Finite Group Databasebeta

← #27287537

Subgroups — #27287537

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279664C211yesmaximal, characteristic
2#20275676C211yesmaximal, characteristic, complemented
3#20275478C211yesmaximal, characteristic, complemented
4#20274716C211yesmaximal, characteristic
5C2×#16810368C211yesmaximal, characteristic, complemented
6C2×#16810352C211yesmaximal, characteristic
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×#13133197—21no—
10C2×#13133197—21no—
11C2×C4×D212C2211yescharacteristic
12C2×#13133148—21no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133139—21no—
15#16810560—22no—
16#16810525—21no—
17#16810493—21no—
18#16810493C2211yescharacteristic
19#16810386—21no—
20#16810386—22no—
21#16810385—21no—
22#16810368—22no—
23#16810352—22no—
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C4×C4⋊C4×C53C2211yescharacteristic
27C22×Dic106—21no—
28C22×D212D411yescharacteristic
29C22×C4×D53—21no—
30C22×#88322—21no—
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35#13133197—42no—
36#13133197—42no—
37#13133197—42no—
38#13133197—42no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212—21no—
42C4×D212—21no—
43C2×C4⋊Dic53—22no—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53D412yes—
50C2×C4⋊Dic53D411yescharacteristic
51C2×#11719921—22no—
52C2×#11719921—22no—
53C2×#11719921—21no—
54#13133148—42no—
55C4⋊#88322—21no—
56C4⋊#88322D412yes—
57#13133139—42no—
58C22×C8×C53—21no—
59C22×C8×C53—21no—
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C42×C53—21no—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68#95×C53—21no—
69C23×D53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—22no—
76C22×Dic53—21no—
77C22×Dic53—41no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212SD1612yes—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×Dic106—41no—
91C2×Dic106—41no—
92C2×Dic106—41no—
93C2×Dic106—41no—
94C2×Dic106—41no—
95C2×Dic106—41no—
96C2×C8×C53—22no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53D812yes—
105C4⋊C4×C53SD1612yes—
106C42×C53—41no—
107C42×C53—41no—
108C42×C53C2×D411yescharacteristic
109C42×C53#4111yescharacteristic
110C42×C53#4111yescharacteristic
111C42×C53#4111yescharacteristic
112#11719921—44no—
113#11719921—44no—
114#11719921—41no—
115#11719921—41no—
116C4⋊Dic53—21no—
117C4⋊Dic53—22no—
118C4⋊Dic53—21no—
119C4⋊Dic53—22no—
120C4⋊Dic53—44no—
121C4⋊Dic53—44no—
122C4⋊Dic53—41no—
123C4⋊Dic53—41no—
124C4⋊Dic53—42no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129C2×#88322—22no—
130C2×#88322—41no—
131C2×#88322—41no—
132C23×C53#12011yescharacteristic
133C22×D53—22no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139C22×D53—41no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12311yescharacteristic
152C2×C4×C53#12311yescharacteristic
153C2×C4×C53#12411yescharacteristic
154C2×C4×C53C2×D811yescharacteristic
155C2×C4×C53C2×SD1611yescharacteristic
156C2×C4×C53C2×SD1611yescharacteristic
157C2×C4×C53#13511yescharacteristic
158C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53#13711yescharacteristic
161C2×C4×C53#13711yescharacteristic
162C2×Dic53—42no—
163C2×Dic53—41no—
164C2×Dic53—42no—
165C2×Dic53—41no—
166C2×Dic53—42no—
167C2×Dic53—41no—
168C2×Dic53—81no—
