Finite Group Databasebeta

← #27276846

Subgroups — #27276846

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133191C211yesmaximal, characteristic
2C2×#16810358C212yesmaximal
3#20272945C211yesmaximal, characteristic, complemented
4#20272595C212yesmaximal, complemented
5C53×#1797C211yesmaximal, characteristic, complemented
6C2×#13133191C2211yescharacteristic
7C2×C4×D212C411yescharacteristic, complemented
8C2×C4×C22⋊D53—21no—
9C22×C4⋊Dic53—21no—
10#16810525C411yescharacteristic, complemented
11#16810521C411yescharacteristic, complemented
12#16810514—21no—
13#16810502—21no—
14#16810483C2211yescharacteristic
15C4×C4⋊Dic53C411yescharacteristic, complemented
16C4×#11719921—21no—
17C2×#13133124—21no—
18C2×#13133124C2212yes—
19#16810358—24no—
20C22×C42×C53C2211yescharacteristic
21C2×C4⋊C8×C53—21no—
22C2×C22⋊C8×C53C2212yes—
23C53×#336—21no—
24C22×D212C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26C23×Dic53—22no—
27C22×C4×D53—21no—
28#13133191—21no—
29#13133191—21no—
30#13133191D412yes—
31#13133191Q812yes—
32C4×D212—42no—
33C4×C22⋊D53—42no—
34C4×C22⋊D53—42no—
35C2×C4⋊Dic53—24no—
36C2×C4⋊Dic53—42no—
37C2×C4⋊Dic53—42no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×#11719919—21no—
42C2×C22⋊Dic53—21no—
43C2×#11719921—21no—
44C2×#11719921—21no—
45C2×#11719921—21no—
46C2×#11719921—21no—
47C2×C4×Dic53—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—22no—
52#13133124—22no—
53#13133124—42no—
54C23×C4×C53D411yescharacteristic
55C23×C4×C53D411yescharacteristic
56C23×C4×C53C2311yescharacteristic
57C22×C8×C53—22no—
58C2×C42×C53—42no—
59C2×C42×C53C2×C411yescharacteristic
60C2×C42×C53C2×C411yescharacteristic
61C2×C42×C53D411yescharacteristic
62C2×C42×C53D411yescharacteristic
63C4⋊C8×C53—42no—
64C22⋊C8×C53—22no—
65C22⋊C8×C53—22no—
66C24×C53C2×D411yescharacteristic
67C23×D53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53D812yes—
81C22×C4×C53SD1612yes—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×Q811yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C2×C22⋊D53—22no—
89C2×C22⋊D53—22no—
90C2×C22⋊D53—41no—
91C2×C22⋊D53—41no—
92C2×C22⋊D53—41no—
93C2×C22⋊D53—41no—
94C22×Dic53—21no—
95C22×Dic53—24no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—42no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C22×Dic53—42no—
102C22×Dic53—42no—
103C22×Dic53—42no—
104C2×D212—21no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212C4⋊C412yescomplemented
108C2×C4×D53—42no—
109C2×C4×D53—42no—
110C2×C4×D53—42no—
111C2×C8×C53—42no—
112C2×C8×C53—42no—
113C2×C8×C53—42no—
114C42×C53—22no—
115C42×C53—22no—
116C42×C53—41no—
117C42×C53—41no—
118C22⋊Dic53—41no—
119C22⋊Dic53—41no—
120#11719921—22no—
121#11719921—22no—
122#11719921—42no—
123#11719921—42no—
124#11719921—42no—
125C4⋊Dic53—21no—
126C4⋊Dic53—48no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129C4⋊Dic53C4⋊C412yescomplemented
130#11719919—42no—
131C4×Dic53—42no—
132C23×C53—21no—
133C23×C53—21no—
134C23×C53—21no—
135C23×C53—21no—
136C23×C53C2×C22⋊C411yescharacteristic
137C23×C53#12111yescharacteristic
138C23×C53C22⋊Q811yescharacteristic
139C23×C53C2×D811yescharacteristic
140C23×C53C2×SD1611yescharacteristic
141C23×C53AGL(1,ℤ/8)11yescharacteristic
142C23×C53#13711yescharacteristic
143C22×D53—22no—
144C22×D53—41no—
145C22×D53—41no—
146C22×D53—41no—
147C22×D53—41no—
148C22×D53—41no—
149C22×D53—41no—
150C2×C4×C53—24no—
151C2×C4×C53—24no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—42no—
