Finite Group Databasebeta

← #27358001

Subgroups — #27358001

Order 13568.

1220 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285472C212yesmaximal, complemented
2#20280528C211yesmaximal, characteristic, complemented
3#20273752C214yesmaximal, complemented
4#20273684C214yesmaximal, complemented
5#20273283C212yesmaximal, complemented
6C2×#16810349C211yesmaximal, characteristic, complemented
7C2×C53×#537C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810953—22no—
10#16810939C2214yescomplemented
11#16810850—24no—
12#16810774—22no—
13#16810761C2214yescomplemented
14C2×C22⋊#88322C2212yes—
15#16810562C2212yescomplemented
16#16810556—22no—
17#16810543C2212yescomplemented
18#16810496—21no—
19#16810486—21no—
20#16810483C2211yescharacteristic
21#16810357C2218yescomplemented
22#16810349C2214yescomplemented
23C22×#41×C53—21no—
24C53×#537—24no—
25C53×#537—22no—
26C53×#537C2214yes—
27C2×C22⋊Q8×C53—22no—
28C2×#121×C53—22no—
29C2×#121×C53—21no—
30C2×#121×C53C2212yes—
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×Dic106C2311yescharacteristic
33C2×#11719923—24no—
34C22×#88322—22no—
35#13133196—22no—
36#13133196—22no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53C2311yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—22no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43#13133148—24no—
44#13133140—24no—
45#13133138—24no—
46C22⋊#88322—22no—
47C22⋊#88322C2314yes—
48#13133124—22no—
49#13133124—22no—
50#13133124—22no—
51#13133124C2312yes—
52C2×#41×C53—24no—
53C2×#41×C53—24no—
54C2×#41×C53—44no—
55C2×#41×C53—44no—
56C22×Q8×C53—21no—
57C22×D4×C53—21no—
58C22×D4×C53—22no—
59C22×D4×C53—21no—
60C22×D4×C53D412yes—
61C23×C4×C53—22no—
62C23×C4×C53D411yescharacteristic
63C23×C4×C53D411yescharacteristic
64C23×C4×C53C2311yescharacteristic
65C22⋊Q8×C53—28no—
66C22⋊Q8×C53—28no—
67#121×C53—28no—
68#121×C53—24no—
69#121×C53—28no—
70#121×C53—24no—
71#121×C53—22no—
72#121×C53—22no—
73#121×C53D414yes—
74#121×C53C2314yes—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53—22no—
77C2×C4⋊C4×C53—22no—
78C2×C4⋊C4×C53D412yes—
79C2×C4⋊C4×C53D412yes—
80C2×C4⋊C4×C53C2312yes—
81C2×C22⋊C4×C53—22no—
82C2×C22⋊C4×C53—22no—
83C24×C53—22no—
84C24×C53C2×D411yescharacteristic
85#41×C53—48no—
86#41×C53—416no—
87#41×C53—48no—
88C2×Q8×C53—24no—
89C2×Q8×C53—44no—
90C2×D4×C53—24no—
91C2×D4×C53—24no—
92C2×D4×C53—24no—
93C2×D4×C53—24no—
94C2×D4×C53—22no—
95C2×D4×C53—44no—
96C2×D4×C53—44no—
97C2×D4×C53—44no—
98C2×D4×C53—82no—
99C2×D4×C53—82no—
100C2×D4×C53C2×D414yes—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—22no—
104C22×C4×C53—28no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—48no—
109C22×C4×C53—81no—
110C22×C4×C53—81no—
111C22×C4×C53SD1614yescomplemented
112C22×C4×C53C2×D412yes—
113C22×C4×C53C2×D412yes—
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D412yes—
116C22×C4×C53C2×D412yes—
117C22×C4×C53C2411yescharacteristic
118C22×Dic53—21no—
119C22×Dic53—22no—
120C22×Dic53—21no—
121C2×Dic106—22no—
122C2×Dic106—81no—
123C4⋊C4×C53—24no—
124C4⋊C4×C53—24no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—24no—
