Finite Group Databasebeta

← #27300055

Subgroups — #27300055

Order 13568.

808 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
1C2×#16810410C212yesmaximal
2#20274707C211yesmaximal, characteristic, complemented
3#20274703C211yesmaximal, characteristic, complemented
4#20272833C212yesmaximal, complemented
5C4×#120×C53C211yesmaximal, characteristic
6#16810647—21no—
7C2×#13133134C2212yes—
8C2×#13133132C2212yes—
9#16810483—21no—
10#16810410—24no—
11#16810392—21no—
12#16810390—21no—
13C24×C4×C53C411yescharacteristic, complemented
14C2×#120×C53C2211yescharacteristic
15C2×C4×D4×C53—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#386—21no—
18C53×#386C2211yescharacteristic
19C53×#385—21no—
20C53×#385C411yescharacteristic, complemented
21C53×#379C411yescharacteristic, complemented
22C4×C22⋊C4×C53—21no—
23C4×C22⋊C4×C53C411yescharacteristic, complemented
24C2×#11719917—22no—
25C2×C4⋊Dic53—41no—
26C2×C22⋊Dic53—22no—
27C4⋊#88322—41no—
28C4⋊#88322—41no—
29#13133134—22no—
30#13133134—22no—
31#13133132—22no—
32#13133132—22no—
33#13133124—41no—
34C25×C53C2×C411yescharacteristic
35C22×D4×C53—21no—
36C22×D4×C53C2311yescharacteristic
37C23×C4×C53—21no—
38C23×C4×C53—21no—
39C23×C4×C53—44no—
40C23×C4×C53—41no—
41C23×C4×C53—41no—
42C23×C4×C53C2×C411yescharacteristic
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45#120×C53—21no—
46#120×C53—21no—
47#120×C53D412yes—
48#120×C53Q812yes—
49C4×D4×C53—41no—
50C4×D4×C53—41no—
51C4×D4×C53—44no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53C2×C411yescharacteristic
59C2×C22⋊C4×C53C2×C411yescharacteristic
60C2×C22⋊C4×C53C2×C411yescharacteristic
61C2×C22⋊C4×C53D411yescharacteristic
62C2×C42×C53—21no—
63C2×C42×C53D411yescharacteristic
64#95×C53—21no—
65#95×C53—21no—
66#95×C53—21no—
67C24×C53—21no—
68C24×C53—21no—
69C24×C53—42no—
70C24×C53—41no—
71C24×C53—42no—
72C24×C53—41no—
73C24×C53C4⋊C412yescomplemented
74C24×C53C22×C411yescharacteristic
75C24×C53C2×D411yescharacteristic
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—41no—
79C2×D4×C53—41no—
80C2×D4×C53—42no—
81C2×D4×C53—42no—
82C2×D4×C53C2×D411yescharacteristic
83C2×D4×C53C2×Q811yescharacteristic
84C2×D4×C53#4111yescharacteristic
85C2×D4×C53#4111yescharacteristic
86C22×C4×C53—24no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—42no—
95C22×C4×C53—42no—
96C22×C4×C53—42no—
97C22×C4×C53—42no—
98C22×C4×C53—42no—
99C22×C4×C53—42no—
100C22×C4×C53—42no—
101C22×C4×C53—44no—
102C22×C4×C53—44no—
103C22×C4×C53—44no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53—41no—
107C22×C4×C53—44no—
108C22×C4×C53C22⋊C411yescharacteristic
109C22×C4×C53C22⋊C411yescharacteristic
110C22×C4×C53C22⋊C411yescharacteristic
111C22×C4×C53C22⋊C411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×Dic53—42no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53—42no—
116C22⋊C4×C53—21no—
117C22⋊C4×C53—22no—
118C22⋊C4×C53—22no—
119C22⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—42no—
122C22⋊C4×C53—42no—
123C22⋊C4×C53—42no—
124C22⋊C4×C53—41no—
125C22⋊C4×C53C4⋊C412yescomplemented
126C42×C53—21no—
127C42×C53—21no—
128C42×C53—42no—
129C22⋊Dic53—42no—
130C22⋊Dic53—42no—
131C4⋊Dic53—82no—
132#11719917—42no—
133#11719917—42no—
134C2×#88322—42no—
135C23×C53—22no—
136C23×C53—22no—
137C23×C53—22no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—41no—
143C23×C53—44no—
144C23×C53—42no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—44no—
149C23×C53—42no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53—42no—
153C23×C53—42no—
