Finite Group Databasebeta

← #27367528

Subgroups — #27367528

Order 13568.

562 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285890C212yesmaximal
2#20285301C211yesmaximal, characteristic, complemented
3#20274243C211yesmaximal, characteristic, complemented
4#20274238C211yesmaximal, characteristic, complemented
5#20274209C211yesmaximal, characteristic, complemented
6#20274204C211yesmaximal, characteristic, complemented
7#20274173C212yesmaximal, complemented
8#20274166C212yesmaximal, complemented
9#20273463C211yesmaximal, characteristic, complemented
10#20273426C212yesmaximal, complemented
11C53×#2966C211yesmaximal, characteristic
12#16811180C2212yes—
13#16811028C2211yescharacteristic
14#16811022—22no—
15#16811003C2211yescharacteristic
16#16810977—22no—
17#16810974C2212yes—
18#16810961—22no—
19#16810944—22no—
20#16810918C2212yes—
21#16810913C2212yes—
22#16810908C2212yes—
23#16810903C2212yes—
24#16810864—21no—
25#16810864—21no—
26C4×Dic212—21no—
27C4×Dic212—21no—
28#16810676C2212yes—
29#16810376C2212yes—
30#16810374C2211yescharacteristic
31#16810374C2211yescharacteristic
32#16810371C2212yes—
33#16810371C2212yes—
34#16810370C2212yes—
35#16810370C2212yes—
36C53×#529C2211yescharacteristic
37C53×#446C2211yescharacteristic
38C8×Q8×C53C2211yescharacteristic
39C53×#436C2211yescharacteristic
40C53×#435—21no—
41C8×D4×C53—21no—
42C8×D4×C53C2211yescharacteristic
43C53×#433—21no—
44C53×#432C2211yescharacteristic
45C53×#405—21no—
46C2×C4×C8×C53C2211yescharacteristic
47C2×Dic212—21no—
48C2×Dic212—21no—
49#13133217C2312yes—
50#13133217C2312yes—
51#13133209—22no—
52#13133196—22no—
53#13133196—22no—
54C4×Dic106—22no—
55C4×Dic106—22no—
56#13133176—22no—
57#13133151—21no—
58#13133151—21no—
59#13133150—22no—
60#13133150—22no—
61#13133148—22no—
62#13133148—22no—
63C4⋊#88322C2312yes—
64C4⋊#88322C2312yes—
65#13133142—22no—
66#13133142—22no—
67#13133141—22no—
68#13133141—22no—
69#13133138—22no—
70#13133138—22no—
71C22⋊#88322—22no—
72C4×#88322—22no—
73C2×M16×C53—21no—
74C22×C8×C53—21no—
75C22×C8×C53D411yescharacteristic
76C22×C8×C53D411yescharacteristic
77#126×C53—21no—
78#125×C53D411yescharacteristic
79#124×C53D411yescharacteristic
80#123×C53—21no—
81C22⋊Q8×C53D411yescharacteristic
82#121×C53D411yescharacteristic
83C4×Q8×C53C2311yescharacteristic
84C4×D4×C53—21no—
85C4×D4×C53C2311yescharacteristic
86#117×C53—21no—
87C2×C42×C53C2311yescharacteristic
88C4⋊C8×C53—21no—
89C4⋊C8×C53—21no—
90C4⋊C8×C53C2311yescharacteristic
91C4⋊C8×C53C2311yescharacteristic
92C22⋊C8×C53—21no—
93C22⋊C8×C53—21no—
94C22⋊C8×C53—21no—
95#97×C53—21no—
96#97×C53—21no—
97C4×C8×C53—21no—
98C4×C8×C53D411yescharacteristic
99C4×C8×C53D411yescharacteristic
100C4×C8×C53C2311yescharacteristic
101C4×C8×C53C2311yescharacteristic
102C2×Q8×C53C2×D411yescharacteristic
103C2×D4×C53—21no—
104C2×D4×C53C2×D411yescharacteristic
105C22×C4×C53—21no—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C2×Dic106—22no—
110C2×Dic106—22no—
111M16×C53—42no—
112C2×C8×C53—22no—
113C2×C8×C53—21no—
114C2×C8×C53—21no—
115C2×C8×C53—41no—
116C2×C8×C53—42no—
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53C2×D411yescharacteristic
