Finite Group Databasebeta

← #27285201

Subgroups — #27285201

Order 13568.

838 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
1C2×#16810910C211yesmaximal, characteristic, complemented
2#20279662C211yesmaximal, characteristic
3#20275664C211yesmaximal, characteristic, complemented
4C2×#16810367C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×#492C211yesmaximal, characteristic, complemented
8C2×#13133213—21no—
9C2×#13133188—22no—
10C2×C4×D212C2211yescharacteristic
11#16810910—22no—
12#16810910—22no—
13C2×#13133148—21no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133137—21no—
17#16810531—22no—
18#16810526—21no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810385—22no—
21#16810367—22no—
22#16810352—22no—
23C22×SD16×C53—22no—
24C2×C4⋊Q8×C53C2211yescharacteristic
25C2×#127×C53C2211yescharacteristic
26C53×#492—22no—
27C53×#492—22no—
28C2×C4×C8×C53C2211yescharacteristic
29C22×D212D411yescharacteristic
30C22×C22⋊D53—24no—
31C22×C4×D53—21no—
32C22×#88322—21no—
33C22×#88322—21no—
34#13133213—42no—
35#13133213—42no—
36#13133188—44no—
37#13133188—44no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212—21no—
42C2×C4⋊Dic53D411yescharacteristic
43C2×C4⋊Dic53D412yes—
44C2×C22⋊Dic53—24no—
45C2×#11719921—21no—
46C2×C4×Dic53—21no—
47#13133148—24no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322D412yescomplemented
52#13133137—24no—
53#13133124—21no—
54C22×Q8×C53—21no—
55C22×D4×C53—21no—
56C22×D4×C53—21no—
57C22×D4×C53D412yes—
58C2×SD16×C53—28no—
59C2×SD16×C53—48no—
60C22×C8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53D412yescomplemented
65#127×C53—21no—
66#127×C53—21no—
67#127×C53—21no—
68#127×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C42×C53C2311yescharacteristic
72C4×C8×C53—21no—
73C4×C8×C53D412yescomplemented
74C24×C53—22no—
75C23×D53—21no—
76C2×Q8×C53—22no—
77C2×Q8×C53—44no—
78C2×Q8×C53—41no—
79C2×D4×C53—22no—
80C2×D4×C53—22no—
81C2×D4×C53—44no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—42no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—42no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C2×C22⋊D53—48no—
97C2×C22⋊D53—48no—
98C2×C22⋊D53—84no—
99C22×Dic53—22no—
100C22×Dic53—21no—
101C2×D212—21no—
102C2×D212—21no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212—21no—
106C2×D212SD1612yescomplemented
107C2×C4×D53—41no—
108C2×C4×D53—41no—
109C2×C4×D53—42no—
110C2×C4×D53—42no—
111SD16×C53—416no—
112C2×C8×C53—24no—
113C2×C8×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122C22⋊Dic53—48no—
123#11719921—41no—
124#11719921—41no—
125C4⋊Dic53—22no—
126C4⋊Dic53—21no—
127C4⋊Dic53—21no—
128C4⋊Dic53SD1614yescomplemented
129C4×Dic53—41no—
130C4×Dic53—41no—
131C2×#88322—22no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—42no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—44no—
139C23×C53—42no—
140C23×C53—44no—
141C23×C53#12011yescharacteristic
142C22×D53—22no—
143C22×D53—41no—
144C22×D53—42no—
145C22×D53—41no—
146C22×D53—42no—
147Q8×C53—44no—
148Q8×C53—82no—
149D4×C53—44no—
150D4×C53—82no—
151D4×C53—81no—
152D4×C53—81no—
153D4×C53—82no—
154D4×C53—81no—
155D4×C53—81no—
156D4×C53—81no—
157D4×C53—81no—
158C2×C4×C53—22no—
159C2×C4×C53—42no—
160C2×C4×C53—82no—
161C2×C4×C53#12011yescharacteristic
