Finite Group Databasebeta

← #27278302

Subgroups — #27278302

Order 13568.

546 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277666C211yesmaximal, characteristic
2#20275474C211yesmaximal, characteristic, complemented
3#20274871C211yesmaximal, characteristic, complemented
4#20274716C211yesmaximal, characteristic
5C2×#16810350C211yesmaximal, characteristic
6#20272554C211yesmaximal, characteristic, complemented
7C53×#1296C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C2×#13133175—21no—
10C2×#13133151—21no—
11C2×#13133148—22no—
12C2×#13133139—21no—
13C2×C4×#88322C2211yescharacteristic
14#16810524—21no—
15#16810513—22no—
16#16810493C2211yescharacteristic
17#16810480—21no—
18C4⋊C4×Dic53—21no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810386—21no—
21#16810386—21no—
22#16810350C414yescomplemented
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C53×#384C2211yescharacteristic
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⋊Dic53D411yescharacteristic
39C2×C4⋊Dic53D411yescharacteristic
40C2×C4⋊Dic53D411yescharacteristic
41C2×#11719921—22no—
42C2×#11719921—22no—
43C2×#11719921—21no—
44C2×C4×Dic53—22no—
45C2×C4×Dic53—21no—
46#13133151—42no—
47#13133148—44no—
48#13133139—42no—
49C4×#88322—21no—
50C4×#88322C2×C412yes—
51#13133124—22no—
52#13133124—21no—
53C22×C8×C53—21no—
54C22×C8×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53D412yes—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53C2×C412yes—
63C23×D53—21no—
64C22×C4×C53—22no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—41no—
72C2×D212—21no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212C4⋊C412yes—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C8×C53—22no—
85C2×C8×C53—41no—
86C2×C8×C53—41no—
87C2×C8×C53—41no—
88C2×C8×C53—41no—
89C2×C8×C53—41no—
90C4⋊C4×C53—42no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—41no—
94C4⋊C4×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—21no—
105C4⋊Dic53—21no—
106C4⋊Dic53—21no—
107C4⋊Dic53—42no—
108C4⋊Dic53—41no—
109C4⋊Dic53—41no—
110C4⋊Dic53—41no—
111C4⋊Dic53—41no—
112C4⋊Dic53C4⋊C412yes—
113C4⋊Dic53SD1612yes—
114C4×Dic53—44no—
115C4×Dic53—41no—
116C4×Dic53—41no—
117C2×#88322—22no—
118C2×#88322—22no—
119C2×#88322—22no—
120C23×C53#12011yescharacteristic
121C22×D53—22no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—41no—
126C22×D53—41no—
127C22×D53—41no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53C2×C4⋊C411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C22⋊Q811yescharacteristic
137C2×C4×C53C22⋊Q811yescharacteristic
138C2×C4×C53#12311yescharacteristic
139C2×C4×C53C2×SD1611yescharacteristic
140C2×C4×C53#13511yescharacteristic
141C2×C4×C53#13511yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53#13711yescharacteristic
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—81no—
157C2×Dic53—81no—
158C2×Dic53—81no—
159D212—22no—
160D212—22no—
161D212—41no—
162D212—41no—
163C4×D53—42no—
164C4×D53—42no—
165C4×D53—42no—
166C4×D53—42no—
167C8×C53—42no—
168C8×C53—81no—
169#88322—24no—
170#8889—531nomaximal, Sylow
171C22×C53#38511yescharacteristic
172C22×C53#44911yescharacteristic
173C22×C53#44911yescharacteristic
174C22×C53#45011yescharacteristic
175C22×C53#45211yescharacteristic
176C22×C53#45311yescharacteristic
177C22×C53#45611yescharacteristic
178C2×D53—42no—
179C2×D53—42no—
180C2×D53—42no—
181C2×D53—81no—
182C2×D53—81no—
183C2×D53—81no—
184C2×D53—81no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—82no—
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—42no—
198Dic53—81no—
199Dic53—161no—
200#2354—531no—
201#1947—531no—
202#1844—531no—
203#1802—531no—
204C2×D4⋊C8—531no—
205#1296D5311yescharacteristic, complemented
206#1279—531no—
207C2×C53#162411yescharacteristic
208C2×C53#163111yescharacteristic
209C2×C53#165211yescharacteristic
210C2×C53#165411yescharacteristic
211C2×C53#185411yescharacteristic
212C2×C53#185911yescharacteristic
213C2×C53#186411yescharacteristic
214D53—82no—
215C2×C4×D4—531no—
216C2×#117—1061no—
217C2×#107—1061no—
218C2×#106—1062no—
219C2×C4⋊C8C2×D5311yescharacteristic
220C2×C4⋊C8C2×D5311yescharacteristic
221C2×#102—1061no—
222C2×C4×C8—531no—
223#390—1061no—
224#388—1062no—
225#384—531no—
