Finite Group Databasebeta

← #27386640

Subgroups — #27386640

Order 13568.

1014 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283304C211yesmaximal, characteristic, complemented
2#20283232C211yesmaximal, characteristic
3C2×#16810620C211yesmaximal, characteristic
4C2×#16810605C211yesmaximal, characteristic
5#20275377C214yesmaximal, complemented
6#20275370C214yesmaximal, complemented
7#20274453C211yesmaximal, characteristic, complemented
8C2×#16810388C211yesmaximal, characteristic, complemented
9C2×C53×#405C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14#16810666C2214yes—
15#16810663—22no—
16#16810659—22no—
17#16810631C2212yes—
18#16810627C2212yes—
19#16810622—22no—
20#16810620—21no—
21#16810620—21no—
22#16810620—22no—
23#16810617—22no—
24C2×#13133142C2212yes—
25C2×#13133139C2212yes—
26#16810605C418yescomplemented
27C2×#13133135C2212yes—
28C2×#13133133C2212yes—
29#16810391C2214yes—
30#16810388C2214yes—
31C22×M16×C53C2211yescharacteristic
32C23×C8×C53C2211yescharacteristic
33C2×#117×C53C2211yescharacteristic
34C53×#405—22no—
35C53×#405C2214yes—
36C2×#97×C53—21no—
37C2×C4×C8×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×#11719917C2×C414yes—
56C2×#11719917C2312yes—
57C2×#11719917C2312yes—
58C22×#88322—21no—
59C22×#88322—21no—
60#13133176—21no—
61#13133176—21no—
62#13133176—21no—
63#13133176—21no—
64C2×C4⋊Dic53C2312yes—
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#13133135C2×C418yescomplemented
76#13133133C2×C418yescomplemented
77C23×C4×C53C2311yescharacteristic
78C2×M16×C53—21no—
79C2×M16×C53—22no—
80C2×M16×C53C2×C414yes—
81C2×M16×C53C2312yes—
82C22×C8×C53—21no—
83C22×C8×C53—22no—
84C22×C8×C53—22no—
85C22×C8×C53D411yescharacteristic
86C22×C8×C53D411yescharacteristic
87C22×C8×C53D412yes—
88C22×C8×C53C2312yes—
89#117×C53—21no—
90#117×C53—21no—
91#117×C53D412yes—
92#117×C53C2312yes—
93C2×C4⋊C4×C53D412yes—
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—22no—
101C22×C4×C53C22×C412yes—
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×C53C2411yescharacteristic
111C2×C22⋊D53—24no—
112C2×C22⋊D53—41no—
113C2×C22⋊D53—41no—
114C2×C22⋊D53—42no—
115C22×Dic53—21no—
116C22×Dic53—22no—
117#11719945—24no—
118#11719945—24no—
119#11719945—24no—
120#11719945—24no—
121#11719945—42no—
122#11719945—42no—
123#11719945—42no—
124#11719945—42no—
125C2×D212—21no—
126C2×D212—21no—
127C2×D212—21no—
128C2×D212—21no—
129C2×D212—21no—
130C2×D212C22×C412yes—
131C2×C4×D53—24no—
132C2×C4×D53—41no—
133C2×C4×D53—41no—
134C2×C4×D53—42no—
135C2×Dic106—21no—
136C2×Dic106—21no—
137C2×Dic106—21no—
138C2×Dic106—21no—
139C2×Dic106—21no—
140C2×Dic106C22×C412yes—
141M16×C53—24no—
142M16×C53—22no—
143M16×C53C22×C414yes—
144C2×C8×C53—24no—
145C2×C8×C53—22no—
146C2×C8×C53—22no—
