Finite Group Databasebeta

← #27394673

Subgroups — #27394673

Order 13568.

786 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810678C211yesmaximal, characteristic
2#20277120C211yesmaximal, characteristic
3#20277108C211yesmaximal, characteristic
4#20275486C212yesmaximal, complemented
5#20275482C212yesmaximal, complemented
6#20274917C212yesmaximal
7#20274915C212yesmaximal
8#20274883C211yesmaximal, characteristic, complemented
9#20274881C211yesmaximal, characteristic, complemented
10#20274848C211yesmaximal, characteristic, complemented
11C53×#1836C211yesmaximal, characteristic, complemented
12C2×#13133175C2212yes—
13C2×C42×D53C2211yescharacteristic
14#16810678C418yescomplemented
15C2×#13133151C2212yes—
16C2×#13133150C2212yes—
17C2×#13133149C2212yes—
18C2×#13133148C2212yes—
19C2×#13133146C2212yes—
20C2×#13133129C2212yes—
21C2×#13133128C2211yescharacteristic
22#16810493C2211yescharacteristic
23#16810490—21no—
24#16810489C2211yescharacteristic
25#16810488C2212yes—
26#16810485—21no—
27#16810480—21no—
28#16810480—21no—
29#16810480C2211yescharacteristic
30#16810480C2211yescharacteristic
31C4⋊C4×Dic53—21no—
32C4⋊C4×Dic53—21no—
33C4⋊C4×Dic53C2211yescharacteristic
34C4⋊C4×Dic53C2211yescharacteristic
35#16810470—21no—
36#16810468—21no—
37#16810385C2212yes—
38#16810385C2212yes—
39C2×#97×C53C2211yescharacteristic
40C53×#384C2211yescharacteristic
41C53×#382C2211yescharacteristic
42C53×#340C2212yes—
43C53×#340C2212yes—
44C22×C4×D53D411yescharacteristic
45C22×C4×D53D411yescharacteristic
46C22×C4×D53C2311yescharacteristic
47C22×#88322C2312yes—
48#13133175—42no—
49#13133175—42no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53C2312yes—
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×#11719921—22no—
56C2×#11719921—21no—
57C2×#11719921—21no—
58C2×C4×Dic53—22no—
59C2×C4×Dic53—21no—
60C2×C4×Dic53—21no—
61C2×C4×Dic53D411yescharacteristic
62C2×C4×Dic53D411yescharacteristic
63C2×C4×Dic53C2311yescharacteristic
64C42×D53C2×C414yes—
65C42×D53Q814yes—
66#13133151—42no—
67#13133150—42no—
68#13133149—42no—
69#13133148—42no—
70#13133146C2×C418yes—
71#13133129C2×C418yes—
72#13133128C2×C414yes—
73#13133124—21no—
74#13133124—21no—
75#13133124—21no—
76#13133124—21no—
77#13133124—22no—
78#13133124—22no—
79#13133124—21no—
80#13133124—21no—
81C22×C8×C53C2312yes—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53C2312yes—
85C2×C4⋊C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C42×C53C2311yescharacteristic
88#97×C53C2×C414yes—
89#95×C53—21no—
90#95×C53—21no—
91C23×D53C2×D411yescharacteristic
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—22no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2411yescharacteristic
98C22×Dic53—21no—
99C22×Dic53—21no—
100C22×Dic53—22no—
101C22×Dic53C2×D411yescharacteristic
102C22×Dic53C2×D411yescharacteristic
103C22×Dic53C2×D411yescharacteristic
104C2×C4×D53—22no—
105C2×C4×D53—22no—
106C2×C4×D53—41no—
107C2×C4×D53—41no—
108C2×C4×D53—41no—
109C2×C4×D53—41no—
110C2×C4×D53C4⋊C414yes—
111C2×C4×D53C4⋊C414yes—
112C2×C4×D53C22×C412yes—
113C2×C4×D53C2×Q812yes—
114C2×C8×C53—22no—
115C2×C8×C53—22no—
116C2×C8×C53C22×C414yes—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—42no—
122C4⋊C4×C53—42no—
123C42×C53C22×C412yes—
124C42×C53C2×Q812yes—
125#11719921—42no—
126#11719921—42no—
127#11719921—44no—
128C4⋊Dic53—22no—
129C4⋊Dic53—22no—
130C4⋊Dic53—22no—
