Finite Group Databasebeta

← #27280192

Subgroups — #27280192

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284497C211yesmaximal, characteristic
2#20283681C211yesmaximal, characteristic
3#20274732C211yesmaximal, characteristic
4#20273214C211yesmaximal, characteristic, complemented
5#20272610C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1493C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810927—21no—
10#16810920—21no—
11#16810915C411yescharacteristic, complemented
12#16810910C411yescharacteristic, complemented
13#16810905C411yescharacteristic, complemented
14#16810900C411yescharacteristic, complemented
15#16810753—21no—
16#16810702—21no—
17#16810694—22no—
18#16810676C2211yescharacteristic
19#16810674—21no—
20#16810659—21no—
21C2×C4⋊#88322C2211yescharacteristic
22C2×#13133143—21no—
23#16810617—21no—
24C4×#11719912—21no—
25C4×C4⋊Dic53C2211yescharacteristic
26#16810384—21no—
27#16810360—22no—
28C53×#533C2211yescharacteristic
29C53×#423C2211yescharacteristic
30C2×C4×C8×C53C2211yescharacteristic
31C53×Q8⋊C8—21no—
32C53×Q8⋊C8—21no—
33C2×#11719945D411yescharacteristic
34C2×#11719912—22no—
35C22×#88322—21no—
36C4×Dic106C2×C411yescharacteristic
37C4×Dic106D411yescharacteristic
38C4×D212C2×C411yescharacteristic
39C4×D212D411yescharacteristic
40C4×C22⋊D53—21no—
41C4×C22⋊D53—21no—
42#13133176D411yescharacteristic
43#13133174—21no—
44C2×C4⋊Dic53—21no—
45C2×C4×Dic53—21no—
46C42×D53—21no—
47#13133150—21no—
48#13133150—21no—
49#13133148—21no—
50#13133148—21no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322C2×C411yescharacteristic
54C4⋊#88322C2×C411yescharacteristic
55#13133146—22no—
56#13133143—42no—
57#13133140—21no—
58#13133140—21no—
59#13133137—21no—
60#13133137—21no—
61C22⋊#88322D412yes—
62#13133129—22no—
63#13133127—22no—
64C4×#88322—21no—
65#13133124—21no—
66C2×M16×C53—21no—
67C22×C8×C53—21no—
68C4⋊Q8×C53C2×C411yescharacteristic
69C4⋊Q8×C53C2×C411yescharacteristic
70#125×C53—21no—
71C22⋊Q8×C53—21no—
72C22⋊Q8×C53—21no—
73C4×Q8×C53—21no—
74C4×Q8×C53—21no—
75#117×C53—21no—
76C2×C42×C53C2311yescharacteristic
77C4⋊C8×C53—21no—
78C4⋊C8×C53—21no—
79C4×C8×C53—21no—
80C4×C8×C53D411yescharacteristic
81C4×C8×C53D411yescharacteristic
82C2×Q8×C53—21no—
83C2×Q8×C53—21no—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C2×C22⋊D53—21no—
88C22×Dic53—41no—
89#11719945—21no—
90#11719945—21no—
91#11719945—21no—
92#11719945—21no—
93C2×D212C22⋊C411yescharacteristic
94C2×C4×D53—21no—
95C2×C4×D53—21no—
96C2×C4×D53—21no—
97C2×Dic106C22⋊C411yescharacteristic
98M16×C53—41no—
99M16×C53—41no—
100C2×C8×C53—21no—
101C2×C8×C53—21no—
102C2×C8×C53—42no—
103C2×C8×C53—41no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C42×C53—41no—
112C42×C53C22×C411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53#4111yescharacteristic
116C22⋊Dic53—21no—
117#11719921—21no—
118#11719921—21no—
119C4⋊Dic53—22no—
120C4⋊Dic53—41no—
121C4⋊Dic53C22⋊C411yescharacteristic
122C4⋊Dic53C22⋊C411yescharacteristic
123#11719919—21no—
124#11719919—21no—
125C4×Dic53—21no—
126C4×Dic53—21no—
127C4×Dic53—21no—
128C4×Dic53—41no—
129C4×Dic53—41no—
130C2×#88322—21no—
