Finite Group Databasebeta

← #27283079

Subgroups — #27283079

Order 13568.

668 conjugacy classes of subgroups.

The subgroup lattice has 444 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#20272942C211yesmaximal, characteristic, complemented
5C4×C22⋊C8×C53C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic, complemented
7C2×#13133196C2211yescharacteristic
8C2×#13133191C2211yescharacteristic
9#16810529C411yescharacteristic, complemented
10#16810525C411yescharacteristic, complemented
11#16810514—21no—
12#16810514—21no—
13#16810502—21no—
14#16810502—21no—
15#16810489C411yescharacteristic, complemented
16C2×#13133124—21no—
17C2×#13133124—21no—
18C2×#13133124C2212yes—
19#16810361—24no—
20#16810358—24no—
21C22×C42×C53C2211yescharacteristic
22C2×C22⋊C8×C53C2212yes—
23C2×C4×C8×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⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43#13133124—24no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—22no—
49#13133124—22no—
50#13133124—42no—
51C23×C4×C53D411yescharacteristic
52C23×C4×C53D411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C22×C8×C53—22no—
55C2×C42×C53—42no—
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53C2×C411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C22⋊C8×C53—22no—
61C22⋊C8×C53D414yes—
62C4×C8×C53—42no—
63C24×C53C2×D411yescharacteristic
64C23×D53—21no—
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53—41no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D412yes—
80C22×C4×C53#4112yes—
81C2×C22⋊D53—41no—
82C2×C22⋊D53—41no—
83C2×C22⋊D53—42no—
84C2×C22⋊D53—42no—
85C22×Dic53—24no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—42no—
90C22×Dic53—42no—
91C22×Dic53—42no—
92C22×Dic53—42no—
93C22×Dic53—42no—
94C22×Dic53—42no—
95C2×D212—22no—
96C2×D212—41no—
97C2×D212—41no—
98C2×Dic106—22no—
99C2×Dic106—41no—
100C2×Dic106—41no—
101C2×C8×C53—24no—
102C2×C8×C53—42no—
103C2×C8×C53—42no—
104C42×C53—22no—
105C42×C53—22no—
106C42×C53—41no—
107C42×C53—41no—
108C22⋊Dic53—41no—
109C22⋊Dic53—41no—
110#11719921—42no—
111#11719921—42no—
112#11719921—42no—
113#11719921—42no—
114#11719921—42no—
115C4⋊Dic53—22no—
116C4⋊Dic53—22no—
117#11719919—42no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53C2×C22⋊C411yescharacteristic
123C23×C53#12411yescharacteristic
124C23×C53#12711yescharacteristic
125C23×C53#13512yes—
126C23×C53#13512yes—
127C22×D53—41no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—41no—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—41no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—44no—
151C2×C4×C53—44no—
152C2×C4×C53#9912yes—
153C2×C4×C53#10412yes—
154C2×C4×C53C4×D412yes—
155C2×C4×C53#12112yes—
156C22⋊D53—82no—
157C22⋊D53—82no—
158C2×Dic53—44no—
159C2×Dic53—44no—
160C2×Dic53—44no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—82no—
171C2×Dic53—82no—
172C2×Dic53—82no—
173C2×Dic53—82no—
174D212—82no—
175Dic106—82no—
176C8×C53—44no—
177#9681—531nomaximal, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—22no—
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53#39011yescharacteristic
187C22×C53C2×#9911yescharacteristic
188C22×C53#41011yescharacteristic
189C22×C53#41611yescharacteristic
190C22×C53#41611yescharacteristic
191C22×C53C2×#10411yescharacteristic
192C22×C53#42111yescharacteristic
193C22×C53#48812yes—
194C22×C53#49512yes—
195C2×D53—81no—
196C2×D53—81no—
197C2×D53—81no—
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—81no—
201C2×D53—81no—
202C4×C53—22no—
203C4×C53—22no—
204C4×C53—22no—
205C4×C53—22no—
206C4×C53—41no—
207C4×C53—41no—
208Dic53—84no—
209Dic53—81no—
210Dic53—81no—
211Dic53—81no—
212#2850—531no—
213C4×C22⋊C8D5311yescharacteristic, complemented
214C2×#328—532no—
215C2×#327—532no—
216#1434—531no—
217C2×C53—22no—
218C2×C53—22no—
219C2×C53#150011yescharacteristic
220C2×C53#150111yescharacteristic
221C2×C53#151611yescharacteristic
222C2×C53#152111yescharacteristic
223C2×C53#195811yescharacteristic
224C2×C53#196411yescharacteristic
225C2×C53#198011yescharacteristic
226D53—161no—
227C2×#124—531no—
228C2×C22⋊Q8—531no—
229C2×#121—531no—
230C22×C42C2×D5311yescharacteristic
231C2×C22⋊C8C2×D5312yes—
232C2×C4×C8—21no—
233#391—531no—
