Finite Group Databasebeta

← #27386649

Subgroups — #27386649

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
1C2×#16810666C211yesmaximal, characteristic, complemented
2#20283232C211yesmaximal, characteristic
3C2×#16810625C211yesmaximal, characteristic
4C2×#16810605C211yesmaximal, characteristic
5#20275377C214yesmaximal, complemented
6#20275370C214yesmaximal, complemented
7C2×#16810391C211yesmaximal, characteristic, complemented
8#20274442C211yesmaximal, characteristic, complemented
9C53×#2873C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14#16810666—22no—
15#16810666C2214yes—
16C2×#13133150—21no—
17C2×#13133148—21no—
18#16810631C2212yes—
19#16810627C2212yes—
20#16810625—22no—
21#16810625—22no—
22#16810622—21no—
23#16810622—21no—
24#16810617—22no—
25C2×#13133142C2212yes—
26C2×#13133139C2212yes—
27#16810605C418yescomplemented
28C2×#13133135C2212yes—
29C2×#13133133C2212yes—
30#16810391C2214yes—
31#16810388C2214yes—
32C22×M16×C53C2211yescharacteristic
33C22×M16×C53C2211yescharacteristic
34C2×#117×C53C2211yescharacteristic
35C53×#405—22no—
36C53×#405C2214yes—
37C4×M16×C53—22no—
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—
49C23×Dic53—21no—
50C22×C4×D53—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—22no—
54C2×#11719917—22no—
55C2×#11719917—22no—
56C2×#11719917C2×C414yes—
57C2×#11719917C2312yes—
58C2×#11719917C2312yes—
59C22×#88322—21no—
60C22×#88322—21no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53C2312yes—
67#13133150—24no—
68#13133148—24no—
69#13133142—24no—
70#13133140—22no—
71#13133140—22no—
72#13133139—24no—
73#13133137—22no—
74#13133137—22no—
75#13133135C2×C418yescomplemented
76#13133133C2×C418yescomplemented
77C23×C4×C53C2311yescharacteristic
78C2×M16×C53—21no—
79C2×M16×C53—22no—
80C2×M16×C53—22no—
81C2×M16×C53—22no—
82C2×M16×C53C2×C414yes—
83C2×M16×C53D412yes—
84C2×M16×C53D412yes—
85C2×M16×C53C2312yes—
86C22×C8×C53—21no—
87C22×C8×C53C2312yes—
88#117×C53—21no—
89#117×C53—21no—
90#117×C53D412yes—
91#117×C53C2312yes—
92C2×C4⋊C4×C53D411yescharacteristic
93C2×C4⋊C4×C53D411yescharacteristic
94C2×C22⋊C4×C53—21no—
95C2×C42×C53—21no—
96#97×C53—24no—
97C4×C8×C53—24no—
98C24×C53#4111yescharacteristic
99C23×D53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53C22×C412yes—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×C4×C53#4112yes—
109C22×C4×C53#4112yes—
110C22×C4×C53#4112yes—
111C22×C4×C53C2411yescharacteristic
112C2×C22⋊D53—24no—
113C2×C22⋊D53—41no—
114C2×C22⋊D53—41no—
115C2×C22⋊D53—42no—
116C22×Dic53—21no—
117C22×Dic53—22no—
118C22×Dic53—41no—
119C22×Dic53—41no—
120C22×Dic53—82no—
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—24no—
147M16×C53—24no—
148M16×C53—22no—
149M16×C53C22×C414yes—
150C2×C8×C53—24no—
151C2×C8×C53—22no—
152C2×C8×C53—22no—
153C2×C8×C53—22no—
154C2×C8×C53C2×D414yes—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53C2×D412yes—
158C4⋊C4×C53C2×D412yes—
159C22⋊C4×C53—22no—
160C22⋊C4×C53—22no—
161C42×C53—22no—
162C42×C53—22no—
163C4⋊Dic53—22no—
164C4⋊Dic53—22no—
165C4⋊Dic53—41no—
166C4⋊Dic53—41no—
167#11719917—24no—
168#11719917—22no—
169#11719917—22no—
170#11719917—22no—
