Finite Group Databasebeta

← #27396395

Subgroups — #27396395

Order 13568.

648 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284503C211yesmaximal, characteristic
2#20283645C211yesmaximal, characteristic, complemented
3#20283359C211yesmaximal, characteristic
4#20275658C211yesmaximal, characteristic, complemented
5#20275488C211yesmaximal, characteristic, complemented
6#20275462C212yesmaximal, complemented
7#20275416C212yesmaximal
8#20275366C212yesmaximal
9#20275317C212yesmaximal, complemented
10C4⋊C4×#88322C211yesmaximal, characteristic, complemented
11C53×#1806C211yesmaximal, characteristic
12#16811180C2211yescharacteristic
13#16810930C412yescomplemented
14#16810923C412yescomplemented
15#16810918C411yescharacteristic, complemented
16#16810913C411yescharacteristic, complemented
17#16810908C411yescharacteristic, complemented
18#16810903C411yescharacteristic, complemented
19#16810750—21no—
20#16810750—21no—
21#16810747—21no—
22#16810747—21no—
23#16810684—21no—
24#16810684—21no—
25#16810676—21no—
26#16810676—21no—
27C2×#13133152C2212yes—
28#16810659C2212yes—
29#16810651C2212yes—
30#16810650C2211yescharacteristic
31#16810650C2211yescharacteristic
32C2×C4⋊#88322C2211yescharacteristic
33#16810619C2212yes—
34#16810590C2211yescharacteristic
35#16810590C2211yescharacteristic
36#16810581C2212yes—
37C4×#11719917C2211yescharacteristic
38C2×C4×#88322C2212yes—
39C2×C4×#88322C2211yescharacteristic
40#16810387C2212yes—
41#16810383C2212yes—
42#16810383C2212yes—
43#16810378C2212yes—
44C4×#41×C53C2211yescharacteristic
45C53×#423C2211yescharacteristic
46C2×#104×C53C2212yes—
47C4×C4⋊C4×C53C2211yescharacteristic
48C53×#339C2212yes—
49C2×#11719917C2311yescharacteristic
50C2×#11719917C2311yescharacteristic
51C22×#88322C2312yes—
52C22×#88322C2311yescharacteristic
53C22×#88322C2311yescharacteristic
54#13133217C2×C411yescharacteristic
55#13133217C2×C411yescharacteristic
56#13133209C2×C412yes—
57#13133196—21no—
58#13133196—21no—
59C4×Dic106—21no—
60C4×Dic106—21no—
61#13133176C2312yes—
62#13133152—42no—
63#13133150C2×C412yes—
64#13133150C2×C412yes—
65#13133148C2×C412yes—
66#13133148C2×C412yes—
67C4⋊#88322—21no—
68C4⋊#88322—21no—
69C4⋊#88322—21no—
70C4⋊#88322—21no—
71C4⋊#88322C2×C411yescharacteristic
72C4⋊#88322C2×C411yescharacteristic
73C4⋊#88322C2×C412yes—
74#13133146—22no—
75#13133146—22no—
76#13133141C2×C412yes—
77#13133141C2×C412yes—
78#13133138C2×C412yes—
79#13133138C2×C412yes—
80C22⋊#88322—21no—
81C22⋊#88322—21no—
82C22⋊#88322—21no—
83C22⋊#88322—21no—
84#13133128—21no—
85#13133127—22no—
86C4×#88322—21no—
87C4×#88322—42no—
88C4×#88322D411yescharacteristic
89C4×#88322D411yescharacteristic
90C4×#88322D411yescharacteristic
91C4×#88322D411yescharacteristic
92C8×Dic53—22no—
93C2×#41×C53C2311yescharacteristic
94C2×M16×C53C2312yes—
95C4×Q8×C53C2×C411yescharacteristic
96C4×Q8×C53C2×C411yescharacteristic
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53C2×C411yescharacteristic
100C4×D4×C53C2×C411yescharacteristic
101#117×C53—21no—
102#117×C53C2311yescharacteristic
103C2×C4⋊C4×C53—21no—
104C2×C42×C53—21no—
105C2×C42×C53C2312yes—
106C2×C42×C53C2311yescharacteristic
107C4⋊C8×C53—21no—
108C4⋊C8×C53—21no—
109#104×C53—42no—
110#95×C53—21no—
111#41×C53—21no—
112#41×C53—21no—
113#41×C53—21no—
114#41×C53—21no—
115C2×Q8×C53C22×C411yescharacteristic
116C2×D4×C53—21no—
117C2×D4×C53C22×C411yescharacteristic
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53#4111yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C22×C4×C53C2411yescharacteristic
125C2×Dic106—21no—
126C2×Dic106—21no—
127M16×C53—22no—
128M16×C53—22no—
129C2×C8×C53—22no—
130C4⋊C4×C53—24no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53C22×C411yescharacteristic
134C4⋊C4×C53C22×C411yescharacteristic
135C22⋊C4×C53—21no—
136C22⋊C4×C53—21no—
137C22⋊C4×C53—21no—
138C42×C53—21no—
139C42×C53—21no—
140C42×C53—21no—
