Finite Group Databasebeta

← #27386648

Subgroups — #27386648

Order 13568.

1022 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
1#20283232C211yesmaximal, characteristic
2C2×#16810622C211yesmaximal, characteristic
3C2×#16810605C211yesmaximal, characteristic
4#20283134C211yesmaximal, characteristic, complemented
5#20275376C214yesmaximal, complemented
6#20275369C214yesmaximal, complemented
7C2×#16810392C211yesmaximal, characteristic, complemented
8#20274441C211yesmaximal, characteristic, complemented
9C2×C53×#428C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14#16810631C2212yes—
15#16810627C2212yes—
16#16810622—22no—
17#16810622—22no—
18#16810622—21no—
19#16810622—21no—
20#16810617—22no—
21C2×#13133140C2212yes—
22C2×#13133137C2212yes—
23#16810605C418yescomplemented
24C2×#13133135C2212yes—
25C2×#13133133C2212yes—
26#16810579—22no—
27#16810579C2214yes—
28C4×#11719917—22no—
29#16810392C2214yes—
30#16810387C2214yes—
31C22×M16×C53C2211yescharacteristic
32C2×#117×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C53×#428—22no—
35C53×#428C2214yes—
36C2×#107×C53—21no—
37C2×#106×C53—21no—
38C2×#11719945—22no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—22no—
44C2×#11719945—22no—
45C2×#11719945C2×C414yes—
46C22×Dic106C2311yescharacteristic
47C22×D212C2311yescharacteristic
48C22×C22⋊D53—21no—
49C22×C4×D53—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—22no—
53C2×#11719917—22no—
54C2×#11719917—22no—
55C2×#11719917—22no—
56C2×#11719917—22no—
57C2×#11719917C2×C414yes—
58C2×#11719917D412yes—
59C2×#11719917D412yes—
60C2×#11719917C2312yes—
61C2×#11719917C2312yes—
62C22×#88322—21no—
63C22×#88322—21no—
64C22×#88322C2312yes—
65#13133140—22no—
66#13133140—22no—
67#13133140—24no—
68#13133137—22no—
69#13133137—22no—
70#13133137—24no—
71#13133135C2×C418yescomplemented
72#13133133C2×C418yescomplemented
73#13133128—24no—
74C4×#88322—24no—
75C23×C4×C53—21no—
76C23×C4×C53C2311yescharacteristic
77C2×M16×C53—21no—
78C2×M16×C53—22no—
79C2×M16×C53C2×C414yes—
80C2×M16×C53C2312yes—
81C22×C8×C53—21no—
82#117×C53—21no—
83#117×C53—21no—
84#117×C53D412yes—
85#117×C53C2312yes—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53—21no—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53D411yescharacteristic
92C2×C4⋊C4×C53D411yescharacteristic
93C2×C4⋊C4×C53C2312yes—
94C2×C22⋊C4×C53—21no—
95C2×C42×C53—21no—
96#107×C53—24no—
97#106×C53—24no—
98C24×C53#4111yescharacteristic
99C23×D53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—22no—
103C22×C4×C53—41no—
104C22×C4×C53—41no—
105C22×C4×C53—82no—
106C22×C4×C53C22×C412yes—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4112yes—
113C22×C4×C53#4112yes—
114C22×C4×C53#4112yes—
115C22×C4×C53C2411yescharacteristic
116C2×C22⋊D53—24no—
117C2×C22⋊D53—41no—
118C2×C22⋊D53—41no—
119C2×C22⋊D53—42no—
120C22×Dic53—21no—
121#11719945—24no—
122#11719945—24no—
123#11719945—24no—
