Finite Group Databasebeta

← #28317228

Subgroups — #28317228

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11719940C211yesmaximal, characteristic
2#20291510C211yesmaximal, characteristic
3#20290355C211yesmaximal, characteristic, complemented
4#20290353C211yesmaximal, characteristic, complemented
5#20290329C211yesmaximal, characteristic, complemented
6#20290328C211yesmaximal, characteristic, complemented
7#20290328C211yesmaximal, characteristic, complemented
8#20290326C211yesmaximal, characteristic, complemented
9#20290302C211yesmaximal, characteristic, complemented
10Q16×#88324C211yesmaximal, characteristic, complemented
11#20290284C211yesmaximal, characteristic
12#20290063C211yesmaximal, characteristic
13C2×Q8⋊#88324C211yesmaximal, characteristic
14#20289916C211yesmaximal, characteristic
15#20284749C211yesmaximal, characteristic
16#16811675—21no—
17#16811674—21no—
18#16811674—21no—
19#16811674C2211yescharacteristic
20#16811674C2211yescharacteristic
21#16811674C2211yescharacteristic
22#16811673—21no—
23#16811673C2211yescharacteristic
24#16811672C2211yescharacteristic
25Q8×#88324C2211yescharacteristic
26Q8×#88324C2211yescharacteristic
27Q8×#88324C2211yescharacteristic
28#16811629—21no—
29#16811626—21no—
30#16811626C2211yescharacteristic
31C2×C4⋊#88324C2211yescharacteristic
32#16811587—21no—
33C4×#11719940—21no—
34C4×#11719940C2211yescharacteristic
35C2×C4×#88324C2211yescharacteristic
36#16811487C2211yescharacteristic
37#16811487C2211yescharacteristic
38#16811486C2211yescharacteristic
39#16811486C2211yescharacteristic
40#16811473C2211yescharacteristic
41#16811473C2211yescharacteristic
42Q8⋊#88324C2211yescharacteristic
43Q8⋊#88324C2211yescharacteristic
44Q8⋊#88324C2211yescharacteristic
45Q8⋊#88324C2211yescharacteristic
46Q8⋊#88324C2211yescharacteristic
47Q8⋊#88324C2211yescharacteristic
48#88324⋊C8C2211yescharacteristic
49#88324⋊C8C2211yescharacteristic
50C8×#88324C2211yescharacteristic
51C8×#88324C2211yescharacteristic
52#16811175C2211yescharacteristic
53C2×Q8×Dic53C2211yescharacteristic
54#16810959C411yescharacteristic
55#16810956C411yescharacteristic
56#16810916C411yescharacteristic
57#16810916C411yescharacteristic
58#16810831C411yescharacteristic
59#16810831C411yescharacteristic
60#16810779C411yescharacteristic
61Q16×Dic53C2211yescharacteristic
62Q16×Dic53C2211yescharacteristic
63#16810715C2211yescharacteristic
64#16810715C2211yescharacteristic
65#16810673C2211yescharacteristic
66C53×#451C411yescharacteristic
67C2×#11719940—21no—
68C2×#11719940D411yescharacteristic
69C2×#11719940D411yescharacteristic
70C22×#88324—21no—
71C22×#88324—21no—
72#13133284—21no—
73#13133284—21no—
74#13133284—21no—
75#13133284—21no—
76#13133284—21no—
77C4⋊#88324—21no—
78C4⋊#88324—21no—
79C4⋊#88324—21no—
80C4⋊#88324—21no—
81C4⋊#88324—21no—
82C4⋊#88324C2311yescharacteristic
83C4⋊#88324C2311yescharacteristic
84C4⋊#88324C2311yescharacteristic
85C4⋊#88324C2311yescharacteristic
86#13133272—21no—
87#13133272—21no—
88#13133272—21no—
89#13133272—21no—
90#13133272D411yescharacteristic
91#13133272D411yescharacteristic
92C4×#88324—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×#11719925C2×C411yescharacteristic
103#13133216C2×C411yescharacteristic
104#13133215C2×C411yescharacteristic
105#13133207—21no—
106#13133196C2×C411yescharacteristic
