Finite Group Databasebeta

← #27367491

Subgroups — #27367491

Order 13568.

584 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286052C212yesmaximal
2#20285301C211yesmaximal, characteristic, complemented
3#20274269C212yesmaximal, complemented
4#20274262C212yesmaximal, complemented
5#20274109C211yesmaximal, characteristic, complemented
6#20274104C211yesmaximal, characteristic, complemented
7#20274085C211yesmaximal, characteristic, complemented
8#20274080C211yesmaximal, characteristic, complemented
9C2×#16810376C211yesmaximal, characteristic, complemented
10#20273416C212yesmaximal, complemented
11C53×#2966C211yesmaximal, characteristic
12#16811180C2212yes—
13#16811028C2211yescharacteristic
14#16811003C2211yescharacteristic
15#16810974C2212yes—
16#16810927—22no—
17#16810908C2212yes—
18#16810903C2212yes—
19#16810864—21no—
20#16810864—21no—
21#16810862—22no—
22#16810850—22no—
23#16810842—22no—
24#16810833C2212yes—
25#16810828C2212yes—
26C4×Dic212—21no—
27C4×Dic212—21no—
28C2×C4⋊#88322C2212yes—
29#16810376C2212yes—
30#16810376C2211yescharacteristic
31#16810376C2211yescharacteristic
32#16810370C2212yes—
33#16810370C2212yes—
34#16810368C2212yes—
35#16810368C2212yes—
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—
49C2×#11719925—22no—
50C2×#11719923—22no—
51C22×#88322—22no—
52#13133217C2312yes—
53#13133217C2312yes—
54#13133209—22no—
55#13133196—22no—
56#13133196—22no—
57C4×Dic106—22no—
58C4×Dic106—22no—
59#13133176—22no—
60#13133151—21no—
61#13133151—21no—
62#13133150—22no—
63#13133148—22no—
64C4⋊#88322—22no—
65C4⋊#88322C2312yes—
66C4⋊#88322C2312yes—
67#13133142—22no—
68#13133142—22no—
69#13133141—22no—
70#13133140—22no—
71#13133140—22no—
72#13133138—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—
138#11719925—82no—
139#11719923—82no—
140C4⋊Dic53—22no—
141C4⋊Dic53—22no—
142C4⋊Dic53—22no—
143C4⋊Dic53—22no—
144#11719919—42no—
145#11719919—42no—
146C4×Dic53—42no—
147C2×#88322—22no—
148C2×#88322—22no—
149C2×#88322—82no—
150Dic212—42no—
151Dic212—42no—
152C23×C53—21no—
153C23×C53#12111yescharacteristic
154Q8×C53—21no—
155Q8×C53—21no—
156D4×C53—21no—
157D4×C53—21no—
158D4×C53—42no—
159C2×C4×C53—22no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53C22×D411yescharacteristic
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C2×#4111yescharacteristic
175C2×Dic53—42no—
176C2×Dic53—42no—
177Dic106—42no—
178Dic106—42no—
179Dic106—42no—
180Dic106—42no—
181C8×C53—21no—
182C8×C53—21no—
183C8×C53—21no—
184C8×C53—41no—
185C8×C53—41no—
186C8×C53#13512yes—
187#88322—42no—
188#88322—42no—
189#19223—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—41no—
193C22×C53—41no—
194C22×C53—41no—
195C22×C53#47312yes—
196C22×C53C2×#12111yescharacteristic
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—41no—
202C4×C53—41no—
203C4×C53—41no—
204C4×C53C2×#13511yescharacteristic
205C4×C53#57511yescharacteristic
206C4×C53#57611yescharacteristic
207C4×C53#57811yescharacteristic
208Dic53—82no—
209Dic53—82no—
210#3248—532no—
211#3170—531no—
212#2966D5311yescharacteristic
213#1700—532no—
214#1699—532no—
215#1674—531no—
216#1673—531no—
217#1670—531no—
218#1669—531no—
219C2×#335—531no—
220#1552—532no—
221C2×C53—22no—
222C2×C53—81no—
223C2×C53#303511yescharacteristic
224C2×C53#306111yescharacteristic
225C2×C53C2×#47311yescharacteristic
226#533—532no—
227#529C2×D5311yescharacteristic
228#500—531no—
229#494—531no—
230#487—532no—
231#479—1062no—
232#475—532no—
233#474—532no—
234#467—1061no—
235#467—1061no—
236#467—1062no—
237#465—1062no—
238#464—1062no—
239#462—532no—
240#461—532no—
241#446C2×D5311yescharacteristic
242C8×Q8C2×D5311yescharacteristic
243C4×Q16—1061no—
244C4×Q16—1061no—
245#436C2×D5311yescharacteristic
246#435—21no—
247C8×D4—21no—
248C8×D4C2×D5311yescharacteristic
249#433—21no—
250#432C2×D5311yescharacteristic
251C2×C4⋊C8—532no—
252#405—21no—
253C2×C4×C8C2×D5311yescharacteristic
254#335—531no—
255#335—531no—
256#335—532no—
257#332—532no—
258#332—532no—
259#332—532no—