124#11719945—24no—
125#11719945—42no—
126#11719945—42no—
127#11719945—42no—
128#11719945—42no—
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—21no—
132C2×D212—21no—
133C2×D212—21no—
134C2×D212C22×C412yes—
135C2×C4×D53—24no—
136C2×C4×D53—41no—
137C2×C4×D53—41no—
138C2×C4×D53—42no—
139C2×Dic106—21no—
140C2×Dic106—21no—
141C2×Dic106—21no—
142C2×Dic106—21no—
143C2×Dic106—21no—
144C2×Dic106C22×C412yes—
145M16×C53—24no—
146M16×C53—22no—
147M16×C53C22×C414yes—
148C2×C8×C53—24no—
149C2×C8×C53—22no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—41no—
155C4⋊C4×C53—41no—
156C4⋊C4×C53C2×D412yes—
157C4⋊C4×C53C2×D412yes—
158C22⋊C4×C53—22no—
159C22⋊C4×C53—22no—
160C42×C53—22no—
161C42×C53—22no—
162#11719917—24no—
163#11719917—24no—
164#11719917—24no—
165#11719917—22no—
166#11719917—22no—
167#11719917—22no—
168#11719917—42no—
169#11719917C22×C414yes—
170C2×#88322—24no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—42no—
176C2×#88322C2×D414yes—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53#11712yes—
180C23×C53#12111yescharacteristic
181C23×C53#12111yescharacteristic
182C23×C53#12612yes—
183C23×C53C2×#4111yescharacteristic
184C22×D53—22no—
185C22×D53—41no—
186C22×D53—41no—
187C22×D53—81no—
188C22×D53—81no—
189C2×C4×C53—22no—
190C2×C4×C53—22no—
191C2×C4×C53—42no—
192C2×C4×C53—81no—
193C2×C4×C53—81no—
194C2×C4×C53—82no—
195C2×C4×C53—82no—
196C2×C4×C53—82no—
197C2×C4×C53#11712yes—
198C2×C4×C53C4×D412yes—
199C2×C4×C53C4×D412yes—
200C2×C4×C53C4×D412yes—
201C2×C4×C53C4×D412yes—
202C2×C4×C53#12111yescharacteristic
203C2×C4×C53#12111yescharacteristic
204C2×C4×C53#12312yes—
205C2×C4×C53#12312yes—
206C2×C4×C53#12412yes—
207C2×C4×C53#12412yes—
208C2×C4×C53#12612yes—
209C2×C4×C53C23×C411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C2×#4111yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C22⋊D53—44no—
216C22⋊D53—44no—
217C2×Dic53—22no—
218C2×Dic53—41no—
219C2×Dic53—41no—
220D212—22no—
221D212—22no—
222D212—41no—
223D212—41no—
224C4×D53—44no—
225C4×D53—44no—
226Dic106—22no—
227Dic106—22no—
228Dic106—41no—
229Dic106—41no—
230C8×C53—24no—
231#88322—24no—
232#88322—24no—
233#88322—42no—
234#21076—531nomaximal, Sylow
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53#38812yes—
244C22×C53#38812yes—
245C22×C53C2×#11711yescharacteristic
246C22×C53C2×#12111yescharacteristic
247C22×C53C2×#12611yescharacteristic
248C2×D53—42no—
249C2×D53—82no—
250C2×D53—81no—
251C2×D53—81no—
252C2×D53—82no—
253C4×C53—42no—
254C4×C53—42no—
255C4×C53—82no—
256C4×C53#38812yes—
257C4×C53#38812yes—
258C4×C53C2×C4×D411yescharacteristic
259C4×C53C2×C4×D411yescharacteristic
260C4×C53C2×#12311yescharacteristic
261C4×C53C2×#12411yescharacteristic
262Dic53—42no—
263C2×#428D5311yescharacteristic, complemented
264#2896—531no—
265C2×#417—531no—
266C2×#413—531no—
267#2873—531no—
268#1931—534no—
269#1930—534no—
270C2×#343—531no—
271#1738—531no—
