Finite Group Databasebeta

← #27279673

Subgroups — #27279673

Order 13568.

612 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810680C211yesmaximal, characteristic
2C2×#16810360C212yesmaximal
3#20272942C211yesmaximal, characteristic
4#20272593C212yesmaximal
5C53×#2855C211yesmaximal, characteristic
6#16810680—24no—
7C2×C4⋊#88322—21no—
8C2×C22⋊#88322C2212yes—
9C2×#13133128—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—
30#13133128—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—21no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
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—41no—
103C22⋊C4×C53—41no—
104C42×C53—22no—
105C42×C53—22no—
106C42×C53—21no—
107C42×C53—21no—
108C42×C53—21no—
109C42×C53—21no—
110C2×#88322—24no—
111C2×#88322—42no—
112C2×#88322—42no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53C2×C22⋊C411yescharacteristic
118C23×C53#12112yes—
119C23×C53AGL(1,ℤ/8)11yescharacteristic
120C23×C53#13711yescharacteristic
121C23×C53#13712yes—
122Q8×C53—82no—
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—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—44no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
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—41no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53#10412yes—
156C2×C4×C53C4×D412yes—
157C2×C4×C53#12011yescharacteristic
158C2×C4×C53#12311yescharacteristic
159C2×Dic53—44no—
160C2×Dic53—44no—
161C2×Dic53—44no—
162C2×Dic53—82no—
163C2×Dic53—82no—
164C2×Dic53—82no—
165C2×Dic53—82no—
166#88322—44no—
167#9116—531nomaximal, Sylow
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—22no—
176C22×C53#38611yescharacteristic
177C22×C53#41011yescharacteristic
178C22×C53#41011yescharacteristic
179C22×C53#41711yescharacteristic
180C22×C53#41911yescharacteristic
181C22×C53C2×#10411yescharacteristic
182C22×C53#42111yescharacteristic
183C22×C53#46812yes—
184C22×C53#47012yes—
185C4×C53—22no—
186C4×C53—21no—
187C4×C53—22no—
188C4×C53—21no—
189C4×C53—41no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—81no—
193C4×C53—81no—
194C4×C53—81no—
195C4×C53—81no—
196Dic53—84no—
197C2×#432—531no—
198#2855D5311yescharacteristic
199C2×#328—532no—
200#1434—531no—
201#1294—532no—
202C2×C53—22no—
203C2×C53—22no—
204C2×C53#150611yescharacteristic
205C2×C53#150911yescharacteristic
206C2×C53#151211yescharacteristic
207C2×C53#151511yescharacteristic
208C2×C53#187111yescharacteristic
209C2×C53#189411yescharacteristic
210C2×C53#189911yescharacteristic
211C2×C22⋊Q8C2×D5311yescharacteristic
212C22×C42C2×D5311yescharacteristic
213#432—1064no—
214C2×C4⋊C8—1061no—
215C2×C22⋊C8—532no—
216C2×#97—1061no—
217#391Dic5311yescharacteristic, complemented
218#391Dic5311yescharacteristic, complemented
219#389Dic5311yescharacteristic, complemented
220#389Dic5311yescharacteristic, complemented
221#387—21no—
222#387—21no—
223#385—21no—
224#381—21no—
225#380C2×D5311yescharacteristic
226C2×#95—532no—
227C2×#95—1061no—
228C2×#95—1061no—
229#328—1064no—
230C53#911611yescharacteristic, complemented, Sylow
231C22×Q8C2×Dic5311yescharacteristic
232C23×C4—1062no—
233C23×C4C22⋊D5311yescharacteristic
234C23×C4C22⋊D5311yescharacteristic
235C23×C4C22×D5311yescharacteristic
236C22×C8—1062no—
237C22⋊Q8—22no—
238C22⋊Q8—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C2×Dic5311yescharacteristic
243C2×C4⋊C4C2×Dic5311yescharacteristic
244C2×C4⋊C4C2×Dic5311yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42C2×Dic5311yescharacteristic
250C2×C42C2×Dic5311yescharacteristic
251C2×C42C22⋊D5311yescharacteristic
252C2×C42C22⋊D5311yescharacteristic
253C4⋊C8—1064no—
254C22⋊C8—534no—
255C22⋊C8—1062no—
256#97—1064no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—1064no—
265#95—1062no—
266#95—1062no—
267#95—2122no—
268C24C2×C22⋊D5311yescharacteristic
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—41no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—1064no—
286C22×C4—2122no—
287C22×C4—2122no—
288C22×C4—2122no—
289C22×C4—2122no—
290C22×C4—2122no—
291C22×C4—2122no—
292C22×C4C22⋊Dic5311yescharacteristic
293C22×C4C22⋊Dic5311yescharacteristic
294C22×C4C22⋊Dic5311yescharacteristic
295C22×C4C22⋊Dic5311yescharacteristic
296C22×C4D4×D5312yes—
297C22×C4#1171994712yes—
298C22×C4C22×Dic5311yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300C2×C8—1064no—
301C2×C8—2122no—
302C2×C8—2122no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—22no—
314C42—22no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23C2×C22⋊Dic5311yescharacteristic
324C23#1313318912yes—
325C23#1313324211yescharacteristic
326C23#1313324711yescharacteristic
327C23#1313324812yes—
328Q8—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
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—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—4242no—
362C2×C4—4242no—
363C2×C4—4242no—
364C2×C4—4242no—
365C2×C4—2124no—
366C2×C4—2124no—
367C2×C4—2124no—
368C2×C4#1313314412yes—
369C2×C4D4×Dic5312yes—
370C2×C4#1313318511yescharacteristic
371C2×C4#1313319811yescharacteristic
372C8—2124no—
373C22—22no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22#1681050311yescharacteristic
382C22#1681059711yescharacteristic
383C22#1681059711yescharacteristic
384C22#1681062411yescharacteristic
385C22#1681063211yescharacteristic
386C22C2×#1313314411yescharacteristic
387C22#1681064011yescharacteristic
388C22#1681087112yes—
389C22#1681087912yes—
390C4—22no—
391C4—21no—
392C4—22no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—4244no—
402C2—22no—
403C2—22no—
404C2#2027322711yescharacteristic
405C2#2027324011yescharacteristic
406C2#2027325311yescharacteristic
407C2#2027326611yescharacteristic
408C2#2027501811yescharacteristic
409C2#2027514911yescharacteristic
410C2#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#9116
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
217. #391Dic53semidirect
218. #391Dic53semidirect
219. #389Dic53semidirect
220. #389Dic53semidirect
230. C53#9116semidirect

Maximal subgroups

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

subgroupindex
C2×#168106802
C2×#168103602
#202729422
#202725932
C53×#28552
#911653