Finite Group Databasebeta

← #27280041

Subgroups — #27280041

Order 13568.

548 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284179C211yesmaximal, characteristic
2#20273274C211yesmaximal, characteristic, complemented
3#20273209C211yesmaximal, characteristic, complemented
4#20273106C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272573C211yesmaximal, characteristic, complemented
7C53×#2357C211yesmaximal, characteristic, complemented
8C2×#13133243—21no—
9#16811173C2211yescharacteristic
10#16810838—22no—
11#16810828C412yescomplemented
12#16810818C412yescomplemented
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810647C2211yescharacteristic
16#16810622—21no—
17#16810356—21no—
18#16810356—21no—
19#16810356—22no—
20D4⋊#88322—22no—
21C2×C4×D4×C53C2211yescharacteristic
22C53×#390—21no—
23C53×#388—22no—
24C53×#382C2211yescharacteristic
25C53×#381—21no—
26C4×C4⋊C4×C53—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C4×C22⋊C4×C53—21no—
29C2×#11719945—21no—
30#13133243—42no—
31#13133243—42no—
32C2×#11719923—22no—
33C2×#11719922—22no—
34C2×#11719917—21no—
35#13133217C2×C411yescharacteristic
36#13133214C2×C411yescharacteristic
37#13133205—21no—
38#13133191—21no—
39#13133191—21no—
40C4×D212—21no—
41C4×D212—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322C2×C412yes—
46#13133140—22no—
47#13133137—22no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52C22×D4×C53D411yescharacteristic
53C23×C4×C53—21no—
54C4×D4×C53—21no—
55C4×D4×C53—21no—
56C4×D4×C53C2×C412yes—
57C4×D4×C53D412yes—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C22⋊C4×C53—22no—
62C2×C22⋊C4×C53—22no—
63C2×C22⋊C4×C53—21no—
64C2×C42×C53—22no—
65C2×C42×C53—21no—
66C2×C42×C53C2311yescharacteristic
67#95×C53—22no—
68#95×C53—21no—
69#95×C53D411yescharacteristic
70#95×C53D411yescharacteristic
71#95×C53D412yes—
72C24×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53C22⋊C412yes—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—21no—
85C22×C4×C53—82no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C2×C22⋊D53—41no—
90#11719945—41no—
91#11719945—41no—
92#11719945—41no—
93#11719945—41no—
94C2×D212—21no—
95C2×D212—21no—
96C2×D212—21no—
97C2×C4×D53—41no—
98C2×Dic106—21no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—42no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53C22⋊C412yes—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—44no—
107C22⋊C4×C53—82no—
108C42×C53—41no—
109C42×C53—41no—
110C42×C53—82no—
111C42×C53C22×C411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53#4111yescharacteristic
114C42×C53#4111yescharacteristic
115#11719923—44no—
116#11719922—44no—
117#11719921—41no—
118#11719921—41no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121#11719917—41no—
122#11719917—41no—
123C2×#88322—22no—
124C2×#88322—41no—
125C2×#88322—41no—
126C23×C53—22no—
127C23×C53—42no—
128C23×C53—42no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53#12011yescharacteristic
132C22×D53—41no—
133D4×C53—22no—
134D4×C53—22no—
135D4×C53—41no—
136D4×C53—41no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—82no—
151C2×C4×C53—82no—
152C2×C4×C53C2×C22⋊C411yescharacteristic
153C2×C4×C53C4×D411yescharacteristic
154C2×C4×C53C4×D411yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12311yescharacteristic
158C2×C4×C53#13511yescharacteristic
159C2×C4×C53#13511yescharacteristic
160C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
162C22⋊D53—81no—
163C22⋊D53—81no—
164C2×Dic53—41no—
165D212—41no—
166D212—41no—
167D212—81no—
168D212—81no—
169C4×D53—81no—
170C4×D53—81no—
171Dic106—41no—
172Dic106—41no—
173#88322—42no—
174#88322—81no—
175#88322—81no—
176#9181—531nomaximal, Sylow
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53—42no—
182C22×C53—82no—
183C22×C53—82no—
184C22×C53#38611yescharacteristic
185C22×C53#44911yescharacteristic
186C22×C53#44911yescharacteristic
187C2×D53—81no—
188C2×D53—81no—
189C2×D53—81no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—42no—
195C4×C53—42no—
196C4×C53—81no—
197C4×C53—82no—
198C4×C53#41611yescharacteristic
199C4×C53#41811yescharacteristic
200C4×C53#48211yescharacteristic
201C4×C53#48211yescharacteristic
202Dic53—81no—
203#3036—531no—
204#2357D5311yescharacteristic, complemented
205#1517—531no—
206#1502—531no—
207#1478—531no—
208#1303—531no—
209#1286—531no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—82no—
