Finite Group Databasebeta

← #27368258

Subgroups — #27368258

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284871C212yesmaximal
2#20283681C211yesmaximal, characteristic
3#20274240C211yesmaximal, characteristic, complemented
4#20274235C211yesmaximal, characteristic, complemented
5#20274230C211yesmaximal, characteristic, complemented
6#20274225C211yesmaximal, characteristic, complemented
7#20274218C212yesmaximal, complemented
8#20274211C212yesmaximal, complemented
9#20273478C211yesmaximal, characteristic, complemented
10#20273425C212yesmaximal, complemented
11C53×#3160C211yesmaximal, characteristic, complemented
12C4×#11719945C2211yescharacteristic
13#16810961—22no—
14#16810915C2212yes—
15#16810910C2212yes—
16#16810830C2212yes—
17#16810825C2212yes—
18#16810783—22no—
19#16810753C2211yescharacteristic
20#16810753C2211yescharacteristic
21#16810728—22no—
22#16810721—22no—
23#16810702C2211yescharacteristic
24#16810702C2211yescharacteristic
25#16810694—21no—
26#16810694—21no—
27#16810680C2212yes—
28#16810676C2211yescharacteristic
29#16810674—21no—
30C4×#11719912—21no—
31#16810376C2212yes—
32#16810374C2211yescharacteristic
33#16810374C2211yescharacteristic
34#16810372C2212yes—
35#16810372C2212yes—
36#16810369C2212yes—
37#16810369C2212yes—
38C53×#533C2212yes—
39C53×#500C2211yescharacteristic
40C53×#500C2211yescharacteristic
41C53×#499—21no—
42C53×#499—21no—
43C53×#498C2212yes—
44C53×#427—21no—
45C53×#427—21no—
46C2×C4×C8×C53C2211yescharacteristic
47C2×#11719945D411yescharacteristic
48C2×#11719925—22no—
49C2×#11719924—22no—
50C2×#11719912—21no—
51C2×#11719912—21no—
52C4×Dic106D411yescharacteristic
53C4×Dic106C2311yescharacteristic
54C4×D212D411yescharacteristic
55C4×D212C2311yescharacteristic
56C4×C22⋊D53—21no—
57C4×C22⋊D53—21no—
58#13133176D411yescharacteristic
59#13133174—21no—
60C42×D53—21no—
61#13133150—22no—
62#13133148—22no—
63C4⋊#88322C2311yescharacteristic
64C4⋊#88322C2311yescharacteristic
65C4⋊#88322C2312yes—
66C4⋊#88322C2312yes—
67#13133146—21no—
68#13133146—21no—
69#13133141—22no—
70#13133141—22no—
71#13133140—22no—
72#13133137—22no—
73C22⋊#88322—22no—
74C22⋊#88322—21no—
75#13133129—21no—
76#13133129—21no—
77#13133128—22no—
78#13133127—21no—
79#13133127—21no—
80C4×#88322—21no—
81C22×C8×C53D411yescharacteristic
82C22×C8×C53D411yescharacteristic
83C4⋊Q8×C53C2312yes—
84C4⋊Q8×C53C2312yes—
85#125×C53—22no—
86C22⋊Q8×C53—22no—
87C22⋊Q8×C53—22no—
88C4×Q8×C53—22no—
89C4×Q8×C53—22no—
90#117×C53—22no—
91C2×C42×C53C2311yescharacteristic
92#107×C53—21no—
93#107×C53—21no—
94#107×C53—21no—
95#107×C53—21no—
96#106×C53—22no—
97#106×C53—22no—
98C4×C8×C53D411yescharacteristic
99C4×C8×C53D411yescharacteristic
100C4×C8×C53C2311yescharacteristic
101C4×C8×C53C2311yescharacteristic
102C2×Q8×C53—22no—
103C2×Q8×C53—22no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C2×C22⋊D53—21no—
108#11719945—42no—
109C2×D212C2×D411yescharacteristic
110C2×C4×D53—21no—
111C2×C4×D53—21no—
112C2×C4×D53—21no—
113C2×Dic106C2×D411yescharacteristic
114C2×C8×C53—22no—
115C2×C8×C53—22no—
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53C2×D411yescharacteristic
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—42no—
125C4⋊C4×C53—42no—
126C22⋊C4×C53—42no—
