Finite Group Databasebeta

← #27276992

Subgroups — #27276992

Order 13568.

524 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810695C211yesmaximal, characteristic
2#20274742C211yesmaximal, characteristic, complemented
3#20274738C211yesmaximal, characteristic, complemented
4#20274703C211yesmaximal, characteristic
5#20272582C211yesmaximal, characteristic, complemented
6#20272582C211yesmaximal, characteristic, complemented
7C53×#1799C211yesmaximal, characteristic
8C22×#11719917C411yescharacteristic, complemented
9C23×#88322C2211yescharacteristic
10#16810695—21no—
11#16810695—21no—
12#16810695—21no—
13#16810695—21no—
14C2×#13133150—21no—
15C2×#13133148—21no—
16C2×C4⋊#88322C411yescharacteristic, complemented
17C2×#13133132—21no—
18C2×C22⋊#88322C411yescharacteristic, complemented
19C2×C22⋊#88322C2211yescharacteristic
20C2×#13133128C411yescharacteristic, complemented
21#16810483C2211yescharacteristic
22#16810483C2211yescharacteristic
23#16810390—21no—
24#16810385—21no—
25#16810385—21no—
26C2×C4×D4×C53C2211yescharacteristic
27C2×#99×C53—21no—
28C53×#380C2211yescharacteristic
29C53×#380C2211yescharacteristic
30C53×#342—21no—
31C2×#11719917—21no—
32C2×#11719917—21no—
33C2×#11719917—42no—
34C22×#88322—42no—
35C22×#88322C2×C411yescharacteristic
36C22×#88322C2×C411yescharacteristic
37C22×#88322C2×C411yescharacteristic
38C22×#88322C2×C411yescharacteristic
39C22×#88322D411yescharacteristic
40C22×#88322D411yescharacteristic
41C22×#88322D411yescharacteristic
42C22×#88322D411yescharacteristic
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C22⋊Dic53—21no—
46C2×C22⋊Dic53—21no—
47#13133150—42no—
48#13133148—42no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51#13133132—42no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56#13133128—21no—
57#13133128—21no—
58#13133124—21no—
59#13133124—21no—
60C22×D4×C53D411yescharacteristic
61C23×C4×C53C2×C411yescharacteristic
62C23×C4×C53C2311yescharacteristic
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53Q811yescharacteristic
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53D411yescharacteristic
73C2×C22⋊C4×C53Q811yescharacteristic
74C2×C42×C53C2×C411yescharacteristic
75#99×C53—42no—
76#95×C53—21no—
77#95×C53—21no—
78C24×C53C4⋊C411yescharacteristic
79C24×C53C2×D411yescharacteristic
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53C4⋊C411yescharacteristic
95C22×C4×C53C4⋊C411yescharacteristic
96C22×C4×C53C4⋊C411yescharacteristic
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×Q811yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×Dic53—41no—
104C22×Dic53—41no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C22⋊C4×C53—21no—
112C22⋊C4×C53—21no—
113C22⋊C4×C53—21no—
114C22⋊C4×C53—21no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C22⋊C4×C53—41no—
118C22⋊C4×C53—41no—
119C42×C53—21no—
120C42×C53—21no—
121C22⋊Dic53—41no—
122C22⋊Dic53—41no—
123C22⋊Dic53—41no—
124C22⋊Dic53—41no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129#11719917—42no—
130#11719917—42no—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322—22no—
134C2×#88322—21no—
135C2×#88322—21no—
136C2×#88322—41no—
137C2×#88322—41no—
138C2×#88322—41no—
139C2×#88322—41no—
140C2×#88322—41no—
141C2×#88322C22⋊C411yescharacteristic
