Finite Group Databasebeta

← #27301572

Subgroups — #27301572

Order 13568.

864 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133134C211yesmaximal, characteristic
2C2×#16810413C212yesmaximal, complemented
3#20274630C211yesmaximal, characteristic
4#20272833C212yesmaximal, complemented
5C53×#2405C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×#13133134—24no—
8C2×#13133134—21no—
9C2×#13133134C2212yes—
10C2×#13133132—21no—
11C2×#13133132C2212yes—
12#16810479—21no—
13#16810413—24no—
14C23×D4×C53C2211yescharacteristic
15C2×#123×C53C411yescharacteristic, complemented
16C2×#123×C53C2211yescharacteristic
17C22×C22⋊C4×C53C411yescharacteristic, complemented
18C53×#394—21no—
19C53×#394—21no—
20C53×#386C411yescharacteristic, complemented
21C53×#386C2211yescharacteristic
22C53×#385—21no—
23C53×#379—21no—
24C53×#379C411yescharacteristic, complemented
25C2×#11719917—24no—
26C2×#11719917—22no—
27C2×#11719917—41no—
28C2×#11719917—42no—
29C22×#88322—41no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—41no—
32C2×C22⋊Dic53—41no—
33#13133134—24no—
34#13133134—22no—
35#13133134D414yes—
36#13133132—22no—
37#13133132—22no—
38#13133132—42no—
39C25×C53C2×C411yescharacteristic
40C25×C53D411yescharacteristic
41C22×D4×C53—22no—
42C22×D4×C53—22no—
43C22×D4×C53—42no—
44C22×D4×C53—41no—
45C22×D4×C53—41no—
46C22×D4×C53C2×C411yescharacteristic
47C22×D4×C53D411yescharacteristic
48C22×D4×C53D411yescharacteristic
49C22×D4×C53C2311yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53C2×C411yescharacteristic
52C23×C4×C53D411yescharacteristic
53#123×C53—22no—
54#123×C53—22no—
55#123×C53—42no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—41no—
67C2×C22⋊C4×C53—41no—
68C2×C22⋊C4×C53C2×C411yescharacteristic
69C2×C22⋊C4×C53C2×C411yescharacteristic
70#95×C53—21no—
71#95×C53—21no—
72C24×C53—21no—
73C24×C53—21no—
74C24×C53—21no—
75C24×C53—21no—
76C24×C53—22no—
77C24×C53—22no—
78C24×C53—41no—
79C24×C53—41no—
80C24×C53—42no—
81C24×C53—42no—
82C24×C53—81no—
83C24×C53—81no—
84C24×C53C22⋊C411yescharacteristic
85C24×C53C22⋊C411yescharacteristic
86C24×C53C22⋊C411yescharacteristic
87C24×C53C22×C411yescharacteristic
88C24×C53C2×D411yescharacteristic
89C2×D4×C53—22no—
90C2×D4×C53—22no—
91C2×D4×C53—22no—
92C2×D4×C53—22no—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—41no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—41no—
101C2×D4×C53—44no—
102C2×D4×C53—44no—
103C2×D4×C53—44no—
104C2×D4×C53—44no—
105C2×D4×C53C2×D412yes—
106C2×D4×C53#4112yes—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—41no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53—81no—
121C22×C4×C53C22⋊C411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×Dic53—42no—
124C4⋊C4×C53—42no—
125C4⋊C4×C53—41no—
126C4⋊C4×C53—41no—
127C22⋊C4×C53—22no—
128C22⋊C4×C53—42no—
129C22⋊C4×C53—42no—
130C22⋊C4×C53—42no—
131C22⋊C4×C53—41no—
132C22⋊C4×C53—41no—
133C22⋊C4×C53—42no—
134C22⋊C4×C53—41no—
135C22⋊C4×C53—81no—
136C22⋊C4×C53—81no—
137C22⋊C4×C53—81no—
138C22⋊Dic53—42no—
139C22⋊Dic53—42no—
140C22⋊Dic53—82no—
141C22⋊Dic53—82no—
142#11719917—24no—
143#11719917—44no—
144#11719917—42no—
145C2×#88322—44no—
146C2×#88322—42no—
147C23×C53—22no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—41no—
