Finite Group Databasebeta

← #27279571

Subgroups — #27279571

Order 13568.

520 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283682C211yesmaximal, characteristic, complemented
2C4×#13133242C211yesmaximal, characteristic
3C2×#16810350C211yesmaximal, characteristic, complemented
4#20273035C211yesmaximal, characteristic, complemented
5#20272565C211yesmaximal, characteristic, complemented
6#20272551C211yesmaximal, characteristic, complemented
7C53×#1477C211yesmaximal, characteristic, complemented
8C2×#13133242C411yescharacteristic
9C2×C4×D212C2211yescharacteristic
10#16810754—21no—
11#16810733—21no—
12#16810721—21no—
13C4×#11719924—21no—
14C4×#11719922—21no—
15#16810701—21no—
16#16810695—22no—
17#16810676—21no—
18#16810676C2211yescharacteristic
19#16810666C411yescharacteristic
20#16810627C411yescharacteristic
21#16810624C411yescharacteristic
22C4×#11719917—21no—
23C4×#11719917C2211yescharacteristic
24C2×C4×#88322C2211yescharacteristic
25#16810350—21no—
26#16810350—21no—
27#16810346—22no—
28C4×#41×C53C2211yescharacteristic
29C53×#431C2211yescharacteristic
30C2×C4⋊C8×C53C2211yescharacteristic
31C53×Q8⋊C8—21no—
32C53×D4⋊C8—21no—
33C22×D212C2×C411yescharacteristic
34C22×C4×D53—21no—
35#13133242—22no—
36#13133242—22no—
37C2×#11719924—21no—
38C2×#11719922—21no—
39C2×#11719917—22no—
40C2×#11719917C2×C411yescharacteristic
41C2×#11719917C2×C411yescharacteristic
42C22×#88322—21no—
43C4×D212—21no—
44C4×D212—21no—
45C4×D212—21no—
46C4×D212—21no—
47C2×C4⋊Dic53C2×C411yescharacteristic
48C2×#11719921—21no—
49#13133150—21no—
50#13133148—21no—
51C4⋊#88322—22no—
52C4⋊#88322—21no—
53#13133141—21no—
54#13133138—21no—
55#13133137—21no—
56#13133137—21no—
57C22⋊#88322—22no—
58C22⋊#88322D412yes—
59#13133128—22no—
60#13133128—21no—
61C4×#88322—21no—
62C4×#88322—21no—
63C4×#88322—21no—
64C4×#88322D411yescharacteristic
65C4×#88322D411yescharacteristic
66C2×#41×C53C2×C411yescharacteristic
67C22×C8×C53—21no—
68C4×Q8×C53D411yescharacteristic
69C4×Q8×C53D411yescharacteristic
70C4×D4×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53D411yescharacteristic
73C4×D4×C53D411yescharacteristic
74#117×C53—21no—
75#117×C53C2×C411yescharacteristic
76C2×C42×C53—21no—
77C2×C42×C53C2311yescharacteristic
78C4⋊C8×C53—21no—
79C4⋊C8×C53—21no—
80C4⋊C8×C53—21no—
81C22⋊C8×C53—21no—
82C4×C8×C53—21no—
83C23×D53—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×C53C22×C411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C22×Dic53—21no—
98C2×D212—21no—
99C2×D212—21no—
100C2×D212—21no—
101C2×D212—21no—
102C2×D212—21no—
103C2×D212M1612yescomplemented
104C2×C4×D53—41no—
105C2×C4×D53—41no—
106C2×C4×D53—81no—
107C2×C4×D53—81no—
108C2×C8×C53—41no—
109C2×C8×C53—41no—
110C2×C8×C53—81no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53C22⋊C411yescharacteristic
114C4⋊C4×C53C22⋊C411yescharacteristic
115C22⋊C4×C53—21no—
116C22⋊C4×C53—21no—
117C22⋊C4×C53—21no—
118C42×C53—21no—
119C42×C53—21no—
120C42×C53—21no—
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53#4111yescharacteristic
125#11719924—24no—
126#11719922—24no—
127#11719921—41no—
128#11719921—41no—
129C4⋊Dic53—21no—
130C4⋊Dic53—21no—
131#11719917—21no—
132#11719917—21no—
133#11719917—21no—
134#11719917—44no—
135#11719917M1612yescomplemented
136C2×#88322—21no—
137C2×#88322—21no—
138C2×#88322—21no—