169C2×Dic53—81no—
170C2×Dic53—81no—
171D212—22no—
172D212—22no—
173D212—41no—
174D212—41no—
175C4×D53—81no—
176C4×D53—81no—
177C4×D53—81no—
178C4×D53—81no—
179Dic106—81no—
180Dic106—81no—
181Dic106—81no—
182Dic106—81no—
183C8×C53—42no—
184C8×C53—81no—
185#88322—42no—
186#10420—531nomaximal, Sylow
187C22×C53#39411yescharacteristic
188C22×C53#44711yescharacteristic
189C22×C53#44811yescharacteristic
190C22×C53#45011yescharacteristic
191C22×C53#45211yescharacteristic
192C22×C53#45411yescharacteristic
193C22×C53#45611yescharacteristic
194C2×D53—42no—
195C2×D53—42no—
196C2×D53—42no—
197C2×D53—81no—
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—81no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—42no—
204C4×C53—42no—
205C4×C53#45911yescharacteristic
206C4×C53#46011yescharacteristic
207C4×C53#46511yescharacteristic
208C4×C53#47011yescharacteristic
209C4×C53#48311yescharacteristic
210C4×C53#48511yescharacteristic
211C4×C53#48611yescharacteristic
212C4×C53#49011yescharacteristic
213Dic53—82no—
214Dic53—81no—
215Dic53—161no—
216#2542—531no—
217#1990—531no—
218#1948—531no—
219#1802—531no—
220C2×#332—531no—
221C2×D4⋊C8—531no—
222#1279D5311yescharacteristic, complemented
223C2×C53#162711yescharacteristic
224C2×C53#163511yescharacteristic
225C2×C53#164711yescharacteristic
226C2×C53#165111yescharacteristic
227C2×C53#203511yescharacteristic
228C2×C53#203811yescharacteristic
229C2×C53#204111yescharacteristic
230D53—82no—
231C2×C4⋊Q8—531no—
232C2×C22⋊Q8—1061no—
233C2×C22⋊Q8—1061no—
234C2×C4×D4—531no—
235C2×#106—1061no—
236C2×C4⋊C8—531no—
237C2×C4⋊C8C2×D5311yescharacteristic
238C2×#102—1061no—
239C2×C4×C8C2×D5311yescharacteristic
240#399—1062no—
241#390—1061no—
242#384—531no—
243#384—1061no—
244C4×C4⋊C4C2×D5311yescharacteristic
245#340—1061no—
246#340—1062no—
247#340—1061no—
248#332—1062no—
249D4⋊C8—1062no—
250C53#1042011yescharacteristic, complemented, Sylow
251C22×Q8—1061no—
252C22×D4—531no—
253C23×C4—1061no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—1061no—
257C4⋊Q8—1061no—
258C4⋊Q8—1061no—
259C4⋊Q8—1061no—
260C4⋊Q8—1061no—
261C22⋊Q8—2122no—
262C22⋊Q8—2122no—
263C22⋊Q8—2122no—
264C22⋊Q8—2122no—
265C4×D4—1061no—
266C4×D4—1061no—
267C4×D4—1061no—
268C4×D4—1061no—
269C2×C4⋊C4—531no—
270C2×C4⋊C4—532no—
271C2×C4⋊C4—1062no—
272C2×C4⋊C4—1062no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4—1061no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4D21211yescharacteristic
278C2×C4⋊C4D21211yescharacteristic
279C2×C22⋊C4—1062no—
280C2×C22⋊C4—1062no—
281C2×C22⋊C4—1061no—
282C2×C42—21no—
283C2×C42C22×D5311yescharacteristic
284#106—2122no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—532no—
288C4⋊C8—1061no—
289#102—2122no—
290C4×C8—21no—
291C4×C8—21no—
292#95—21no—
293C24—1061no—
294C2×Q8—2121no—
295C2×Q8—2121no—
296C2×Q8—2121no—
297C2×Q8—2121no—
298C2×Q8—2121no—
299C2×Q8—2121no—
300C2×D4—532no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—1061no—
304C2×D4—1061no—
305C2×D4—1061no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—1062no—
309C22×C4—1061no—
310C22×C4—2121no—
311C22×C4—2121no—
312C22×C4—2121no—
313C22×C4—2121no—
314C22×C4—2121no—
315C22×C4—2121no—
316C22×C4—2121no—
317C22×C4C2×D21211yescharacteristic
