Finite Group Databasebeta

← #27281162

Subgroups — #27281162

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284655C211yesmaximal, characteristic
2#20274810C211yesmaximal, characteristic, complemented
3#20273284C211yesmaximal, characteristic
4#20273076C211yesmaximal, characteristic
5#20272622C211yesmaximal, characteristic, complemented
6#20272603C211yesmaximal, characteristic, complemented
7C53×#2924C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9#16810947—21no—
10#16810947—21no—
11#16810941—21no—
12#16810936—21no—
13#16810666C411yescharacteristic, complemented
14#16810666C411yescharacteristic, complemented
15#16810627C411yescharacteristic, complemented
16#16810627C411yescharacteristic, complemented
17#16810587C2211yescharacteristic
18C2×C22⋊#88322C2211yescharacteristic
19#16810483C2211yescharacteristic
20#16810385—21no—
21#16810385—21no—
22#16810358—21no—
23#16810358—21no—
24#16810346—22no—
25C2×#117×C53C2211yescharacteristic
26C53×#433—22no—
27C53×#432—21no—
28C53×#431—21no—
29C53×#424—21no—
30C53×#407C2211yescharacteristic
31C2×C22⋊C8×C53C2211yescharacteristic
32C53×#405—21no—
33C22×D212C2×C411yescharacteristic
34C22×C22⋊D53—21no—
35C2×#11719917C2×C411yescharacteristic
36C2×#11719917C2×C411yescharacteristic
37C22×#88322—21no—
38#13133191—21no—
39#13133191—21no—
40#13133191—21no—
41#13133191—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53C2×C411yescharacteristic
44C2×C22⋊Dic53—21no—
45C2×#11719921—21no—
46#13133150—21no—
47#13133150—21no—
48#13133148—21no—
49#13133148—21no—
50#13133137—21no—
51#13133137—21no—
52#13133137—21no—
53#13133137—21no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56C22⋊#88322—21no—
57C22⋊#88322—21no—
58#13133124—21no—
59C23×C4×C53C2311yescharacteristic
60C2×M16×C53—21no—
61C22×C8×C53—21no—
62#117×C53—21no—
63#117×C53—21no—
64#117×C53C2×C411yescharacteristic
65#117×C53C2×C411yescharacteristic
66#117×C53D411yescharacteristic
67#117×C53D411yescharacteristic
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53—21no—
71C2×C42×C53—21no—
72C4⋊C8×C53—22no—
73C4⋊C8×C53—22no—
74C22⋊C8×C53—21no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53D412yes—
77C22⋊C8×C53D411yescharacteristic
78C22⋊C8×C53D411yescharacteristic
79#97×C53—22no—
80C4×C8×C53—22no—
81C24×C53#4111yescharacteristic
82C23×D53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53C22×C411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C2×C22⋊D53—41no—
94C2×C22⋊D53—41no—
95C2×C22⋊D53—81no—
96C2×C22⋊D53—81no—
97C22×Dic53—21no—
98C22×Dic53—41no—
99C2×D212—21no—
100C2×D212—21no—
101C2×D212—41no—
102C2×D212—41no—
103M16×C53—41no—
104M16×C53—41no—
105C2×C8×C53—21no—
106C2×C8×C53—22no—
107C2×C8×C53—21no—
108C2×C8×C53—41no—
109C2×C8×C53—41no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53C22⋊C411yescharacteristic
113C4⋊C4×C53C22⋊C411yescharacteristic
114C4⋊C4×C53C22⋊C411yescharacteristic
115C4⋊C4×C53C22⋊C411yescharacteristic
116C22⋊C4×C53—21no—
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C42×C53—21no—
121C42×C53—21no—
122C42×C53—21no—
123C42×C53—21no—
124C22⋊Dic53—81no—
125#11719921—41no—
126#11719921—41no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129C4⋊Dic53—41no—
130C4⋊Dic53—41no—
131#11719917—41no—
132#11719917—41no—
133C2×#88322—21no—
134C2×#88322—21no—
135C2×#88322—41no—
136C2×#88322—41no—
137C2×#88322—41no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53C4×D411yescharacteristic
143C23×C53#12111yescharacteristic
144C23×C53#12311yescharacteristic
145C22×D53—41no—
146C22×D53—42no—
147C22×D53—81no—
148C22×D53—81no—
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—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—81no—
165C2×C4×C53C2×C22⋊C411yescharacteristic
166C2×C4×C53C4×D411yescharacteristic
167C2×C4×C53#12011yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C22⋊D53—82no—
170C22⋊D53—81no—
171C22⋊D53—81no—
172C2×Dic53—41no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175C2×Dic53—81no—
176C2×Dic53—81no—
177D212—41no—
178D212—41no—
179D212—81no—
180C8×C53—41no—
181C8×C53—41no—
182C8×C53—42no—
183#88322—41no—
184#88322—41no—
185#88322—81no—
186#9348—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53#38611yescharacteristic
198C22×C53#47111yescharacteristic
199C22×C53#47211yescharacteristic
200C2×D53—81no—
201C2×D53—81no—
202C2×D53—81no—
203C2×D53—162no—
204C4×C53—21no—
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—41no—
208C4×C53—41no—
209C4×C53—41no—
210C4×C53—41no—
211C4×C53#41011yescharacteristic
212C4×C53#41011yescharacteristic
213Dic53—81no—
214Dic53—161no—
215#3094—531no—
216#2924D5311yescharacteristic, complemented
217#1825—531no—
218#1519—531no—
219#1471—531no—
220#1305—531no—
