Finite Group Databasebeta

← #27366849

Subgroups — #27366849

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285301C211yesmaximal, characteristic, complemented
2#20284786C212yesmaximal
3#20274269C212yesmaximal, complemented
4#20274262C212yesmaximal, complemented
5#20274233C211yesmaximal, characteristic, complemented
6#20274228C211yesmaximal, characteristic, complemented
7#20274209C211yesmaximal, characteristic, complemented
8#20274204C211yesmaximal, characteristic, complemented
9#20273463C211yesmaximal, characteristic, complemented
10#20273379C212yesmaximal, complemented
11C53×#2965C211yesmaximal, characteristic
12#16811180C2212yes—
13#16811028C2211yescharacteristic
14#16811003C2211yescharacteristic
15#16810974C2212yes—
16#16810964—22no—
17#16810908C2212yes—
18#16810903C2212yes—
19#16810864—21no—
20#16810864—21no—
21#16810833C2212yes—
22#16810828C2212yes—
23#16810785—22no—
24C4×Dic212—21no—
25C4×Dic212—21no—
26C4×#11719925—22no—
27C4×#11719923—22no—
28#16810676C2212yes—
29#16810376C2212yes—
30#16810374C2211yescharacteristic
31#16810374C2211yescharacteristic
32#16810370C2212yes—
33#16810370C2212yes—
34#16810368C2212yes—
35#16810368C2212yes—
36C4×#41×C53C2211yescharacteristic
37C53×#446C2211yescharacteristic
38C53×#446C2211yescharacteristic
39C53×#436C2211yescharacteristic
40C53×#436C2211yescharacteristic
41C53×#435—21no—
42C53×#435—21no—
43C53×#431—21no—
44C53×#431C2211yescharacteristic
45C4×M16×C53—21no—
46C2×C4×C8×C53C2211yescharacteristic
47C2×Dic212—21no—
48C2×Dic212—21no—
49C2×#11719925—22no—
50C2×#11719923—22no—
51#13133217C2312yes—
52#13133217C2312yes—
53#13133209—22no—
54#13133196—22no—
55#13133196—22no—
56C4×Dic106—22no—
57C4×Dic106—22no—
58#13133176—22no—
59#13133151—21no—
60#13133151—21no—
61#13133150—22no—
62#13133148—22no—
63C4⋊#88322C2312yes—
64C4⋊#88322C2312yes—
65#13133142—22no—
66#13133142—22no—
67#13133141—22no—
68#13133140—22no—
69#13133140—22no—
70#13133138—22no—
71C22⋊#88322—22no—
72C4×#88322—22no—
73C2×#41×C53D411yescharacteristic
74C2×M16×C53—21no—
75C2×M16×C53—21no—
76C22×C8×C53D411yescharacteristic
77C22×C8×C53D411yescharacteristic
78C4×Q8×C53D411yescharacteristic
79C4×Q8×C53C2311yescharacteristic
80C4×D4×C53—21no—
81C4×D4×C53—21no—
82C4×D4×C53D411yescharacteristic
83C4×D4×C53C2311yescharacteristic
84#117×C53—21no—
85#117×C53D411yescharacteristic
86C2×C42×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
102#41×C53—42no—
103C2×Q8×C53C2×D411yescharacteristic
104C2×D4×C53—21no—
105C2×D4×C53C2×D411yescharacteristic
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C2×Dic106—22no—
113C2×Dic106—22no—
114M16×C53—42no—
115M16×C53—42no—
116C2×C8×C53—22no—
117C2×C8×C53—21no—
118C2×C8×C53—21no—
119C2×C8×C53—41no—
120C2×C8×C53C2×D411yescharacteristic
121C2×C8×C53C2×D411yescharacteristic
122C2×C8×C53C2×D411yescharacteristic
123C2×C8×C53C2×D411yescharacteristic
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53C2×D411yescharacteristic
127C4⋊C4×C53C2×D411yescharacteristic
128C22⋊C4×C53—21no—
129C22⋊C4×C53—21no—
130C22⋊C4×C53—21no—
131C42×C53—21no—
132C42×C53—21no—
133C42×C53—21no—
134C42×C53C2×D411yescharacteristic