161C23×C53—42no—
162C23×C53—42no—
163C23×C53—42no—
164C23×C53—42no—
165C23×C53—41no—
166C23×C53—41no—
167C23×C53—41no—
168C23×C53—41no—
169C23×C53—41no—
170C23×C53—41no—
171C23×C53—42no—
172C23×C53—42no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—81no—
178C23×C53—81no—
179C23×C53—81no—
180C23×C53—81no—
181C23×C53—84no—
182C23×C53—84no—
183C23×C53—84no—
184C23×C53—84no—
185C23×C53#9911yescharacteristic
186C23×C53#9911yescharacteristic
187C23×C53C2×C22⋊C411yescharacteristic
188C23×C53C4×D412yes—
189C23×C53#12112yes—
190D4×C53—42no—
191D4×C53—42no—
192D4×C53—44no—
193D4×C53—44no—
194D4×C53—42no—
195D4×C53—42no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—82no—
217C2×C4×C53—82no—
218C2×C4×C53#9912yes—
219C2×C4×C53#12011yescharacteristic
220C2×C4×C53#12311yescharacteristic
221C2×Dic53—82no—
222C2×Dic53—82no—
223C2×Dic53—82no—
224#88322—44no—
225#13080—531nomaximal, Sylow
226C22×C53—21no—
227C22×C53—21no—
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—22no—
233C22×C53—21no—
234C22×C53—22no—
235C22×C53—42no—
236C22×C53—42no—
237C22×C53—42no—
238C22×C53—42no—
239C22×C53—42no—
240C22×C53—42no—
241C22×C53—41no—
242C22×C53—41no—
243C22×C53—41no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53—41no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—81no—
252C22×C53—84no—
253C22×C53—84no—
254C22×C53—81no—
255C22×C53—81no—
256C22×C53—81no—
257C22×C53—84no—
258C22×C53—84no—
259C22×C53#35212yescomplemented
260C22×C53#38611yescharacteristic
261C22×C53C2×#9911yescharacteristic
262C22×C53C2×#9911yescharacteristic
263C4×C53—21no—
264C4×C53—22no—
265C4×C53—21no—
266C4×C53—81no—
267C4×C53—81no—
268C4×C53—81no—
269C4×C53—81no—
270Dic53—162no—
271C22×#100—531no—
272#2405D5311yescharacteristic
273C2×#352—532no—
274#1781—531no—
275#1390—532no—
276C2×C53—21no—
277C2×C53—21no—
278C2×C53—22no—
279C2×C53—42no—
280C2×C53—41no—
281C2×C53—42no—
282C2×C53—81no—
283C2×C53—81no—
284C2×C53#189811yescharacteristic
285C2×C53C2×#35211yescharacteristic
286C2×C53#212111yescharacteristic
287C23×D4C2×D5311yescharacteristic
288C22×M16—1061no—
289C2×#123Dic5311yescharacteristic, complemented
290C2×#123C2×D5311yescharacteristic
291C22×C22⋊C4Dic5311yescharacteristic, complemented
292C2×#100—532no—
293C2×#100—1064no—
294C2×#100—1061no—
295C2×#99—532no—
296C2×#99—1061no—
297#394—21no—
298#394—21no—
299#386Dic5311yescharacteristic, complemented
300#386C2×D5311yescharacteristic
301#385—21no—
302#379—21no—
303#379—1061no—
304#379Dic5311yescharacteristic, complemented
305#352—1064no—
306C53#1308011yescharacteristic, complemented, Sylow
307C25C2×Dic5311yescharacteristic
308C25C22⋊D5311yescharacteristic
309C22×D4—22no—
310C22×D4—22no—
311C22×D4—42no—
312C22×D4—41no—
313C22×D4—41no—
314C22×D4C2×Dic5311yescharacteristic
315C22×D4C22⋊D5311yescharacteristic
316C22×D4C22⋊D5311yescharacteristic
317C22×D4C22×D5311yescharacteristic
318C23×C4C2×Dic5311yescharacteristic
319C23×C4C2×Dic5311yescharacteristic
320C23×C4C22⋊D5311yescharacteristic
321C2×M16—1064no—
322C2×M16—1062no—
323C2×M16—2121no—
324C2×M16—2122no—
325C22×C8—2121no—
326#123—22no—
327#123—22no—
328#123—42no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—41no—
