Finite Group Databasebeta

← #27283084

Subgroups — #27283084

Order 13568.

700 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283016C211yesmaximal, characteristic
2C2×#16810361C212yesmaximal, complemented
3C2×#16810358C212yesmaximal
4#20272945C211yesmaximal, characteristic, complemented
5C53×#1749C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic, complemented
7C2×#13133196C2211yescharacteristic
8C2×#13133191C2211yescharacteristic
9C22×C4⋊Dic53—21no—
10#16810529C411yescharacteristic, complemented
11#16810525C411yescharacteristic, complemented
12#16810514—21no—
13#16810514—21no—
14#16810502—21no—
15#16810502—21no—
16#16810489C411yescharacteristic, complemented
17C2×#13133124—21no—
18C2×#13133124C2212yes—
19#16810361—24no—
20#16810358—24no—
21C22×C42×C53C2211yescharacteristic
22C2×C22⋊C8×C53C2212yes—
23C2×#97×C53—21no—
24C53×#336—21no—
25C22×Dic106C2×C411yescharacteristic
26C22×D212C2×C411yescharacteristic
27C22×C22⋊D53—21no—
28C23×Dic53—22no—
29#13133205—42no—
30#13133196—22no—
31#13133196—22no—
32#13133191—22no—
33#13133191—22no—
34C2×C4⋊Dic53—24no—
35C2×C4⋊Dic53—42no—
36C2×C4⋊Dic53—42no—
37C2×C4⋊Dic53C2×C411yescharacteristic
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719919—21no—
40C2×C22⋊Dic53—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×#11719921—21no—
44C2×#11719921—21no—
45C2×#11719921—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—22no—
51#13133124—22no—
52#13133124—42no—
53C23×C4×C53D411yescharacteristic
54C23×C4×C53D411yescharacteristic
55C23×C4×C53C2311yescharacteristic
56C22×C8×C53—22no—
57C2×C42×C53—42no—
58C2×C42×C53C2×C411yescharacteristic
59C2×C42×C53C2×C411yescharacteristic
60C2×C42×C53D411yescharacteristic
61C2×C42×C53D411yescharacteristic
62C22⋊C8×C53—22no—
63C22⋊C8×C53D414yes—
64#97×C53—42no—
65C24×C53C2×D411yescharacteristic
66C23×D53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—22no—
70C22×C4×C53—22no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D412yes—
84C22×C4×C53#4112yes—
85C2×C22⋊D53—41no—
86C2×C22⋊D53—41no—
87C2×C22⋊D53—41no—
88C2×C22⋊D53—41no—
89C2×C22⋊D53—41no—
90C2×C22⋊D53—41no—
91C22×Dic53—24no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—21no—
95C22×Dic53—42no—
96C22×Dic53—42no—
97C22×Dic53—42no—
98C22×Dic53—42no—
99C22×Dic53—42no—
100C22×Dic53—42no—
101C2×D212—22no—
102C2×D212—41no—
103C2×D212—41no—
104C2×Dic106—22no—
105C2×Dic106—41no—
106C2×Dic106—41no—
107C2×C8×C53—24no—
108C2×C8×C53—42no—
109C2×C8×C53—42no—
110C42×C53—22no—
111C42×C53—22no—
112C42×C53—41no—
113C42×C53—41no—
114C22⋊Dic53—41no—
115C22⋊Dic53—41no—
116#11719921—42no—
117#11719921—42no—
118#11719921—42no—
119#11719921—42no—
120#11719921—42no—
121C4⋊Dic53—22no—
122C4⋊Dic53—22no—
123C4⋊Dic53—48no—
124#11719919—42no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53C2×C22⋊C411yescharacteristic
130C23×C53#12411yescharacteristic
131C23×C53#12711yescharacteristic
132C23×C53AGL(1,ℤ/8)11yescharacteristic
133C23×C53AGL(1,ℤ/8)11yescharacteristic
134C23×C53#13711yescharacteristic
135C23×C53#13711yescharacteristic
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139C22×D53—41no—
140C22×D53—41no—
141C22×D53—41no—
142C22×D53—41no—
143C2×C4×C53—24no—