131C2×#88322—21no—
132C2×#88322—22no—
133C2×#88322—41no—
134C2×#88322—41no—
135#11719912—44no—
136C23×C53#12111yescharacteristic
137C22×D53—21no—
138Q8×C53—41no—
139Q8×C53—41no—
140Q8×C53—81no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53C2×C22⋊C411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53#12011yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12311yescharacteristic
152C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C53#13711yescharacteristic
154C2×C4×C53#13711yescharacteristic
155C2×C4×C53#13711yescharacteristic
156C22⋊D53—41no—
157C22⋊D53—41no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—42no—
162C2×Dic53—81no—
163C2×Dic53—81no—
164D212—21no—
165D212—21no—
166C4×D53—42no—
167C4×D53—41no—
168C4×D53—41no—
169C4×D53—41no—
170C4×D53—41no—
171Dic106—21no—
172Dic106—21no—
173C8×C53—22no—
174C8×C53—41no—
175C8×C53—81no—
176#88322—42no—
177#88322—41no—
178#88322—41no—
179#9209—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53#38611yescharacteristic
183C22×C53#46811yescharacteristic
184C22×C53#47011yescharacteristic
185C2×D53—41no—
186C2×D53—41no—
187C2×D53—41no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—81no—
192C4×C53—81no—
193C4×C53#41711yescharacteristic
194C4×C53#41911yescharacteristic
195C4×C53#44311yescharacteristic
196C4×C53#44411yescharacteristic
197C4×C53#46811yescharacteristic
198C4×C53#47011yescharacteristic
199Dic53—41no—
200Dic53—41no—
201Dic53—41no—
202Dic53—82no—
203#3076—531no—
204#2965—531no—
205#1806—531no—
206#1503—531no—
207#1493D5311yescharacteristic, complemented
208#1301—531no—
209#1279—531no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53#187611yescharacteristic
213C2×C53#189411yescharacteristic
214C2×C53#190611yescharacteristic
215D53—81no—
216#533C2×D5311yescharacteristic
217C4×#41—531no—
218#479—1061no—
219#478—1061no—
220#477—531no—
221#476—531no—
222#475—531no—
223#474—531no—
224#446—1061no—
225#436—1061no—
226#435—1062no—
227#431—531no—
228#431—1061no—
229#427—1061no—
230#423C2×D5311yescharacteristic
231C2×C4⋊C8—531no—
232C2×#104—1061no—
233#416—1061no—
234C4×M16—1061no—
235C2×C4×C8C2×D5311yescharacteristic
236C4×C4⋊C4—531no—
237#339—1061no—
238#328—1062no—
239Q8⋊C8—21no—
240Q8⋊C8—21no—
241C53#920911yescharacteristic, complemented, Sylow
242C2×#41—531no—
243C2×M16—21no—
244C2×M16—1062no—
245C22×C8—21no—
246C22×C8—1061no—
247C4⋊Q8C4×D5311yescharacteristic
248C4⋊Q8C4×D5311yescharacteristic
249#125—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—531no—
255C4×Q8—531no—
256C4×D4—531no—
257C4×D4—531no—
258C4×D4—1061no—
259C4×D4—1061no—
260#117—21no—
261#117—531no—
262#117—1061no—
263C2×C4⋊C4—1061no—
264C2×C42—1061no—
265C2×C42—1061no—
266C2×C42C22×D5311yescharacteristic
267#107—1061no—
268#107—1061no—
269#106—1061no—
270#106—1061no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—531no—
274C4⋊C8—531no—
275C4⋊C8—1062no—
276C4⋊C8—1061no—
277C4⋊C8—1061no—
278#104—2122no—
279#103—1061no—
280#103—1061no—
281#102—1061no—
282#102—1061no—
283C22⋊C8—532no—
284C22⋊C8—1062no—
285#97—1062no—
286C4×C8—21no—
287C4×C8—1061no—
288C4×C8C22⋊D5311yescharacteristic
