Finite Group Databasebeta

← #27280198

Subgroups — #27280198

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#20283682C211yesmaximal, characteristic, complemented
2#20274735C211yesmaximal, characteristic, complemented
3#20273210C211yesmaximal, characteristic, complemented
4#20273166C211yesmaximal, characteristic, complemented
5#20272612C211yesmaximal, characteristic, complemented
6#20272554C211yesmaximal, characteristic, complemented
7C53×#3035C211yesmaximal, characteristic
8#16810754—21no—
9#16810701—21no—
10#16810695—22no—
11#16810676—21no—
12#16810676C2211yescharacteristic
13#16810647C2211yescharacteristic
14C2×#13133144—21no—
15C4×#11719917—21no—
16C2×C4×#88322C2211yescharacteristic
17C4×C4⋊Dic53C2211yescharacteristic
18#16810382—21no—
19#16810357—22no—
20D4⋊#88322—21no—
21D4⋊#88322—21no—
22C2×#135×C53—21no—
23C53×#532C2211yescharacteristic
24C4×#41×C53C2211yescharacteristic
25C53×#464—21no—
26C53×#463—21no—
27C53×#462C411yescharacteristic, complemented
28C53×#461C411yescharacteristic, complemented
29C53×#460C411yescharacteristic, complemented
30C53×#459C411yescharacteristic, complemented
31C2×C4⋊C8×C53C2211yescharacteristic
32C53×#416—21no—
33C2×#11719917—22no—
34C2×#11719917—21no—
35C22×#88322—21no—
36C2×C4⋊Dic53—21no—
37C2×C4×Dic53—21no—
38C4⋊#88322—22no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42#13133144—42no—
43C22⋊#88322—22no—
44C22⋊#88322D412yes—
45#13133128—22no—
46C4×#88322—21no—
47C4×#88322—21no—
48C4×#88322D411yescharacteristic
49C4×#88322D411yescharacteristic
50#13133124—21no—
51C2×#41×C53—21no—
52C2×#41×C53D411yescharacteristic
53#135×C53—42no—
54#135×C53—42no—
55C2×Q16×C53—21no—
56C2×SD16×C53—21no—
57C2×SD16×C53—21no—
58C2×D8×C53—21no—
59C22×C8×C53—21no—
60C4⋊Q8×C53C2×C411yescharacteristic
61#127×C53C2×C411yescharacteristic
62#124×C53—21no—
63#121×C53—21no—
64#121×C53—21no—
65C4×Q8×C53C2×C411yescharacteristic
66C4×Q8×C53D411yescharacteristic
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70C4×D4×C53—21no—
71C4×D4×C53C2×C411yescharacteristic
72C4×D4×C53D411yescharacteristic
73#117×C53—21no—
74#117×C53D411yescharacteristic
75C2×C42×C53—21no—
76C2×C42×C53C2311yescharacteristic
77C4⋊C8×C53—21no—
78C4⋊C8×C53C2×C411yescharacteristic
79C4⋊C8×C53C2×C411yescharacteristic
80#103×C53—21no—
81#103×C53—21no—
82#102×C53—21no—
83#102×C53—21no—
84#41×C53—21no—
85#41×C53—21no—
86#41×C53—21no—
87#41×C53—21no—
88#41×C53—41no—
89#41×C53—41no—
90#41×C53—41no—
91#41×C53—41no—
92C2×Q8×C53—21no—
93C2×Q8×C53C22⋊C411yescharacteristic
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—21no—
97C2×D4×C53—21no—
98C2×D4×C53—41no—
99C2×D4×C53C22⋊C411yescharacteristic
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—41no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×Dic53—41no—
108Q16×C53—42no—
109SD16×C53—42no—
110SD16×C53—42no—
111D8×C53—42no—
112C2×C8×C53—21no—
113C2×C8×C53—21no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53C22⋊C411yescharacteristic
121C4⋊C4×C53C22⋊C411yescharacteristic
122C22⋊C4×C53—21no—
123C22⋊C4×C53—21no—
124C22⋊C4×C53—21no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C42×C53—21no—
128C42×C53—21no—
129C42×C53—21no—
130C42×C53C22×C411yescharacteristic
131C42×C53C2×D411yescharacteristic
132C42×C53C2×D411yescharacteristic
133C42×C53#4111yescharacteristic
134C4⋊Dic53—22no—
135C4⋊Dic53—41no—
