Finite Group Databasebeta

← #27367407

Subgroups — #27367407

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286027C211yesmaximal, characteristic
2#20285040C211yesmaximal, characteristic
3#20283650C211yesmaximal, characteristic, complemented
4#20274299C212yesmaximal, complemented
5#20274285C212yesmaximal, complemented
6#20274147C212yesmaximal, complemented
7#20274133C212yesmaximal, complemented
8C2×#16810373C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10#20273390C211yesmaximal, characteristic
11C53×#3249C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16810980—21no—
14#16810980—21no—
15#16810975C2211yescharacteristic
16#16810975C2211yescharacteristic
17#16810971C2211yescharacteristic
18#16810971C2211yescharacteristic
19#16810910—21no—
20#16810908C2212yes—
21#16810854—22no—
22#16810825—21no—
23#16810823C2212yes—
24#16810754—21no—
25#16810754C2212yes—
26Q8×#88322—21no—
27Q8×#88322C2212yes—
28#16810650—21no—
29#16810647C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31#16810617—21no—
32#16810617—21no—
33#16810579—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810373C2212yes—
36#16810373C2212yes—
37#16810372C2212yes—
38#16810369C2212yes—
39#16810368C2212yes—
40#16810366C2212yes—
41C53×#533C2211yescharacteristic
42C53×#533C2211yescharacteristic
43C53×#477—21no—
44C53×#477C2212yes—
45C53×#467—22no—
46C53×#462—21no—
47C53×#462C2212yes—
48C2×C4⋊C8×C53C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719924—22no—
51C2×#11719917—21no—
52C22×#88322—21no—
53C22×#88322D411yescharacteristic
54C22×#88322D411yescharacteristic
55#13133217C2311yescharacteristic
56#13133214C2311yescharacteristic
57#13133205—21no—
58#13133191—21no—
59#13133191—21no—
60C4×D212—21no—
61C4×D212—21no—
62#13133148—22no—
63C4⋊#88322—21no—
64C4⋊#88322—21no—
65C4⋊#88322—22no—
66C4⋊#88322—22no—
67C4⋊#88322C2312yes—
68C4⋊#88322C2312yes—
69#13133141—22no—
70#13133140—21no—
71#13133140—21no—
72#13133140—21no—
73#13133140—21no—
74#13133137—22no—
75#13133137—21no—
76#13133137—21no—
77#13133137—21no—
78#13133137—21no—
79#13133128—22no—
80C4×#88322—22no—
81C4×#88322D411yescharacteristic
82C4×#88322D411yescharacteristic
83C4×#88322C2311yescharacteristic
84C4×#88322C2311yescharacteristic
85C2×Q16×C53—22no—
86C22×C8×C53—21no—
87C4⋊Q8×C53D411yescharacteristic
88C4⋊Q8×C53D411yescharacteristic
89C4⋊Q8×C53C2311yescharacteristic
90C4⋊Q8×C53C2311yescharacteristic
91#125×C53—21no—
92#125×C53—21no—
93C22⋊Q8×C53—21no—
94C22⋊Q8×C53—21no—
95C22⋊Q8×C53—21no—
96C22⋊Q8×C53D412yes—
97C4×Q8×C53—21no—
98C4×Q8×C53—21no—
99C4×Q8×C53—21no—
100C4×Q8×C53C2312yes—
101#117×C53—21no—
102#117×C53—21no—
103C2×C42×C53C2311yescharacteristic
104#107×C53—22no—
105C4⋊C8×C53—21no—
106C4⋊C8×C53C2312yes—
107#103×C53—22no—
108#103×C53—22no—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53C2×D412yes—
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C2×C22⋊D53—41no—
117#11719945—42no—
118#11719945—42no—
119C2×D212—21no—
120C2×D212—21no—
121C2×D212—21no—
122C2×C4×D53—41no—
123C2×Dic106—21no—
124Q16×C53—82no—
125C2×C8×C53—22no—
126C2×C8×C53—81no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53C2×D412yes—
139C22⋊C4×C53—22no—
140C22⋊C4×C53—41no—
141C42×C53—22no—
