Finite Group Databasebeta

← #28316792

Subgroups — #28316792

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291630C211yesmaximal, characteristic
2#20291619C211yesmaximal, characteristic, complemented
3#20290340C211yesmaximal, characteristic, complemented
4#20290330C211yesmaximal, characteristic, complemented
5#20290326C211yesmaximal, characteristic, complemented
6#20290320C211yesmaximal, characteristic, complemented
7#20290309C211yesmaximal, characteristic, complemented
8SD16⋊#88324C211yesmaximal, characteristic, complemented
9Q16×#88324C211yesmaximal, characteristic, complemented
10#20290298C211yesmaximal, characteristic, complemented
11#20290290C211yesmaximal, characteristic, complemented
12#20290046C211yesmaximal, characteristic
13#20290023C211yesmaximal, characteristic, complemented
14#20289936C211yesmaximal, characteristic, complemented
15#20284779C211yesmaximal, characteristic
16#16811675—21no—
17#16811674C2211yescharacteristic
18#16811673—21no—
19#16811673C2211yescharacteristic
20#16811672C2211yescharacteristic
21Q8×#88324C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24D4×#88324—21no—
25D4×#88324—21no—
26D4×#88324C2211yescharacteristic
27D4×#88324C2211yescharacteristic
28#16811647—21no—
29#16811647—21no—
30#16811644C2211yescharacteristic
31#16811640C2211yescharacteristic
32#16811639—21no—
33#16811639C2211yescharacteristic
34C2×C4⋊#88324—21no—
35C2×C4⋊#88324C2211yescharacteristic
36#16811486C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811485C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811175C2211yescharacteristic
53#41×Dic53C2211yescharacteristic
54#16810965C411yescharacteristic
55#16810961C411yescharacteristic
56#16810906C411yescharacteristic
57#16810901C411yescharacteristic
58#16810831C411yescharacteristic
59#16810826C411yescharacteristic
60#16810785C411yescharacteristic
61Q16×Dic53C2211yescharacteristic
62#16810715C2211yescharacteristic
63#16810714C2211yescharacteristic
64SD16×Dic53C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#452C411yescharacteristic
67C22×#88324—21no—
68C22×#88324—21no—
69C22×#88324—21no—
70#13133284—21no—
71#13133284—21no—
72#13133284—21no—
73C4⋊#88324—21no—
74C4⋊#88324—21no—
75C4⋊#88324—21no—
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324D411yescharacteristic
79C4⋊#88324D411yescharacteristic
80C4⋊#88324C2311yescharacteristic
81C4⋊#88324C2311yescharacteristic
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84C22⋊#88324—21no—
85C22⋊#88324—21no—
86C22⋊#88324—21no—
87C22⋊#88324—21no—
88C22⋊#88324D411yescharacteristic
89C22⋊#88324D411yescharacteristic
90#13133272—21no—
91#13133272—21no—
92#13133272—21no—
93C4×#88324—21no—
94C4×#88324—21no—
95C4×#88324—21no—
96C4×#88324—21no—
97C4×#88324C2311yescharacteristic
98C4×#88324C2311yescharacteristic
99C4×#88324C2311yescharacteristic
100C4×#88324C2311yescharacteristic
101C2×#11719925C2×C411yescharacteristic
102C2×#11719923C2×C411yescharacteristic
103#13133216C2×C411yescharacteristic
104#13133215C2×C411yescharacteristic
105#13133207—21no—
106#13133196C2×C411yescharacteristic
107#13133194—21no—
108Q8×Dic53C2×C411yescharacteristic
109Q8×Dic53C2311yescharacteristic
110Q8×Dic53C2311yescharacteristic
111#13133182—21no—
112#13133182C2311yescharacteristic
113D4×Dic53—21no—
114D4×Dic53—21no—
115D4×Dic53C2×C411yescharacteristic
116D4×Dic53C2311yescharacteristic
117#13133176—21no—
118#13133176C2×C411yescharacteristic
119#13133173—21no—
120C2×C4×Dic53—21no—
121C2×C4×Dic53C2311yescharacteristic
122#13133151C2×C411yescharacteristic
123#13133150C2×C411yescharacteristic
124#13133149C2×C411yescharacteristic
125#13133148C2×C411yescharacteristic
126#13133146C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133142C2×C411yescharacteristic
129#13133142C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133141C2×C411yescharacteristic
132#13133141C2×C411yescharacteristic
133#13133138C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×#41×C53C2×C411yescharacteristic
138C2×Q16×C53C2×C411yescharacteristic
139C2×SD16×C53C2×C411yescharacteristic
140C22⋊Q8×C53C2×C411yescharacteristic
141#103×C53C2×C411yescharacteristic
