Finite Group Databasebeta

← #27280194

Subgroups — #27280194

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284170C211yesmaximal, characteristic
2#20283682C211yesmaximal, characteristic, complemented
3#20273209C211yesmaximal, characteristic, complemented
4#20273165C211yesmaximal, characteristic, complemented
5#20272610C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1806C211yesmaximal, characteristic, complemented
8C2×#13133237—21no—
9#16811173C2211yescharacteristic
10#16810845—21no—
11#16810838—21no—
12#16810833C411yescharacteristic, complemented
13#16810828C411yescharacteristic, complemented
14#16810823C411yescharacteristic, complemented
15#16810818C411yescharacteristic, complemented
16#16810754—21no—
17#16810701—21no—
18#16810695—22no—
19#16810676—21no—
20#16810676C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22#16810617—21no—
23C4×#11719917—21no—
24C2×C4×#88322C2211yescharacteristic
25#16810356—22no—
26#16810350—21no—
27#16810350—21no—
28C4×#41×C53C2211yescharacteristic
29C53×#423C2211yescharacteristic
30C2×#104×C53—21no—
31C4×C4⋊C4×C53C2211yescharacteristic
32C53×#339—21no—
33C2×#11719945—21no—
34#13133237—42no—
35#13133237—42no—
36C2×#11719925—21no—
37C2×#11719923—21no—
38C2×#11719924—21no—
39C2×#11719922—21no—
40C2×#11719917—22no—
41C22×#88322—21no—
42C22×#88322—21no—
43#13133217C2×C411yescharacteristic
44#13133214C2×C411yescharacteristic
45#13133205—21no—
46#13133191—21no—
47#13133191—21no—
48C4×D212—21no—
49C4×D212—21no—
50C4⋊#88322—22no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322C2×C411yescharacteristic
54C4⋊#88322C2×C411yescharacteristic
55#13133140—21no—
56#13133140—21no—
57#13133137—21no—
58#13133137—21no—
59C22⋊#88322—22no—
60C22⋊#88322D412yes—
61#13133128—22no—
62C4×#88322—21no—
63C4×#88322—21no—
64C4×#88322D411yescharacteristic
65C4×#88322D411yescharacteristic
66C2×#41×C53D411yescharacteristic
67C2×M16×C53—21no—
68C4×Q8×C53C2×C411yescharacteristic
69C4×Q8×C53D411yescharacteristic
70C4×D4×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53C2×C411yescharacteristic
73C4×D4×C53D411yescharacteristic
74#117×C53—21no—
75#117×C53D411yescharacteristic
76C2×C4⋊C4×C53—21no—
77C2×C42×C53—21no—
78C2×C42×C53—21no—
79C2×C42×C53C2311yescharacteristic
80C4⋊C8×C53—21no—
81C4⋊C8×C53—21no—
82#104×C53—42no—
83#95×C53—21no—
84#41×C53—21no—
85#41×C53—21no—
86#41×C53—21no—
87#41×C53—21no—
88C2×Q8×C53C22⋊C411yescharacteristic
89C2×D4×C53—21no—
90C2×D4×C53C22⋊C411yescharacteristic
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—41no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53#4111yescharacteristic
98C2×C22⋊D53—41no—
99#11719945—41no—
100#11719945—41no—
101#11719945—41no—
102#11719945—41no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212—21no—
106C2×C4×D53—41no—
107C2×Dic106—21no—
108M16×C53—41no—
109M16×C53—41no—
110C2×C8×C53—41no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53C22⋊C411yescharacteristic
116C4⋊C4×C53C22⋊C411yescharacteristic
117C22⋊C4×C53—21no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C42×C53—21no—
121C42×C53—21no—
122C42×C53—21no—
123C42×C53—41no—
124C42×C53—41no—
125C42×C53C22×C411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53C2×D411yescharacteristic
128C42×C53#4111yescharacteristic
129#11719925—42no—
130#11719924—42no—
131#11719923—42no—
132#11719922—42no—
133#11719921—41no—
134#11719921—41no—
135C4⋊Dic53—21no—
