Finite Group Databasebeta

← #27279342

Subgroups — #27279342

Order 13568.

690 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810970C211yesmaximal, characteristic
2#20277779C211yesmaximal, characteristic
3#20275666C211yesmaximal, characteristic, complemented
4#20275638C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×#13133214C2211yescharacteristic
10C2×C4×D212—21no—
11C2×#13133179—21no—
12#16810970—22no—
13#16810970C414yescomplemented
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133140—21no—
16C2×#13133140—21no—
17C2×#13133139—22no—
18#16810525—21no—
19#16810524—21no—
20#16810501—22no—
21C4×C4⋊Dic53C2211yescharacteristic
22#16810386—21no—
23#16810386—21no—
24#16810366—22no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4×C8×C53C2211yescharacteristic
27C4×C4⋊C4×C53C2211yescharacteristic
28C22×Dic106—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212D412yes—
32C22×C4×D53—22no—
33C22×#88322—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217C2×C412yescomplemented
38#13133214—21no—
39#13133214—21no—
40#13133214C2×C412yes—
41#13133214D412yes—
42C4×D212—42no—
43C4×D212—42no—
44#13133179—42no—
45#13133179—42no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53D411yescharacteristic
49C2×C4⋊Dic53D411yescharacteristic
50C2×#11719921—22no—
51C2×#11719921—22no—
52C2×#11719921—22no—
53C2×C4×Dic53—21no—
54C4⋊#88322—21no—
55C4⋊#88322—21no—
56#13133140—42no—
57#13133140—42no—
58#13133139—28no—
59#13133124—21no—
60C22×C8×C53—21no—
61C22×C8×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C42×C53—21no—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C4×C8×C53—21no—
69C4×C8×C53C2×C412yescomplemented
70#95×C53—21no—
71C23×D53—22no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—41no—
80C2×D212—22no—
81C2×D212—22no—
82C2×D212—22no—
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—42no—
94C2×D212—42no—
95C2×D212C22⋊C414yes—
96C2×C4×D53—44no—
97C2×C4×D53—44no—
98C2×C4×D53—82no—
99C2×Dic106—22no—
100C2×Dic106—41no—
101C2×Dic106—41no—
102C2×Dic106—41no—
103C2×Dic106—41no—
104C2×Dic106—41no—
105C2×C8×C53—22no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C2×C8×C53—41no—
110C2×C8×C53—41no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53D812yes—
115C42×C53—41no—
116C42×C53—41no—
117C42×C53C22×C411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53#4111yescharacteristic
120C42×C53#4111yescharacteristic
121#11719921—44no—
122#11719921—44no—
123#11719921—82no—
124C4⋊Dic53—21no—
125C4⋊Dic53—22no—
126C4⋊Dic53—22no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129C4⋊Dic53—41no—
130C4⋊Dic53—41no—
131C4⋊Dic53SD1612yes—
132C4×Dic53—41no—
133C4×Dic53—41no—
134C2×#88322—41no—
135C2×#88322—41no—
136C2×#88322—41no—
137C23×C53#12011yescharacteristic
138C22×D53—24no—
139C22×D53—42no—
140C22×D53—42no—
141C22×D53—42no—
142C22×D53—42no—
143C22×D53—42no—
144C22×D53—42no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—22no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53C2×C22⋊C411yescharacteristic
154C2×C4×C53C4×D411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12311yescharacteristic
159C2×C4×C53C2×D811yescharacteristic
160C2×C4×C53C2×SD1611yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53#13511yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C53#13711yescharacteristic
