic
231C2×C53#176211yescharacteristic
232C2×#121C2×D5311yescharacteristic
233C2×C4×D4—21no—
234C2×C4×D4Dic5311yescharacteristic, complemented
235C22×C42C2×D5311yescharacteristic
236C2×C22⋊C8—532no—
237C2×#97—1061no—
238#390Dic5311yescharacteristic, complemented
239#389Dic5311yescharacteristic, complemented
240#388—21no—
241#386—21no—
242#380C2×D5311yescharacteristic
243C4×C4⋊C4Dic5311yescharacteristic, complemented
244C4×C22⋊C4—21no—
245C2×#95—532no—
246C2×#95—1061no—
247C2×#95—1061no—
248#336—1061no—
249#327—538no—
250C53#821611yescharacteristic, complemented, Sylow
251C22×D4—21no—
252C22×D4C2×Dic5311yescharacteristic
253C23×C4—21no—
254C23×C4—1062no—
255C23×C4C22⋊D5311yescharacteristic
256C23×C4C22⋊D5311yescharacteristic
257C23×C4C22×D5311yescharacteristic
258C22×C8—1062no—
259#121—22no—
260#121C4×D5314yes—
261C4×D4—42no—
262C4×D4—44no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C2×Dic5311yescharacteristic
265C2×C4⋊C4C2×Dic5311yescharacteristic
266C2×C4⋊C4C2×Dic5311yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C42—21no—
273C2×C42—42no—
274C2×C42C2×Dic5311yescharacteristic
275C2×C42C2×Dic5311yescharacteristic
276C2×C42C22⋊D5311yescharacteristic
277C2×C42C22⋊D5311yescharacteristic
278C22⋊C8—534no—
279C22⋊C8—1062no—
280#97—2122no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—534no—
285#95—1062no—
286#95—2122no—
287#95—2122no—
288C24—21no—
289C24C2×C22⋊D5311yescharacteristic
290C2×D4—24no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4C4×Dic5312yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—1064no—
314C22×C4—2122no—
315C22×C4—2122no—
316C22×C4—2122no—
317C22×C4—2122no—
318C22×C4—2122no—
319C22×C4—2122no—
320C22×C4C22⋊Dic5311yescharacteristic
321C22×C4C22⋊Dic5311yescharacteristic
322C22×C4C22⋊Dic5311yescharacteristic
323C22×C4C22⋊Dic5311yescharacteristic
324C22×C4C2×C4×D5312yes—
325C22×C4#1171994512yes—
326C22×C4C22×Dic5311yescharacteristic
327C22×C4C2×C22⋊D5311yescharacteristic
328C2×C8—1064no—
329C2×C8—2122no—
330C2×C8—2122no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4C4×Dic5312yes—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42—41no—
345C42—42no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—22no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23C2×C22⋊Dic5311yescharacteristic
358C23C4×C22⋊D5312yes—
359C23#1313324211yescharacteristic
360C23#1313324312yes—
361C23#1313324711yescharacteristic
362D4—42no—
363D4—42no—
364D4—44no—
365D4—44no—
366C2×C4—21no—
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—22no—
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—84no—
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×C4C2×C4×Dic5311yescharacteristic
405C2×C4#1313317611yescharacteristic
406C2×C4C4×C22⋊D5312yes—
407C8—2124no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—22no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22C4×C22⋊Dic5311yescharacteristic
424C22C2×#1313313211yescharacteristic
425C22#1681059711yescharacteristic
426C22#1681062411yescharacteristic
427C22#1681062811yescharacteristic
428C22#1681064011yescharacteristic
429C22#1681064011yescharacteristic
430C22#1681072312yes—
431C22#1681073312yes—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—42no—
440C4—81no—
441C4—81no—
442C4—4244no—
443C2—22no—
444C2—22no—
445C2—82no—
446C2#2027324411yescharacteristic
447C2#2027325311yescharacteristic
448C2#2027325811yescharacteristic
449C2#2027326211yescharacteristic
450C2C4×#1313313211yescharacteristic
451C2#2027452511yescharacteristic
452C2#2027454011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC4×#121×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenterC4×#121D53
hypocenterC53#8216
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. #20274477C2semidirect
3. #20272942C2semidirect
4. #20272593C2semidirect
12. #16810357C4semidirect
15. C2×C4×D4×C53C4semidirect
17. C53×#390C4semidirect
18. C53×#389C4semidirect
22. C4×C4⋊C4×C53C4semidirect
27. C22⋊#88322C2×C4semidirect
32. #13133124C2×C4semidirect
234. C2×C4×D4Dic53semidirect
238. #390Dic53semidirect
239. #389Dic53semidirect
243. C4×C4⋊C4Dic53semidirect
250. C53#8216semidirect

Maximal subgroups

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

subgroupindex
#202744772
C2×#168103572
#202729422
#202725932
C4×#121×C532
#821653