Finite Group Databasebeta

← #27278874

Subgroups — #27278874

Order 13568.

692 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810818C211yesmaximal, characteristic
2#20275642C211yesmaximal, characteristic, complemented
3#20275520C211yesmaximal, characteristic, complemented
4C2×D4⋊#88322C211yesmaximal, characteristic, complemented
5#20272598C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2354C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×#13133214C2211yescharacteristic
10#16810818—22no—
11#16810818C414yescomplemented
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133137—21no—
15C2×#13133137—22no—
16C2×#13133137—21no—
17C2×#13133137—21no—
18C2×C4×#88322C2211yescharacteristic
19D4⋊#88322—22no—
20C2×C4×D4×C53C2211yescharacteristic
21C2×#117×C53—21no—
22C53×#390—21no—
23C53×#388—22no—
24C53×#384C2211yescharacteristic
25C53×#380—21no—
26C4×C4⋊C4×C53—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C2×#11719922—24no—
32C2×#11719922—42no—
33C2×#11719922—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36C22×#88322—21no—
37#13133214—21no—
38#13133214—21no—
39#13133214—21no—
40#13133214C2×C412yes—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322C2×C412yes—
45#13133137—24no—
46#13133137—44no—
47#13133137—42no—
48#13133137—42no—
49C4×#88322—21no—
50C4×#88322—21no—
51C22×D4×C53D411yescharacteristic
52C23×C4×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53—21no—
55C4×D4×C53C2×C412yes—
56C4×D4×C53D412yes—
57#117×C53—42no—
58#117×C53—42no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C4⋊C4×C53D412yes—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—22no—
67C2×C22⋊C4×C53—21no—
68C2×C42×C53—22no—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71#95×C53—22no—
72#95×C53—21no—
73C24×C53—21no—
74C23×D53—41no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53C22⋊C412yes—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—22no—
84C22×C4×C53—22no—
85C22×C4×C53—21no—
86C22×C4×C53—22no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C2×D212—22no—
95C2×D212—22no—
96C2×D212—22no—
97C2×D212—42no—
98C2×D212—42no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—41no—
109C2×D212—41no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—42no—
115C4⋊C4×C53—41no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53—41no—
119C4⋊C4×C53C22⋊C412yes—
120C4⋊C4×C53D812yescomplemented
121C22⋊C4×C53—22no—
122C22⋊C4×C53—22no—
123C22⋊C4×C53—44no—
124C22⋊C4×C53—82no—
125C42×C53—44no—
126C42×C53—41no—
127C42×C53—41no—
128C42×C53C22×C411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131C42×C53#4111yescharacteristic
132#11719922—48no—
133C2×#88322—22no—
134C2×#88322—22no—
135C2×#88322—41no—
136C2×#88322—41no—
137C2×#88322—41no—
138C2×#88322—41no—
139C2×#88322—41no—
140C2×#88322—41no—
141C2×#88322—41no—
142C23×C53—22no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53#12011yescharacteristic
150C22×D53—42no—
151C22×D53—81no—
152C22×D53—81no—
153C22×D53—81no—
154C22×D53—81no—
155C22×D53—81no—
156C22×D53—81no—
157D4×C53—22no—
158D4×C53—22no—
159D4×C53—41no—
160D4×C53—41no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—22no—
164C2×C4×C53—22no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—42no—
177C2×C4×C53—41no—
178C2×C4×C53—42no—
179C2×C4×C53C2×C22⋊C411yescharacteristic
180C2×C4×C53C4×D411yescharacteristic
181C2×C4×C53C4×D411yescharacteristic
182C2×C4×C53#12111yescharacteristic
