Finite Group Databasebeta

← #27279663

Subgroups — #27279663

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810676C211yesmaximal, characteristic, complemented
2C2×#16810357C212yesmaximal
3#20272942C211yesmaximal, characteristic, complemented
4#20272593C212yesmaximal, complemented
5C53×#2854C211yesmaximal, characteristic
6#16810676—24no—
7C2×C4⋊#88322—21no—
8C2×C22⋊#88322C2212yes—
9C2×C4×#88322—21no—
10C2×#13133124—21no—
11C2×#13133124—21no—
12C2×#13133124C2212yes—
13#16810357—24no—
14C2×#123×C53—21no—
15C2×#121×C53C2211yescharacteristic
16C22×C42×C53C2211yescharacteristic
17C53×#390C411yescharacteristic, complemented
18C53×#390C411yescharacteristic, complemented
19C53×#389C411yescharacteristic, complemented
20C53×#389C411yescharacteristic, complemented
21C53×#388—21no—
22C53×#388—21no—
23C53×#386—21no—
24C53×#380C2211yescharacteristic
25C23×Dic53—22no—
26C22×#88322—22no—
27C4⋊#88322—24no—
28C22⋊#88322—22no—
29C22⋊#88322D414yes—
30C4×#88322—24no—
31#13133124—24no—
32#13133124—22no—
33#13133124—22no—
34#13133124—42no—
35C22×D4×C53—21no—
36C22×D4×C53C2×C411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40#123×C53—42no—
41#123×C53—42no—
42#121×C53—22no—
43#121×C53—22no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
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×C22⋊C4×C53—21no—
55C2×C42×C53—22no—
56C2×C42×C53—22no—
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—
64C24×C53—21no—
65C24×C53C2×D411yescharacteristic
66C2×D4×C53—22no—
67C2×D4×C53—41no—
68C2×D4×C53—41no—
69C2×D4×C53—42no—
70C2×D4×C53—41no—
71C2×D4×C53—41no—
72C2×D4×C53—42no—
73C22×C4×C53—22no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—22no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53D812yes—
88C22×C4×C53SD1612yes—
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D412yes—
92C22×C4×C53#4112yes—
93C22×Dic53—24no—
94C22×Dic53—42no—
95C22×Dic53—42no—
96C22×Dic53—42no—
97C22×Dic53—42no—
98C22×Dic53—42no—
99C22×Dic53—42no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—42no—
104C4⋊C4×C53—42no—
105C22⋊C4×C53—42no—
106C22⋊C4×C53—41no—
107C22⋊C4×C53—41no—
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—21no—
116C42×C53—21no—
117C42×C53—21no—
118C42×C53—21no—
119C2×#88322—24no—
120C2×#88322—42no—
121C2×#88322—42no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53C2×C22⋊C411yescharacteristic
134C23×C53#12112yes—
135C23×C53C2×D811yescharacteristic
136C23×C53C2×SD1611yescharacteristic
137C23×C53#13512yes—
138D4×C53—82no—
139D4×C53—82no—
140D4×C53—82no—
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—24no—
152C2×C4×C53—24no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—44no—
159C2×C4×C53—41no—
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#10212yes—
169C2×C4×C53#10212yes—
170C2×C4×C53#10412yes—
171C2×C4×C53C4×D412yes—
172C2×C4×C53#12011yescharacteristic
173C2×C4×C53#12311yescharacteristic
174C2×Dic53—44no—
175C2×Dic53—44no—
176C2×Dic53—44no—
177C2×Dic53—82no—
178C2×Dic53—82no—
179C2×Dic53—82no—
180C2×Dic53—82no—
181#88322—44no—
182#9114—531nomaximal, Sylow
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—22no—
191C22×C53—81no—
192C22×C53—81no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53#38611yescharacteristic
199C22×C53#41011yescharacteristic
200C22×C53#41011yescharacteristic
201C22×C53C2×#10211yescharacteristic
202C22×C53#41611yescharacteristic
203C22×C53C2×#10411yescharacteristic
204C22×C53#42111yescharacteristic
205C22×C53#46512yes—
206C22×C53#46612yes—
207C4×C53—22no—
208C4×C53—22no—
209C4×C53—22no—
210C4×C53—21no—
211C4×C53—22no—
212C4×C53—21no—
213C4×C53—41no—
214C4×C53—81no—
215C4×C53—81no—
216C4×C53—81no—
217Dic53—84no—
218C2×#431—531no—
219#2854D5311yescharacteristic
220C2×#327—532no—
221#1434—531no—
222#1294—532no—
223C2×C53—22no—
224C2×C53—22no—
225C2×C53—161no—
226C2×C53#150011yescharacteristic
227C2×C53#150211yescharacteristic
228C2×C53#151711yescharacteristic
229C2×C53#152111yescharacteristic
230C2×C53#187111yescharacteristic