141C42×C53—22no—
142C42×C53—22no—
143C42×C53C22×C411yescharacteristic
144C42×C53C22×C411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53C2×D411yescharacteristic
147C22⋊Dic53—22no—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53C22×C412yes—
151C4⋊Dic53C22×C412yes—
152#11719919—22no—
153#11719919—22no—
154C4×Dic53—22no—
155#11719917—21no—
156#11719917—21no—
157#11719917—21no—
158#11719917—21no—
159C2×#88322—21no—
160C2×#88322—22no—
161C2×#88322—22no—
162C2×#88322—21no—
163C2×#88322—21no—
164C2×#88322—21no—
165C2×#88322—21no—
166C2×#88322—21no—
167C2×#88322C22×C412yes—
168C2×#88322C22×C412yes—
169C2×#88322C2×D411yescharacteristic
170C2×#88322C2×D411yescharacteristic
171C2×#88322C2×D411yescharacteristic
172C2×#88322C2×D411yescharacteristic
173C23×C53—21no—
174C23×C53C2×#4111yescharacteristic
175Q8×C53—21no—
176Q8×C53—21no—
177D4×C53—21no—
178D4×C53—21no—
179D4×C53—41no—
180D4×C53—41no—
181C2×C4×C53—22no—
182C2×C4×C53—24no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—21no—
186C2×C4×C53—41no—
187C2×C4×C53—42no—
188C2×C4×C53—42no—
189C2×C4×C53—42no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—42no—
193C2×C4×C53C4×D411yescharacteristic
194C2×C4×C53C4×D411yescharacteristic
195C2×C4×C53C4×D411yescharacteristic
196C2×C4×C53C4×D411yescharacteristic
197C2×C4×C53C23×C411yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53H(5,𝔽2)11yescharacteristic
200C2×C4×C532−1+411yescharacteristic
201C2×C4×C532−1+411yescharacteristic
202C2×C4×C532−1+411yescharacteristic
203C2×Dic53—22no—
204C2×Dic53—22no—
205Dic106—42no—
206Dic106—42no—
207C8×C53—42no—
208#88322—22no—
209#88322—22no—
210#88322—21no—
211#88322—21no—
212#88322—21no—
213#88322—21no—
214#88322—41no—
215#88322—41no—
216#22130—531nomaximal, Sylow
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53C2×C4×D411yescharacteristic
223C22×C53#54111yescharacteristic
224C22×C53#54211yescharacteristic
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—44no—
229C4×C53—41no—
230C4×C53—41no—
231C4×C53—41no—
232C4×C53#44312yes—
233C4×C53#51911yescharacteristic
234C4×C53#52011yescharacteristic
235C4×C53#53611yescharacteristic
236C4×C53#53711yescharacteristic
237Dic53—42no—
238Dic53—42no—
239#3076—531no—
240#2960—531no—
241#2922—531no—
242#1986—531no—
243#1950—531no—
244#1945—532no—
245#1937—532no—
246#1929—532no—
247#1922—532no—
248#1806D5311yescharacteristic
249C8×C4⋊C4—531no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53—81no—
253C2×C53#232311yescharacteristic
254C2×C53C2×#44311yescharacteristic
255C2×C53#295511yescharacteristic
256#533—531no—
257C4×#41C2×D5311yescharacteristic
258#479—532no—
259#478—532no—
260#477—531no—
261#476—531no—
262#475—531no—
263#474—531no—
264#446—1061no—
265#446—1061no—
266C8×Q8—1061no—
267C8×Q8—1061no—
268#433—1061no—
269#433—1061no—
270#431—1061no—
271#431—1061no—
272C2×#108—532no—
273#427—532no—
274#424—532no—
275#424—531no—
276#424—531no—
277#423C2×D5311yescharacteristic
278C2×C4⋊C8—531no—
279C2×#104C2×D5312yes—
280#416—532no—
281#408—531no—
282#408—531no—
283#405—532no—
284C4×M16—531no—
285C2×C4×C8—531no—
286C2×C4×C8—532no—
287C4×C4⋊C4C2×D5311yescharacteristic
288#341—532no—
289#339—532no—
290#339—532no—
291#339C2×D5312yes—
292#336—532no—
293C53#2213011yescharacteristic, complemented, Sylow
294C2×#41C22×D5311yescharacteristic
295C2×M16—531no—
296C2×M16—531no—
297C2×M16C22×D5312yes—
298C22×C8—531no—
299C22×C8—532no—
300C22×C8—531no—
301C4⋊Q8—531no—
302C4⋊Q8—531no—
303#125—532no—
304C22⋊Q8—1061no—
305C22⋊Q8—1061no—
306C4×Q8—1061no—
307C4×Q8—1061no—
308C4×Q8C4×D5311yescharacteristic
309C4×Q8C4×D5311yescharacteristic
310C4×D4—21no—
311C4×D4—21no—
312C4×D4C4×D5311yescharacteristic
313C4×D4C4×D5311yescharacteristic
314#117—21no—
315#117—532no—
316#117C22×D5311yescharacteristic
317C2×C4⋊C4—21no—
318C2×C42—21no—
319C2×C42C22×D5312yes—
320C2×C42C22×D5311yescharacteristic