142C42×C53—41no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53#4111yescharacteristic
146C42×C53C2411yescharacteristic
147#11719924—82no—
148#11719921—41no—
149#11719921—41no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152#11719917—42no—
153C2×#88322—22no—
154C2×#88322—22no—
155C2×#88322—22no—
156C2×#88322—22no—
157C2×#88322—81no—
158C2×#88322C2×D411yescharacteristic
159C2×#88322C2×D411yescharacteristic
160C2×#88322C2×D411yescharacteristic
161C2×#88322C2×D411yescharacteristic
162C23×C53#12111yescharacteristic
163C22×D53—41no—
164Q8×C53—22no—
165Q8×C53—22no—
166Q8×C53—42no—
167Q8×C53—41no—
168Q8×C53—41no—
169Q8×C53—81no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—22no—
173C2×C4×C53—22no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53#13512yes—
181C2×C4×C53#13712yes—
182C2×C4×C53C22×D411yescharacteristic
183C2×C4×C53C22×D411yescharacteristic
184C2×C4×C53C2×#4111yescharacteristic
185C22⋊D53—82no—
186C2×Dic53—41no—
187D212—41no—
188D212—41no—
189D212—41no—
190D212—41no—
191D212—81no—
192C4×D53—82no—
193Dic106—41no—
194Dic106—41no—
195C8×C53—42no—
196#88322—21no—
197#88322—21no—
198#88322—21no—
199#88322—21no—
200#88322—42no—
201#88322—42no—
202#19216—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53#46412yes—
206C22×C53C2×#12111yescharacteristic
207C2×D53—81no—
208C2×D53—81no—
209C2×D53—81no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—42no—
214C4×C53—42no—
215C4×C53—81no—
216C4×C53—81no—
217C4×C53#46412yes—
218C4×C53C2×#13511yescharacteristic
219C4×C53C2×#13711yescharacteristic
220C4×C53#57711yescharacteristic
221C4×C53#57711yescharacteristic
222Dic53—81no—
223#3249D5311yescharacteristic, complemented
224#3246—531no—
225#3131—531no—
226#2960—531no—
227#1704—532no—
228#1702—532no—
229#1680—532no—
230#1678—532no—
231C2×#334—531no—
232#1550—531no—
233#1547—531no—
234C2×C53—22no—
235C2×C53C2×#46411yescharacteristic
236C2×C53#306011yescharacteristic
237C2×C53#306711yescharacteristic
238D53—161no—
239#533C2×D5311yescharacteristic
240#533C2×D5311yescharacteristic
241#532—531no—
242#488—1061no—
243#488—1061no—
244#487—531no—
245#487—531no—
246#486—531no—
247#486—531no—
248#477—21no—
249#477C2×D5312yes—
250#476—1061no—
251#475—532no—
252#467—22no—
253#465—1062no—
254#462—21no—
255#462C2×D5312yes—
256#461—1061no—
257#460—532no—
258#446—532no—
259#446—1061no—
260C8×Q8—532no—
261C8×Q8—1061no—
262#424—1061no—
263#423—531no—
264C2×C4⋊C8—531no—
265C2×C4⋊C8C2×D5311yescharacteristic
266#416—1061no—
267#416—1061no—
268#405—1061no—
269C2×C4×C8—531no—
270#334—532no—
271#334—532no—
272#333—532no—
273#332—532no—
274#332—532no—
275#331—532no—
276C53#1921611yescharacteristic, complemented, Sylow
277C2×#41—1061no—
278C2×Q16—22no—
279C2×SD16—1062no—
280C2×M16—1061no—
281C22×C8—21no—
282C22×C8—531no—
283C22×C8—531no—
284C22×C8—1061no—
285C4⋊Q8—531no—
286C4⋊Q8C22⋊D5311yescharacteristic
287C4⋊Q8C22⋊D5311yescharacteristic
288C4⋊Q8C22×D5311yescharacteristic
289C4⋊Q8C22×D5311yescharacteristic
290#127—531no—
291#125—21no—
292#125—21no—
293#124—1061no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297C22⋊Q8C22⋊D5312yes—
298#121—1061no—
299#121—1061no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8C22×D5312yes—
304C4×D4—1061no—
305C4×D4—1061no—
306#117—21no—
307#117—21no—
