Finite Group Databasebeta

← #27279324

Subgroups — #27279324

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285275C211yesmaximal, characteristic
2#20283461C211yesmaximal, characteristic
3#20273209C211yesmaximal, characteristic
4#20273118C211yesmaximal, characteristic, complemented
5#20272556C211yesmaximal, characteristic, complemented
6#20272553C211yesmaximal, characteristic, complemented
7C53×#1986C211yesmaximal, characteristic, complemented
8C2×#13133225—21no—
9#16811173C2211yescharacteristic
10#16811019C411yescharacteristic, complemented
11#16810997C411yescharacteristic, complemented
12#16810973C411yescharacteristic, complemented
13#16810971C411yescharacteristic, complemented
14#16810854—22no—
15C4×#11719924—22no—
16#16810702—21no—
17#16810696—22no—
18C8×D212—21no—
19#16810676C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21#16810591—21no—
22C2×C4×#88322C2211yescharacteristic
23#16810356—22no—
24Q8⋊#88322—21no—
25Q8⋊#88322—21no—
26C53×#533C2211yescharacteristic
27C4×M16×C53C2211yescharacteristic
28C4×C4⋊C4×C53C2211yescharacteristic
29C53×#339—21no—
30C53×#339—21no—
31C2×#11719945—21no—
32C2×#11719924—22no—
33#13133225—42no—
34#13133225—42no—
35C2×#11719912—22no—
36C22×#88322—21no—
37C22×#88322—21no—
38C2×C8×D53—22no—
39#13133217C2×C411yescharacteristic
40#13133214C2×C411yescharacteristic
41#13133205—21no—
42#13133191—21no—
43#13133191D412yes—
44C4×D212—21no—
45C4×D212D412yes—
46#13133148—22no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322D411yescharacteristic
50C4⋊#88322D411yescharacteristic
51#13133141—22no—
52#13133137—22no—
53C22⋊#88322—21no—
54#13133129—22no—
55#13133129—22no—
56C4×#88322—21no—
57C4×#88322—21no—
58C4×#88322C2×C411yescharacteristic
59C4×#88322C2×C411yescharacteristic
60C2×M16×C53D411yescharacteristic
61C2×M16×C53D411yescharacteristic
62C4⋊Q8×C53C2×C411yescharacteristic
63C4⋊Q8×C53C2×C411yescharacteristic
64#125×C53—21no—
65C22⋊Q8×C53—21no—
66C22⋊Q8×C53—21no—
67C4×Q8×C53—21no—
68C4×Q8×C53—21no—
69#117×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73#97×C53—21no—
74C4×C8×C53—21no—
75#95×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C22×C4×C53—41no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C2×C22⋊D53—22no—
83#11719945—41no—
84#11719945—41no—
85#11719945—41no—
86#11719945—41no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212C22⋊C412yes—
90C2×C4×D53—22no—
91C2×Dic106—21no—
92M16×C53—21no—
93M16×C53—21no—
94M16×C53—21no—
95M16×C53SD1612yescomplemented
96C2×C8×C53—21no—
97C2×C8×C53—21no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C42×C53—41no—
108C42×C53—41no—
109C42×C53—41no—
110C42×C53C22×C411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53#4111yescharacteristic
114#11719924—44no—
115#11719921—22no—
116#11719921—22no—
117C4⋊Dic53—21no—
118C4⋊Dic53C22⋊C412yes—
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—42no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
127#11719912—44no—
128C8×D53—44no—
129C23×C53#12111yescharacteristic
130C22×D53—22no—
131Q8×C53—42no—
132Q8×C53—41no—
133Q8×C53—41no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—81no—
141C2×C4×C53—81no—
142C2×C4×C53C2×C22⋊C411yescharacteristic
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12311yescharacteristic
148C2×C4×C53C2×SD1611yescharacteristic
149C2×C4×C53#13511yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C53#13711yescharacteristic
152C22⋊D53—42no—
153C22⋊D53—42no—
154C2×Dic53—22no—
155D212—42no—
156D212—41no—
157D212—41no—
158D212—41no—
159D212—41no—
160C4×D53—42no—
161C4×D53—42no—
162Dic106—41no—
163Dic106—41no—
164C8×C53—22no—
165C8×C53—41no—
166#88322—22no—
167#88322—41no—
168#88322—41no—
169#88322—41no—
170#88322—81no—
171#9055—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53#38611yescharacteristic
175C22×C53#46111yescharacteristic
176C22×C53#46311yescharacteristic
177C2×D53—42no—
178C2×D53—42no—
179C2×D53—42no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—82no—
184C4×C53—81no—
185C4×C53—81no—
186C4×C53C4×SD1611yescharacteristic
187C4×C53#44011yescharacteristic
188C4×C53#44311yescharacteristic
189C4×C53#44411yescharacteristic
190C4×C53#45011yescharacteristic
191C4×C53#45211yescharacteristic
192Dic53—42no—
193#3167—531no—
194#2935—531no—
195#1986D5311yescharacteristic, complemented
196#1502—531no—
197#1481—531no—
198#1279—531no—
199#1278—531no—
200C2×C53—21no—
201C2×C53—21no—
202C2×C53#187611yescharacteristic
