Finite Group Databasebeta

← #27357932

Subgroups — #27357932

Order 13568.

1408 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1408 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×#16811185C211yesmaximal, characteristic
2#20285468C212yesmaximal, complemented
3#20273734C214yesmaximal, complemented
4#20273666C214yesmaximal, complemented
5#20273282C212yesmaximal, complemented
6C2×#16810347C211yesmaximal, characteristic, complemented
7C53×#2626C211yesmaximal, characteristic, complemented
8C23×C22⋊D53—21no—
9#16811185—24no—
10#16811185—22no—
11#16811185C2214yes—
12C2×#13133201—22no—
13C2×#13133191—21no—
14C2×#13133191C2212yes—
15C2×#13133187—22no—
16#16810951—22no—
17#16810941C2214yescomplemented
18#16810854—24no—
19#16810773—22no—
20#16810762C2214yescomplemented
21C2×C22⋊#88322C2212yescomplemented
22#16810356C2218yescomplemented
23#16810347C2214yescomplemented
24C2×C22⋊Q8×C53C2211yescharacteristic, complemented
25C22×C22⋊C4×C53C2211yescharacteristic
26C53×#399C2212yescomplemented
27C53×#396—22no—
28C53×#393C2212yescomplemented
29C53×#385—21no—
30C53×#380C2211yescharacteristic, complemented
31C53×#379—21no—
32C24×D53—21no—
33C22×D212—21no—
34C22×D212D412yes—
35C22×C22⋊D53—22no—
36C22×C22⋊D53—24no—
37C22×C22⋊D53—22no—
38C22×C22⋊D53—24no—
39C22×C22⋊D53—44no—
40C22×C22⋊D53—44no—
41C23×Dic53—21no—
42C2×#11719924—24no—
43C22×#88322—22no—
44#13133201—28no—
45#13133201—28no—
46#13133191—24no—
47#13133191—24no—
48#13133191—22no—
49#13133191—22no—
50#13133191D414yes—
51#13133191C2314yes—
52#13133187—28no—
53#13133187—28no—
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⋊#88322C2314yescomplemented
63C25×C53D411yescharacteristic
64C22×Q8×C53C2311yescharacteristic, complemented
65C23×C4×C53D411yescharacteristic
66C23×C4×C53C2311yescharacteristic
67C22⋊Q8×C53—22no—
68C22⋊Q8×C53—22no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53C2311yescharacteristic, complemented
73C2×C22⋊C4×C53—21no—
74C2×C22⋊C4×C53—21no—
75C2×C22⋊C4×C53—22no—
76C2×C22⋊C4×C53—22no—
77C2×C22⋊C4×C53—22no—
78C2×C22⋊C4×C53—22no—
79C2×C22⋊C4×C53D412yes—
80C2×C22⋊C4×C53C2312yes—
81#95×C53—22no—
82#95×C53C2312yescomplemented
83C24×C53—21no—
84C24×C53—21no—
85C24×C53—22no—
86C24×C53—81no—
87C24×C53—81no—
88C24×C53SD1614yescomplemented
89C24×C53C2×D412yes—
90C24×C53C2×D411yescharacteristic
91C23×D53—22no—
92C23×D53—24no—
93C23×D53—44no—
94C23×D53—82no—
95C23×D53—82no—
96C2×Q8×C53—22no—
97C2×Q8×C53—81no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—21no—
103C22×C4×C53—81no—
104C22×C4×C53—81no—
105C22×C4×C53C2×D412yes—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D412yes—
108C22×C4×C53C2×D412yes—
109C22×C4×C53C2411yescharacteristic
110C2×C22⋊D53—28no—
111C2×C22⋊D53—24no—
112C2×C22⋊D53—24no—
113C2×C22⋊D53—48no—
114C2×C22⋊D53—416no—
115C2×C22⋊D53—48no—
116C2×C22⋊D53—48no—
117C2×C22⋊D53—44no—
118C2×C22⋊D53—44no—
119C22×Dic53—24no—
120C22×Dic53—22no—
121C22×Dic53—44no—
122C2×D212—24no—
123C2×D212—22no—
124C2×D212—82no—
125C2×D212—82no—
126C2×D212C2×D414yes—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—42no—
129C4⋊C4×C53—81no—
130C4⋊C4×C53—82no—
131C22⋊C4×C53—24no—
132C22⋊C4×C53—24no—
133C22⋊C4×C53—41no—
134C22⋊C4×C53—41no—
135C22⋊C4×C53—42no—
136C22⋊C4×C53—82no—
137C22⋊C4×C53—82no—
138#11719924—84no—
139#11719921—28no—
140#11719921—24no—
141#11719921—24no—
142C4⋊Dic53—24no—
143C4⋊Dic53—22no—
144C4⋊Dic53C2×D414yes—
145C2×#88322—24no—
146C2×#88322—82no—
147C23×C53—22no—