321#108—2122no—
322#107—532no—
323#107—532no—
324#106—532no—
325#106—532no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—531no—
329C4⋊C8—531no—
330C4⋊C8—532no—
331C4⋊C8—1061no—
332C4⋊C8—1061no—
333C4⋊C8—1061no—
334C4⋊C8—1061no—
335C4⋊C8—1062no—
336C4⋊C8—1062no—
337#104—42no—
338#103—532no—
339#103—532no—
340#102—532no—
341#102—532no—
342C22⋊C8—1061no—
343C22⋊C8—1061no—
344C22⋊C8—1061no—
345C22⋊C8—1061no—
346#97—1062no—
347#97—1061no—
348C4×C8—531no—
349C4×C8—531no—
350C4×C8—531no—
351C4×C8—531no—
352C4×C8—1062no—
353C4×C8—1061no—
354C4×C8—2122no—
355#95—21no—
356#41—21no—
357#41—21no—
358#41—21no—
359#41—21no—
360C2×Q8—1061no—
361C2×Q8—1061no—
362C2×Q8C2×C4×D5311yescharacteristic
363C2×D4—21no—
364C2×D4C2×C4×D5311yescharacteristic
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4#1171994711yescharacteristic
370C22×C4#1171994711yescharacteristic
371C22×C4C23×D5311yescharacteristic
372M16—22no—
373M16—22no—
374M16—1061no—
375M16—1061no—
376M16—1061no—
377M16—1061no—
378C2×C8—22no—
379C2×C8—532no—
380C2×C8—532no—
381C2×C8—531no—
382C2×C8—531no—
383C2×C8—531no—
384C2×C8—531no—
385C2×C8—1061no—
386C2×C8—1062no—
387C2×C8—1061no—
388C2×C8—1061no—
389C2×C8—1062no—
390C2×C8—1061no—
391C2×C8—1061no—
392C2×C8—1061no—
393C4⋊C4—24no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—532no—
397C4⋊C4—532no—
398C4⋊C4—1062no—
399C4⋊C4—1062no—
400C4⋊C4—1061no—
401C4⋊C4—1061no—
402C4⋊C4C2×C4×D5311yescharacteristic
403C4⋊C4C2×C4×D5311yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—1062no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—22no—
412C42—22no—
413C42—1062no—
414C42C2×C4×D5311yescharacteristic
415C42C2×C4×D5311yescharacteristic
416C42D4×D5311yescharacteristic
417C42D4×D5311yescharacteristic
418C23—21no—
419C23C2×#1171994711yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—2122no—
423Q8—2122no—
424D4—21no—
425D4—21no—
426D4—41no—
427D4—41no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—1062no—
441C2×C4—1062no—
442C2×C4#1313317711yescharacteristic
443C2×C4#1313317711yescharacteristic
444C2×C4#1313317711yescharacteristic
445C2×C4#1313317711yescharacteristic
446C2×C4C22×C4×D5311yescharacteristic
447C2×C4C2×D4×D5311yescharacteristic
448C2×C4#1313326211yescharacteristic
449C2×C4#1313326411yescharacteristic
450C2×C4#1313326411yescharacteristic
451C2×C4#1313326511yescharacteristic
452C8—42no—
453C8—1062no—
454C8—1062no—
455C8—1061no—
456C8—1061no—
457C8—1061no—
458C8—1061no—
459C8—2121no—
460C8—2121no—
461C22—21no—
462C22—21no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22C2×#1313317711yescharacteristic
467C22#1681123311yescharacteristic
468C22#1681124011yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—44no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—2122no—
477C4—2122no—
478C4#1681073712yes—
479C4#1681110011yescharacteristic
480C4#1681110411yescharacteristic
481C4#1681119611yescharacteristic
482C4#1681120211yescharacteristic
483C2—21no—
484C2—21no—
485C2—81no—
486C2#2027739311yescharacteristic
487C2C2×#1681073711yescharacteristic
488C2#2028361411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1806C2
socleC22×C53C2×C4×D4
prime radicalC2×#41×C53C23
cosocleC22×C4×C53C24
hypercenter#1806D53
hypocenterC53#22130
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #20283645C2semidirect
4. #20275658C2semidirect
5. #20275488C2semidirect
6. #20275462C2semidirect
9. #20275317C2semidirect
10. C4⋊C4×#88322C2semidirect
13. #16810930C4semidirect
14. #16810923C4semidirect
15. #16810918C4semidirect
16. #16810913C4semidirect
17. #16810908C4semidirect
18. #16810903C4semidirect
293. C53#22130semidirect

Maximal subgroups

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

subgroupindex
#202845032
#202836452
#202833592
#202756582
#202754882
#202754622
#202754162
#202753662
#202753172
C4⋊C4×#883222
C53×#18062
#2213053