107#13133194—21no—
108Q8×Dic53—21no—
109Q8×Dic53—21no—
110Q8×Dic53C2×C411yescharacteristic
111Q8×Dic53C2×C411yescharacteristic
112Q8×Dic53C2311yescharacteristic
113Q8×Dic53C2311yescharacteristic
114Q8×Dic53C2311yescharacteristic
115#13133182—21no—
116#13133182C2311yescharacteristic
117#13133173—21no—
118C2×C4⋊Dic53—21no—
119C2×C4⋊Dic53C2×C411yescharacteristic
120C2×C4×Dic53—21no—
121C2×C4×Dic53C2311yescharacteristic
122#13133151C2×C411yescharacteristic
123#13133151C2×C411yescharacteristic
124#13133150C2×C411yescharacteristic
125#13133150C2×C411yescharacteristic
126#13133146C2×C411yescharacteristic
127#13133146C2×C411yescharacteristic
128#13133142C2×C411yescharacteristic
129#13133142C2×C411yescharacteristic
130#13133141C2×C411yescharacteristic
131#13133141C2×C411yescharacteristic
132#13133141C2×C411yescharacteristic
133#13133141C2×C411yescharacteristic
134C22⋊#88322C2×C411yescharacteristic
135C8×Dic53C2311yescharacteristic
136C8×Dic53C2311yescharacteristic
137C22×Q8×C53C2×C411yescharacteristic
138C2×Q16×C53C2×C411yescharacteristic
139C2×Q16×C53C2×C411yescharacteristic
140C22⋊Q8×C53C2×C411yescharacteristic
141#103×C53C2×C411yescharacteristic
142#103×C53C2×C411yescharacteristic
143C22⋊C8×C53C2×C411yescharacteristic
144C2×Q8×C53—21no—
145C2×Q8×C53—22no—
146C2×Q8×C53—41no—
147C2×Q8×C53C22×C411yescharacteristic
148C2×Q8×C53C22×C411yescharacteristic
149C2×Q8×C53C22×C411yescharacteristic
150C22×C4×C53—21no—
151C22×C4×C53C22×C411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53#4111yescharacteristic
155C2×Dic106—21no—
156C2×Dic106C22×C411yescharacteristic
157#11719940—22no—
158#11719940—42no—
159#11719940Q1614yescomplemented
160C2×#88324—21no—
161C2×#88324—21no—
162C2×#88324—21no—
163C2×#88324—21no—
164C2×#88324—21no—
165C2×#88324—21no—
166C2×#88324—42no—
167C2×#88324—42no—
168C2×#88324C2×D411yescharacteristic
169C2×#88324C2×D411yescharacteristic
170C2×#88324C2×D411yescharacteristic
171C2×#88324C2×D411yescharacteristic
172Q16×C53—22no—
173Q16×C53—41no—
174C2×C8×C53C22×C411yescharacteristic
175C2×C8×C53C22×C411yescharacteristic
176C4⋊C4×C53—21no—
177C4⋊C4×C53C22×C411yescharacteristic
178C22⋊C4×C53—21no—
179C22⋊Dic53—21no—
180#11719925—22no—
181#11719925—41no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53C22×C411yescharacteristic
189C4⋊Dic53C22×C411yescharacteristic
190C4⋊Dic53C22×C411yescharacteristic
191#11719919—21no—
192#11719919—21no—
193#11719919—21no—
194#11719919—21no—
195#11719919—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—21no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—21no—
202C4×Dic53C22×C411yescharacteristic
203C4×Dic53C2×D411yescharacteristic
204C4×Dic53#4111yescharacteristic
205C4×Dic53C2411yescharacteristic
206C2×#88322C22×C411yescharacteristic
207C2×#88322C22×C411yescharacteristic
208C23×C53C4×D411yescharacteristic
209Q8×C53—21no—
210Q8×C53—21no—
211Q8×C53—21no—
212Q8×C53—21no—
213Q8×C53—21no—
214Q8×C53—42no—
215Q8×C53#13112yes—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—42no—
223C2×C4×C53C4×D411yescharacteristic
224C2×C4×C53C23×C411yescharacteristic
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—41no—
230C2×Dic53—81no—
231C2×Dic53C4×D411yescharacteristic
232C2×Dic53C4×D411yescharacteristic
233C2×Dic53C2×Q1612yes—
234C2×Dic53#13512yes—