171#11719917—42no—
172#11719917C22×C414yes—
173C2×#88322—24no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—42no—
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—81no—
192C2×C4×C53—81no—
193C2×C4×C53#11712yes—
194C2×C4×C53C4×D412yes—
195C2×C4×C53C4×D412yes—
196C2×C4×C53C4×D412yes—
197C2×C4×C53C4×D412yes—
198C2×C4×C53#12111yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12312yes—
201C2×C4×C53#12312yes—
202C2×C4×C53#12412yes—
203C2×C4×C53#12412yes—
204C2×C4×C53#12612yes—
205C2×C4×C53C23×C411yescharacteristic
206C2×C4×C53C22×D411yescharacteristic
207C2×C4×C53C22×D411yescharacteristic
208C2×C4×C53C2×#4111yescharacteristic
209C2×C4×C53C2×#4111yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C22⋊D53—44no—
212C22⋊D53—44no—
213C2×Dic53—22no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216C2×Dic53—42no—
217C2×Dic53—82no—
218C2×Dic53—82no—
219C2×Dic53—82no—
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—
231C8×C53—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#38812yes—
256C4×C53#38812yes—
257C4×C53C2×C4×D411yescharacteristic
258C4×C53C2×C4×D411yescharacteristic
259C4×C53C2×#12311yescharacteristic
260C4×C53C2×#12411yescharacteristic
261Dic53—42no—
262Dic53—82no—
263C2×#428—531no—
264#2896—531no—
265C2×#417—531no—
266C2×#413—531no—
267#2873D5311yescharacteristic, complemented
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×M16C2×D5311yescharacteristic
284C22×M16C2×D5311yescharacteristic
285C2×#117C2×D5311yescharacteristic
286C22×C4⋊C4—531no—
287#428—534no—
288#428—1062no—
289C2×#107—1061no—
290C2×#106—1061no—
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—22no—
304#405C2×D5314yes—
305C4×M16—22no—
306#343—534no—
307#341—534no—
308C53#2107611yescharacteristic, complemented, Sylow
309C2×#41—534no—
310C2×#41—1062no—
311C2×#41—1061no—
312C2×#41—1061no—
313C2×#41—1062no—
314C2×#41—1062no—
315C2×#41—1061no—
316C2×#41—1061no—
317C22×Q8—531no—
318C22×D4—531no—
319C22×D4—1061no—
320C23×C4—1061no—
321C23×C4—1061no—
322C23×C4C22×D5311yescharacteristic
323C2×M16—22no—
324C2×M16—22no—
325C2×M16—22no—
326C2×M16—21no—
327C2×M16—534no—
328C2×M16—532no—
329C2×M16—532no—
330C2×M16—1061no—
331C2×M16—1062no—
332C2×M16—1061no—
333C2×M16—1062no—
334C2×M16—1062no—
335C2×M16C4×D5314yes—
336C2×M16C22⋊D5312yes—
337C2×M16C22⋊D5312yes—
338C2×M16C22×D5312yes—
339C22×C8—21no—
340C22×C8—1061no—
341C22×C8—1061no—
342C22×C8C22×D5312yes—
343#117—21no—
344#117—21no—
345#117D21212yes—
346#117C22×D5312yes—
347C2×C4⋊C4—532no—
348C2×C4⋊C4—1061no—
349C2×C4⋊C4—1061no—
350C2×C4⋊C4—1061no—
351C2×C4⋊C4—1061no—
352C2×C4⋊C4—1061no—
353C2×C4⋊C4D21211yescharacteristic
354C2×C4⋊C4D21211yescharacteristic
355C2×C22⋊C4—21no—
356C2×C42—21no—
357#107—1064no—
358#106—1064no—
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—24no—
368C4×C8—24no—
369C24—1061no—
370C24#1171994511yescharacteristic
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—1061no—
398C22×C4—1064no—
399C22×C4—1062no—
400C22×C4—4242no—
401C22×C4—2121no—
402C22×C4—2121no—
403C22×C4—2121no—
404C22×C4—2121no—
405C22×C4—2122no—
406C22×C4C2×C4×D5312yes—
407C22×C4C2×D21211yescharacteristic
