Finite Group Databasebeta

← #27413003

Subgroups — #27413003

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
1#20283232C211yesmaximal, characteristic
2C2×#16810622C211yesmaximal, characteristic
3#20283159C211yesmaximal, characteristic
4#20276161C214yesmaximal, complemented
5#20276156C214yesmaximal, complemented
6#20274454C211yesmaximal, characteristic, complemented
7C2×#16810392C211yesmaximal, characteristic, complemented
8#20274442C211yesmaximal, characteristic, complemented
9C2×C53×#428C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14#16810631—21no—
15#16810627—21no—
16#16810622—22no—
17#16810622C2214yes—
18#16810622C2212yes—
19#16810622C2212yes—
20#16810617—22no—
21C2×#13133140—21no—
22C2×#13133137—21no—
23#16810591—22no—
24#16810589C2214yes—
25#16810586—22no—
26#16810392C2214yescomplemented
27#16810392C2212yescomplemented
28#16810392C2212yescomplemented
29#16810388C2214yescomplemented
30C22×M16×C53C2211yescharacteristic
31C2×#117×C53C2211yescharacteristic
32C22×C4⋊C4×C53C2211yescharacteristic
33C53×#428—22no—
34C53×#428—22no—
35C2×#107×C53C2212yescomplemented
36C2×#106×C53C2212yescomplemented
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—21no—
51C2×#11719917—21no—
52C2×#11719917—22no—
53C2×#11719917—22no—
54C2×#11719917—22no—
55C2×#11719917—22no—
56C2×#11719917—22no—
57C2×#11719917—22no—
58C2×#11719917C2312yes—
59C2×#11719917C2312yes—
60C2×#11719917C2312yes—
61C22×#88322—21no—
62C22×#88322—21no—
63C22×#88322—21no—
64#13133140—24no—
65#13133140—22no—
66#13133140—22no—
67#13133137—24no—
68#13133137—22no—
69#13133137—22no—
70#13133129—28no—
71C23×C4×C53—21no—
72C23×C4×C53C2311yescharacteristic
73C2×M16×C53—22no—
74C2×M16×C53—22no—
75C2×M16×C53D412yes—
76C2×M16×C53D412yes—
77C22×C8×C53C2312yes—
78#117×C53—21no—
79#117×C53—21no—
80#117×C53—21no—
81#117×C53—21no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53C2312yes—
88C2×C4⋊C4×C53C2311yescharacteristic
89C2×C4⋊C4×C53C2311yescharacteristic
90C2×C22⋊C4×C53—21no—
91C2×C42×C53—21no—
92#107×C53—24no—
93#106×C53—24no—
94C24×C53#4111yescharacteristic
95C23×D53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—82no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4112yes—
108C22×C4×C53#4112yes—
109C22×C4×C53#4112yes—
110C22×C4×C53#4112yes—
111C22×C4×C53C2411yescharacteristic
112C2×C22⋊D53—22no—
113C2×C22⋊D53—22no—
114C2×C22⋊D53—42no—
115C2×C22⋊D53—42no—
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—24no—
140M16×C53—24no—
141C2×C8×C53—22no—
142C2×C8×C53—22no—
143C2×C8×C53C2×D414yes—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—41no—
151C4⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C42×C53—41no—
155C42×C53—41no—
156#11719917—22no—
157#11719917—22no—
158#11719917—22no—
159#11719917—22no—
160#11719917—22no—
161#11719917—22no—
162#11719917—42no—
163#11719917—42no—
164#11719917—42no—
165C2×#88322—22no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322—42no—
169C2×#88322—42no—
170C2×#88322—42no—
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—41no—
184C2×C4×C53—41no—
