Finite Group Databasebeta

← #27278872

Subgroups — #27278872

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#20277664C211yesmaximal, characteristic, complemented
3#20275641C211yesmaximal, characteristic, complemented
4#20275521C211yesmaximal, characteristic, complemented
5#20272599C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×#13133214C2211yescharacteristic
10C2×#13133176—21no—
11#16810818—22no—
12#16810818C414yescomplemented
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133139—22no—
15C2×#13133139—21no—
16C2×#13133139—21no—
17C2×#13133137—21no—
18#16810526—21no—
19#16810516—22no—
20#16810493C2211yescharacteristic
21#16810483—21no—
22C4⋊C4×Dic53—21no—
23C4×C4⋊Dic53C2211yescharacteristic
24C2×C4×D4×C53C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4×C8×C53C2211yescharacteristic
27C53×D4⋊C8—22no—
28C22×D212—21no—
29C22×D212—21no—
30C22×D212—21no—
31C2×#11719922—24no—
32C2×#11719922—42no—
33C2×#11719922—42no—
34C22×#88322—21no—
35#13133214—21no—
36#13133214—21no—
37#13133214—21no—
38#13133214C2×C412yes—
39#13133176—42no—
40#13133176—42no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53D411yescharacteristic
44C2×C4⋊Dic53D411yescharacteristic
45C2×C4⋊Dic53D412yes—
46C2×C22⋊Dic53—22no—
47C2×C22⋊Dic53—22no—
48C2×C4×Dic53—22no—
49C2×C4×Dic53—21no—
50C4⋊#88322—21no—
51C4⋊#88322C2×C412yes—
52#13133139—42no—
53#13133139—44no—
54#13133139—42no—
55#13133137—24no—
56#13133124—22no—
57#13133124—21no—
58C22×D4×C53D411yescharacteristic
59C23×C4×C53—21no—
60C22×C8×C53—21no—
61C22×C8×C53—21no—
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53C2×C412yes—
65C4×D4×C53D412yes—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4×C8×C53—21no—
72C4×C8×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—22no—
83C22×C4×C53—22no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—22no—
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—
110C2×C8×C53—22no—
111C2×C8×C53—41no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53C22⋊C412yes—
118C22⋊C4×C53—22no—
119C22⋊C4×C53—22no—
120C42×C53C22×C411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53#4111yescharacteristic
123C42×C53#4111yescharacteristic
124C22⋊Dic53—44no—
125C22⋊Dic53—82no—
126#11719922—48no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129C4⋊Dic53—21no—
130C4⋊Dic53—41no—
131C4⋊Dic53—41no—
132C4⋊Dic53—41no—
133C4⋊Dic53—41no—
134C4⋊Dic53—42no—
135C4⋊Dic53D812yescomplemented
136C4×Dic53—44no—
137C4×Dic53—41no—
138C4×Dic53—41no—
139C2×#88322—22no—
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—41no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53—42no—
170C2×C4×C53C2×C22⋊C411yescharacteristic
171C2×C4×C53C4×D411yescharacteristic
172C2×C4×C53C4×D411yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12111yescharacteristic
175C2×C4×C53#12311yescharacteristic
176C2×C4×C53C2×D811yescharacteristic
177C2×C4×C53#13511yescharacteristic
178C2×C4×C53#13511yescharacteristic
179C2×C4×C53#13511yescharacteristic
180C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
182C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
184C2×Dic53—22no—
185C2×Dic53—41no—
186C2×Dic53—41no—
187C2×Dic53—42no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—42no—
191C2×Dic53—42no—
192C2×Dic53—41no—
193D212—44no—
194D212—42no—
195D212—42no—
196D212—42no—
197D212—42no—
198D212—82no—
199D212—81no—
200D212—81no—
201D212—81no—
202D212—81no—
203C8×C53—42no—
204C8×C53—81no—
205#88322—42no—
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#41611yescharacteristic
232C4×C53#41811yescharacteristic
233C4×C53C4×D811yescharacteristic
234C4×C53#44211yescharacteristic
235C4×C53#45911yescharacteristic
236C4×C53#46311yescharacteristic
237C4×C53#48211yescharacteristic
238C4×C53#48211yescharacteristic
239Dic53—42no—
240Dic53—82no—
241Dic53—81no—
242C2×#459—531no—
243#2354—531no—
244#1982—531no—
245#1957—531no—
246C2×D4⋊C8D5311yescharacteristic, complemented
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—1061no—
262#459—534no—
263#459—1062no—
264C2×C4⋊C8—531no—
265C2×C4⋊C8C2×D5311yescharacteristic
266C2×#102—1061no—
267C2×#102—1062no—
268C2×#102—1061no—
269C2×#102—1061no—
270C2×C4×C8C2×D5311yescharacteristic
271#390—1061no—
