Finite Group Databasebeta

← #27281418

Subgroups — #27281418

Order 13568.

548 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283744C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic
3#20273287C212yesmaximal, complemented
4#20272594C212yesmaximal, complemented
5C53×#2564C211yesmaximal, characteristic
6#16810695—24no—
7#16810680—21no—
8C2×#13133132—21no—
9#16810587—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
25C2×#11719917—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—
34#13133128—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—21no—
69C2×D4×C53—21no—
70C2×D4×C53—22no—
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—41no—
79C22×C4×C53—41no—
80C22×C4×C53—81no—
81C22×C4×C53—81no—
82C22×C4×C53—81no—
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22×C411yescharacteristic
87C22×C4×C53C2×D412yes—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4112yes—
90C22×Dic53—42no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—42no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—81no—
97C22⋊C4×C53—22no—
98C22⋊C4×C53—22no—
99C22⋊C4×C53—42no—
100C22⋊C4×C53—42no—
101C22⋊C4×C53—81no—
102C42×C53—21no—
103C42×C53—21no—
104C22⋊Dic53—42no—
105C22⋊Dic53—42no—
106C4⋊Dic53—82no—
107#11719917—82no—
108C2×#88322—24no—
109C2×#88322—82no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53C2×C22⋊C411yescharacteristic
118C23×C53#12112yes—
119C23×C53AGL(1,ℤ/8)11yescharacteristic
120C23×C53#13711yescharacteristic
121C23×C53#13712yes—
122Q8×C53—82no—
123D4×C53—42no—
124D4×C53—42no—
125C2×C4×C53—22no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
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—41no—
148C2×C4×C53—41no—
149C2×C4×C53—81no—
150C2×C4×C53—82no—
151C2×C4×C53—82no—
152C2×C4×C53C4×D412yes—
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12311yescharacteristic
155C2×Dic53—82no—
156C2×Dic53—82no—
157C2×Dic53—82no—
158#88322—44no—
159#9385—531nomaximal, Sylow
160C22×C53—22no—
161C22×C53—21no—
162C22×C53—22no—
163C22×C53—21no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—81no—
172C22×C53#38611yescharacteristic
173C22×C53#41711yescharacteristic
174C22×C53#41911yescharacteristic
175C22×C53#46812yes—
176C22×C53#47012yes—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—22no—
180C4×C53—42no—
181C4×C53—41no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53—81no—
185C4×C53—81no—
186C4×C53—81no—
187Dic53—162no—
188#2973—531no—
189#2564D5311yescharacteristic
190#1799—531no—
191#1520—532no—
192#1294—532no—
193C2×C53—22no—
194C2×C53—21no—
195C2×C53—21no—
196C2×C53—22no—
197C2×C53—81no—
198C2×C53#188411yescharacteristic
199C2×C53#189411yescharacteristic
200C2×C53#191311yescharacteristic
201C2×C22⋊Q8C2×D5311yescharacteristic
202C2×C4×D4C2×D5311yescharacteristic
203#435—1064no—
204#432—1061no—
205C2×#99—1061no—
206#407—1061no—
207C2×C22⋊C8—532no—
208#400Dic5311yescharacteristic, complemented
209#399Dic5311yescharacteristic, complemented
210#395Dic5311yescharacteristic, complemented
211#393Dic5311yescharacteristic, complemented
212#387—21no—
213#386—21no—
214#385—21no—
215#380—21no—
216#380—532no—
217C2×#95C2×D5311yescharacteristic
218#342—1061no—
219#328—1064no—
220C53#938511yescharacteristic, complemented, Sylow
221C22×Q8C2×Dic5311yescharacteristic
222C22×D4C22⋊D5311yescharacteristic
223C23×C4—21no—
224C23×C4C22⋊D5311yescharacteristic
225C23×C4C22×D5311yescharacteristic
226C2×M16—1062no—
227C22×C8—1062no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—1062no—
236C2×C4⋊C4C2×Dic5311yescharacteristic
237C2×C4⋊C4C2×Dic5311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—1062no—
242C2×C22⋊C4C2×Dic5311yescharacteristic
243C2×C22⋊C4C22⋊D5311yescharacteristic
244C2×C42C22⋊D5311yescharacteristic
245C4⋊C8—1062no—
246#99—2122no—
247C22⋊C8—534no—
248C22⋊C8—1062no—
249C22⋊C8—1062no—
250#97—1062no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—1062no—
257#95C2×Dic5311yescharacteristic
258#95C2×Dic5311yescharacteristic
259C24C22⋊Dic5311yescharacteristic
260C24C2×C22⋊D5311yescharacteristic
261C2×Q8—22no—
262C2×Q8—81no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—2122no—
280C22×C4C22⋊Dic5311yescharacteristic
281C22×C4C22⋊Dic5311yescharacteristic
282C22×C4C22⋊Dic5311yescharacteristic
283C22×C4D4×D5312yes—
284C22×C4#1171994712yes—
285C22×C4C22×Dic5311yescharacteristic
286C22×C4C2×C22⋊D5311yescharacteristic
287M16—4242no—
288C2×C8—1064no—
289C2×C8—4242no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—81no—
296C4⋊C4—4242no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—81no—
302C22⋊C4—2122no—
303C22⋊C4—2122no—
304C42—21no—
305C42—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23C2×C22⋊Dic5311yescharacteristic
314C23#1313318912yes—
315C23#1313324211yescharacteristic
316C23#1313324711yescharacteristic
317C23#1313324812yes—
318Q8—82no—
319D4—42no—
320D4—42no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
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—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—4242no—
349C2×C4—4242no—
350C2×C4—4242no—
351C2×C4D4×Dic5312yes—
352C2×C4#1313318511yescharacteristic
353C2×C4#1313319811yescharacteristic
354C8—2124no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22#1681050311yescharacteristic
368C22#1681062411yescharacteristic
369C22#1681063211yescharacteristic
370C22#1681087112yes—
371C22#1681087912yes—
372C4—21no—
373C4—21no—
374C4—22no—
375C4—42no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—8482no—
383C2—21no—
384C2—22no—
385C2—21no—
386C2—22no—
387C2—81no—
388C2#2027509911yescharacteristic
389C2#2027514911yescharacteristic
390C2#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#9385
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
208. #400Dic53semidirect
209. #399Dic53semidirect
210. #395Dic53semidirect
211. #393Dic53semidirect
220. C53#9385semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202747032
#202732872
#202725942
C53×#25642
#938553