Finite Group Databasebeta

← #28317624

Subgroups — #28317624

Order 13568.

624 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291647C211yesmaximal, characteristic
2#20291618C211yesmaximal, characteristic, complemented
3#20290358C211yesmaximal, characteristic, complemented
4#20290351C211yesmaximal, characteristic, complemented
5#20290344C211yesmaximal, characteristic, complemented
6#20290340C211yesmaximal, characteristic, complemented
7#20290328C211yesmaximal, characteristic, complemented
8#20290324C211yesmaximal, characteristic, complemented
9#20290308C211yesmaximal, characteristic, complemented
10Q16×#88324C211yesmaximal, characteristic, complemented
11#20290279C211yesmaximal, characteristic, complemented
12#20290046C211yesmaximal, characteristic
13#20290043C211yesmaximal, characteristic, complemented
14#20289934C211yesmaximal, characteristic, complemented
15#137×Dic53C211yesmaximal, characteristic
16#16811674—21no—
17#16811674—21no—
18#16811674C2211yescharacteristic
19#16811674C2211yescharacteristic
20Q8×#88324C2211yescharacteristic
21Q8×#88324C2211yescharacteristic
22Q8×#88324C2211yescharacteristic
23Q8×#88324C2211yescharacteristic
24#16811658—21no—
25#16811658—21no—
26#16811658C2211yescharacteristic
27#16811658C2211yescharacteristic
28#16811644—21no—
29#16811644C2211yescharacteristic
30#16811640—21no—
31#16811640—21no—
32#16811640C2211yescharacteristic
33#16811640C2211yescharacteristic
34#16811626—21no—
35#16811626C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811486C2211yescharacteristic
38#16811485C2211yescharacteristic
39#16811484C2211yescharacteristic
40Q8⋊#88324C2211yescharacteristic
41Q8⋊#88324C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46D4⋊#88324C2211yescharacteristic
47D4⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50#16811464C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#41×Dic53C2211yescharacteristic
53C2×Q8×Dic53C2211yescharacteristic
54#16810730C411yescharacteristic
55#16810730C411yescharacteristic
56#16810724C411yescharacteristic
57#16810724C411yescharacteristic
58Q16×Dic53C2211yescharacteristic
59Q16×Dic53C2211yescharacteristic
60SD16×Dic53C2211yescharacteristic
61SD16×Dic53C2211yescharacteristic
62#16810665C411yescharacteristic
63#16810632C411yescharacteristic
64#16810624C411yescharacteristic
65M16×Dic53C2211yescharacteristic
66C2×#137×C53C411yescharacteristic
67C2×#11719940—21no—
68C2×#11719940—21no—
69C2×#11719940—21no—
70#13133284—21no—
71#13133284—21no—
72#13133284—21no—
73#13133284—21no—
74#13133284D411yescharacteristic
75#13133284D411yescharacteristic
76C4⋊#88324—21no—
77C4⋊#88324—21no—
78C4⋊#88324—21no—
79C4⋊#88324—21no—
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×#11719917C2×C411yescharacteristic
102Q8×Dic53—21no—
103Q8×Dic53—21no—
104Q8×Dic53C2×C411yescharacteristic
105Q8×Dic53C2×C411yescharacteristic
106Q8×Dic53C2×C411yescharacteristic
107Q8×Dic53C2311yescharacteristic
108Q8×Dic53C2311yescharacteristic
109Q8×Dic53C2311yescharacteristic
110D4×Dic53—21no—
111D4×Dic53—21no—
112D4×Dic53C2×C411yescharacteristic
113D4×Dic53C2311yescharacteristic
114#13133176—21no—
115#13133176C2×C411yescharacteristic