120C2×C8×C53C2×D411yescharacteristic
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53C2×D411yescharacteristic
126C4⋊C4×C53C2×D411yescharacteristic
127C22⋊C4×C53—21no—
128C22⋊C4×C53—21no—
129C22⋊C4×C53—21no—
130C22⋊C4×C53—21no—
131C22⋊C4×C53—21no—
132C42×C53—21no—
133C42×C53C2×D411yescharacteristic
134C42×C53C2×D411yescharacteristic
135C42×C53#4111yescharacteristic
136C42×C53C2411yescharacteristic
137C22⋊Dic53—42no—
138C4⋊Dic53—22no—
139C4⋊Dic53—22no—
140C4⋊Dic53—22no—
141C4⋊Dic53—22no—
142#11719919—42no—
143#11719919—42no—
144C4×Dic53—42no—
145C2×#88322—22no—
146C2×#88322—22no—
147Dic212—42no—
148Dic212—42no—
149C23×C53—21no—
150C23×C53#12111yescharacteristic
151Q8×C53—22no—
152D4×C53—22no—
153D4×C53—42no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—22no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53C22×D411yescharacteristic
167C2×C4×C53C22×D411yescharacteristic
168C2×C4×C53C2×#4111yescharacteristic
169C2×Dic53—42no—
170C2×Dic53—42no—
171Dic106—42no—
172Dic106—42no—
173Dic106—82no—
174C8×C53—21no—
175C8×C53—21no—
176C8×C53—21no—
177C8×C53—41no—
178C8×C53—41no—
179C8×C53#13512yes—
180#88322—42no—
181#88322—42no—
182#19226—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53C2×#12111yescharacteristic
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—41no—
195C4×C53—41no—
196C4×C53—41no—
197C4×C53C2×#13511yescharacteristic
198C4×C53#57511yescharacteristic
199C4×C53#57812yes—
200Dic53—82no—
201Dic53—82no—
202#3233—532no—
203#3170—531no—
204#2966D5311yescharacteristic
205#1696—531no—
206#1695—531no—
207#1690—531no—
208#1689—531no—
209#1684—532no—
210#1683—532no—
211#1564—531no—
212#1554—532no—
213C2×C53—41no—
214C2×C53—81no—
215C2×C53#303511yescharacteristic
216C2×C53#305911yescharacteristic
217C2×C53#306711yescharacteristic
218#533—532no—
219#529C2×D5311yescharacteristic
220#500—531no—
221#499—1062no—
222#494—531no—
223#488—1062no—
224#487—532no—
225#485—1062no—
226#482—1062no—
227#477—532no—
228#476—532no—
229#475—532no—
230#474—532no—
231#467—1061no—
232#467—1061no—
233#446C2×D5311yescharacteristic
234C8×Q8C2×D5311yescharacteristic
235C4×Q16—1061no—
236C4×Q16—1061no—
237#436C2×D5311yescharacteristic
238#435—21no—
239C8×D4—21no—
240C8×D4C2×D5311yescharacteristic
241#433—21no—
242#432C2×D5311yescharacteristic
243#431—532no—
244#405—21no—
245C2×C4×C8C2×D5311yescharacteristic
246#335—532no—
247#334—531no—
248#334—531no—
249#333—532no—
250#333—532no—
251#332—532no—
252#332—532no—
253C53#1922611yescharacteristic, complemented, Sylow
254C2×Q16—1061no—
255C2×Q16—1061no—
256C2×M16—21no—
257C22×C8—21no—
258C22×C8C22⋊D5311yescharacteristic
259C22×C8C22⋊D5311yescharacteristic
260C4⋊Q8—532no—
261C4⋊Q8—532no—
262#126—21no—
263#125—1062no—
264#125D21211yescharacteristic
265#124D21211yescharacteristic
266#123—21no—
267C22⋊Q8—1062no—
268C22⋊Q8—1062no—
269C22⋊Q8D21211yescharacteristic
270#121D21211yescharacteristic
271C4×Q8—1062no—
272C4×Q8—1062no—
273C4×Q8C22×D5311yescharacteristic
274C4×D4—21no—
275C4×D4C22×D5311yescharacteristic
276#117—21no—
277#117—1062no—
278C2×C42C22×D5311yescharacteristic
279#107—1061no—