260#332—532no—
261C53#1922311yescharacteristic, complemented, Sylow
262C2×Q16—1062no—
263C2×Q16—1061no—
264C2×Q16—1061no—
265C2×SD16—1062no—
266C2×M16—21no—
267C22×C8—21no—
268C22×C8—1062no—
269C22×C8C22⋊D5311yescharacteristic
270C22×C8C22⋊D5311yescharacteristic
271C4⋊Q8—532no—
272C4⋊Q8—532no—
273#126—21no—
274#125—1062no—
275#125D21211yescharacteristic
276#124D21211yescharacteristic
277#123—21no—
278C22⋊Q8—1062no—
279C22⋊Q8—1062no—
280C22⋊Q8D21211yescharacteristic
281#121D21211yescharacteristic
282C4×Q8—1062no—
283C4×Q8—1062no—
284C4×Q8C22×D5311yescharacteristic
285C4×D4—21no—
286C4×D4C22×D5311yescharacteristic
287#117—21no—
288#117—1062no—
289C2×C42C22×D5311yescharacteristic
290#107—1061no—
291#107—1061no—
292#107—1062no—
293#106—1062no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—532no—
297C4⋊C8—532no—
298C4⋊C8—1062no—
299C4⋊C8C22×D5311yescharacteristic
300C4⋊C8C22×D5311yescharacteristic
301#103—1062no—
302#103—1062no—
303#103—1062no—
304#103—1062no—
305#103—1062no—
306#102—1062no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310#97—21no—
311#97—21no—
312C4×C8—21no—
313C4×C8C22⋊D5311yescharacteristic
314C4×C8C22⋊D5311yescharacteristic
315C4×C8C22×D5311yescharacteristic
316C4×C8C22×D5311yescharacteristic
317C2×Q8—1062no—
318C2×Q8—1062no—
319C2×Q8C2×D21211yescharacteristic
320C2×D4—21no—
321C2×D4C2×D21211yescharacteristic
322C22×C4—21no—
323C22×C4C2×D21211yescharacteristic
324C22×C4#1171994511yescharacteristic
325C22×C4C2×C22⋊D5311yescharacteristic
326Q16—4242no—
327Q16—2122no—
328Q16—2122no—
329SD16—4242no—
330M16—42no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—1062no—
337C2×C8—1062no—
338C2×C8—4242no—
339C2×C8C2×C22⋊D5311yescharacteristic
340C2×C8C2×C22⋊D5311yescharacteristic
341C2×C8C2×C22⋊D5311yescharacteristic
342C2×C8C2×C22⋊D5311yescharacteristic
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—1062no—
348C4⋊C4—1062no—
349C4⋊C4—1062no—
350C4⋊C4—1062no—
351C4⋊C4—2122no—
352C4⋊C4—2122no—
353C4⋊C4C2×D21211yescharacteristic
354C4⋊C4C2×D21211yescharacteristic
355C22⋊C4—21no—
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—21no—
360C22⋊C4—2122no—
361C42—21no—
362C42—2122no—
363C42C2×D21211yescharacteristic
364C42#1171994511yescharacteristic
365C42C2×C22⋊D5311yescharacteristic
366C42C23×D5311yescharacteristic
367C23—21no—
368C23#1313319111yescharacteristic
369Q8—21no—
370Q8—21no—
371Q8—2122no—
372Q8—2122no—
373Q8—2122no—
374Q8—2122no—
375D4—21no—
376D4—21no—
377D4—42no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—2122no—
389C2×C4—2122no—
390C2×C4#1313319111yescharacteristic
391C2×C4#1313319111yescharacteristic
392C2×C4#1313319111yescharacteristic
393C2×C4C22×C22⋊D5311yescharacteristic
394C2×C4C22×D21211yescharacteristic
395C2×C4C2×#1171994511yescharacteristic
396C8—21no—
397C8—21no—
398C8—21no—
399C8—41no—
400C8—41no—
401C8—2122no—
402C8—2122no—
403C8#1313323712yes—
404C22—21no—
405C22—21no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22#1681089812yes—
410C22C2×#1313319111yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—4242no—
419C4—4242no—
420C4C2×#1313323711yescharacteristic
421C4#1681142111yescharacteristic
422C4#1681142811yescharacteristic
423C4#1681144011yescharacteristic
424C2—22no—
425C2—81no—
426C2#2028417011yescharacteristic
427C2#2028438511yescharacteristic
428C2C2×#1681089811yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2966C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#2966D53
hypocenterC53#19223
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. #20274269C2semidirect
4. #20274262C2semidirect
5. #20274109C2semidirect
6. #20274104C2semidirect
7. #20274085C2semidirect
8. #20274080C2semidirect
9. C2×#16810376C2semidirect
10. #20273416C2semidirect
261. C53#19223semidirect

Maximal subgroups

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

subgroupindex
#202860522
#202853012
#202742692
#202742622
#202741092
#202741042
#202740852
#202740802
C2×#168103762
#202734162
C53×#29662
#1922353