131C4⋊Dic53—22no—
132C4⋊Dic53—42no—
133C4⋊Dic53—42no—
134C4×Dic53—42no—
135C4×Dic53—42no—
136C4×Dic53—44no—
137C4×Dic53C4⋊C414yes—
138C4×Dic53C4⋊C414yes—
139C4×Dic53C22×C412yes—
140C4×Dic53C2×Q812yes—
141C2×#88322—22no—
142C2×#88322—22no—
143C2×#88322C22×C414yes—
144C23×C53C22×D411yescharacteristic
145C22×D53—21no—
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—21no—
149C22×D53—21no—
150C22×D53—21no—
151C22×D53C2×C4⋊C412yes—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—42no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53C2×C4⋊C412yes—
166C2×C4×C53C2×C4⋊C412yes—
167C2×C4×C53C23×C411yescharacteristic
168C2×C4×C53C22×Q811yescharacteristic
169C2×C4×C53H(5,𝔽2)12yes—
170C2×C4×C532−1+412yes—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—42no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—42no—
183C2×Dic53—42no—
184C2×Dic53—41no—
185C2×Dic53—41no—
186C2×Dic53C2×C4⋊C412yes—
187C2×Dic53C2×C4⋊C412yes—
188C2×Dic53C2×C4⋊C412yes—
189C4×D53—24no—
190C4×D53—24no—
191C4×D53—44no—
192C4×D53—44no—
193C8×C53—24no—
194#88322—24no—
195#21989—531nomaximal, Sylow
196C22×C53C22×C4⋊C411yescharacteristic
197C22×C53#53511yescharacteristic
198C22×C53#53611yescharacteristic
199C22×C53#57511yescharacteristic
200C22×C53#57511yescharacteristic
201C22×C53#57511yescharacteristic
202C22×C53#57511yescharacteristic
203C2×D53—22no—
204C2×D53—22no—
205C2×D53—22no—
206C2×D53—41no—
207C2×D53—41no—
208C2×D53—41no—
209C2×D53—41no—
210C4×C53—22no—
211C4×C53—22no—
212C4×C53—82no—
213C4×C53—81no—
214C4×C53—81no—
215C4×C53#52012yes—
216C4×C53#54412yes—
217Dic53—22no—
218Dic53—22no—
219Dic53—22no—
220Dic53—81no—
221Dic53—82no—
222Dic53—81no—
223C2×#432—531no—
224#2285—531no—
225#2283—531no—
226#1949—532no—
227#1949—532no—
228#1854—532no—
229#1854—532no—
230#1847—531no—
231#1847—531no—
232#1836—531no—
233#1836D5311yescharacteristic, complemented
234C2×C53#231711yescharacteristic
235C2×C53#291111yescharacteristic
236C2×C53#291111yescharacteristic
237C2×C53#321911yescharacteristic
238C2×C53#321911yescharacteristic
239C2×C53#324011yescharacteristic
240C2×C53#324011yescharacteristic
241D53—42no—
242C2×#117—532no—
243C22×C42—531no—
244#432—538no—
245C2×#107—532no—
246C2×#107—532no—
247C2×#106—532no—
248C2×#106—532no—
249C2×C4⋊C8—532no—
250C2×C22⋊C8—532no—
251C2×#97—531no—
252C2×#97C2×D5311yescharacteristic
253#384—531no—
254#384C2×D5311yescharacteristic
255#383—1061no—
256#382—532no—
257#382—531no—
258#382C2×D5311yescharacteristic
259#381—1061no—
260#380—531no—
261#380—531no—
262#380—1061no—
263#380—1061no—
264C4×C4⋊C4—531no—
265C4×C4⋊C4—531no—
266C4×C4⋊C4—1061no—
267C4×C4⋊C4—1061no—
268C4×C22⋊C4—1061no—
269#376—1061no—
270#340—532no—
271#340—532no—
272#340C2×D5312yes—
273#340C2×D5312yes—
274C53#2198911yescharacteristic, complemented, Sylow
275C23×C4—531no—
276C23×C4—531no—
277C23×C4—531no—
278C22×C8—532no—
279C22×C8C22×D5312yes—
280#117—2122no—
281#117—2122no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—532no—
285C2×C4⋊C4—531no—
286C2×C4⋊C4—531no—
287C2×C4⋊C4—1061no—
288C2×C4⋊C4—1061no—
289C2×C4⋊C4C22×D5312yes—
290C2×C4⋊C4C22×D5311yescharacteristic
291C2×C4⋊C4C22×D5311yescharacteristic
292C2×C22⋊C4—1062no—
293C2×C22⋊C4—1061no—
294C2×C22⋊C4—1061no—
295C2×C42—531no—
296C2×C42—531no—
297C2×C42—531no—
298C2×C42—534no—
299C2×C42—534no—
300C2×C42—1062no—
301C2×C42—1061no—
302C2×C42—1061no—
