Finite Group Databasebeta

← #27286395

Subgroups — #27286395

Order 13568.

744 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810823C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic
3#20275520C211yesmaximal, characteristic, complemented
4C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272598C211yesmaximal, characteristic, complemented
7C53×#2538C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C22×#11719922—21no—
10C2×#13133214C2211yescharacteristic
11#16810823—22no—
12#16810823—22no—
13#16810818—22no—
14#16810818—22no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133137—21no—
18C2×#13133137—21no—
19C2×#13133137—22no—
20C2×C4×#88322C2211yescharacteristic
21Q8⋊#88322—22no—
22D4⋊#88322—22no—
23C2×C22⋊Q8×C53—21no—
24C2×C4×Q8×C53C2211yescharacteristic
25C2×C4×D4×C53C2211yescharacteristic
26C53×#396—22no—
27C53×#390—21no—
28C53×#389—21no—
29C53×#384C2211yescharacteristic
30C53×#380—21no—
31C22×D212—21no—
32C22×D212—21no—
33C22×D212—21no—
34C2×#11719924—24no—
35C2×#11719924—42no—
36C2×#11719924—42no—
37C2×#11719922—24no—
38C2×#11719922—42no—
39C2×#11719922—42no—
40C22×#88322—21no—
41C22×#88322—21no—
42C22×#88322—21no—
43#13133214—21no—
44#13133214—21no—
45#13133214—21no—
46#13133214D412yes—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322D412yes—
50C4⋊#88322D412yes—
51#13133137—24no—
52#13133137—24no—
53#13133137—44no—
54C4×#88322—21no—
55C4×#88322—21no—
56C22×Q8×C53D411yescharacteristic
57C22×D4×C53D411yescharacteristic
58C23×C4×C53—21no—
59C22⋊Q8×C53—42no—
60C22⋊Q8×C53—42no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53D411yescharacteristic
74C2×C4⋊C4×C53D411yescharacteristic
75C2×C4⋊C4×C53D412yes—
76C2×C22⋊C4×C53—22no—
77C2×C22⋊C4×C53—22no—
78C2×C22⋊C4×C53—21no—
79C2×C42×C53—21no—
80C2×C42×C53C2311yescharacteristic
81#95×C53—22no—
82C24×C53—21no—
83C23×D53—41no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C2×Q8×C53—21no—
89C2×Q8×C53D812yescomplemented
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53SD1612yescomplemented
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—21no—
99C22×C4×C53—22no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C2×D212—22no—
110C2×D212—22no—
111C2×D212—22no—
112C2×D212—42no—
113C2×D212—42no—
114C2×D212—41no—
115C2×D212—41no—
116C2×D212—41no—
117C2×D212—41no—
118C2×D212—41no—
119C2×D212—41no—
120C2×D212—41no—
121C2×D212—41no—
122C2×D212—41no—
123C2×D212—41no—
124C2×D212—41no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—44no—
130C4⋊C4×C53—42no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C22⋊C4×C53—44no—
138C22⋊C4×C53—41no—
139C22⋊C4×C53—41no—
140C22⋊C4×C53—82no—
141C42×C53—41no—
142C42×C53—41no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53#4111yescharacteristic
147#11719924—48no—
148#11719922—48no—
149C2×#88322—22no—
150C2×#88322—22no—
151C2×#88322—22no—
152C2×#88322—41no—
153C2×#88322—41no—
154C2×#88322—41no—
155C2×#88322—41no—
156C2×#88322—41no—
157C2×#88322—41no—
158C23×C53—22no—
159C23×C53—41no—
160C23×C53—41no—
161C23×C53—41no—
162C23×C53—41no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53#12011yescharacteristic
166C22×D53—42no—
167C22×D53—81no—
168C22×D53—81no—
169C22×D53—81no—
170C22×D53—81no—
171C22×D53—81no—
172C22×D53—81no—
173Q8×C53—22no—
174Q8×C53—22no—
175Q8×C53—41no—
176Q8×C53—41no—
177D4×C53—22no—
178D4×C53—22no—
179D4×C53—41no—
180D4×C53—41no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—22no—
184C2×C4×C53—41no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—42no—
