Finite Group Databasebeta

← #27413007

Subgroups — #27413007

Order 13568.

914 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810663C211yesmaximal, characteristic, complemented
2#20283235C211yesmaximal, characteristic
3C2×#16810622C211yesmaximal, characteristic
4#20283159C211yesmaximal, characteristic
5#20276164C214yesmaximal, complemented
6#20276159C214yesmaximal, complemented
7#20274453C211yesmaximal, characteristic, complemented
8#20274441C211yesmaximal, characteristic, complemented
9C2×C53×#343C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14#16810663—22no—
15#16810663—22no—
16C2×#13133150C2212yescomplemented
17C2×#13133148C2212yescomplemented
18#16810633—21no—
19#16810629—21no—
20#16810625C2212yes—
21#16810625C2212yes—
22#16810622—22no—
23#16810622C2214yes—
24#16810620—22no—
25C2×#13133140—21no—
26C2×#13133137—21no—
27#16810591C2214yes—
28#16810589—22no—
29#16810586—22no—
30#16810391C2212yescomplemented
31#16810391C2212yescomplemented
32#16810387C2214yescomplemented
33C22×M16×C53C2211yescharacteristic
34C22×M16×C53C2211yescharacteristic
35C22×C4⋊C4×C53C2211yescharacteristic
36C53×#343C2214yescomplemented
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—22no—
40C2×#11719945—22no—
41C2×#11719945—22no—
42C2×#11719945—22no—
43C2×#11719945D412yes—
44C2×#11719945C2312yes—
45C22×Dic106D411yescharacteristic
46C22×D212D411yescharacteristic
47C22×C22⋊D53—21no—
48C22×C4×D53—21no—
49C2×#11719917—21no—
50C2×#11719917—22no—
51C2×#11719917—22no—
52C2×#11719917—22no—
53C2×#11719917—22no—
54C2×#11719917D412yes—
55C2×#11719917D412yes—
56C2×#11719917C2312yes—
57C22×#88322—21no—
58C22×#88322C2312yes—
59#13133176—21no—
60#13133176—21no—
61#13133176—21no—
62#13133176—21no—
63C2×C4⋊Dic53C2311yescharacteristic
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C22⋊Dic53—21no—
66C2×C4×Dic53—21no—
67#13133150—24no—
68#13133148—24no—
69#13133142—22no—
70#13133142—22no—
71#13133140—24no—
72#13133139—22no—
73#13133139—22no—
74#13133137—24no—
75#13133129—28no—
76C23×C4×C53—21no—
77C23×C4×C53C2311yescharacteristic
78C2×M16×C53—21no—
79C2×M16×C53—21no—
80C2×M16×C53—22no—
81C2×M16×C53—22no—
82C2×M16×C53—22no—
83C2×M16×C53—22no—
84C2×M16×C53C2312yes—
85C2×M16×C53C2312yes—
86C22×C8×C53—21no—
87C22×C8×C53—21no—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53C2312yes—
94C24×C53#4111yescharacteristic
95C23×D53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—82no—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×C4×C53#4112yes—
106C22×C4×C53#4112yes—
107C22×C4×C53#4112yes—
108C22×C4×C53#4112yes—
109C22×C4×C53C2411yescharacteristic
110C2×C22⋊D53—22no—
111C2×C22⋊D53—22no—
112C2×C22⋊D53—42no—
113C2×C22⋊D53—42no—
114C22×Dic53—21no—
115C22×Dic53—21no—
116C22×Dic53—21no—
117#11719945—24no—
118#11719945—24no—
119#11719945—22no—
120#11719945—22no—
121#11719945—22no—
122#11719945—22no—
123#11719945—42no—