142C2×#88322C22⋊C411yescharacteristic
143C2×#88322C22⋊C411yescharacteristic
144C2×#88322C22⋊C411yescharacteristic
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53C2×C4⋊C411yescharacteristic
153C23×C53#12111yescharacteristic
154C23×C53#12111yescharacteristic
155C23×C53AGL(1,ℤ/8)11yescharacteristic
156C23×C53AGL(1,ℤ/8)11yescharacteristic
157C23×C53#13711yescharacteristic
158C23×C53#13711yescharacteristic
159D4×C53—41no—
160D4×C53—41no—
161D4×C53—41no—
162D4×C53—41no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—41no—
182C2×C4×C53—81no—
183C2×C4×C53—81no—
184C2×C4×C53—81no—
185C2×C4×C53—81no—
186C2×C4×C53—81no—
187C2×C4×C53—81no—
188C2×C4×C53C2×C22⋊C411yescharacteristic
189C2×C4×C53#11711yescharacteristic
190C2×C4×C53C22⋊Q811yescharacteristic
191C2×C4×C53C22⋊Q811yescharacteristic
192C2×Dic53—81no—
193C2×Dic53—81no—
194C2×Dic53—81no—
195C2×Dic53—81no—
196C2×Dic53—81no—
197C2×Dic53—81no—
198#88322—21no—
199#88322—21no—
200#88322—21no—
201#88322—41no—
202#88322—41no—
203#88322#9912yescomplemented
204#8610—531nomaximal, Sylow
205C22×C53—21no—
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—41no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—81no—
219C22×C53—81no—
220C22×C53#38011yescharacteristic
221C22×C53#42811yescharacteristic
222C22×C53#42811yescharacteristic
223C22×C53#46811yescharacteristic
224C22×C53#46811yescharacteristic
225C22×C53#46911yescharacteristic
226C22×C53#47011yescharacteristic
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53—41no—
233C4×C53—161no—
234C4×C53—161no—
235C4×C53C2×#9911yescharacteristic
236C4×C53#41011yescharacteristic
237Dic53—161no—
238Dic53—161no—
239C2×#435—531no—
240#1808—531no—
241#1807—531no—
242#1799—531no—
243#1799D5311yescharacteristic
244#1289—531no—
245#1289—531no—
246C2×C53—21no—
247C2×C53—21no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53—81no—
253C2×C53#179911yescharacteristic
254C2×C53#180911yescharacteristic
255C2×C53#181211yescharacteristic
256C22×M16—531no—
257C23×C8—531no—
258C2×C4×D4C2×D5311yescharacteristic
259#435—1061no—
260#435—1061no—
261#435—1061no—
262#435—1061no—
263C2×#107—1061no—
264C2×#106—1061no—
265C2×C4⋊C8—531no—
266C2×#99—21no—
267C2×#99—1061no—
268C2×C22⋊C8—531no—
269C2×C22⋊C8—531no—
270C2×#97—531no—
271#380—531no—
272#380—531no—
273#380C2×D5311yescharacteristic
274#380C2×D5311yescharacteristic
275#342—21no—
276#342—1061no—
277#340—1061no—
278#340—1061no—
279C53#861011yescharacteristic, complemented, Sylow
280C22×D4C22⋊D5311yescharacteristic
281C23×C4C4×D5311yescharacteristic
282C23×C4C22×D5311yescharacteristic
283C2×M16—1061no—
284C2×M16—1061no—
285C2×M16—2122no—
286C22×C8—531no—
287C22×C8—531no—
288C22×C8—531no—
289C22×C8—531no—
290C22×C8—531no—
291C22×C8—531no—
292C22×C8—531no—
293C22×C8—531no—
294C22×C8—2122no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—1061no—
302C2×C4⋊C4—1061no—
303C2×C4⋊C4Dic10611yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—1061no—
307C2×C22⋊C4—1061no—
308C2×C22⋊C4Dic10611yescharacteristic
309C2×C22⋊C4C22⋊D5311yescharacteristic
310C2×C42C4×D5311yescharacteristic
311#107—2122no—
312#106—2122no—
313C4⋊C8—1061no—
314C4⋊C8—1061no—
315#99—42no—
316#99—2122no—
317C22⋊C8—1061no—
318C22⋊C8—1061no—
319C22⋊C8—1061no—
320C22⋊C8—1061no—
321#97—1061no—