272C2×C53—21no—
273C2×C53—21no—
274C2×C53—22no—
275C2×C53—22no—
276C2×C53C2×#38811yescharacteristic
277C2×C53#306811yescharacteristic
278C2×C53#306911yescharacteristic
279D53—82no—
280C22×#41—531no—
281C22×M16—531no—
282C22×M16—531no—
283C22×M16—531no—
284C22×M16C2×D5311yescharacteristic
285C2×#117C2×D5311yescharacteristic
286C22×C4⋊C4C2×D5311yescharacteristic
287#428—22no—
288#428C2×D5314yes—
289C2×#107—21no—
290C2×#106—21no—
291#419—532no—
292#418—532no—
293#417—1062no—
294#417—1062no—
295#417—1061no—
296#417—1061no—
297#416—1062no—
298C2×#103—532no—
299C2×#102—532no—
300#413—538no—
301C2×#101—532no—
302C2×#100—532no—
303#405—534no—
304#405—1062no—
305C4×M16—1062no—
306#343—534no—
307#341—534no—
308C53#2107611yescharacteristic, complemented, Sylow
309C2×#41—534no—
310C2×#41—1062no—
311C2×#41—1062no—
312C2×#41—1061no—
313C2×#41—1061no—
314C2×#41—1062no—
315C2×#41—1061no—
316C2×#41—1061no—
317C22×Q8—531no—
318C22×D4—531no—
319C22×D4—1061no—
320C23×C4—21no—
321C23×C4—1061no—
322C23×C4C22×D5311yescharacteristic
323C2×M16—22no—
324C2×M16—21no—
325C2×M16—532no—
326C2×M16—532no—
327C2×M16—534no—
328C2×M16—532no—
329C2×M16—532no—
330C2×M16—1062no—
331C2×M16—1061no—
332C2×M16—1062no—
333C2×M16—1062no—
334C2×M16—1061no—
335C2×M16—1062no—
336C2×M16—1062no—
337C2×M16C4×D5314yes—
338C2×M16C22×D5312yes—
339C22×C8—21no—
340C22×C8—532no—
341C22×C8—1061no—
342C22×C8—1061no—
343#117—21no—
344#117—21no—
345#117C22⋊D5312yes—
346#117C22×D5312yes—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4C22⋊D5311yescharacteristic
353C2×C4⋊C4C22⋊D5311yescharacteristic
354C2×C4⋊C4C22×D5312yes—
355C2×C22⋊C4—21no—
356C2×C42—21no—
357#107—24no—
358#106—24no—
359#103—1064no—
360#103—1062no—
361#103—1062no—
362#102—1064no—
363#102—1062no—
364#102—1062no—
365#101—538no—
366#100—538no—
367#97—1064no—
368C4×C8—1064no—
369C24—1061no—
370C24#1171994711yescharacteristic
371#41—1064no—
372#41—1064no—
373#41—1064no—
374#41—1064no—
375#41—2122no—
376#41—2122no—
377#41—2122no—
378#41—2122no—
379C2×Q8—532no—
380C2×Q8—1061no—
381C2×Q8—1061no—
382C2×Q8—1061no—
383C2×Q8—1061no—
384C2×Q8—1061no—
385C2×D4—532no—
386C2×D4—1064no—
387C2×D4—1061no—
388C2×D4—1061no—
389C2×D4—1061no—
390C2×D4—1061no—
391C2×D4—1061no—
392C2×D4—2121no—
393C2×D4—2121no—
394C2×D4—2122no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—82no—
401C22×C4—1061no—
402C22×C4—1064no—
403C22×C4—2121no—
404C22×C4—2121no—
405C22×C4—2122no—
406C22×C4C2×C4×D5312yes—
407C22×C4#1171994512yes—
408C22×C4D4×D5311yescharacteristic
409C22×C4D4×D5311yescharacteristic
410C22×C4#1171994711yescharacteristic
411C22×C4#1171994912yes—
412C22×C4#1171994912yes—
413C22×C4C2×C22⋊D5311yescharacteristic
414C22×C4C2×C22⋊D5311yescharacteristic
415C22×C4C23×D5311yescharacteristic
416M16—24no—
417M16—22no—
418M16—534no—
419M16—1064no—
420M16—1064no—
421M16—1064no—
422M16—1062no—
423M16—1062no—
424M16—1062no—
425M16—2122no—
