Finite Group Databasebeta

← #27358000

Subgroups — #27358000

Order 13568.

1220 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C2×#16811204C211yesmaximal, characteristic
2#20285468C212yesmaximal, complemented
3#20273751C214yesmaximal, complemented
4#20273683C214yesmaximal, complemented
5#20273282C212yesmaximal, complemented
6C2×#16810349C211yesmaximal, characteristic, complemented
7C53×#2627C211yesmaximal, characteristic, complemented
8C22×#11719945—21no—
9#16811204—24no—
10#16811204—22no—
11#16811204C2214yes—
12C2×#13133197—22no—
13C2×#13133192—22no—
14C2×#13133191—21no—
15C2×#13133191C2212yes—
16#16810951—22no—
17#16810941C2214yescomplemented
18#16810854—24no—
19#16810773—22no—
20#16810762C2214yescomplemented
21C2×C22⋊#88322C2212yes—
22#16810356C2218yescomplemented
23#16810349C2214yescomplemented
24C2×C22⋊Q8×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#400C2212yescomplemented
27C53×#398—22no—
28C53×#395C2212yescomplemented
29C53×#385—21no—
30C53×#381—21no—
31C53×#380C2211yescharacteristic
32C2×#11719945—24no—
33C2×#11719945—24no—
34C2×#11719945—44no—
35C2×#11719945—44no—
36C22×Dic106—21no—
37C22×D212—21no—
38C22×D212—21no—
39C22×D212D412yes—
40C22×C22⋊D53—22no—
41C22×C4×D53—22no—
42C2×#11719924—24no—
43C22×#88322—22no—
44#13133197—28no—
45#13133197—28no—
46#13133192—28no—
47#13133192—28no—
48#13133191—24no—
49#13133191—24no—
50#13133191—22no—
51#13133191—22no—
52#13133191D414yes—
53#13133191C2314yes—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53D412yes—
56C2×#11719921—22no—
57C2×#11719921—22no—
58#13133148—24no—
59#13133141—24no—
60#13133137—24no—
61C22⋊#88322—22no—
62C22⋊#88322C2314yes—
63C22×Q8×C53C2311yescharacteristic
64C23×C4×C53D411yescharacteristic
65C23×C4×C53D411yescharacteristic
66C23×C4×C53C2311yescharacteristic
67C22⋊Q8×C53—22no—
68C22⋊Q8×C53—22no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—22no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53—22no—
74C2×C4⋊C4×C53D412yes—
75C2×C4⋊C4×C53C2311yescharacteristic
76C2×C4⋊C4×C53C2312yes—
77C2×C22⋊C4×C53—21no—
78C2×C22⋊C4×C53—21no—
79#95×C53—22no—
80#95×C53—22no—
81#95×C53—22no—
82#95×C53C2312yes—
83C24×C53C2×D411yescharacteristic
84C23×D53—22no—
85C2×Q8×C53—22no—
86C2×Q8×C53—81no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—81no—
96C22×C4×C53—81no—
97C22×C4×C53SD1614yescomplemented
98C22×C4×C53C2×D412yes—
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D412yes—
102C22×C4×C53C2×D412yes—
103C22×C4×C53C2411yescharacteristic
104C2×C22⋊D53—24no—
105C2×C22⋊D53—24no—
106C2×C22⋊D53—44no—
107C2×C22⋊D53—44no—
108C22×Dic53—22no—
109#11719945—48no—
110#11719945—416no—
111#11719945—48no—
112C2×D212—24no—
113C2×D212—24no—
114C2×D212—22no—
115C2×D212—44no—
116C2×D212—82no—
117C2×D212—82no—
118C2×D212C2×D414yes—
119C2×C4×D53—28no—
120C2×C4×D53—48no—
121C2×Dic106—24no—
122C2×Dic106—44no—
123C4⋊C4×C53—24no—
124C4⋊C4×C53—24no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—42no—
127C4⋊C4×C53—81no—
128C4⋊C4×C53—82no—
129C22⋊C4×C53—41no—