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53#12311yescharacteristic
185C2×C4×C53C2×D811yescharacteristic
186C2×C4×C53#13511yescharacteristic
187C2×C4×C53#13511yescharacteristic
188C2×C4×C53#13511yescharacteristic
189C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
190C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
193D212—44no—
194D212—42no—
195D212—42no—
196D212—42no—
197D212—42no—
198D212—82no—
199D212—81no—
200D212—81no—
201D212—81no—
202D212—81no—
203#88322—42no—
204#88322—42no—
205#88322—81no—
206#8980—531nomaximal, Sylow
207C22×C53—42no—
208C22×C53—42no—
209C22×C53—42no—
210C22×C53—81no—
211C22×C53—81no—
212C22×C53—81no—
213C22×C53—81no—
214C22×C53#38611yescharacteristic
215C22×C53#44711yescharacteristic
216C22×C53#44911yescharacteristic
217C22×C53#44911yescharacteristic
218C22×C53#44911yescharacteristic
219C22×C53#45311yescharacteristic
220C22×C53#45411yescharacteristic
221C2×D53—82no—
222C2×D53—82no—
223C2×D53—82no—
224C2×D53—161no—
225C2×D53—161no—
226C2×D53—161no—
227C2×D53—161no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—42no—
231C4×C53—42no—
232C4×C53—82no—
233C4×C53—81no—
234C4×C53#41611yescharacteristic
235C4×C53#41811yescharacteristic
236C4×C53C4×D811yescharacteristic
237C4×C53#44211yescharacteristic
238C4×C53#45911yescharacteristic
239C4×C53#46311yescharacteristic
240C4×C53#48211yescharacteristic
241C4×C53#48211yescharacteristic
242C2×#459—531no—
243#2354D5311yescharacteristic, complemented
244#1982—531no—
245#1957—531no—
246C2×D4⋊C8—531no—
247#1296—531no—
248#1279—531no—
249C2×C53—82no—
250C2×C53#162111yescharacteristic
251C2×C53#162411yescharacteristic
252C2×C53#162511yescharacteristic
253C2×C53#162811yescharacteristic
254C2×C53#187511yescharacteristic
255C2×C53#187911yescharacteristic
256C2×C53#188311yescharacteristic
257D53—162no—
258C22×D8—1061no—
259C2×#127—531no—
260C2×C4×D4C2×D5311yescharacteristic
261C2×#117—21no—
262#459—534no—
263#459—1062no—
264C2×C4⋊C8—531no—
265C2×C4⋊C8—531no—
266C2×#102—1061no—
267C2×#102—1062no—
268C2×#102—1061no—
269C2×#102—1061no—
270C2×C4×C8—531no—
271#390—21no—
272#388—22no—
273#384C2×D5311yescharacteristic
274#380—21no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4C2×D5311yescharacteristic
277D4⋊C8—1062no—
278C53#898011yescharacteristic, complemented, Sylow
279C22×D4—1061no—
280C22×D4—1061no—
281C22×D4—1061no—
282C22×D4C22⋊D5311yescharacteristic
283C23×C4—21no—
284C2×D8—1064no—
285C2×D8—2122no—
286C2×D8—2122no—
287C22×C8—1061no—
288C22×C8—1061no—
289C22×C8—1061no—
290#127—532no—
291#127—1061no—
292#127—1061no—
293#127—1061no—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4C4×D5312yes—
297C4×D4D21212yes—
298#117—42no—
299#117—42no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4C22⋊D5311yescharacteristic
303C2×C4⋊C4C22⋊D5311yescharacteristic
304C2×C4⋊C4C22⋊D5311yescharacteristic
305C2×C4⋊C4C22⋊D5312yes—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C42—22no—
310C2×C42—21no—
311C2×C42C22×D5311yescharacteristic
312C4⋊C8—532no—
313C4⋊C8—1061no—
314C4⋊C8—1061no—
315C4⋊C8—1061no—
316#102—1064no—
317#102—2122no—
318#102—2124no—
319#102—2122no—
320C4×C8—1061no—
321C4×C8—1061no—
322#95—22no—
323#95—21no—
324C24—21no—
325C24—2121no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—1062no—
332C2×D4—1062no—
333C2×D4—1062no—
334C2×D4—2122no—
335C2×D4—2122no—
336C2×D4—2121no—
337C2×D4—2121no—
338C2×D4—2121no—
339C2×D4—2121no—
340C2×D4—2121no—
