Finite Group Databasebeta

← #27278332

Subgroups — #27278332

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277662C211yesmaximal, characteristic
2#20275478C211yesmaximal, characteristic, complemented
3#20274843C211yesmaximal, characteristic, complemented
4#20274716C211yesmaximal, characteristic
5C2×#16810352C211yesmaximal, characteristic
6#20272599C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C2×#13133175—21no—
10C2×#13133148—21no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133139—21no—
13#16810525—21no—
14#16810515—22no—
15#16810493C2211yescharacteristic
16#16810493C2211yescharacteristic
17#16810480—21no—
18C4×C4⋊Dic53—21no—
19#16810386—21no—
20#16810386—22no—
21#16810385—21no—
22#16810352C414yescomplemented
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C4×C8×C53C2211yescharacteristic
25C4×C4⋊C4×C53C2211yescharacteristic
26C22×D212D411yescharacteristic
27C22×C4×D53—21no—
28C22×#88322—21no—
29C4×D212—21no—
30C4×D212—21no—
31C4×D212C2×C412yes—
32C4×D212Q812yes—
33#13133175—42no—
34#13133175—42no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53D411yescharacteristic
41C2×C4⋊Dic53D412yes—
42C2×#11719921—22no—
43C2×#11719921—22no—
44C2×#11719921—21no—
45C2×C4×Dic53—22no—
46#13133148—42no—
47C4⋊#88322—21no—
48C4⋊#88322C2×C412yes—
49#13133139—42no—
50#13133124—22no—
51C22×C8×C53—21no—
52C22×C8×C53—21no—
53C2×C4⋊C4×C53D411yescharacteristic
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C42×C53—21no—
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—21no—
58C4⋊C8×C53—21no—
59C4×C8×C53—21no—
60C4×C8×C53C2×C412yes—
61#95×C53—21no—
62C23×D53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—22no—
70C22×Dic53—41no—
71C2×D212—21no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212C4⋊C412yes—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C8×C53—22no—
84C2×C8×C53—41no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C4⋊C4×C53—21no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53SD1612yes—
93C42×C53—41no—
94C42×C53—41no—
95C42×C53C22×C411yescharacteristic
96C42×C53C2×Q811yescharacteristic
97C42×C53#4111yescharacteristic
98C42×C53#4111yescharacteristic
99#11719921—22no—
100#11719921—22no—
101#11719921—44no—
102#11719921—82no—
103C4⋊Dic53—21no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106C4⋊Dic53—41no—
107C4⋊Dic53—41no—
108C4⋊Dic53—42no—
109C4⋊Dic53—42no—
110C4⋊Dic53—41no—
111C4⋊Dic53—41no—
112C4⋊Dic53—41no—
113C4⋊Dic53—41no—
114C4⋊Dic53C4⋊C412yes—
115C4×Dic53—44no—
116C2×#88322—22no—
117C2×#88322—41no—
118C2×#88322—41no—
119C23×C53#12011yescharacteristic
120C22×D53—22no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—41no—
126C22×D53—41no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C2×C4⋊C411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53C22⋊Q811yescharacteristic
139C2×C4×C53C22⋊Q811yescharacteristic
140C2×C4×C53#12311yescharacteristic
141C2×C4×C53C2×SD1611yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53#13511yescharacteristic
144C2×C4×C53#13511yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×Dic53—22no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—81no—
156C2×Dic53—81no—
157C2×Dic53—81no—
158D212—22no—
159D212—22no—
160D212—41no—
161D212—41no—
162C4×D53—42no—
163C4×D53—42no—
164C4×D53—42no—
165C4×D53—42no—
166C8×C53—42no—
167C8×C53—81no—
168#88322—42no—
169#8893—531nomaximal, Sylow
170C22×C53#38511yescharacteristic
171C22×C53#44911yescharacteristic
172C22×C53#44911yescharacteristic
173C22×C53#45011yescharacteristic
174C22×C53#45211yescharacteristic
175C22×C53#45411yescharacteristic
176C22×C53#45511yescharacteristic
177C2×D53—42no—
178C2×D53—42no—
179C2×D53—42no—
180C2×D53—81no—
181C2×D53—81no—
182C2×D53—81no—
183C2×D53—81no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—42no—
187C4×C53—81no—
188C4×C53#42711yescharacteristic
189C4×C53#42811yescharacteristic
190C4×C53C4×SD1611yescharacteristic
191C4×C53#44211yescharacteristic
192C4×C53#47611yescharacteristic
193C4×C53#47811yescharacteristic
194C4×C53#48211yescharacteristic
195C4×C53#48511yescharacteristic
196Dic53—42no—
197Dic53—82no—
198Dic53—161no—
199#2354—531no—
200#1948—531no—
201#1834—531no—
202#1802—531no—
203C2×D4⋊C8—531no—
204#1296—531no—
205#1279D5311yescharacteristic, complemented
206C2×C53#162811yescharacteristic
207C2×C53#163511yescharacteristic
208C2×C53#164211yescharacteristic
209C2×C53#164611yescharacteristic
210C2×C53#185411yescharacteristic
211C2×C53#186011yescharacteristic
212C2×C53#186411yescharacteristic
213D53—82no—
