Finite Group Databasebeta

← #27367016

Subgroups — #27367016

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285111C211yesmaximal, characteristic
2#20284746C211yesmaximal, characteristic
3Q8×#11719917C211yesmaximal, characteristic, complemented
4#20274299C212yesmaximal, complemented
5#20274257C212yesmaximal, complemented
6#20274175C212yesmaximal, complemented
7#20274133C212yesmaximal, complemented
8#20273465C211yesmaximal, characteristic, complemented
9#20273396C211yesmaximal, characteristic, complemented
10#20273390C211yesmaximal, characteristic
11C2×C53×#477C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16810985—22no—
14#16810975C2212yes—
15#16810971C2212yes—
16#16810941—21no—
17#16810908C2211yescharacteristic
18#16810908C2211yescharacteristic
19#16810823C2211yescharacteristic
20#16810823C2211yescharacteristic
21#16810762—21no—
22#16810754—22no—
23#16810754C2212yes—
24Q8×#88322C2212yes—
25C4×#11719924—21no—
26C4×#11719924—21no—
27#16810676C2211yescharacteristic
28#16810647—21no—
29#16810647C2211yescharacteristic
30#16810622—21no—
31#16810622—21no—
32C4×#11719917—21no—
33C4×#11719917C2211yescharacteristic
34#16810375C2212yes—
35#16810373C2212yes—
36#16810372C2212yes—
37#16810372C2212yes—
38#16810368C2212yes—
39#16810366C2212yes—
40C2×C4⋊Q8×C53C2211yescharacteristic
41C2×C4×Q8×C53C2211yescharacteristic
42C53×#477—21no—
43C53×#477—21no—
44C53×#477C2212yes—
45C53×#477C2212yes—
46C2×#107×C53—21no—
47C2×C4⋊C8×C53C2211yescharacteristic
48C2×#103×C53—21no—
49C2×#11719945—21no—
50C2×#11719924—21no—
51C2×#11719924—21no—
52C2×#11719917—21no—
53C2×#11719917D411yescharacteristic
54C2×#11719917D411yescharacteristic
55#13133217C2311yescharacteristic
56#13133214C2311yescharacteristic
57#13133205—21no—
58#13133191—21no—
59#13133191—21no—
60C4×D212—21no—
61C4×D212—21no—
62#13133148—21no—
63#13133148—21no—
64C4⋊#88322—21no—
65C4⋊#88322—22no—
66C4⋊#88322—22no—
67C4⋊#88322C2312yes—
68C4⋊#88322C2311yescharacteristic
69C4⋊#88322C2311yescharacteristic
70#13133141—21no—
71#13133141—21no—
72#13133140—22no—
73#13133140—22no—
74#13133137—21no—
75#13133137—21no—
76#13133137—22no—
77#13133137—22no—
78C22⋊#88322—21no—
79#13133128—22no—
80#13133128D412yes—
81C4×#88322—21no—
82C4×#88322—22no—
83C4×#88322C2312yes—
84C22×Q8×C53—21no—
85C22×Q8×C53D411yescharacteristic
86C22×C8×C53—21no—
87C4⋊Q8×C53—21no—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53—21no—
90C4⋊Q8×C53C2312yes—
91C4×Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53D411yescharacteristic
94C4×Q8×C53D411yescharacteristic
95C4×Q8×C53C2311yescharacteristic
96C4×Q8×C53C2311yescharacteristic
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C42×C53—21no—
102C2×C42×C53C2311yescharacteristic
103#107×C53—22no—
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—22no—
111C2×Q8×C53—22no—
112C2×Q8×C53—41no—
113C2×Q8×C53—41no—
114C2×Q8×C53—82no—
115C2×Q8×C53C2×D411yescharacteristic
116C2×Q8×C53C2×D411yescharacteristic
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—41no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C2×C22⋊D53—41no—
124#11719945—42no—
125#11719945—42no—
126C2×D212—21no—
127C2×D212—21no—
128C2×D212—21no—
129C2×C4×D53—41no—
130C2×Dic106—21no—
131C2×C8×C53—22no—
132C2×C8×C53—42no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—41no—
141C4⋊C4×C53—41no—
142C4⋊C4×C53C2×D411yescharacteristic
143C4⋊C4×C53C2×D411yescharacteristic
