Finite Group Databasebeta

← #27279344

Subgroups — #27279344

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×#16810971C211yesmaximal, characteristic
2#20277779C211yesmaximal, characteristic
3#20275639C211yesmaximal, characteristic, complemented
4C2×#16810366C211yesmaximal, characteristic
5#20272556C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C53×#1988C211yesmaximal, characteristic, complemented
8C2×#13133214C2211yescharacteristic
9C2×C4×D212—21no—
10C2×#13133179—21no—
11#16810971—22no—
12#16810971C414yescomplemented
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133141—21no—
15C2×#13133141—21no—
16C2×#13133137—22no—
17C2×C4×#88322C2211yescharacteristic
18#16810525—21no—
19#16810524—21no—
20#16810501—22no—
21C4×C4⋊Dic53C2211yescharacteristic
22#16810366—22no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C4×C4⋊C4×C53C2211yescharacteristic
26C53×#340—21no—
27C53×#340—21no—
28C22×D212—21no—
29C22×D212—21no—
30C22×D212D412yes—
31C22×C4×D53—22no—
32C22×#88322—21no—
33C22×#88322—21no—
34#13133214—21no—
35#13133214—21no—
36#13133214C2×C412yes—
37#13133214D412yes—
38C4×D212—42no—
39C4×D212—42no—
40#13133179—42no—
41#13133179—42no—
42C2×C4⋊Dic53D411yescharacteristic
43C2×C4⋊Dic53D411yescharacteristic
44C2×#11719921—22no—
45C2×#11719921—22no—
46C2×#11719921—22no—
47C2×C4×Dic53—21no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50#13133141—42no—
51#13133141—42no—
52#13133137—28no—
53C4×#88322—21no—
54C4×#88322C2×C412yescomplemented
55#13133124—21no—
56C22×Q8×C53—21no—
57C22×C8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53C2×C412yescomplemented
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70#95×C53—21no—
71C23×D53—22no—
72C2×Q8×C53—22no—
73C2×Q8×C53—41no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—41no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×Dic53—21no—
85C22×Dic53—21no—
86C2×D212—22no—
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—22no—
90C2×D212—22no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—42no—
100C2×D212—42no—
101C2×D212C22⋊C414yes—
102C2×C4×D53—44no—
103C2×C4×D53—44no—
104C2×C4×D53—82no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53SD1612yes—
116C42×C53—41no—
117C42×C53—41no—
118C42×C53C22×C411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121C42×C53#4111yescharacteristic
122#11719921—44no—
123#11719921—44no—
124#11719921—82no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127C4⋊Dic53—21no—
128C4⋊Dic53D812yes—
129C4×Dic53—41no—
130C4×Dic53—41no—
131C2×#88322—22no—
132C2×#88322—41no—
133C2×#88322—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—
145Q8×C53—42no—
146Q8×C53—42no—
147Q8×C53—81no—
148Q8×C53—81no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—22no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—81no—
159C2×C4×C53—81no—
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12311yescharacteristic
166C2×C4×C53C2×D811yescharacteristic
167C2×C4×C53C2×SD1611yescharacteristic
168C2×C4×C53#13511yescharacteristic
169C2×C4×C53#13511yescharacteristic
170C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C53#13711yescharacteristic
174C2×Dic53—22no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180D212—44no—
181D212—42no—
182D212—42no—
183D212—42no—
184D212—42no—
185D212—42no—
186D212—42no—
187D212—42no—
188D212—42no—
189D212—82no—
190C4×D53—88no—
191C8×C53—81no—
192#88322—42no—
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—82no—
213C4×C53—81no—
214C4×C53C4×D811yescharacteristic
215C4×C53C4×SD1611yescharacteristic
216C4×C53#44011yescharacteristic
217C4×C53#44211yescharacteristic
218C4×C53#45911yescharacteristic
219C4×C53#46111yescharacteristic
220C4×C53#46311yescharacteristic
221C4×C53#46311yescharacteristic
222Dic53—42no—
223Dic53—81no—
224C2×#486—531no—
225#2366—531no—
226#1988D5311yescharacteristic, complemented
227#1981—531no—
228C2×#331—531no—
229#1279—531no—
230#1279—531no—
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⋊Q8C2×D5311yescharacteristic