142#102×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144#41×C53—22no—
145#41×C53—22no—
146C2×Q8×C53C22×C411yescharacteristic
147C2×Q8×C53C22×C411yescharacteristic
148C2×D4×C53—21no—
149C2×D4×C53C22×C411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53C22×C411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53#4111yescharacteristic
156C2×Dic106—21no—
157C2×Dic106C22×C411yescharacteristic
158C2×#88324—21no—
159C2×#88324—21no—
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—81no—
165C2×#88324—81no—
166C2×#88324—81no—
167C2×#88324C2×D411yescharacteristic
168C2×#88324C2×D411yescharacteristic
169C2×#88324C2×D411yescharacteristic
170C2×#88324C2×D411yescharacteristic
171Q16×C53—22no—
172SD16×C53—22no—
173C2×C8×C53C22×C411yescharacteristic
174C2×C8×C53C22×C411yescharacteristic
175C4⋊C4×C53—21no—
176C4⋊C4×C53C22×C411yescharacteristic
177C22⋊C4×C53—21no—
178C22⋊Dic53—21no—
179C22⋊Dic53—21no—
180C22⋊Dic53—21no—
181C22⋊Dic53—21no—
182#11719925—22no—
183#11719923—22no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53C22×C411yescharacteristic
188C4⋊Dic53C22×C411yescharacteristic
189C4⋊Dic53C22×C411yescharacteristic
190#11719919—21no—
191#11719919—21no—
192#11719919—21no—
193#11719919—21no—
194#11719919—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53C22×C411yescharacteristic
201C4×Dic53C2×D411yescharacteristic
202C4×Dic53#4111yescharacteristic
203C4×Dic53C2411yescharacteristic
204C2×#88322C22×C411yescharacteristic
205C2×#88322C22×C411yescharacteristic
206C23×C53—21no—
207C23×C53C4×D411yescharacteristic
208Q8×C53—21no—
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53#13112yes—
212D4×C53—24no—
213D4×C53—21no—
214D4×C53#13112yes—
215C2×C4×C53—24no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53C4×D411yescharacteristic
222C2×C4×C53C23×C411yescharacteristic
223C2×Dic53—21no—
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—81no—
233C2×Dic53C4×D411yescharacteristic
234C2×Dic53C4×D411yescharacteristic
235C2×Dic53C22×D411yescharacteristic
236C2×Dic53C2×#4111yescharacteristic
237Dic106—21no—
238Dic106—21no—
239Dic106—81no—
240#88324—21no—
241#88324—21no—
242#88324—21no—
243#88324—21no—
244#88324—41no—
245#88324—41no—
246#88324—41no—
247#88324—41no—
248#88324—41no—
249#88324—41no—
250C8×C53—21no—
251C8×C53—21no—
252#88322—21no—
253#88322—21no—
254#17921—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—22no—
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×C4×D411yescharacteristic
262C4×C53—21no—
263C4×C53—22no—
264C4×C53—41no—
265C4×C53—41no—
266C4×C53C2×#13111yescharacteristic
267C4×C53C2×#13111yescharacteristic
268Dic53—21no—
269Dic53—21no—
270Dic53—21no—
271Dic53—41no—
272Dic53—41no—
273Dic53—41no—
274Dic53—41no—
275Dic53#57711yescharacteristic
276Dic53#57811yescharacteristic
277#3107—531no—
278#2988—531no—
279#2985—531no—
280#1587—531no—
281#1585—531no—
282#1584—531no—
283#1582—531no—
284#1579—531no—
285SD16⋊C8—531no—
286C8×Q16—531no—
287C8×SD16—531no—
288#1573—531no—
289#1488—531no—
290#1481—531no—
291#1454—531no—
292C2×C53—41no—
293C2×C53—42no—
294C2×C53#292911yescharacteristic
295C2×C53#294711yescharacteristic
296C2×C53#295711yescharacteristic
297#533—531no—
298C4×#41—531no—
299#485—531no—
300#485—531no—
301#475—531no—
302#474—531no—
303#462—531no—
304#461—531no—
305#452—531no—
306#452#615711yescharacteristic
307#446—531no—
308#446—531no—
309#446—1061no—
310#446—1061no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—531no—
314C8×Q8—531no—
315C4×Q16—531no—
316C4×Q16—531no—
317C4×SD16—531no—
318C4×SD16—531no—
319C8×D4—531no—
320C8×D4—531no—
321C8×D4—1061no—
322C8×D4—1061no—
323#433—1061no—
324#433—1061no—
325#432—531no—
326#431—531no—
327#431—531no—
328#431—531no—
329#431—1061no—
330C2×C4⋊C8—531no—
331C2×C4⋊C8—1061no—
332#335—531no—
333#335—531no—
334#334—531no—
335#334—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341Q8⋊C8—531no—
342D4⋊C8—531no—
343D4⋊C8—531no—
344#322—531no—
345#322—531no—
346#322—531no—
347C82—531no—
348C53#1792111yescharacteristic, complemented, Sylow
349C2×#41C2×#615711yescharacteristic
350C2×Q16—531no—
351C2×Q16C2×#615711yescharacteristic
352C2×SD16—531no—
353C2×SD16C2×#615711yescharacteristic
354C22×C8—1061no—
