Finite Group Databasebeta

← #27280043

Subgroups — #27280043

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#20277688C211yesmaximal, characteristic, complemented
2#20273275C211yesmaximal, characteristic, complemented
3#20273210C211yesmaximal, characteristic, complemented
4#20273106C211yesmaximal, characteristic, complemented
5#20272616C211yesmaximal, characteristic, complemented
6#20272575C211yesmaximal, characteristic, complemented
7C53×#3036C211yesmaximal, characteristic
8#16810676C2211yescharacteristic
9#16810676C2211yescharacteristic
10#16810526—21no—
11#16810516—22no—
12#16810489C2211yescharacteristic
13#16810486—21no—
14C4⋊C4×Dic53—21no—
15C4×C4⋊Dic53C2211yescharacteristic
16C4×C22⋊Dic53—21no—
17#16810357—21no—
18#16810357—21no—
19#16810357—22no—
20D4⋊#88322—22no—
21C2×AGL(1,ℤ/8)×C53—21no—
22C53×#532C2211yescharacteristic
23C2×C4×D4×C53C2211yescharacteristic
24C53×#463—22no—
25C53×#461C412yescomplemented
26C53×#459C412yescomplemented
27C53×#423C2211yescharacteristic
28C53×#417—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—22no—
33C2×C4×Dic53—22no—
34C2×C4×Dic53—21no—
35C4⋊#88322—21no—
36C4⋊#88322—21no—
37C22⋊#88322—21no—
38C22⋊#88322—21no—
39C4×#88322—21no—
40C4×#88322—21no—
41#13133124—22no—
42#13133124—21no—
43#13133124D412yes—
44#13133124D411yescharacteristic
45#13133124D411yescharacteristic
46C2×#41×C53—21no—
47C22×D4×C53D411yescharacteristic
48C23×C4×C53—21no—
49AGL(1,ℤ/8)×C53—42no—
50AGL(1,ℤ/8)×C53—42no—
51C2×SD16×C53—22no—
52C2×D8×C53—22no—
53C2×M16×C53—21no—
54C4⋊Q8×C53C2×C411yescharacteristic
55#127×C53C2×C411yescharacteristic
56#124×C53—21no—
57#121×C53—21no—
58#121×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53C2×C412yes—
64C4×D4×C53D412yes—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C22⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53C2×C412yes—
70#103×C53—22no—
71#102×C53—22no—
72C24×C53—21no—
73#41×C53—41no—
74#41×C53—41no—
75#41×C53—41no—
76#41×C53—41no—
77C2×Q8×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—41no—
87C2×D4×C53C22⋊C412yes—
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—41no—
92C22×C4×C53—82no—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×Dic53—21no—
97C22×Dic53—22no—
98C22×Dic53—21no—
99SD16×C53—44no—
100D8×C53—44no—
101M16×C53—41no—
102M16×C53—41no—
103C2×C8×C53—22no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53C22⋊C412yes—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C42×C53C22×C411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53#4111yescharacteristic
115C42×C53#4111yescharacteristic
116C22⋊Dic53—44no—
117C22⋊Dic53—82no—
118C4⋊Dic53—42no—
119C4⋊Dic53—41no—
120C4⋊Dic53—41no—
121C4×Dic53—41no—
122C4×Dic53—41no—
123C4×Dic53—82no—
124C2×#88322—41no—
125C2×#88322—41no—
126C23×C53—22no—
127C23×C53—42no—
128C23×C53—42no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53#12011yescharacteristic
133Q8×C53—41no—
134Q8×C53—41no—
135D4×C53—22no—
136D4×C53—22no—
137D4×C53—41no—
138D4×C53—41no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53—81no—
142D4×C53—81no—
143D4×C53—81no—
144D4×C53—81no—
145C2×C4×C53—22no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—41no—
150C2×C4×C53—82no—
151C2×C4×C53—82no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×C4×C53C2×C22⋊C411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12311yescharacteristic
160C2×C4×C53#13511yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×Dic53—22no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—42no—
168C2×Dic53—41no—
169C2×Dic53—42no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—42no—
173C8×C53—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—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53#38611yescharacteristic
188C22×C53#44911yescharacteristic
189C22×C53#44911yescharacteristic
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—42no—
195C4×C53—81no—
196C4×C53#41611yescharacteristic
197C4×C53#41811yescharacteristic
198C4×C53#48211yescharacteristic
199C4×C53#48211yescharacteristic
200Dic53—42no—
201Dic53—81no—
202Dic53—82no—
203#3036D5311yescharacteristic
204#2357—531no—
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—161no—
214C2×C53#187511yescharacteristic
215C2×C53#190411yescharacteristic
216C2×C53#190611yescharacteristic
217C2×AGL(1,ℤ/8)—21no—
218#532C2×D5311yescharacteristic