130C22⋊C4×C53—41no—
131C22⋊C4×C53—42no—
132#11719924—84no—
133#11719921—28no—
134#11719921—24no—
135#11719921—24no—
136C4⋊Dic53—24no—
137C4⋊Dic53—22no—
138C4⋊Dic53C2×D414yes—
139C2×#88322—24no—
140C2×#88322—82no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53#12011yescharacteristic
144C23×C53#12011yescharacteristic
145C23×C53C2×SD1612yes—
146C23×C53AGL(1,ℤ/8)12yes—
147C23×C53C22×D411yescharacteristic
148C22×D53—24no—
149C22×D53—44no—
150C22×D53—44no—
151C22×D53—44no—
152Q8×C53—44no—
153C2×C4×C53—22no—
154C2×C4×C53—22no—
155C2×C4×C53—24no—
156C2×C4×C53—24no—
157C2×C4×C53—22no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—42no—
171C2×C4×C53—81no—
172C2×C4×C53#12012yes—
173C2×C4×C53C2×SD1614yes—
174C2×C4×C53C22×D412yes—
175C22⋊D53—44no—
176C22⋊D53—44no—
177C22⋊D53—48no—
178C2×Dic53—24no—
179C2×Dic53—44no—
180D212—44no—
181D212—48no—
182D212—82no—
183D212—84no—
184C4×D53—416no—
185Dic106—48no—
186#88322—44no—
187#18293—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—42no—
195C22×C53—42no—
196C22×C53#45012yes—
197C22×C53#45012yes—
198C22×C53C2×#12011yescharacteristic
199C22×C53C22×SD1611yescharacteristic
200C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
201C2×D53—44no—
202C2×D53—44no—
203C2×D53—44no—
204C2×D53—88no—
205C4×C53—24no—
206C4×C53—22no—
207C4×C53—22no—
208C4×C53—42no—
209C4×C53—82no—
210C4×C53—81no—
211C4×C53—81no—
212Dic53—44no—
213C2×#537—531no—
214#3187—532no—
215#2627D5311yescharacteristic, complemented
216#1616—534no—
217#1604—534no—
218#1519—532no—
219C2×#324—531no—
220C2×C53—21no—
221C2×C53—22no—
222C2×C53—21no—
223C2×C53—22no—
224C2×C53C2×#45011yescharacteristic
225C2×C53#302011yescharacteristic
226C2×C53#302611yescharacteristic
227D53—84no—
228C22×#41—1061no—
229#537—534no—
230#537—1064no—
231#537—1062no—
232C2×C22⋊Q8—1062no—
233C2×C22⋊Q8C2×D5311yescharacteristic
234C2×#121—532no—
235C2×#121—1062no—
236C2×#121—1061no—
237C22×C4⋊C4C2×D5311yescharacteristic
238#483—1062no—
239#481—534no—
240#465—1064no—
241#450—1062no—
242#448—534no—
243C2×C22⋊C8—532no—
244#400C2×D5312yescomplemented
245#398—22no—
246#395C2×D5312yescomplemented
247#385—21no—
248#381—21no—
249#380C2×D5311yescharacteristic
250#327—538no—
251#324—534no—
252C53#1829311yescharacteristic, complemented, Sylow
253C2×#41—1064no—
254C2×#41—1064no—
255C2×#41—2124no—
256C2×#41—2124no—
257C22×Q8—1061no—
258C22×Q8C22×D5311yescharacteristic
259C22×D4—532no—
260C22×D4—1061no—
261C22×D4—1061no—
262C22×D4—1062no—
263C23×C4—1062no—
264C23×C4C22⋊D5311yescharacteristic
265C23×C4C22⋊D5311yescharacteristic
266C23×C4C22×D5311yescharacteristic
267C2×SD16—1064no—
268C22×C8—1062no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—1068no—
272C22⋊Q8—1068no—
273#121—534no—
274#121—534no—
275#121—1068no—
276#121—1068no—
277#121—1064no—
278#121—1062no—
279#121—1062no—
280#121—1064no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—532no—
287C2×C4⋊C4—1061no—
288C2×C4⋊C4C22⋊D5312yes—
289C2×C4⋊C4C22×D5311yescharacteristic
290C2×C4⋊C4C22×D5312yes—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—1062no—