235C2×Dic53C22×D411yescharacteristic
236C2×Dic53C2×#4111yescharacteristic
237Dic106—21no—
238Dic106—21no—
239Dic106—81no—
240#88324—22no—
241#88324—21no—
242#88324—21no—
243#88324—41no—
244#88324—41no—
245#88324—41no—
246#88324—41no—
247#88324—41no—
248#88324C2×Q1612yes—
249#88324C2×Q1612yes—
250C8×C53—21no—
251C8×C53—21no—
252#88322—21no—
253#88322—21no—
254#17998—531nomaximal, Sylow
255C22×C53—21no—
256C22×C53—21no—
257C22×C53C4×Q1612yes—
258C22×C53C2×C4×D411yescharacteristic
259C4×C53—21no—
260C4×C53—22no—
261C4×C53—41no—
262C4×C53—41no—
263C4×C53—41no—
264C4×C53C2×#13111yescharacteristic
265C4×C53#56811yescharacteristic
266Dic53—21no—
267Dic53—21no—
268Dic53—41no—
269Dic53—41no—
270Dic53—41no—
271Dic53—41no—
272Dic53C4×Q1612yes—
273Dic53C22×Q1611yescharacteristic
274Dic53C2×#13511yescharacteristic
275#3104—531no—
276Q8×M16—531no—
277#2962—531no—
278#1591—531no—
279#1591—531no—
280#1584—531no—
281#1584—531no—
282#1584—531no—
283#1584—531no—
284C8×Q16—531no—
285C8×Q16—531no—
286#1571—531no—
287#1493—531no—
288C2×Q8⋊C8—531no—
289C8×M16—531no—
290C2×C53—22no—
291C2×C53#293311yescharacteristic
292C2×C53C2×C4×Q1611yescharacteristic
293C2×C53#295611yescharacteristic
294#533—531no—
295C2×C4×Q8—531no—
296#484—531no—
297#484—531no—
298#477—531no—
299#477—531no—
300#462—531no—
301#462—531no—
302#451—531no—
303#451#615711yescharacteristic
304#446—531no—
305#446—531no—
306#446—531no—
307#446—531no—
308#446—1061no—
309#446—1061no—
310#446—1061no—
311#446—1061no—
312C8×Q8—531no—
313C8×Q8—531no—
314C8×Q8—531no—
315C8×Q8—531no—
316C4×Q16—531no—
317C4×Q16—531no—
318C4×Q16—531no—
319C4×Q16—531no—
320#431—531no—
321#424—1061no—
322#423—531no—
323#423—1061no—
324C2×C4⋊C8—531no—
325#405—1061no—
326C4×M16—531no—
327C4×M16—1061no—
328C2×C4×C8—531no—
329#335—531no—
330#335—531no—
331#335—531no—
332#335—531no—
333Q8⋊C8—531no—
334Q8⋊C8—531no—
335Q8⋊C8—531no—
336Q8⋊C8—531no—
337Q8⋊C8—531no—
338Q8⋊C8—531no—
339Q8⋊C8—531no—
340Q8⋊C8—531no—
341#322—531no—
342#322—531no—
343C82—531no—
344C82—531no—
345C53#1799811yescharacteristic, complemented, Sylow
346C22×Q8C2×#615711yescharacteristic
347C2×Q16—531no—
348C2×Q16—531no—
349C2×Q16C2×#615711yescharacteristic
350C2×Q16C2×#615711yescharacteristic
351C2×M16—531no—
352C2×M16—531no—
353C2×M16—1061no—
354C22×C8—1061no—
355C22×C8—1061no—
356C4⋊Q8—531no—
357C4⋊Q8—531no—
358#125—1061no—
359C22⋊Q8—531no—
360C22⋊Q8—1061no—
361C22⋊Q8C2×#615711yescharacteristic
362C4×Q8—531no—
363C4×Q8—531no—
364C4×Q8—531no—
365C4×Q8—531no—
366C4×Q8—531no—
367C4×Q8—531no—
368C4×Q8—1061no—
369C4×Q8—1061no—
370C4×Q8—1061no—
371#117—1061no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—1061no—
374C2×C42—531no—
375C2×C42—1061no—
376#107—531no—
377#107—531no—
378#107—531no—
379#107—531no—
380C4⋊C8—531no—
381C4⋊C8—531no—
382C4⋊C8—531no—
383C4⋊C8—531no—
384C4⋊C8—531no—
385C4⋊C8—531no—
386C4⋊C8—1061no—
387C4⋊C8—1061no—
388C4⋊C8—1061no—
389C4⋊C8—1061no—
390C4⋊C8—1061no—
391C4⋊C8—1061no—
392C4⋊C8—1061no—
393C4⋊C8—1061no—
394C4⋊C8—1061no—
395C4⋊C8—1061no—
396#103—531no—
397#103—531no—
398#103—531no—
399#103—531no—
400#103—531no—
401#103—531no—
402#103C2×#615711yescharacteristic
403#103C2×#615711yescharacteristic
404C22⋊C8—531no—
405C22⋊C8C2×#615711yescharacteristic
406#97—531no—
407#97—531no—
408#97—1061no—
409#97—1061no—