124#11719945—42no—
125#11719945—42no—
126#11719945—42no—
127C2×D212—22no—
128C2×D212—22no—
129C2×D212—21no—
130C2×D212C2×D412yes—
131C2×C4×D53—22no—
132C2×C4×D53—22no—
133C2×C4×D53—42no—
134C2×C4×D53—42no—
135C2×Dic106—22no—
136C2×Dic106—22no—
137C2×Dic106—21no—
138C2×Dic106C2×D412yes—
139M16×C53—22no—
140M16×C53—22no—
141M16×C53—22no—
142M16×C53—22no—
143M16×C53—42no—
144M16×C53—42no—
145C2×C8×C53—22no—
146C2×C8×C53—22no—
147C2×C8×C53—42no—
148C2×C8×C53—42no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—41no—
152C4⋊C4×C53—41no—
153C22⋊Dic53—41no—
154C22⋊Dic53—41no—
155C4⋊Dic53—21no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4×Dic53—41no—
160C4×Dic53—41no—
161#11719917—24no—
162#11719917—24no—
163#11719917—22no—
164#11719917—22no—
165#11719917—42no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—22no—
169C2×#88322—42no—
170C2×#88322C2×D414yes—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53#12111yescharacteristic
174C23×C53#12111yescharacteristic
175C23×C53#12612yes—
176C23×C53#12612yes—
177C23×C53C2×#4111yescharacteristic
178C22×D53—22no—
179C22×D53—42no—
180C22×D53—81no—
181C22×D53—81no—
182C2×C4×C53—42no—
183C2×C4×C53—82no—
184C2×C4×C53—82no—
185C2×C4×C53—82no—
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12312yes—
189C2×C4×C53#12312yes—
190C2×C4×C53#12312yes—
191C2×C4×C53#12312yes—
192C2×C4×C53#12312yes—
193C2×C4×C53#12312yes—
194C2×C4×C53#12412yes—
195C2×C4×C53#12412yes—
196C2×C4×C53#12612yes—
197C2×C4×C53#12612yes—
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C2×C4×C53C2×#4111yescharacteristic
201C2×C4×C53C2×#4111yescharacteristic
202C2×C4×C53C2×#4111yescharacteristic
203C2×C4×C53C2×#4111yescharacteristic
204C22⋊D53—42no—
205C22⋊D53—42no—
206C22⋊D53—44no—
207C2×Dic53—22no—
208C2×Dic53—42no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—81no—
212C2×Dic53—81no—
213D212—22no—
214D212—22no—
215D212—42no—
216C4×D53—42no—
217C4×D53—42no—
218C4×D53—44no—
219Dic106—22no—
220Dic106—22no—
221Dic106—42no—
222C8×C53—42no—
223C8×C53—42no—
224#88322—24no—
225#88322—42no—
226#23647—531nomaximal, Sylow
227C22×C53—21no—
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—42no—
234C22×C53—42no—
235C22×C53#39712yes—
236C22×C53#39712yes—
237C22×C53C2×#12111yescharacteristic
238C22×C53C2×#12611yescharacteristic
239C22×C53C2×#12611yescharacteristic
240C2×D53—42no—
241C2×D53—42no—
242C2×D53—42no—
243C2×D53—84no—
244C4×C53—82no—
245C4×C53#39712yes—
246C4×C53#39712yes—
247C4×C53C2×#12311yescharacteristic
248C4×C53C2×#12311yescharacteristic
249C4×C53C2×#12311yescharacteristic
250C4×C53C2×#12411yescharacteristic
251Dic53—42no—
252Dic53—81no—
253Dic53—81no—
254C2×#428—531no—
255#2896—531no—
256C2×#417—531no—
257#2879—531no—
258#2079—534no—
259#2078—534no—
260#1742—531no—
261C2×#343D5311yescharacteristic, complemented
262#1738—531no—
263C2×C53—22no—
264C2×C53—21no—
265C2×C53—21no—
