Finite Group Databasebeta

← #27280648

Subgroups — #27280648

Order 13568.

606 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 107 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#20285468C211yesmaximal, characteristic
2#20277762C211yesmaximal, characteristic
3#20275701C211yesmaximal, characteristic, complemented
4#20273284C211yesmaximal, characteristic
5#20272594C211yesmaximal, characteristic, complemented
6#20272562C211yesmaximal, characteristic, complemented
7C2×C53×#328C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9C2×C4×C22⋊D53—21no—
10#16810951—21no—
11#16810941C412yescomplemented
12#16810854—22no—
13#16810773—21no—
14#16810762C412yescomplemented
15C2×C22⋊#88322C2211yescharacteristic
16#16810512—21no—
17#16810506—21no—
18#16810502—21no—
19#16810501—21no—
20#16810483C2211yescharacteristic
21C4×C22⋊Dic53C2211yescharacteristic
22#16810470—21no—
23#16810390—21no—
24#16810390—21no—
25#16810358—22no—
26C2×C22⋊Q8×C53C2211yescharacteristic
27C2×C22⋊C8×C53C2211yescharacteristic
28C2×#95×C53C2211yescharacteristic
29C53×#328—21no—
30C53×#328—21no—
31C22×D212D411yescharacteristic
32C22×C22⋊D53—21no—
33C22×C4×D53—21no—
34C2×#11719924—22no—
35C22×#88322—21no—
36#13133191—21no—
37#13133191—21no—
38#13133191C2×C412yes—
39#13133191D412yes—
40C4×C22⋊D53—42no—
41C4×C22⋊D53—42no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53D411yescharacteristic
44C2×#11719919—21no—
45C2×C22⋊Dic53—21no—
46C2×C22⋊Dic53D411yescharacteristic
47C2×C22⋊Dic53D411yescharacteristic
48C2×#11719921—21no—
49C2×#11719921—21no—
50C2×#11719921—21no—
51C2×#11719921—21no—
52C2×C4×Dic53—21no—
53C2×C4×Dic53—21no—
54#13133148—22no—
55#13133141—22no—
56#13133137—22no—
57C22⋊#88322—21no—
58C22⋊#88322C2×C412yescomplemented
59#13133124—21no—
60#13133124—21no—
61#13133124—21no—
62#13133124—21no—
63C22×Q8×C53C2×C411yescharacteristic, complemented
64C23×C4×C53—21no—
65C23×C4×C53C2311yescharacteristic
66C22×C8×C53—21no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53—21no—
70C22⋊Q8×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
73C2×C22⋊C4×C53—21no—
74C22⋊C8×C53—21no—
75C22⋊C8×C53—21no—
76#95×C53—21no—
77#95×C53D411yescharacteristic
78#95×C53D411yescharacteristic
79C24×C53C2×D411yescharacteristic
80C23×D53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—81no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—42no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—81no—
92C22×C4×C53—81no—
93C22×C4×C53C22×C411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53#4111yescharacteristic
98C22×C4×C53#4111yescharacteristic
99C2×C22⋊D53—22no—
100C2×C22⋊D53—22no—
101C2×C22⋊D53—41no—
102C2×C22⋊D53—41no—
103C2×C22⋊D53—41no—
104C2×C22⋊D53—41no—
105C22×Dic53—21no—
106C22×Dic53—21no—
107C22×Dic53—21no—
108C22×Dic53—41no—
109C2×D212—21no—
110C2×D212—81no—
111C2×D212C22⋊C412yes—
112C2×C4×D53—42no—
113C2×C4×D53—42no—
114C2×C4×D53—81no—
115C2×C8×C53—41no—
116C2×C8×C53—81no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C22⋊C4×C53—41no—
123C22⋊Dic53—21no—
124C22⋊Dic53—21no—
125C22⋊Dic53—21no—
126C22⋊Dic53—41no—
127C22⋊Dic53—41no—
128C22⋊Dic53SD1612yescomplemented
129#11719924—82no—
130#11719921—22no—
131#11719921—22no—
132#11719921—42no—
133#11719921—42no—
134#11719921—81no—
135C4⋊Dic53—21no—
136C4⋊Dic53—81no—
137C4⋊Dic53C22⋊C412yes—
138#11719919—42no—
139C4×Dic53—42no—
140C4×Dic53—81no—
141C2×#88322—22no—