167C2×Dic53—22no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—42no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176D212—44no—
177D212—42no—
178D212—42no—
179D212—42no—
180D212—42no—
181D212—42no—
182D212—42no—
183D212—42no—
184D212—42no—
185D212—82no—
186C4×D53—88no—
187Dic106—42no—
188Dic106—42no—
189Dic106—81no—
190Dic106—81no—
191C8×C53—42no—
192C8×C53—81no—
193#88322—81no—
194#9057—531nomaximal, Sylow
195C22×C53#38611yescharacteristic
196C22×C53#44711yescharacteristic
197C22×C53#44911yescharacteristic
198C22×C53#45011yescharacteristic
199C22×C53#45211yescharacteristic
200C22×C53#45511yescharacteristic
201C22×C53#45511yescharacteristic
202C2×D53—44no—
203C2×D53—44no—
204C2×D53—44no—
205C2×D53—82no—
206C2×D53—82no—
207C2×D53—82no—
208C2×D53—82no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—42no—
212C4×C53—81no—
213C4×C53C4×D811yescharacteristic
214C4×C53C4×SD1611yescharacteristic
215C4×C53#44011yescharacteristic
216C4×C53#44211yescharacteristic
217C4×C53#45911yescharacteristic
218C4×C53#46111yescharacteristic
219C4×C53#46311yescharacteristic
220C4×C53#46311yescharacteristic
221Dic53—42no—
222Dic53—82no—
223Dic53—81no—
224C2×#486—531no—
225#2366—531no—
226#1988—531no—
227#1981—531no—
228C2×#331—531no—
229#1279—531no—
230#1279D5311yescharacteristic, complemented
231C2×C53#163911yescharacteristic
232C2×C53#164011yescharacteristic
233C2×C53#164111yescharacteristic
234C2×C53#164211yescharacteristic
235C2×C53#187711yescharacteristic
236C2×C53#187911yescharacteristic
237C2×C53#188511yescharacteristic
238D53—84no—
239C2×C4⋊Q8—531no—
240C2×#127—531no—
241C2×C4×D4—1061no—
242C2×C4×D4—1061no—
243#486—534no—
244#486—1062no—
245C2×C4⋊C8—531no—
246C2×C4⋊C8C2×D5311yescharacteristic
247C2×#103—1061no—
248C2×#103—1061no—
249C2×#102—1062no—
250C2×C4×C8C2×D5311yescharacteristic
251#390—1061no—
252#390—1061no—
253#386—1062no—
254C4×C4⋊C4—531no—
255C4×C4⋊C4C2×D5311yescharacteristic
256#340—1061no—
257#340—1061no—
258#331—1062no—
259C53#905711yescharacteristic, complemented, Sylow
260C22×Q8—1061no—
261C22×D4—532no—
262C22×D4—1061no—
263C22×D4—1061no—
264C23×C4—1062no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—1061no—
268C4⋊Q8—532no—
269C4⋊Q8—1061no—
270C4⋊Q8—1061no—
271C4⋊Q8—1061no—
272#127—532no—
273#127—532no—
274#127—1061no—
275#127—1061no—
276C4×D4—2122no—
277C4×D4—2122no—
278C4×D4—2122no—
279C4×D4—2122no—
280C2×C4⋊C4—531no—
281C2×C4⋊C4—531no—
282C2×C4⋊C4—1061no—
283C2×C4⋊C4—1061no—
284C2×C4⋊C4D21211yescharacteristic
285C2×C4⋊C4D21211yescharacteristic
286C2×C22⋊C4—1062no—
287C2×C22⋊C4—1062no—
288C2×C22⋊C4—1062no—
289C2×C42—21no—
290C2×C42—1061no—
291C2×C42C22×D5311yescharacteristic
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—1061no—
295C4⋊C8—1061no—
296#103—2122no—
297#103—2122no—
298#102—1068no—
299C4×C8—21no—
300C4×C8C4×D5312yescomplemented
301#95—21no—
302#95—1061no—
303C24—1062no—
304C2×Q8—1062no—
305C2×Q8—2121no—
306C2×Q8—2121no—
307C2×Q8—2121no—
308C2×Q8—2121no—
309C2×Q8—2121no—
310C2×D4—534no—
311C2×D4—1062no—
312C2×D4—1062no—
313C2×D4—1062no—
314C2×D4—1062no—
315C2×D4—1062no—
316C2×D4—2122no—
317C2×D4—2122no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2121no—
323C2×D4—2121no—
324C2×D4—2121no—
325C2×D4—2121no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—1061no—
329C22×C4—1061no—