280#107—1061no—
281#107—1062no—
282#107—1062no—
283#106—1062no—
284#106—1062no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—532no—
288C4⋊C8—532no—
289C4⋊C8C22×D5311yescharacteristic
290C4⋊C8C22×D5311yescharacteristic
291#103—1062no—
292#103—1062no—
293#103—1062no—
294#103—1062no—
295#102—1062no—
296#102—1062no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—1062no—
301#97—21no—
302#97—21no—
303C4×C8—21no—
304C4×C8—1062no—
305C4×C8C22⋊D5311yescharacteristic
306C4×C8C22⋊D5311yescharacteristic
307C4×C8C22×D5311yescharacteristic
308C4×C8C22×D5311yescharacteristic
309C2×Q8—1062no—
310C2×Q8—1062no—
311C2×Q8C2×D21211yescharacteristic
312C2×D4—21no—
313C2×D4C2×D21211yescharacteristic
314C22×C4—21no—
315C22×C4C2×D21211yescharacteristic
316C22×C4#1171994511yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318Q16—2122no—
319Q16—2122no—
320M16—42no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—1062no—
327C2×C8—1062no—
328C2×C8C2×C22⋊D5311yescharacteristic
329C2×C8C2×C22⋊D5311yescharacteristic
330C2×C8C2×C22⋊D5311yescharacteristic
331C2×C8C2×C22⋊D5311yescharacteristic
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—1062no—
337C4⋊C4—1062no—
338C4⋊C4—1062no—
339C4⋊C4—1062no—
340C4⋊C4—2122no—
341C4⋊C4—2122no—
342C4⋊C4C2×D21211yescharacteristic
343C4⋊C4C2×D21211yescharacteristic
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—2122no—
350C42—21no—
351C42—2122no—
352C42C2×D21211yescharacteristic
353C42#1171994511yescharacteristic
354C42C2×C22⋊D5311yescharacteristic
355C42C23×D5311yescharacteristic
356C23—21no—
357C23#1313319111yescharacteristic
358Q8—22no—
359Q8—4242no—
360Q8—2122no—
361Q8—2122no—
362D4—22no—
363D4—42no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—2122no—
374C2×C4—2122no—
375C2×C4#1313319111yescharacteristic
376C2×C4#1313319111yescharacteristic
377C2×C4#1313319111yescharacteristic
378C2×C4C22×C22⋊D5311yescharacteristic
379C2×C4C22×D21211yescharacteristic
380C2×C4C2×#1171994511yescharacteristic
381C8—21no—
382C8—21no—
383C8—21no—
384C8—41no—
385C8—41no—
386C8—2122no—
387C8—2122no—
388C8#1313323712yes—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22C2×#1313319111yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—4242no—
404C4—4242no—
405C4C2×#1313323711yescharacteristic
406C4#1681142111yescharacteristic
407C4#1681144012yes—
408C2—41no—
409C2—81no—
410C2#2028417011yescharacteristic
411C2#2028437011yescharacteristic
412C2#2028443511yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2966C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#2966D53
hypocenterC53#19226
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
2. #20285301C2semidirect
3. #20274243C2semidirect
4. #20274238C2semidirect
5. #20274209C2semidirect
6. #20274204C2semidirect
7. #20274173C2semidirect
8. #20274166C2semidirect
9. #20273463C2semidirect
10. #20273426C2semidirect
253. C53#19226semidirect

Maximal subgroups

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

subgroupindex
#202858902
#202853012
#202742432
#202742382
#202742092
#202742042
#202741732
#202741662
#202734632
#202734262
C53×#29662
#1922653