Finite Group Databasebeta

← #27281408

Subgroups — #27281408

Order 13568.

570 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283739C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic
3#20273287C212yesmaximal, complemented
4#20272594C212yesmaximal, complemented
5C53×#2564C211yesmaximal, characteristic
6D4×#88322—24no—
7#16810676—21no—
8C2×#13133132—21no—
9C2×C22⋊#88322—21no—
10C2×C22⋊#88322C2212yes—
11#16810483C2212yes—
12#16810390—21no—
13#16810360—24no—
14C2×C22⋊Q8×C53C2211yescharacteristic
15C2×C4×D4×C53C2211yescharacteristic
16C53×#400C411yescharacteristic, complemented
17C53×#399C411yescharacteristic, complemented
18C53×#395C411yescharacteristic, complemented
19C53×#393C411yescharacteristic, complemented
20C53×#387—21no—
21C53×#386—21no—
22C53×#385—21no—
23C53×#380—21no—
24C2×#95×C53C2211yescharacteristic
25C22×#88322—22no—
26C22×#88322—22no—
27C2×C4⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C4⋊#88322—22no—
30#13133132—42no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33C22⋊#88322D414yes—
34C4×#88322—22no—
35#13133124—22no—
36C22×Q8×C53C2×C411yescharacteristic
37C22×D4×C53D411yescharacteristic
38C23×C4×C53—21no—
39C23×C4×C53D411yescharacteristic
40C23×C4×C53C2311yescharacteristic
41C22⋊Q8×C53—22no—
42C22⋊Q8×C53—22no—
43C4×D4×C53—22no—
44C4×D4×C53—22no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53C2×C411yescharacteristic
62#95×C53C2×C411yescharacteristic
63C24×C53C22⋊C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C2×Q8×C53—22no—
66C2×Q8×C53—81no—
67C2×D4×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53SD1612yes—
70C2×D4×C53Q1612yes—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53—81no—
80C22×C4×C53—81no—
81C22×C4×C53—81no—
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D412yes—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4112yes—
89C22×Dic53—42no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—42no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—81no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53—42no—
99C22⋊C4×C53—42no—
100C22⋊C4×C53—81no—
101C42×C53—21no—
102C42×C53—21no—
103C22⋊Dic53—42no—
104C22⋊Dic53—42no—
105C4⋊Dic53—82no—
106C2×#88322—24no—
107C2×#88322—82no—
108C2×#88322—82no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53#10312yes—
116C23×C53C2×C22⋊C411yescharacteristic
117C23×C53#12112yes—
118C23×C53C2×SD1611yescharacteristic
119C23×C53C2×Q1611yescharacteristic
120C23×C53#13512yes—
121Q8×C53—82no—
122D4×C53—24no—
123D4×C53—24no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—22no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
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—81no—
148C2×C4×C53—84no—
149C2×C4×C53#10312yes—
150C2×C4×C53C4×D412yes—
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12311yescharacteristic
153C2×Dic53—82no—
154C2×Dic53—82no—
155C2×Dic53—82no—
156#88322—44no—
157#9383—531nomaximal, Sylow
158C22×C53—22no—
159C22×C53—21no—
160C22×C53—22no—
161C22×C53—21no—
162C22×C53—22no—
163C22×C53—42no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—42no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—81no—
170C22×C53#38611yescharacteristic
171C22×C53C2×#10311yescharacteristic
172C22×C53#41611yescharacteristic
173C22×C53#46512yes—
174C22×C53#46712yes—
175C4×C53—22no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—22no—
179C4×C53—42no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—81no—
183C4×C53—81no—
184C4×C53—81no—
185Dic53—162no—
186#2972—531no—
187#2564D5311yescharacteristic
188#1799—531no—
189#1520—532no—
190#1294—532no—
191C2×C53—22no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—22no—
195C2×C53—42no—
196C2×C53#188411yescharacteristic
197C2×C53#189211yescharacteristic