162C2×C4×C53#12011yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12711yescharacteristic
167C2×C4×C53C2×SD1611yescharacteristic
168C2×C4×C53C2×SD1612yes—
169C2×C4×C53#13511yescharacteristic
170C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)12yes—
172C2×C4×C53#13711yescharacteristic
173C22⋊D53—816no—
174C2×Dic53—24no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177C2×Dic53—41no—
178C2×Dic53—42no—
179D212—22no—
180D212—22no—
181D212—41no—
182D212—41no—
183C4×D53—82no—
184C4×D53—81no—
185C4×D53—81no—
186C8×C53—24no—
187#88322—42no—
188#88322—81no—
189#10054—531nomaximal, Sylow
190C22×C53—84no—
191C22×C53—84no—
192C22×C53—82no—
193C22×C53—82no—
194C22×C53—82no—
195C22×C53#39211yescharacteristic
196C22×C53#45011yescharacteristic
197C22×C53#45012yes—
198C22×C53#45211yescharacteristic
199C22×C53#45312yes—
200C2×D53—44no—
201C2×D53—42no—
202C2×D53—82no—
203C2×D53—81no—
204C2×D53—81no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—82no—
208C4×C53#44811yescharacteristic
209C4×C53#44911yescharacteristic
210C4×C53#46012yes—
211C4×C53#46511yescharacteristic
212C4×C53#46811yescharacteristic
213C4×C53#49211yescharacteristic
214C4×C53#49611yescharacteristic
215Dic53—44no—
216Dic53—81no—
217C2×#492D5311yescharacteristic, complemented
218C2×#476—531no—
219#2541—531no—
220#1987—531no—
221C2×#331—531no—
222C2×D4⋊C8—531no—
223#1279—531no—
224C2×C53—162no—
225C2×C53#162912yes—
226C2×C53#163112yes—
227C2×C53#200011yescharacteristic
228C2×C53#200211yescharacteristic
229C2×C53#201411yescharacteristic
230D53—82no—
231C22×SD16—22no—
232C2×C4⋊Q8C2×D5311yescharacteristic
233C2×#127—1061no—
234C2×#127C2×D5311yescharacteristic
235C2×#121—1062no—
236C2×C4×D4—531no—
237#492—22no—
238#492—22no—
239#476—1062no—
240#476—1062no—
241C2×#106—1061no—
242C2×C4⋊C8—531no—
243C2×C4⋊C8—531no—
244C2×#102—1061no—
245C2×C4×C8C2×D5311yescharacteristic
246#392—1062no—
247#390—1061no—
248C4×C4⋊C4—531no—
249#340—1062no—
250#331—1062no—
251D4⋊C8—1062no—
252C53#1005411yescharacteristic, complemented, Sylow
253C22×Q8—21no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4—531no—
257C22×D4—1064no—
258C22×D4C22⋊D5312yes—
259C23×C4—1061no—
260C2×SD16—28no—
261C2×SD16—48no—
262C22×C8—21no—
263C22×C8—1061no—
264C22×C8—1061no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8C22⋊D5312yescomplemented
269#127—21no—
270#127—21no—
271#127—21no—
272#127—21no—
273#127—2122no—
274#127—2122no—
275#121—2124no—
276#121—2124no—
277C4×D4—1061no—
278C4×D4—1061no—
279C4×D4—1061no—
280C4×D4—1061no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—532no—
284C2×C4⋊C4—531no—
285C2×C22⋊C4—1064no—
286C2×C22⋊C4—1061no—
287C2×C42—1061no—
288C2×C42C22×D5311yescharacteristic
289#106—1064no—
290C4⋊C8—532no—
291C4⋊C8—1061no—
292C4⋊C8—1061no—
293C4⋊C8—1061no—
294#102—1064no—
295C4×C8—21no—
296C4×C8C22⋊D5312yescomplemented
297#95—1061no—
298C24—22no—
299C24—1061no—
300C2×Q8—22no—
301C2×Q8—44no—
302C2×Q8—41no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—44no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—42no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—532no—
317C2×D4—1061no—
318C2×D4—1061no—
319C2×D4—1061no—
320C2×D4—1061no—
321C2×D4—1061no—
322C2×D4—4244no—