116C2×C4⋊Dic53—21no—
117C2×C4⋊Dic53C2×C411yescharacteristic
118C2×C4×Dic53—21no—
119C2×C4×Dic53—21no—
120C2×C4×Dic53C2311yescharacteristic
121#13133151C2×C411yescharacteristic
122#13133151C2×C411yescharacteristic
123#13133149C2×C411yescharacteristic
124#13133149C2×C411yescharacteristic
125#13133141C2×C411yescharacteristic
126#13133141C2×C411yescharacteristic
127#13133141C2×C411yescharacteristic
128#13133141C2×C411yescharacteristic
129#13133141C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133138C2×C411yescharacteristic
132#13133138C2×C411yescharacteristic
133#13133127C2×C411yescharacteristic
134#13133127C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C2×#41×C53C2×C411yescharacteristic
138C22×Q8×C53C2×C411yescharacteristic
139#137×C53—24no—
140C2×Q16×C53C2×C411yescharacteristic
141C2×Q16×C53C2×C411yescharacteristic
142C2×SD16×C53C2×C411yescharacteristic
143C2×SD16×C53C2×C411yescharacteristic
144C2×M16×C53C2×C411yescharacteristic
145#41×C53—22no—
146#41×C53—41no—
147C2×Q8×C53—21no—
148C2×Q8×C53—22no—
149C2×Q8×C53—41no—
150C2×Q8×C53C22×C411yescharacteristic
151C2×Q8×C53C22×C411yescharacteristic
152C2×Q8×C53C22×C411yescharacteristic
153C2×D4×C53—21no—
154C2×D4×C53C22×C411yescharacteristic
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53C22×C411yescharacteristic
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53#4111yescharacteristic
163#11719940—81no—
164#11719940—81no—
165#11719940—81no—
166C2×#88324—21no—
167C2×#88324—21no—
168C2×#88324—21no—
169C2×#88324—21no—
170C2×#88324—21no—
171C2×#88324—21no—
172C2×#88324C2×D411yescharacteristic
173C2×#88324C2×D411yescharacteristic
174C2×#88324C2×D411yescharacteristic
175C2×#88324C2×D411yescharacteristic
176Q16×C53—22no—
177Q16×C53M1614yescomplemented
178SD16×C53—22no—
179SD16×C53—22no—
180M16×C53—22no—
181C2×C8×C53C22×C411yescharacteristic
182C2×C8×C53C22×C411yescharacteristic
183C22⋊Dic53—21no—
184C22⋊Dic53—21no—
185C22⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53C22×C411yescharacteristic
194C4⋊Dic53C22×C411yescharacteristic
195C4⋊Dic53C22×C411yescharacteristic
196C4⋊Dic53C22×C411yescharacteristic
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53C22×C411yescharacteristic
205C4×Dic53C2×D411yescharacteristic
206C4×Dic53#4111yescharacteristic
207C4×Dic53C2411yescharacteristic
208#11719917—41no—
209C2×#88322C22×C411yescharacteristic
210C2×#88322C22×C411yescharacteristic
211C23×C53—21no—
212C23×C53C4×D411yescharacteristic
213Q8×C53—21no—
214Q8×C53—21no—
215Q8×C53—21no—
216Q8×C53—21no—
217Q8×C53—42no—
218Q8×C53C2×M1612yes—
219Q8×C53C2×M1612yes—
220D4×C53—21no—
221D4×C53—21no—
222D4×C53—42no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C23×C411yescharacteristic
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—41no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53C4×D411yescharacteristic
244C2×Dic53C4×D411yescharacteristic
245C2×Dic53C22×D411yescharacteristic
246C2×Dic53C2×#4111yescharacteristic
247#88324—21no—
248#88324—21no—
249#88324—21no—
250#88324—21no—
251#88324—41no—
252#88324—41no—
253#88324—41no—
254#88324—41no—
255#88324—41no—
256#88324—41no—
257C8×C53—21no—
258C8×C53C2×M1612yes—
