Finite Group Databasebeta

← #27396402

Subgroups — #27396402

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#20284497C211yesmaximal, characteristic
2#20283650C211yesmaximal, characteristic, complemented
3#20283360C211yesmaximal, characteristic
4#20275490C211yesmaximal, characteristic, complemented
5#20275462C212yesmaximal, complemented
6#20275420C212yesmaximal
7#20275361C212yesmaximal
8#20275312C212yesmaximal, complemented
9#20274736C211yesmaximal, characteristic
10C4⋊C4×#88322C211yesmaximal, characteristic, complemented
11C53×#1986C211yesmaximal, characteristic, complemented
12C4×#11719945C2211yescharacteristic
13#16810927C412yescomplemented
14#16810920C412yescomplemented
15#16810915C411yescharacteristic, complemented
16#16810910C411yescharacteristic, complemented
17#16810905C411yescharacteristic, complemented
18#16810900C411yescharacteristic, complemented
19#16810754—21no—
20#16810754—21no—
21Q8×#88322—21no—
22Q8×#88322—21no—
23#16810685—21no—
24#16810685—21no—
25#16810674—21no—
26#16810674—21no—
27C2×#13133152C2212yes—
28#16810659C2212yes—
29#16810651C2211yescharacteristic
30#16810651C2211yescharacteristic
31#16810650C2212yes—
32#16810647C2211yescharacteristic
33C2×C4⋊#88322C2211yescharacteristic
34C2×#13133145C2212yes—
35#16810617C2212yes—
36#16810591C2211yescharacteristic
37#16810591C2211yescharacteristic
38#16810579C2212yes—
39C2×C4×#88322C2212yes—
40C2×C4×#88322C2211yescharacteristic
41#16810387C2212yes—
42#16810383C2212yes—
43#16810378C2212yes—
44C53×#533C2211yescharacteristic
45C4×M16×C53C2211yescharacteristic
46C4×C4⋊C4×C53C2211yescharacteristic
47C53×#339C2212yes—
48C53×#339C2212yes—
49C2×#11719945C2311yescharacteristic
50C2×#11719917C2312yes—
51C22×#88322C2312yes—
52C22×#88322C2311yescharacteristic
53C22×#88322C2311yescharacteristic
54C4×Dic106C2×C411yescharacteristic
55C4×Dic106C2×C411yescharacteristic
56C4×D212C2×C411yescharacteristic
57C4×D212C2×C411yescharacteristic
58C4×C22⋊D53—21no—
59C4×C22⋊D53—21no—
60#13133176C2311yescharacteristic
61#13133174—21no—
62C42×D53—21no—
63#13133152—42no—
64#13133150C2×C412yes—
65#13133150C2×C412yes—
66#13133148C2×C412yes—
67#13133148C2×C412yes—
68C4⋊#88322—21no—
69C4⋊#88322—21no—
70C4⋊#88322—22no—
71C4⋊#88322—22no—
72C4⋊#88322C2×C411yescharacteristic
73C4⋊#88322C2×C411yescharacteristic
74C4⋊#88322C2×C412yes—
75#13133146—21no—
76#13133146—21no—
77#13133146—21no—
78#13133146—21no—
79#13133145—42no—
80#13133140C2×C412yes—
81#13133140C2×C412yes—
82#13133137C2×C412yes—
83#13133137C2×C412yes—
84#13133129—21no—
85#13133129—21no—
86#13133129—21no—
87#13133129—21no—
88#13133128—22no—
89C4×#88322—22no—
90C4×#88322—42no—
91C4×#88322D411yescharacteristic
92C4×#88322D411yescharacteristic
93C4×#88322D411yescharacteristic
94C4×#88322D411yescharacteristic
95C2×M16×C53C2311yescharacteristic
96C2×M16×C53C2311yescharacteristic
97C4⋊Q8×C53C2×C411yescharacteristic
98C4⋊Q8×C53C2×C411yescharacteristic
99#125×C53C2×C412yes—
100C22⋊Q8×C53—21no—
101C22⋊Q8×C53—21no—
102C4×Q8×C53—21no—
103C4×Q8×C53—21no—
104#117×C53C2312yes—
105C2×C4⋊C4×C53—21no—
106C2×C42×C53C2312yes—
107C2×C42×C53C2311yescharacteristic
108#97×C53—21no—
109C4×C8×C53—21no—
110#95×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×C4×C53C2411yescharacteristic
117C2×C22⋊D53—21no—
118#11719945—21no—
119#11719945—21no—
120#11719945—21no—
121#11719945—21no—
122C2×D212C22×C411yescharacteristic
123C2×C4×D53—21no—
124C2×C4×D53—21no—
125C2×C4×D53—21no—
126C2×Dic106C22×C411yescharacteristic
127M16×C53—21no—
128M16×C53—21no—
129M16×C53—21no—
130M16×C53—21no—
131C2×C8×C53—21no—
132C2×C8×C53—21no—
133C4⋊C4×C53—24no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53C22×C412yes—
139C4⋊C4×C53C22×C412yes—
140C22⋊C4×C53—22no—
141C42×C53—22no—
142C42×C53—22no—
