Finite Group Databasebeta

← #27279653

Subgroups — #27279653

Order 13568.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810676C211yesmaximal, characteristic
2C2×#16810360C212yesmaximal
3#20272942C211yesmaximal, characteristic
4#20272593C212yesmaximal
5C53×#2855C211yesmaximal, characteristic
6#16810676—24no—
7C2×C4⋊#88322—21no—
8C2×C22⋊#88322C2212yes—
9C2×C4×#88322—21no—
10C2×#13133124—21no—
11C2×#13133124—21no—
12C2×#13133124C2212yes—
13#16810360—24no—
14C2×C22⋊Q8×C53C2211yescharacteristic
15C22×C42×C53C2211yescharacteristic
16C53×#391C411yescharacteristic, complemented
17C53×#391C411yescharacteristic, complemented
18C53×#389C411yescharacteristic, complemented
19C53×#389C411yescharacteristic, complemented
20C53×#387—21no—
21C53×#387—21no—
22C53×#385—21no—
23C53×#381—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×Q8×C53C2×C411yescharacteristic
36C23×C4×C53D411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53C2311yescharacteristic
39C22⋊Q8×C53—22no—
40C22⋊Q8×C53—22no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
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×C42×C53—22no—
50C2×C42×C53—22no—
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53—21no—
62C24×C53C2×D411yescharacteristic
63C2×Q8×C53—22no—
64C2×Q8×C53—41no—
65C2×Q8×C53—41no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53SD1612yes—
82C22×C4×C53Q1612yes—
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D412yes—
86C22×C4×C53#4112yes—
87C22×Dic53—24no—
88C22×Dic53—42no—
89C22×Dic53—42no—
90C22×Dic53—42no—
91C22×Dic53—42no—
92C22×Dic53—42no—
93C22×Dic53—42no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—42no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—42no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C22⋊C4×C53—42no—
102C22⋊C4×C53—42no—
103C42×C53—22no—
104C42×C53—22no—
105C42×C53—21no—
106C42×C53—21no—
107C42×C53—21no—
108C42×C53—21no—
109C2×#88322—24no—
110C2×#88322—42no—
111C2×#88322—42no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53C2×C22⋊C411yescharacteristic
117C23×C53#12112yes—
118C23×C53C2×SD1611yescharacteristic
119C23×C53C2×Q1611yescharacteristic
120C23×C53#13512yes—
121Q8×C53—82no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—24no—
133C2×C4×C53—24no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—44no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53#10312yes—
153C2×C4×C53#10312yes—
154C2×C4×C53#10412yes—
155C2×C4×C53C4×D412yes—
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12311yescharacteristic
158C2×Dic53—44no—
159C2×Dic53—44no—
160C2×Dic53—44no—
161C2×Dic53—82no—
162C2×Dic53—82no—
163C2×Dic53—82no—
164C2×Dic53—82no—
165#88322—44no—
166#9112—531nomaximal, Sylow
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—22no—
175C22×C53#38611yescharacteristic
176C22×C53#41011yescharacteristic
177C22×C53#41011yescharacteristic
178C22×C53C2×#10311yescharacteristic
179C22×C53#41611yescharacteristic
180C22×C53C2×#10411yescharacteristic
181C22×C53#42111yescharacteristic
182C22×C53#46512yes—
183C22×C53#46712yes—
184C4×C53—22no—
185C4×C53—22no—
186C4×C53—22no—
187C4×C53—21no—
188C4×C53—22no—
189C4×C53—21no—
190C4×C53—41no—
191C4×C53—81no—
192C4×C53—81no—
193C4×C53—81no—
194C4×C53—81no—
195Dic53—84no—
196C2×#431—531no—
197#2855D5311yescharacteristic
198C2×#328—532no—
199#1434—531no—
200#1294—532no—
201C2×C53—22no—
202C2×C53—22no—
203C2×C53#150011yescharacteristic
204C2×C53#150311yescharacteristic
205C2×C53#151811yescharacteristic
206C2×C53#152111yescharacteristic
207C2×C53#187111yescharacteristic
208C2×C53#189211yescharacteristic
209C2×C53#189911yescharacteristic
210C2×C22⋊Q8C2×D5311yescharacteristic