408C22×C4C2×D21211yescharacteristic
409C22×C4#1171994511yescharacteristic
410C22×C4#1171994512yes—
411C22×C4#1171994512yes—
412C22×C4#1171994512yes—
413C22×C4C2×C22⋊D5311yescharacteristic
414C22×C4C2×C22⋊D5311yescharacteristic
415C22×C4C23×D5311yescharacteristic
416M16—24no—
417M16—24no—
418M16—24no—
419M16—22no—
420M16—534no—
421M16—1064no—
422M16—1062no—
423M16—1062no—
424M16—1062no—
425M16—2122no—
426M16C2×C4×D5314yes—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—1064no—
432C2×C8—1062no—
433C2×C8—1062no—
434C2×C8—2122no—
435C2×C8C2×C22⋊D5314yes—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—1062no—
439C4⋊C4—1062no—
440C4⋊C4—2121no—
441C4⋊C4—2121no—
442C4⋊C4C2×D21212yes—
443C4⋊C4C2×D21212yes—
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#1313317412yes—
456C23#1313319111yescharacteristic
457C23#1313319111yescharacteristic
458C23#1313321112yes—
459C23C2×#1171994511yescharacteristic
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—81no—
473C2×C4—81no—
474C2×C4—1062no—
475C2×C4—4242no—
476C2×C4—4242no—
477C2×C4—4242no—
478C2×C4—2122no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4—2124no—
482C2×C4—2124no—
483C2×C4#1313317412yes—
484C2×C4C4×C22⋊D5312yes—
485C2×C4C4×C22⋊D5312yes—
486C2×C4C4×D21212yes—
487C2×C4C4×D21212yes—
488C2×C4#1313319111yescharacteristic
489C2×C4#1313319111yescharacteristic
490C2×C4#1313320012yes—
491C2×C4#1313320012yes—
492C2×C4#1313320512yes—
493C2×C4#1313320512yes—
494C2×C4#1313321112yes—
495C2×C4C22×C4×D5311yescharacteristic
496C2×C4C22×C22⋊D5311yescharacteristic
497C2×C4C22×D21211yescharacteristic
498C2×C4C2×#1171994511yescharacteristic
499C2×C4C2×#1171994511yescharacteristic
500C2×C4C2×#1171994511yescharacteristic
501C8—24no—
502C8—24no—
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#1681051412yes—
519C22#1681051412yes—
520C22C2×#1313317411yescharacteristic
521C22C2×#1313319111yescharacteristic
522C22C2×#1313321111yescharacteristic
523C4—42no—
524C4—42no—
525C4—4242no—
526C4—2122no—
527C4#1681051412yes—
528C4#1681051412yes—
529C4C2×C4×C22⋊D5311yescharacteristic
530C4C2×C4×D21211yescharacteristic
531C4C2×#1313320011yescharacteristic
532C4C2×#1313320511yescharacteristic
533C2—21no—
534C2—21no—
535C2—22no—
536C2—22no—
537C2—4242no—
538C2C2×#1681051411yescharacteristic
539C2#2028444311yescharacteristic
540C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2873C2
socleC22×C53C2×#121
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#2873D53
hypocenterC53#21076
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2873

Complements

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

NH ≅ G/Ntype
1. C2×#16810666C2semidirect
5. #20275377C2semidirect
6. #20275370C2semidirect
7. C2×#16810391C2semidirect
8. #20274442C2semidirect
9. C53×#2873C2semidirect
27. #16810605C4semidirect
75. #13133135C2×C4semidirect
76. #13133133C2×C4semidirect
267. #2873D53semidirect
308. C53#21076semidirect

Maximal subgroups

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

subgroupindex
C2×#168106662
#202832322
C2×#168106252
C2×#168106052
#202753772
#202753702
C2×#168103912
#202744422
C53×#28732
#2107653