322#97—1061no—
323#95—21no—
324#95—21no—
325#95—1061no—
326#95—1061no—
327C24#1171991911yescharacteristic
328C24C2×C22⋊D5311yescharacteristic
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—2121no—
344C22×C4—2121no—
345C22×C4#1171991911yescharacteristic
346C22×C4#1171991911yescharacteristic
347C22×C4#1171991911yescharacteristic
348C22×C4C2×Dic10611yescharacteristic
349C22×C4C2×C4×D5311yescharacteristic
350C22×C4D4×D5311yescharacteristic
351C22×C4D4×D5311yescharacteristic
352C22×C4#1171994711yescharacteristic
353C22×C4#1171994711yescharacteristic
354M16—2122no—
355M16—2122no—
356C2×C8—531no—
357C2×C8—531no—
358C2×C8—531no—
359C2×C8—531no—
360C2×C8—1062no—
361C2×C8—1062no—
362C2×C8—1062no—
363C2×C8—1061no—
364C2×C8—1061no—
365C2×C8—2121no—
366C2×C8—2121no—
367C2×C8—2121no—
368C2×C8—2121no—
369C2×C8—2121no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—2121no—
377C4⋊C4—2121no—
378C4⋊C4—2121no—
379C4⋊C4—2121no—
380C22⋊C4—21no—
381C22⋊C4—21no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—2121no—
389C22⋊C4—2121no—
390C22⋊C4—2121no—
391C22⋊C4—2121no—
392C42—21no—
393C42—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23C2×#1171991911yescharacteristic
402C23#1313318911yescharacteristic
403C23#1313318911yescharacteristic
404C23#1313324211yescharacteristic
405C23#1313324311yescharacteristic
406C23#1313324711yescharacteristic
407C23#1313324811yescharacteristic
408D4—41no—
409D4—41no—
410D4—41no—
411D4—41no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
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—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—4241no—
438C2×C4—4241no—
439C2×C4—4241no—
440C2×C4—4241no—
441C2×C4—4241no—
442C2×C4—4241no—
443C2×C4C22⋊C4×D5311yescharacteristic
444C2×C4#1313317311yescharacteristic
445C2×C4#1313319411yescharacteristic
446C2×C4#1313319411yescharacteristic
447C8—532no—
448C8—1061no—
449C8—1061no—
450C8—1061no—
451C8—2121no—
452C8—2121no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—81no—
467C22—81no—
468C22#1681048111yescharacteristic
469C22#1681066311yescharacteristic
470C22#1681066611yescharacteristic
471C22#1681086711yescharacteristic
472C22#1681087111yescharacteristic
473C22#1681087411yescharacteristic
474C22#1681087911yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—161no—
482C4—161no—
483C4—8481no—
484C4—8481no—
485C4#99×D5311yescharacteristic
486C4#1681059511yescharacteristic
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—81no—
494C2#2027470111yescharacteristic
495C2#2027474611yescharacteristic
496C2#2027475911yescharacteristic
subgroupG/H
centerC22#16810481
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1799C2
socleC22×C53#380
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#1799D53
hypocenterC53#8610
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #20274742C2semidirect
3. #20274738C2semidirect
5. #20272582C2semidirect
6. #20272582C2semidirect
8. C22×#11719917C4semidirect
16. C2×C4⋊#88322C4semidirect
18. C2×C22⋊#88322C4semidirect
20. C2×#13133128C4semidirect
203. #88322#99semidirect
279. C53#8610semidirect

Maximal subgroups

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

subgroupindex
C2×#168106952
#202747422
#202747382
#202747032
#202725822
C53×#17992
#861053