226#384C2×D5311yescharacteristic
227#380—1061no—
228C4×C4⋊C4—531no—
229C4×C4⋊C4—1061no—
230#340—1061no—
231#340—1061no—
232D4⋊C8—534no—
233C53#888911yescharacteristic, complemented, Sylow
234C22×D4—531no—
235C23×C4—1061no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—1061no—
239C4×D4—532no—
240C4×D4—532no—
241C4×D4—1061no—
242C4×D4—1061no—
243#117—2122no—
244#117—2122no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4—531no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4—1061no—
252C2×C4⋊C4—1061no—
253C2×C4⋊C4D21212yes—
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1062no—
256C2×C22⋊C4—1062no—
257C2×C42—1062no—
258C2×C42—1061no—
259C2×C42C22×D5311yescharacteristic
260#107—2122no—
261#106—2124no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8C4×D5312yes—
266#102—2122no—
267C4×C8—532no—
268C4×C8—1061no—
269#95—1062no—
270#95—1061no—
271C24—1061no—
272C2×D4—532no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C22×C4—22no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1062no—
282C22×C4—1062no—
283C22×C4—1061no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4—2121no—
288C22×C4—2121no—
289C22×C4C2×D21211yescharacteristic
290C22×C4D4×D5311yescharacteristic
291C22×C4D4×D5311yescharacteristic
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—1062no—
299C2×C8—1062no—
300C2×C8—1062no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—532no—
307C4⋊C4—532no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—2122no—
313C4⋊C4—2121no—
314C4⋊C4—2121no—
315C4⋊C4—2121no—
316C4⋊C4—2121no—
317C22⋊C4—1062no—
318C22⋊C4—1062no—
319C22⋊C4—4242no—
320C22⋊C4—2124no—
321C42—2124no—
322C42—2121no—
323C42—2121no—
324C42C2×C4×D5311yescharacteristic
325C42#1171994511yescharacteristic
326C42#1171994711yescharacteristic
327C42Q8×D5311yescharacteristic
328C23—1062no—
329C23—2121no—
330C23—2121no—
331C23—2121no—
332C23—2121no—
333C23—2121no—
334C23—2121no—
335C23#1313318711yescharacteristic
336D4—1062no—
337D4—1062no—
338D4—2121no—
339D4—2121no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—1062no—
346C2×C4—1062no—
347C2×C4—4241no—
348C2×C4—4241no—
349C2×C4—4241no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4C4⋊C4×D5311yescharacteristic
362C2×C4#1313317711yescharacteristic
363C2×C4C4×D21211yescharacteristic
364C2×C4#1313319511yescharacteristic
365C2×C4#1313319711yescharacteristic
366C2×C4#1313319911yescharacteristic
367C2×C4SD16×D5311yescharacteristic
368C2×C4#1313323611yescharacteristic
369C2×C4#1313324011yescharacteristic
370C2×C4#1313324011yescharacteristic
371C2×C4#1313324111yescharacteristic
372C2×C4#1313324411yescharacteristic
373C2×C4#1313324411yescharacteristic
374C2×C4#1313324611yescharacteristic
375C8—42no—
376C8—81no—
377C8—1064no—
378C22—4241no—
379C22—4241no—
380C22—4241no—
381C22—4241no—
382C22—2122no—
383C22—2122no—
384C22—2122no—
385C22#1681049711yescharacteristic
386C22#1681077111yescharacteristic
387C22#1681077111yescharacteristic
388C22#1681077511yescharacteristic
389C22#1681078611yescharacteristic
390C22#1681079311yescharacteristic
391C22#1681080811yescharacteristic
392C4—21no—
393C4—21no—
394C4—82no—
395C4—8481no—
396C4—4241no—
397C4—2122no—
398C4—2122no—
399C4#1681066011yescharacteristic
400C4#1681066211yescharacteristic
401C4#1681070811yescharacteristic
402C4#1681073211yescharacteristic
403C4#1681091211yescharacteristic
404C4#1681092611yescharacteristic
405C4#1681094511yescharacteristic
406C4#1681096311yescharacteristic
407C2—4242no—
408C2#2027379911yescharacteristic
409C2#2027384311yescharacteristic
410C2#2027398611yescharacteristic
411C2#2027400011yescharacteristic
412C2#2027491711yescharacteristic
413C2#2027494411yescharacteristic
414C2#2027497711yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1296D53
hypocenterC53#8889
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #20275474C2semidirect
3. #20274871C2semidirect
6. #20272554C2semidirect
7. C53×#1296C2semidirect
22. #16810350C4semidirect
205. #1296D53semidirect
233. C53#8889semidirect

Maximal subgroups

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

subgroupindex
#202776662
#202754742
#202748712
#202747162
C2×#168103502
#202725542
C53×#12962
#888953