Finite Group Databasebeta

← #27279562

Subgroups — #27279562

Order 13568.

546 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×#11719945C211yesmaximal, characteristic, complemented
2C4×#13133243C211yesmaximal, characteristic
3#20273100C211yesmaximal, characteristic, complemented
4#20273036C211yesmaximal, characteristic, complemented
5#20272565C211yesmaximal, characteristic, complemented
6#20272552C211yesmaximal, characteristic, complemented
7C2×C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#13133243C411yescharacteristic
9C4×#11719945C2211yescharacteristic
10C8×Dic106—21no—
11#16810733—21no—
12#16810723—21no—
13C4×#11719923—21no—
14C4×#11719922—21no—
15C8×D212—21no—
16C8×C22⋊D53—22no—
17#16810676C2211yescharacteristic
18#16810676C2211yescharacteristic
19#16810674—21no—
20#16810663C411yescharacteristic
21#16810628C411yescharacteristic
22#16810622C411yescharacteristic
23C4×#11719917C2211yescharacteristic
24C4×C8×D53—21no—
25#16810353—21no—
26#16810350—21no—
27#16810347—22no—
28C2×C4×D4×C53C2211yescharacteristic
29C2×C4⋊C8×C53C2211yescharacteristic
30C2×C4×C8×C53C2211yescharacteristic
31C53×D4⋊C8—21no—
32C53×D4⋊C8—21no—
33C2×#11719945C2×C411yescharacteristic
34#13133243—22no—
35#13133243C814yescomplemented
36C2×#11719923—21no—
37C2×#11719922—21no—
38C2×#11719917C2×C411yescharacteristic
39C2×#11719917C2×C411yescharacteristic
40C2×C8×D53—22no—
41C4×Dic106D411yescharacteristic
42C4×Dic106D411yescharacteristic
43C4×D212D411yescharacteristic
44C4×D212D411yescharacteristic
45C4×C22⋊D53—21no—
46C4×C22⋊D53—21no—
47#13133176C2×C411yescharacteristic
48#13133174—21no—
49C42×D53—21no—
50#13133150—21no—
51#13133148—21no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54#13133146—22no—
55#13133140—21no—
56#13133138—21no—
57#13133138—21no—
58#13133137—21no—
59C22⋊#88322—21no—
60C22⋊#88322D412yes—
61#13133129—22no—
62#13133128—21no—
63C4×#88322—21no—
64C4×#88322—21no—
65C4×#88322—21no—
66C8×Dic53—22no—
67C22×D4×C53C2×C411yescharacteristic
68C23×C4×C53—21no—
69C22×C8×C53—21no—
70C22×C8×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53—21no—
74C4×D4×C53—21no—
75C2×C4⋊C4×C53C2×C411yescharacteristic
76C2×C22⋊C4×C53—21no—
77C2×C42×C53C2311yescharacteristic
78C4⋊C8×C53—21no—
79C4⋊C8×C53—21no—
80C4×C8×C53—21no—
81C4×C8×C53D411yescharacteristic
82C4×C8×C53D411yescharacteristic
83C24×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53C2×C812yescomplemented
90C22×C4×C53—21no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—81no—
94C22×C4×C53—81no—
95C22×C4×C53C22×C411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53#4111yescharacteristic
98C2×C22⋊D53—21no—
99#11719945—21no—
100#11719945—21no—
101#11719945C2×C812yes—
102#11719945M1612yes—
103C2×D212C22⋊C411yescharacteristic
104C2×C4×D53—21no—
105C2×C4×D53—21no—
106C2×C4×D53—21no—
107C2×Dic106C22⋊C411yescharacteristic
108C2×C8×C53—21no—
109C2×C8×C53—21no—
110C2×C8×C53—42no—
111C2×C8×C53—41no—
112C2×C8×C53—81no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121C22⋊Dic53—21no—
122#11719923—24no—
123#11719922—24no—
124#11719921—21no—
125#11719921—21no—
126C4⋊Dic53C22⋊C411yescharacteristic
127C4⋊Dic53C22⋊C411yescharacteristic
128#11719919—21no—
129#11719919—21no—
130C4×Dic53—21no—
131C4×Dic53—21no—
132C4×Dic53—21no—
133#11719917—21no—
134#11719917—21no—
135#11719917—21no—
136#11719917C2×C812yescomplemented
137C2×#88322—21no—
138C2×#88322—21no—