259#88322—21no—
260#88322—21no—
261#18053—531nomaximal, Sylow
262C22×C53—21no—
263C22×C53—21no—
264C22×C53—21no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53C2×C4×D411yescharacteristic
269C4×C53—21no—
270C4×C53—22no—
271C4×C53—22no—
272C4×C53—41no—
273C4×C53C22×M1611yescharacteristic
274C4×C53#56811yescharacteristic
275Dic53—21no—
276Dic53—21no—
277Dic53—21no—
278Dic53—41no—
279Dic53—41no—
280Dic53—41no—
281Dic53—41no—
282Dic53#57711yescharacteristic
283Dic53#57811yescharacteristic
284#2991—531no—
285#2984—531no—
286C4×#137—531no—
287#1592—531no—
288#1590—531no—
289#1588—531no—
290#1587—531no—
291#1584—531no—
292#1583—531no—
293#1579—531no—
294C8×Q16—531no—
295#1569—531no—
296#1488—531no—
297#1487—531no—
298#1453—531no—
299C2×C53—41no—
300C2×C53—81no—
301C2×C53D4×M1611yescharacteristic
302C2×C53#294711yescharacteristic
303C2×C53#295511yescharacteristic
304C2×#137#615711yescharacteristic
305C4×#41—531no—
306C2×C4×Q8—531no—
307#446—531no—
308#446—531no—
309#446—1061no—
310#446—1061no—
311C8×Q8—531no—
312C8×Q8—531no—
313C8×Q8—531no—
314C8×Q8—531no—
315#441—531no—
316#441—531no—
317#440—531no—
318#440—531no—
319C4×Q16—531no—
320C4×Q16—531no—
321C4×SD16—531no—
322C4×SD16—531no—
323#436—531no—
324#436—531no—
325#436—1061no—
326#436—1061no—
327#432—531no—
328#432—1061no—
329#431—531no—
330#431—531no—
331#431—1061no—
332#431—1061no—
333#428—531no—
334#423—531no—
335#423—1061no—
336#419—531no—
337#417—531no—
338C4×M16—531no—
339#335—531no—
340#335—531no—
341#334—531no—
342#334—531no—
343Q8⋊C8—531no—
344Q8⋊C8—531no—
345Q8⋊C8—531no—
346Q8⋊C8—531no—
347Q8⋊C8—531no—
348Q8⋊C8—531no—
349D4⋊C8—531no—
350D4⋊C8—531no—
351#322—531no—
352#322—531no—
353#322—531no—
354C82—531no—
355C53#1805311yescharacteristic, complemented, Sylow
356C2×#41C2×#615711yescharacteristic
357C22×Q8C2×#615711yescharacteristic
358#137—24no—
359C2×Q16C2×#615711yescharacteristic
360C2×Q16C2×#615711yescharacteristic
361C2×SD16C2×#615711yescharacteristic
362C2×SD16C2×#615711yescharacteristic
363C2×M16—531no—
364C2×M16—1061no—
365C2×M16—1061no—
366C2×M16—1061no—
367C2×M16C2×#615711yescharacteristic
368C4×Q8—531no—
369C4×Q8—531no—
370C4×Q8—531no—
371C4×Q8—531no—
372C4×Q8—531no—
373C4×Q8—531no—
374C4×Q8—1061no—
375C4×Q8—1061no—
376C4×D4—531no—
377C4×D4—531no—
378C4×D4—1061no—
379C4×D4—1061no—
380#117—531no—
381#117—1061no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—1061no—
384C2×C42—531no—
385C2×C42—1061no—
386C2×C42—1061no—
387#107—531no—
388#107—531no—
389#106—531no—
390#106—531no—
391C4⋊C8—531no—
392C4⋊C8—531no—
393C4⋊C8—531no—
394C4⋊C8—531no—
395C4⋊C8—531no—
396C4⋊C8—531no—
397C4⋊C8—1061no—
398C4⋊C8—1061no—
399C4⋊C8—1061no—
400C4⋊C8—1061no—
401C4⋊C8—1061no—
402C4⋊C8—1061no—
403C4⋊C8—1061no—
404C4⋊C8—1061no—
405#103—531no—
406#103—531no—
407#103—531no—
408#103—531no—
409#103—531no—
410#103—531no—
411#102—531no—
412#102—531no—
413C22⋊C8—531no—
414C22⋊C8—531no—
415C22⋊C8—1061no—
416C22⋊C8—1061no—
417C22⋊C8—1061no—
418C22⋊C8—1061no—
419#97—531no—
420#97—531no—
421#97—1061no—
422#97—1061no—
423#97—1061no—
424C4×C8—531no—
425C4×C8—531no—
426C4×C8—531no—
427C4×C8—531no—
428C4×C8—531no—
429C4×C8—531no—
