Finite Group Databasebeta

← #27286407

Subgroups — #27286407

Order 13568.

728 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810903C211yesmaximal, characteristic, complemented
2C2×#16810828C211yesmaximal, characteristic, complemented
3#20275675C211yesmaximal, characteristic, complemented
4C2×D4⋊#88322C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272598C211yesmaximal, characteristic, complemented
7C53×#2537C211yesmaximal, characteristic
8C22×#11719923—21no—
9C2×#13133217C2211yescharacteristic
10#16810903—22no—
11#16810903—22no—
12#16810828—22no—
13#16810828—22no—
14C2×#13133150—21no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133140—21no—
18C2×#13133138—21no—
19C2×C4×#88322C2211yescharacteristic
20#16810385—22no—
21D4⋊#88322—22no—
22D4⋊#88322—22no—
23C2×#121×C53—21no—
24C2×#121×C53—21no—
25C2×C4×D4×C53C2211yescharacteristic
26C2×C4×D4×C53C2211yescharacteristic
27C53×#394—22no—
28C53×#384C2211yescharacteristic
29C53×#380—21no—
30C53×#380—21no—
31C22×Dic106—21no—
32C2×#11719923—24no—
33C2×#11719923—42no—
34C2×#11719923—42no—
35C22×#88322—21no—
36C22×#88322—21no—
37C22×#88322—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217—21no—
41#13133217D412yescomplemented
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44#13133150—24no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322D412yescomplemented
48C4⋊#88322D412yescomplemented
49#13133140—24no—
50#13133138—24no—
51C4×#88322—21no—
52C4×#88322—21no—
53C22×D4×C53—22no—
54C22×D4×C53D411yescharacteristic
55C22×D4×C53D411yescharacteristic
56C23×C4×C53—21no—
57C23×C4×C53—21no—
58#121×C53—42no—
59#121×C53—42no—
60#121×C53—42no—
61#121×C53—42no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53D411yescharacteristic
73C2×C4⋊C4×C53D411yescharacteristic
74C2×C4⋊C4×C53D412yes—
75C2×C22⋊C4×C53—22no—
76C2×C22⋊C4×C53—22no—
77C2×C22⋊C4×C53—21no—
78C2×C22⋊C4×C53—21no—
79C2×C42×C53C2311yescharacteristic
80#95×C53—22no—
81C24×C53—21no—
82C24×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—44no—
94C2×D4×C53—44no—
95C2×D4×C53—82no—
96C2×D4×C53D812yes—
97C2×D4×C53SD1612yescomplemented
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53—41no—
107C22×C4×C53—41no—
108C22×C4×C53—41no—
109C22×C4×C53—41no—
110C22×C4×C53—41no—
111C22×C4×C53—41no—
112C22×C4×C53—41no—
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×Dic53—41no—
117C2×Dic106—22no—
118C2×Dic106—41no—
119C2×Dic106—41no—
120C2×Dic106—41no—
121C2×Dic106—41no—
122C2×Dic106—41no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—42no—
128C4⋊C4×C53—41no—
129C4⋊C4×C53—41no—
130C4⋊C4×C53—41no—
131C4⋊C4×C53—41no—
132C22⋊C4×C53—44no—
133C22⋊C4×C53—44no—
134C22⋊C4×C53—41no—
135C22⋊C4×C53—41no—
136C22⋊C4×C53—41no—
137C22⋊C4×C53—41no—
138C42×C53C2×D411yescharacteristic
139C42×C53C2×D411yescharacteristic
140C42×C53C2×D411yescharacteristic
141C42×C53#4111yescharacteristic
142#11719923—48no—
143C4⋊Dic53—22no—
144C4⋊Dic53—22no—
145C4⋊Dic53—41no—
146C4⋊Dic53—41no—
147C2×#88322—22no—
148C2×#88322—22no—
149C2×#88322—22no—
150C2×#88322—41no—
151C2×#88322—41no—
152C2×#88322—41no—
153C2×#88322—41no—
154C2×#88322—41no—
155C2×#88322—41no—
156C23×C53—22no—
157C23×C53—22no—
158C23×C53—41no—
159C23×C53—41no—
160C23×C53—41no—
161C23×C53—41no—
162C23×C53—41no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—41no—
166C23×C53—41no—
167C23×C53—41no—
168C23×C53—41no—
169C23×C53—41no—
170C23×C53#12011yescharacteristic
171D4×C53—22no—
172D4×C53—22no—
173D4×C53—22no—
174D4×C53—22no—
175D4×C53—41no—
176D4×C53—41no—
177D4×C53—41no—
178D4×C53—41no—