139C2×#88322—21no—
140C2×#88322—22no—
141C2×#88322—42no—
142C23×C53—21no—
143C23×C53C4×D411yescharacteristic
144C22×D53—22no—
145C22×D53—41no—
146C22×D53—41no—
147C22×D53—81no—
148C22×D53—81no—
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×C53C2×C22⋊C411yescharacteristic
167C2×C4×C53C4×D411yescharacteristic
168C2×C4×C53#12011yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12311yescharacteristic
172C2×C4×C53C2×M1611yescharacteristic
173C2×C4×C53C2×M1611yescharacteristic
174C2×C4×C53#13111yescharacteristic
175C2×C4×C53#13111yescharacteristic
176C2×Dic53—41no—
177C2×Dic53—81no—
178D212—22no—
179D212—22no—
180D212—41no—
181D212—41no—
182C4×D53—82no—
183C4×D53—81no—
184C4×D53—81no—
185C8×C53—81no—
186C8×C53—81no—
187#88322—22no—
188#88322—22no—
189#88322—41no—
190#88322—42no—
191#88322—41no—
192#9097—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—41no—
196C22×C53—41no—
197C22×C53—41no—
198C22×C53#38611yescharacteristic
199C22×C53#43511yescharacteristic
200C22×C53#43611yescharacteristic
201C2×D53—42no—
202C2×D53—82no—
203C2×D53—81no—
204C2×D53—81no—
205C2×D53—82no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—21no—
209C4×C53—41no—
210C4×C53—41no—
211C4×C53—41no—
212C4×C53#40711yescharacteristic
213C4×C53#40811yescharacteristic
214C4×C53#43511yescharacteristic
215C4×C53#43611yescharacteristic
216C4×C53#47111yescharacteristic
217C4×C53#47211yescharacteristic
218Dic53—81no—
219#2965—531no—
220C4×AGL(1,ℤ/8)—531no—
221#1477D5311yescharacteristic, complemented
222C2×D4⋊C8—531no—
223#1460—531no—
224#1283—531no—
225#1278—531no—
226C2×C53—21no—
227C2×C53—21no—
228C2×C53—81no—
229C2×C53#187011yescharacteristic
230C2×C53#190611yescharacteristic
231C2×C53#190711yescharacteristic
232D53—82no—
233C2×AGL(1,ℤ/8)—531no—
234C4×#41C2×D5311yescharacteristic
235C2×C4×D4—531no—
236#446—1061no—
237#442—1061no—
238#440—1061no—
239C4×SD16—1061no—
240C4×D8—1061no—
241#436—1061no—
242#435—1062no—
243#431—531no—
244#431—1061no—
245#431C2×D5311yescharacteristic
246#428—531no—
247C2×C4⋊C8C2×D5311yescharacteristic
248#418—531no—
249#417—531no—
250C4×M16—531no—
251C4×M16—1061no—
252C2×C4×C8—531no—
253Q8⋊C8—21no—
254D4⋊C8—21no—
255D4⋊C8—1061no—
256D4⋊C8—1061no—
257#323—1062no—
258C53#909711yescharacteristic, complemented, Sylow
259C2×#41C4×D5311yescharacteristic
260C22×D4—531no—
261C23×C4—1061no—
262AGL(1,ℤ/8)—1062no—
263AGL(1,ℤ/8)—1062no—
264C2×SD16—1061no—
265C2×D8—1061no—
266C2×M16—531no—
267C2×M16—531no—
268C2×M16—1062no—
269C22×C8—21no—
270C22×C8—1061no—
271C4×Q8D21211yescharacteristic
272C4×Q8C22⋊D5311yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—1061no—
276C4×D4—1061no—
277C4×D4—1061no—
278C4×D4—1061no—
279C4×D4D21211yescharacteristic
280C4×D4C22⋊D5311yescharacteristic
281#117—21no—
282#117C4×D5311yescharacteristic
283C2×C4⋊C4—531no—
284C2×C22⋊C4—1061no—
285C2×C42—21no—
286C2×C42C22×D5311yescharacteristic
287#107—1061no—
288#106—1061no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—1062no—
293C4⋊C8—1061no—
294#103—1061no—
295#102—1061no—
296#102—1061no—
297#102—1061no—
298C22⋊C8—21no—
299C22⋊C8—532no—
300C22⋊C8—1062no—
301#97—1062no—
302#97—1061no—
303C4×C8—21no—
304C4×C8—531no—
305C4×C8—531no—
306C4×C8—1061no—
307C4×C8—1061no—
308C4×C8—1061no—
309C24—1061no—
310#41—21no—
311#41—21no—