135C42×C53C2×D411yescharacteristic
136C42×C53#4111yescharacteristic
137C42×C53C2411yescharacteristic
138C22⋊Dic53—42no—
139#11719925—44no—
140#11719923—44no—
141C4⋊Dic53—22no—
142C4⋊Dic53—22no—
143C4⋊Dic53—22no—
144C4⋊Dic53—22no—
145#11719919—42no—
146#11719919—42no—
147C4×Dic53—42no—
148C2×#88322—22no—
149C2×#88322—22no—
150Dic212—42no—
151Dic212—42no—
152C23×C53—21no—
153C23×C53#12111yescharacteristic
154Q8×C53—22no—
155D4×C53—22no—
156D4×C53—42no—
157C2×C4×C53—21no—
158C2×C4×C53—22no—
159C2×C4×C53—21no—
160C2×C4×C53—22no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—42no—
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53C22×D411yescharacteristic
172C2×C4×C53C22×D411yescharacteristic
173C2×C4×C53C2×#4111yescharacteristic
174C2×Dic53—42no—
175C2×Dic53—42no—
176Dic106—42no—
177Dic106—42no—
178Dic106—42no—
179Dic106—42no—
180C8×C53—21no—
181C8×C53—21no—
182C8×C53—21no—
183C8×C53—41no—
184C8×C53—41no—
185C8×C53#13512yes—
186#88322—42no—
187#88322—42no—
188#19161—531nomaximal, Sylow
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53C2×#12111yescharacteristic
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—41no—
202C4×C53#47312yes—
203C4×C53C2×#13511yescharacteristic
204C4×C53#57511yescharacteristic
205C4×C53#57512yes—
206Dic53—82no—
207Dic53—82no—
208#3170—531no—
209#3107—532no—
210#2965D5311yescharacteristic
211#1700—532no—
212#1699—532no—
213#1694—531no—
214#1693—531no—
215#1690—531no—
216#1689—531no—
217#1564—531no—
218#1544—532no—
219C2×C53—41no—
220C2×C53—81no—
221C2×C53#303511yescharacteristic
222C2×C53#305411yescharacteristic
223C2×C53C2×#47311yescharacteristic
224#533—532no—
225C4×#41C2×D5311yescharacteristic
226#500—531no—
227#494—531no—
228#487—532no—
229#485—1062no—
230#475—532no—
231#474—532no—
232#467—1061no—
233#467—1061no—
234#462—532no—
235#461—532no—
236#452—1062no—
237#446C2×D5311yescharacteristic
238#446C2×D5311yescharacteristic
239C4×Q16—1062no—
240C4×Q16—1061no—
241C4×Q16—1061no—
242C4×SD16—1062no—
243#436C2×D5311yescharacteristic
244#436C2×D5311yescharacteristic
245#435—21no—
246#435—21no—
247#431—21no—
248#431—532no—
249#431C2×D5311yescharacteristic
250C4×M16—21no—
251C2×C4×C8C2×D5311yescharacteristic
252#335—532no—
253#334—531no—
254#334—531no—
255#332—532no—
256#332—532no—
257#332—532no—
258#332—532no—
259C53#1916111yescharacteristic, complemented, Sylow
260C2×#41D21211yescharacteristic
261C2×Q16—1062no—
262C2×Q16—1061no—
263C2×Q16—1061no—
264C2×SD16—1062no—
265C2×M16—21no—
266C2×M16—21no—
267C22×C8C22⋊D5311yescharacteristic
268C22×C8C22⋊D5311yescharacteristic
269C4⋊Q8—532no—
270C4⋊Q8—532no—
271#125—1062no—
272C22⋊Q8—1062no—
273C22⋊Q8—1062no—
274C4×Q8—1062no—
275C4×Q8—1062no—
276C4×Q8D21211yescharacteristic
277C4×Q8C22×D5311yescharacteristic
278C4×D4—21no—
279C4×D4—21no—
280C4×D4D21211yescharacteristic
281C4×D4C22×D5311yescharacteristic
282#117—21no—
283#117—1062no—
284#117D21211yescharacteristic
285C2×C42—21no—
286C2×C42C22×D5311yescharacteristic
287#107—1061no—
288#107—1061no—
289#107—1062no—
290#106—1062no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—532no—
294C4⋊C8—532no—