143C42×C53—22no—
144C42×C53C22×C411yescharacteristic
145C42×C53C22×C411yescharacteristic
146C42×C53C2×D411yescharacteristic
147C42×C53C2×D411yescharacteristic
148C22⋊Dic53—21no—
149#11719921—21no—
150#11719921—21no—
151C4⋊Dic53C22×C411yescharacteristic
152C4⋊Dic53C22×C411yescharacteristic
153#11719919—21no—
154#11719919—21no—
155C4×Dic53—21no—
156C4×Dic53—21no—
157C4×Dic53—21no—
158#11719917—22no—
159#11719917—22no—
160C2×#88322—21no—
161C2×#88322—22no—
162C2×#88322—22no—
163C2×#88322—21no—
164C2×#88322—21no—
165C2×#88322—21no—
166C2×#88322—22no—
167C2×#88322C22×C412yes—
168C2×#88322C22×C412yes—
169C2×#88322C2×D411yescharacteristic
170C2×#88322C2×D411yescharacteristic
171C2×#88322C2×D411yescharacteristic
172C2×#88322C2×D411yescharacteristic
173C23×C53C2×#4111yescharacteristic
174C22×D53—21no—
175Q8×C53—42no—
176Q8×C53—42no—
177C2×C4×C53—22no—
178C2×C4×C53—24no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—41no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53C4×D411yescharacteristic
185C2×C4×C53C4×D411yescharacteristic
186C2×C4×C53C4×D411yescharacteristic
187C2×C4×C53C4×D411yescharacteristic
188C2×C4×C53C23×C411yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53H(5,𝔽2)11yescharacteristic
191C2×C4×C532−1+411yescharacteristic
192C2×C4×C532−1+411yescharacteristic
193C2×C4×C532−1+411yescharacteristic
194C22⋊D53—41no—
195C22⋊D53—41no—
196C2×Dic53—21no—
197C2×Dic53—21no—
198C2×Dic53—21no—
199D212—21no—
200D212—21no—
201C4×D53—42no—
202C4×D53—41no—
203C4×D53—41no—
204C4×D53—42no—
205Dic106—21no—
206Dic106—21no—
207C8×C53—41no—
208C8×C53—41no—
209#88322—22no—
210#88322—22no—
211#88322—21no—
212#88322—21no—
213#88322—21no—
214#88322—21no—
215#88322—42no—
216#22130—531nomaximal, Sylow
217C22×C53—21no—
218C22×C53—21no—
219C22×C53C2×C4×D411yescharacteristic
220C22×C53#54111yescharacteristic
221C22×C53#54211yescharacteristic
222C2×D53—41no—
223C2×D53—41no—
224C2×D53—41no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—44no—
229C4×C53—42no—
230C4×C53—42no—
231C4×C53#44312yes—
232C4×C53#51911yescharacteristic
233C4×C53#52011yescharacteristic
234C4×C53#53611yescharacteristic
235C4×C53#53711yescharacteristic
236Dic53—41no—
237Dic53—41no—
238Dic53—41no—
239#3076—531no—
240#2960—531no—
241#2922—531no—
242#1986D5311yescharacteristic, complemented
243#1950—531no—
244#1945—532no—
245#1937—532no—
246#1929—532no—
247#1922—532no—
248#1806—531no—
249C8×C4⋊C4—531no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53#232311yescharacteristic
253C2×C53C2×#44311yescharacteristic
254C2×C53#295511yescharacteristic
255D53—81no—
256#533C2×D5311yescharacteristic
257C4×#41—531no—
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—531no—
275#424—531no—
276#424—532no—
277#423—531no—
278C2×C4⋊C8—531no—
279C2×#104—532no—
280#416—532no—
281#408—531no—
282#408—531no—
283#405—532no—
284C4×M16C2×D5311yescharacteristic
285C2×C4×C8—531no—
286C2×C4×C8—532no—
287C4×C4⋊C4C2×D5311yescharacteristic
288#341—532no—
289#339—532no—
290#339C2×D5312yes—
291#339C2×D5312yes—
292#336—532no—
293C53#2213011yescharacteristic, complemented, Sylow
294C2×#41—531no—
295C2×M16—532no—
296C2×M16C22×D5311yescharacteristic
297C2×M16C22×D5311yescharacteristic
298C22×C8—531no—
299C22×C8—532no—
300C22×C8—531no—
301C4⋊Q8C4×D5311yescharacteristic
302C4⋊Q8C4×D5311yescharacteristic
303#125C4×D5312yes—
304C22⋊Q8—21no—
305C22⋊Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—531no—
309C4×Q8—531no—
310C4×D4—531no—
311C4×D4—531no—
312C4×D4—1061no—
313C4×D4—1061no—
314#117—531no—
315#117—1061no—
316#117C22×D5312yes—
317C2×C4⋊C4—21no—
318C2×C42—1061no—
319C2×C42C22×D5312yes—
320C2×C42C22×D5311yescharacteristic
321#108—2122no—
322#107—532no—
323#107—532no—
324#106—532no—
325#106—532no—
326C4⋊C8—531no—
