Finite Group Databasebeta

← #27287103

Subgroups — #27287103

Order 13568.

588 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279741C211yesmaximal, characteristic, complemented
2#20275642C211yesmaximal, characteristic
3#20275641C211yesmaximal, characteristic
4#20275521C211yesmaximal, characteristic
5#20272599C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×#13133217—21no—
9C2×#13133216—21no—
10C2×#13133214C2211yescharacteristic
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133139—22no—
13C2×#13133139—21no—
14C2×#13133139—21no—
15C2×#13133137—21no—
16C2×#13133137—21no—
17#16810544—22no—
18#16810522—21no—
19#16810521—21no—
20#16810493C2211yescharacteristic
21C4×C4⋊Dic53C2211yescharacteristic
22C2×C4⋊C8×C53C2211yescharacteristic
23C2×C4×C8×C53C2211yescharacteristic
24C4×C4⋊C4×C53C2211yescharacteristic
25C22×Dic106—22no—
26C22×D212—21no—
27C22×D212—21no—
28C22×D212—21no—
29C22×#88322—21no—
30#13133217—42no—
31#13133217—42no—
32#13133216—42no—
33#13133216—42no—
34#13133214—21no—
35#13133214—21no—
36#13133214—21no—
37#13133214Q812yescomplemented
38C2×C4⋊Dic53—22no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×C4⋊Dic53D411yescharacteristic
43C2×C4⋊Dic53D412yes—
44C2×#11719919—22no—
45C2×C4×Dic53—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48#13133139—44no—
49#13133139—42no—
50#13133139—42no—
51#13133137—42no—
52#13133137—42no—
53#13133124—22no—
54#13133124—21no—
55C22×C8×C53—21no—
56C22×C8×C53—21no—
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C4⋊C4×C53D411yescharacteristic
59C2×C42×C53—21no—
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4×C8×C53—21no—
64C4×C8×C53—21no—
65#95×C53—21no—
66C23×D53—41no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—22no—
75C2×D212—22no—
76C2×D212—22no—
77C2×D212—22no—
78C2×D212—42no—
79C2×D212—42no—
80C2×D212—41no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×Dic106—44no—
92C2×Dic106—44no—
93C2×Dic106—82no—
94C2×C8×C53—22no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53D812yes—
103C4⋊C4×C53SD1612yes—
104C42×C53—41no—
105C42×C53—41no—
106C42×C53C2×Q811yescharacteristic
107C42×C53#4111yescharacteristic
108C42×C53#4111yescharacteristic
109C42×C53#4111yescharacteristic
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112C4⋊Dic53—44no—
113C4⋊Dic53—41no—
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116C4⋊Dic53—41no—
117C4⋊Dic53—42no—
118C4⋊Dic53D812yes—
119C4⋊Dic53SD1612yes—
120#11719919—44no—
121C4×Dic53—41no—
122C4×Dic53—41no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322—41no—
126C23×C53#12011yescharacteristic
127C22×D53—42no—
128C22×D53—81no—
129C22×D53—81no—
130C22×D53—81no—
131C22×D53—81no—
132C22×D53—81no—
133C22×D53—81no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53C22⋊Q811yescharacteristic
143C2×C4×C53C22⋊Q811yescharacteristic
144C2×C4×C53C22⋊Q811yescharacteristic
145C2×C4×C53#12411yescharacteristic
146C2×C4×C53#12411yescharacteristic
147C2×C4×C53#12411yescharacteristic
148C2×C4×C53C2×D811yescharacteristic
149C2×C4×C53C2×D811yescharacteristic
150C2×C4×C53C2×SD1611yescharacteristic
151C2×C4×C53C2×SD1611yescharacteristic
152C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C53#13711yescharacteristic
