Finite Group Databasebeta

← #27290610

Subgroups — #27290610

Order 13568.

1630 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133144C212yesmaximal, complemented
2C2×#16810587C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
4C2×#16810382C212yesmaximal, complemented
5C2×C53×#525C211yesmaximal, characteristic
6C22×#11719917C2212yes—
7C22×C22⋊Dic53—21no—
8C22×C4×Dic53C2212yes—
9C2×#13133144—24no—
10C2×#13133144—24no—
11C2×#13133144—24no—
12#16810587—24no—
13C2×C22⋊#88322—22no—
14C4×C22⋊Dic53—24no—
15C2×#13133124—21no—
16#16810382—24no—
17C22×#41×C53C2211yescharacteristic
18C24×C4×C53C2211yescharacteristic
19C53×#525—24no—
20C53×#525—24no—
21C2×#123×C53—21no—
22C2×C22⋊Q8×C53C412yescomplemented
23C2×#121×C53C412yescomplemented
24C2×#120×C53—21no—
25C2×C4×D4×C53—22no—
26C2×#117×C53C2211yescharacteristic
27C23×Dic53—22no—
28C2×#11719917—22no—
29C2×#11719917—22no—
30C2×#11719917—22no—
31C2×#11719917—22no—
32C2×#11719917—22no—
33C2×#11719917D414yescomplemented
34C22×#88322—22no—
35C2×C22⋊Dic53—28no—
36C2×C22⋊Dic53—24no—
37C2×C4×Dic53—22no—
38C2×C4×Dic53—22no—
39C2×C4×Dic53—22no—
40C2×C4×Dic53—22no—
41C2×C4×Dic53—22no—
42C2×C4×Dic53D414yescomplemented
43#13133144—216no—
44C22⋊#88322—28no—
45#13133124—24no—
46C25×C53D411yescharacteristic
47C2×#41×C53—22no—
48C2×#41×C53—22no—
49C2×#41×C53—22no—
50C2×#41×C53—22no—
51C2×#41×C53—22no—
52C2×#41×C53D414yes—
53C22×Q8×C53C2×C411yescharacteristic
54C22×D4×C53—22no—
55C22×D4×C53—21no—
56C22×D4×C53C2×C411yescharacteristic
57C23×C4×C53—21no—
58C23×C4×C53—22no—
59C23×C4×C53—22no—
60C23×C4×C53—22no—
61C23×C4×C53—22no—
62C23×C4×C53—22no—
63C23×C4×C53—22no—
64C23×C4×C53C2×C412yes—
65C23×C4×C53D411yescharacteristic
66C23×C4×C53D412yes—
67C23×C4×C53C2311yescharacteristic
68#123×C53—28no—
69C22⋊Q8×C53—24no—
70C22⋊Q8×C53—24no—
71#121×C53—24no—
72#121×C53—24no—
73#120×C53—28no—
74C4×D4×C53—216no—
75#117×C53—22no—
76#117×C53D414yes—
77C2×C4⋊C4×C53—22no—
78C2×C4⋊C4×C53C2×C412yes—
79C2×C22⋊C4×C53—22no—
80C2×C22⋊C4×C53—21no—
81C2×C22⋊C4×C53—22no—
82C2×C42×C53—21no—
83C24×C53—24no—
84C24×C53—21no—
85C24×C53—21no—
86C24×C53—21no—
87C24×C53—22no—
88C24×C53—22no—
89C24×C53—42no—
90C24×C53C22⋊C412yes—
91C24×C53C2×D411yescharacteristic
92#41×C53—24no—
93#41×C53—24no—
94#41×C53—24no—
95#41×C53—24no—
96#41×C53—42no—
97#41×C53—42no—
98#41×C53—42no—
99#41×C53—42no—
100C2×Q8×C53—21no—
101C2×Q8×C53—22no—
102C2×Q8×C53—22no—
103C2×Q8×C53C22⋊C412yescomplemented
104C2×D4×C53—28no—
105C2×D4×C53—24no—
106C2×D4×C53—21no—
107C2×D4×C53—22no—
108C2×D4×C53—22no—
109C2×D4×C53—44no—
110C2×D4×C53—44no—
111C2×D4×C53—41no—
112C2×D4×C53—42no—
113C2×D4×C53—41no—
114C2×D4×C53C22⋊C412yescomplemented
115C22×C4×C53—22no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—22no—
120C22×C4×C53—21no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—24no—
124C22×C4×C53—24no—
125C22×C4×C53—24no—