148C23×C53—24no—
149C23×C53—22no—
150C23×C53—24no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53—42no—
154C23×C53—42no—
155C23×C53—42no—
156C23×C53—42no—
157C23×C53—81no—
158C23×C53—81no—
159C23×C53—84no—
160C23×C53—84no—
161C23×C53#12011yescharacteristic
162C23×C53#12011yescharacteristic
163C23×C53C2×SD1614yes—
164C23×C53C2×SD1612yes—
165C23×C53AGL(1,ℤ/8)12yes—
166C23×C53C22×D411yescharacteristic
167C22×D53—24no—
168C22×D53—44no—
169C22×D53—44no—
170C22×D53—48no—
171C22×D53—44no—
172C22×D53—84no—
173C22×D53—84no—
174C22×D53—816no—
175Q8×C53—44no—
176C2×C4×C53—22no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—42no—
190C2×C4×C53—81no—
191C2×C4×C53#12012yes—
192C2×C4×C53C22×D412yes—
193C22⋊D53—416no—
194C22⋊D53—44no—
195C22⋊D53—44no—
196C22⋊D53—48no—
197C2×Dic53—24no—
198C2×Dic53—48no—
199C2×Dic53—44no—
200D212—44no—
201D212—82no—
202D212—84no—
203#88322—44no—
204#18285—531nomaximal, Sylow
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—24no—
212C22×C53—42no—
213C22×C53—42no—
214C22×C53—44no—
215C22×C53—48no—
216C22×C53—42no—
217C22×C53—42no—
218C22×C53—82no—
219C22×C53—84no—
220C22×C53#45012yes—
221C22×C53#45012yes—
222C22×C53C2×#12011yescharacteristic
223C22×C53C22×SD1611yescharacteristic
224C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
225C2×D53—44no—
226C2×D53—44no—
227C2×D53—44no—
228C2×D53—88no—
229C2×D53—816no—
230C4×C53—22no—
231C4×C53—22no—
232C4×C53—42no—
233C4×C53—82no—
234C4×C53—81no—
235C4×C53—81no—
236Dic53—44no—
237C2×#534—531no—
238#3187—532no—
239#2626D5311yescharacteristic, complemented
240#1613—534no—
241#1601—534no—
242#1519—532no—
243C2×#323—531no—
244C2×C53—21no—
245C2×C53—22no—
246C2×C53—21no—
247C2×C53—22no—
248C2×C53—44no—
249C2×C53C2×#45011yescharacteristic
250C2×C53#301911yescharacteristic
251C2×C53#302611yescharacteristic
252D53—84no—
253C23×D4—1061no—
254#534—534no—
255#534—1064no—
256#534—1062no—
257C2×#123—1062no—
258C2×C22⋊Q8C2×D5311yescharacteristic, complemented
259C2×#121—532no—
260C2×#121—1061no—
261C2×#120—1062no—
262C22×C22⋊C4C2×D5311yescharacteristic
263#483—1062no—
264#481—534no—
265#465—1064no—
266#450—1062no—
267#448—534no—
268C2×C22⋊C8—532no—
269#399C2×D5312yescomplemented
270#396—22no—
271#393C2×D5312yescomplemented
272#385—21no—
273#380C2×D5311yescharacteristic, complemented
274#379—21no—
275#327—538no—
276#323—534no—
277C53#1828511yescharacteristic, complemented, Sylow
278C25—1061no—
279C25C22⋊D5311yescharacteristic
280C22×Q8C22×D5311yescharacteristic, complemented
281C22×D4—532no—
282C22×D4—1061no—
283C22×D4—1062no—
284C22×D4—1064no—
285C22×D4—1062no—
286C22×D4—1064no—
287C22×D4—2124no—
288C22×D4—2124no—
289C23×C4—1061no—
290C23×C4C22⋊D5311yescharacteristic
291C23×C4C22×D5311yescharacteristic
292C2×SD16—1064no—
293C22×C8—1062no—
294#123—1068no—
295#123—1068no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298#121—534no—
299#121—534no—
300#121—1064no—
301#121—1062no—
302#121—1062no—
303#121—1064no—
304#120—1068no—
305#120—1068no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—532no—
310C2×C4⋊C4—1061no—
311C2×C4⋊C4C22×D5311yescharacteristic, complemented
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—1062no—
319C2×C22⋊C4—1062no—
320C2×C22⋊C4C22⋊D5312yes—
321C2×C22⋊C4C22×D5312yes—
322#106—1064no—
323#103—1064no—
324#102—1064no—
325C22⋊C8—534no—
326C22⋊C8—1062no—
327#95—22no—
328#95C22×D5312yescomplemented
329C24—21no—
330C24—21no—
331C24—22no—
332C24—81no—
333C24—81no—
334C24—1064no—
