Finite Group Databasebeta

← #27279662

Subgroups — #27279662

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283046C211yesmaximal, characteristic
2C2×#16810358C212yesmaximal
3#20272942C211yesmaximal, characteristic, complemented
4#20272595C212yesmaximal, complemented
5C2×C53×#431C211yesmaximal, characteristic, complemented
6C2×#13133200—21no—
7C2×#13133191C2211yescharacteristic
8#16810525C411yescharacteristic, complemented
9#16810525C411yescharacteristic, complemented
10#16810521C411yescharacteristic, complemented
11#16810521C411yescharacteristic, complemented
12#16810514—21no—
13#16810514—21no—
14#16810502—21no—
15#16810483C2211yescharacteristic
16C2×#13133124—21no—
17C2×#13133124—21no—
18C2×#13133124C2212yes—
19#16810358—24no—
20C22×C42×C53C2211yescharacteristic
21C53×#431—24no—
22C2×C4⋊C8×C53—21no—
23C2×C22⋊C8×C53C2212yes—
24C2×C4×C8×C53—21no—
25C22×D212C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C23×Dic53—22no—
28#13133200—42no—
29#13133200—42no—
30#13133191—22no—
31#13133191—22no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
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—22no—
48#13133124—22no—
49#13133124—42no—
50C23×C4×C53D411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C23×C4×C53C2311yescharacteristic
53C22×C8×C53—22no—
54C2×C42×C53—22no—
55C2×C42×C53—22no—
56C2×C42×C53C2×C411yescharacteristic
57C2×C42×C53C2×C411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C4⋊C8×C53—24no—
61C22⋊C8×C53—22no—
62C22⋊C8×C53D414yes—
63C4×C8×C53—24no—
64C24×C53C2×D411yescharacteristic
65C23×D53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—22no—
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×C53D812yes—
78C22×C4×C53SD1612yes—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D412yes—
82C22×C4×C53#4112yes—
83C2×C22⋊D53—42no—
84C2×C22⋊D53—41no—
85C2×C22⋊D53—41no—
86C2×C22⋊D53—42no—
87C22×Dic53—24no—
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—42no—
92C22×Dic53—42no—
93C22×Dic53—42no—
94C22×Dic53—42no—
95C22×Dic53—42no—
96C22×Dic53—42no—
97C2×D212—22no—
98C2×D212—41no—
99C2×D212—41no—
100C2×C8×C53—24no—
101C2×C8×C53—42no—
102C2×C8×C53—42no—
103C42×C53—22no—
104C42×C53—22no—
105C42×C53—21no—
106C42×C53—21no—
107C42×C53—21no—
108C42×C53—21no—
109C22⋊Dic53—41no—
110C22⋊Dic53—41no—
111#11719921—42no—
112#11719921—42no—
113#11719921—42no—
114#11719921—42no—
115#11719921—41no—
116#11719921—41no—
117C4⋊Dic53—22no—
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120#11719919—42no—
121#11719919—42no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53C2×C22⋊C411yescharacteristic
127C23×C53#12112yes—
128C23×C53C2×D811yescharacteristic
129C23×C53C2×SD1611yescharacteristic
130C23×C53#13512yes—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
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—24no—
149C2×C4×C53—24no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—44no—
156C2×C4×C53#10212yes—
157C2×C4×C53#10212yes—
158C2×C4×C53#10412yes—
159C2×C4×C53C4×D412yes—
160C2×C4×C53#12011yescharacteristic
161C2×C4×C53#12311yescharacteristic
162C22⋊D53—82no—
163C22⋊D53—82no—
164C2×Dic53—44no—
165C2×Dic53—44no—
166C2×Dic53—44no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—82no—
177C2×Dic53—82no—
178C2×Dic53—82no—
179C2×Dic53—82no—
180D212—82no—
181C8×C53—44no—
182#9114—531nomaximal, Sylow
183C22×C53—22no—
184C22×C53—22no—
185C22×C53—22no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—22no—
191C22×C53#38611yescharacteristic
192C22×C53#41011yescharacteristic
193C22×C53#41011yescharacteristic
194C22×C53C2×#10211yescharacteristic
195C22×C53#41611yescharacteristic
196C22×C53C2×#10411yescharacteristic
197C22×C53#42111yescharacteristic
198C22×C53#46512yes—
199C22×C53#46612yes—
200C2×D53—81no—
201C2×D53—81no—
202C2×D53—81no—
203C2×D53—81no—
204C2×D53—81no—
205C2×D53—81no—
206C2×D53—81no—
207C4×C53—22no—
208C4×C53—22no—
209C4×C53—22no—
210C4×C53—21no—
211C4×C53—22no—
212C4×C53—21no—
213C4×C53—41no—
214Dic53—84no—
215Dic53—81no—
216Dic53—81no—
217Dic53—81no—
218C2×#431D5311yescharacteristic, complemented
219#2854—531no—
220C2×#327—532no—
221#1434—531no—
222#1294—532no—
223C2×C53—22no—
224C2×C53—22no—
225C2×C53#150011yescharacteristic
226C2×C53#150211yescharacteristic
227C2×C53#151711yescharacteristic
228C2×C53#152111yescharacteristic
229C2×C53#187111yescharacteristic
230C2×C53#189311yescharacteristic
231C2×C53#189911yescharacteristic
232D53—161no—
233C2×#123—1061no—
234C2×#121—531no—