327C4⋊C8—531no—
328C4⋊C8—532no—
329C4⋊C8—1062no—
330C4⋊C8—1062no—
331C4⋊C8—1061no—
332C4⋊C8—1061no—
333C4⋊C8—1061no—
334C4⋊C8—1061no—
335C4⋊C8—1061no—
336C4⋊C8—1061no—
337#104—2122no—
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—21no—
347#97—1062no—
348C4×C8—21no—
349C4×C8—531no—
350C4×C8—531no—
351C4×C8—531no—
352C4×C8—531no—
353C4×C8—1062no—
354C4×C8—2122no—
355#95—21no—
356#41—1061no—
357#41—1061no—
358#41—1061no—
359#41—1061no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—531no—
363C2×D4—531no—
364C2×D4—1061no—
365C22×C4—22no—
366C22×C4—1061no—
367C22×C4—1061no—
368C22×C4—1061no—
369C22×C4#1171994711yescharacteristic
370C22×C4#1171994711yescharacteristic
371C22×C4C23×D5311yescharacteristic
372M16—21no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—1062no—
377M16—1062no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—532no—
381C2×C8—532no—
382C2×C8—531no—
383C2×C8—531no—
384C2×C8—531no—
385C2×C8—531no—
386C2×C8—1061no—
387C2×C8—1062no—
388C2×C8—1061no—
389C2×C8—1062no—
390C2×C8—1061no—
391C2×C8—1061no—
392C2×C8—1062no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—531no—
399C4⋊C4—531no—
400C4⋊C4—1061no—
401C4⋊C4—1061no—
402C4⋊C4C2×C4×D5312yes—
403C4⋊C4C2×C4×D5312yes—
404C22⋊C4—22no—
405C22⋊C4—1061no—
406C22⋊C4—1061no—
407C22⋊C4—1061no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—1061no—
412C42—1061no—
413C42—1061no—
414C42C2×C4×D5311yescharacteristic
415C42C2×C4×D5311yescharacteristic
416C42D4×D5311yescharacteristic
417C42D4×D5311yescharacteristic
418C23—1061no—
419C23C2×#1171994711yescharacteristic
420Q8—42no—
421Q8—42no—
422Q8—1061no—
423Q8—1061no—
424D4—1061no—
425D4—1061no—
426D4—2121no—
427D4—2121no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—1061no—
436C2×C4—1061no—
437C2×C4—1061no—
438C2×C4—2122no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4—2122no—
442C2×C4#1313317711yescharacteristic
443C2×C4#1313317711yescharacteristic
444C2×C4#1313317711yescharacteristic
445C2×C4#1313317711yescharacteristic
446C2×C4C22×C4×D5311yescharacteristic
447C2×C4C2×D4×D5311yescharacteristic
448C2×C4#1313326311yescharacteristic
449C2×C4#1313326411yescharacteristic
450C2×C4#1313326511yescharacteristic
451C2×C4#1313326511yescharacteristic
452C8—41no—
453C8—41no—
454C8—1062no—
455C8—1062no—
456C8—1061no—
457C8—1061no—
458C8—1061no—
459C8—1061no—
460C8—2122no—
461C22—21no—
462C22—21no—
463C22—2121no—
464C22—2121no—
465C22—2121no—
466C22C2×#1313317711yescharacteristic
467C22#1681123411yescharacteristic
468C22#1681124211yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—44no—
473C4—42no—
474C4—42no—
475C4—2121no—
476C4—2121no—
477C4—2121no—
478C4#1681073712yes—
479C4#1681109911yescharacteristic
480C4#1681110511yescharacteristic
481C4#1681119611yescharacteristic
482C4#1681120311yescharacteristic
483C2—21no—
484C2—21no—
485C2—4241no—
486C2#2027740011yescharacteristic
487C2C2×#1681073711yescharacteristic
488C2#2028361411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1986C2
socleC22×C53C2×C4×D4
prime radicalC2×#11719945C23
cosocleC22×C4×C53C24
hypercenter#1986D53
hypocenterC53#22130
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
2. #20283650C2semidirect
4. #20275490C2semidirect
5. #20275462C2semidirect
8. #20275312C2semidirect
10. C4⋊C4×#88322C2semidirect
11. C53×#1986C2semidirect
13. #16810927C4semidirect
14. #16810920C4semidirect
15. #16810915C4semidirect
16. #16810910C4semidirect
17. #16810905C4semidirect
18. #16810900C4semidirect
242. #1986D53semidirect
293. C53#22130semidirect

Maximal subgroups

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

subgroupindex
#202844972
#202836502
#202833602
#202754902
#202754622
#202754202
#202753612
#202753122
#202747362
C4⋊C4×#883222
C53×#19862
#2213053