Finite Group Databasebeta

← #27278876

Subgroups — #27278876

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
1#20277664C211yesmaximal, characteristic, complemented
2#20275643C211yesmaximal, characteristic, complemented
3#20275522C211yesmaximal, characteristic, complemented
4C2×D4⋊#88322C211yesmaximal, characteristic, complemented
5#20272599C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C2×C53×#459C211yesmaximal, characteristic
8C2×#13133176—21no—
9C2×C4⋊#88322C2211yescharacteristic
10C2×#13133138—22no—
11C2×#13133138—21no—
12C2×#13133138—21no—
13C2×C4×#88322C2211yescharacteristic
14#16810526—21no—
15#16810516—22no—
16#16810493C2211yescharacteristic
17#16810483—21no—
18C4⋊C4×Dic53—21no—
19C4×C4⋊Dic53C2211yescharacteristic
20D4⋊#88322—22no—
21C22×D8×C53—21no—
22C2×#127×C53C2211yescharacteristic
23C2×C4×D4×C53C2211yescharacteristic
24C53×#459—22no—
25C53×#459C414yescomplemented
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×#102×C53—21no—
28C22×#88322—21no—
29C22×#88322—21no—
30#13133176—42no—
31#13133176—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53D411yescharacteristic
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D412yes—
37C2×C22⋊Dic53—22no—
38C2×C22⋊Dic53—22no—
39C2×C4×Dic53—22no—
40C2×C4×Dic53—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43#13133138—44no—
44#13133138—42no—
45#13133138—42no—
46C4×#88322—21no—
47C4×#88322—21no—
48#13133124—22no—
49#13133124—21no—
50C22×D4×C53—21no—
51C22×D4×C53—21no—
52C22×D4×C53—21no—
53C22×D4×C53D411yescharacteristic
54C23×C4×C53—21no—
55C2×D8×C53—24no—
56C2×D8×C53—42no—
57C2×D8×C53—42no—
58C22×C8×C53—21no—
59#127×C53—21no—
60#127×C53—21no—
61#127×C53—21no—
62#127×C53C2×C412yes—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53C2×C412yes—
66C4×D4×C53D412yes—
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53C2×C412yes—
72#102×C53—24no—
73C24×C53—21no—
74C24×C53—41no—
75C2×D4×C53—22no—
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—42no—
84C2×D4×C53—42no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53C22⋊C412yes—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×Dic53—21no—
108C22×Dic53—21no—
109C22×Dic53—22no—
110D8×C53—48no—
111C2×C8×C53—22no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53C22⋊C412yes—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C42×C53C22×C411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121C42×C53#4111yescharacteristic
122C22⋊Dic53—44no—
123C22⋊Dic53—82no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129C4⋊Dic53—41no—
130C4⋊Dic53—41no—
131C4⋊Dic53—42no—
132C4⋊Dic53D812yescomplemented
133C4×Dic53—44no—
134C4×Dic53—41no—
135C4×Dic53—41no—
136C2×#88322—22no—
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—42no—
150C23×C53—81no—
151C23×C53—81no—
152C23×C53—81no—
153C23×C53—81no—
154C23×C53—81no—
155C23×C53—81no—
156C23×C53#12011yescharacteristic
157D4×C53—22no—
158D4×C53—22no—
159D4×C53—44no—
160D4×C53—42no—
161D4×C53—42no—
162D4×C53—42no—
163D4×C53—42no—
164D4×C53—41no—
165D4×C53—41no—
166D4×C53—82no—
167D4×C53—81no—
168D4×C53—81no—
169D4×C53—81no—
170D4×C53—81no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—22no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53C2×C22⋊C411yescharacteristic
181C2×C4×C53C4×D411yescharacteristic
182C2×C4×C53C4×D411yescharacteristic
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#12311yescharacteristic
186C2×C4×C53C2×D811yescharacteristic
187C2×C4×C53#13511yescharacteristic
188C2×C4×C53#13511yescharacteristic
189C2×C4×C53#13511yescharacteristic
190C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
191C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
193C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
194C2×Dic53—22no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197C2×Dic53—42no—
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—42no—
201C2×Dic53—42no—
202C2×Dic53—41no—
203C8×C53—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—82no—
215C22×C53—82no—
216C22×C53—82no—
217C22×C53—161no—
218C22×C53—161no—
219C22×C53—161no—
220C22×C53—161no—
221C22×C53#38611yescharacteristic
222C22×C53#44711yescharacteristic
223C22×C53#44911yescharacteristic
224C22×C53#44911yescharacteristic
225C22×C53#44911yescharacteristic
226C22×C53#45311yescharacteristic
227C22×C53#45411yescharacteristic
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×#459D5311yescharacteristic
243#2354—531no—
244#1982—531no—
245#1957—531no—
246C2×D4⋊C8—531no—
247#1296—531no—
248#1279—531no—
249C2×C53—82no—
250C2×C53—162no—