235C22×C42C2×D5311yescharacteristic
236#431—24no—
237C2×C4⋊C8—21no—
238C2×C22⋊C8C2×D5312yes—
239C2×C4×C8—21no—
240#390—531no—
241#390—531no—
242#389—531no—
243#389—531no—
244#388—1061no—
245#388—1061no—
246#386—1061no—
247#380—531no—
248C2×#95—532no—
249C2×#95—1061no—
250C2×#95—1061no—
251#327—1064no—
252C53#911411yescharacteristic, complemented, Sylow
253C22×D4—531no—
254C22×D4—1061no—
255C23×C4—1062no—
256C23×C4D21211yescharacteristic
257C23×C4D21211yescharacteristic
258C23×C4C22×D5311yescharacteristic
259C22×C8—22no—
260#123—2122no—
261#123—2122no—
262#121—1062no—
263#121—1062no—
264C2×C4⋊C4—531no—
265C2×C4⋊C4—531no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4—1061no—
268C2×C4⋊C4—1061no—
269C2×C22⋊C4—1061no—
270C2×C22⋊C4—1061no—
271C2×C22⋊C4—1061no—
272C2×C22⋊C4—1061no—
273C2×C22⋊C4—1061no—
274C2×C22⋊C4—1061no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C4×D5311yescharacteristic
278C2×C42C4×D5311yescharacteristic
279C2×C42C22⋊D5311yescharacteristic
280C2×C42C22⋊D5311yescharacteristic
281C4⋊C8—24no—
282C22⋊C8—22no—
283C22⋊C8C22⋊D5314yes—
284C4×C8—24no—
285#95—1064no—
286#95—1061no—
287#95—1061no—
288#95—1061no—
289#95—1062no—
290#95—1062no—
291#95—2122no—
292C24—1061no—
293C24C2×D21211yescharacteristic
294C2×D4—1062no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2122no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2122no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—1064no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4—1061no—
312C22×C4—2122no—
313C22×C4—2122no—
314C22×C4—2122no—
315C22×C4—2122no—
316C22×C4—2122no—
317C22×C4—2122no—
318C22×C4#1171991312yes—
319C22×C4D42412yes—
320C22×C4#1171992111yescharacteristic
321C22×C4#1171992111yescharacteristic
322C22×C4#1171992111yescharacteristic
323C22×C4#1171992111yescharacteristic
324C22×C4C2×C4×D5311yescharacteristic
325C22×C4#1171994512yes—
326C22×C4C2×C22⋊D5311yescharacteristic
327C22×C4C2×C22⋊D5312yes—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—1062no—
332C4⋊C4—2121no—
333C4⋊C4—2121no—
334C4⋊C4—2122no—
335C4⋊C4—2122no—
336C22⋊C4—2122no—
337C22⋊C4—2121no—
338C22⋊C4—2121no—
339C22⋊C4—2122no—
340C22⋊C4—2122no—
341C22⋊C4—2122no—
342C22⋊C4—2121no—
343C22⋊C4—2121no—
344C42—22no—
345C42—22no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—2121no—
355C23—2121no—
356C23—2121no—
357C23—2121no—
358C23—2121no—
359C23—2121no—
360C23—2121no—
361C23C2×#1171992111yescharacteristic
362C23#1313319112yes—
363C23C2×D42411yescharacteristic
364C23C2×#1171991311yescharacteristic
365C23#1313323912yes—
366D4—4242no—
367D4—4242no—
368D4—4242no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—4242no—
388C2×C4—4242no—
389C2×C4—4242no—
390C2×C4—4242no—
391C2×C4—2124no—
392C2×C4—2124no—
393C2×C4—2124no—
394C2×C4—2121no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4#1313313912yes—
404C2×C4#1313313912yes—
405C2×C4#1313314312yes—
406C2×C4C4×C22⋊D5312yes—
407C2×C4#1313318611yescharacteristic
408C2×C4#1313320011yescharacteristic
409C8—44no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—4241no—
419C22—4241no—
420C22—4241no—
421C22—4241no—
422C22—4241no—
423C22—4241no—
424C22—4241no—
425C22#1681050211yescharacteristic
426C22#1681059611yescharacteristic
427C22#1681059611yescharacteristic
428C22C2×#1313313911yescharacteristic
429C22#1681062011yescharacteristic
430C22C2×#1313314311yescharacteristic
431C22#1681063911yescharacteristic
432C22#1681085212yes—
433C22#1681085912yes—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—21no—
438C4—22no—
439C4—21no—
440C4—41no—
441C4—4244no—
442C4—4241no—
443C4—4241no—
444C4—4241no—
445C2—22no—
446C2—22no—
447C2—8481no—
448C2#2027320211yescharacteristic
449C2#2027321111yescharacteristic
450C2#2027327611yescharacteristic
451C2#2027329311yescharacteristic
452C2#2027501711yescharacteristic
453C2#2027514311yescharacteristic
454C2#2027517311yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#431C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenterC2×#431D53
hypocenterC53#9114
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
3. #20272942C2semidirect
4. #20272595C2semidirect
5. C2×C53×#431C2semidirect
8. #16810525C4semidirect
9. #16810525C4semidirect
10. #16810521C4semidirect
11. #16810521C4semidirect
218. C2×#431D53semidirect
252. C53#9114semidirect

Maximal subgroups

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

subgroupindex
#202830462
C2×#168103582
#202729422
#202725952
C2×C53×#4312
#911453