126C22×C4×C53—24no—
127C22×C4×C53—22no—
128C22×C4×C53—24no—
129C22×C4×C53—22no—
130C22×C4×C53—42no—
131C22×C4×C53—41no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53—41no—
139C22×C4×C53—42no—
140C22×C4×C53—42no—
141C22×C4×C53C22⋊C412yes—
142C22×C4×C53C22×C411yescharacteristic
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D412yes—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53#4112yes—
149C22×C4×C53#4112yes—
150C22×Dic53—28no—
151C22×Dic53—24no—
152C22×Dic53—44no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C4⋊C4×C53—28no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53C22⋊C414yescomplemented
158C22⋊C4×C53—28no—
159C22⋊C4×C53—28no—
160C22⋊C4×C53—24no—
161C42×C53—24no—
162C22⋊Dic53—216no—
163C4×Dic53—24no—
164C4×Dic53—24no—
165C4×Dic53—42no—
166C4×Dic53—42no—
167#11719917—24no—
168#11719917—24no—
169#11719917—42no—
170#11719917—42no—
171C2×#88322—24no—
172C2×#88322—42no—
173C2×#88322—42no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—24no—
177C23×C53—24no—
178C23×C53—22no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—48no—
186C23×C53—48no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—42no—
194C23×C53—42no—
195C23×C53#10414yescomplemented
196C23×C53C2×C22⋊C411yescharacteristic
197C23×C53#12011yescharacteristic
198C23×C53#12012yes—
199C23×C53#12112yes—
200C23×C53#12311yescharacteristic
201Q8×C53—24no—
202Q8×C53—42no—
203D4×C53—24no—
204D4×C53—416no—
205D4×C53—44no—
206D4×C53—44no—
207D4×C53—42no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—24no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—44no—
220C2×C4×C53—44no—
221C2×C4×C53—44no—
222C2×C4×C53—44no—
223C2×C4×C53—44no—
224C2×C4×C53—44no—
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53C2×C22⋊C411yescharacteristic
227C2×C4×C53C4×D412yes—
228C2×C4×C53C4×D412yes—
229C2×C4×C53#12012yes—
230C2×C4×C53#12011yescharacteristic
231C2×C4×C53#12012yes—
232C2×C4×C53#12112yes—
233C2×C4×C53#12112yes—
234C2×C4×C53#12112yes—
235C2×C4×C53#12112yes—
236C2×C4×C53#12112yes—
237C2×C4×C53#12312yes—
238C2×C4×C53#12311yescharacteristic
239C2×C4×C53#12412yes—
240C2×C4×C53#12712yes—
241C2×Dic53—28no—
242C2×Dic53—48no—
243C2×Dic53—44no—
244C2×Dic53—48no—
245C2×Dic53—44no—
246#88322—44no—
247#10971—531nomaximal, Sylow
248C22×C53—24no—
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—24no—
254C22×C53—24no—
255C22×C53—24no—
256C22×C53—24no—
257C22×C53—48no—
258C22×C53—48no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—44no—
265C22×C53—82no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53#37911yescharacteristic
269C22×C53#38611yescharacteristic
270C22×C53#38611yescharacteristic
271C22×C53#39212yes—
272C22×C53#39412yes—
273C22×C53C2×#10412yes—
274C22×C53#42112yes—
275C4×C53—42no—
276C4×C53—42no—
277C4×C53—44no—
278C4×C53#38612yes—
279C4×C53#39012yes—
280C4×C53#39212yes—
281C4×C53#39412yes—