295C4⋊C8C22×D5311yescharacteristic
296C4⋊C8C22×D5311yescharacteristic
297#103—1062no—
298#103—1062no—
299#103—1062no—
300#103—1062no—
301#103—1062no—
302#102—1062no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—1062no—
307#97—21no—
308#97—21no—
309C4×C8—21no—
310C4×C8—1062no—
311C4×C8C22⋊D5311yescharacteristic
312C4×C8C22⋊D5311yescharacteristic
313C4×C8C22×D5311yescharacteristic
314C4×C8C22×D5311yescharacteristic
315#41—42no—
316C2×Q8—1062no—
317C2×Q8—1062no—
318C2×Q8C2×D21211yescharacteristic
319C2×D4—21no—
320C2×D4C2×D21211yescharacteristic
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4C2×D21211yescharacteristic
325C22×C4#1171994511yescharacteristic
326C22×C4C2×C22⋊D5311yescharacteristic
327Q16—2122no—
328Q16—2124no—
329Q16—2122no—
330SD16—2124no—
331M16—42no—
332M16—42no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—41no—
337C2×C8—1062no—
338C2×C8—1062no—
339C2×C8C2×C22⋊D5311yescharacteristic
340C2×C8C2×C22⋊D5311yescharacteristic
341C2×C8C2×C22⋊D5311yescharacteristic
342C2×C8C2×C22⋊D5311yescharacteristic
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—1062no—
346C4⋊C4—1062no—
347C4⋊C4—1062no—
348C4⋊C4—1062no—
349C4⋊C4—2122no—
350C4⋊C4—2122no—
351C4⋊C4C2×D21211yescharacteristic
352C4⋊C4C2×D21211yescharacteristic
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—2122no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—2122no—
361C42C2×D21211yescharacteristic
362C42#1171994511yescharacteristic
363C42C2×C22⋊D5311yescharacteristic
364C42C23×D5311yescharacteristic
365C23—21no—
366C23#1313319111yescharacteristic
367Q8—22no—
368Q8—2122no—
369Q8—2122no—
370Q8—2122no—
371Q8—2122no—
372D4—22no—
373D4—42no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—2122no—
386C2×C4—2122no—
387C2×C4#1313319111yescharacteristic
388C2×C4#1313319111yescharacteristic
389C2×C4#1313319111yescharacteristic
390C2×C4C22×C22⋊D5311yescharacteristic
391C2×C4C22×D21211yescharacteristic
392C2×C4C2×#1171994511yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8—41no—
397C8—41no—
398C8—2122no—
399C8—2122no—
400C8#1313323712yes—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22C2×#1313319111yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—4242no—
415C4—4242no—
416C4#1681089812yes—
417C4C2×#1313323711yescharacteristic
418C4#1681142111yescharacteristic
419C4#1681142312yes—
420C2—41no—
421C2—81no—
422C2#2028417011yescharacteristic
423C2#2028432411yescharacteristic
424C2C2×#1681089811yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2965C2
socleC22×C53C2×#121
prime radicalC2×#41×C53D4
hypercenter#2965D53
hypocenterC53#19161
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. #20285301C2semidirect
3. #20274269C2semidirect
4. #20274262C2semidirect
5. #20274233C2semidirect
6. #20274228C2semidirect
7. #20274209C2semidirect
8. #20274204C2semidirect
9. #20273463C2semidirect
10. #20273379C2semidirect
259. C53#19161semidirect

Maximal subgroups

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

subgroupindex
#202853012
#202847862
#202742692
#202742622
#202742332
#202742282
#202742092
#202742042
#202734632
#202733792
C53×#29652
#1916153