127C4⋊C4×C53C2×D414yes—
128C22⋊C4×C53—28no—
129C22⋊C4×C53—24no—
130C22⋊C4×C53—24no—
131C22⋊Dic53—41no—
132C22⋊Dic53—41no—
133C22⋊Dic53—42no—
134#11719923—84no—
135C4⋊Dic53—22no—
136C4⋊Dic53—81no—
137#11719919—42no—
138#11719919—82no—
139C2×#88322—24no—
140C2×#88322—82no—
141C23×C53—24no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—44no—
145C23×C53—44no—
146C23×C53—44no—
147C23×C53#12011yescharacteristic
148C23×C53#12011yescharacteristic
149C23×C53C2×SD1612yes—
150C23×C53AGL(1,ℤ/8)12yes—
151C23×C53C22×D411yescharacteristic
152Q8×C53—48no—
153D4×C53—44no—
154D4×C53—48no—
155D4×C53—44no—
156D4×C53—44no—
157D4×C53—48no—
158D4×C53—82no—
159D4×C53—84no—
160C2×C4×C53—22no—
161C2×C4×C53—22no—
162C2×C4×C53—24no—
163C2×C4×C53—24no—
164C2×C4×C53—24no—
165C2×C4×C53—22no—
166C2×C4×C53—41no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53—41no—
170C2×C4×C53—416no—
171C2×C4×C53—42no—
172C2×C4×C53—44no—
173C2×C4×C53—42no—
174C2×C4×C53—81no—
175C2×C4×C53#12012yes—
176C2×C4×C53C2×SD1614yes—
177C2×C4×C53C22×D412yes—
178C2×Dic53—42no—
179C2×Dic53—42no—
180C2×Dic53—42no—
181C2×Dic53—42no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—42no—
185Dic106—44no—
186#88322—44no—
187#18293—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—42no—
195C22×C53—42no—
196C22×C53—44no—
197C22×C53—44no—
198C22×C53—44no—
199C22×C53—88no—
200C22×C53#45012yes—
201C22×C53#45012yes—
202C22×C53C2×#12011yescharacteristic
203C22×C53C22×SD1611yescharacteristic
204C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
205C4×C53—24no—
206C4×C53—22no—
207C4×C53—22no—
208C4×C53—44no—
209C4×C53—42no—
210Dic53—82no—
211Dic53—81no—
212Dic53—81no—
213C2×#537D5311yescharacteristic
214#3187—532no—
215#2627—531no—
216#1616—534no—
217#1604—534no—
218#1519—532no—
219C2×#324—531no—
220C2×C53—21no—
221C2×C53—22no—
222C2×C53—21no—
223C2×C53—22no—
224C2×C53—84no—
225C2×C53C2×#45011yescharacteristic
226C2×C53#302011yescharacteristic
227C2×C53#302611yescharacteristic
228C22×#41—21no—
229#537—24no—
230#537—22no—
231#537C2×D5314yes—
232C2×C22⋊Q8—22no—
233C2×C22⋊Q8—531no—
234C2×#121—22no—
235C2×#121—21no—
236C2×#121C2×D5312yes—
237C22×C4⋊C4C2×D5311yescharacteristic
238#483—1062no—
239#481—534no—
240#465—1064no—
241#450—1062no—
242#448—534no—
243C2×C22⋊C8—532no—
244#400—532no—
245#398—1062no—
246#395—532no—
247#385—1061no—
248#381—1061no—
249#380—531no—
250#327—538no—
251#324—534no—
252C53#1829311yescharacteristic, complemented, Sylow
253C2×#41—24no—
254C2×#41—24no—
255C2×#41—44no—
256C2×#41—44no—
257C22×Q8—21no—
258C22×Q8—531no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—21no—
262C22×D4C22⋊D5312yes—
263C23×C4—22no—
264C23×C4C22⋊D5311yescharacteristic
265C23×C4C22⋊D5311yescharacteristic
266C23×C4C22×D5311yescharacteristic
267C2×SD16—1064no—
268C22×C8—1062no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—1062no—
272C22⋊Q8—1062no—
273#121—24no—
274#121—28no—
275#121—28no—
276#121—24no—
277#121—22no—
278#121—22no—
279#121C22⋊D5314yes—
280#121C22×D5314yes—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—531no—
285C2×C4⋊C4—1062no—
286C2×C4⋊C4—1061no—