430C4×C8—1061no—
431C4×C8—1061no—
432C4×C8—1061no—
433C4×C8—1061no—
434#41—22no—
435#41—41no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—41no—
439C2×Q8C22×#615711yescharacteristic
440C2×Q8C22×#615711yescharacteristic
441C2×Q8C22×#615711yescharacteristic
442C2×D4—21no—
443C2×D4C22×#615711yescharacteristic
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4C22×#615711yescharacteristic
452Q16—22no—
453Q16#1171994014yescomplemented
454SD16—22no—
455SD16—22no—
456M16—22no—
457M16—4241no—
458M16—4241no—
459M16—4241no—
460M16—2121no—
461C2×C8—531no—
462C2×C8—531no—
463C2×C8—531no—
464C2×C8—531no—
465C2×C8—531no—
466C2×C8—531no—
467C2×C8—1061no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C2×C8—1061no—
473C2×C8C22×#615711yescharacteristic
474C2×C8C22×#615711yescharacteristic
475C4⋊C4—531no—
476C4⋊C4—531no—
477C4⋊C4—531no—
478C4⋊C4—531no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C22⋊C4—1061no—
487C22⋊C4—1061no—
488C22⋊C4—1061no—
489C42—531no—
490C42—531no—
491C42—531no—
492C42—531no—
493C42—1061no—
494C42—1061no—
495C42—1061no—
496C42—1061no—
497C42—1061no—
498C42—1061no—
499C42—1061no—
500C23—21no—
501C23D4×#615711yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8—21no—
506Q8—42no—
507Q8C2×#1171994012yes—
508Q8C2×#1171994012yes—
509D4—21no—
510D4—21no—
511D4—42no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—531no—
520C2×C4—531no—
521C2×C4—531no—
522C2×C4—531no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—1061no—
528C2×C4—1061no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4D4×#615711yescharacteristic
535C2×C4C23×#615711yescharacteristic
536C8—21no—
537C8—1061no—
538C8—1061no—
539C8—1061no—
540C8—1061no—
541C8—1061no—
542C8—1061no—
543C8—2121no—
544C8—2121no—
545C8—2121no—
546C8—2121no—
547C8—2121no—
548C8—2121no—
549C8C2×#1171994012yes—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22C2×D4×#615711yescharacteristic
557C4—21no—
558C4—22no—
559C4—22no—
560C4—41no—
561C4—531no—
562C4—531no—
563C4—1061no—
564C4—1061no—
565C4—1061no—
566C4—2121no—
567C4—2121no—
568C4—2121no—
569C4—2121no—
570C4C22×#1171994011yescharacteristic
571C4#1681170111yescharacteristic
572C2—41no—
573C2—81no—
574C2D4×#1171994011yescharacteristic
575C2#2029147311yescharacteristic
576C2#2029148811yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×#137×C53C4
socleC22×C53C2×C4×D4
prime radicalC2×#41×C53C2×C4
cosocleC4×Dic53C24
hypercenterC2×#137#6157
hypocenterC53#18053
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. #20291618C2semidirect
3. #20290358C2semidirect
4. #20290351C2semidirect
5. #20290344C2semidirect
6. #20290340C2semidirect
7. #20290328C2semidirect
8. #20290324C2semidirect
9. #20290308C2semidirect
10. Q16×#88324C2semidirect
11. #20290279C2semidirect
13. #20290043C2semidirect
14. #20289934C2semidirect
177. Q16×C53M16semidirect
355. C53#18053semidirect
453. Q16#11719940semidirect

Maximal subgroups

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

subgroupindex
#202916472
#202916182
#202903582
#202903512
#202903442
#202903402
#202903282
#202903242
#202903082
Q16×#883242
#202902792
#202900462
#202900432
#202899342
#137×Dic532
#1805353