341C2×D4—2121no—
342C2×D4—2121no—
343C2×D4—2121no—
344C2×D4—2121no—
345C2×D4—2121no—
346C2×D4—2121no—
347C2×D4#1171992112yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4C2×C22⋊D5311yescharacteristic
359C22×C4C2×C22⋊D5311yescharacteristic
360C22×C4C2×C22⋊D5311yescharacteristic
361D8—2128no—
362C2×C8—1062no—
363C2×C8—1062no—
364C2×C8—2121no—
365C2×C8—2121no—
366C2×C8—2121no—
367C2×C8—2121no—
368C2×C8—2121no—
369C2×C8—2121no—
370C2×C8—2121no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4#1171992112yes—
381C4⋊C4#1171992212yescomplemented
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—44no—
385C22⋊C4—82no—
386C42—44no—
387C42—41no—
388C42—41no—
389C42C2×C4×D5311yescharacteristic
390C42C2×D21211yescharacteristic
391C42#1171994511yescharacteristic
392C42#1171994511yescharacteristic
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—4241no—
401C23—4241no—
402C23—4241no—
403C23—4241no—
404C23—4241no—
405C23—4241no—
406C23—2122no—
407C23#1313318611yescharacteristic
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412D4—4242no—
413D4—4241no—
414D4—4241no—
415D4—4241no—
416D4—4241no—
417D4—2124no—
418D4—2122no—
419D4—2122no—
420D4—2122no—
421D4—2122no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4C2×#1171992111yescharacteristic
441C2×C4C4×C22⋊D5311yescharacteristic
442C2×C4C4×C22⋊D5311yescharacteristic
443C2×C4#1313319111yescharacteristic
444C2×C4#1313319111yescharacteristic
445C2×C4#1313320011yescharacteristic
446C2×C4C2×#1171992211yescharacteristic
447C2×C4#1313323711yescharacteristic
448C2×C4#1313323711yescharacteristic
449C2×C4#1313323711yescharacteristic
450C2×C4#1313324211yescharacteristic
451C2×C4#1313324211yescharacteristic
452C2×C4#1313324211yescharacteristic
453C2×C4#1313324311yescharacteristic
454C8—4241no—
455C8—2122no—
456C8—2122no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—8481no—
465C22—8481no—
466C22—8481no—
467C22—8481no—
468C22—4242no—
469C22—4242no—
470C22—4242no—
471C22#1681050211yescharacteristic
472C22#1681075711yescharacteristic
473C22#1681076811yescharacteristic
474C22#1681076811yescharacteristic
475C22#1681076811yescharacteristic
476C22#1681079111yescharacteristic
477C22#1681079611yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—42no—
482C4—82no—
483C4—81no—
484C4#1681061711yescharacteristic
485C4#1681062711yescharacteristic
486C4C4×#1171992211yescharacteristic
487C4#1681073311yescharacteristic
488C4#1681081811yescharacteristic
489C4#1681083811yescharacteristic
490C4#1681094711yescharacteristic
491C4#1681094711yescharacteristic
492C2—82no—
493C2—8482no—
494C2#2027378011yescharacteristic
495C2#2027379811yescharacteristic
496C2#2027380411yescharacteristic
497C2#2027382511yescharacteristic
498C2#2027503711yescharacteristic
499C2#2027506311yescharacteristic
500C2#2027509111yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2354C2
socleC23×C53#120
prime radicalC2×#16810818C2
cosocleC2×C42×C53C23
hypercenter#2354D53
hypocenterC53#8980
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #20275642C2semidirect
3. #20275520C2semidirect
4. C2×D4⋊#88322C2semidirect
5. #20272598C2semidirect
6. #20272556C2semidirect
7. C53×#2354C2semidirect
11. #16810818C4semidirect
120. C4⋊C4×C53D8semidirect
243. #2354D53semidirect
278. C53#8980semidirect
381. C4⋊C4#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168108182
#202756422
#202755202
C2×D4⋊#883222
#202725982
#202725562
C53×#23542
#898053