127C42×C53—42no—
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131C42×C53C2411yescharacteristic
132C22⋊Dic53—21no—
133#11719925—44no—
134#11719924—44no—
135#11719921—21no—
136#11719921—21no—
137C4⋊Dic53C2×D411yescharacteristic
138C4⋊Dic53C2×D411yescharacteristic
139#11719919—21no—
140#11719919—21no—
141C4×Dic53—21no—
142C4×Dic53—21no—
143C4×Dic53—21no—
144C2×#88322—21no—
145C2×#88322—21no—
146C2×#88322—22no—
147C2×#88322—22no—
148#11719912—42no—
149#11719912—42no—
150C23×C53#12111yescharacteristic
151C22×D53—21no—
152Q8×C53—42no—
153Q8×C53—42no—
154Q8×C53—82no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—22no—
158C2×C4×C53—41no—
159C2×C4×C53—42no—
160C2×C4×C53—42no—
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53C22×D411yescharacteristic
165C2×C4×C53C22×D411yescharacteristic
166C2×C4×C53C2×#4111yescharacteristic
167C22⋊D53—42no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—21no—
171D212—22no—
172C4×D53—42no—
173C4×D53—42no—
174C4×D53—41no—
175C4×D53—41no—
176Dic106—22no—
177C8×C53—21no—
178C8×C53—21no—
179C8×C53#13712yes—
180C8×C53#13712yes—
181#88322—41no—
182#88322—41no—
183#88322—42no—
184#88322—42no—
185#19298—531nomaximal, Sylow
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53C2×#12111yescharacteristic
190C2×D53—41no—
191C2×D53—41no—
192C2×D53—41no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—82no—
198C4×C53—82no—
199C4×C53C2×#13711yescharacteristic
200C4×C53C2×#13711yescharacteristic
201C4×C53#57512yes—
202Dic53—41no—
203Dic53—41no—
204Dic53—41no—
205#3160D5311yescharacteristic, complemented
206#3116—532no—
207#2965—531no—
208#1696—531no—
209#1695—531no—
210#1694—531no—
211#1693—531no—
212#1692—532no—
213#1691—532no—
214#1568—531no—
215#1554—532no—
216C2×C53—41no—
217C2×C53#303911yescharacteristic
218C2×C53#305411yescharacteristic
219C2×C53#306711yescharacteristic
220D53—81no—
221#533C2×D5312yes—
222C4×#41—531no—
223#500C2×D5311yescharacteristic
224#500C2×D5311yescharacteristic
225#499—21no—
226#499—21no—
227#498C2×D5312yes—
228#485—1062no—
229#477—532no—
230#476—532no—
231#462—532no—
232#461—532no—
233#452—1062no—
234#446—531no—
235#446—531no—
236#441—1062no—
237#440—1062no—
238#436—531no—
239#436—531no—
240#435—1061no—
241#435—1061no—
242#432—532no—
243#431—531no—
244#431—1061no—
245#427—21no—
246#427—21no—
247C4×M16—1061no—
248C2×C4×C8C2×D5311yescharacteristic
249#335—532no—
250#334—531no—
251#334—531no—
252#333—532no—
253#333—532no—
254#332—532no—
255#332—532no—
256C53#1929811yescharacteristic, complemented, Sylow
257C2×#41—531no—
258C2×Q16—1062no—
259C2×SD16—1062no—
260C2×M16—1061no—
261C2×M16—1061no—
262C22×C8C22⋊D5311yescharacteristic
263C22×C8C22⋊D5311yescharacteristic
264C4⋊Q8C22×D5312yes—
265C4⋊Q8C22×D5312yes—
266#125—22no—
267C22⋊Q8—22no—
268C22⋊Q8—22no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—531no—
272C4×Q8—531no—
273C4×D4—531no—
274C4×D4—531no—
275C4×D4—1061no—
276C4×D4—1061no—
277#117—22no—
278#117—531no—
279#117—1061no—
280C2×C42—1061no—
281C2×C42C22×D5311yescharacteristic
282#107—21no—
283#107—21no—
284#107—21no—
285#107—21no—
286#107—1062no—
287#106—22no—
288#106—22no—
289#106—1062no—
290C4⋊C8—531no—
291C4⋊C8—531no—
292C4⋊C8—532no—
293C4⋊C8—532no—
294C4⋊C8—1061no—