185C2×C4×C53—82no—
186C2×C4×C53—82no—
187C2×C4×C53—81no—
188C2×C4×C53—81no—
189C2×C4×C53—82no—
190C2×C4×C53#12111yescharacteristic
191C2×C4×C53#12111yescharacteristic
192C2×C4×C53#12312yes—
193C2×C4×C53#12312yes—
194C2×C4×C53#12312yes—
195C2×C4×C53#12312yes—
196C2×C4×C53#12312yes—
197C2×C4×C53#12312yes—
198C2×C4×C53#12412yes—
199C2×C4×C53#12412yes—
200C2×C4×C53#12612yes—
201C2×C4×C53#12612yes—
202C2×C4×C53C22×D411yescharacteristic
203C2×C4×C53C22×D411yescharacteristic
204C2×C4×C53C2×#4111yescharacteristic
205C2×C4×C53C2×#4111yescharacteristic
206C2×C4×C53C2×#4111yescharacteristic
207C2×C4×C53C2×#4111yescharacteristic
208C22⋊D53—42no—
209C22⋊D53—42no—
210C22⋊D53—44no—
211C2×Dic53—22no—
212C2×Dic53—42no—
213D212—22no—
214D212—22no—
215D212—42no—
216C4×D53—42no—
217C4×D53—42no—
218C4×D53—44no—
219Dic106—22no—
220Dic106—22no—
221Dic106—42no—
222C8×C53—24no—
223#88322—42no—
224#88322—42no—
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—81no—
245C4×C53—82no—
246C4×C53—81no—
247C4×C53#39712yes—
248C4×C53#39712yes—
249C4×C53C2×#12311yescharacteristic
250C4×C53C2×#12311yescharacteristic
251C4×C53C2×#12311yescharacteristic
252C4×C53C2×#12411yescharacteristic
253Dic53—42no—
254C2×#428D5311yescharacteristic, complemented
255#2896—531no—
256C2×#417—531no—
257#2879—531no—
258#2079—534no—
259#2078—534no—
260#1742—531no—
261C2×#343—531no—
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×M16—531no—
275C22×M16C2×D5311yescharacteristic
276C2×#117C2×D5311yescharacteristic
277C22×C4⋊C4C2×D5311yescharacteristic
278#428—22no—
279#428—22no—
280C2×#107C2×D5312yescomplemented
281C2×#106C2×D5312yescomplemented
282#419—1061no—
283#418—1061no—
284#417—532no—
285#417—534no—
286#417—532no—
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—534no—
296#343—532no—
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—22no—
314C2×M16—22no—
315C2×M16—532no—
316C2×M16—532no—
317C2×M16—532no—
318C2×M16—1061no—
319C2×M16—1061no—
320C2×M16—1061no—
321C2×M16—1062no—
322C2×M16—1062no—
323C2×M16—1062no—
324C2×M16—1062no—
325C2×M16—1062no—
326C2×M16—1062no—
327C2×M16C22⋊D5312yes—
328C2×M16C22⋊D5312yes—
329C22×C8—1061no—
330C22×C8—1061no—
331C22×C8—1061no—
332C22×C8C22×D5312yes—
333#117—21no—
334#117—21no—
335#117—21no—
336#117—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4C22×D5312yes—
343C2×C4⋊C4C22×D5311yescharacteristic
344C2×C4⋊C4C22×D5311yescharacteristic
345C2×C22⋊C4—21no—
346C2×C42—21no—
347#107—24no—
348#106—24no—
349#103—1064no—
350#103—1062no—
351#103—1062no—
352#102—1064no—
353#102—1062no—
354#102—1062no—
355C22⋊C8—1068no—
356C24—1061no—
357C24#1171994711yescharacteristic
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—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—82no—
386C22×C4—1061no—
387C22×C4—1062no—
388C22×C4—1062no—
389C22×C4—2122no—
390C22×C4—2122no—
391C22×C4#1171994512yes—
392C22×C4#1171994512yes—
393C22×C4D4×D5311yescharacteristic
394C22×C4D4×D5311yescharacteristic
395C22×C4#1171994711yescharacteristic
396C22×C4#1171994912yes—
397C22×C4#1171994912yes—
