Finite Group Databasebeta

← #27279138

Subgroups — #27279138

Order 13568.

768 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810549C211yesmaximal, characteristic
2#20277328C211yesmaximal, characteristic
3#20274801C211yesmaximal, characteristic, complemented
4C2×#16810346C211yesmaximal, characteristic
5#20272595C211yesmaximal, characteristic, complemented
6#20272593C211yesmaximal, characteristic, complemented
7C53×#1823C211yesmaximal, characteristic, complemented
8C22×#11719921—21no—
9C22×#11719921C2211yescharacteristic
10C2×C22⋊C4×D53—21no—
11C2×C22⋊#88322C2211yescharacteristic
12#16810549—22no—
13#16810549C414yescomplemented
14#16810497—22no—
15#16810494—22no—
16#16810483C2211yescharacteristic
17#16810481—21no—
18C2×#13133124—21no—
19C2×#13133124C2211yescharacteristic
20#16810385—22no—
21#16810346—22no—
22C22×C4⋊C4×C53C2211yescharacteristic
23C2×C22⋊C8×C53C2211yescharacteristic
24C53×#380C2211yescharacteristic
25C53×#340—22no—
26C24×D53D411yescharacteristic
27C23×Dic53—21no—
28C23×Dic53D411yescharacteristic
29C22×C4×D53—22no—
30C22×#88322—21no—
31C2×C4⋊Dic53D412yes—
32C2×#11719919—22no—
33C2×C22⋊Dic53—21no—
34C2×#11719921—22no—
35C2×#11719921—24no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—42no—
41C2×#11719921—42no—
42C2×#11719921C2×C412yes—
43C2×#11719921D412yes—
44C22⋊C4×D53—42no—
45C22⋊C4×D53—42no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133124—22no—
49#13133124—21no—
50#13133124—21no—
51#13133124—24no—
52#13133124C2×C412yescomplemented
53C23×C4×C53—21no—
54C23×C4×C53C2311yescharacteristic
55C22×C8×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53C2×C412yescomplemented
62C2×C4⋊C4×C53D412yes—
63C2×C22⋊C4×C53—21no—
64C22⋊C8×C53—21no—
65C22⋊C8×C53—21no—
66#95×C53—21no—
67C24×C53C2×D411yescharacteristic
68C23×D53—21no—
69C23×D53—21no—
70C23×D53—21no—
71C23×D53—21no—
72C23×D53—42no—
73C23×D53—42no—
74C23×D53C22⋊C412yes—
75C23×D53D812yes—
76C23×D53SD1612yes—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—42no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53C22×C411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53#4111yescharacteristic
91C22×C4×C53#4111yescharacteristic
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—22no—
95C22×Dic53—22no—
96C22×Dic53—42no—
97C22×Dic53—42no—
98C22×Dic53—42no—
99C22×Dic53—41no—
100C22×Dic53—41no—
101C22×Dic53C22⋊C412yes—
102C2×C4×D53—44no—
103C2×C4×D53—82no—
104C2×C4×D53—82no—
105C2×C8×C53—42no—
106C2×C8×C53—41no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—42no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C22⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C22⋊Dic53—41no—
115C22⋊Dic53—41no—
116#11719921—22no—
117#11719921—22no—
118#11719921—22no—
119#11719921—22no—
120#11719921—44no—
121#11719921—48no—
122C4⋊Dic53—42no—
123#11719919—82no—
124C2×#88322—42no—
125C2×#88322—41no—
126C23×C53—22no—
127C23×C53—22no—
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53#12011yescharacteristic
130C23×C53#12311yescharacteristic
131C23×C53C2×D811yescharacteristic
132C23×C53C2×SD1611yescharacteristic
133C23×C53AGL(1,ℤ/8)11yescharacteristic
134C23×C53#13711yescharacteristic
