Finite Group Databasebeta

← #27368264

Subgroups — #27368264

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#20285229C211yesmaximal, characteristic, complemented
2#20284877C212yesmaximal
3#20274243C211yesmaximal, characteristic, complemented
4#20274238C211yesmaximal, characteristic, complemented
5#20274233C211yesmaximal, characteristic, complemented
6#20274228C211yesmaximal, characteristic, complemented
7#20274221C212yesmaximal, complemented
8#20274214C212yesmaximal, complemented
9#20273478C211yesmaximal, characteristic, complemented
10#20273423C212yesmaximal, complemented
11C53×#2965C211yesmaximal, characteristic
12#16811180C2212yes—
13#16811028C2211yescharacteristic
14#16811028C2211yescharacteristic
15#16811025—21no—
16#16811025—21no—
17#16811021C2212yes—
18#16810964—22no—
19#16810918C2212yes—
20#16810913C2212yes—
21#16810833C2212yes—
22#16810828C2212yes—
23#16810785—22no—
24#16810728—22no—
25#16810723—22no—
26#16810680C2212yes—
27#16810661—21no—
28#16810661—21no—
29#16810376C2212yes—
30#16810374C2211yescharacteristic
31#16810374C2211yescharacteristic
32#16810371C2212yes—
33#16810371C2212yes—
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×#11719925—22no—
48C2×#11719923—22no—
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#13133151—21no—
60#13133151—21no—
61#13133150—22no—
62#13133149—22no—
63#13133149—22no—
64#13133148—22no—
65C4⋊#88322C2312yes—
66C4⋊#88322C2312yes—
67#13133141—22no—
68#13133140—22no—
69#13133140—22no—
70#13133138—22no—
71C22⋊#88322—22no—
72#13133128—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—22no—
118C2×C8×C53—21no—
119C2×C8×C53—21no—
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—
150C23×C53—21no—
151C23×C53#12111yescharacteristic
152Q8×C53—22no—
153D4×C53—22no—
154D4×C53—42no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—22no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—41no—
162C2×C4×C53—42no—
163C2×C4×C53—42no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53C22×D411yescharacteristic
170C2×C4×C53C22×D411yescharacteristic
171C2×C4×C53C2×#4111yescharacteristic
172C2×Dic53—42no—
173C2×Dic53—42no—
174Dic106—42no—
175Dic106—42no—
176Dic106—82no—
177C8×C53—21no—
178C8×C53—21no—
179C8×C53—41no—
180C8×C53—41no—
181C8×C53#13712yes—
182C8×C53#13712yes—
183#88322—42no—
184#88322—42no—
185#19298—531nomaximal, Sylow
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53C2×#12111yescharacteristic
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53C2×#13711yescharacteristic
201C4×C53C2×#13711yescharacteristic
202C4×C53#57512yes—
203Dic53—82no—
204Dic53—82no—
205#3160—531no—
206#3116—532no—
207#2965D5311yescharacteristic
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—81no—
218C2×C53#303911yescharacteristic
219C2×C53#305411yescharacteristic
220C2×C53#306711yescharacteristic
221#533—532no—
222C4×#41C2×D5311yescharacteristic
223#500—531no—
224#500—531no—
225#499—1061no—
226#499—1061no—
227#498—532no—
228#485—1062no—
229#477—532no—
230#476—532no—
231#462—532no—
232#461—532no—
233#452—1062no—
234#446C2×D5311yescharacteristic
235#446C2×D5311yescharacteristic
236#441—1062no—
237#440—1062no—
238#436C2×D5311yescharacteristic
239#436C2×D5311yescharacteristic
240#435—21no—
241#435—21no—
242#432—532no—
243#431—21no—
244#431C2×D5311yescharacteristic
245#427—1061no—
246#427—1061no—
247C4×M16—21no—
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×#41D21211yescharacteristic
258C2×Q16—1062no—
259C2×SD16—1062no—
260C2×M16—21no—
261C2×M16—21no—
262C22×C8C22⋊D5311yescharacteristic
263C22×C8C22⋊D5311yescharacteristic
264C4⋊Q8—532no—
265C4⋊Q8—532no—