323C2×D4—2128no—
324C2×D4—2128no—
325C22×C4—41no—
326C22×C4—1062no—
327C22×C4—1061no—
328C22×C4—2121no—
329C22×C4—2121no—
330C22×C4—2122no—
331C22×C4—2122no—
332C22×C4D4×D5311yescharacteristic
333C22×C4D4×D5311yescharacteristic
334C22×C4C2×C22⋊D5311yescharacteristic
335SD16—416no—
336C2×C8—24no—
337C2×C8—22no—
338C2×C8—1062no—
339C2×C8—2121no—
340C2×C8—2122no—
341C2×C8—2121no—
342C2×C8—2121no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—534no—
348C4⋊C4—1062no—
349C4⋊C4—1061no—
350C4⋊C4—1061no—
351C22⋊C4—2128no—
352C22⋊C4—2121no—
353C22⋊C4—2121no—
354C42—2121no—
355C42—2121no—
356C42D4×D5311yescharacteristic
357C42#1171994711yescharacteristic
358C42C2×C22⋊D5311yescharacteristic
359C42C2×C22⋊D5311yescharacteristic
360C23—44no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—44no—
365C23—1062no—
366C23—2122no—
367C23—2121no—
368C23—2121no—
369C23—2122no—
370C23#1313318511yescharacteristic
371Q8—44no—
372Q8—82no—
373D4—44no—
374D4—82no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—82no—
379D4—81no—
380D4—81no—
381D4—81no—
382D4—1062no—
383D4—1062no—
384D4—42416no—
385D4—2121no—
386D4—2121no—
387C2×C4—22no—
388C2×C4—42no—
389C2×C4—82no—
390C2×C4—1064no—
391C2×C4—4241no—
392C2×C4—4242no—
393C2×C4—4241no—
394C2×C4—2122no—
395C2×C4—2121no—
396C2×C4—2122no—
397C2×C4—2122no—
398C2×C4#1313318511yescharacteristic
399C2×C4#1313318611yescharacteristic
400C2×C4#1313318811yescharacteristic
401C2×C4#1313318911yescharacteristic
402C2×C4#1313318911yescharacteristic
403C2×C4#1313321311yescharacteristic
404C2×C4SD16×D5311yescharacteristic
405C2×C4SD16×D5312yes—
406C2×C4#1313323811yescharacteristic
407C2×C4#1313324412yes—
408C2×C4#1313324411yescharacteristic
409C2×C4#1313324611yescharacteristic
410C8—24no—
411C8—4241no—
412C8—2122no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—84no—
417C22—84no—
418C22—4241no—
419C22—4241no—
420C22—4242no—
421C22—2122no—
422C22—2124no—
423C22#1681053111yescharacteristic
424C22#1681077211yescharacteristic
425C22#1681077212yes—
426C22#1681078211yescharacteristic
427C22#1681079012yes—
428C4—21no—
429C4—21no—
430C4—82no—
431C4—4241no—
432C4—2124no—
433C4#1681076011yescharacteristic
434C4#1681076711yescharacteristic
435C4#1681082112yes—
436C4#1681084911yescharacteristic
437C4#1681086611yescharacteristic
438C4#1681099411yescharacteristic
439C4#1681100911yescharacteristic
440C2—162no—
441C2—4242no—
442C2#2027382912yes—
443C2#2027383912yes—
444C2#2027571511yescharacteristic
445C2#2027573011yescharacteristic
446C2#2027580111yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#492C2
socleC23×C53#120
prime radical#20279662C2
hypercenterC2×#492D53
hypocenterC53#10054
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
1. C2×#16810910C2semidirect
3. #20275664C2semidirect
4. C2×#16810367C2semidirect
5. C2×#16810352C2semidirect
6. #20272555C2semidirect
7. C2×C53×#492C2semidirect
51. C4⋊#88322D4semidirect
64. C4⋊Q8×C53D4semidirect
73. C4×C8×C53D4semidirect
106. C2×D212SD16semidirect
128. C4⋊Dic53SD16semidirect
217. C2×#492D53semidirect
252. C53#10054semidirect
268. C4⋊Q8C22⋊D53semidirect
296. C4×C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
#202796622
#202756642
C2×#168103672
C2×#168103522
#202725552
C2×C53×#4922
#1005453