Finite Group Databasebeta

← #27287451

Subgroups — #27287451

Order 13568.

708 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279661C211yesmaximal, characteristic
2#20275521C211yesmaximal, characteristic, complemented
3#20275474C211yesmaximal, characteristic, complemented
4#20274715C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×#13133213—21no—
10C2×#13133191—21no—
11C2×#13133191—21no—
12C2×#13133151—21no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133139—22no—
15C2×#13133138—21no—
16#16810532—22no—
17#16810524—21no—
18#16810493C2211yescharacteristic
19C4×C4⋊Dic53C2211yescharacteristic
20#16810386—21no—
21#16810385—21no—
22#16810366—22no—
23C2×C4×D4×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C53×D4⋊C8—22no—
27C22×D212—21no—
28C22×D212—21no—
29C22×D212D412yes—
30C22×C22⋊D53—22no—
31C22×C22⋊D53—22no—
32C22×#88322—21no—
33#13133214—21no—
34#13133214—21no—
35#13133214—21no—
36#13133214—21no—
37#13133213—42no—
38#13133213—42no—
39#13133191—42no—
40#13133191—42no—
41#13133191—42no—
42#13133191—42no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53D411yescharacteristic
47C2×C4⋊Dic53D411yescharacteristic
48C2×#11719921—22no—
49C2×#11719921—22no—
50C2×C4×Dic53—21no—
51#13133151—42no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54#13133139—44no—
55#13133138—42no—
56#13133124—21no—
57C22×D4×C53D411yescharacteristic
58C23×C4×C53—21no—
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C22⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53D412yescomplemented
70C4×C8×C53—21no—
71C4×C8×C53—21no—
72C24×C53—21no—
73C23×D53—22no—
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53D812yes—
80C22×C4×C53—21no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C2×C22⋊D53—44no—
91C2×C22⋊D53—44no—
92C2×C22⋊D53—44no—
93C2×C22⋊D53—44no—
94C2×C22⋊D53—82no—
95C2×C22⋊D53—82no—
96C22×Dic53—21no—
97C22×Dic53—21no—
98C22×Dic53—41no—
99C2×D212—22no—
100C2×D212—22no—
101C2×D212—42no—
102C2×D212—42no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—41no—
109C2×D212—41no—
110C2×D212—41no—
111C2×D212—41no—
112C2×D212—41no—
113C2×D212—41no—
114C2×D212—41no—
115C2×C8×C53—22no—
116C2×C8×C53—22no—
117C2×C8×C53—41no—
118C2×C8×C53—41no—
119C2×C8×C53—41no—
120C2×C8×C53—41no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C22⋊C4×C53—41no—
124C22⋊C4×C53—41no—
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127C42×C53#4111yescharacteristic
128C42×C53#4111yescharacteristic
129#11719921—44no—
130#11719921—44no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133C4⋊Dic53—42no—
134C4⋊Dic53—41no—
135C4⋊Dic53—41no—
136C4⋊Dic53—41no—
137C4⋊Dic53—41no—
138C4⋊Dic53D812yes—
139C4⋊Dic53SD1612yes—
140C4×Dic53—41no—
141C4×Dic53—41no—
142C2×#88322—41no—
143C2×#88322—41no—
144C2×#88322—41no—
145C23×C53—22no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53#12011yescharacteristic
153C22×D53—42no—
154C22×D53—42no—
155C22×D53—42no—
156C22×D53—42no—
157C22×D53—42no—
158C22×D53—42no—
159C22×D53—42no—
160D4×C53—22no—
161D4×C53—22no—
162D4×C53—41no—
163D4×C53—41no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—81no—
170C2×C4×C53—81no—
171C2×C4×C53—81no—
172C2×C4×C53—81no—
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12311yescharacteristic
177C2×C4×C53#12311yescharacteristic
178C2×C4×C53#12411yescharacteristic
179C2×C4×C53C2×D811yescharacteristic
180C2×C4×C53C2×D811yescharacteristic
181C2×C4×C53C2×SD1611yescharacteristic
182C2×C4×C53#13511yescharacteristic