214C2×C4×D4—531no—
215C2×#117—1061no—
216C2×#106—1061no—
217C2×C4⋊C8—531no—
218C2×C4⋊C8C2×D5311yescharacteristic
219C2×#102—1061no—
220C2×C4×C8C2×D5311yescharacteristic
221#390—1061no—
222#388—1062no—
223#384—531no—
224#384—531no—
225#380—1061no—
226C4×C4⋊C4—1061no—
227C4×C4⋊C4C2×D5311yescharacteristic
228#340—1061no—
229#340—1062no—
230#340—1061no—
231D4⋊C8—534no—
232C53#889311yescharacteristic, complemented, Sylow
233C22×D4—531no—
234C23×C4—1061no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—1061no—
238C4×D4—532no—
239C4×D4—532no—
240C4×D4—1061no—
241C4×D4—1061no—
242#117—2122no—
243#117—2122no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—532no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4—1061no—
249C2×C4⋊C4—1061no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4D21211yescharacteristic
252C2×C4⋊C4D21211yescharacteristic
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1062no—
255C2×C22⋊C4—1062no—
256C2×C42—21no—
257C2×C42—1062no—
258C2×C42C22×D5311yescharacteristic
259#106—2122no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—532no—
263C4⋊C8—1061no—
264#102—2122no—
265C4×C8—21no—
266C4×C8C4×D5312yes—
267#95—21no—
268#95—1062no—
269C24—1061no—
270C2×D4—532no—
271C2×D4—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—1061no—
279C22×C4—1062no—
280C22×C4—1062no—
281C22×C4—1062no—
282C22×C4—2121no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4C2×D21211yescharacteristic
288C22×C4D4×D5311yescharacteristic
289C22×C4D4×D5311yescharacteristic
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—1062no—
297C2×C8—2121no—
298C2×C8—2121no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—532no—
303C4⋊C4—1061no—
304C4⋊C4—2122no—
305C4⋊C4—2122no—
306C4⋊C4—2121no—
307C4⋊C4—2121no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C4⋊C4—2121no—
311C4⋊C4—2121no—
312C4⋊C4—2121no—
313C4⋊C4—2121no—
314C4⋊C4#1171991312yes—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—4242no—
318C22⋊C4—2124no—
319C42—41no—
320C42—41no—
321C42—2124no—
322C42C2×C4×D5311yescharacteristic
323C42#1171994511yescharacteristic
324C42#1171994711yescharacteristic
325C42Q8×D5311yescharacteristic
326C23—1062no—
327C23—2121no—
328C23—2121no—
329C23—2121no—
330C23—2121no—
331C23—2121no—
332C23—2121no—
333C23#1313318711yescharacteristic
334D4—1062no—
335D4—1062no—
336D4—2121no—
337D4—2121no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—1062no—
347C2×C4—4241no—
348C2×C4—4241no—
349C2×C4—4241no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4C4⋊C4×D5311yescharacteristic
360C2×C4#1313317711yescharacteristic
361C2×C4C4×D21211yescharacteristic
362C2×C4#1313319511yescharacteristic
363C2×C4#1313319711yescharacteristic
364C2×C4#1313319911yescharacteristic
365C2×C4C2×#1171991311yescharacteristic
366C2×C4#1313323611yescharacteristic
367C2×C4#1313323911yescharacteristic
368C2×C4#1313324011yescharacteristic
369C2×C4#1313324411yescharacteristic
370C2×C4#1313324511yescharacteristic
371C2×C4#1313324511yescharacteristic
372C2×C4#1313324611yescharacteristic
373C8—42no—
374C8—81no—
375C8—2122no—
376C22—4241no—
377C22—4241no—
378C22—4241no—
379C22—4241no—
380C22—2122no—
381C22—2122no—
382C22—2122no—
383C22#1681049711yescharacteristic
384C22#1681077011yescharacteristic
385C22#1681077111yescharacteristic
386C22#1681077611yescharacteristic
387C22#1681078611yescharacteristic
388C22#1681079711yescharacteristic
389C22#1681080211yescharacteristic
390C4—21no—
391C4—21no—
392C4—42no—
393C4—81no—
394C4—8481no—
395C4—4242no—
396C4—2122no—
397C4#1681066011yescharacteristic
398C4#1681066211yescharacteristic
399C4C4×#1171991311yescharacteristic
400C4#1681073611yescharacteristic
401C4#1681091411yescharacteristic
402C4#1681092411yescharacteristic
403C4#1681094511yescharacteristic
404C4#1681096311yescharacteristic
405C2—4242no—
406C2#2027382811yescharacteristic
407C2#2027386811yescharacteristic
408C2#2027391811yescharacteristic
409C2#2027394511yescharacteristic
410C2#2027491711yescharacteristic
411C2#2027494911yescharacteristic
412C2#2027497911yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1279D53
hypocenterC53#8893
LCSG ▷ 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. #20275478C2semidirect
3. #20274843C2semidirect
6. #20272599C2semidirect
7. C53×#1279C2semidirect
22. #16810352C4semidirect
205. #1279D53semidirect
232. C53#8893semidirect

Maximal subgroups

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

subgroupindex
#202776622
#202754782
#202748432
#202747162
C2×#168103522
#202725992
C53×#12792
#889353