190C2×C4×C53—41no—
191C2×C4×C53—42no—
192C2×C4×C53—41no—
193C2×C4×C53—42no—
194C2×C4×C53—41no—
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
213D212—44no—
214D212—42no—
215D212—42no—
216D212—42no—
217D212—42no—
218D212—82no—
219D212—81no—
220D212—81no—
221D212—81no—
222D212—81no—
223#88322—42no—
224#88322—42no—
225#88322—42no—
226#10229—531nomaximal, Sylow
227C22×C53—42no—
228C22×C53—42no—
229C22×C53—42no—
230C22×C53—81no—
231C22×C53—81no—
232C22×C53—81no—
233C22×C53—81no—
234C22×C53#39211yescharacteristic
235C22×C53#44711yescharacteristic
236C22×C53#44811yescharacteristic
237C22×C53#44911yescharacteristic
238C22×C53#44911yescharacteristic
239C22×C53#45311yescharacteristic
240C22×C53#45411yescharacteristic
241C2×D53—82no—
242C2×D53—82no—
243C2×D53—82no—
244C2×D53—161no—
245C2×D53—161no—
246C2×D53—161no—
247C2×D53—161no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—42no—
251C4×C53—82no—
252C4×C53—81no—
253C4×C53—81no—
254C4×C53#44711yescharacteristic
255C4×C53#44911yescharacteristic
256C4×C53#45011yescharacteristic
257C4×C53#45211yescharacteristic
258C4×C53#46511yescharacteristic
259C4×C53#46611yescharacteristic
260C4×C53#46811yescharacteristic
261C4×C53#46911yescharacteristic
262C2×#460—531no—
263C2×#459—531no—
264#2538D5311yescharacteristic, complemented
265#1957—531no—
266C2×Q8⋊C8—531no—
267C2×D4⋊C8—531no—
268#1296—531no—
269C2×C53—82no—
270C2×C53#162111yescharacteristic
271C2×C53#162211yescharacteristic
272C2×C53#162711yescharacteristic
273C2×C53#162811yescharacteristic
274C2×C53#199911yescharacteristic
275C2×C53#200011yescharacteristic
276C2×C53#200611yescharacteristic
277D53—162no—
278C22×SD16—1061no—
279C22×D8—1061no—
280C2×#127—531no—
281C2×C22⋊Q8—21no—
282C2×C4×Q8C2×D5311yescharacteristic
283C2×C4×D4C2×D5311yescharacteristic
284#460—1062no—
285#460—1062no—
286#459—1062no—
287#459—1062no—
288C2×C4⋊C8—531no—
289C2×C4⋊C8—531no—
290C2×#102—1061no—
291C2×#102—1061no—
292C2×#102—1062no—
293C2×C4×C8—531no—
294#396—22no—
295#390—21no—
296#389—21no—
297#384C2×D5311yescharacteristic
298#380—21no—
299Q8⋊C8—1062no—
300D4⋊C8—1062no—
301C53#1022911yescharacteristic, complemented, Sylow
302C22×Q8C22⋊D5311yescharacteristic
303C22×D4—1061no—
304C22×D4—1061no—
305C22×D4—1061no—
306C22×D4C22⋊D5311yescharacteristic
307C23×C4—21no—
308C2×SD16—1064no—
309C2×SD16—2122no—
310C2×SD16—2122no—
311C2×D8—1064no—
312C2×D8—2122no—
313C2×D8—2122no—
314C22×C8—1061no—
315C22×C8—1061no—
316C22×C8—1061no—
317#127—532no—
318#127—1061no—
319#127—1061no—
320#127—1061no—
321C22⋊Q8—42no—
322C22⋊Q8—42no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×D4—21no—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4C22⋊D5311yescharacteristic
336C2×C4⋊C4C22⋊D5311yescharacteristic
337C2×C4⋊C4C22⋊D5312yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C42—21no—
342C2×C42C22×D5311yescharacteristic
343C4⋊C8—532no—
344C4⋊C8—532no—
345C4⋊C8—1061no—
346C4⋊C8—1061no—
347#102—1064no—
348#102—1064no—
349#102—2124no—
350C4×C8—1061no—
351C4×C8—1061no—
352#95—22no—
353C24—21no—
354C24—2121no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8#1171992212yescomplemented
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—1062no—
367C2×D4—1062no—
368C2×D4—1062no—
369C2×D4—2122no—
370C2×D4—2122no—
371C2×D4—2121no—
372C2×D4—2121no—
373C2×D4—2121no—
374C2×D4—2121no—
375C2×D4—2121no—
376C2×D4—2121no—
377C2×D4—2121no—
378C2×D4—2121no—
379C2×D4—2121no—
380C2×D4—2121no—
381C2×D4—2121no—