410#97—1061no—
411#97—1061no—
412C4×C8—531no—
413C4×C8—531no—
414C4×C8—531no—
415C4×C8—531no—
416C4×C8—531no—
417C4×C8—531no—
418C4×C8—1061no—
419C4×C8—1061no—
420C4×C8—1061no—
421C4×C8—1061no—
422C4×C8—1061no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—41no—
426C2×Q8—531no—
427C2×Q8—1061no—
428C2×Q8C22×#615711yescharacteristic
429C2×Q8C22×#615711yescharacteristic
430C2×Q8C22×#615711yescharacteristic
431C22×C4—21no—
432C22×C4—531no—
433C22×C4—531no—
434C22×C4—1061no—
435C22×C4C22×#615711yescharacteristic
436Q16—22no—
437Q16—41no—
438Q16—1062no—
439Q16—2121no—
440M16—534no—
441M16—1062no—
442M16—2122no—
443C2×C8—531no—
444C2×C8—531no—
445C2×C8—531no—
446C2×C8—531no—
447C2×C8—531no—
448C2×C8—531no—
449C2×C8—1061no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1061no—
453C2×C8—1061no—
454C2×C8—1061no—
455C2×C8—2122no—
456C2×C8—2122no—
457C2×C8C22×#615711yescharacteristic
458C2×C8C22×#615711yescharacteristic
459C4⋊C4—21no—
460C4⋊C4—531no—
461C4⋊C4—531no—
462C4⋊C4—531no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
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⋊C4—1061no—
474C4⋊C4C22×#615711yescharacteristic
475C22⋊C4—21no—
476C22⋊C4—1061no—
477C42—531no—
478C42—531no—
479C42—531no—
480C42—531no—
481C42—1061no—
482C42—1061no—
483C42—1061no—
484C42—1061no—
485C42—1061no—
486C42—1061no—
487C23D4×#615711yescharacteristic
488Q8—21no—
489Q8—21no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—42no—
494Q8—1061no—
495Q8—1061no—
496Q8—4241no—
497Q8#1313330412yes—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—42no—
505C2×C4—532no—
506C2×C4—532no—
507C2×C4—531no—
508C2×C4—531no—
509C2×C4—531no—
510C2×C4—531no—
511C2×C4—1061no—
512C2×C4—1061no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—4241no—
516C2×C4—2121no—
517C2×C4D4×#615711yescharacteristic
518C2×C4C23×#615711yescharacteristic
519C8—21no—
520C8—21no—
521C8—532no—
522C8—532no—
523C8—1061no—
524C8—1061no—
525C8—1062no—
526C8—1061no—
527C8—1061no—
528C8—2121no—
529C8—2121no—
530C8—2121no—
531C8—2121no—
532C8—2121no—
533C22—21no—
534C22—21no—
535C22Q16×#615712yes—
536C22C2×D4×#615711yescharacteristic
537C4—21no—
538C4—22no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—531no—
543C4—531no—
544C4—532no—
545C4—1061no—
546C4—1061no—
547C4—2121no—
548C4—2121no—
549C4—2121no—
550C4—2121no—
551C4C2×#1313330411yescharacteristic
552C4#1681170111yescharacteristic
553C2—22no—
554C2#2029144711yescharacteristic
555C2C2×Q16×#615711yescharacteristic
556C2#2029149011yescharacteristic
subgroupG/H
centerC22C2×D4×#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC53×#451C4
socleC22×C53C2×C4×D4
prime radicalC23×C53C4×D4
cosocleC4×Dic53C24
hypercenter#451#6157
hypocenterC53#17998
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #20290355C2semidirect
4. #20290353C2semidirect
5. #20290329C2semidirect
6. #20290328C2semidirect
7. #20290328C2semidirect
8. #20290326C2semidirect
9. #20290302C2semidirect
10. Q16×#88324C2semidirect
159. #11719940Q16semidirect
345. C53#17998semidirect

Maximal subgroups

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

subgroupindex
Q8×#117199402
#202915102
#202903552
#202903532
#202903292
#202903282
#202903262
#202903022
Q16×#883242
#202902842
#202900632
C2×Q8⋊#883242
#202899162
#202847492
#1799853