142C2×#88322—81no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53C4×D411yescharacteristic
146C23×C53#12011yescharacteristic
147C23×C53#12111yescharacteristic
148C23×C53C2×SD1611yescharacteristic
149C23×C53#13511yescharacteristic
150C23×C53AGL(1,ℤ/8)11yescharacteristic
151C23×C53#13711yescharacteristic
152C22×D53—22no—
153C22×D53—41no—
154C22×D53—41no—
155C22×D53—41no—
156C22×D53—41no—
157C22×D53—41no—
158C22×D53—41no—
159Q8×C53—42no—
160Q8×C53—81no—
161C2×C4×C53—22no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—82no—
176C2×C4×C53—82no—
177C2×C4×C53—81no—
178C2×C4×C53—81no—
179C2×C4×C53C2×C22⋊C411yescharacteristic
180C2×C4×C53C4×D411yescharacteristic
181C2×C4×C53#12111yescharacteristic
182C2×C4×C53#12311yescharacteristic
183C22⋊D53—42no—
184C22⋊D53—42no—
185C22⋊D53—42no—
186C22⋊D53—42no—
187C2×Dic53—22no—
188C2×Dic53—22no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—81no—
197C2×Dic53—81no—
198C2×Dic53—81no—
199D212—42no—
200D212—81no—
201C4×D53—84no—
202C8×C53—81no—
203#88322—42no—
204#9275—531nomaximal, Sylow
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53#38611yescharacteristic
216C22×C53C4×SD1611yescharacteristic
217C22×C53#44011yescharacteristic
218C22×C53#45011yescharacteristic
219C22×C53#45211yescharacteristic
220C22×C53#46111yescharacteristic
221C22×C53#46311yescharacteristic
222C2×D53—42no—
223C2×D53—42no—
224C2×D53—42no—
225C2×D53—81no—
226C2×D53—81no—
227C2×D53—81no—
228C2×D53—81no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—42no—
234C4×C53—81no—
235C4×C53—81no—
236C4×C53—81no—
237Dic53—42no—
238Dic53—42no—
239Dic53—81no—
240Dic53—161no—
241#3187—531no—
242#2364—531no—
243#1996—531no—
244#1519—531no—
245C2×#328D5311yescharacteristic, complemented
246#1294—531no—
247#1282—531no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53—21no—
253C2×C53—21no—
254C2×C53#187711yescharacteristic
255C2×C53#190611yescharacteristic
256C2×C53#191311yescharacteristic
257D53—82no—
258C2×C22⋊Q8C2×D5311yescharacteristic
259C2×#121—531no—
260C2×C4×D4—1061no—
261#483—1061no—
262#481—532no—
263#465—1062no—
264#450—1061no—
265#448—532no—
266C2×C22⋊C8—531no—
267C2×C22⋊C8C2×D5311yescharacteristic
268#388—1061no—
269#387—1061no—
270#386—1061no—
271#386—1061no—
272#380—531no—
273C4×C22⋊C4—531no—
274C4×C22⋊C4—1061no—
275C2×#95C2×D5311yescharacteristic
276#342—1061no—
277#342—1061no—
278#328—21no—
279#328—21no—
280#327—1062no—
281C53#927511yescharacteristic, complemented, Sylow
282C22×Q8C4×D5311yescharacteristic, complemented
283C22×D4—531no—
284C22×D4—1061no—
285C23×C4—21no—
286C23×C4—1061no—
287C23×C4C22×D5311yescharacteristic
288C2×SD16—1062no—
289C22×C8—21no—
290C22×C8—1061no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295#121—532no—
296#121—532no—
297#121—1061no—
298#121—1061no—
299C4×D4—2122no—
300C4×D4—2122no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—531no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4C4×D5311yescharacteristic, complemented
306C2×C22⋊C4—21no—
307C2×C22⋊C4—531no—
308C2×C22⋊C4—531no—
309C2×C22⋊C4—1061no—
310C2×C22⋊C4—1061no—
311C2×C22⋊C4—1061no—
312C2×C22⋊C4—1061no—
313C2×C22⋊C4—1061no—
314C2×C42—1061no—
315C2×C42—1061no—
316#106—1062no—
317#103—1062no—
318#102—1062no—
319C22⋊C8—21no—
320C22⋊C8—21no—
321C22⋊C8—532no—
322C22⋊C8—1061no—
323#95—21no—
324#95—1061no—
325#95—1061no—
326#95—1061no—
327#95—1061no—
328#95C22⋊D5311yescharacteristic
329#95C22⋊D5311yescharacteristic
330C24—1061no—