382C2×D4#1171992412yescomplemented
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4C2×C22⋊D5311yescharacteristic
394C22×C4C2×C22⋊D5311yescharacteristic
395C22×C4C2×C22⋊D5311yescharacteristic
396SD16—2128no—
397D8—2128no—
398C2×C8—1062no—
399C2×C8—1062no—
400C2×C8—1062no—
401C2×C8—2121no—
402C2×C8—2121no—
403C2×C8—2121no—
404C2×C8—2121no—
405C2×C8—2121no—
406C2×C8—2121no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C22⋊C4—44no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—82no—
423C42—41no—
424C42—41no—
425C42D4×D5311yescharacteristic
426C42D4×D5311yescharacteristic
427C42D4×D5311yescharacteristic
428C42#1171994711yescharacteristic
429C23—22no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—4241no—
437C23—4241no—
438C23—4241no—
439C23—4241no—
440C23—4241no—
441C23—4241no—
442C23—2122no—
443C23#1313318611yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—41no—
447Q8—41no—
448D4—22no—
449D4—22no—
450D4—41no—
451D4—41no—
452D4—4242no—
453D4—4241no—
454D4—4241no—
455D4—4241no—
456D4—4241no—
457D4—2124no—
458D4—2122no—
459D4—2122no—
460D4—2122no—
461D4—2122no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4#1313318511yescharacteristic
481C2×C4#1313318511yescharacteristic
482C2×C4#1313318811yescharacteristic
483C2×C4#1313318911yescharacteristic
484C2×C4#1313318911yescharacteristic
485C2×C4#1313321311yescharacteristic
486C2×C4C2×#1171992211yescharacteristic
487C2×C4C2×#1171992411yescharacteristic
488C2×C4#1313323711yescharacteristic
489C2×C4#1313323711yescharacteristic
490C2×C4#1313324211yescharacteristic
491C2×C4#1313324211yescharacteristic
492C2×C4#1313324311yescharacteristic
493C2×C4#1313324811yescharacteristic
494C8—2122no—
495C8—2122no—
496C8—2122no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22—8481no—
505C22—8481no—
506C22—8481no—
507C22—8481no—
508C22—4242no—
509C22—4242no—
510C22—4242no—
511C22#1681053111yescharacteristic
512C22#1681075711yescharacteristic
513C22#1681076211yescharacteristic
514C22#1681076811yescharacteristic
515C22#1681076811yescharacteristic
516C22#1681079111yescharacteristic
517C22#1681079611yescharacteristic
518C4—21no—
519C4—21no—
520C4—42no—
521C4—81no—
522C4—82no—
523C4—81no—
524C4#1681075611yescharacteristic
525C4#1681076611yescharacteristic
526C4#1681077311yescharacteristic
527C4#1681078311yescharacteristic
528C4#1681085411yescharacteristic
529C4#1681085711yescharacteristic
530C4#1681087111yescharacteristic
531C4#1681087411yescharacteristic
532C2—82no—
533C2—8482no—
534C2#2027378011yescharacteristic
535C2#2027378711yescharacteristic
536C2#2027381811yescharacteristic
537C2#2027382511yescharacteristic
538C2#2027571111yescharacteristic
539C2#2027571611yescharacteristic
540C2#2027575711yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2538C2
socleC23×C53#120
prime radicalC2×#16810818C2
hypercenter#2538D53
hypocenterC53#10229
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2538

Complements

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

NH ≅ G/Ntype
1. C2×#16810823C2semidirect
3. #20275520C2semidirect
4. C2×Q8⋊#88322C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272598C2semidirect
7. C53×#2538C2semidirect
89. C2×Q8×C53D8semidirect
95. C2×D4×C53SD16semidirect
264. #2538D53semidirect
301. C53#10229semidirect
360. C2×Q8#11719922semidirect
382. C2×D4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168108232
C2×#168108182
#202755202
C2×Q8⋊#883222
C2×D4⋊#883222
#202725982
C53×#25382
#1022953