155C2×C4×C53#13711yescharacteristic
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—42no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165D212—44no—
166D212—42no—
167D212—42no—
168D212—42no—
169D212—42no—
170D212—82no—
171D212—81no—
172D212—81no—
173D212—81no—
174D212—81no—
175Dic106—88no—
176C8×C53—42no—
177C8×C53—81no—
178#88322—81no—
179#10348—531nomaximal, Sylow
180C22×C53#39311yescharacteristic
181C22×C53#44711yescharacteristic
182C22×C53#44711yescharacteristic
183C22×C53#44811yescharacteristic
184C22×C53#44811yescharacteristic
185C22×C53#45311yescharacteristic
186C22×C53#45411yescharacteristic
187C2×D53—82no—
188C2×D53—82no—
189C2×D53—82no—
190C2×D53—161no—
191C2×D53—161no—
192C2×D53—161no—
193C2×D53—161no—
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—42no—
197C4×C53—42no—
198C4×C53#47411yescharacteristic
199C4×C53#47411yescharacteristic
200C4×C53#47611yescharacteristic
201C4×C53#47611yescharacteristic
202C4×C53#48611yescharacteristic
203C4×C53#48611yescharacteristic
204C4×C53#48911yescharacteristic
205C4×C53#48911yescharacteristic
206Dic53—42no—
207Dic53—42no—
208Dic53—82no—
209#2551—531no—
210#1982—531no—
211#1982—531no—
212#1957—531no—
213#1296—531no—
214#1279—531no—
215#1279D5311yescharacteristic, complemented
216C2×C53#161911yescharacteristic
217C2×C53#162311yescharacteristic
218C2×C53#162711yescharacteristic
219C2×C53#162711yescharacteristic
220C2×C53#202311yescharacteristic
221C2×C53#202311yescharacteristic
222C2×C53#202711yescharacteristic
223D53—162no—
224C2×C4⋊Q8—1061no—
225C2×C4⋊Q8—1061no—
226C2×#127—531no—
227C2×C4⋊C8—531no—
228C2×C4⋊C8C2×D5311yescharacteristic
229C2×#102—1062no—
230C2×#102—1061no—
231C2×#102—1061no—
232C2×#102—1061no—
233C2×#102—1061no—
234C2×C4×C8C2×D5311yescharacteristic
235#395—1062no—
236#389—1061no—
237#389—1061no—
238#384—531no—
239C4×C4⋊C4—531no—
240C4×C4⋊C4C2×D5311yescharacteristic
241C53#1034811yescharacteristic, complemented, Sylow
242C22×Q8—1062no—
243C22×D4—1061no—
244C22×D4—1061no—
245C22×D4—1061no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—1061no—
249C4⋊Q8—2122no—
250C4⋊Q8—2122no—
251C4⋊Q8—2122no—
252C4⋊Q8—2122no—
253#127—532no—
254#127—1061no—
255#127—1061no—
256#127—1061no—
257C2×C4⋊C4—531no—
258C2×C4⋊C4—531no—
259C2×C4⋊C4—532no—
260C2×C4⋊C4—1062no—
261C2×C4⋊C4—1062no—
262C2×C4⋊C4—1061no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4D21211yescharacteristic
265C2×C4⋊C4D21211yescharacteristic
266C2×C42—21no—
267C2×C42—1061no—
268C2×C42C22×D5311yescharacteristic
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—1061no—
272C4⋊C8—1061no—
273#102—2122no—
274#102—2122no—
275#102—2124no—
276#102—2122no—
277#102—2122no—
278C4×C8—21no—
279C4×C8—21no—
280#95—21no—
281#95—1062no—
282#95—1061no—
283C24—2121no—
284C2×Q8—4242no—
285C2×Q8—2124no—
286C2×Q8—2124no—
287C2×D4—1062no—
288C2×D4—1062no—
289C2×D4—1062no—
290C2×D4—2122no—
291C2×D4—2122no—
292C2×D4—2121no—
293C2×D4—2121no—
294C2×D4—2121no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2121no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—1061no—
306C22×C4—1061no—
307C22×C4—1062no—
308C22×C4C2×D21211yescharacteristic