144C42×C53—21no—
145C42×C53—21no—
146C42×C53—21no—
147C42×C53—21no—
148C42×C53C2×D411yescharacteristic
149C42×C53C2×D411yescharacteristic
150C42×C53#4111yescharacteristic
151C42×C53C2411yescharacteristic
152#11719924—42no—
153#11719924—42no—
154#11719921—41no—
155#11719921—41no—
156C4⋊Dic53—21no—
157C4⋊Dic53—21no—
158#11719917—22no—
159#11719917—22no—
160#11719917—42no—
161C2×#88322—22no—
162C2×#88322—21no—
163C2×#88322—21no—
164C2×#88322C2×D412yes—
165C2×#88322C2×D412yes—
166C23×C53#12111yescharacteristic
167C22×D53—41no—
168Q8×C53—21no—
169Q8×C53—21no—
170Q8×C53—21no—
171Q8×C53—42no—
172Q8×C53—82no—
173Q8×C53—82no—
174Q8×C53#13712yes—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—41no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—81no—
185C2×C4×C53—82no—
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#13512yes—
190C2×C4×C53#13712yes—
191C2×C4×C53C22×D411yescharacteristic
192C2×C4×C53C22×D411yescharacteristic
193C2×C4×C53C2×#4111yescharacteristic
194C22⋊D53—82no—
195C2×Dic53—41no—
196D212—41no—
197D212—41no—
198D212—41no—
199D212—41no—
200D212—81no—
201C4×D53—82no—
202Dic106—41no—
203Dic106—41no—
204C8×C53—42no—
205#88322—22no—
206#88322—22no—
207#88322—42no—
208#88322—41no—
209#88322—41no—
210#19176—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53#46412yes—
214C22×C53C2×#12111yescharacteristic
215C2×D53—81no—
216C2×D53—81no—
217C2×D53—81no—
218C4×C53—21no—
219C4×C53—22no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53—41no—
223C4×C53—41no—
224C4×C53—41no—
225C4×C53—82no—
226C4×C53#46412yes—
227C4×C53C2×#13511yescharacteristic
228C4×C53C2×#13711yescharacteristic
229C4×C53C2×#13711yescharacteristic
230C4×C53#57511yescharacteristic
231Dic53—81no—
232#3141—531no—
233#3103—531no—
234C2×#477D5311yescharacteristic, complemented
235Q8×M16—531no—
236#1704—532no—
237#1698—532no—
238#1684—532no—
239#1678—532no—
240#1565—531no—
241#1548—531no—
242#1547—531no—
243C2×C53—22no—
244C2×C53C2×#46411yescharacteristic
245C2×C53#305611yescharacteristic
246C2×C53#306811yescharacteristic
247D53—161no—
248#532—531no—
249C2×C4⋊Q8C2×D5311yescharacteristic
250C2×C4×Q8C2×D5311yescharacteristic
251#489—1062no—
252#487—532no—
253#486—532no—
254#481—1061no—
255#477—21no—
256#477—21no—
257#477C2×D5312yes—
258#477C2×D5312yes—
259#475—531no—
260#475—531no—
261#460—531no—
262#460—531no—
263#448—1061no—
264#446—532no—
265#446—1062no—
266C8×Q8—532no—
267C4×SD16—1061no—
268C4×SD16—1061no—
269#431—531no—
270C2×#107—21no—
271#423—531no—
272#423—1061no—
273C2×C4⋊C8C2×D5311yescharacteristic
274#417—1061no—
275#417—1061no—
276C2×#103—21no—
277C4×M16—531no—
278C4×M16—1061no—
279#335—532no—
280#334—532no—
281#333—532no—
282#333—532no—
283#332—532no—
284#331—532no—
285C53#1917611yescharacteristic, complemented, Sylow
286C2×#41—1061no—
287C22×Q8—21no—
288C22×Q8C22⋊D5311yescharacteristic
289C2×SD16—1061no—
290C2×SD16—1061no—
291C2×M16—531no—
292C2×M16—531no—
293C2×M16—1061no—
294C22×C8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—531no—
299C4⋊Q8C22×D5312yes—
300#127—531no—
301#124—1061no—
302#121—1061no—
303#121—1061no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8C22⋊D5311yescharacteristic
307C4×Q8C22⋊D5311yescharacteristic
308C4×Q8C22×D5311yescharacteristic
309C4×Q8C22×D5311yescharacteristic
310C4×D4—1061no—
311C4×D4—1061no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4C22⋊D5311yescharacteristic
316C2×C42—21no—