426M16C2×C4×D5314yes—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—534no—
430C2×C8—1064no—
431C2×C8—1062no—
432C2×C8—1062no—
433C2×C8—1062no—
434C2×C8—1062no—
435C2×C8—2122no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4C2×C22⋊D5312yes—
443C4⋊C4C2×C22⋊D5312yes—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C42—22no—
447C42—22no—
448C23—41no—
449C23—41no—
450C23—1062no—
451C23—4241no—
452C23—4241no—
453C23—2121no—
454C23—2121no—
455C23#1313317512yes—
456C23#1313318911yescharacteristic
457C23#1313318911yescharacteristic
458C23#1313321212yes—
459C23C2×#1171994711yescharacteristic
460Q8—1062no—
461Q8—1062no—
462Q8—2121no—
463Q8—2121no—
464D4—1062no—
465D4—1062no—
466D4—2124no—
467D4—2124no—
468D4—2121no—
469D4—2121no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—1062no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4—2124no—
482C2×C4—2124no—
483C2×C4#1313317512yes—
484C2×C4C4×C22⋊D5312yes—
485C2×C4C4×C22⋊D5312yes—
486C2×C4#1313317912yes—
487C2×C4#1313317912yes—
488C2×C4#1313318911yescharacteristic
489C2×C4#1313318911yescharacteristic
490C2×C4#1313320212yes—
491C2×C4#1313320212yes—
492C2×C4#1313320612yes—
493C2×C4#1313320612yes—
494C2×C4#1313321212yes—
495C2×C4C22×C4×D5311yescharacteristic
496C2×C4C2×D4×D5311yescharacteristic
497C2×C4C22×C22⋊D5311yescharacteristic
498C2×C4C2×#1171994511yescharacteristic
499C2×C4C2×#1171994911yescharacteristic
500C2×C4C2×#1171994911yescharacteristic
501C8—24no—
502C8—1064no—
503C8—1064no—
504C8—2122no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—42no—
512C22—42no—
513C22—4242no—
514C22—4242no—
515C22—4241no—
516C22—4241no—
517C22—2122no—
518C22#1681051312yes—
519C22#1681051312yes—
520C22C2×#1313317511yescharacteristic
521C22C2×#1313318911yescharacteristic
522C22C2×#1313321211yescharacteristic
523C4—42no—
524C4—42no—
525C4—82no—
526C4—2122no—
527C4#1681051312yes—
528C4#1681051312yes—
529C4C2×C4×C22⋊D5311yescharacteristic
530C4C2×#1313317911yescharacteristic
531C4C2×#1313320211yescharacteristic
532C4C2×#1313320611yescharacteristic
533C2—21no—
534C2—21no—
535C2—22no—
536C2—22no—
537C2—4242no—
538C2C2×#1681051311yescharacteristic
539C2#2028444011yescharacteristic
540C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC22×C53C2×#121
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#428D53
hypocenterC53#21076
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. #20283134C2semidirect
5. #20275376C2semidirect
6. #20275369C2semidirect
7. C2×#16810392C2semidirect
8. #20274441C2semidirect
9. C2×C53×#428C2semidirect
23. #16810605C4semidirect
71. #13133135C2×C4semidirect
72. #13133133C2×C4semidirect
263. C2×#428D53semidirect
308. C53#21076semidirect

Maximal subgroups

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

subgroupindex
#202832322
C2×#168106222
C2×#168106052
#202831342
#202753762
#202753692
C2×#168103922
#202744412
C2×C53×#4282
#2107653