147C2×C8×C53—24no—
148C2×C8×C53—22no—
149C2×C8×C53—22no—
150C2×C8×C53—22no—
151C2×C8×C53C2×D412yes—
152C2×C8×C53C2×D412yes—
153C4⋊C4×C53—22no—
154C4⋊C4×C53C2×D414yes—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—22no—
157C42×C53—22no—
158C42×C53—22no—
159C22⋊Dic53—41no—
160C22⋊Dic53—41no—
161C4⋊Dic53—22no—
162C4⋊Dic53—22no—
163C4×Dic53—41no—
164C4×Dic53—41no—
165#11719917—24no—
166#11719917—22no—
167#11719917—22no—
168#11719917—22no—
169#11719917—42no—
170#11719917C22×C414yes—
171C2×#88322—24no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—42no—
175C23×C53—42no—
176C23×C53#11712yes—
177C23×C53#12111yescharacteristic
178C23×C53#12111yescharacteristic
179C23×C53#12612yes—
180C23×C53C2×#4111yescharacteristic
181C22×D53—22no—
182C22×D53—41no—
183C22×D53—41no—
184C22×D53—82no—
185C2×C4×C53—24no—
186C2×C4×C53—44no—
187C2×C4×C53#11712yes—
188C2×C4×C53C4×D412yes—
189C2×C4×C53C4×D412yes—
190C2×C4×C53C4×D412yes—
191C2×C4×C53C4×D412yes—
192C2×C4×C53#12111yescharacteristic
193C2×C4×C53#12111yescharacteristic
194C2×C4×C53#12312yes—
195C2×C4×C53#12312yes—
196C2×C4×C53#12412yes—
197C2×C4×C53#12412yes—
198C2×C4×C53#12612yes—
199C2×C4×C53C23×C411yescharacteristic
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C22×D411yescharacteristic
202C2×C4×C53C2×#4111yescharacteristic
203C2×C4×C53C2×#4111yescharacteristic
204C2×C4×C53C2×#4111yescharacteristic
205C22⋊D53—44no—
206C22⋊D53—44no—
207C2×Dic53—22no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—42no—
211C2×Dic53—82no—
212D212—22no—
213D212—22no—
214D212—41no—
215D212—41no—
216C4×D53—44no—
217C4×D53—44no—
218Dic106—22no—
219Dic106—22no—
220Dic106—41no—
221Dic106—41no—
222C8×C53—24no—
223C8×C53—22no—
224C8×C53—22no—
225#88322—24no—
226#88322—42no—
227#21075—531nomaximal, Sylow
228C22×C53—22no—
229C22×C53—22no—
230C22×C53—22no—
231C22×C53—42no—
232C22×C53—42no—
233C22×C53#38812yes—
234C22×C53#38812yes—
235C22×C53C2×#11711yescharacteristic
236C22×C53C2×#12111yescharacteristic
237C22×C53C2×#12611yescharacteristic
238C2×D53—42no—
239C2×D53—82no—
240C2×D53—82no—
241C2×D53—82no—
242C4×C53—44no—
243C4×C53#38812yes—
244C4×C53#38812yes—
245C4×C53C2×C4×D411yescharacteristic
246C4×C53C2×C4×D411yescharacteristic
247C4×C53C2×#12311yescharacteristic
248C4×C53C2×#12411yescharacteristic
249Dic53—42no—
250Dic53—82no—
251#2911—531no—
252#2896—531no—
253C2×#416—531no—
254C2×#413—531no—
255C2×#405D5311yescharacteristic, complemented
256#1931—534no—
257#1930—534no—
258#1742—531no—
259C2×#341—531no—
260C2×C53—21no—
261C2×C53—21no—
262C2×C53—22no—
263C2×C53—22no—
264C2×C53C2×#38811yescharacteristic
265C2×C53#306712yes—
266D53—82no—
267C22×#41—531no—
268C22×M16—531no—
269C22×M16—531no—
270C22×M16C2×D5311yescharacteristic
271C23×C8C2×D5311yescharacteristic