272#388—1062no—
273#384—531no—
274#380—1061no—
275C4×C4⋊C4—531no—
276C4×C4⋊C4—1061no—
277D4⋊C8—22no—
278C53#898011yescharacteristic, complemented, Sylow
279C22×D4—1061no—
280C22×D4—1061no—
281C22×D4—1061no—
282C22×D4D21211yescharacteristic
283C23×C4—21no—
284C2×D8—1064no—
285C2×D8—2122no—
286C2×D8—2122no—
287C22×C8—21no—
288C22×C8—21no—
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×D4C22⋊D5312yes—
298#117—2122no—
299#117—2122no—
300C2×C4⋊C4—531no—
301C2×C4⋊C4—531no—
302C2×C4⋊C4—532no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4D21211yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—1062no—
308C2×C22⋊C4—1062no—
309C2×C42—1062no—
310C2×C42—1061no—
311C2×C42C22×D5311yescharacteristic
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—532no—
315C4⋊C8—1061no—
316#102—1064no—
317#102—2122no—
318#102—2122no—
319#102—2124no—
320C4×C8—21no—
321C4×C8—21no—
322#95—1062no—
323#95—1061no—
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—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—1061no—
356C22×C4—1061no—
357C22×C4—1062no—
358C22×C4C2×D21211yescharacteristic
359C22×C4D4×D5311yescharacteristic
360C22×C4D4×D5311yescharacteristic
361D8—2128no—
362C2×C8—22no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—1062no—
369C2×C8—2121no—
370C2×C8—2121no—
371C4⋊C4—21no—
372C4⋊C4—532no—
373C4⋊C4—1061no—
374C4⋊C4—1061no—
375C4⋊C4—1061no—
376C4⋊C4—2122no—
377C4⋊C4—2121no—
378C4⋊C4—2121no—
379C4⋊C4—2121no—
380C4⋊C4—2121no—
381C4⋊C4#1171992112yes—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—4242no—
385C22⋊C4—2124no—
386C42—2124no—
387C42—2121no—
388C42—2121no—
389C42C2×C4×D5311yescharacteristic
390C42#1171994711yescharacteristic
391C42#1171994711yescharacteristic
392C42C2×C22⋊D5311yescharacteristic
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#1313318711yescharacteristic
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—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—1062no—
432C2×C4—2122no—
433C2×C4—2121no—
434C2×C4—2121no—
435C2×C4—2122no—
436C2×C4—2121no—
437C2×C4—2122no—
438C2×C4—2121no—
439C2×C4—2121no—
440C2×C4C2×#1171992111yescharacteristic
441C2×C4#1313317711yescharacteristic
442C2×C4#1313317711yescharacteristic
443C2×C4#1313318911yescharacteristic
444C2×C4#1313318911yescharacteristic
445C2×C4#1313320111yescharacteristic
446C2×C4D8×D5311yescharacteristic
447C2×C4#1313323811yescharacteristic
448C2×C4#1313323911yescharacteristic
449C2×C4#1313323911yescharacteristic
450C2×C4#1313324111yescharacteristic
451C2×C4#1313324411yescharacteristic
452C2×C4#1313324511yescharacteristic
453C2×C4#1313324511yescharacteristic
454C8—42no—
455C8—81no—
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#1681050011yescharacteristic
472C22#1681075811yescharacteristic
473C22#1681076911yescharacteristic
474C22#1681077011yescharacteristic
475C22#1681077011yescharacteristic
476C22#1681079211yescharacteristic
477C22#1681079711yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—4242no—
482C4—4241no—
483C4—2122no—
484C4#1681062011yescharacteristic
485C4#1681062911yescharacteristic
486C4#1681070311yescharacteristic
487C4#1681073211yescharacteristic
488C4#1681081611yescharacteristic
489C4#1681084011yescharacteristic
490C4#1681094811yescharacteristic
491C4#1681094811yescharacteristic
492C2—82no—
493C2—8482no—
494C2#2027378311yescharacteristic
495C2#2027380011yescharacteristic
496C2#2027380711yescharacteristic
497C2#2027382711yescharacteristic
498C2#2027503511yescharacteristic
499C2#2027506111yescharacteristic
500C2#2027508911yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×D4⋊C8C2
socleC23×C53#120
prime radicalC2×#16810818C2
cosocleC2×C42×C53C23
hypercenterC2×D4⋊C8D53
hypocenterC53#8980
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20277664C2semidirect
3. #20275641C2semidirect
4. #20275521C2semidirect
5. #20272599C2semidirect
6. #20272555C2semidirect
7. C2×C53×D4⋊C8C2semidirect
12. #16810818C4semidirect
135. C4⋊Dic53D8semidirect
246. C2×D4⋊C8D53semidirect
278. C53#8980semidirect

Maximal subgroups

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

subgroupindex
C2×#168108182
#202776642
#202756412
#202755212
#202725992
#202725552
C2×C53×D4⋊C82
#898053