266C2×C53—22no—
267C2×C53C2×#39711yescharacteristic
268C2×C53#306811yescharacteristic
269C2×C53#306911yescharacteristic
270D53—82no—
271C22×#41—531no—
272C22×M16—531no—
273C22×M16—531no—
274C22×M16C2×D5311yescharacteristic
275C22×M16C2×D5311yescharacteristic
276C2×#117—531no—
277C22×C4⋊C4C2×D5311yescharacteristic
278#428—1062no—
279#428—1062no—
280C2×#107—532no—
281C2×#106—532no—
282#419—1061no—
283#418—1061no—
284#417—532no—
285#417—532no—
286#417—534no—
287#417—1062no—
288#416—1062no—
289C2×#103—1061no—
290C2×#102—1061no—
291#408—534no—
292#408—1062no—
293#407—1062no—
294#343—532no—
295#343—532no—
296#343C2×D5314yescomplemented
297#341—534no—
298C53#2364711yescharacteristic, complemented, Sylow
299C2×#41—532no—
300C2×#41—532no—
301C2×#41—1061no—
302C2×#41—1061no—
303C2×#41—1062no—
304C2×#41—1062no—
305C2×#41—1062no—
306C2×#41—1062no—
307C22×Q8—531no—
308C22×D4—531no—
309C22×D4—1061no—
310C23×C4—21no—
311C23×C4—1061no—
312C23×C4C22×D5311yescharacteristic
313C2×M16—21no—
314C2×M16—21no—
315C2×M16—22no—
316C2×M16—22no—
317C2×M16—22no—
318C2×M16—22no—
319C2×M16—532no—
320C2×M16—532no—
321C2×M16—532no—
322C2×M16—1062no—
323C2×M16—1062no—
324C2×M16—1061no—
325C2×M16—1062no—
326C2×M16—1062no—
327C2×M16C22×D5312yes—
328C2×M16C22×D5312yes—
329C22×C8—21no—
330C22×C8—21no—
331C22×C8—532no—
332C22×C8—1061no—
333#117—1061no—
334#117—1061no—
335#117—1061no—
336#117—1061no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—531no—
343C2×C4⋊C4—531no—
344C2×C4⋊C4C22×D5312yes—
345C2×C22⋊C4—1061no—
346C2×C42—1061no—
347#107—1064no—
348#106—1064no—
349#103—1064no—
350#103—1062no—
351#103—1062no—
352#102—1064no—
353#102—1062no—
354#102—1062no—
355C22⋊C8—1068no—
356C24—1061no—
357C24#1171994511yescharacteristic
358#41—1064no—
359#41—1064no—
360#41—1062no—
361#41—1062no—
362#41—1062no—
363#41—1062no—
364#41—2122no—
365#41—2122no—
366#41—2122no—
367#41—2122no—
368C2×Q8—532no—
369C2×Q8—1061no—
370C2×Q8—1062no—
371C2×Q8—1062no—
372C2×D4—532no—
373C2×D4—1062no—
374C2×D4—1062no—
375C2×D4—1061no—
376C2×D4—1062no—
377C2×D4—1062no—
378C2×D4—2122no—
379C2×D4—2122no—
380C22×C4—22no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—82no—
384C22×C4—1061no—
385C22×C4—1062no—
386C22×C4—1062no—
387C22×C4—1061no—
388C22×C4—1061no—
389C22×C4—2122no—
390C22×C4—2122no—
391C22×C4#1171994511yescharacteristic
392C22×C4#1171994512yes—
393C22×C4#1171994512yes—
394C22×C4D4×D5311yescharacteristic
395C22×C4D4×D5311yescharacteristic
396C22×C4D4×D5311yescharacteristic
397C22×C4D4×D5311yescharacteristic
398C22×C4#1171994712yes—
399C22×C4#1171994912yes—
400C22×C4C23×D5311yescharacteristic
401M16—22no—
402M16—22no—
403M16—22no—
404M16—22no—
405M16—42no—
406M16—42no—
407M16—1064no—
408M16—1064no—
409M16—1062no—
410M16—1062no—
411M16—2122no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—534no—
417C2×C8—1062no—
418C2×C8—1062no—
419C2×C8—1062no—