340C2×C22⋊C4—41no—
341C2×C22⋊C4—1062no—
342C2×C22⋊C4—2121no—
343C2×C22⋊C4—2121no—
344C2×C22⋊C4C2×Dic5311yescharacteristic
345C2×C22⋊C4C2×Dic5311yescharacteristic
346#100—534no—
347#100—1064no—
348#100—1062no—
349#99—1062no—
350#99—1062no—
351#99—2122no—
352#95—21no—
353#95—21no—
354C24—21no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—41no—
361C24—41no—
362C24—42no—
363C24—42no—
364C24—81no—
365C24—81no—
366C24C22⋊Dic5311yescharacteristic
367C24C22⋊Dic5311yescharacteristic
368C24C22⋊Dic5311yescharacteristic
369C24C22×Dic5311yescharacteristic
370C24C2×C22⋊D5311yescharacteristic
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4D4×D5312yes—
388C2×D4#1171994712yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—81no—
403C22×C4—2122no—
404C22×C4C22⋊Dic5311yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406M16—1064no—
407M16—2124no—
408M16—2122no—
409C2×C8—2124no—
410C2×C8—2122no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C22⋊C4—22no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—42no—
421C22⋊C4—41no—
422C22⋊C4—81no—
423C22⋊C4—81no—
424C22⋊C4—81no—
425C22⋊C4—4242no—
426C22⋊C4—4242no—
427C22⋊C4—2122no—
428C22⋊C4—2122no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—22no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23#1313313211yescharacteristic
468C23#1313313211yescharacteristic
469C23C2×C22⋊Dic5311yescharacteristic
470C23D4×Dic5312yes—
471C23#1313318912yes—
472D4—42no—
473D4—42no—
474D4—44no—
475D4—44no—
476D4—42no—
477D4—42no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—4242no—
501C2×C4—4242no—
502C2×C4—4242no—
503C2×C4#1313313212yes—
504C2×C4#1313318511yescharacteristic
505C2×C4#1313319811yescharacteristic
506C8—2124no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—22no—
515C22—22no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—81no—
533C22—84no—
534C22—84no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—84no—
539C22—84no—
540C22#1681041312yescomplemented
541C22#1681050311yescharacteristic
542C22C2×#1313313211yescharacteristic
543C22C2×#1313313211yescharacteristic
544C4—21no—
545C4—22no—
546C4—21no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—8482no—
552C2—21no—
553C2—21no—
554C2—22no—
555C2—42no—
556C2—41no—
557C2—42no—
558C2—81no—
559C2—81no—
560C2#2027516911yescharacteristic
561C2C2×#1681041311yescharacteristic
562C2#2027636111yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2405C2
socleC22×C53C2×#99
prime radicalC23×D4×C53C22
cosocleC22×D4×C53C23
hypercenter#2405D53
hypocenterC53#13080
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2405

Complements

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

NH ≅ G/Ntype
2. C2×#16810413C2semidirect
4. #20272833C2semidirect
15. C2×#123×C53C4semidirect
17. C22×C22⋊C4×C53C4semidirect
20. C53×#386C4semidirect
24. C53×#379C4semidirect
259. C22×C53#352semidirect
289. C2×#123Dic53semidirect
291. C22×C22⋊C4Dic53semidirect
299. #386Dic53semidirect
304. #379Dic53semidirect
306. C53#13080semidirect
540. C22#16810413semidirect

Maximal subgroups

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

subgroupindex
C22×#131331342
C2×#168104132
#202746302
#202728332
C53×#24052
#1308053