308C2×C42C22×D5311yescharacteristic
309#107—22no—
310#106—1062no—
311C4⋊C8—21no—
312C4⋊C8—532no—
313C4⋊C8—532no—
314C4⋊C8—1062no—
315C4⋊C8—1062no—
316C4⋊C8—1061no—
317C4⋊C8—1061no—
318C4⋊C8C22×D5312yes—
319#103—22no—
320#103—22no—
321#103—1061no—
322#103—1061no—
323#103—1061no—
324#103—1061no—
325#103—1062no—
326#102—1061no—
327#102—1061no—
328#102—1061no—
329#102—1061no—
330#102—1062no—
331#97—1062no—
332C4×C8—531no—
333C4×C8—531no—
334C4×C8—531no—
335C4×C8—531no—
336C4×C8—1062no—
337#41—2122no—
338#41—2122no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—1061no—
343C2×Q8C2×C22⋊D5312yes—
344C2×D4—1061no—
345C2×D4—1061no—
346C2×D4—1061no—
347C2×D4—2121no—
348C22×C4—2121no—
349C22×C4D4×D5311yescharacteristic
350C22×C4#1171994711yescharacteristic
351C22×C4C2×C22⋊D5311yescharacteristic
352Q16—82no—
353SD16—4242no—
354M16—2122no—
355C2×C8—22no—
356C2×C8—81no—
357C2×C8—531no—
358C2×C8—531no—
359C2×C8—531no—
360C2×C8—531no—
361C2×C8—1062no—
362C2×C8—1062no—
363C2×C8—1062no—
364C2×C8—1062no—
365C2×C8—4241no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—1061no—
378C4⋊C4—1061no—
379C4⋊C4C2×C22⋊D5312yes—
380C22⋊C4—22no—
381C22⋊C4—41no—
382C22⋊C4—2121no—
383C22⋊C4—2121no—
384C42—22no—
385C42—41no—
386C42D4×D5311yescharacteristic
387C42#1171994711yescharacteristic
388C42C2×C22⋊D5311yescharacteristic
389C42C23×D5311yescharacteristic
390C23—2121no—
391C23#1313318911yescharacteristic
392Q8—22no—
393Q8—22no—
394Q8—42no—
395Q8—41no—
396Q8—41no—
397Q8—81no—
398Q8—2121no—
399Q8—2121no—
400D4—4241no—
401D4—4242no—
402D4—2121no—
403D4—2121no—
404D4—2121no—
405D4—2121no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—4242no—
414C2×C4—2121no—
415C2×C4#1313318911yescharacteristic
416C2×C4#1313318911yescharacteristic
417C2×C4#1313318911yescharacteristic
418C2×C4#1313324012yes—
419C2×C4#1313324912yes—
420C2×C4C2×D4×D5311yescharacteristic
421C2×C4C22×C22⋊D5311yescharacteristic
422C2×C4C2×#1171994711yescharacteristic
423C8—42no—
424C8—1061no—
425C8—1061no—
426C8—1061no—
427C8—1061no—
428C8—2122no—
429C8—2122no—
430C22—21no—
431C22—21no—
432C22—4241no—
433C22—4241no—
434C22—4241no—
435C22#1681084712yes—
436C22C2×#1313318911yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—42no—
441C4—42no—
442C4—81no—
443C4—81no—
444C4—4241no—
445C4#1681084712yes—
446C4C2×#1313324011yescharacteristic
447C4C2×#1313324911yescharacteristic
448C4#1681143111yescharacteristic
449C4#1681143411yescharacteristic
450C2—22no—
451C2—8481no—
452C2C2×#1681084711yescharacteristic
453C2#2028437411yescharacteristic
454C2#2028443211yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3249C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#3249D53
hypocenterC53#19216
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
3. #20283650C2semidirect
4. #20274299C2semidirect
5. #20274285C2semidirect
6. #20274147C2semidirect
7. #20274133C2semidirect
8. C2×#16810373C2semidirect
9. #20273405C2semidirect
11. C53×#3249C2semidirect
223. #3249D53semidirect
276. C53#19216semidirect

Maximal subgroups

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

subgroupindex
#202860272
#202850402
#202836502
#202742992
#202742852
#202741472
#202741332
C2×#168103732
#202734052
#202733902
C53×#32492
#1921653