Finite Group Databasebeta

← #27357925

Subgroups — #27357925

Order 13568.

1486 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1486 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, complemented
2#20285463C212yesmaximal, complemented
3#20273700C214yesmaximal, complemented
4#20273632C214yesmaximal, complemented
5#20273282C212yesmaximal, complemented
6C2×#16810347C211yesmaximal, characteristic, complemented
7C53×#2617C211yesmaximal, 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#16810936C2214yescomplemented
17#16810934—22no—
18#16810857—24no—
19#16810757C2214yescomplemented
20#16810756—22no—
21C2×C22⋊#88322C2212yescomplemented
22#16810356C2218yescomplemented
23#16810347C2214yescomplemented
24C2×#121×C53C2211yescharacteristic, complemented
25C2×#120×C53—21no—
26C22×C22⋊C4×C53C2211yescharacteristic
27C53×#397C2212yescomplemented
28C53×#394—22no—
29C53×#392C2212yescomplemented
30C53×#386—21no—
31C53×#380C2211yescharacteristic, complemented
32C24×D53—21no—
33C22×D212—21no—
34C22×D212D412yescomplemented
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×#11719922—24no—
43C22×#88322—22no—
44#13133201—28no—
45#13133201—28no—
46#13133191—24no—
47#13133191—24no—
48#13133191—22no—
49#13133191—22no—
50#13133191D414yescomplemented
51#13133191C2314yes—
52#13133187—28no—
53#13133187—28no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53D412yescomplemented
56C2×#11719921—22no—
57C2×#11719921—22no—
58#13133150—24no—
59#13133138—24no—
60#13133137—24no—
61C22⋊#88322—22no—
62C22⋊#88322C2314yescomplemented
63C25×C53D411yescharacteristic
64C22×D4×C53—21no—
65C22×D4×C53—22no—
66C22×D4×C53C2311yescharacteristic, complemented
67C23×C4×C53D411yescharacteristic
68C23×C4×C53C2311yescharacteristic
69#121×C53—22no—
70#121×C53—22no—
71#120×C53—44no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53C2311yescharacteristic, complemented
74C2×C22⋊C4×C53—21no—
75C2×C22⋊C4×C53—22no—
76C2×C22⋊C4×C53—22no—
77C2×C22⋊C4×C53—22no—
78C2×C22⋊C4×C53—21no—
79C2×C22⋊C4×C53—22no—
80C2×C22⋊C4×C53D412yes—
81C2×C22⋊C4×C53C2312yes—
82#95×C53—22no—
83#95×C53C2312yescomplemented
84C24×C53—21no—
85C24×C53—21no—
86C24×C53—22no—
87C24×C53—21no—
88C24×C53—81no—
89C24×C53—81no—
90C24×C53D814yescomplemented
91C24×C53C2×D412yes—
92C24×C53C2×D411yescharacteristic
93C23×D53—22no—
94C23×D53—24no—
95C23×D53—44no—
96C23×D53—82no—
97C23×D53—82no—
98C2×D4×C53—22no—
99C2×D4×C53—41no—
100C2×D4×C53—41no—
101C2×D4×C53—44no—
102C2×D4×C53—42no—
103C2×D4×C53—42no—
104C2×D4×C53—81no—
105C2×D4×C53—82no—
106C2×D4×C53—82no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—81no—
112C22×C4×C53—81no—
113C22×C4×C53C2×D412yes—
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D412yes—
116C22×C4×C53C2×D412yes—
117C22×C4×C53C2411yescharacteristic
118C2×C22⋊D53—28no—
119C2×C22⋊D53—24no—
120C2×C22⋊D53—24no—
121C2×C22⋊D53—48no—
122C2×C22⋊D53—416no—
123C2×C22⋊D53—48no—
124C2×C22⋊D53—48no—
125C2×C22⋊D53—44no—
126C2×C22⋊D53—44no—
127C22×Dic53—24no—
128C22×Dic53—22no—
129C22×Dic53—44no—
130C2×D212—24no—
131C2×D212—22no—
132C2×D212—82no—
133C2×D212—82no—
134C2×D212C2×D414yes—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—81no—
137C22⋊C4×C53—24no—
138C22⋊C4×C53—24no—
139C22⋊C4×C53—44no—
140C22⋊C4×C53—42no—
141C22⋊C4×C53—42no—
142C22⋊C4×C53—82no—
143#11719922—84no—
144#11719921—28no—
145#11719921—24no—
146#11719921—24no—
147C4⋊Dic53—24no—
148C4⋊Dic53—22no—
149C4⋊Dic53C2×D414yes—
150C2×#88322—24no—
151C2×#88322—82no—
152C23×C53—22no—
153C23×C53—24no—
154C23×C53—22no—
155C23×C53—24no—
156C23×C53—41no—
157C23×C53—41no—
158C23×C53—42no—
159C23×C53—42no—
160C23×C53—42no—
161C23×C53—42no—
162C23×C53—42no—
163C23×C53—42no—
164C23×C53—41no—
165C23×C53—41no—
166C23×C53—41no—