154C23×C53—42no—
155C23×C53—48no—
156C23×C53—42no—
157C23×C53—41no—
158C23×C53—41no—
159C23×C53C2×C22⋊C411yescharacteristic
160C23×C53C2×C4⋊C411yescharacteristic
161C23×C53#11711yescharacteristic
162C23×C53#12111yescharacteristic
163C23×C53C22⋊Q811yescharacteristic
164D4×C53—41no—
165D4×C53—41no—
166D4×C53—41no—
167D4×C53—41no—
168D4×C53—44no—
169D4×C53—44no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—22no—
173C2×C4×C53—22no—
174C2×C4×C53—22no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—44no—
179C2×C4×C53—42no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—48no—
184C2×C4×C53—42no—
185C2×C4×C53—41no—
186C2×C4×C53—44no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—42no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—81no—
197C2×C4×C53#9911yescharacteristic
198C2×C4×C53#9911yescharacteristic
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53C22⋊Q811yescharacteristic
201C2×Dic53—82no—
202C2×Dic53—82no—
203C2×Dic53—82no—
204#88322—82no—
205#12793—531nomaximal, Sylow
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—24no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—42no—
218C22×C53—42no—
219C22×C53—42no—
220C22×C53—42no—
221C22×C53—44no—
222C22×C53—44no—
223C22×C53—44no—
224C22×C53—44no—
225C22×C53—42no—
226C22×C53—41no—
227C22×C53—42no—
228C22×C53—41no—
229C22×C53—41no—
230C22×C53—42no—
231C22×C53—41no—
232C22×C53—81no—
233C22×C53#38011yescharacteristic
234C22×C53C2×#9911yescharacteristic
235C22×C53#41011yescharacteristic
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—42no—
240C4×C53—42no—
241C4×C53—41no—
242C4×C53—41no—
243C4×C53—41no—
244C4×C53—81no—
245C4×C53AΣL(1,𝔽16)12yescomplemented
246Dic53—162no—
247C4×#120D5311yescharacteristic
248C2×AΣL(1,𝔽16)—532no—
249#1800—531no—
250#1799—531no—
251#1390—532no—
252C2×C53—21no—
253C2×C53—21no—
254C2×C53—41no—
255C2×C53—44no—
256C2×C53—41no—
257C2×C53—41no—
258C2×C53—41no—
259C2×C53#179911yescharacteristic
260C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
261C2×C53#212011yescharacteristic
262C24×C4Dic5311yescharacteristic, complemented
263C2×#120C2×D5311yescharacteristic
264C2×C4×D4—21no—
265C2×C4×D4C2×D5311yescharacteristic
266#423—1061no—
267C2×#100—532no—
268C2×#99—532no—
269#386—21no—
270#386C2×D5311yescharacteristic
271#385—21no—
272#385Dic5311yescharacteristic, complemented
273#380—1061no—
274#379Dic5311yescharacteristic, complemented
275C4×C22⋊C4—21no—
276C4×C22⋊C4Dic5311yescharacteristic, complemented
277AΣL(1,𝔽16)—1064no—
278#343—1061no—
279#342—1061no—
280C53#1279311yescharacteristic, complemented, Sylow
281C25C2×Dic5311yescharacteristic
282C22×D4—21no—
283C22×D4C22×D5311yescharacteristic
284C23×C4—21no—
285C23×C4—21no—
286C23×C4—44no—
287C23×C4—41no—
288C23×C4—41no—
289C23×C4C2×Dic5311yescharacteristic
290C23×C4C2×Dic5311yescharacteristic
291C23×C4C22⋊D5311yescharacteristic
292C2×M16—1062no—
293#120—21no—
294#120—21no—
295#120Dic10612yes—
296#120D21212yes—
297C4×D4—41no—
298C4×D4—41no—
299C4×D4—44no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—2121no—
302C2×C4⋊C4C22⋊D5311yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—1062no—
308C2×C22⋊C4C2×Dic5311yescharacteristic
309C2×C22⋊C4C2×Dic5311yescharacteristic
310C2×C22⋊C4C2×Dic5311yescharacteristic
311C2×C22⋊C4C22⋊D5311yescharacteristic
312C2×C42—21no—
313C2×C42C22⋊D5311yescharacteristic
314C4⋊C8—2121no—
315C4⋊C8—2121no—
316#100—1062no—
317#100—1062no—
318#99—1062no—
319#99—1062no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—2121no—