331C24D4×D5311yescharacteristic
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—81no—
335C2×D4—532no—
336C2×D4—1062no—
337C2×D4—1062no—
338C2×D4—1061no—
339C2×D4—4241no—
340C2×D4—2121no—
341C2×D4—2121no—
342C2×D4—2121no—
343C2×D4—2121no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—1061no—
354C22×C4—1061no—
355C22×C4—1061no—
356C22×C4—4241no—
357C22×C4—2122no—
358C22×C4—2122no—
359C22×C4—2121no—
360C22×C4C2×C4×D5311yescharacteristic
361C22×C4#1171994511yescharacteristic
362C22×C4D4×D5311yescharacteristic
363C22×C4D4×D5311yescharacteristic
364C22×C4#1171994711yescharacteristic
365C22×C4C2×C22⋊D5311yescharacteristic
366SD16—4242no—
367C2×C8—41no—
368C2×C8—81no—
369C2×C8—1062no—
370C2×C8—4241no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—532no—
376C4⋊C4—1061no—
377C4⋊C4—4241no—
378C4⋊C4—2122no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—532no—
382C22⋊C4—1061no—
383C22⋊C4—1062no—
384C22⋊C4—1062no—
385C22⋊C4—1061no—
386C22⋊C4—1061no—
387C22⋊C4—4241no—
388C22⋊C4—2122no—
389C22⋊C4—2122no—
390C22⋊C4—2121no—
391C22⋊C4—2121no—
392C42—4241no—
393C42—2122no—
394C23—41no—
395C23—41no—
396C23—1062no—
397C23—2121no—
398C23—2121no—
399C23—2121no—
400C23—2121no—
401C23—2121no—
402C23—2121no—
403C23#1313317711yescharacteristic
404C23#1313318511yescharacteristic
405C23#1313319011yescharacteristic
406C23SD16×D5311yescharacteristic
407C23#1313324011yescharacteristic
408C23#1313324411yescharacteristic
409C23#1313324911yescharacteristic
410Q8—42no—
411Q8—81no—
412D4—4241no—
413D4—2122no—
414D4—2122no—
415D4—2122no—
416D4—2122no—
417D4—2122no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—1062no—
437C2×C4—1062no—
438C2×C4—4244no—
439C2×C4—4241no—
440C2×C4—4241no—
441C2×C4—4241no—
442C2×C4—2121no—
443C2×C4—2121no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4—2121no—
448C2×C4—2121no—
449C2×C4C22⋊C4×D5311yescharacteristic
450C2×C4C4×C22⋊D5311yescharacteristic
451C2×C4#1313318811yescharacteristic
452C2×C4#1313319911yescharacteristic
453C8—81no—
454C8—2122no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—4241no—
466C22—4241no—
467C22—4241no—
468C22—4241no—
469C22—2122no—
470C22—2122no—
471C22—2122no—
472C22#1681049911yescharacteristic
473C22#1681070811yescharacteristic
474C22#1681072611yescharacteristic
475C22#1681077211yescharacteristic
476C22#1681078411yescharacteristic
477C22#1681082711yescharacteristic
478C22#1681084111yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—81no—
485C4—81no—
486C4—81no—
487C4—8481no—
488C4—4241no—
489C4—2122no—
490C4—2122no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2—4242no—
498C2#2027504611yescharacteristic
499C2#2027521911yescharacteristic
500C2#2027525411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#328C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenterC2×#328D53
hypocenterC53#9275
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
3. #20275701C2semidirect
5. #20272594C2semidirect
6. #20272562C2semidirect
7. C2×C53×#328C2semidirect
11. #16810941C4semidirect
14. #16810762C4semidirect
58. C22⋊#88322C2×C4semidirect
63. C22×Q8×C53C2×C4semidirect
72. C2×C4⋊C4×C53C2×C4semidirect
128. C22⋊Dic53SD16semidirect
245. C2×#328D53semidirect
281. C53#9275semidirect
282. C22×Q8C4×D53semidirect
305. C2×C4⋊C4C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202854682
#202777622
#202757012
#202732842
#202725942
#202725622
C2×C53×#3282
#927553