167C23×C53—81no—
168C23×C53—81no—
169C23×C53—84no—
170C23×C53—84no—
171C23×C53#12011yescharacteristic
172C23×C53#12011yescharacteristic
173C23×C53C2×D814yes—
174C23×C53C2×D812yes—
175C23×C53AGL(1,ℤ/8)12yes—
176C23×C53C22×D411yescharacteristic
177C22×D53—24no—
178C22×D53—44no—
179C22×D53—44no—
180C22×D53—48no—
181C22×D53—44no—
182C22×D53—84no—
183C22×D53—84no—
184C22×D53—816no—
185D4×C53—44no—
186D4×C53—88no—
187D4×C53—82no—
188D4×C53—82no—
189C2×C4×C53—22no—
190C2×C4×C53—42no—
191C2×C4×C53—42no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—41no—
199C2×C4×C53—42no—
200C2×C4×C53—41no—
201C2×C4×C53—41no—
202C2×C4×C53—81no—
203C2×C4×C53#12012yes—
204C2×C4×C53C22×D412yes—
205C22⋊D53—416no—
206C22⋊D53—44no—
207C22⋊D53—44no—
208C22⋊D53—48no—
209C2×Dic53—24no—
210C2×Dic53—48no—
211C2×Dic53—44no—
212D212—44no—
213D212—82no—
214D212—84no—
215#88322—44no—
216#18284—531nomaximal, Sylow
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—24no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—44no—
227C22×C53—48no—
228C22×C53—42no—
229C22×C53—42no—
230C22×C53—82no—
231C22×C53—84no—
232C22×C53—82no—
233C22×C53—82no—
234C22×C53—81no—
235C22×C53—81no—
236C22×C53—81no—
237C22×C53#44712yes—
238C22×C53#44712yes—
239C22×C53C2×#12011yescharacteristic
240C22×C53C22×D811yescharacteristic
241C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
242C2×D53—44no—
243C2×D53—44no—
244C2×D53—44no—
245C2×D53—88no—
246C2×D53—816no—
247C4×C53—22no—
248C4×C53—22no—
249C4×C53—42no—
250C4×C53—82no—
251C4×C53—81no—
252Dic53—44no—
253C2×#534—531no—
254#3186—532no—
255#2617D5311yescharacteristic, complemented
256#1607—534no—
257#1595—534no—
258#1519—532no—
259C2×#323—531no—
260C2×C53—21no—
261C2×C53—22no—
262C2×C53—21no—
263C2×C53—22no—
264C2×C53—44no—
265C2×C53—161no—
266C2×C53C2×#44711yescharacteristic
267C2×C53#301911yescharacteristic
268C2×C53#302611yescharacteristic
269D53—84no—
270C23×D4—1061no—
271#534—534no—
272#534—1064no—
273#534—1062no—
274C2×#123—1062no—
275C2×#121—532no—
276C2×#121—1061no—
277C2×#121C2×D5311yescharacteristic, complemented
278C2×#120—21no—
279C2×#120—1062no—
280C22×C22⋊C4C2×D5311yescharacteristic
281#480—534no—
282#480—1062no—
283#466—1064no—
284#447—534no—
285#447—1062no—
286C2×C22⋊C8—532no—
287#397C2×D5312yescomplemented
288#394—22no—
289#392C2×D5312yescomplemented
290#386—21no—
291#380C2×D5311yescharacteristic, complemented
292#327—538no—
293#323—534no—
294C53#1828411yescharacteristic, complemented, Sylow
295C25—1061no—
296C25C22⋊D5311yescharacteristic
297C22×D4—21no—
298C22×D4—22no—
299C22×D4—532no—
300C22×D4—1061no—
301C22×D4—1062no—
302C22×D4—1064no—
303C22×D4—1062no—
304C22×D4—1064no—
305C22×D4—2124no—
306C22×D4—2124no—
307C22×D4C22×D5311yescharacteristic, complemented
308C23×C4—1061no—
309C23×C4C22⋊D5311yescharacteristic
310C23×C4C22×D5311yescharacteristic
311C2×D8—1064no—
312C22×C8—1062no—
313#123—1068no—
314#123—1068no—
315#121—22no—
316#121—22no—
317#121—534no—
318#121—534no—
319#121—1064no—
320#121—1062no—
321#121—1062no—
322#121—1064no—
323#120—44no—
324#120—1068no—
325#120—1068no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—532no—
328C2×C4⋊C4—1061no—
329C2×C4⋊C4C22×D5311yescharacteristic, complemented
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—1062no—
337C2×C22⋊C4—1062no—
338C2×C22⋊C4C22⋊D5312yes—
339C2×C22⋊C4C22×D5312yes—
340#107—1064no—
341#102—1064no—
342#102—1064no—
343C22⋊C8—534no—
344C22⋊C8—1062no—
345#95—22no—
346#95C22×D5312yescomplemented
347C24—21no—
348C24—21no—
349C24—22no—
350C24—21no—
351C24—81no—
352C24—81no—
353C24—1064no—
354C24—1062no—
355C24—4242no—
356C24—4242no—
357C24—2124no—
358C24#1171992214yescomplemented
359C24C2×C22⋊D5312yes—