198C2×C53#191311yescharacteristic
199C2×C22⋊Q8C2×D5311yescharacteristic
200C2×C4×D4C2×D5311yescharacteristic
201C8×D4—1064no—
202#431—1061no—
203C2×#99—1061no—
204C2×C22⋊C8—532no—
205C2×C22⋊C8—1061no—
206#400Dic5311yescharacteristic, complemented
207#399Dic5311yescharacteristic, complemented
208#395Dic5311yescharacteristic, complemented
209#393Dic5311yescharacteristic, complemented
210#387—21no—
211#386—21no—
212#385—21no—
213#380—21no—
214#380—532no—
215C2×#95C2×D5311yescharacteristic
216#342—1061no—
217#328—1064no—
218C53#938311yescharacteristic, complemented, Sylow
219C22×Q8C2×Dic5311yescharacteristic
220C22×D4C22⋊D5311yescharacteristic
221C23×C4—21no—
222C23×C4C22⋊D5311yescharacteristic
223C23×C4C22×D5311yescharacteristic
224C22×C8—1062no—
225C22×C8—1062no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C4×D4—22no—
229C4×D4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—1062no—
234C2×C4⋊C4C2×Dic5311yescharacteristic
235C2×C4⋊C4C2×Dic5311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—1062no—
240C2×C22⋊C4C2×Dic5311yescharacteristic
241C2×C22⋊C4C22⋊D5311yescharacteristic
242C2×C42C22⋊D5311yescharacteristic
243C4⋊C8—1062no—
244#99—2122no—
245C22⋊C8—534no—
246C22⋊C8—1062no—
247C22⋊C8—1062no—
248C4×C8—1062no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—1062no—
255#95C2×Dic5311yescharacteristic
256#95C2×Dic5311yescharacteristic
257C24C22⋊Dic5311yescharacteristic
258C24C2×C22⋊D5311yescharacteristic
259C2×Q8—22no—
260C2×Q8—81no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4#1171992412yes—
264C2×D4#1171992512yes—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—2122no—
277C22×C4C22⋊Dic5311yescharacteristic
278C22×C4C22⋊Dic5311yescharacteristic
279C22×C4C22⋊Dic5311yescharacteristic
280C22×C4D4×D5312yes—
281C22×C4#1171994712yes—
282C22×C4C22×Dic5311yescharacteristic
283C22×C4C2×C22⋊D5311yescharacteristic
284C2×C8—1064no—
285C2×C8—4242no—
286C2×C8—4242no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—81no—
293C4⋊C4—4242no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—81no—
299C22⋊C4—2122no—
300C22⋊C4—2122no—
301C42—21no—
302C42—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#1313314112yes—
310C23C2×C22⋊Dic5311yescharacteristic
311C23#1313318912yes—
312C23C2×#1171992411yescharacteristic
313C23C2×#1171992511yescharacteristic
314C23#1313323712yes—
315Q8—82no—
316D4—24no—
317D4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—84no—
343C2×C4—4242no—
344C2×C4—4242no—
345C2×C4—4242no—
346C2×C4#1313314112yes—
347C2×C4D4×Dic5312yes—
348C2×C4#1313318511yescharacteristic
349C2×C4#1313319811yescharacteristic
350C8—2124no—
351C22—21no—
352C22—22no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—81no—
363C22#1681050311yescharacteristic
364C22C2×#1313314111yescharacteristic
365C22#1681061911yescharacteristic
366C22#1681085412yes—
367C22#1681086212yes—
368C4—21no—
369C4—21no—
370C4—22no—
371C4—22no—
372C4—42no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—8482no—
379C2—21no—
380C2—22no—
381C2—21no—
382C2—22no—
383C2—42no—
384C2#2027509911yescharacteristic
385C2#2027513911yescharacteristic
386C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2564C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2564D53
hypocenterC53#9383
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #20273287C2semidirect
4. #20272594C2semidirect
16. C53×#400C4semidirect
17. C53×#399C4semidirect
18. C53×#395C4semidirect
19. C53×#393C4semidirect
206. #400Dic53semidirect
207. #399Dic53semidirect
208. #395Dic53semidirect
209. #393Dic53semidirect
218. C53#9383semidirect

Maximal subgroups

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

subgroupindex
#202837392
#202747032
#202732872
#202725942
C53×#25642
#938353