179D4×C53—88no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—41no—
185C2×C4×C53—42no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—42no—
189C2×C4×C53—41no—
190C2×C4×C53—42no—
191C2×C4×C53—81no—
192C2×C4×C53—81no—
193C2×C4×C53—81no—
194C2×C4×C53—81no—
195C2×C4×C53—81no—
196C2×C4×C53—81no—
197C2×C4×C53—81no—
198C2×C4×C53—81no—
199C2×C4×C53#12011yescharacteristic
200C2×C4×C53#12011yescharacteristic
201C2×C4×C53#12111yescharacteristic
202C2×C4×C53#12111yescharacteristic
203C2×C4×C53#12111yescharacteristic
204C2×C4×C53#12711yescharacteristic
205C2×C4×C53C2×D811yescharacteristic
206C2×C4×C53C2×SD1611yescharacteristic
207C2×C4×C53#13511yescharacteristic
208C2×C4×C53#13511yescharacteristic
209C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
210C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
211C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
212C2×C4×C53#13711yescharacteristic
213C2×Dic53—42no—
214C2×Dic53—81no—
215C2×Dic53—81no—
216Dic106—42no—
217Dic106—42no—
218Dic106—81no—
219Dic106—81no—
220#88322—42no—
221#88322—42no—
222#88322—42no—
223#10231—531nomaximal, Sylow
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—42no—
227C22×C53—42no—
228C22×C53—42no—
229C22×C53—42no—
230C22×C53—81no—
231C22×C53—81no—
232C22×C53—81no—
233C22×C53—81no—
234C22×C53—81no—
235C22×C53—81no—
236C22×C53—81no—
237C22×C53—81no—
238C22×C53#39211yescharacteristic
239C22×C53#44711yescharacteristic
240C22×C53#44911yescharacteristic
241C22×C53#45011yescharacteristic
242C22×C53#45211yescharacteristic
243C22×C53#45511yescharacteristic
244C22×C53#45611yescharacteristic
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—82no—
248C4×C53—81no—
249C4×C53—81no—
250C4×C53#44711yescharacteristic
251C4×C53#44811yescharacteristic
252C4×C53#44911yescharacteristic
253C4×C53#44911yescharacteristic
254C4×C53#46511yescharacteristic
255C4×C53#46611yescharacteristic
256C4×C53#46811yescharacteristic
257C4×C53#46911yescharacteristic
258Dic53—82no—
259C2×#474—531no—
260C2×#461—531no—
261#2537D5311yescharacteristic
262#1990—531no—
263C2×D4⋊C8—531no—
264C2×D4⋊C8—531no—
265#1296—531no—
266C2×C53—82no—
267C2×C53—82no—
268C2×C53#163911yescharacteristic
269C2×C53#164611yescharacteristic
270C2×C53#164711yescharacteristic
271C2×C53#165411yescharacteristic
272C2×C53#199911yescharacteristic
273C2×C53#200011yescharacteristic
274C2×C53#200711yescharacteristic
275C22×SD16—1061no—
276C2×C4⋊Q8—531no—
277C2×#121—21no—
278C2×#121—21no—
279C2×C4×D4C2×D5311yescharacteristic
280C2×C4×D4C2×D5311yescharacteristic
281#474—1062no—
282#474—1062no—
283#461—1062no—
284#461—1062no—
285C2×#107—1061no—
286C2×C4⋊C8—531no—
287C2×C4⋊C8—531no—
288C2×#103—1061no—
289C2×#102—1061no—
290C2×C4×C8—531no—
291#394—22no—
292#384C2×D5311yescharacteristic
293#380—21no—
294#380—21no—
295#340—1062no—
296D4⋊C8—1062no—
297D4⋊C8—1062no—
298C53#1023111yescharacteristic, complemented, Sylow
299C22×Q8—1061no—
300C22×D4—22no—
301C22×D4C22⋊D5311yescharacteristic
302C22×D4C22⋊D5311yescharacteristic
303C23×C4—21no—
304C23×C4—21no—
305C2×SD16—1064no—
306C2×SD16—2122no—
307C2×SD16—2122no—
308C22×C8—1061no—
309C22×C8—1061no—
310C22×C8—1061no—
311C4⋊Q8—532no—
312C4⋊Q8—1061no—
313C4⋊Q8—1061no—
314C4⋊Q8—1061no—
315#121—42no—
316#121—42no—
317#121—42no—
318#121—42no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—1061no—
330C2×C4⋊C4—1061no—
331C2×C4⋊C4C22⋊D5311yescharacteristic
332C2×C4⋊C4C22⋊D5311yescharacteristic
333C2×C4⋊C4C22⋊D5312yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C42C22×D5311yescharacteristic
339#107—1064no—