398C22×C4C2×C22⋊D5311yescharacteristic
399C22×C4C2×C22⋊D5311yescharacteristic
400C22×C4C23×D5311yescharacteristic
401M16—24no—
402M16—24no—
403M16—1062no—
404M16—1062no—
405M16—1062no—
406M16—1062no—
407M16—1062no—
408M16—1062no—
409M16—2122no—
410M16—2122no—
411M16—2122no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—1062no—
415C2×C8—1062no—
416C2×C8—1062no—
417C2×C8—2122no—
418C2×C8—2122no—
419C2×C8—2122no—
420C2×C8C2×C22⋊D5314yes—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C42—41no—
432C42—41no—
433C23—41no—
434C23—41no—
435C23—1062no—
436C23—4241no—
437C23—4241no—
438C23—2122no—
439C23#1313318811yescharacteristic
440C23#1313318811yescharacteristic
441C23#1313321212yes—
442C23#1313321212yes—
443C23C2×#1171994711yescharacteristic
444Q8—1062no—
445Q8—1062no—
446Q8—2122no—
447D4—1062no—
448D4—1062no—
449D4—2124no—
450D4—2122no—
451D4—2122no—
452D4—2122no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—1062no—
462C2×C4—2122no—
463C2×C4—2124no—
464C2×C4—2122no—
465C2×C4—2122no—
466C2×C4#1313318811yescharacteristic
467C2×C4#1313318811yescharacteristic
468C2×C4#1313320012yes—
469C2×C4#1313320012yes—
470C2×C4#1313320212yes—
471C2×C4#1313320212yes—
472C2×C4#1313320212yes—
473C2×C4#1313320212yes—
474C2×C4#1313320612yes—
475C2×C4#1313320612yes—
476C2×C4#1313321212yes—
477C2×C4#1313321212yes—
478C2×C4C2×D4×D5311yescharacteristic
479C2×C4C22×C22⋊D5311yescharacteristic
480C2×C4C2×#1171994511yescharacteristic
481C2×C4C2×#1171994511yescharacteristic
482C2×C4C2×#1171994911yescharacteristic
483C2×C4C2×#1171994911yescharacteristic
484C8—24no—
485C8—2122no—
486C8—2122no—
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#1681054912yes—
501C22#1681054912yes—
502C22C2×#1313318811yescharacteristic
503C22C2×#1313321211yescharacteristic
504C22C2×#1313321211yescharacteristic
505C4—82no—
506C4—81no—
507C4—81no—
508C4—2122no—
509C4#1681054912yes—
510C4#1681054912yes—
511C4C2×#1313320011yescharacteristic
512C4C2×#1313320211yescharacteristic
513C4C2×#1313320211yescharacteristic
514C4C2×#1313320611yescharacteristic
515C2—22no—
516C2—22no—
517C2—21no—
518C2—21no—
519C2—4242no—
520C2C2×#1681054911yescharacteristic
521C2#2028443911yescharacteristic
522C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC22×C53C2×#121
prime radicalC22×#11719945C22
hypercenterC2×#428D53
hypocenterC53#23647
LCSG ▷ C22×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. #20276161C2semidirect
5. #20276156C2semidirect
6. #20274454C2semidirect
7. C2×#16810392C2semidirect
8. #20274442C2semidirect
9. C2×C53×#428C2semidirect
26. #16810392C22semidirect
27. #16810392C22semidirect
28. #16810392C22semidirect
29. #16810388C22semidirect
35. C2×#107×C53C22semidirect
36. C2×#106×C53C22semidirect
254. C2×#428D53semidirect
280. C2×#107C2×D53semidirect
281. C2×#106C2×D53semidirect
298. C53#23647semidirect

Maximal subgroups

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

subgroupindex
#202832322
C2×#168106222
#202831592
#202761612
#202761562
#202744542
C2×#168103922
#202744422
C2×C53×#4282
#2364753