135C22×D53—22no—
136C22×D53—22no—
137C22×D53—22no—
138C22×D53—22no—
139C22×D53—22no—
140C22×D53—22no—
141C22×D53—42no—
142C22×D53—42no—
143C22×D53—42no—
144C22×D53—42no—
145C22×D53—44no—
146C22×D53—44no—
147C22×D53—82no—
148C22×D53—82no—
149C22×D53—81no—
150C22×D53—81no—
151C22×D53#10214yes—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—81no—
165C2×C4×C53—81no—
166C2×C4×C53—81no—
167C2×C4×C53—81no—
168C2×C4×C53C4×D411yescharacteristic
169C2×C4×C53C4×D411yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×Dic53—22no—
173C2×Dic53—42no—
174C2×Dic53—42no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—42no—
181C2×Dic53—81no—
182C2×Dic53—82no—
183C2×Dic53—82no—
184C4×D53—88no—
185C8×C53—81no—
186#88322—81no—
187#9025—531nomaximal, Sylow
188C22×C53—22no—
189C22×C53—22no—
190C22×C53—22no—
191C22×C53—22no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53#38611yescharacteristic
197C22×C53C2×#10211yescharacteristic
198C22×C53#41711yescharacteristic
199C22×C53#44711yescharacteristic
200C22×C53#44811yescharacteristic
201C22×C53#45512yes—
202C22×C53#45812yes—
203C22×C53#48011yescharacteristic
204C22×C53#48111yescharacteristic
205C2×D53—24no—
206C2×D53—44no—
207C2×D53—44no—
208C2×D53—42no—
209C2×D53—42no—
210C2×D53—44no—
211C2×D53—82no—
212C2×D53—82no—
213C2×D53—82no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—21no—
217C4×C53—21no—
218C4×C53—82no—
219C4×C53—82no—
220Dic53—42no—
221Dic53—82no—
222Dic53—82no—
223C2×#397—531no—
224#2315—531no—
225#1823—531no—
226#1823D5311yescharacteristic, complemented
227C2×#323—531no—
228#1294—531no—
229#1294—531no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—21no—
233C2×C53—21no—
234C2×C53#160712yes—
235C2×C53#160812yes—
236C2×C53#187411yescharacteristic
237C2×C53#188511yescharacteristic
238C2×C53#189011yescharacteristic
239D53—44no—
240C22×C4⋊C4C2×D5311yescharacteristic
241C22×C22⋊C4—531no—
242C22×C22⋊C4—1061no—
243C22×C22⋊C4—1061no—
244C2×C22⋊C8—531no—
245C2×C22⋊C8C2×D5311yescharacteristic
246#397—534no—
247#397—1062no—
248#385—1062no—
249#385—1062no—
250#380—531no—
251#380—1061no—
252#380C2×D5311yescharacteristic
253C2×#95—531no—
254C2×#95—1061no—
255#340—22no—
256#340—1062no—
257#323—1062no—
258C53#902511yescharacteristic, complemented, Sylow
259C25—531no—
260C23×C4—21no—
261C23×C4—531no—
262C23×C4—1062no—
263C23×C4—1061no—
264C23×C4C22×D5311yescharacteristic
265C22×C8—21no—
266C22×C8—1061no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—532no—
273C2×C4⋊C4—1062no—
274C2×C4⋊C4C4×D5312yescomplemented
275C2×C4⋊C4C22⋊D5312yes—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—532no—
278C2×C22⋊C4—532no—
279C2×C22⋊C4—1061no—
280C2×C22⋊C4—1061no—
281C2×C22⋊C4—1061no—
282C2×C22⋊C4—1061no—
283C2×C22⋊C4—1064no—
284C2×C22⋊C4—1062no—
285C2×C22⋊C4—1061no—
286C2×C22⋊C4—2122no—
287C2×C22⋊C4—2122no—
288C2×C22⋊C4—2122no—
289C2×C22⋊C4—2122no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—1061no—
293C22⋊C8—1061no—
294#95—21no—
295#95—532no—
296#95—1064no—
297#95—1062no—
298#95—1061no—
299#95—1061no—
300C24—532no—
301C24—532no—
302C24—532no—
303C24—1061no—
304C24—1061no—
305C24—1061no—
306C24—1061no—
307C24—2122no—