287C2×C4⋊C4—1061no—
288C2×C4⋊C4C22⋊D5312yes—
289C2×C4⋊C4C22⋊D5312yes—
290C2×C4⋊C4C22×D5312yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—1061no—
294C2×C22⋊C4—1061no—
295#106—1064no—
296#103—1064no—
297#102—1064no—
298C22⋊C8—534no—
299C22⋊C8—1062no—
300#95—532no—
301#95—1062no—
302#95—1062no—
303#95—1062no—
304C24—22no—
305C24C2×C22⋊D5311yescharacteristic
306#41—416no—
307#41—48no—
308#41—48no—
309C2×Q8—24no—
310C2×Q8—44no—
311C2×Q8—1062no—
312C2×Q8—4241no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4C2×C22⋊D5314yes—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—48no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—1062no—
335C22×C4—1061no—
336C22×C4—1061no—
337C22×C4#1171992314yescomplemented
338C22×C4C2×C22⋊D5312yes—
339C22×C4C2×C22⋊D5312yes—
340C22×C4C2×C22⋊D5311yescharacteristic
341C22×C4C2×C22⋊D5312yes—
342C22×C4C2×C22⋊D5312yes—
343C22×C4C23×D5311yescharacteristic
344SD16—4244no—
345C2×C8—1064no—
346C2×C8—4242no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—1062no—
352C4⋊C4—4242no—
353C4⋊C4—4241no—
354C4⋊C4—2122no—
355C4⋊C4C2×C22⋊D5314yes—
356C22⋊C4—28no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C22⋊C4—2122no—
362C23—24no—
363C23—41no—
364C23—41no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23#1313318611yescharacteristic
369C23#1313318611yescharacteristic
370C23C2×#1171992312yes—
371C23#1313324312yes—
372C23C22×C22⋊D5311yescharacteristic
373Q8—48no—
374Q8—2124no—
375D4—44no—
376D4—48no—
377D4—48no—
378D4—44no—
379D4—44no—
380D4—82no—
381D4—84no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—416no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—81no—
397C2×C4—2122no—
398C2×C4—2122no—
399C2×C4—2122no—
400C2×C4—2122no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4—2122no—
404C2×C4#1313318612yes—
405C2×C4C2×#1171992314yes—
406C2×C4C22×C22⋊D5312yes—
407C8—2124no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—88no—
420C22#1681077412yes—
421C22#1681077412yes—
422C22C2×#1313318611yescharacteristic
423C22C22×#1171992311yescharacteristic
424C22C2×#1313324311yescharacteristic
425C4—22no—
426C4—22no—
427C4—24no—
428C4—42no—
429C4—44no—
430C4—4242no—
431C4—4241no—
432C4—4241no—
433C2—21no—
434C2—22no—
435C2—22no—
436C2—21no—
437C2—84no—
438C2C2×#1681077411yescharacteristic
439C2#2028407911yescharacteristic
440C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#537C2
socleC22×C53C2×#120
hypercenterC2×#537D53
hypocenterC53#18293
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #20285472C2semidirect
2. #20280528C2semidirect
3. #20273752C2semidirect
4. #20273684C2semidirect
5. #20273283C2semidirect
6. C2×#16810349C2semidirect
10. #16810939C22semidirect
13. #16810761C22semidirect
15. #16810562C22semidirect
17. #16810543C22semidirect
21. #16810357C22semidirect
22. #16810349C22semidirect
111. C22×C4×C53SD16semidirect
252. C53#18293semidirect
337. C22×C4#11719923semidirect

Maximal subgroups

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

subgroupindex
#202854722
#202805282
#202737522
#202736842
#202732832
C2×#168103492
C2×C53×#5372
#1829353