335C24—1062no—
336C24—4242no—
337C24—4242no—
338C24—2124no—
339C24#1171992414yescomplemented
340C24C2×C22⋊D5312yes—
341C24C2×C22⋊D5311yescharacteristic
342C2×Q8—22no—
343C2×Q8—81no—
344C2×D4—534no—
345C2×D4—1064no—
346C2×D4—1068no—
347C2×D4—1064no—
348C2×D4—1064no—
349C2×D4—1062no—
350C2×D4—4242no—
351C2×D4—4242no—
352C2×D4—21216no—
353C2×D4—2128no—
354C2×D4—2128no—
355C2×D4—2128no—
356C2×D4—2124no—
357C2×D4—2124no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—1064no—
366C22×C4—1062no—
367C22×C4—2124no—
368C22×C4D4×D5312yes—
369C22×C4D4×D5312yes—
370C22×C4C2×C22⋊D5312yes—
371C22×C4C2×C22⋊D5311yescharacteristic
372C22×C4C23×D5311yescharacteristic
373SD16—4244no—
374C2×C8—1064no—
375C2×C8—4242no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4—534no—
381C4⋊C4—1064no—
382C4⋊C4—1062no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—42no—
388C22⋊C4—82no—
389C22⋊C4—82no—
390C22⋊C4—1068no—
391C22⋊C4—1064no—
392C22⋊C4—1064no—
393C23—24no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—81no—
404C23—81no—
405C23—84no—
406C23—84no—
407C23—1064no—
408C23—42416no—
409C23—4244no—
410C23—4244no—
411C23—2124no—
412C23—2124no—
413C23—2128no—
414C23—2124no—
415C23#1313318511yescharacteristic
416C23#1313318511yescharacteristic
417C23C2×#1171992414yes—
418C23C2×#1171992412yes—
419C23#1313324212yes—
420C23C22×C22⋊D5311yescharacteristic
421Q8—44no—
422D4—4242no—
423D4—4244no—
424D4—2124no—
425D4—21216no—
426D4—2128no—
427D4—2124no—
428D4—2124no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—81no—
444C2×C4—1064no—
445C2×C4—2128no—
446C2×C4—2124no—
447C2×C4#1313318512yes—
448C2×C4C2×D4×D5312yes—
449C8—2124no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—24no—
457C22—42no—
458C22—42no—
459C22—44no—
460C22—48no—
461C22—42no—
462C22—42no—
463C22—82no—
464C22—84no—
465C22—4248no—
466C22—42416no—
467C22—2124no—
468C22—2124no—
469C22—2124no—
470C22#1681077312yes—
471C22#1681077312yes—
472C22C2×#1313318511yescharacteristic
473C22C22×#1171992411yescharacteristic
474C22C2×#1313324211yescharacteristic
475C4—22no—
476C4—22no—
477C4—42no—
478C4—82no—
479C4—81no—
480C4—81no—
481C4—2124no—
482C2—21no—
483C2—22no—
484C2—21no—
485C2—22no—
486C2—44no—
487C2—4244no—
488C2C2×#1681077311yescharacteristic
489C2#2028407111yescharacteristic
490C2#2028411111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2626C2
socleC22×C53C2×#120
prime radicalC2×#16811185C2
hypercenter#2626D53
hypocenterC53#18285
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. #20285468C2semidirect
3. #20273734C2semidirect
4. #20273666C2semidirect
5. #20273282C2semidirect
6. C2×#16810347C2semidirect
7. C53×#2626C2semidirect
17. #16810941C22semidirect
20. #16810762C22semidirect
21. C2×C22⋊#88322C22semidirect
22. #16810356C22semidirect
23. #16810347C22semidirect
24. C2×C22⋊Q8×C53C22semidirect
26. C53×#399C22semidirect
28. C53×#393C22semidirect
30. C53×#380C22semidirect
62. C22⋊#88322C23semidirect
64. C22×Q8×C53C23semidirect
72. C2×C4⋊C4×C53C23semidirect
82. #95×C53C23semidirect
88. C24×C53SD16semidirect
239. #2626D53semidirect
258. C2×C22⋊Q8C2×D53semidirect
269. #399C2×D53semidirect
271. #393C2×D53semidirect
273. #380C2×D53semidirect
277. C53#18285semidirect
280. C22×Q8C22×D53semidirect
311. C2×C4⋊C4C22×D53semidirect
328. #95C22×D53semidirect
339. C24#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168111852
#202854682
#202737342
#202736662
#202732822
C2×#168103472
C53×#26262
#1828553