Finite Group Databasebeta

← #27280791

Subgroups — #27280791

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285656C211yesmaximal, characteristic
2#20275696C211yesmaximal, characteristic, complemented
3#20273383C211yesmaximal, characteristic
4#20273278C211yesmaximal, characteristic, complemented
5#20272573C211yesmaximal, characteristic, complemented
6#20272544C211yesmaximal, characteristic, complemented
7C53×#2328C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810995C412yescomplemented
10#16810969C412yescomplemented
11#16810820—22no—
12#16810774—22no—
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810647C2211yescharacteristic
16#16810383—21no—
17#16810383—21no—
18#16810367—22no—
19#16810359—22no—
20C2×#127×C53C2211yescharacteristic
21C2×C4×D4×C53—21no—
22C53×#390—21no—
23C53×#386—22no—
24C53×#386—22no—
25C53×#382C2211yescharacteristic
26C53×#376C2211yescharacteristic
27C2×#11719923—24no—
28C2×#11719917—21no—
29#13133217C2×C411yescharacteristic, complemented
30#13133217C2×C411yescharacteristic, complemented
31#13133209—21no—
32#13133196—21no—
33#13133196—21no—
34C4×Dic106—21no—
35C4×Dic106—21no—
36#13133176—21no—
37C4⋊#88322—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41#13133138—24no—
42C22⋊#88322—21no—
43C22⋊#88322—21no—
44C4×#88322—21no—
45C4×#88322C2×C412yescomplemented
46C22×D4×C53—21no—
47C22×D4×C53—21no—
48C22×D4×C53D412yes—
49C23×C4×C53—22no—
50#127×C53—21no—
51#127×C53—21no—
52#127×C53C2×C412yes—
53#127×C53D412yes—
54C4×D4×C53—42no—
55C4×D4×C53—42no—
56C2×C4⋊C4×C53—21no—
57C2×C22⋊C4×C53—22no—
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53—22no—
60C2×C42×C53D411yescharacteristic
61C2×C42×C53D411yescharacteristic
62C2×C42×C53C2311yescharacteristic
63#95×C53—21no—
64#95×C53—21no—
65#95×C53—21no—
66#95×C53D412yes—
67C24×C53—22no—
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—22no—
72C2×D4×C53—22no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C2×D4×C53—82no—
78C2×D4×C53—82no—
79C2×D4×C53C22⋊C414yes—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—22no—
83C22×C4×C53—44no—
84C22×C4×C53—82no—
85C22×C4×C53—82no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C2×Dic106—21no—
90C2×Dic106—21no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C22⋊C4×C53—44no—
94C22⋊C4×C53—82no—
95C22⋊C4×C53—82no—
96C42×C53—21no—
97C42×C53—21no—
98C42×C53—21no—
99C42×C53SD1612yescomplemented
100C42×C53C22×C411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53#4111yescharacteristic
103C42×C53#4111yescharacteristic
104C22⋊Dic53—41no—
105#11719923—84no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110#11719919—41no—
111#11719919—41no—
112C4×Dic53—41no—
113#11719917—41no—
114#11719917—41no—
115C2×#88322—22no—
116C2×#88322—41no—
117C2×#88322—41no—
118C23×C53—24no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—42no—
125C23×C53#12011yescharacteristic
126D4×C53—44no—
127D4×C53—42no—
128D4×C53—42no—
129D4×C53—42no—
130D4×C53—42no—
131D4×C53—82no—
132D4×C53—82no—
133D4×C53—82no—
134C2×C4×C53—22no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—41no—
145C2×C4×C53—88no—
146C2×C4×C53C2×C22⋊C411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12311yescharacteristic
152C2×C4×C53C2×SD1611yescharacteristic
153C2×C4×C53#13511yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C53#13711yescharacteristic
156C2×Dic53—41no—
157C2×Dic53—41no—
158Dic106—42no—
159Dic106—81no—
160#88322—42no—
161#88322—81no—
162#88322—81no—
163#9295—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—44no—
167C22×C53—44no—
168C22×C53—44no—
169C22×C53—82no—
170C22×C53—82no—
171C22×C53—82no—
172C22×C53—82no—
173C22×C53#38611yescharacteristic
174C22×C53#45011yescharacteristic
175C22×C53#45211yescharacteristic
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—42no—
181C4×C53—81no—
182C4×C53—82no—
183C4×C53C4×SD1611yescharacteristic
184C4×C53#44011yescharacteristic
185C4×C53#46111yescharacteristic
186C4×C53#46311yescharacteristic
187Dic53—81no—
188Dic53—81no—
189#3208—531no—
190#2328D5311yescharacteristic
191#1995—531no—
192#1545—531no—
193#1518—531no—