360C24C2×C22⋊D5311yescharacteristic
361C2×D4—22no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—44no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—81no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—534no—
371C2×D4—1064no—
372C2×D4—1068no—
373C2×D4—1064no—
374C2×D4—1064no—
375C2×D4—1062no—
376C2×D4—4242no—
377C2×D4—4242no—
378C2×D4—21216no—
379C2×D4—2128no—
380C2×D4—2128no—
381C2×D4—2128no—
382C2×D4—2124no—
383C2×D4—2124no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—1064no—
391C22×C4—1062no—
392C22×C4—2124no—
393C22×C4D4×D5312yes—
394C22×C4D4×D5312yes—
395C22×C4C2×C22⋊D5312yes—
396C22×C4C2×C22⋊D5311yescharacteristic
397C22×C4C23×D5311yescharacteristic
398D8—4244no—
399C2×C8—1064no—
400C2×C8—4242no—
401C4⋊C4—22no—
402C4⋊C4—81no—
403C4⋊C4—534no—
404C4⋊C4—1064no—
405C4⋊C4—1062no—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—44no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—82no—
412C22⋊C4—1068no—
413C22⋊C4—1064no—
414C22⋊C4—1064no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—22no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—81no—
431C23—81no—
432C23—84no—
433C23—84no—
434C23—1064no—
435C23—42416no—
436C23—4244no—
437C23—4244no—
438C23—2124no—
439C23—2124no—
440C23—2128no—
441C23—2124no—
442C23#1313318511yescharacteristic
443C23#1313318511yescharacteristic
444C23C2×#1171992214yes—
445C23C2×#1171992212yes—
446C23#1313324212yes—
447C23C22×C22⋊D5311yescharacteristic
448D4—44no—
449D4—88no—
450D4—82no—
451D4—82no—
452D4—4242no—
453D4—4244no—
454D4—2124no—
455D4—21216no—
456D4—2128no—
457D4—2124no—
458D4—2124no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—81no—
473C2×C4—1064no—
474C2×C4—2128no—
475C2×C4—2124no—
476C2×C4#1313318512yes—
477C2×C4C2×D4×D5312yes—
478C8—2124no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—24no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—48no—
490C22—42no—
491C22—42no—
492C22—82no—
493C22—84no—
494C22—82no—
495C22—82no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—4248no—
500C22—42416no—
501C22—2124no—
502C22—2124no—
503C22—2124no—
504C22#1681075612yes—
505C22#1681075612yes—
506C22C2×#1313318511yescharacteristic
507C22C22×#1171992211yescharacteristic
508C22C2×#1313324211yescharacteristic
509C4—22no—
510C4—22no—
511C4—42no—
512C4—82no—
513C4—81no—
514C4—2124no—
515C2—21no—
516C2—22no—
517C2—21no—
518C2—22no—
519C2—44no—
520C2—161no—
521C2—4244no—
522C2C2×#1681075611yescharacteristic
523C2#2028407111yescharacteristic
524C2#2028411111yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2617C2
socleC22×C53C2×#120
hypercenter#2617D53
hypocenterC53#18284
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
1. C2×#16811185C2semidirect
2. #20285463C2semidirect
3. #20273700C2semidirect
4. #20273632C2semidirect
5. #20273282C2semidirect
6. C2×#16810347C2semidirect
7. C53×#2617C2semidirect
16. #16810936C22semidirect
19. #16810757C22semidirect
21. C2×C22⋊#88322C22semidirect
22. #16810356C22semidirect
23. #16810347C22semidirect
24. C2×#121×C53C22semidirect
27. C53×#397C22semidirect
29. C53×#392C22semidirect
31. C53×#380C22semidirect
34. C22×D212D4semidirect
50. #13133191D4semidirect
55. C2×C4⋊Dic53D4semidirect
62. C22⋊#88322C23semidirect
66. C22×D4×C53C23semidirect
73. C2×C4⋊C4×C53C23semidirect
83. #95×C53C23semidirect
90. C24×C53D8semidirect
255. #2617D53semidirect
277. C2×#121C2×D53semidirect
287. #397C2×D53semidirect
289. #392C2×D53semidirect
291. #380C2×D53semidirect
294. C53#18284semidirect
307. C22×D4C22×D53semidirect
329. C2×C4⋊C4C22×D53semidirect
346. #95C22×D53semidirect
358. C24#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168111852
#202854632
#202737002
#202736322
#202732822
C2×#168103472
C53×#26172
#1828453