136C4⋊Dic53—21no—
137#11719917—44no—
138C2×#88322—21no—
139C2×#88322—21no—
140C2×#88322—21no—
141C2×#88322—21no—
142C2×#88322—22no—
143C2×#88322—42no—
144C2×#88322—41no—
145C2×#88322—41no—
146C23×C53—21no—
147C23×C53#12111yescharacteristic
148C22×D53—41no—
149Q8×C53—21no—
150Q8×C53—21no—
151D4×C53—21no—
152D4×C53—21no—
153D4×C53—41no—
154D4×C53—41no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—41no—
161C2×C4×C53—42no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—81no—
168C2×C4×C53—81no—
169C2×C4×C53C2×C22⋊C411yescharacteristic
170C2×C4×C53C4×D411yescharacteristic
171C2×C4×C53C4×D411yescharacteristic
172C2×C4×C53#12011yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#12311yescharacteristic
175C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C53#13711yescharacteristic
179C22⋊D53—81no—
180C22⋊D53—81no—
181C2×Dic53—41no—
182D212—41no—
183D212—41no—
184D212—41no—
185D212—41no—
186D212—81no—
187C4×D53—81no—
188C4×D53—81no—
189Dic106—41no—
190Dic106—41no—
191C8×C53—81no—
192#88322—22no—
193#88322—41no—
194#88322—42no—
195#88322—41no—
196#88322—41no—
197#9210—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53#38611yescharacteristic
204C22×C53#46811yescharacteristic
205C22×C53#46911yescharacteristic
206C2×D53—81no—
207C2×D53—81no—
208C2×D53—81no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53—41no—
215C4×C53—82no—
216C4×C53#41711yescharacteristic
217C4×C53#41811yescharacteristic
218C4×C53#44311yescharacteristic
219C4×C53#44411yescharacteristic
220C4×C53#46811yescharacteristic
221C4×C53#46911yescharacteristic
222Dic53—81no—
223#3035—531no—
224#2965—531no—
225#1806D5311yescharacteristic, complemented
226#1502—531no—
227#1492—531no—
228#1301—531no—
229#1279—531no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—81no—
233C2×C53#187611yescharacteristic
234C2×C53#189511yescharacteristic
235C2×C53#190611yescharacteristic
236D53—161no—
237C2×#135—1061no—
238#532—531no—
239C4×#41C2×D5311yescharacteristic
240#464—1061no—
241#463—1061no—
242#462—531no—
243#461—531no—
244#460—531no—
245#459—531no—
246#446—1061no—
247#436—1061no—
248#435—1062no—
249#431—531no—
250#431—1061no—
251#423C2×D5311yescharacteristic
252C2×C4⋊C8—531no—
253C2×#104—21no—
254#416—1061no—
255C4×M16—1061no—
256C2×C4×C8—531no—
257C4×C4⋊C4C2×D5311yescharacteristic
258#339—21no—
259#327—1062no—
260D4⋊C8—1061no—
261D4⋊C8—1061no—
262C53#921011yescharacteristic, complemented, Sylow
263C2×#41—1061no—
264C2×#41C22⋊D5311yescharacteristic
265#135—2122no—
266#135—2122no—
267C2×Q16—1061no—
268C2×SD16—1061no—
269C2×SD16—1061no—
270C2×D8—1061no—
271C2×M16—21no—
272C2×M16—1062no—
273C22×C8—1061no—
274C22×C8—1061no—
275C4⋊Q8—531no—
276#127—531no—
277#124—1061no—
278#121—1061no—
279#121—1061no—
280C4×Q8C4×D5311yescharacteristic
281C4×Q8D21211yescharacteristic
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—1061no—
285C4×D4—1061no—
286C4×D4C4×D5311yescharacteristic
287C4×D4D21211yescharacteristic
288#117—21no—
289#117C22⋊D5311yescharacteristic
290C2×C4⋊C4—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C22×D5311yescharacteristic
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—531no—
297C4⋊C8—531no—
298C4⋊C8—1062no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301#104—42no—
302#103—1061no—
303#103—1061no—
304#102—1061no—
305#102—1061no—
306C22⋊C8—532no—
307C22⋊C8—1062no—
308#97—1062no—
309C4×C8—531no—
310C4×C8—531no—