136C4×Dic53—41no—
137C4×Dic53—41no—
138#11719917—44no—
139#11719917—41no—
140#11719917—41no—
141C2×#88322—21no—
142C2×#88322—21no—
143C2×#88322—22no—
144C2×#88322—42no—
145C2×#88322—41no—
146C23×C53—21no—
147C23×C53—41no—
148C23×C53#12111yescharacteristic
149Q8×C53—21no—
150Q8×C53—21no—
151Q8×C53—41no—
152Q8×C53—41no—
153D4×C53—21no—
154D4×C53—21no—
155D4×C53—41no—
156D4×C53—41no—
157D4×C53—41no—
158D4×C53—41no—
159D4×C53—41no—
160D4×C53—41no—
161D4×C53—81no—
162D4×C53—81no—
163D4×C53—81no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—41no—
170C2×C4×C53—42no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—81no—
177C2×C4×C53—81no—
178C2×C4×C53C2×C22⋊C411yescharacteristic
179C2×C4×C53C4×D411yescharacteristic
180C2×C4×C53C4×D411yescharacteristic
181C2×C4×C53#12011yescharacteristic
182C2×C4×C53#12111yescharacteristic
183C2×C4×C53#12311yescharacteristic
184C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C53#13711yescharacteristic
188C2×Dic53—42no—
189C2×Dic53—81no—
190C2×Dic53—81no—
191C8×C53—41no—
192C8×C53—41no—
193#88322—22no—
194#88322—41no—
195#88322—42no—
196#88322—81no—
197#9210—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53—81no—
206C22×C53#38611yescharacteristic
207C22×C53#46811yescharacteristic
208C22×C53#46911yescharacteristic
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—41no—
213C4×C53—41no—
214C4×C53—41no—
215C4×C53—81no—
216C4×C53#41711yescharacteristic
217C4×C53#41811yescharacteristic
218C4×C53#44311yescharacteristic
219C4×C53#44411yescharacteristic
220C4×C53#46811yescharacteristic
221C4×C53#46911yescharacteristic
222Dic53—82no—
223#3035D5311yescharacteristic
224#2965—531no—
225#1806—531no—
226#1502—531no—
227#1492—531no—
228#1301—531no—
229#1279—531no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—81no—
233C2×C53—161no—
234C2×C53#187611yescharacteristic
235C2×C53#189511yescharacteristic
236C2×C53#190611yescharacteristic
237C2×#135—21no—
238#532C2×D5311yescharacteristic
239C4×#41C2×D5311yescharacteristic
240#464—21no—
241#463—21no—
242#462Dic5311yescharacteristic, complemented
243#461Dic5311yescharacteristic, complemented
244#460Dic5311yescharacteristic, complemented
245#459Dic5311yescharacteristic, complemented
246#446—1061no—
247#436—1061no—
248#435—1062no—
249#431—531no—
250#431—1061no—
251#423—531no—
252C2×C4⋊C8C2×D5311yescharacteristic
253C2×#104—1061no—
254#416—21no—
255C4×M16—1061no—
256C2×C4×C8—531no—
257C4×C4⋊C4—531no—
258#339—1061no—
259#327—1062no—
260D4⋊C8—1061no—
261D4⋊C8—1061no—
262C53#921011yescharacteristic, complemented, Sylow
263C2×#41—21no—
264C2×#41C22⋊D5311yescharacteristic
265#135—42no—
266#135—42no—
267C2×Q16—21no—
268C2×SD16—21no—
269C2×SD16—21no—
270C2×D8—21no—
271C2×M16—1062no—
272C2×M16—1061no—
273C22×C8—21no—
274C22×C8—1061no—
275C4⋊Q8C2×Dic5311yescharacteristic
276#127C2×Dic5311yescharacteristic
277#124—21no—
278#121—21no—
279#121—21no—
280C4×Q8C2×Dic5311yescharacteristic
281C4×Q8C22⋊D5311yescharacteristic
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4C2×Dic5311yescharacteristic
287C4×D4C22⋊D5311yescharacteristic
288#117—21no—
289#117C22⋊D5311yescharacteristic
290C2×C4⋊C4—1061no—
291C2×C42—21no—
292C2×C42—1061no—
293C2×C42C22×D5311yescharacteristic
294C4⋊C8—21no—
295C4⋊C8—1062no—
296C4⋊C8—1061no—
297C4⋊C8—1061no—
298C4⋊C8—1061no—
299C4⋊C8C2×Dic5311yescharacteristic