309C22×C4D4×D5311yescharacteristic
310C22×C4D4×D5311yescharacteristic
311C2×C8—22no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—2121no—
318C2×C8—2121no—
319C2×C8—2121no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—532no—
323C4⋊C4—532no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C4⋊C4—2122no—
327C4⋊C4—2124no—
328C4⋊C4—2124no—
329C4⋊C4—2121no—
330C4⋊C4—2121no—
331C4⋊C4—2121no—
332C4⋊C4—2121no—
333C4⋊C4#1171991312yes—
334C4⋊C4D42412yes—
335C42—41no—
336C42—41no—
337C42—2121no—
338C42—2121no—
339C42#1171994511yescharacteristic
340C42#1171994511yescharacteristic
341C42#1171994711yescharacteristic
342C42Q8×D5311yescharacteristic
343C23—4241no—
344C23—4241no—
345C23—4241no—
346C23—4241no—
347C23—4241no—
348C23—4241no—
349C23—2122no—
350C23#1313318711yescharacteristic
351Q8—4248no—
352D4—4242no—
353D4—4241no—
354D4—4241no—
355D4—4241no—
356D4—4241no—
357D4—2124no—
358D4—2122no—
359D4—2122no—
360D4—2122no—
361D4—2122no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—1062no—
371C2×C4—1062no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4—2122no—
375C2×C4—2121no—
376C2×C4—2121no—
377C2×C4—2121no—
378C2×C4—2121no—
379C2×C4#1313319511yescharacteristic
380C2×C4#1313319511yescharacteristic
381C2×C4#1313319711yescharacteristic
382C2×C4#1313320311yescharacteristic
383C2×C4#1313320311yescharacteristic
384C2×C4#1313320511yescharacteristic
385C2×C4D8×D5311yescharacteristic
386C2×C4C2×D42411yescharacteristic
387C2×C4SD16×D5311yescharacteristic
388C2×C4C2×#1171991311yescharacteristic
389C2×C4#1313324411yescharacteristic
390C2×C4#1313324511yescharacteristic
391C2×C4#1313324911yescharacteristic
392C2×C4#1313325011yescharacteristic
393C8—42no—
394C8—81no—
395C8—4241no—
396C22—8481no—
397C22—8481no—
398C22—8481no—
399C22—8481no—
400C22—4242no—
401C22—4242no—
402C22—4242no—
403C22#1681053611yescharacteristic
404C22#1681075811yescharacteristic
405C22#1681075911yescharacteristic
406C22#1681076311yescharacteristic
407C22#1681076411yescharacteristic
408C22#1681079211yescharacteristic
409C22#1681079711yescharacteristic
410C4—21no—
411C4—21no—
412C4—42no—
413C4—42no—
414C4—4242no—
415C4—2122no—
416C4—2122no—
417C4#1681090211yescharacteristic
418C4#1681090411yescharacteristic
419C4#1681091211yescharacteristic
420C4#1681091411yescharacteristic
421C4#1681096811yescharacteristic
422C4#1681097011yescharacteristic
423C4#1681098311yescharacteristic
424C4#1681098411yescharacteristic
425C2—8482no—
426C2#2027377111yescharacteristic
427C2#2027379511yescharacteristic
428C2#2027382011yescharacteristic
429C2#2027382111yescharacteristic
430C2#2027585011yescharacteristic
431C2#2027585211yescharacteristic
432C2#2027587011yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenter#1279D53
hypocenterC53#10348
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. #20279741C2semidirect
5. #20272599C2semidirect
6. #20272555C2semidirect
7. C53×#1279C2semidirect
37. #13133214Q8semidirect
215. #1279D53semidirect
241. C53#10348semidirect

Maximal subgroups

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

subgroupindex
#202797412
#202756422
#202756412
#202755212
#202725992
#202725552
C53×#12792
#1034853