165C2×C4×C53—42no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53#9912yes—
171C2×C4×C53#10212yes—
172C2×C4×C53#10212yes—
173C2×C4×C53#10412yes—
174C2×C4×C53C2×C4⋊C411yescharacteristic
175C2×C4×C53#11711yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53C22⋊Q811yescharacteristic
178C22⋊D53—42no—
179C22⋊D53—42no—
180C22⋊D53—42no—
181C22⋊D53—42no—
182C2×Dic53—22no—
183C2×Dic53—41no—
184C2×Dic53—41no—
185C2×Dic53—44no—
186C2×Dic53—44no—
187C2×Dic53—44no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—82no—
195C2×Dic53—82no—
196C2×Dic53—82no—
197C2×Dic53—82no—
198D212—42no—
199D212—42no—
200C4×D53—84no—
201C8×C53—82no—
202#8577—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—21no—
215C22×C53#38011yescharacteristic
216C22×C53C2×#9911yescharacteristic
217C22×C53#41011yescharacteristic
218C22×C53C2×#10211yescharacteristic
219C22×C53#41711yescharacteristic
220C22×C53C2×#10411yescharacteristic
221C22×C53#42111yescharacteristic
222C22×C53#45911yescharacteristic
223C22×C53#46111yescharacteristic
224C22×C53#47411yescharacteristic
225C22×C53#47611yescharacteristic
226C2×D53—42no—
227C2×D53—42no—
228C2×D53—42no—
229C2×D53—81no—
230C2×D53—81no—
231C2×D53—81no—
232C2×D53—81no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—22no—
238C4×C53—22no—
239C4×C53—22no—
240C4×C53—22no—
241Dic53—42no—
242Dic53—84no—
243Dic53—81no—
244Dic53—81no—
245C4×#121—531no—
246#1797D5311yescharacteristic, complemented
247C2×#327—532no—
248C4⋊#95—531no—
249#1294—532no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53—21no—
253C2×C53—21no—
254C2×C53#150211yescharacteristic
255C2×C53#150411yescharacteristic
256C2×C53#151211yescharacteristic
257C2×C53#151911yescharacteristic
258C2×C53#179911yescharacteristic
259C2×C53#180211yescharacteristic
260C2×C53#180611yescharacteristic
261D53—82no—
262C2×#121—531no—
263C2×C4×D4—531no—
264C2×C4×D4—1061no—
265C22×C4⋊C4—1061no—
266C22×C42C2×D5311yescharacteristic
267C2×C4⋊C8—21no—
268C2×C22⋊C8C2×D5312yes—
269#390—531no—
270#389—531no—
271#388—1061no—
272#386—1061no—
273#380—531no—
274C4×C4⋊C4—531no—
275C4×C22⋊C4—1061no—
276C2×#95—532no—
277C2×#95—1061no—
278#336—21no—
279#327—1064no—
280C53#857711yescharacteristic, complemented, Sylow
281C22×D4—531no—
282C22×D4—1061no—
283C23×C4—1062no—
284C23×C4—1061no—
285C23×C4D21211yescharacteristic
286C23×C4D21211yescharacteristic
287C23×C4C22×D5311yescharacteristic
288C22×C8—22no—
289#121—532no—
290#121—532no—
291#121—1061no—
292#121—1061no—
293C4×D4—2122no—
294C4×D4—2122no—
295C4×D4—2122no—
296C2×C4⋊C4—531no—
297C2×C4⋊C4—531no—
298C2×C4⋊C4—531no—
299C2×C4⋊C4—1064no—
300C2×C4⋊C4—1061no—
301C2×C4⋊C4—2122no—
302C2×C4⋊C4—2122no—
303C2×C22⋊C4—1061no—
304C2×C22⋊C4—1061no—
305C2×C22⋊C4—1061no—
306C2×C22⋊C4—1061no—
307C2×C22⋊C4—1061no—
308C2×C42—42no—
309C2×C42—1061no—
310C2×C42C4×D5311yescharacteristic
311C2×C42C4×D5311yescharacteristic
312C2×C42C22⋊D5311yescharacteristic
313C2×C42C22⋊D5311yescharacteristic
314C4⋊C8—42no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317#95—1061no—
318#95—1061no—
319#95—1061no—
320#95—1062no—
321#95—1062no—
322#95—2122no—
323C24—1061no—
324C24C2×D21211yescharacteristic
325C2×D4—532no—
326C2×D4—1062no—
327C2×D4—1062no—
328C2×D4—1061no—
329C2×D4—2121no—
330C2×D4—2121no—
331C2×D4—2121no—
332C2×D4—2121no—
333C2×D4—2121no—
334C2×D4—2121no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—1061no—
344C22×C4—1064no—
345C22×C4—1061no—
346C22×C4—1061no—
347C22×C4—2122no—
348C22×C4—2122no—
349C22×C4—2122no—
350C22×C4—2122no—