300C4⋊C8C2×Dic5311yescharacteristic
301#104—2122no—
302#103—21no—
303#103—21no—
304#102—21no—
305#102—21no—
306C22⋊C8—532no—
307C22⋊C8—1062no—
308#97—1062no—
309C4×C8—531no—
310C4×C8—531no—
311C4×C8—1061no—
312C4×C8—1061no—
313#95—1061no—
314#41—21no—
315#41—21no—
316#41—21no—
317#41—21no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322C2×Q8—21no—
323C2×Q8C22⋊Dic5311yescharacteristic
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—41no—
329C2×D4C22⋊Dic5311yescharacteristic
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—42no—
339SD16—42no—
340SD16—42no—
341D8—42no—
342M16—2124no—
343M16—2121no—
344M16—2121no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—1061no—
350C2×C8—1061no—
351C2×C8—1062no—
352C2×C8—2122no—
353C2×C8—2121no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—1062no—
359C4⋊C4—2121no—
360C4⋊C4C22⋊Dic5311yescharacteristic
361C4⋊C4C22⋊Dic5311yescharacteristic
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—2121no—
371C42—2121no—
372C42D4×D5311yescharacteristic
373C42#1171994711yescharacteristic
374C42C22×Dic5311yescharacteristic
375C42C2×C22⋊D5311yescharacteristic
376C23—21no—
377C23—41no—
378C23#1313318911yescharacteristic
379Q8—21no—
380Q8—21no—
381Q8—41no—
382Q8—41no—
383D4—21no—
384D4—21no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—41no—
391D4—81no—
392D4—81no—
393D4—81no—
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—41no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—4241no—
409C2×C4—4241no—
410C2×C4—2122no—
411C2×C4C2×C22⋊Dic5311yescharacteristic
412C2×C4D4×Dic5311yescharacteristic
413C2×C4D4×Dic5311yescharacteristic
414C2×C4#1313318511yescharacteristic
415C2×C4#1313318911yescharacteristic
416C2×C4#1313319811yescharacteristic
417C2×C4#1313324211yescharacteristic
418C2×C4#1313324311yescharacteristic
419C2×C4#1313324311yescharacteristic
420C2×C4#1313324711yescharacteristic
421C8—41no—
422C8—41no—
423C8—1062no—
424C8—4241no—
425C8—2121no—
426C8—2122no—
427C22—21no—
428C22—21no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22#1681050311yescharacteristic
436C22#1681086711yescharacteristic
437C22#1681087411yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—41no—
443C4—41no—
444C4—81no—
445C4—4242no—
446C4#1681062411yescharacteristic
447C4#1681062811yescharacteristic
448C4#1681074011yescharacteristic
449C4#1681074511yescharacteristic
450C4#1681086711yescharacteristic
451C4#1681087411yescharacteristic
452C2—21no—
453C2—21no—
454C2—81no—
455C2—161no—
456C2#2027504411yescharacteristic
457C2#2027515411yescharacteristic
458C2#2027521811yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3035C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3035D53
hypocenterC53#9210
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #20283682C2semidirect
2. #20274735C2semidirect
3. #20273210C2semidirect
4. #20273166C2semidirect
5. #20272612C2semidirect
6. #20272554C2semidirect
27. C53×#462C4semidirect
28. C53×#461C4semidirect
29. C53×#460C4semidirect
30. C53×#459C4semidirect
242. #462Dic53semidirect
243. #461Dic53semidirect
244. #460Dic53semidirect
245. #459Dic53semidirect
262. C53#9210semidirect

Maximal subgroups

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

subgroupindex
#202836822
#202747352
#202732102
#202731662
#202726122
#202725542
C53×#30352
#921053