234#390—531no—
235#388—1061no—
236#388—1061no—
237#386—1061no—
238#386—1061no—
239#382—531no—
240C2×#95—532no—
241C2×#95—1061no—
242C2×#95—1061no—
243#336—21no—
244#328—1064no—
245#327—1064no—
246C53#968111yescharacteristic, complemented, Sylow
247C22×Q8—531no—
248C22×D4—531no—
249C22×D4—1061no—
250C23×C4—1062no—
251C23×C4D21211yescharacteristic
252C23×C4D21211yescharacteristic
253C23×C4C22×D5311yescharacteristic
254C22×C8—22no—
255#124—2122no—
256C22⋊Q8—1062no—
257C22⋊Q8—1062no—
258#121—1062no—
259#121—1062no—
260C2×C4⋊C4—531no—
261C2×C4⋊C4—531no—
262C2×C4⋊C4—1061no—
263C2×C22⋊C4—1061no—
264C2×C22⋊C4—1061no—
265C2×C22⋊C4—1061no—
266C2×C22⋊C4—1061no—
267C2×C22⋊C4—1061no—
268C2×C22⋊C4—1061no—
269C2×C42—42no—
270C2×C42C4×D5311yescharacteristic
271C2×C42C4×D5311yescharacteristic
272C2×C42C22⋊D5311yescharacteristic
273C2×C42C22⋊D5311yescharacteristic
274C22⋊C8—22no—
275C22⋊C8D21214yes—
276C4×C8—42no—
277#95—1064no—
278#95—1061no—
279#95—1061no—
280#95—1061no—
281#95—1061no—
282#95—1062no—
283#95—1062no—
284#95—2122no—
285C24—1061no—
286C24C2×D21211yescharacteristic
287C2×Q8—1062no—
288C2×Q8—2121no—
289C2×Q8—2121no—
290C2×D4—1062no—
291C2×D4—2121no—
292C2×D4—2121no—
293C2×D4—2121no—
294C2×D4—2121no—
295C2×D4—2122no—
296C2×D4—2122no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—1064no—
306C22×C4—1061no—
307C22×C4—1061no—
308C22×C4—1061no—
309C22×C4—2122no—
310C22×C4—2122no—
311C22×C4—2122no—
312C22×C4—2122no—
313C22×C4—2122no—
314C22×C4—2122no—
315C22×C4#1171992111yescharacteristic
316C22×C4#1171992111yescharacteristic
317C22×C4#1171992111yescharacteristic
318C22×C4#1171992111yescharacteristic
319C22×C4C2×C4×D5311yescharacteristic
320C22×C4C2×D21212yes—
321C22×C4#1171994512yes—
322C22×C4C2×C22⋊D5311yescharacteristic
323C2×C8—24no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—1062no—
327C4⋊C4—1062no—
328C4⋊C4—2122no—
329C22⋊C4—2122no—
330C22⋊C4—2122no—
331C22⋊C4—2121no—
332C22⋊C4—2121no—
333C22⋊C4—2122no—
334C22⋊C4—2122no—
335C22⋊C4—2122no—
336C42—22no—
337C42—22no—
338C42—41no—
339C42—41no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—2121no—
345C23—2121no—
346C23—2121no—
347C23—2121no—
348C23—2121no—
349C23—2121no—
350C23—2121no—
351C23C2×#1171992111yescharacteristic
352C23#1313320511yescharacteristic
353C23#1313321411yescharacteristic
354C23#1313323912yes—
355C23#1313323912yes—
356Q8—4242no—
357D4—4242no—
358D4—4242no—
359D4—4242no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—4242no—
379C2×C4—4242no—
380C2×C4—4242no—
381C2×C4—4242no—
382C2×C4—2124no—
383C2×C4—2124no—
384C2×C4—2124no—
385C2×C4—2121no—
386C2×C4—2121no—
387C2×C4—2121no—
388C2×C4—2121no—
389C2×C4—2121no—
390C2×C4—2121no—
391C2×C4—2121no—
392C2×C4—2121no—
393C2×C4—2121no—
394C2×C4#1313313112yes—
395C2×C4#1313314312yes—
396C2×C4C4×D21212yes—
397C2×C4#1313319112yes—
398C8—44no—
399C22—22no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—4241no—
408C22—4241no—
409C22—4241no—
410C22—4241no—
411C22—4241no—
412C22—4241no—
413C22—4241no—
414C22#1681052511yescharacteristic
415C22C2×#1313313111yescharacteristic
416C22#1681059611yescharacteristic
417C22#1681062011yescharacteristic
418C22#1681062011yescharacteristic
419C22C2×#1313314311yescharacteristic
420C22#1681063911yescharacteristic
421C22#1681097812yes—
422C22#1681100712yes—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—4244no—
430C4—4241no—
431C4—4241no—
432C4—4241no—
433C2—22no—
434C2—22no—
435C2—8481no—
436C2#2027320211yescharacteristic
437C2#2027320711yescharacteristic
438C2#2027327211yescharacteristic
439C2#2027329311yescharacteristic
440C2#2027552611yescharacteristic
441C2#2027555111yescharacteristic
442C2#2027563411yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC4×C22⋊C8×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenterC4×C22⋊C8D53
hypocenterC53#9681
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16810361C2semidirect
4. #20272942C2semidirect
5. C4×C22⋊C8×C53C2semidirect
6. C2×#13133205C4semidirect
9. #16810529C4semidirect
10. #16810525C4semidirect
15. #16810489C4semidirect
213. C4×C22⋊C8D53semidirect
246. C53#9681semidirect

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
#202729422
C4×C22⋊C8×C532
#968153