289C4×C8C22⋊D5311yescharacteristic
290#95—1061no—
291#41—1061no—
292#41—1061no—
293#41—1061no—
294#41—1061no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—531no—
298C2×D4—531no—
299C2×D4—1061no—
300C22×C4—1061no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4—2121no—
304C22×C4#1171994511yescharacteristic
305C22×C4D4×D5311yescharacteristic
306C22×C4D4×D5311yescharacteristic
307M16—41no—
308M16—41no—
309M16—2124no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—1061no—
315C2×C8—1061no—
316C2×C8—1062no—
317C2×C8—2121no—
318C2×C8—2121no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—531no—
326C4⋊C4—531no—
327C4⋊C4—1062no—
328C4⋊C4—1061no—
329C4⋊C4—1061no—
330C4⋊C4—2121no—
331C22⋊C4—41no—
332C22⋊C4—1061no—
333C22⋊C4—1061no—
334C22⋊C4—1061no—
335C42—41no—
336C42—1061no—
337C42—1061no—
338C42—1061no—
339C42—2121no—
340C42—2121no—
341C42C2×C4×D5311yescharacteristic
342C42D4×D5311yescharacteristic
343C42#1171994711yescharacteristic
344C42C2×C22⋊D5311yescharacteristic
345C23—1061no—
346C23#1313319011yescharacteristic
347Q8—41no—
348Q8—41no—
349Q8—81no—
350Q8—1061no—
351Q8—1061no—
352D4—1061no—
353D4—1061no—
354D4—2121no—
355D4—2121no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—1061no—
364C2×C4—4241no—
365C2×C4—4241no—
366C2×C4—2122no—
367C2×C4—2122no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4C22⋊C4×D5311yescharacteristic
373C2×C4#1313317711yescharacteristic
374C2×C4C4×C22⋊D5311yescharacteristic
375C2×C4#1313318511yescharacteristic
376C2×C4#1313318811yescharacteristic
377C2×C4#1313319911yescharacteristic
378C2×C4#1313324411yescharacteristic
379C2×C4#1313324611yescharacteristic
380C2×C4#1313324911yescharacteristic
381C2×C4#1313324911yescharacteristic
382C8—22no—
383C8—41no—
384C8—81no—
385C8—2122no—
386C8—2121no—
387C8—2121no—
388C22—21no—
389C22—21no—
390C22—2121no—
391C22—2121no—
392C22—2121no—
393C22#1681049911yescharacteristic
394C22#1681087211yescharacteristic
395C22#1681088011yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—81no—
400C4—81no—
401C4—4242no—
402C4—2121no—
403C4—2121no—
404C4—2121no—
405C4#1681062311yescharacteristic
406C4#1681063011yescharacteristic
407C4#1681073811yescharacteristic
408C4#1681074311yescharacteristic
409C4#1681086611yescharacteristic
410C4#1681087811yescharacteristic
411C2—21no—
412C2—21no—
413C2—4241no—
414C2#2027504511yescharacteristic
415C2#2027514511yescharacteristic
416C2#2027521911yescharacteristic
subgroupG/H
centerC2×C4#13133188
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1493C2
socleC22×C53#386
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1493D53
hypocenterC53#9209
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
4. #20273214C2semidirect
5. #20272610C2semidirect
6. #20272555C2semidirect
7. C53×#1493C2semidirect
11. #16810915C4semidirect
12. #16810910C4semidirect
13. #16810905C4semidirect
14. #16810900C4semidirect
207. #1493D53semidirect
241. C53#9209semidirect

Maximal subgroups

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

subgroupindex
#202844972
#202836812
#202747322
#202732142
#202726102
#202725552
C53×#14932
#920953