272C2×#117—531no—
273C2×#117C2×D5311yescharacteristic
274#428—534no—
275#428—1062no—
276#427—1062no—
277#419—532no—
278#418—532no—
279#417—1062no—
280#416—1061no—
281#416—1061no—
282#416—1062no—
283#416—1062no—
284C2×#103—532no—
285C2×#102—532no—
286#413—538no—
287C2×#101—532no—
288C2×#100—532no—
289#405—22no—
290#405C2×D5314yes—
291C2×#97—21no—
292C2×C4×C8—21no—
293#343—534no—
294#341—534no—
295C53#2107511yescharacteristic, complemented, Sylow
296C2×#41—534no—
297C2×#41—1062no—
298C2×#41—1061no—
299C2×#41—1061no—
300C2×#41—1062no—
301C2×#41—1062no—
302C2×#41—1061no—
303C2×#41—1061no—
304C22×Q8—531no—
305C22×D4—531no—
306C22×D4—1061no—
307C23×C4—1061no—
308C23×C4C22×D5311yescharacteristic
309C2×M16—22no—
310C2×M16—21no—
311C2×M16—534no—
312C2×M16—532no—
313C2×M16—532no—
314C2×M16—1061no—
315C2×M16—1062no—
316C2×M16—1061no—
317C2×M16—1062no—
318C2×M16—1062no—
319C2×M16C4×D5314yes—
320C2×M16C22×D5312yes—
321C22×C8—22no—
322C22×C8—22no—
323C22×C8—21no—
324C22×C8—1061no—
325C22×C8—1061no—
326C22×C8C22⋊D5311yescharacteristic
327C22×C8C22⋊D5311yescharacteristic
328C22×C8C22⋊D5312yes—
329C22×C8C22×D5312yes—
330#117—21no—
331#117—21no—
332#117—1061no—
333#117—1061no—
334#117—1061no—
335#117—1061no—
336#117D21212yes—
337#117C22×D5312yes—
338C2×C4⋊C4—532no—
339C2×C4⋊C4D21212yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—1061no—
342C2×C42—21no—
343C2×C42—1061no—
344#107—1064no—
345#106—1064no—
346#103—1062no—
347#103—1062no—
348#103—1064no—
349#102—1062no—
350#102—1062no—
351#102—1064no—
352#101—538no—
353#100—538no—
354#97—24no—
355C4×C8—24no—
356C24—1061no—
357C24#1171994511yescharacteristic
358#41—1064no—
359#41—1064no—
360#41—1064no—
361#41—1064no—
362#41—2122no—
363#41—2122no—
364#41—2122no—
365#41—2122no—
366C2×Q8—532no—
367C2×Q8—1061no—
368C2×Q8—1061no—
369C2×Q8—1061no—
370C2×Q8—1061no—
371C2×Q8—1061no—
372C2×D4—532no—
373C2×D4—1064no—
374C2×D4—1061no—
375C2×D4—1061no—
376C2×D4—1061no—
377C2×D4—1061no—
378C2×D4—1061no—
379C2×D4—2121no—
380C2×D4—2121no—
381C2×D4—2122no—
382C22×C4—22no—
383C22×C4—1061no—
384C22×C4—1064no—
385C22×C4—1062no—
386C22×C4—2121no—
387C22×C4—2121no—
388C22×C4—2122no—
389C22×C4C2×C4×D5312yes—
390C22×C4C2×D21211yescharacteristic
391C22×C4C2×D21211yescharacteristic
392C22×C4#1171994511yescharacteristic
393C22×C4#1171994512yes—
394C22×C4#1171994512yes—
395C22×C4#1171994512yes—
396C22×C4C2×C22⋊D5311yescharacteristic
397C22×C4C2×C22⋊D5311yescharacteristic
398C22×C4C23×D5311yescharacteristic
399M16—24no—
400M16—22no—
401M16—534no—
402M16—1064no—
403M16—1062no—
404M16—1062no—
405M16—1062no—
406M16—2122no—
407M16C2×C4×D5314yes—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—1064no—
416C2×C8—1062no—
417C2×C8—1062no—
418C2×C8—2122no—