194#1286—531no—
195#1274—531no—
196C2×C53—21no—
197C2×C53—21no—
198C2×C53—84no—
199C2×C53#187711yescharacteristic
200C2×C53#190411yescharacteristic
201C2×C53#191111yescharacteristic
202#533—531no—
203C2×#127C2×D5311yescharacteristic
204C2×C4×D4—21no—
205#492—532no—
206#486—532no—
207#460—1062no—
208#450—1062no—
209#431—531no—
210#431—531no—
211#423—531no—
212#390—21no—
213#386—22no—
214#386—22no—
215#382C2×D5311yescharacteristic
216#376C2×D5311yescharacteristic
217#339—1061no—
218#339—1061no—
219#331—1062no—
220#328—1062no—
221C53#929511yescharacteristic, complemented, Sylow
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C22⋊D5312yes—
225C23×C4—22no—
226C2×SD16—1064no—
227C2×M16—1061no—
228C4⋊Q8—531no—
229C4⋊Q8—531no—
230#127—21no—
231#127—21no—
232#127C4×D5312yes—
233#127D21212yes—
234#125—1061no—
235C22⋊Q8—1061no—
236C22⋊Q8—1061no—
237C4×Q8—1061no—
238C4×Q8—1061no—
239C4×D4—42no—
240C4×D4—42no—
241#117—1061no—
242C2×C4⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C42C22⋊D5311yescharacteristic
247C2×C42C22⋊D5311yescharacteristic
248C2×C42C22×D5311yescharacteristic
249C4⋊C8—1061no—
250C4⋊C8—1061no—
251C4⋊C8—1061no—
252C4⋊C8—1061no—
253#102—1064no—
254C22⋊C8—1061no—
255C22⋊C8—1061no—
256C4×C8—532no—
257C4×C8—1061no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95C22⋊D5312yes—
262C24—22no—
263C2×Q8—1061no—
264C2×Q8—1061no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4#1171992114yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—44no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4C2×C22⋊D5311yescharacteristic
284C22×C4C2×C22⋊D5311yescharacteristic
285C22×C4C2×C22⋊D5311yescharacteristic
286SD16—4244no—
287M16—2121no—
288M16—2121no—
289C2×C8—1062no—
290C2×C8—2121no—
291C2×C8—2121no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—1061no—
295C4⋊C4—1061no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4—2121no—
299C4⋊C4—2121no—
300C22⋊C4—44no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—2121no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—2121no—
308C42#1171992312yescomplemented
309C42C2×C4×D5311yescharacteristic
310C42C2×D21211yescharacteristic
311C42#1171994511yescharacteristic
312C42#1171994511yescharacteristic
313C23—24no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23#1313318611yescharacteristic
321Q8—4241no—
322Q8—2122no—
323D4—44no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—88no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4C2×#1171992111yescharacteristic
346C2×C4C4×C22⋊D5311yescharacteristic
347C2×C4C4×C22⋊D5311yescharacteristic
348C2×C4#1313319111yescharacteristic
349C2×C4#1313319111yescharacteristic
350C2×C4#1313320011yescharacteristic
351C2×C4C2×#1171992311yescharacteristic
352C2×C4#1313323711yescharacteristic
353C2×C4#1313324311yescharacteristic
354C2×C4#1313324711yescharacteristic
355C8—4241no—
356C8—4241no—
357C8—2122no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22#1681050211yescharacteristic
368C22#1681077411yescharacteristic
369C22#1681078511yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—82no—
376C4—81no—
377C4—4241no—
378C4—4241no—
379C4C4×#1171992311yescharacteristic
380C4#1681072311yescharacteristic
381C4#1681082811yescharacteristic
382C4#1681083811yescharacteristic
383C2—21no—
384C2—21no—
385C2—84no—
386C2#2027504911yescharacteristic
387C2#2027520211yescharacteristic
388C2#2027524711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2328C2
socleC22×C53#386
prime radicalC2×#127×C53C22
cosocleC2×C42×C53C23
hypercenter#2328D53
hypocenterC53#9295
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #20275696C2semidirect
4. #20273278C2semidirect
5. #20272573C2semidirect
6. #20272544C2semidirect
9. #16810995C4semidirect
10. #16810969C4semidirect
29. #13133217C2×C4semidirect
30. #13133217C2×C4semidirect
45. C4×#88322C2×C4semidirect
99. C42×C53SD16semidirect
221. C53#9295semidirect
308. C42#11719923semidirect

Maximal subgroups

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

subgroupindex
#202856562
#202756962
#202733832
#202732782
#202725732
#202725442
C53×#23282
#929553