221#1298—531no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53—41no—
225C2×C53—41no—
226C2×C53#188111yescharacteristic
227C2×C53#189911yescharacteristic
228C2×C53#190711yescharacteristic
229D53—161no—
230C2×#121—531no—
231C2×#117C2×D5311yescharacteristic
232#482—1061no—
233#482—1061no—
234#481—1061no—
235#480—1061no—
236#433—22no—
237#432—21no—
238#431—21no—
239#428—531no—
240#428—531no—
241#424—21no—
242#418—531no—
243#418—531no—
244#407—531no—
245#407C2×D5311yescharacteristic
246C2×C22⋊C8—531no—
247C2×C22⋊C8C2×D5311yescharacteristic
248#405—21no—
249#380—531no—
250#340—1061no—
251#340—1061no—
252#327—1061no—
253#327—1061no—
254#323—1062no—
255C53#934811yescharacteristic, complemented, Sylow
256C22×D4—531no—
257C22×D4—1061no—
258C23×C4C22×D5311yescharacteristic
259C2×M16—21no—
260C2×M16—531no—
261C2×M16—531no—
262C22×C8—21no—
263C22×C8—1061no—
264#121—1061no—
265#121—1061no—
266#121—1061no—
267#121—1061no—
268#117—21no—
269#117—21no—
270#117C4×D5311yescharacteristic
271#117C4×D5311yescharacteristic
272#117C22⋊D5311yescharacteristic
273#117C22⋊D5311yescharacteristic
274C2×C4⋊C4—531no—
275C2×C4⋊C4—1061no—
276C2×C4⋊C4D21211yescharacteristic
277C2×C4⋊C4D21211yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—1061no—
280C2×C22⋊C4—1061no—
281C2×C42—21no—
282#107—1061no—
283#107—1061no—
284#106—1061no—
285#106—1061no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288#102—1061no—
289#102—1061no—
290#102—1061no—
291#102—1061no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—1061no—
295C22⋊C8—1061no—
296C22⋊C8—1061no—
297C22⋊C8—1061no—
298C22⋊C8C22⋊D5312yes—
299C22⋊C8C22⋊D5311yescharacteristic
300C22⋊C8C22⋊D5311yescharacteristic
301#97—22no—
302C4×C8—22no—
303#95—1061no—
304C24—1061no—
305C24#1171994511yescharacteristic
306C2×D4—1061no—
307C2×D4—1061no—
308C2×D4—4241no—
309C2×D4—4241no—
310C2×D4—2121no—
311C2×D4—2121no—
312C2×D4—2121no—
313C2×D4—2121no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—1061no—
319C22×C4—2121no—
320C22×C4C2×C4×D5311yescharacteristic
321C22×C4C2×D21211yescharacteristic
322C22×C4#1171994511yescharacteristic
323C22×C4C2×C22⋊D5311yescharacteristic
324C22×C4C2×C22⋊D5311yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326M16—41no—
327M16—41no—
328M16—2121no—
329M16—2121no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—1061no—
336C2×C8—1061no—
337C2×C8—2121no—
338C2×C8—2121no—
339C2×C8—2121no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—1061no—
343C4⋊C4—1061no—
344C4⋊C4—2121no—
345C4⋊C4—2121no—
346C4⋊C4#1171992111yescharacteristic
347C4⋊C4#1171992111yescharacteristic
348C4⋊C4#1171992111yescharacteristic
349C4⋊C4#1171992111yescharacteristic
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—4241no—
355C22⋊C4—2121no—
356C22⋊C4—2121no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—4241no—
366C23—4241no—
367C23—2122no—
368C23—2121no—
369C23C4×C22⋊D5311yescharacteristic
370C23#1313319111yescharacteristic
371C23#1313320011yescharacteristic
372D4—4241no—
373D4—4242no—
374D4—4241no—
375D4—4241no—
376D4—2121no—
377D4—2121no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—81no—
394C2×C4—4241no—
395C2×C4—4241no—
396C2×C4—4241no—
397C2×C4—4241no—
398C2×C4—2121no—
399C2×C4C2×#1171992111yescharacteristic
400C2×C4C4×C22⋊D5311yescharacteristic
401C2×C4#1313318611yescharacteristic
402C2×C4#1313319111yescharacteristic
403C8—41no—
404C8—41no—
405C8—42no—
406C8—4241no—
407C8—2121no—
408C8—2121no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—81no—
418C22—81no—
419C22—8482no—
420C22—4241no—
421C22—4241no—
422C22—4241no—
423C22#1681050211yescharacteristic
424C22#1681088611yescharacteristic
425C22#1681089311yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—8481no—
434C4—4241no—
435C4#1681059611yescharacteristic
436C4#1681059611yescharacteristic
437C2—21no—
438C2—21no—
439C2—41no—
440C2—41no—
441C2—8481no—
442C2#2027507711yescharacteristic
443C2#2027517311yescharacteristic
444C2#2027522311yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2924C2
socleC22×C53#386
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#2924D53
hypocenterC53#9348
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2924

Complements

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

NH ≅ G/Ntype
2. #20274810C2semidirect
5. #20272622C2semidirect
6. #20272603C2semidirect
7. C53×#2924C2semidirect
13. #16810666C4semidirect
14. #16810666C4semidirect
15. #16810627C4semidirect
16. #16810627C4semidirect
216. #2924D53semidirect
255. C53#9348semidirect

Maximal subgroups

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

subgroupindex
#202846552
#202748102
#202732842
#202730762
#202726222
#202726032
C53×#29242
#934853