139C2×#88322—22no—
140C2×#88322—41no—
141C8×D53—44no—
142C23×C53—22no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—81no—
146C23×C53—81no—
147C23×C53C4×D411yescharacteristic
148C22×D53—21no—
149D4×C53—22no—
150D4×C53—22no—
151D4×C53—41no—
152D4×C53—41no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—82no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53C2×C22⋊C411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53#12311yescharacteristic
167C2×C4×C53C22×C811yescharacteristic
168C2×C4×C53C2×M1611yescharacteristic
169C2×C4×C53#13111yescharacteristic
170C2×C4×C53#13111yescharacteristic
171C22⋊D53—22no—
172C22⋊D53—22no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176D212—21no—
177D212C22⋊C812yes—
178C4×D53—22no—
179C4×D53—22no—
180C4×D53—42no—
181C4×D53—41no—
182C4×D53—41no—
183Dic106—21no—
184Dic106C22⋊C812yes—
185C8×C53—22no—
186C8×C53—41no—
187C8×C53—81no—
188#88322—22no—
189#88322—42no—
190#88322—41no—
191#88322—81no—
192#9095—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—42no—
196C22×C53—82no—
197C22×C53—82no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53#38611yescharacteristic
201C22×C53C8×D411yescharacteristic
202C22×C53#43511yescharacteristic
203C2×D53—22no—
204C2×D53—41no—
205C2×D53—41no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—81no—
210C4×C53C2×C22⋊C811yescharacteristic
211C4×C53#40811yescharacteristic
212C4×C53C8×D411yescharacteristic
213C4×C53#43511yescharacteristic
214C4×C53#47111yescharacteristic
215C4×C53#47211yescharacteristic
216Dic53—22no—
217Dic53—41no—
218Dic53—41no—
219C8×#41—531no—
220C4×AGL(1,ℤ/8)—531no—
221#1477—531no—
222C2×D4⋊C8D5311yescharacteristic, complemented
223#1460—531no—
224#1283—531no—
225#1278—531no—
226C2×C53—21no—
227C2×C53—21no—
228C2×C53—82no—
229C2×C53#186911yescharacteristic
230C2×C53#190611yescharacteristic
231C2×C53#190711yescharacteristic
232D53—42no—
233C2×AGL(1,ℤ/8)—531no—
234C4×#41—531no—
235C2×C4×D4C2×D5311yescharacteristic
236C8×Q8—1061no—
237#442—1061no—
238#440—1061no—
239C4×SD16—1061no—
240C4×D8—1061no—
241C8×D4—1061no—
242C8×D4—1062no—
243#431—531no—
244#431—531no—
245#431—1061no—
246#428—531no—
247C2×C4⋊C8C2×D5311yescharacteristic
248#418—531no—
249#417—531no—
250C4×M16—531no—
251C2×C4×C8—1061no—
252C2×C4×C8C2×D5311yescharacteristic
253Q8⋊C8—1061no—
254D4⋊C8—21no—
255D4⋊C8—21no—
256D4⋊C8—1061no—
257#323—1062no—
258C53#909511yescharacteristic, complemented, Sylow
259C2×#41—531no—
260C22×D4C4×D5311yescharacteristic
261C23×C4—21no—
262AGL(1,ℤ/8)—534no—
263AGL(1,ℤ/8)—1062no—
264C2×SD16—1061no—
265C2×D8—1061no—
266C2×M16—531no—
267C2×M16—531no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—1062no—
271C4×Q8—531no—
272C4×Q8—531no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—531no—
278C4×D4—531no—
279C4×D4—1061no—
280C4×D4—1061no—
281#117—531no—
282#117—1061no—
283C2×C4⋊C4C4×D5311yescharacteristic
284C2×C22⋊C4—21no—
285C2×C42—1061no—
286C2×C42C22×D5311yescharacteristic
287#107—1061no—
288#106—1061no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—1062no—
292C4⋊C8—1061no—
293C4⋊C8—1061no—
294#103—1061no—
295#102—1061no—
296#102—1061no—
297#102—1061no—
298C22⋊C8—532no—
299C22⋊C8—1062no—
300C22⋊C8—1061no—
301#97—1061no—
302C4×C8—21no—
303C4×C8—1062no—
304C4×C8—1061no—
305C4×C8—1061no—
306C4×C8—1061no—
307C4×C8C22⋊D5311yescharacteristic
308C4×C8C22⋊D5311yescharacteristic
309C24—21no—
310#41—532no—