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×C8—531no—
251#390—1061no—
252#390—1061no—
253#386—1062no—
254C4×C4⋊C4—531no—
255C4×C4⋊C4C2×D5311yescharacteristic
256#340—21no—
257#340—21no—
258#331—1062no—
259C53#905711yescharacteristic, complemented, Sylow
260C22×Q8—21no—
261C22×D4—532no—
262C22×D4—1061no—
263C22×D4—1061no—
264C23×C4—1062no—
265C22×C8—21no—
266C22×C8—1061no—
267C22×C8—1061no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8C4×D5312yescomplemented
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—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—531no—
283C2×C4⋊C4—531no—
284C2×C4⋊C4C22⋊D5311yescharacteristic
285C2×C4⋊C4C22⋊D5311yescharacteristic
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—532no—
300C4×C8—1061no—
301#95—21no—
302#95—1061no—
303C24—1062no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
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—41no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4—4242no—
332C22×C4—2124no—
333C22×C4—2124no—
334C22×C4D4×D5311yescharacteristic
335C22×C4D4×D5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—1062no—
341C2×C8—2121no—
342C2×C8—2121no—
343C2×C8—2121no—
344C2×C8—2121no—
345C2×C8—2121no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—532no—
354C4⋊C4—1061no—
355C4⋊C4—1061no—
356C4⋊C4—1061no—
357C4⋊C4#1171992412yes—
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#1171994711yescharacteristic
369C23—1064no—
370C23—2122no—
371C23—2122no—
372C23—2122no—
373C23—2122no—
374C23—2122no—
375C23—2122no—
376C23#1313318511yescharacteristic
377Q8—42no—
378Q8—42no—
379Q8—81no—
380Q8—81no—
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—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—1062no—
403C2×C4—4248no—
404C2×C4—2121no—
405C2×C4—2121no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4C22⋊C4×D5311yescharacteristic
410C2×C4#1313317711yescharacteristic
411C2×C4C4×C22⋊D5311yescharacteristic
412C2×C4#1313318811yescharacteristic
413C2×C4#1313319011yescharacteristic
414C2×C4#1313319911yescharacteristic
415C2×C4D8×D5311yescharacteristic
416C2×C4C2×#1171992411yescharacteristic
417C2×C4#1313323711yescharacteristic
418C2×C4#1313323811yescharacteristic
419C2×C4#1313324111yescharacteristic
420C2×C4#1313324211yescharacteristic
421C2×C4#1313324411yescharacteristic
422C2×C4#1313324811yescharacteristic
423C8—81no—
424C8—4241no—
425C8—2122no—
426C22—4242no—
427C22—4242no—
428C22—4242no—
429C22—4242no—
430C22—2124no—
431C22—2124no—
432C22—2124no—
433C22#1681049911yescharacteristic
434C22#1681075511yescharacteristic
435C22#1681076711yescharacteristic
436C22#1681077311yescharacteristic
437C22#1681078311yescharacteristic
438C22#1681079911yescharacteristic
439C22#1681079911yescharacteristic
440C4—21no—
441C4—21no—
442C4—42no—
443C4—82no—
444C4—81no—
445C4—4241no—
446C4—2122no—
447C4#1681070311yescharacteristic
448C4C4×#1171992411yescharacteristic
449C4#1681072111yescharacteristic
450C4#1681073211yescharacteristic
451C4#1681081711yescharacteristic
452C4#1681082511yescharacteristic
453C4#1681083511yescharacteristic
454C4#1681083711yescharacteristic
455C2—4244no—
456C2#2027389311yescharacteristic
457C2#2027390011yescharacteristic
458C2#2027390711yescharacteristic
459C2#2027391411yescharacteristic
460C2#2027505211yescharacteristic
461C2#2027506011yescharacteristic
462C2#2027510011yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1988C2
socleC23×C53#120
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1988D53
hypocenterC53#9057
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
3. #20275639C2semidirect
5. #20272556C2semidirect
6. #20272554C2semidirect
7. C53×#1988C2semidirect
12. #16810971C4semidirect
54. C4×#88322C2×C4semidirect
61. C4⋊Q8×C53C2×C4semidirect
226. #1988D53semidirect
259. C53#9057semidirect
271. C4⋊Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
#202777792
#202756392
C2×#168103662
#202725562
#202725542
C53×#19882
#905753