419C2×C8C2×C22⋊D5312yes—
420C2×C8C2×C22⋊D5312yes—
421C4⋊C4—22no—
422C4⋊C4—1062no—
423C4⋊C4—1062no—
424C4⋊C4C2×D21214yes—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—2121no—
428C22⋊C4—2121no—
429C42—22no—
430C42—22no—
431C42—2121no—
432C42—2121no—
433C23—42no—
434C23—1062no—
435C23—4242no—
436C23—2121no—
437C23—2121no—
438C23#1313317412yes—
439C23#1313319111yescharacteristic
440C23#1313319111yescharacteristic
441C23#1313321112yes—
442C23C2×#1171994511yescharacteristic
443Q8—1062no—
444Q8—1062no—
445Q8—2121no—
446Q8—2121no—
447D4—1062no—
448D4—1062no—
449D4—2124no—
450D4—2124no—
451D4—2121no—
452D4—2121no—
453C2×C4—24no—
454C2×C4—44no—
455C2×C4—1062no—
456C2×C4—4242no—
457C2×C4—2122no—
458C2×C4—2121no—
459C2×C4—2121no—
460C2×C4—2124no—
461C2×C4—2124no—
462C2×C4#1313317412yes—
463C2×C4C4×C22⋊D5312yes—
464C2×C4C4×C22⋊D5312yes—
465C2×C4C4×D21212yes—
466C2×C4C4×D21212yes—
467C2×C4#1313319111yescharacteristic
468C2×C4#1313319111yescharacteristic
469C2×C4#1313320012yes—
470C2×C4#1313320012yes—
471C2×C4#1313320512yes—
472C2×C4#1313320512yes—
473C2×C4#1313321112yes—
474C2×C4C22×C4×D5311yescharacteristic
475C2×C4C22×C22⋊D5311yescharacteristic
476C2×C4C22×D21211yescharacteristic
477C2×C4C2×#1171994511yescharacteristic
478C2×C4C2×#1171994511yescharacteristic
479C2×C4C2×#1171994511yescharacteristic
480C8—24no—
481C8—22no—
482C8—22no—
483C8—1064no—
484C8—2122no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—4242no—
491C22—4242no—
492C22—4242no—
493C22—2122no—
494C22#1681051412yes—
495C22#1681051412yes—
496C22C2×#1313317411yescharacteristic
497C22C2×#1313319111yescharacteristic
498C22C2×#1313321111yescharacteristic
499C4—44no—
500C4—4242no—
501C4—2122no—
502C4#1681051412yes—
503C4#1681051412yes—
504C4C2×C4×C22⋊D5311yescharacteristic
505C4C2×C4×D21211yescharacteristic
506C4C2×#1313320011yescharacteristic
507C4C2×#1313320511yescharacteristic
508C2—21no—
509C2—21no—
510C2—22no—
511C2—22no—
512C2—4242no—
513C2C2×#1681051411yescharacteristic
514C2#2028443512yes—
subgroupG/H
centerC2×C4#13133191
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#405C2
socleC22×C53C2×#121
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#405D53
hypocenterC53#21075
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
1. #20283304C2semidirect
5. #20275377C2semidirect
6. #20275370C2semidirect
7. #20274453C2semidirect
8. C2×#16810388C2semidirect
9. C2×C53×#405C2semidirect
26. #16810605C4semidirect
75. #13133135C2×C4semidirect
76. #13133133C2×C4semidirect
255. C2×#405D53semidirect
295. C53#21075semidirect

Maximal subgroups

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

subgroupindex
#202833042
#202832322
C2×#168106202
C2×#168106052
#202753772
#202753702
#202744532
C2×#168103882
C2×C53×#4052
#2107553