282Dic53—48no—
283C2×#525D5311yescharacteristic
284C22×#104—532no—
285C2×#407—531no—
286C2×C4×C22⋊C4—531no—
287C2×#339—532no—
288C2×C53—22no—
289C2×C53—22no—
290C2×C53—44no—
291C2×C53—82no—
292C2×C53#143811yescharacteristic
293C2×C53#178912yes—
294C2×C53#187112yes—
295C2×C53#189611yescharacteristic
296C2×C53#189711yescharacteristic
297C22×#41C2×D5311yescharacteristic
298C24×C4C2×D5311yescharacteristic
299C22×M16—532no—
300#525—24no—
301#525—24no—
302C2×#123—21no—
303C2×C22⋊Q8Dic5312yescomplemented
304C2×#121Dic5312yescomplemented
305C2×#120—21no—
306C2×C4×D4—22no—
307C2×#117C2×D5311yescharacteristic
308C22×C22⋊C4—1061no—
309C22×C42—532no—
310C2×#104—1064no—
311C2×#104—1064no—
312C2×#104—1064no—
313#407—1064no—
314C2×C22⋊C8—1062no—
315C4×C22⋊C4—1064no—
316C2×#95—1061no—
317#339—1064no—
318C53#1097111yescharacteristic, complemented, Sylow
319C25C22⋊D5311yescharacteristic
320C2×#41—22no—
321C2×#41—22no—
322C2×#41—22no—
323C2×#41—22no—
324C2×#41—22no—
325C2×#41C22⋊D5314yes—
326C22×Q8C2×Dic5311yescharacteristic
327C22×D4—21no—
328C22×D4—22no—
329C22×D4C2×Dic5311yescharacteristic
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—21no—
337C23×C4—1062no—
338C23×C4C2×Dic5312yes—
339C23×C4C22⋊D5312yes—
340C23×C4C22⋊D5311yescharacteristic
341C23×C4C22×D5311yescharacteristic
342C2×M16—534no—
343C2×M16—1062no—
344C2×M16—1062no—
345C2×M16—1062no—
346C2×M16—1062no—
347C2×M16—1062no—
348C22×C8—1062no—
349#123—28no—
350C22⋊Q8—24no—
351C22⋊Q8—24no—
352#121—24no—
353#121—24no—
354#120—28no—
355C4×D4—216no—
356#117—22no—
357#117C22⋊D5314yes—
358C2×C4⋊C4—22no—
359C2×C4⋊C4C2×Dic5312yes—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—1068no—
364C2×C22⋊C4—1064no—
365C2×C42—21no—
366C2×C42—534no—
367C2×C42—1062no—
368C2×C42—1062no—
369C2×C42—1062no—
370C2×C42—1062no—
371C2×C42—1062no—
372#104—10616no—
373C22⋊C8—1068no—
374#95—1064no—
375C24—24no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—22no—
381C24—42no—
382C24C22⋊Dic5312yes—
383C24C2×C22⋊D5311yescharacteristic
384#41—24no—
385#41—24no—
386#41—24no—
387#41—24no—
388#41—42no—
389#41—42no—
390#41—42no—
391#41—42no—
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8C22⋊Dic5312yescomplemented
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—41no—
404C2×D4—42no—
405C2×D4—41no—
406C2×D4C22⋊Dic5312yescomplemented
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—1068no—
434C22×C4—1064no—
435C22×C4—2124no—
436C22×C4—2122no—
437C22×C4—2122no—
438C22×C4C22⋊Dic5312yes—
439C22×C4D4×D5312yes—
440C22×C4D4×D5312yes—
441C22×C4#1171994712yes—
442C22×C4#1171994712yes—
443C22×C4C22×Dic5311yescharacteristic
444C22×C4C2×C22⋊D5312yes—
445C22×C4C2×C22⋊D5312yes—
446C22×C4C2×C22⋊D5311yescharacteristic
447M16—1064no—
448M16—1064no—
449M16—2122no—
450M16—2122no—
451C2×C8—1064no—
452C2×C8—2122no—
453C2×C8—2122no—
454C4⋊C4—28no—
455C4⋊C4—22no—