340C4⋊C8—532no—
341C4⋊C8—532no—
342C4⋊C8—1061no—
343C4⋊C8—1061no—
344#103—1064no—
345#102—1064no—
346C4×C8—1061no—
347C4×C8—1061no—
348#95—22no—
349C24—21no—
350C24—21no—
351C2×Q8—1062no—
352C2×Q8—2121no—
353C2×Q8—2121no—
354C2×Q8—2121no—
355C2×Q8—2121no—
356C2×Q8—2121no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—82no—
370C2×D4#1171992212yes—
371C2×D4#1171992312yescomplemented
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—2121no—
388C22×C4C2×C22⋊D5311yescharacteristic
389C22×C4C2×C22⋊D5311yescharacteristic
390C22×C4C2×C22⋊D5311yescharacteristic
391SD16—2128no—
392C2×C8—1062no—
393C2×C8—1062no—
394C2×C8—1062no—
395C2×C8—2121no—
396C2×C8—2121no—
397C2×C8—2121no—
398C2×C8—2121no—
399C2×C8—2121no—
400C2×C8—2121no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—1062no—
411C4⋊C4—1062no—
412C4⋊C4—2121no—
413C4⋊C4—2121no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C42D4×D5311yescharacteristic
421C42D4×D5311yescharacteristic
422C42D4×D5311yescharacteristic
423C42#1171994711yescharacteristic
424C23—22no—
425C23—22no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23#1313318611yescharacteristic
439Q8—4241no—
440Q8—4241no—
441Q8—2122no—
442Q8—2122no—
443D4—22no—
444D4—22no—
445D4—22no—
446D4—22no—
447D4—41no—
448D4—41no—
449D4—41no—
450D4—41no—
451D4—88no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—4241no—
472C2×C4—4241no—
473C2×C4—2122no—
474C2×C4#1313318511yescharacteristic
475C2×C4#1313318511yescharacteristic
476C2×C4#1313318811yescharacteristic
477C2×C4#1313318911yescharacteristic
478C2×C4#1313318911yescharacteristic
479C2×C4#1313321311yescharacteristic
480C2×C4C2×#1171992211yescharacteristic
481C2×C4C2×#1171992311yescharacteristic
482C2×C4#1313323711yescharacteristic
483C2×C4#1313323711yescharacteristic
484C2×C4#1313324211yescharacteristic
485C2×C4#1313324311yescharacteristic
486C2×C4#1313324311yescharacteristic
487C2×C4#1313324711yescharacteristic
488C8—2122no—
489C8—2122no—
490C8—2122no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22#1681053111yescharacteristic
506C22#1681075711yescharacteristic
507C22#1681076811yescharacteristic
508C22#1681077411yescharacteristic
509C22#1681078511yescharacteristic
510C22#1681080111yescharacteristic
511C22#1681080611yescharacteristic
512C4—21no—
513C4—21no—
514C4—81no—
515C4—82no—
516C4—81no—
517C4—4242no—
518C4#1681075611yescharacteristic
519C4#1681076111yescharacteristic
520C4#1681076611yescharacteristic
521C4#1681076611yescharacteristic
522C4#1681085011yescharacteristic
523C4#1681085711yescharacteristic
524C4#1681086711yescharacteristic
525C4#1681087411yescharacteristic
526C2—82no—
527C2—82no—
528C2#2027389411yescharacteristic
529C2#2027394311yescharacteristic
530C2#2027395011yescharacteristic
531C2#2027399911yescharacteristic
532C2#2027571111yescharacteristic
533C2#2027571811yescharacteristic
534C2#2027576211yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#2537C2
socleC23×C53#120
hypercenter#2537D53
hypocenterC53#10231
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#16810903C2semidirect
2. C2×#16810828C2semidirect
3. #20275675C2semidirect
4. C2×D4⋊#88322C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272598C2semidirect
41. #13133217D4semidirect
47. C4⋊#88322D4semidirect
48. C4⋊#88322D4semidirect
97. C2×D4×C53SD16semidirect
298. C53#10231semidirect
371. C2×D4#11719923semidirect

Maximal subgroups

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

subgroupindex
C2×#168109032
C2×#168108282
#202756752
C2×D4⋊#883222
#202725982
C53×#25372
#1023153