251C2×C53#162111yescharacteristic
252C2×C53#162411yescharacteristic
253C2×C53#162511yescharacteristic
254C2×C53#162811yescharacteristic
255C2×C53#187511yescharacteristic
256C2×C53#187911yescharacteristic
257C2×C53#188311yescharacteristic
258C22×D8—21no—
259C2×#127C2×D5311yescharacteristic
260C2×C4×D4C2×D5311yescharacteristic
261C2×#117—1061no—
262#459—22no—
263#459Dic5314yescomplemented
264C2×C4⋊C8—531no—
265C2×C4⋊C8C2×D5311yescharacteristic
266C2×#102—21no—
267C2×#102—1062no—
268C2×#102—1061no—
269C2×#102—1061no—
270C2×C4×C8—531no—
271#390—1061no—
272#388—1062no—
273#384—531no—
274#380—1061no—
275C4×C4⋊C4—531no—
276C4×C4⋊C4—1061no—
277D4⋊C8—1062no—
278C53#898011yescharacteristic, complemented, Sylow
279C22×D4—21no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C22⋊D5311yescharacteristic
283C23×C4—21no—
284C2×D8—24no—
285C2×D8—42no—
286C2×D8—42no—
287C22×C8—21no—
288C22×C8—1061no—
289C22×C8—1061no—
290#127—21no—
291#127—21no—
292#127—21no—
293#127C2×Dic5312yes—
294C4×D4—21no—
295C4×D4—21no—
296C4×D4C2×Dic5312yes—
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⋊C4C22⋊D5311yescharacteristic
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—1061no—
314C4⋊C8—1061no—
315C4⋊C8C2×Dic5312yes—
316#102—24no—
317#102—2122no—
318#102—2122no—
319#102—2124no—
320C4×C8—1061no—
321C4×C8—1061no—
322#95—1062no—
323#95—1061no—
324C24—21no—
325C24—41no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4C22⋊Dic5312yes—
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×C4D4×D5311yescharacteristic
359C22×C4D4×D5311yescharacteristic
360C22×C4C2×C22⋊D5311yescharacteristic
361D8—48no—
362C2×C8—22no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—1062no—
366C2×C8—2121no—
367C2×C8—2121no—
368C2×C8—2121no—
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⋊C4C22⋊Dic5312yes—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—4242no—
385C22⋊C4—2124no—
386C42—2124no—
387C42—2121no—
388C42—2121no—
389C42#1171994711yescharacteristic
390C42#1171994711yescharacteristic
391C42C22×Dic5311yescharacteristic
392C42C2×C22⋊D5311yescharacteristic
393C23—22no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23#1313318511yescharacteristic
408D4—22no—
409D4—22no—
410D4—44no—
411D4—42no—
412D4—42no—
413D4—41no—
414D4—41no—
415D4—42no—
416D4—42no—
417D4—82no—
418D4—81no—
419D4—81no—
420D4—81no—
421D4—81no—
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×C22⋊Dic5311yescharacteristic
441C2×C4D4×Dic5311yescharacteristic
442C2×C4D4×Dic5311yescharacteristic
443C2×C4#1313318911yescharacteristic
444C2×C4#1313318911yescharacteristic
445C2×C4#1313319811yescharacteristic
446C2×C4D8×D5311yescharacteristic
447C2×C4#1313323611yescharacteristic
448C2×C4#1313323711yescharacteristic
449C2×C4#1313323711yescharacteristic
450C2×C4#1313324111yescharacteristic
451C2×C4#1313324111yescharacteristic
452C2×C4#1313324311yescharacteristic
453C2×C4#1313324311yescharacteristic
454C8—42no—
455C8—4241no—
456C8—2122no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—161no—
468C22—161no—
469C22—161no—
470C22—161no—
471C22#1681050311yescharacteristic
472C22#1681075511yescharacteristic
473C22#1681076511yescharacteristic
474C22#1681076611yescharacteristic
475C22#1681076611yescharacteristic
476C22#1681078911yescharacteristic
477C22#1681079411yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—4242no—
482C4—4241no—
483C4—2122no—
484C4#1681061911yescharacteristic
485C4#1681062811yescharacteristic
486C4D8×Dic5311yescharacteristic
487C4#1681073511yescharacteristic
488C4#1681081611yescharacteristic
489C4#1681083611yescharacteristic
490C4#1681094411yescharacteristic
491C4#1681094411yescharacteristic
492C2—82no—
493C2—162no—
494C2#2027377911yescharacteristic
495C2#2027379711yescharacteristic
496C2#2027380311yescharacteristic
497C2#2027382311yescharacteristic
498C2#2027503811yescharacteristic
499C2#2027506511yescharacteristic
500C2#2027509311yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#459C2
socleC23×C53#120
cosocleC2×C42×C53C23
hypercenterC2×#459D53
hypocenterC53#8980
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. #20277664C2semidirect
2. #20275643C2semidirect
3. #20275522C2semidirect
4. C2×D4⋊#88322C2semidirect
5. #20272599C2semidirect
6. #20272554C2semidirect
25. C53×#459C4semidirect
132. C4⋊Dic53D8semidirect
263. #459Dic53semidirect
278. C53#8980semidirect

Maximal subgroups

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

subgroupindex
#202776642
#202756432
#202755222
C2×D4⋊#883222
#202725992
#202725542
C2×C53×#4592
#898053