203C2×C53#187711yescharacteristic
204C2×C53#188411yescharacteristic
205D53—82no—
206C2×#131—1061no—
207#533C2×D5311yescharacteristic
208#532—531no—
209#498—531no—
210#492—531no—
211#487—531no—
212#486—531no—
213#465—1062no—
214C4×SD16—1062no—
215#436—1061no—
216#435—1062no—
217C8×D4—1061no—
218#431—531no—
219C2×C4⋊C8—531no—
220#408—1061no—
221C4×M16C2×D5311yescharacteristic
222C2×C4×C8—531no—
223C4×C4⋊C4C2×D5311yescharacteristic
224#339—21no—
225#339—21no—
226#327—1062no—
227Q8⋊C8—1061no—
228Q8⋊C8—1061no—
229C53#905511yescharacteristic, complemented, Sylow
230C2×#41—1061no—
231C2×SD16—1062no—
232#131—2122no—
233#131—2122no—
234C2×M16—1062no—
235C2×M16C22⋊D5311yescharacteristic
236C2×M16C22⋊D5311yescharacteristic
237C22×C8—1062no—
238C22×C8—1061no—
239C22×C8—1061no—
240C4⋊Q8—531no—
241C4⋊Q8C4×D5311yescharacteristic
242C4⋊Q8C4×D5311yescharacteristic
243#127—531no—
244#125—21no—
245#124—1061no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248#121—532no—
249#121—1061no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×D4—532no—
253C4×D4—1061no—
254#117—21no—
255C2×C4⋊C4—21no—
256C2×C42—21no—
257C2×C42C22×D5311yescharacteristic
258#106—1062no—
259C4⋊C8—531no—
260C4⋊C8—531no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263#103—1062no—
264#102—1062no—
265C22⋊C8—1062no—
266C22⋊C8—1062no—
267C22⋊C8—1061no—
268#97—21no—
269C4×C8—21no—
270C4×C8—531no—
271C4×C8—531no—
272C4×C8—1061no—
273C4×C8—1061no—
274#95—21no—
275#41—2121no—
276#41—2121no—
277#41—2121no—
278#41—2121no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—1061no—
282C2×D4—532no—
283C2×D4—1062no—
284C2×D4—1061no—
285C2×D4—1061no—
286C22×C4—41no—
287C22×C4—1062no—
288C22×C4D4×D5311yescharacteristic
289C22×C4#1171994711yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291SD16—2124no—
292M16—21no—
293M16—21no—
294M16—21no—
295M16—2124no—
296M16#1171992412yescomplemented
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—1061no—
300C2×C8—1061no—
301C2×C8—1061no—
302C2×C8—1061no—
303C2×C8—2124no—
304C2×C8—2122no—
305C2×C8—2121no—
306C2×C8—2121no—
307C2×C8—2121no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—532no—
317C4⋊C4—1061no—
318C22⋊C4—41no—
319C22⋊C4—1062no—
320C22⋊C4—1062no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42C2×C4×D5311yescharacteristic
325C42#1171994511yescharacteristic
326C42D4×D5311yescharacteristic
327C42D4×D5311yescharacteristic
328C23—1062no—
329C23#1313318811yescharacteristic
330Q8—42no—
331Q8—41no—
332Q8—41no—
333Q8—2121no—
334Q8—2121no—
335D4—2122no—
336D4—2122no—
337D4—2122no—
338D4—2121no—
339D4—2121no—
340D4—2121no—
341D4—2121no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—1062no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4C22⋊C4×D5311yescharacteristic
354C2×C4#1313317711yescharacteristic
355C2×C4C4×C22⋊D5311yescharacteristic
356C2×C4#1313318511yescharacteristic
357C2×C4#1313319011yescharacteristic
358C2×C4#1313319911yescharacteristic
359C2×C4C2×#1171992411yescharacteristic
360C2×C4#1313323711yescharacteristic
361C2×C4#1313324211yescharacteristic
362C2×C4#1313324811yescharacteristic
363C8—22no—
364C8—41no—
365C8—1062no—
366C8—4241no—
367C8—2121no—
368C8—2121no—
369C8—2121no—
370C22—21no—
371C22—21no—
372C22—2122no—
373C22—2122no—
374C22—2122no—
375C22#1681049911yescharacteristic
376C22#1681082511yescharacteristic
377C22#1681083511yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4—2122no—
385C4C4×#1171992411yescharacteristic
386C4#1681072111yescharacteristic
387C4#1681073711yescharacteristic
388C4#1681074211yescharacteristic
389C4#1681077311yescharacteristic
390C4#1681078311yescharacteristic
391C2—21no—
392C2—21no—
393C2—4242no—
394C2#2027503911yescharacteristic
395C2#2027505211yescharacteristic
396C2#2027509511yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1986C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#1986D53
hypocenterC53#9055
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
4. #20273118C2semidirect
5. #20272556C2semidirect
6. #20272553C2semidirect
7. C53×#1986C2semidirect
10. #16811019C4semidirect
11. #16810997C4semidirect
12. #16810973C4semidirect
13. #16810971C4semidirect
95. M16×C53SD16semidirect
195. #1986D53semidirect
229. C53#9055semidirect
296. M16#11719924semidirect

Maximal subgroups

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

subgroupindex
#202852752
#202834612
#202732092
#202731182
#202725562
#202725532
C53×#19862
#905553