231C2×C53#189311yescharacteristic
232C2×C53#189911yescharacteristic
233C2×#123—21no—
234C2×#121C2×D5311yescharacteristic
235C22×C42C2×D5311yescharacteristic
236#431—1064no—
237C2×C4⋊C8—1061no—
238C2×C22⋊C8—532no—
239C2×C4×C8—1061no—
240#390Dic5311yescharacteristic, complemented
241#390Dic5311yescharacteristic, complemented
242#389Dic5311yescharacteristic, complemented
243#389Dic5311yescharacteristic, complemented
244#388—21no—
245#388—21no—
246#386—21no—
247#380C2×D5311yescharacteristic
248C2×#95—532no—
249C2×#95—1061no—
250C2×#95—1061no—
251#327—1064no—
252C53#911411yescharacteristic, complemented, Sylow
253C22×D4—21no—
254C22×D4C2×Dic5311yescharacteristic
255C23×C4—1062no—
256C23×C4C22⋊D5311yescharacteristic
257C23×C4C22⋊D5311yescharacteristic
258C23×C4C22×D5311yescharacteristic
259C22×C8—1062no—
260#123—42no—
261#123—42no—
262#121—22no—
263#121—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C2×Dic5311yescharacteristic
267C2×C4⋊C4C2×Dic5311yescharacteristic
268C2×C4⋊C4C2×Dic5311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C2×Dic5311yescharacteristic
278C2×C42C2×Dic5311yescharacteristic
279C2×C42C22⋊D5311yescharacteristic
280C2×C42C22⋊D5311yescharacteristic
281C4⋊C8—1064no—
282C22⋊C8—534no—
283C22⋊C8—1062no—
284C4×C8—1064no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—1064no—
289#95—1062no—
290#95—1062no—
291#95—2122no—
292C24—21no—
293C24C2×C22⋊D5311yescharacteristic
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—1064no—
312C22×C4—2122no—
313C22×C4—2122no—
314C22×C4—2122no—
315C22×C4—2122no—
316C22×C4—2122no—
317C22×C4—2122no—
318C22×C4#1171992212yes—
319C22×C4#1171992312yes—
320C22×C4C22⋊Dic5311yescharacteristic
321C22×C4C22⋊Dic5311yescharacteristic
322C22×C4C22⋊Dic5311yescharacteristic
323C22×C4C22⋊Dic5311yescharacteristic
324C22×C4D4×D5312yes—
325C22×C4#1171994712yes—
326C22×C4C22×Dic5311yescharacteristic
327C22×C4C2×C22⋊D5311yescharacteristic
328C2×C8—1064no—
329C2×C8—2122no—
330C2×C8—2122no—
331C4⋊C4—22no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C42—22no—
345C42—22no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23C2×C22⋊Dic5311yescharacteristic
362C23#1313318912yes—
363C23C2×#1171992211yescharacteristic
364C23C2×#1171992311yescharacteristic
365C23#1313323712yes—
366D4—82no—
367D4—82no—
368D4—82no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
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#1313313812yes—
404C2×C4#1313313812yes—
405C2×C4#1313314412yes—
406C2×C4D4×Dic5312yes—
407C2×C4#1313318511yescharacteristic
408C2×C4#1313319811yescharacteristic
409C8—2124no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22#1681050311yescharacteristic
426C22#1681059711yescharacteristic
427C22#1681059711yescharacteristic
428C22C2×#1313313811yescharacteristic
429C22#1681061911yescharacteristic
430C22C2×#1313314411yescharacteristic
431C22#1681064011yescharacteristic
432C22#1681085012yes—
433C22#1681085712yes—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—21no—
438C4—22no—
439C4—21no—
440C4—41no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—4244no—
445C2—22no—
446C2—22no—
447C2—161no—
448C2#2027320111yescharacteristic
449C2#2027321011yescharacteristic
450C2#2027327511yescharacteristic
451C2#2027329211yescharacteristic
452C2#2027501811yescharacteristic
453C2#2027514411yescharacteristic
454C2#2027517411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2854C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#2854D53
hypocenterC53#9114
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#16810676C2semidirect
3. #20272942C2semidirect
4. #20272593C2semidirect
17. C53×#390C4semidirect
18. C53×#390C4semidirect
19. C53×#389C4semidirect
20. C53×#389C4semidirect
240. #390Dic53semidirect
241. #390Dic53semidirect
242. #389Dic53semidirect
243. #389Dic53semidirect
252. C53#9114semidirect

Maximal subgroups

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

subgroupindex
C2×#168106762
C2×#168103572
#202729422
#202725932
C53×#28542
#911453