144C2×C4×C53—24no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53—42no—
162C2×C4×C53—42no—
163C2×C4×C53#9912yes—
164C2×C4×C53#10412yes—
165C2×C4×C53C4×D412yes—
166C2×C4×C53#12112yes—
167C22⋊D53—81no—
168C22⋊D53—81no—
169C22⋊D53—81no—
170C22⋊D53—81no—
171C2×Dic53—44no—
172C2×Dic53—44no—
173C2×Dic53—44no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—41no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—41no—
182C2×Dic53—41no—
183C2×Dic53—82no—
184C2×Dic53—82no—
185C2×Dic53—82no—
186C2×Dic53—82no—
187D212—82no—
188Dic106—82no—
189C8×C53—44no—
190#9682—531nomaximal, Sylow
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53#39011yescharacteristic
204C22×C53C2×#9911yescharacteristic
205C22×C53#41011yescharacteristic
206C22×C53#41711yescharacteristic
207C22×C53#41711yescharacteristic
208C22×C53C2×#10411yescharacteristic
209C22×C53#42111yescharacteristic
210C22×C53#48911yescharacteristic
211C22×C53#48911yescharacteristic
212C22×C53#49611yescharacteristic
213C22×C53#49711yescharacteristic
214C2×D53—81no—
215C2×D53—81no—
216C2×D53—81no—
217C2×D53—81no—
218C2×D53—81no—
219C2×D53—81no—
220C2×D53—81no—
221C4×C53—22no—
222C4×C53—22no—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—41no—
226C4×C53—41no—
227Dic53—84no—
228Dic53—81no—
229Dic53—81no—
230Dic53—81no—
231#2850—531no—
232#1749D5311yescharacteristic, complemented
233C2×#328—532no—
234C2×#327—532no—
235C4⋊#95—531no—
236C2×C53—21no—
237C2×C53—21no—
238C2×C53—21no—
239C2×C53—21no—
240C2×C53#150411yescharacteristic
241C2×C53#150911yescharacteristic
242C2×C53#151211yescharacteristic
243C2×C53#151311yescharacteristic
244C2×C53#195911yescharacteristic
245C2×C53#196411yescharacteristic
246C2×C53#198011yescharacteristic
247D53—161no—
248C2×#124—531no—
249C2×C22⋊Q8—531no—
250C2×#121—531no—
251C22×C4⋊C4—1061no—
252C22×C42C2×D5311yescharacteristic
253C2×C22⋊C8C2×D5312yes—
254C2×#97—21no—
255#391—531no—
256#390—531no—
257#388—1061no—
258#388—1061no—
259#386—1061no—
260#386—1061no—
261#382—531no—
262C2×#95—532no—
263C2×#95—1061no—
264#336—21no—
265#328—1064no—
266#327—1064no—
267C53#968211yescharacteristic, complemented, Sylow
268C22×Q8—531no—
269C22×D4—531no—
270C22×D4—1061no—
271C23×C4—1062no—
272C23×C4D21211yescharacteristic
273C23×C4D21211yescharacteristic
274C23×C4C22×D5311yescharacteristic
275C22×C8—22no—
276#124—2122no—
277C22⋊Q8—1062no—
278C22⋊Q8—1062no—
279#121—1062no—
280#121—1062no—
281C2×C4⋊C4—531no—
282C2×C4⋊C4—531no—
283C2×C4⋊C4—1064no—
284C2×C4⋊C4—1061no—
285C2×C4⋊C4—2122no—
286C2×C4⋊C4—2122no—
287C2×C22⋊C4—1061no—
288C2×C22⋊C4—1061no—
289C2×C22⋊C4—1061no—
290C2×C22⋊C4—1061no—
291C2×C22⋊C4—1061no—
292C2×C22⋊C4—1061no—
293C2×C42—42no—
294C2×C42C4×D5311yescharacteristic
295C2×C42C4×D5311yescharacteristic
296C2×C42C22⋊D5311yescharacteristic
297C2×C42C22⋊D5311yescharacteristic
298C22⋊C8—22no—
299C22⋊C8D21214yes—
300#97—42no—
301#95—1061no—
302#95—1061no—
303#95—1061no—
304#95—1061no—
305#95—1062no—
306#95—1062no—
307#95—2122no—
308C24—1061no—
309C24C2×D21211yescharacteristic
310C2×Q8—1062no—
311C2×Q8—2121no—
312C2×Q8—2121no—
313C2×D4—1062no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C2×D4—2121no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—1064no—
333C22×C4—1061no—