213C2×C53#187511yescharacteristic
214C2×C53#190411yescharacteristic
215C2×C53#190611yescharacteristic
216D53—161no—
217C2×AGL(1,ℤ/8)—1061no—
218#532—531no—
219C2×C4×D4C2×D5311yescharacteristic
220#463—1062no—
221#461—532no—
222#459—532no—
223#431—531no—
224#431—531no—
225#423—531no—
226#417—1061no—
227#390—21no—
228#388—22no—
229#382C2×D5311yescharacteristic
230#381—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4C2×D5311yescharacteristic
233C4×C22⋊C4—21no—
234#327—1061no—
235#327—1061no—
236#327—1062no—
237D4⋊C8—1062no—
238C53#918111yescharacteristic, complemented, Sylow
239C2×#41—1061no—
240C22×D4C22⋊D5311yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—2122no—
243AGL(1,ℤ/8)—2122no—
244C2×SD16—1062no—
245C2×D8—1062no—
246C2×M16—1061no—
247C4⋊Q8—531no—
248#127—531no—
249#124—1061no—
250#121—1061no—
251#121—1061no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—1061no—
255C4×D4—1061no—
256C4×D4C4×D5312yes—
257C4×D4D21212yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22⋊D5311yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42C22×D5311yescharacteristic
267C4⋊C8—532no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270C4⋊C8—1061no—
271#103—1062no—
272#102—1062no—
273C22⋊C8—1061no—
274C22⋊C8—1061no—
275C4×C8—1061no—
276C4×C8—1061no—
277#95—22no—
278#95—21no—
279#95C22⋊D5311yescharacteristic
280#95C22⋊D5311yescharacteristic
281#95C22⋊D5312yes—
282C24—21no—
283#41—2121no—
284#41—2121no—
285#41—2121no—
286#41—2121no—
287C2×Q8—1061no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—1061no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—2121no—
297C2×D4#1171992112yes—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—82no—
305C22×C4—2121no—
306C22×C4C2×C22⋊D5311yescharacteristic
307C22×C4C2×C22⋊D5311yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309SD16—2124no—
310D8—2124no—
311M16—2121no—
312M16—2121no—
313C2×C8—1062no—
314C2×C8—2121no—
315C2×C8—2121no—
316C4⋊C4—21no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—1061no—
321C4⋊C4—1061no—
322C4⋊C4#1171992112yes—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C22⋊C4—2121no—
328C22⋊C4—2121no—
329C42—41no—
330C42—41no—
331C42—82no—
332C42C2×C4×D5311yescharacteristic
333C42C2×D21211yescharacteristic
334C42#1171994511yescharacteristic
335C42#1171994511yescharacteristic
336C23—22no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—2121no—
342C23#1313318611yescharacteristic
343Q8—2121no—
344Q8—2121no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349D4—4241no—
350D4—4241no—
351D4—4241no—
352D4—4241no—
353D4—2121no—
354D4—2121no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—4241no—
371C2×C4—4241no—
372C2×C4—2121no—
373C2×C4C2×#1171992111yescharacteristic
374C2×C4C4×C22⋊D5311yescharacteristic
375C2×C4C4×C22⋊D5311yescharacteristic
376C2×C4#1313319111yescharacteristic
377C2×C4#1313319111yescharacteristic
378C2×C4#1313320011yescharacteristic
379C2×C4#1313323711yescharacteristic
380C2×C4#1313323711yescharacteristic
381C2×C4#1313324211yescharacteristic
382C2×C4#1313324211yescharacteristic
383C8—4241no—
384C8—4241no—
385C8—2122no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—82no—
392C22—82no—
393C22—4241no—
394C22—4241no—
395C22—4241no—
396C22#1681050211yescharacteristic
397C22#1681076811yescharacteristic
398C22#1681076811yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—82no—
407C4—4241no—
408C4#1681061711yescharacteristic
409C4#1681062711yescharacteristic
410C4#1681094711yescharacteristic
411C4#1681094711yescharacteristic
412C2—21no—
413C2—21no—
414C2—82no—
415C2—8481no—
416C2#2027503711yescharacteristic
417C2#2027520211yescharacteristic
418C2#2027521611yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2357C2
socleC22×C53#386
prime radical#20284179C2
cosocleC2×C42×C53C23
hypercenter#2357D53
hypocenterC53#9181
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. #20273274C2semidirect
3. #20273209C2semidirect
4. #20273106C2semidirect
5. #20272618C2semidirect
6. #20272573C2semidirect
7. C53×#2357C2semidirect
11. #16810828C4semidirect
12. #16810818C4semidirect
204. #2357D53semidirect
238. C53#9181semidirect

Maximal subgroups

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

subgroupindex
#202841792
#202732742
#202732092
#202731062
#202726182
#202725732
C53×#23572
#918153