355C22×C8—1061no—
356C22×C8—1061no—
357C4⋊Q8—531no—
358C4⋊Q8—531no—
359#125—1061no—
360C22⋊Q8—531no—
361C22⋊Q8—1061no—
362C22⋊Q8C2×#615711yescharacteristic
363C4×Q8—531no—
364C4×Q8—531no—
365C4×Q8—531no—
366C4×Q8—531no—
367C4×Q8—1061no—
368C4×D4—531no—
369C4×D4—531no—
370C4×D4—1061no—
371C4×D4—1061no—
372#117—531no—
373#117—1061no—
374#117—1061no—
375C2×C42—531no—
376C2×C42—1061no—
377#107—531no—
378#107—531no—
379#106—531no—
380#106—531no—
381C4⋊C8—531no—
382C4⋊C8—531no—
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—531no—
387C4⋊C8—531no—
388C4⋊C8—531no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396C4⋊C8—1061no—
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103—531no—
402#103C2×#615711yescharacteristic
403#102—531no—
404#102C2×#615711yescharacteristic
405C22⋊C8—531no—
406C22⋊C8—531no—
407C22⋊C8—531no—
408C22⋊C8—1061no—
409C22⋊C8—1061no—
410C22⋊C8—1061no—
411C22⋊C8—1061no—
412C22⋊C8C2×#615711yescharacteristic
413#97—1061no—
414#97—1061no—
415#97—1061no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—531no—
419C4×C8—531no—
420C4×C8—531no—
421C4×C8—531no—
422C4×C8—1061no—
423C4×C8—1061no—
424C4×C8—1061no—
425C4×C8—1061no—
426#41—22no—
427#41—22no—
428C2×Q8—531no—
429C2×Q8—1061no—
430C2×Q8C22×#615711yescharacteristic
431C2×Q8C22×#615711yescharacteristic
432C2×D4—21no—
433C2×D4C22×#615711yescharacteristic
434C22×C4—21no—
435C22×C4—531no—
436C22×C4—531no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4C22×#615711yescharacteristic
440Q16—22no—
441Q16—1062no—
442SD16—22no—
443SD16—1062no—
444C2×C8—531no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—531no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—1061no—
456C2×C8—4241no—
457C2×C8—4241no—
458C2×C8—4241no—
459C2×C8C22×#615711yescharacteristic
460C2×C8C22×#615711yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—531no—
463C4⋊C4—531no—
464C4⋊C4—531no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4C22×#615711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—1061no—
476C22⋊C4—1061no—
477C22⋊C4—1061no—
478C22⋊C4—1061no—
479C42—531no—
480C42—531no—
481C42—531no—
482C42—531no—
483C42—1061no—
484C42—1061no—
485C42—1061no—
486C42—1061no—
487C42—1061no—
488C23—21no—
489C23D4×#615711yescharacteristic
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—1061no—
494Q8—1061no—
495Q8—4241no—
496Q8#1313330412yes—
497D4—24no—
498D4—21no—
499D4#1313330412yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—531no—
507C2×C4—531no—
508C2×C4—531no—
509C2×C4—531no—
510C2×C4—1061no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—4241no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4D4×#615711yescharacteristic
521C2×C4C23×#615711yescharacteristic
522C8—21no—
523C8—21no—
524C8—1061no—
525C8—1061no—
526C8—1061no—
527C8—1061no—
528C8—1061no—
529C8—1061no—
530C8—2121no—
531C8—2121no—
532C8—2121no—
533C8—2121no—
534C8—2121no—
535C8—2121no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22C2×D4×#615711yescharacteristic
543C4—21no—
544C4—22no—
545C4—41no—
546C4—41no—
547C4—531no—
548C4—531no—
549C4—1061no—
550C4—1061no—
551C4—1061no—
552C4—2121no—
553C4—2121no—
554C4—2121no—
555C4—2121no—
556C4C2×#1313330411yescharacteristic
557C4C2×#1313330411yescharacteristic
558C2—41no—
559C2—42no—
560C2#2029143011yescharacteristic
561C2#2029147311yescharacteristic
562C2#2029149211yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#452C4
socleC22×C53C2×C4×D4
prime radicalC2×#41×C53C2×C4
cosocleC4×Dic53C24
hypercenter#452#6157
hypocenterC53#17921
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #20291619C2semidirect
3. #20290340C2semidirect
4. #20290330C2semidirect
5. #20290326C2semidirect
6. #20290320C2semidirect
7. #20290309C2semidirect
8. SD16⋊#88324C2semidirect
9. Q16×#88324C2semidirect
10. #20290298C2semidirect
11. #20290290C2semidirect
13. #20290023C2semidirect
14. #20289936C2semidirect
348. C53#17921semidirect

Maximal subgroups

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

subgroupindex
#202916302
#202916192
#202903402
#202903302
#202903262
#202903202
#202903092
SD16⋊#883242
Q16×#883242
#202902982
#202902902
#202900462
#202900232
#202899362
#202847792
#1792153