266#125—1062no—
267C22⋊Q8—1062no—
268C22⋊Q8—1062no—
269C4×Q8—1062no—
270C4×Q8—1062no—
271C4×Q8D21211yescharacteristic
272C4×Q8C22×D5311yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C4×D4D21211yescharacteristic
276C4×D4C22×D5311yescharacteristic
277#117—21no—
278#117—1062no—
279#117D21211yescharacteristic
280C2×C42—21no—
281C2×C42C22×D5311yescharacteristic
282#107—1061no—
283#107—1061no—
284#107—1061no—
285#107—1061no—
286#107—1062no—
287#106—1062no—
288#106—1062no—
289#106—1062no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—532no—
293C4⋊C8—532no—
294C4⋊C8C22×D5311yescharacteristic
295C4⋊C8C22×D5311yescharacteristic
296#103—1062no—
297#103—1062no—
298#103—1062no—
299#102—1062no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—1062no—
304#97—21no—
305#97—21no—
306#97—1062no—
307C4×C8—21no—
308C4×C8C22⋊D5311yescharacteristic
309C4×C8C22⋊D5311yescharacteristic
310C4×C8C22×D5311yescharacteristic
311C4×C8C22×D5311yescharacteristic
312#41—42no—
313C2×Q8—1062no—
314C2×Q8—1062no—
315C2×Q8C2×D21211yescharacteristic
316C2×D4—21no—
317C2×D4C2×D21211yescharacteristic
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4C2×D21211yescharacteristic
322C22×C4#1171994511yescharacteristic
323C22×C4C2×C22⋊D5311yescharacteristic
324Q16—2124no—
325SD16—2124no—
326M16—42no—
327M16—42no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—1062no—
333C2×C8—1062no—
334C2×C8C2×C22⋊D5311yescharacteristic
335C2×C8C2×C22⋊D5311yescharacteristic
336C2×C8C2×C22⋊D5311yescharacteristic
337C2×C8C2×C22⋊D5311yescharacteristic
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—1062no—
341C4⋊C4—1062no—
342C4⋊C4—1062no—
343C4⋊C4—1062no—
344C4⋊C4—2122no—
345C4⋊C4—2122no—
346C4⋊C4C2×D21211yescharacteristic
347C4⋊C4C2×D21211yescharacteristic
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—2122no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—2122no—
356C42C2×D21211yescharacteristic
357C42#1171994511yescharacteristic
358C42C2×C22⋊D5311yescharacteristic
359C42C23×D5311yescharacteristic
360C23—21no—
361C23#1313319111yescharacteristic
362Q8—22no—
363Q8—4242no—
364Q8—2122no—
365Q8—2122no—
366D4—22no—
367D4—42no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—2122no—
380C2×C4—2122no—
381C2×C4#1313319111yescharacteristic
382C2×C4#1313319111yescharacteristic
383C2×C4#1313319111yescharacteristic
384C2×C4C22×C22⋊D5311yescharacteristic
385C2×C4C22×D21211yescharacteristic
386C2×C4C2×#1171994511yescharacteristic
387C8—21no—
388C8—21no—
389C8—41no—
390C8—41no—
391C8—2122no—
392C8—2122no—
393C8#1313324712yes—
394C8#1313324712yes—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22C2×#1313319111yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—4242no—
410C4—4242no—
411C4C2×#1313324711yescharacteristic
412C4C2×#1313324711yescharacteristic
413C4#1681142312yes—
414C2—41no—
415C2—81no—
416C2#2028420211yescharacteristic
417C2#2028432411yescharacteristic
418C2#2028443511yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2965C2
socleC22×C53C2×#121
prime radicalC2×#41×C53D4
hypercenter#2965D53
hypocenterC53#19298
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. #20285229C2semidirect
3. #20274243C2semidirect
4. #20274238C2semidirect
5. #20274233C2semidirect
6. #20274228C2semidirect
7. #20274221C2semidirect
8. #20274214C2semidirect
9. #20273478C2semidirect
10. #20273423C2semidirect
256. C53#19298semidirect

Maximal subgroups

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

subgroupindex
#202852292
#202848772
#202742432
#202742382
#202742332
#202742282
#202742212
#202742142
#202734782
#202734232
C53×#29652
#1929853