294C2×C22⋊C4—1062no—
295#106—1064no—
296#103—1064no—
297#102—1064no—
298C22⋊C8—534no—
299C22⋊C8—1062no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95C22×D5312yes—
304C24—1062no—
305C24C2×C22⋊D5311yescharacteristic
306#41—21216no—
307#41—2128no—
308#41—2128no—
309C2×Q8—22no—
310C2×Q8—81no—
311C2×Q8—1064no—
312C2×Q8—2124no—
313C2×D4—534no—
314C2×D4—1064no—
315C2×D4—1064no—
316C2×D4—1064no—
317C2×D4—1064no—
318C2×D4—1062no—
319C2×D4—4242no—
320C2×D4—4242no—
321C2×D4—2124no—
322C2×D4—2124no—
323C2×D4—2124no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—1068no—
335C22×C4—1062no—
336C22×C4—2128no—
337C22×C4#1171992414yescomplemented
338C22×C4D4×D5312yes—
339C22×C4D4×D5312yes—
340C22×C4C2×C22⋊D5312yes—
341C22×C4C2×C22⋊D5312yes—
342C22×C4C2×C22⋊D5311yescharacteristic
343C22×C4C23×D5311yescharacteristic
344SD16—4244no—
345C2×C8—1064no—
346C2×C8—4242no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—81no—
352C4⋊C4—82no—
353C4⋊C4—534no—
354C4⋊C4—1064no—
355C4⋊C4—1062no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—42no—
359C22⋊C4—1068no—
360C22⋊C4—1064no—
361C22⋊C4—1064no—
362C23—41no—
363C23—41no—
364C23—1064no—
365C23—2124no—
366C23—2124no—
367C23—2124no—
368C23#1313318511yescharacteristic
369C23#1313318511yescharacteristic
370C23C2×#1171992412yes—
371C23#1313324212yes—
372C23C22×C22⋊D5311yescharacteristic
373Q8—44no—
374Q8—2128no—
375D4—4242no—
376D4—4244no—
377D4—2124no—
378D4—2128no—
379D4—2128no—
380D4—2124no—
381D4—2124no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—1064no—
402C2×C4—21216no—
403C2×C4—2124no—
404C2×C4#1313318512yes—
405C2×C4C2×#1171992414yes—
406C2×C4C2×D4×D5312yes—
407C8—2124no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—4248no—
417C22—2124no—
418C22—2124no—
419C22—2124no—
420C22#1681077312yes—
421C22#1681077312yes—
422C22C2×#1313318511yescharacteristic
423C22C22×#1171992411yescharacteristic
424C22C2×#1313324211yescharacteristic
425C4—22no—
426C4—22no—
427C4—24no—
428C4—42no—
429C4—82no—
430C4—81no—
431C4—81no—
432C4—2124no—
433C2—21no—
434C2—22no—
435C2—22no—
436C2—21no—
437C2—4244no—
438C2C2×#1681077311yescharacteristic
439C2#2028407811yescharacteristic
440C2#2028411111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2627C2
socleC22×C53C2×#120
prime radicalC2×#16811204C2
hypercenter#2627D53
hypocenterC53#18293
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
2. #20285468C2semidirect
3. #20273751C2semidirect
4. #20273683C2semidirect
5. #20273282C2semidirect
6. C2×#16810349C2semidirect
7. C53×#2627C2semidirect
17. #16810941C22semidirect
20. #16810762C22semidirect
22. #16810356C22semidirect
23. #16810349C22semidirect
26. C53×#400C22semidirect
28. C53×#395C22semidirect
97. C22×C4×C53SD16semidirect
215. #2627D53semidirect
244. #400C2×D53semidirect
246. #395C2×D53semidirect
252. C53#18293semidirect
337. C22×C4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168112042
#202854682
#202737512
#202736832
#202732822
C2×#168103492
C53×#26272
#1829353