420C2×C8—2122no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—1061no—
426C4⋊C4—1061no—
427C4⋊C4—1061no—
428C4⋊C4—1061no—
429C22⋊C4—2121no—
430C22⋊C4—2121no—
431C42—2121no—
432C42—2121no—
433C23—41no—
434C23—41no—
435C23—1062no—
436C23—4241no—
437C23—4241no—
438C23—2122no—
439C23#1313319011yescharacteristic
440C23#1313319011yescharacteristic
441C23#1313321112yes—
442C23#1313321212yes—
443C23C2×#1171994511yescharacteristic
444Q8—1062no—
445Q8—1062no—
446Q8—2122no—
447D4—1062no—
448D4—1062no—
449D4—2124no—
450D4—2122no—
451D4—2122no—
452D4—2122no—
453C2×C4—42no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—1062no—
458C2×C4—4241no—
459C2×C4—4241no—
460C2×C4—2121no—
461C2×C4—2121no—
462C2×C4—2122no—
463C2×C4—2124no—
464C2×C4—2122no—
465C2×C4—2122no—
466C2×C4#1313319011yescharacteristic
467C2×C4#1313319011yescharacteristic
468C2×C4#1313319912yes—
469C2×C4#1313319912yes—
470C2×C4#1313320212yes—
471C2×C4#1313320212yes—
472C2×C4#1313320212yes—
473C2×C4#1313320212yes—
474C2×C4#1313320312yes—
475C2×C4#1313320312yes—
476C2×C4#1313321112yes—
477C2×C4#1313321212yes—
478C2×C4C2×D4×D5311yescharacteristic
479C2×C4C2×D4×D5311yescharacteristic
480C2×C4C2×#1171994711yescharacteristic
481C2×C4C2×#1171994511yescharacteristic
482C2×C4C2×#1171994511yescharacteristic
483C2×C4C2×#1171994911yescharacteristic
484C8—42no—
485C8—42no—
486C8—1064no—
487C8—2122no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—42no—
495C22—42no—
496C22—4244no—
497C22—2122no—
498C22—2122no—
499C22—2122no—
500C22#1681055212yes—
501C22#1681055212yes—
502C22C2×#1313319011yescharacteristic
503C22C2×#1313321111yescharacteristic
504C22C2×#1313321211yescharacteristic
505C4—82no—
506C4—4241no—
507C4—4241no—
508C4—2122no—
509C4#1681055212yes—
510C4#1681055212yes—
511C4C2×#1313319911yescharacteristic
512C4C2×#1313320211yescharacteristic
513C4C2×#1313320211yescharacteristic
514C4C2×#1313320311yescharacteristic
515C2—22no—
516C2—22no—
517C2—21no—
518C2—21no—
519C2—4242no—
520C2C2×#1681055211yescharacteristic
521C2#2028444211yescharacteristic
522C2#2028445111yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#343C2
socleC22×C53C2×#121
prime radicalC22×#11719945C22
hypercenterC2×#343D53
hypocenterC53#23647
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
1. C2×#16810663C2semidirect
5. #20276164C2semidirect
6. #20276159C2semidirect
7. #20274453C2semidirect
8. #20274441C2semidirect
9. C2×C53×#343C2semidirect
16. C2×#13133150C22semidirect
17. C2×#13133148C22semidirect
30. #16810391C22semidirect
31. #16810391C22semidirect
32. #16810387C22semidirect
36. C53×#343C22semidirect
261. C2×#343D53semidirect
296. #343C2×D53semidirect
298. C53#23647semidirect

Maximal subgroups

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

subgroupindex
C2×#168106632
#202832352
C2×#168106222
#202831592
#202761642
#202761592
#202744532
#202744412
C2×C53×#3432
#2364753