183C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C53#13711yescharacteristic
187C22⋊D53—88no—
188C22⋊D53—88no—
189C2×Dic53—22no—
190C2×Dic53—22no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—81no—
196C2×Dic53—81no—
197C2×Dic53—81no—
198D212—44no—
199D212—82no—
200D212—81no—
201D212—81no—
202D212—81no—
203D212—81no—
204D212—81no—
205D212—81no—
206D212—81no—
207D212—81no—
208C8×C53—42no—
209C8×C53—42no—
210#88322—81no—
211#10408—531nomaximal, Sylow
212C22×C53—42no—
213C22×C53—42no—
214C22×C53—42no—
215C22×C53—81no—
216C22×C53—81no—
217C22×C53—81no—
218C22×C53—81no—
219C22×C53#39411yescharacteristic
220C22×C53#44711yescharacteristic
221C22×C53#44711yescharacteristic
222C22×C53#45011yescharacteristic
223C22×C53#45211yescharacteristic
224C22×C53#45311yescharacteristic
225C22×C53#45511yescharacteristic
226C2×D53—82no—
227C2×D53—82no—
228C2×D53—82no—
229C2×D53—82no—
230C2×D53—82no—
231C2×D53—82no—
232C2×D53—82no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—81no—
236C4×C53#45911yescharacteristic
237C4×C53#46011yescharacteristic
238C4×C53#46611yescharacteristic
239C4×C53#46811yescharacteristic
240C4×C53#48011yescharacteristic
241C4×C53#48211yescharacteristic
242C4×C53#48611yescharacteristic
243C4×C53#49011yescharacteristic
244Dic53—42no—
245Dic53—42no—
246Dic53—161no—
247#2541—531no—
248#1957—531no—
249#1947—531no—
250#1802—531no—
251C2×#331—531no—
252C2×D4⋊C8D5311yescharacteristic, complemented
253#1279—531no—
254C2×C53—82no—
255C2×C53#161911yescharacteristic
256C2×C53#163111yescharacteristic
257C2×C53#163911yescharacteristic
258C2×C53#164111yescharacteristic
259C2×C53#203311yescharacteristic
260C2×C53#203711yescharacteristic
261C2×C53#204111yescharacteristic
262D53—162no—
263C2×#127—531no—
264C2×#127—1061no—
265C2×#121—1061no—
266C2×#121—1061no—
267C2×C4×D4C2×D5311yescharacteristic
268C2×#107—1061no—
269C2×C4⋊C8—531no—
270C2×C4⋊C8C2×D5311yescharacteristic
271C2×#102—1062no—
272C2×#102—1061no—
273C2×C4×C8C2×D5311yescharacteristic
274#392—1062no—
275#390—1061no—
276#384—531no—
277C4×C4⋊C4—531no—
278#340—1061no—
279#340—1061no—
280#331—1062no—
281D4⋊C8—22no—
282C53#1040811yescharacteristic, complemented, Sylow
283C22×D4—532no—
284C22×D4—1062no—
285C22×D4—1062no—
286C22×D4—1061no—
287C22×D4—1061no—
288C22×D4D21211yescharacteristic
289C23×C4—21no—
290C22×C8—21no—
291C22×C8—21no—
292C22×C8—1061no—
293#127—1061no—
294#127—1061no—
295#127—1061no—
296#127—1061no—
297#127—2122no—
298#127—2122no—
299#121—2122no—
300#121—2122no—
301#121—2122no—
302#121—2122no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—21no—
307C2×C4⋊C4—531no—
308C2×C4⋊C4—531no—
309C2×C4⋊C4—1061no—
310C2×C4⋊C4—1061no—
311C2×C4⋊C4—1061no—
312C2×C4⋊C4D21211yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—1062no—
315C2×C22⋊C4—1062no—
316C2×C42—1061no—
317C2×C42C22×D5311yescharacteristic
318#107—2122no—
319C4⋊C8—21no—
320C4⋊C8—1061no—
321C4⋊C8—1061no—
322C4⋊C8C22⋊D5312yescomplemented
323#102—2124no—
324#102—2122no—
325C4×C8—21no—
326C4×C8—21no—
327#95—1061no—
328C24—21no—
329C24—1062no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—1062no—
336C2×D4—1062no—
337C2×D4—4242no—
338C2×D4—4242no—
339C2×D4—2122no—
340C2×D4—2122no—
341C2×D4—2124no—
342C2×D4—2124no—
343C2×D4—2124no—
344C2×D4—2124no—
345C2×D4—2121no—
346C2×D4—2121no—
347C2×D4—2121no—
348C2×D4—2121no—
349C2×D4—2121no—
350C2×D4—2121no—
351C2×D4—2121no—
352C2×D4—2121no—
353C2×D4—2121no—
354C2×D4—2121no—
355C2×D4—2121no—
356C2×D4—2121no—