219C2×C4×D4C2×D5311yescharacteristic
220#463—22no—
221#461Dic5312yescomplemented
222#459Dic5312yescomplemented
223#431—531no—
224#431—531no—
225#423C2×D5311yescharacteristic
226#417—21no—
227#390—1061no—
228#388—1062no—
229#382—531no—
230#381—1061no—
231C4×C4⋊C4—531no—
232C4×C4⋊C4—1061no—
233C4×C22⋊C4—1061no—
234#327—1061no—
235#327—1061no—
236#327—1062no—
237D4⋊C8—1062no—
238C53#918111yescharacteristic, complemented, Sylow
239C2×#41—21no—
240C22×D4C22⋊D5311yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—42no—
243AGL(1,ℤ/8)—42no—
244C2×SD16—22no—
245C2×D8—22no—
246C2×M16—21no—
247C4⋊Q8C2×Dic5311yescharacteristic
248#127C2×Dic5311yescharacteristic
249#124—21no—
250#121—21no—
251#121—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4C2×Dic5312yes—
257C4×D4C22⋊D5312yes—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4—1061no—
260C2×C4⋊C4C22⋊D5311yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—1062no—
263C2×C22⋊C4—1062no—
264C2×C42—1062no—
265C2×C42—1061no—
266C2×C42C22×D5311yescharacteristic
267C4⋊C8—21no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270C4⋊C8C2×Dic5312yes—
271#103—22no—
272#102—22no—
273C22⋊C8—1061no—
274C22⋊C8—1061no—
275C4×C8—1061no—
276C4×C8—1061no—
277#95—531no—
278#95—531no—
279#95—532no—
280#95—1062no—
281#95—1061no—
282C24—21no—
283#41—41no—
284#41—41no—
285#41—41no—
286#41—41no—
287C2×Q8—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4C22⋊Dic5312yes—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—82no—
303C22×C4—1061no—
304C22×C4—1062no—
305C22×C4—1061no—
306C22×C4D4×D5311yescharacteristic
307C22×C4D4×D5311yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309SD16—44no—
310D8—44no—
311M16—41no—
312M16—41no—
313C2×C8—22no—
314C2×C8—2121no—
315C2×C8—2121no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—2122no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C4⋊C4C22⋊Dic5312yes—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—4242no—
328C22⋊C4—2124no—
329C42—4242no—
330C42—2121no—
331C42—2121no—
332C42#1171994711yescharacteristic
333C42#1171994711yescharacteristic
334C42C22×Dic5311yescharacteristic
335C42C2×C22⋊D5311yescharacteristic
336C23—22no—
337C23—42no—
338C23—42no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23#1313318511yescharacteristic
343Q8—41no—
344Q8—41no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349D4—41no—
350D4—41no—
351D4—81no—
352D4—81no—
353D4—81no—
354D4—81no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—1062no—
365C2×C4—2122no—
366C2×C4—2121no—
367C2×C4—2122no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2122no—
371C2×C4—2121no—
372C2×C4—2121no—
373C2×C4C2×C22⋊Dic5311yescharacteristic
374C2×C4D4×Dic5311yescharacteristic
375C2×C4D4×Dic5311yescharacteristic
376C2×C4#1313318911yescharacteristic
377C2×C4#1313318911yescharacteristic
378C2×C4#1313319811yescharacteristic
379C2×C4#1313323711yescharacteristic
380C2×C4#1313323711yescharacteristic
381C2×C4#1313324311yescharacteristic
382C2×C4#1313324311yescharacteristic
383C8—42no—
384C8—4241no—
385C8—4241no—
386C22—21no—
387C22—21no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—82no—
392C22—82no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#1681050311yescharacteristic
397C22#1681076611yescharacteristic
398C22#1681076611yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—81no—
405C4—4242no—
406C4—4241no—
407C4—2122no—
408C4#1681061911yescharacteristic
409C4#1681062811yescharacteristic
410C4#1681094411yescharacteristic
411C4#1681094411yescharacteristic
412C2—21no—
413C2—21no—
414C2—82no—
415C2—161no—
416C2#2027503811yescharacteristic
417C2#2027520411yescharacteristic
418C2#2027521811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3036C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3036D53
hypocenterC53#9181
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #20277688C2semidirect
2. #20273275C2semidirect
3. #20273210C2semidirect
4. #20273106C2semidirect
5. #20272616C2semidirect
6. #20272575C2semidirect
25. C53×#461C4semidirect
26. C53×#459C4semidirect
221. #461Dic53semidirect
222. #459Dic53semidirect
238. C53#9181semidirect

Maximal subgroups

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

subgroupindex
#202776882
#202732752
#202732102
#202731062
#202726162
#202725752
C53×#30362
#918153