456C4⋊C4C22⋊Dic5314yescomplemented
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—24no—
460C22⋊C4—10616no—
461C42—24no—
462C42—1064no—
463C42—1064no—
464C42—2122no—
465C42—2122no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—48no—
478C23—48no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23#1313314414yescomplemented
488C23C2×C22⋊Dic5311yescharacteristic
489C23#1313318511yescharacteristic
490C23#1313318612yes—
491C23#1313318912yes—
492C23#1313319811yescharacteristic
493Q8—24no—
494Q8—42no—
495D4—24no—
496D4—416no—
497D4—44no—
498D4—44no—
499D4—42no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—1068no—
518C2×C4—2128no—
519C2×C4—2124no—
520C2×C4—2128no—
521C2×C4—2124no—
522C2×C4C2×C22⋊Dic5311yescharacteristic
523C2×C4C2×C22⋊Dic5311yescharacteristic
524C2×C4D4×Dic5312yes—
525C2×C4D4×Dic5312yes—
526C2×C4#1313318512yes—
527C2×C4#1313318511yescharacteristic
528C2×C4#1313318612yes—
529C2×C4#1313318812yes—
530C2×C4#1313318812yes—
531C2×C4#1313318912yes—
532C2×C4#1313318912yes—
533C2×C4#1313318912yes—
534C2×C4#1313319812yes—
535C2×C4#1313319811yescharacteristic
536C2×C4#1313320412yes—
537C2×C4#1313321312yes—
538C8—2124no—
539C22—24no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—48no—
549C22—48no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—44no—
556C22—82no—
557C22—81no—
558C22—81no—
559C22#1681047911yescharacteristic
560C22#1681050311yescharacteristic
561C22#1681050311yescharacteristic
562C22#1681053112yes—
563C22#1681053712yes—
564C22C2×#1313314412yes—
565C22#1681064012yes—
566C4—42no—
567C4—42no—
568C4—44no—
569C4—2128no—
570C4#1681050312yes—
571C4#1681052612yes—
572C4#1681053112yes—
573C4#1681053712yes—
574C2—22no—
575C2—22no—
576C2—44no—
577C2—82no—
578C2#2027295611yescharacteristic
579C2#2027465912yes—
580C2#2027501812yes—
581C2#2027515911yescharacteristic
582C2#2027516411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC2×C53×#525C2
socleC23×C53C2×C22⋊C4
cosocleC23×C4×C53C23
hypercenterC2×#525D53
hypocenterC53#10971
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ C2×#525

Complements

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

NH ≅ G/Ntype
1. C22×#13133144C2semidirect
2. C2×#16810587C2semidirect
3. C2×C4×C22⋊Dic53C2semidirect
4. C2×#16810382C2semidirect
22. C2×C22⋊Q8×C53C4semidirect
23. C2×#121×C53C4semidirect
33. C2×#11719917D4semidirect
42. C2×C4×Dic53D4semidirect
103. C2×Q8×C53C22⋊C4semidirect
114. C2×D4×C53C22⋊C4semidirect
157. C4⋊C4×C53C22⋊C4semidirect
195. C23×C53#104semidirect
303. C2×C22⋊Q8Dic53semidirect
304. C2×#121Dic53semidirect
318. C53#10971semidirect
395. C2×Q8C22⋊Dic53semidirect
406. C2×D4C22⋊Dic53semidirect
456. C4⋊C4C22⋊Dic53semidirect
487. C23#13133144semidirect

Maximal subgroups

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

subgroupindex
C22×#131331442
C2×#168105872
C2×C4×C22⋊Dic532
C2×#168103822
C2×C53×#5252
#1097153