312#41—21no—
313#41—21no—
314C2×Q8#1171992111yescharacteristic
315C2×D4—21no—
316C2×D4—532no—
317C2×D4—1061no—
318C2×D4—1061no—
319C2×D4—1061no—
320C2×D4—1061no—
321C2×D4—1061no—
322C2×D4#1171992111yescharacteristic
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—1061no—
327C22×C4—4241no—
328C22×C4—4241no—
329C22×C4—2121no—
330C22×C4—2121no—
331C22×C4C2×C4×D5311yescharacteristic
332C22×C4D4×D5311yescharacteristic
333C22×C4#1171994711yescharacteristic
334SD16—1064no—
335D8—1064no—
336M16—532no—
337M16—1061no—
338M16—1061no—
339M16—1061no—
340M16—2124no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—81no—
344C2×C8—1061no—
345C2×C8—1061no—
346C2×C8—1061no—
347C2×C8—1061no—
348C2×C8—1062no—
349C2×C8—2122no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—1061no—
353C4⋊C4—1061no—
354C4⋊C4#1171992111yescharacteristic
355C4⋊C4#1171992111yescharacteristic
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—2121no—
360C22⋊C4—2121no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42C2×D21211yescharacteristic
365C42D4×D5311yescharacteristic
366C42#1171994711yescharacteristic
367C42C2×C22⋊D5311yescharacteristic
368C23—21no—
369C23—1062no—
370C23—4241no—
371C23—4241no—
372C23—2121no—
373C23—2121no—
374C23#1313317711yescharacteristic
375Q8—21no—
376Q8—21no—
377D4—21no—
378D4—21no—
379D4—41no—
380D4—41no—
381D4—1062no—
382D4—1062no—
383D4—2121no—
384D4—2121no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—4242no—
397C2×C4—4241no—
398C2×C4—4241no—
399C2×C4—4241no—
400C2×C4—2121no—
401C2×C4C2×#1171992111yescharacteristic
402C2×C4#1313317711yescharacteristic
403C2×C4#1313318711yescharacteristic
404C2×C4#1313318911yescharacteristic
405C2×C4#1313318911yescharacteristic
406C2×C4#1313320111yescharacteristic
407C2×C4M16×D5311yescharacteristic
408C2×C4M16×D5311yescharacteristic
409C2×C4#1313322311yescharacteristic
410C2×C4#1313322311yescharacteristic
411C8—81no—
412C8—81no—
413C8—1062no—
414C8—1062no—
415C8—2121no—
416C8—2122no—
417C8—2121no—
418C22—21no—
419C22—21no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—4242no—
424C22—4242no—
425C22—4241no—
426C22—4241no—
427C22—2122no—
428C22#1681050011yescharacteristic
429C22#1681069711yescharacteristic
430C22#1681069811yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—4241no—
438C4#1681058611yescharacteristic
439C4#1681058911yescharacteristic
440C4#1681069711yescharacteristic
441C4#1681069811yescharacteristic
442C4#1681088811yescharacteristic
443C4#1681089111yescharacteristic
444C2—21no—
445C2—21no—
446C2—81no—
447C2—4242no—
448C2#2027501011yescharacteristic
449C2#2027521411yescharacteristic
450C2#2027522111yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1477C2
socleC22×C53#386
prime radicalC2×#13133242C4
cosocleC2×C42×C53C23
hypercenter#1477D53
hypocenterC53#9097
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
1. #20283682C2semidirect
3. C2×#16810350C2semidirect
4. #20273035C2semidirect
5. #20272565C2semidirect
6. #20272551C2semidirect
7. C53×#1477C2semidirect
103. C2×D212M16semidirect
135. #11719917M16semidirect
221. #1477D53semidirect
258. C53#9097semidirect

Maximal subgroups

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

subgroupindex
#202836822
C4×#131332422
C2×#168103502
#202730352
#202725652
#202725512
C53×#14772
#909753