324C24—21no—
325C24—21no—
326C24—42no—
327C24—41no—
328C24—42no—
329C24—41no—
330C24C4⋊Dic5312yescomplemented
331C24C22×Dic5311yescharacteristic
332C24C2×C22⋊D5311yescharacteristic
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4C2×Dic10611yescharacteristic
340C2×D4C2×D21211yescharacteristic
341C2×D4#1171994511yescharacteristic
342C2×D4#1171994511yescharacteristic
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—44no—
365C22×C4—2122no—
366C22×C4C22⋊Dic5311yescharacteristic
367C22×C4C22⋊Dic5311yescharacteristic
368C22×C4C22⋊Dic5311yescharacteristic
369C22×C4C22⋊Dic5311yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371M16—2122no—
372M16—2122no—
373C2×C8—2122no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—4242no—
377C22⋊C4—21no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—41no—
381C22⋊C4—41no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—41no—
386C22⋊C4—2122no—
387C22⋊C4—2122no—
388C22⋊C4C4⋊Dic5312yescomplemented
389C42—21no—
390C42—21no—
391C42—42no—
392C23—22no—
393C23—22no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—21no—
399C23—41no—
400C23—41no—
401C23—44no—
402C23—42no—
403C23—41no—
404C23—41no—
405C23—44no—
406C23—42no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—48no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23C2×C22⋊Dic5311yescharacteristic
417C23C2×C4⋊Dic5311yescharacteristic
418C23#1313317611yescharacteristic
419C23#1313319111yescharacteristic
420C23#1313319611yescharacteristic
421D4—41no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—44no—
426D4—44no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—81no—
454C2×C4—4242no—
455C2×C4—4242no—
456C2×C4—4242no—
457C2×C4#1313313211yescharacteristic
458C2×C4#1313313211yescharacteristic
459C2×C4#1313319111yescharacteristic
460C2×C4#1313319611yescharacteristic
461C8—4242no—
462C22—21no—
463C22—21no—
464C22—24no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—44no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—42no—
482C22—42no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—42no—
487C22—41no—
488C22—81no—
489C22#1681048311yescharacteristic
490C22C2×#1313313211yescharacteristic
491C22#1681059711yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—42no—
496C4—42no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—81no—
501C4—8482no—
502C4#1681041012yescomplemented
503C2—21no—
504C2—21no—
505C2—41no—
506C2—44no—
507C2—41no—
508C2—41no—
509C2—41no—
510C2#2027470311yescharacteristic
511C2C2×#1681041011yescharacteristic
512C2#2027635611yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC4×#120×C53C2
socleC22×C53C2×#99
prime radicalC2×#120×C53C22
cosocleC22×D4×C53C23
hypercenterC4×#120D53
hypocenterC53#12793
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#120

Complements

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

NH ≅ G/Ntype
2. #20274707C2semidirect
3. #20274703C2semidirect
4. #20272833C2semidirect
13. C24×C4×C53C4semidirect
20. C53×#385C4semidirect
21. C53×#379C4semidirect
23. C4×C22⋊C4×C53C4semidirect
73. C24×C53C4⋊C4semidirect
125. C22⋊C4×C53C4⋊C4semidirect
245. C4×C53AΣL(1,𝔽16)semidirect
262. C24×C4Dic53semidirect
272. #385Dic53semidirect
274. #379Dic53semidirect
276. C4×C22⋊C4Dic53semidirect
280. C53#12793semidirect
330. C24C4⋊Dic53semidirect
388. C22⋊C4C4⋊Dic53semidirect
502. C4#16810410semidirect

Maximal subgroups

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

subgroupindex
C2×#168104102
#202747072
#202747032
#202728332
C4×#120×C532
#1279353