295C4⋊C8—1061no—
296#103—1062no—
297#103—1062no—
298#103—1062no—
299#102—1062no—
300C22⋊C8—1062no—
301C22⋊C8—1061no—
302C22⋊C8—1061no—
303C22⋊C8—1061no—
304#97—1061no—
305#97—1061no—
306#97—1062no—
307C4×C8—1061no—
308C4×C8C22⋊D5311yescharacteristic
309C4×C8C22⋊D5311yescharacteristic
310C4×C8C22×D5311yescharacteristic
311C4×C8C22×D5311yescharacteristic
312#41—2122no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—531no—
316C2×D4—531no—
317C2×D4—1061no—
318C22×C4—1061no—
319C22×C4—1061no—
320C22×C4—1061no—
321C22×C4D4×D5311yescharacteristic
322C22×C4#1171994711yescharacteristic
323C22×C4C2×C22⋊D5311yescharacteristic
324Q16—2124no—
325SD16—2124no—
326M16—2122no—
327M16—2122no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—1062no—
331C2×C8—1062no—
332C2×C8—1061no—
333C2×C8—1061no—
334C2×C8C2×C22⋊D5311yescharacteristic
335C2×C8C2×C22⋊D5311yescharacteristic
336C2×C8C2×C22⋊D5311yescharacteristic
337C2×C8C2×C22⋊D5311yescharacteristic
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—531no—
345C4⋊C4—531no—
346C4⋊C4—1061no—
347C4⋊C4—1061no—
348C22⋊C4—42no—
349C22⋊C4—1061no—
350C22⋊C4—1061no—
351C22⋊C4—1061no—
352C42—42no—
353C42—1061no—
354C42—1061no—
355C42—1061no—
356C42D4×D5311yescharacteristic
357C42#1171994711yescharacteristic
358C42C2×C22⋊D5311yescharacteristic
359C42C23×D5311yescharacteristic
360C23—1061no—
361C23#1313318811yescharacteristic
362Q8—42no—
363Q8—42no—
364Q8—82no—
365Q8—1062no—
366D4—1062no—
367D4—2122no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—1061no—
375C2×C4—1061no—
376C2×C4—1061no—
377C2×C4—2122no—
378C2×C4—2122no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4#1313318811yescharacteristic
382C2×C4#1313318811yescharacteristic
383C2×C4#1313318811yescharacteristic
384C2×C4C2×D4×D5311yescharacteristic
385C2×C4C22×C22⋊D5311yescharacteristic
386C2×C4C2×#1171994711yescharacteristic
387C8—21no—
388C8—21no—
389C8—2121no—
390C8—2121no—
391C8—2122no—
392C8—2122no—
393C8#1313324812yes—
394C8#1313324812yes—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—2121no—
399C22—2121no—
400C22—2121no—
401C22C2×#1313318811yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—82no—
407C4—82no—
408C4—2121no—
409C4—2121no—
410C4—2121no—
411C4C2×#1313324811yescharacteristic
412C4C2×#1313324811yescharacteristic
413C4#1681142012yes—
414C2—41no—
415C2—4241no—
416C2#2028419811yescharacteristic
417C2#2028432011yescharacteristic
418C2#2028443111yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3160C2
socleC22×C53C2×#121
prime radicalC2×#11719945D4
hypercenter#3160D53
hypocenterC53#19298
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3160

Complements

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

NH ≅ G/Ntype
3. #20274240C2semidirect
4. #20274235C2semidirect
5. #20274230C2semidirect
6. #20274225C2semidirect
7. #20274218C2semidirect
8. #20274211C2semidirect
9. #20273478C2semidirect
10. #20273425C2semidirect
11. C53×#3160C2semidirect
205. #3160D53semidirect
256. C53#19298semidirect

Maximal subgroups

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

subgroupindex
#202848712
#202836812
#202742402
#202742352
#202742302
#202742252
#202742182
#202742112
#202734782
#202734252
C53×#31602
#1929853