334C22×C4—1061no—
335C22×C4—1061no—
336C22×C4—2122no—
337C22×C4—2122no—
338C22×C4—2122no—
339C22×C4—2122no—
340C22×C4—2122no—
341C22×C4—2122no—
342C22×C4#1171992111yescharacteristic
343C22×C4#1171992111yescharacteristic
344C22×C4#1171992111yescharacteristic
345C22×C4#1171992111yescharacteristic
346C22×C4C2×C4×D5311yescharacteristic
347C22×C4C2×D21212yes—
348C22×C4#1171994512yes—
349C22×C4C2×C22⋊D5311yescharacteristic
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—1062no—
354C4⋊C4—1062no—
355C4⋊C4—2128no—
356C4⋊C4—2122no—
357C22⋊C4—2122no—
358C22⋊C4—2122no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C22⋊C4—2122no—
362C22⋊C4—2122no—
363C22⋊C4—2122no—
364C42—22no—
365C42—22no—
366C42—41no—
367C42—41no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—2121no—
373C23—2121no—
374C23—2121no—
375C23—2121no—
376C23—2121no—
377C23—2121no—
378C23—2121no—
379C23C2×#1171992111yescharacteristic
380C23#1313320511yescharacteristic
381C23#1313321411yescharacteristic
382C23#1313324511yescharacteristic
383C23#1313324511yescharacteristic
384C23#1313325011yescharacteristic
385C23#1313325011yescharacteristic
386Q8—4242no—
387D4—4242no—
388D4—4241no—
389D4—4241no—
390D4—4241no—
391D4—4241no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—4242no—
413C2×C4—4242no—
414C2×C4—4242no—
415C2×C4—4242no—
416C2×C4—2124no—
417C2×C4—2124no—
418C2×C4—2124no—
419C2×C4—2121no—
420C2×C4—2121no—
421C2×C4—2121no—
422C2×C4—2121no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4—2121no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4#1313313112yes—
429C2×C4#1313314312yes—
430C2×C4C4×D21212yes—
431C2×C4#1313319112yes—
432C8—44no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—4241no—
446C22—4241no—
447C22—4241no—
448C22—4241no—
449C22—4241no—
450C22—4241no—
451C22—4241no—
452C22#1681052511yescharacteristic
453C22C2×#1313313111yescharacteristic
454C22#1681059611yescharacteristic
455C22#1681062511yescharacteristic
456C22#1681062511yescharacteristic
457C22C2×#1313314311yescharacteristic
458C22#1681063911yescharacteristic
459C22#1681098311yescharacteristic
460C22#1681098311yescharacteristic
461C22#1681101211yescharacteristic
462C22#1681101711yescharacteristic
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—41no—
468C4—41no—
469C4—4244no—
470C4—4241no—
471C4—4241no—
472C4—4241no—
473C2—21no—
474C2—21no—
475C2—21no—
476C2—21no—
477C2—8481no—
478C2#2027322011yescharacteristic
479C2#2027324111yescharacteristic
480C2#2027325411yescharacteristic
481C2#2027325911yescharacteristic
482C2#2027553111yescharacteristic
483C2#2027555111yescharacteristic
484C2#2027563411yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1749C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1749D53
hypocenterC53#9682
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C2×#16810361C2semidirect
4. #20272945C2semidirect
5. C53×#1749C2semidirect
6. C2×#13133205C4semidirect
10. #16810529C4semidirect
11. #16810525C4semidirect
16. #16810489C4semidirect
232. #1749D53semidirect
267. C53#9682semidirect

Maximal subgroups

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

subgroupindex
#202830162
C2×#168103612
C2×#168103582
#202729452
C53×#17492
#968253