357C2×D4D42412yes—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—1061no—
366C22×C4—1061no—
367C22×C4—2121no—
368C22×C4C2×D21211yescharacteristic
369C22×C4D4×D5311yescharacteristic
370C22×C4D4×D5311yescharacteristic
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—2121no—
378C2×C8—2121no—
379C2×C8—2121no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—532no—
383C4⋊C4—532no—
384C4⋊C4—1061no—
385C4⋊C4—1061no—
386C4⋊C4—2122no—
387C4⋊C4—2121no—
388C4⋊C4—2121no—
389C4⋊C4—2121no—
390C4⋊C4—2121no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—2124no—
394C22⋊C4—2124no—
395C42—2121no—
396C42—2121no—
397C42#1171994511yescharacteristic
398C42#1171994711yescharacteristic
399C42#1171994711yescharacteristic
400C42C2×C22⋊D5311yescharacteristic
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—2122no—
409C23—2122no—
410C23—2122no—
411C23—2122no—
412C23—2122no—
413C23—2122no—
414C23—2122no—
415C23#1313318711yescharacteristic
416D4—22no—
417D4—22no—
418D4—41no—
419D4—41no—
420D4—4242no—
421D4—4248no—
422D4—4248no—
423D4—4241no—
424D4—4241no—
425D4—4241no—
426D4—4241no—
427D4—4241no—
428D4—4241no—
429D4—4241no—
430D4—4241no—
431D4—2124no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—1062no—
442C2×C4—1062no—
443C2×C4—4241no—
444C2×C4—4241no—
445C2×C4—4241no—
446C2×C4—2121no—
447C2×C4—2121no—
448C2×C4—2121no—
449C2×C4—2121no—
450C2×C4#1313318811yescharacteristic
451C2×C4#1313318911yescharacteristic
452C2×C4#1313319111yescharacteristic
453C2×C4#1313319911yescharacteristic
454C2×C4#1313320111yescharacteristic
455C2×C4#1313320311yescharacteristic
456C2×C4D8×D5311yescharacteristic
457C2×C4C2×D42411yescharacteristic
458C2×C4SD16×D5311yescharacteristic
459C2×C4#1313323911yescharacteristic
460C2×C4#1313324111yescharacteristic
461C2×C4#1313324411yescharacteristic
462C2×C4#1313324511yescharacteristic
463C2×C4#1313325011yescharacteristic
464C8—42no—
465C8—42no—
466C8—4241no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—4242no—
475C22—4242no—
476C22—4242no—
477C22—4242no—
478C22—4242no—
479C22—4242no—
480C22—4242no—
481C22#1681053911yescharacteristic
482C22#1681075811yescharacteristic
483C22#1681075911yescharacteristic
484C22#1681077511yescharacteristic
485C22#1681078711yescharacteristic
486C22#1681079211yescharacteristic
487C22#1681080211yescharacteristic
488C4—21no—
489C4—21no—
490C4—81no—
491C4—8481no—
492C4—2122no—
493C4—2122no—
494C4#1681081611yescharacteristic
495C4#1681082111yescharacteristic
496C4#1681085911yescharacteristic
497C4#1681086911yescharacteristic
498C4#1681093711yescharacteristic
499C4#1681094811yescharacteristic
500C4#1681096811yescharacteristic
501C4#1681098611yescharacteristic
502C2—82no—
503C2—8482no—
504C2#2027377111yescharacteristic
505C2#2027384511yescharacteristic
506C2#2027389611yescharacteristic
507C2#2027391111yescharacteristic
508C2#2027589811yescharacteristic
509C2#2027592411yescharacteristic
510C2#2027594811yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×D4⋊C8C2
socleC23×C53#120
prime radical#20279661C2
hypercenterC2×D4⋊C8D53
hypocenterC53#10408
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20275521C2semidirect
3. #20275474C2semidirect
4. #20274715C2semidirect
5. C2×#16810366C2semidirect
6. #20272555C2semidirect
7. C2×C53×D4⋊C8C2semidirect
69. C4⋊C8×C53D4semidirect
252. C2×D4⋊C8D53semidirect
282. C53#10408semidirect
322. C4⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202796612
#202755212
#202754742
#202747152
C2×#168103662
#202725552
C2×C53×D4⋊C82
#1040853