351C22×C4—2122no—
352C22×C4—2122no—
353C22×C4—2122no—
354C22×C4—2122no—
355C22×C4—2122no—
356C22×C4#1171991312yes—
357C22×C4D42412yes—
358C22×C4#1171992111yescharacteristic
359C22×C4#1171992111yescharacteristic
360C22×C4#1171992111yescharacteristic
361C22×C4#1171992111yescharacteristic
362C22×C4C2×C4×D5311yescharacteristic
363C22×C4D4×D5311yescharacteristic
364C22×C4#1171994711yescharacteristic
365C22×C4Q8×D5311yescharacteristic
366C22×C4#1171994911yescharacteristic
367C22×C4C2×C22⋊D5311yescharacteristic
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—532no—
372C4⋊C4—1061no—
373C4⋊C4—2121no—
374C4⋊C4—2121no—
375C4⋊C4—2128no—
376C4⋊C4—2122no—
377C22⋊C4—1062no—
378C22⋊C4—1062no—
379C22⋊C4—2121no—
380C22⋊C4—2121no—
381C22⋊C4—2122no—
382C22⋊C4—2122no—
383C22⋊C4—2122no—
384C42—22no—
385C42—22no—
386C42—41no—
387C42—41no—
388C42—2122no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—1062no—
394C23—2121no—
395C23—2121no—
396C23—2121no—
397C23—2121no—
398C23—2121no—
399C23—2121no—
400C23C2×#1171992111yescharacteristic
401C23#1313319211yescharacteristic
402C23#1313319711yescharacteristic
403C23C2×D42411yescharacteristic
404C23C2×#1171991311yescharacteristic
405C23#1313324511yescharacteristic
406C23#1313325011yescharacteristic
407D4—2122no—
408D4—2122no—
409D4—2122no—
410D4—2122no—
411D4—2122no—
412D4—2122no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—1062no—
434C2×C4—4244no—
435C2×C4—4242no—
436C2×C4—4242no—
437C2×C4—4242no—
438C2×C4—4242no—
439C2×C4—2124no—
440C2×C4—2124no—
441C2×C4—2124no—
442C2×C4—2121no—
443C2×C4—2121no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4—2121no—
448C2×C4—2121no—
449C2×C4—2121no—
450C2×C4#1313313112yes—
451C2×C4#1313313912yes—
452C2×C4#1313313912yes—
453C2×C4#1313314312yes—
454C2×C4C4⋊C4×D5311yescharacteristic
455C2×C4#1313317511yescharacteristic
456C2×C4#1313318811yescharacteristic
457C2×C4#1313319311yescharacteristic
458C8—82no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—4241no—
472C22—4241no—
473C22—4241no—
474C22—4241no—
475C22—2122no—
476C22—2122no—
477C22—2122no—
478C22#1681048011yescharacteristic
479C22C2×#1313313111yescharacteristic
480C22#1681059611yescharacteristic
481C22C2×#1313313911yescharacteristic
482C22#1681062511yescharacteristic
483C22C2×#1313314311yescharacteristic
484C22#1681063911yescharacteristic
485C22#1681081911yescharacteristic
486C22#1681082911yescharacteristic
487C22#1681090411yescharacteristic
488C22#1681091411yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—4244no—
498C4—4241no—
499C4—4241no—
500C4—2122no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—4242no—
506C2#2027321111yescharacteristic
507C2#2027322011yescharacteristic
508C2#2027325411yescharacteristic
509C2#2027328411yescharacteristic
510C2#2027470011yescharacteristic
511C2#2027471611yescharacteristic
512C2#2027473211yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1797C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1797D53
hypocenterC53#8577
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #20272945C2semidirect
4. #20272595C2semidirect
5. C53×#1797C2semidirect
7. C2×C4×D212C4semidirect
10. #16810525C4semidirect
11. #16810521C4semidirect
15. C4×C4⋊Dic53C4semidirect
107. C2×D212C4⋊C4semidirect
129. C4⋊Dic53C4⋊C4semidirect
246. #1797D53semidirect
280. C53#8577semidirect

Maximal subgroups

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

subgroupindex
C4×#131331912
C2×#168103582
#202729452
#202725952
C53×#17972
#857753