211C22×C42C2×D5311yescharacteristic
212#431—1064no—
213C2×C4⋊C8—1061no—
214C2×C22⋊C8—532no—
215C2×C4×C8—1061no—
216#391Dic5311yescharacteristic, complemented
217#391Dic5311yescharacteristic, complemented
218#389Dic5311yescharacteristic, complemented
219#389Dic5311yescharacteristic, complemented
220#387—21no—
221#387—21no—
222#385—21no—
223#381—21no—
224#380C2×D5311yescharacteristic
225C2×#95—532no—
226C2×#95—1061no—
227C2×#95—1061no—
228#328—1064no—
229C53#911211yescharacteristic, complemented, Sylow
230C22×Q8C2×Dic5311yescharacteristic
231C23×C4—1062no—
232C23×C4C22⋊D5311yescharacteristic
233C23×C4C22⋊D5311yescharacteristic
234C23×C4C22×D5311yescharacteristic
235C22×C8—1062no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C2×Dic5311yescharacteristic
242C2×C4⋊C4C2×Dic5311yescharacteristic
243C2×C4⋊C4C2×Dic5311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42C2×Dic5311yescharacteristic
249C2×C42C2×Dic5311yescharacteristic
250C2×C42C22⋊D5311yescharacteristic
251C2×C42C22⋊D5311yescharacteristic
252C4⋊C8—1064no—
253C22⋊C8—534no—
254C22⋊C8—1062no—
255C4×C8—1064no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—1064no—
264#95—1062no—
265#95—1062no—
266#95—2122no—
267C24C2×C22⋊D5311yescharacteristic
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—1064no—
283C22×C4—2122no—
284C22×C4—2122no—
285C22×C4—2122no—
286C22×C4—2122no—
287C22×C4—2122no—
288C22×C4—2122no—
289C22×C4#1171992412yes—
290C22×C4#1171992512yes—
291C22×C4C22⋊Dic5311yescharacteristic
292C22×C4C22⋊Dic5311yescharacteristic
293C22×C4C22⋊Dic5311yescharacteristic
294C22×C4C22⋊Dic5311yescharacteristic
295C22×C4D4×D5312yes—
296C22×C4#1171994712yes—
297C22×C4C22×Dic5311yescharacteristic
298C22×C4C2×C22⋊D5311yescharacteristic
299C2×C8—1064no—
300C2×C8—2122no—
301C2×C8—2122no—
302C4⋊C4—22no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C22⋊Dic5311yescharacteristic
322C23#1313318912yes—
323C23C2×#1171992411yescharacteristic
324C23C2×#1171992511yescharacteristic
325C23#1313323712yes—
326Q8—82no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—4242no—
358C2×C4—4242no—
359C2×C4—4242no—
360C2×C4—4242no—
361C2×C4—2124no—
362C2×C4—2124no—
363C2×C4—2124no—
364C2×C4#1313314112yes—
365C2×C4#1313314112yes—
366C2×C4#1313314412yes—
367C2×C4D4×Dic5312yes—
368C2×C4#1313318511yescharacteristic
369C2×C4#1313319811yescharacteristic
370C8—2124no—
371C22—22no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—22no—
379C22#1681050311yescharacteristic
380C22#1681059711yescharacteristic
381C22#1681059711yescharacteristic
382C22C2×#1313314111yescharacteristic
383C22#1681061911yescharacteristic
384C22C2×#1313314411yescharacteristic
385C22#1681064011yescharacteristic
386C22#1681085412yes—
387C22#1681086212yes—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—4244no—
400C2—22no—
401C2—22no—
402C2#2027320111yescharacteristic
403C2#2027321411yescharacteristic
404C2#2027327911yescharacteristic
405C2#2027329211yescharacteristic
406C2#2027501811yescharacteristic
407C2#2027513911yescharacteristic
408C2#2027517411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2855C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2855D53
hypocenterC53#9112
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
16. C53×#391C4semidirect
17. C53×#391C4semidirect
18. C53×#389C4semidirect
19. C53×#389C4semidirect
216. #391Dic53semidirect
217. #391Dic53semidirect
218. #389Dic53semidirect
219. #389Dic53semidirect
229. C53#9112semidirect

Maximal subgroups

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

subgroupindex
C2×#168106762
C2×#168103602
#202729422
#202725932
C53×#28552
#911253