308C24—2122no—
309C24D4×D5311yescharacteristic
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—532no—
320C22×C4—1061no—
321C22×C4—1061no—
322C22×C4—1062no—
323C22×C4—1062no—
324C22×C4—4242no—
325C22×C4—4242no—
326C22×C4—2124no—
327C22×C4—2122no—
328C22×C4—2122no—
329C22×C4—2121no—
330C22×C4—2121no—
331C22×C4—2122no—
332C22×C4C2×C4×D5311yescharacteristic
333C22×C4#1171994511yescharacteristic
334C22×C4D4×D5311yescharacteristic
335C22×C4D4×D5311yescharacteristic
336C22×C4#1171994711yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—2122no—
341C2×C8—2121no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—4242no—
348C4⋊C4—2122no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—1062no—
352C22⋊C4—1062no—
353C22⋊C4—1062no—
354C22⋊C4—1062no—
355C22⋊C4—2128no—
356C22⋊C4—2124no—
357C22⋊C4—2121no—
358C22⋊C4—2121no—
359C23—22no—
360C23—22no—
361C23—534no—
362C23—1062no—
363C23—1062no—
364C23—1062no—
365C23—1062no—
366C23—1062no—
367C23—1062no—
368C23—4242no—
369C23—4242no—
370C23—4241no—
371C23—4241no—
372C23—2124no—
373C23—2124no—
374C23—2122no—
375C23—2122no—
376C23—2122no—
377C23—2122no—
378C23C22⋊C4×D5311yescharacteristic
379C23#1313318511yescharacteristic
380C23#1313319911yescharacteristic
381C23D8×D5311yescharacteristic
382C23SD16×D5311yescharacteristic
383C23#1313324111yescharacteristic
384C23#1313324611yescharacteristic
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—1062no—
402C2×C4—4248no—
403C2×C4—4242no—
404C2×C4—4242no—
405C2×C4—4241no—
406C2×C4—2122no—
407C2×C4—2122no—
408C2×C4—2122no—
409C2×C4—2122no—
410C2×C4—2122no—
411C2×C4—2122no—
412C2×C4—2122no—
413C2×C4—2122no—
414C2×C4#1313317711yescharacteristic
415C2×C4C4×C22⋊D5311yescharacteristic
416C2×C4#1313318811yescharacteristic
417C2×C4#1313319011yescharacteristic
418C8—81no—
419C8—4241no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—1064no—
429C22—4242no—
430C22—4242no—
431C22—4242no—
432C22—2124no—
433C22—2124no—
434C22—2124no—
435C22—2122no—
436C22—2122no—
437C22#1681049911yescharacteristic
438C22#102×D5311yescharacteristic
439C22#1681062111yescharacteristic
440C22#1681075511yescharacteristic
441C22#1681076011yescharacteristic
442C22#1681079912yes—
443C22#1681081212yes—
444C22#1681093511yescharacteristic
445C22#1681094011yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—82no—
451C4—82no—
452C4—4242no—
453C4—4242no—
454C4—2122no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—2124no—
460C2#2027369912yes—
461C2#2027370412yes—
462C2#2027502911yescharacteristic
463C2#2027510011yescharacteristic
464C2#2027512511yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1823C2
socleC23×C53#120
prime radicalC24×D53D4
cosocleC23×C4×C53C23
hypercenter#1823D53
hypocenterC53#9025
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. #20274801C2semidirect
5. #20272595C2semidirect
6. #20272593C2semidirect
7. C53×#1823C2semidirect
13. #16810549C4semidirect
52. #13133124C2×C4semidirect
61. C2×C4⋊C4×C53C2×C4semidirect
226. #1823D53semidirect
258. C53#9025semidirect
274. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105492
#202773282
#202748012
C2×#168103462
#202725952
#202725932
C53×#18232
#902553