303C2×C42C22×D5311yescharacteristic
304#107—2122no—
305#107—2122no—
306#106—2122no—
307#106—2122no—
308C4⋊C8—538no—
309C22⋊C8—538no—
310#97—534no—
311#97C4×D5314yes—
312#95—21no—
313#95—21no—
314#95—1061no—
315#95—1061no—
316#95—1061no—
317#95—1061no—
318#95—1062no—
319#95—1062no—
320#95—1061no—
321#95—1061no—
322C24—531no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—531no—
329C22×C4—532no—
330C22×C4—532no—
331C22×C4—534no—
332C22×C4—534no—
333C22×C4—1062no—
334C22×C4—1062no—
335C22×C4—1061no—
336C22×C4—1061no—
337C22×C4—1062no—
338C22×C4—2121no—
339C22×C4—2121no—
340C22×C4—2121no—
341C22×C4—2121no—
342C22×C4D4×D5311yescharacteristic
343C22×C4D4×D5311yescharacteristic
344C22×C4C23×D5311yescharacteristic
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—534no—
348C2×C8—1062no—
349C2×C8—1062no—
350C2×C8C2×C4×D5314yes—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—1062no—
358C4⋊C4—1062no—
359C4⋊C4—1062no—
360C4⋊C4—1062no—
361C4⋊C4—2122no—
362C4⋊C4—2122no—
363C22⋊C4—2124no—
364C22⋊C4—2122no—
365C22⋊C4—2122no—
366C42—534no—
367C42—534no—
368C42—532no—
369C42—532no—
370C42—2124no—
371C42—2122no—
372C42—2122no—
373C42C2×C4×D5312yes—
374C42Q8×D5312yes—
375C23—532no—
376C23—1061no—
377C23—1061no—
378C23—1061no—
379C23—1061no—
380C23—1061no—
381C23—1061no—
382C23C2×D4×D5311yescharacteristic
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—532no—
397C2×C4—532no—
398C2×C4—532no—
399C2×C4—1061no—
400C2×C4—1061no—
401C2×C4—1061no—
402C2×C4—1061no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—1064no—
406C2×C4—1064no—
407C2×C4—2124no—
408C2×C4—2124no—
409C2×C4—2121no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4—2122no—
414C2×C4—2122no—
415C2×C4—2122no—
416C2×C4—2121no—
417C2×C4—2121no—
418C2×C4C4⋊C4×D5312yes—
419C2×C4C4⋊C4×D5312yes—
420C2×C4C22×C4×D5311yescharacteristic
421C2×C4C2×Q8×D5311yescharacteristic
422C2×C4#1313326312yes—
423C2×C4#1313326412yes—
424C8—24no—
425C8—1064no—
426C22—1062no—
427C22—1062no—
428C22—1062no—
429C22—2121no—
430C22—2121no—
431C22—2121no—
432C22—2121no—
433C22C2×C4⋊C4×D5311yescharacteristic
434C22#1681119111yescharacteristic
435C22#1681119311yescharacteristic
436C22#1681141911yescharacteristic
437C22#1681141911yescharacteristic
438C22#1681142211yescharacteristic
439C22#1681142211yescharacteristic
440C4—22no—
441C4—22no—
442C4—81no—
443C4—82no—
444C4—81no—
445C4—1062no—
446C4—1062no—
447C4—1062no—
448C4—4241no—
449C4—4241no—
450C4—4242no—
451C4#1681110312yes—
452C4#1681125312yes—
453C2—2122no—
454C2#2027734011yescharacteristic
455C2#2028330211yescharacteristic
456C2#2028330211yescharacteristic
457C2#2028575311yescharacteristic
458C2#2028575511yescharacteristic
459C2#2028595011yescharacteristic
460C2#2028595211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1836C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1836D53
hypocenterC53#21989
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
4. #20275486C2semidirect
5. #20275482C2semidirect
8. #20274883C2semidirect
9. #20274881C2semidirect
10. #20274848C2semidirect
11. C53×#1836C2semidirect
14. #16810678C4semidirect
233. #1836D53semidirect
274. C53#21989semidirect

Maximal subgroups

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

subgroupindex
C2×#168106782
#202771202
#202771082
#202754862
#202754822
#202749172
#202749152
#202748832
#202748812
#202748482
C53×#18362
#2198953