311C4×C8—1061no—
312C4×C8—1061no—
313#95—21no—
314#41—21no—
315#41—21no—
316#41—21no—
317#41—21no—
318#41—2121no—
319#41—2121no—
320#41—2121no—
321#41—2121no—
322C2×Q8—1061no—
323C2×Q8#1171992111yescharacteristic
324C2×D4—21no—
325C2×D4—1061no—
326C2×D4—1061no—
327C2×D4—1061no—
328C2×D4—2121no—
329C2×D4#1171992111yescharacteristic
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—41no—
334C22×C4—2121no—
335C22×C4D4×D5311yescharacteristic
336C22×C4#1171994711yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338Q16—2122no—
339SD16—2122no—
340SD16—2122no—
341D8—2122no—
342M16—41no—
343M16—41no—
344M16—2124no—
345C2×C8—41no—
346C2×C8—1061no—
347C2×C8—1061no—
348C2×C8—1061no—
349C2×C8—1061no—
350C2×C8—1062no—
351C2×C8—2122no—
352C2×C8—2121no—
353C2×C8—2121no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—41no—
358C4⋊C4—1061no—
359C4⋊C4—1061no—
360C4⋊C4#1171992111yescharacteristic
361C4⋊C4#1171992111yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—2121no—
366C22⋊C4—2121no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42C2×C4×D5311yescharacteristic
373C42C2×D21211yescharacteristic
374C42D4×D5311yescharacteristic
375C42#1171994711yescharacteristic
376C23—21no—
377C23—2121no—
378C23#1313318911yescharacteristic
379Q8—21no—
380Q8—21no—
381Q8—2121no—
382Q8—2121no—
383D4—21no—
384D4—21no—
385D4—41no—
386D4—41no—
387D4—4241no—
388D4—4241no—
389D4—4241no—
390D4—2121no—
391D4—2121no—
392D4—2121no—
393D4—2121no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—4241no—
409C2×C4—4241no—
410C2×C4—2121no—
411C2×C4C2×#1171992111yescharacteristic
412C2×C4#1313317711yescharacteristic
413C2×C4#1313317711yescharacteristic
414C2×C4#1313318711yescharacteristic
415C2×C4#1313318911yescharacteristic
416C2×C4#1313320111yescharacteristic
417C2×C4#1313324211yescharacteristic
418C2×C4#1313324211yescharacteristic
419C2×C4#1313324311yescharacteristic
420C2×C4#1313324811yescharacteristic
421C8—81no—
422C8—1062no—
423C8—2121no—
424C8—2122no—
425C8—2121no—
426C8—2121no—
427C22—21no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—4241no—
433C22—4241no—
434C22—4241no—
435C22#1681050011yescharacteristic
436C22#1681087111yescharacteristic
437C22#1681087411yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—82no—
445C4—4241no—
446C4#1681062211yescharacteristic
447C4#1681062711yescharacteristic
448C4#1681073711yescharacteristic
449C4#1681074211yescharacteristic
450C4#1681087111yescharacteristic
451C4#1681087411yescharacteristic
452C2—21no—
453C2—21no—
454C2—81no—
455C2—8481no—
456C2#2027504011yescharacteristic
457C2#2027515111yescharacteristic
458C2#2027521411yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1806C2
socleC22×C53#386
prime radical#20284170C2
cosocleC2×C42×C53C23
hypercenter#1806D53
hypocenterC53#9210
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

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

NH ≅ G/Ntype
2. #20283682C2semidirect
3. #20273209C2semidirect
4. #20273165C2semidirect
5. #20272610C2semidirect
6. #20272556C2semidirect
7. C53×#1806C2semidirect
12. #16810833C4semidirect
13. #16810828C4semidirect
14. #16810823C4semidirect
15. #16810818C4semidirect
225. #1806D53semidirect
262. C53#9210semidirect

Maximal subgroups

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

subgroupindex
#202841702
#202836822
#202732092
#202731652
#202726102
#202725562
C53×#18062
#921053