311#41—532no—
312#41—1061no—
313#41—1061no—
314C2×Q8—531no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—531no—
321C2×D4—1061no—
322C2×D4C8×D5312yescomplemented
323C22×C4—21no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4—1061no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4C2×C4×D5311yescharacteristic
332C22×C4D4×D5311yescharacteristic
333C22×C4#1171994711yescharacteristic
334SD16—1064no—
335D8—1064no—
336M16—532no—
337M16—1061no—
338M16—1061no—
339M16—1061no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—81no—
345C2×C8—1061no—
346C2×C8—1061no—
347C2×C8—1062no—
348C2×C8—2124no—
349C2×C8—2121no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—531no—
353C4⋊C4—531no—
354C4⋊C4—1061no—
355C4⋊C4—1061no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—1061no—
359C22⋊C4—1061no—
360C22⋊C4—1061no—
361C42—1061no—
362C42—1061no—
363C42—1061no—
364C42#1171994511yescharacteristic
365C42D4×D5311yescharacteristic
366C42D4×D5311yescharacteristic
367C42C2×C22⋊D5311yescharacteristic
368C23—22no—
369C23—41no—
370C23—41no—
371C23—81no—
372C23—81no—
373C23—1061no—
374C23#1313317711yescharacteristic
375Q8—532no—
376Q8—1061no—
377D4—22no—
378D4—22no—
379D4—41no—
380D4—41no—
381D4—532no—
382D4—1062no—
383D4—1062no—
384D4—1061no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—1061no—
394C2×C4—1062no—
395C2×C4—1062no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—2122no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4C22⋊C4×D5311yescharacteristic
402C2×C4C4×C22⋊D5311yescharacteristic
403C2×C4#1313318511yescharacteristic
404C2×C4#1313318811yescharacteristic
405C2×C4#1313319011yescharacteristic
406C2×C4#1313319911yescharacteristic
407C2×C4C2×C8×D5311yescharacteristic
408C2×C4M16×D5311yescharacteristic
409C2×C4#1313322311yescharacteristic
410C2×C4#1313322511yescharacteristic
411C8—22no—
412C8—41no—
413C8—81no—
414C8—1062no—
415C8—4241no—
416C8—2122no—
417C8—2121no—
418C22—21no—
419C22—21no—
420C22—42no—
421C22—82no—
422C22—82no—
423C22—81no—
424C22—81no—
425C22—1062no—
426C22—2121no—
427C22—2121no—
428C22#1681049911yescharacteristic
429C22#1681068611yescharacteristic
430C22#1681069711yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—81no—
435C4—1062no—
436C4—2121no—
437C4—2121no—
438C4C22⋊C8×D5311yescharacteristic
439C4#1681058811yescharacteristic
440C4C8×C22⋊D5311yescharacteristic
441C4#1681069211yescharacteristic
442C4#1681088311yescharacteristic
443C4#1681089011yescharacteristic
444C2—21no—
445C2—21no—
446C2—82no—
447C2—2122no—
448C2#2027500411yescharacteristic
449C2#2027521911yescharacteristic
450C2#2027522611yescharacteristic
subgroupG/H
centerC2×C4#13133188
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC2×C53×D4⋊C8C2
socleC22×C53#386
prime radicalC2×#13133243C4
cosocleC2×C42×C53C23
hypercenterC2×D4⋊C8D53
hypocenterC53#9095
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
1. C8×#11719945C2semidirect
3. #20273100C2semidirect
4. #20273036C2semidirect
5. #20272565C2semidirect
6. #20272552C2semidirect
7. C2×C53×D4⋊C8C2semidirect
35. #13133243C8semidirect
89. C2×D4×C53C2×C8semidirect
136. #11719917C2×C8semidirect
222. C2×D4⋊C8D53semidirect
258. C53#9095semidirect
322. C2×D4C8×D53semidirect

Maximal subgroups

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

subgroupindex
C8×#117199452
C4×#131332432
#202731002
#202730362
#202725652
#202725522
C2×C53×D4⋊C82
#909553