318C22×C4D4×D5311yescharacteristic
319C22×C4D4×D5311yescharacteristic
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—1062no—
327C2×C8—2121no—
328C2×C8—2121no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—1061no—
332C4⋊C4—1062no—
333C4⋊C4—1061no—
334C4⋊C4—1062no—
335C4⋊C4—2122no—
336C4⋊C4—2124no—
337C4⋊C4—2124no—
338C4⋊C4—2121no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C4⋊C4—2121no—
342C4⋊C4—2121no—
343C4⋊C4—2121no—
344C4⋊C4#1171991312yes—
345C4⋊C4D42412yes—
346C22⋊C4—2124no—
347C22⋊C4—2124no—
348C22⋊C4—2121no—
349C22⋊C4—2121no—
350C42—41no—
351C42—41no—
352C42#1171994511yescharacteristic
353C42#1171994511yescharacteristic
354C42D4×D5311yescharacteristic
355C42#1171994911yescharacteristic
356C23—1062no—
357C23—2121no—
358C23—2121no—
359C23—2121no—
360C23—2121no—
361C23—2121no—
362C23—2121no—
363C23#1313318711yescharacteristic
364Q8—4241no—
365Q8—4241no—
366Q8—4241no—
367Q8—4241no—
368D4—1062no—
369D4—1062no—
370D4—2121no—
371D4—2121no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—4241no—
381C2×C4—4241no—
382C2×C4—4241no—
383C2×C4—4241no—
384C2×C4—4241no—
385C2×C4—4241no—
386C2×C4—4241no—
387C2×C4—2122no—
388C2×C4—2121no—
389C2×C4—2122no—
390C2×C4—2121no—
391C2×C4—2121no—
392C2×C4—2122no—
393C2×C4#1313319011yescharacteristic
394C2×C4#1313319011yescharacteristic
395C2×C4#1313319211yescharacteristic
396C2×C4#1313320211yescharacteristic
397C2×C4#1313320211yescharacteristic
398C2×C4#1313320511yescharacteristic
399C2×C4C2×D42411yescharacteristic
400C2×C4SD16×D5311yescharacteristic
401C2×C4C2×#1171991311yescharacteristic
402C2×C4#1313323611yescharacteristic
403C2×C4#1313324511yescharacteristic
404C2×C4#1313324511yescharacteristic
405C2×C4#1313324611yescharacteristic
406C2×C4#1313324911yescharacteristic
407C8—42no—
408C8—81no—
409C8—2122no—
410C22—4241no—
411C22—4241no—
412C22—4241no—
413C22—4241no—
414C22—2122no—
415C22—2122no—
416C22—2122no—
417C22#1681054111yescharacteristic
418C22#1681075911yescharacteristic
419C22#1681076311yescharacteristic
420C22#1681077611yescharacteristic
421C22#1681078611yescharacteristic
422C22#1681079711yescharacteristic
423C22#1681080711yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—42no—
428C4—8481no—
429C4—4241no—
430C4—4242no—
431C4#1681081911yescharacteristic
432C4#1681082411yescharacteristic
433C4#1681085111yescharacteristic
434C4#1681088011yescharacteristic
435C4#1681095511yescharacteristic
436C4#1681096611yescharacteristic
437C4#1681097011yescharacteristic
438C4#1681098811yescharacteristic
439C2—4242no—
440C2#2027382011yescharacteristic
441C2#2027386811yescharacteristic
442C2#2027395211yescharacteristic
443C2#2027398111yescharacteristic
444C2#2027591411yescharacteristic
445C2#2027593511yescharacteristic
446C2#2027595011yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenter#1279D53
hypocenterC53#10420
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #20275676C2semidirect
3. #20275478C2semidirect
5. C2×#16810368C2semidirect
7. C53×#1279C2semidirect
222. #1279D53semidirect
250. C53#10420semidirect

Maximal subgroups

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

subgroupindex
#202796642
#202756762
#202754782
#202747162
C2×#168103682
C2×#168103522
C53×#12792
#1042053