317C2×C42C22×D5311yescharacteristic
318#107—22no—
319#107—22no—
320#106—1061no—
321#106—1061no—
322C4⋊C8—21no—
323C4⋊C8—532no—
324C4⋊C8—531no—
325C4⋊C8—531no—
326C4⋊C8—1062no—
327C4⋊C8—1062no—
328C4⋊C8—1061no—
329C4⋊C8C22×D5312yes—
330#103—22no—
331#103—22no—
332#103—1062no—
333#103—1062no—
334#103—1061no—
335#103—1061no—
336#102—1062no—
337#102—1062no—
338#102—1061no—
339#102—1061no—
340C22⋊C8—1061no—
341#97—532no—
342#97—1062no—
343C4×C8—532no—
344C4×C8—1062no—
345C4×C8—1061no—
346#41—2122no—
347#41—2122no—
348C2×Q8—21no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—82no—
354C2×Q8—1061no—
355C2×Q8C2×C22⋊D5311yescharacteristic
356C2×Q8C2×C22⋊D5311yescharacteristic
357C2×D4—1061no—
358C2×D4—1061no—
359C2×D4—1061no—
360C2×D4—2121no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—41no—
364C22×C4—2121no—
365C22×C4D4×D5311yescharacteristic
366C22×C4#1171994711yescharacteristic
367C22×C4C2×C22⋊D5311yescharacteristic
368SD16—2122no—
369SD16—2122no—
370M16—1062no—
371M16—1062no—
372M16—2122no—
373C2×C8—22no—
374C2×C8—42no—
375C2×C8—532no—
376C2×C8—532no—
377C2×C8—1062no—
378C2×C8—1061no—
379C2×C8—1061no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—1061no—
390C4⋊C4—1061no—
391C4⋊C4C2×C22⋊D5311yescharacteristic
392C4⋊C4C2×C22⋊D5311yescharacteristic
393C22⋊C4—2121no—
394C22⋊C4—2121no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42D4×D5311yescharacteristic
400C42#1171994711yescharacteristic
401C42C2×C22⋊D5311yescharacteristic
402C42C23×D5311yescharacteristic
403C23—2121no—
404C23#1313318911yescharacteristic
405Q8—21no—
406Q8—21no—
407Q8—21no—
408Q8—42no—
409Q8—82no—
410Q8—82no—
411Q8—2121no—
412Q8—2121no—
413Q8#1313324812yes—
414D4—4241no—
415D4—4242no—
416D4—2121no—
417D4—2121no—
418D4—2121no—
419D4—2121no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—4242no—
432C2×C4—2121no—
433C2×C4#1313318911yescharacteristic
434C2×C4#1313318911yescharacteristic
435C2×C4#1313318911yescharacteristic
436C2×C4#1313324012yes—
437C2×C4#1313324912yes—
438C2×C4C2×D4×D5311yescharacteristic
439C2×C4C22×C22⋊D5311yescharacteristic
440C2×C4C2×#1171994711yescharacteristic
441C8—42no—
442C8—1062no—
443C8—1062no—
444C8—2122no—
445C8—2121no—
446C8—2121no—
447C22—21no—
448C22—21no—
449C22—4241no—
450C22—4241no—
451C22—4241no—
452C22#1681084712yes—
453C22C2×#1313318911yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—22no—
458C4—41no—
459C4—41no—
460C4—41no—
461C4—82no—
462C4—4241no—
463C4#1681084712yes—
464C4C2×#1313324011yescharacteristic
465C4C2×#1313324811yescharacteristic
466C4C2×#1313324911yescharacteristic
467C4#1681142111yescharacteristic
468C2—22no—
469C2—8481no—
470C2C2×#1681084711yescharacteristic
471C2#2028434111yescharacteristic
472C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#477C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterC2×#477D53
hypocenterC53#19176
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. Q8×#11719917C2semidirect
4. #20274299C2semidirect
5. #20274257C2semidirect
6. #20274175C2semidirect
7. #20274133C2semidirect
8. #20273465C2semidirect
9. #20273396C2semidirect
11. C2×C53×#477C2semidirect
234. C2×#477D53semidirect
285. C53#19176semidirect

Maximal subgroups

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

subgroupindex
#202851112
#202847462
Q8×#117199172
#202742992
#202742572
#202741752
#202741332
#202734652
#202733962
#202733902
C2×C53×#4772
#1917653