330C22×C4—4242no—
331C22×C4—2124no—
332C22×C4—2124no—
333C22×C4—2121no—
334C22×C4C2×D21211yescharacteristic
335C22×C4D4×D5311yescharacteristic
336C22×C4D4×D5311yescharacteristic
337C2×C8—22no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—2121no—
344C2×C8—2121no—
345C2×C8—2121no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—532no—
350C4⋊C4—1061no—
351C4⋊C4—1062no—
352C4⋊C4—1062no—
353C4⋊C4—1061no—
354C4⋊C4—1061no—
355C4⋊C4—2121no—
356C4⋊C4—2121no—
357C4⋊C4D42412yes—
358C22⋊C4—4242no—
359C22⋊C4—2124no—
360C22⋊C4—2124no—
361C42—41no—
362C42—41no—
363C42—2121no—
364C42—2121no—
365C42C2×C4×D5311yescharacteristic
366C42#1171994511yescharacteristic
367C42D4×D5311yescharacteristic
368C42#1171994911yescharacteristic
369C23—1064no—
370C23—2122no—
371C23—2122no—
372C23—2122no—
373C23—2122no—
374C23—2122no—
375C23—2122no—
376C23#1313318711yescharacteristic
377Q8—4241no—
378Q8—4241no—
379Q8—2122no—
380Q8—2122no—
381D4—4242no—
382D4—2124no—
383D4—2122no—
384D4—2122no—
385D4—2122no—
386D4—2122no—
387D4—2122no—
388D4—2122no—
389D4—2122no—
390D4—2122no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—1062no—
400C2×C4—4248no—
401C2×C4—4241no—
402C2×C4—4241no—
403C2×C4—2122no—
404C2×C4—2121no—
405C2×C4—2121no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4C22⋊C4×D5311yescharacteristic
410C2×C4#1313317911yescharacteristic
411C2×C4C4×D21211yescharacteristic
412C2×C4#1313319011yescharacteristic
413C2×C4#1313319211yescharacteristic
414C2×C4#1313320211yescharacteristic
415C2×C4C2×D42411yescharacteristic
416C2×C4SD16×D5311yescharacteristic
417C2×C4#1313323811yescharacteristic
418C2×C4#1313323911yescharacteristic
419C2×C4#1313324111yescharacteristic
420C2×C4#1313324511yescharacteristic
421C2×C4#1313324511yescharacteristic
422C2×C4#1313324911yescharacteristic
423C8—42no—
424C8—81no—
425C8—4241no—
426C22—4242no—
427C22—4242no—
428C22—4242no—
429C22—4242no—
430C22—2124no—
431C22—2124no—
432C22—2124no—
433C22#1681050111yescharacteristic
434C22#1681075911yescharacteristic
435C22#1681077011yescharacteristic
436C22#1681077511yescharacteristic
437C22#1681078811yescharacteristic
438C22#1681080211yescharacteristic
439C22#1681080211yescharacteristic
440C4—21no—
441C4—21no—
442C4—42no—
443C4—81no—
444C4—4242no—
445C4—4241no—
446C4—2122no—
447C4C4×D42411yescharacteristic
448C4#1681071011yescharacteristic
449C4#1681072211yescharacteristic
450C4#1681073611yescharacteristic
451C4#1681081911yescharacteristic
452C4#1681082711yescharacteristic
453C4#1681083711yescharacteristic
454C4#1681083911yescharacteristic
455C2—4244no—
456C2#2027389611yescharacteristic
457C2#2027390311yescharacteristic
458C2#2027391011yescharacteristic
459C2#2027391711yescharacteristic
460C2#2027505011yescharacteristic
461C2#2027506211yescharacteristic
462C2#2027510211yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1279D53
hypocenterC53#9057
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
3. #20275666C2semidirect
4. #20275638C2semidirect
6. #20272555C2semidirect
7. C53×#1279C2semidirect
13. #16810970C4semidirect
37. #13133217C2×C4semidirect
69. C4×C8×C53C2×C4semidirect
230. #1279D53semidirect
259. C53#9057semidirect
300. C4×C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109702
#202777792
#202756662
#202756382
C2×#168103662
#202725552
C53×#12792
#905753