Finite Group Databasebeta

← #27277992

Subgroups — #27277992

Order 13568.

440 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283833C211yesmaximal, characteristic
2#20274866C211yesmaximal, characteristic, complemented
3C4×#13133143C211yesmaximal, characteristic
4#20273467C211yesmaximal, characteristic, complemented
5#20273103C211yesmaximal, characteristic
6#20272547C211yesmaximal, characteristic, complemented
7C53×#1283C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810750—21no—
10#16810700—21no—
11#16810694—22no—
12#16810679—21no—
13#16810676C2211yescharacteristic
14#16810645—21no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133146—22no—
17C2×#13133143C411yescharacteristic
18C2×#13133143C411yescharacteristic
19C42×Dic53C2211yescharacteristic
20#16810384C411yescharacteristic
21#16810384C411yescharacteristic
22#16810376—21no—
23#16810374—21no—
24#16810355—21no—
25#16810352—21no—
26C53×#431C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C4×M16×C53C2211yescharacteristic
29C2×#11719945C2×C411yescharacteristic
30C2×#11719912—22no—
31C22×#88322—21no—
32C4×Dic106D411yescharacteristic
33C4×Dic106Q811yescharacteristic
34C4×D212D411yescharacteristic
35C4×D212Q811yescharacteristic
36C4×C22⋊D53—21no—
37C4×C22⋊D53—21no—
38#13133176C2×C411yescharacteristic
39#13133174—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53C2×C411yescharacteristic
44C42×D53—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322D411yescharacteristic
48C4⋊#88322D411yescharacteristic
49#13133146—22no—
50#13133146—44no—
51#13133143—22no—
52#13133143—22no—
53C22⋊#88322—21no—
54#13133129—22no—
55#13133127—22no—
56C4×#88322—21no—
57C2×M16×C53C2×C411yescharacteristic
58C2×M16×C53C2×C411yescharacteristic
59C22×C8×C53D412yes—
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65#97×C53—21no—
66C4×C8×C53—21no—
67C4×C8×C53—21no—
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4111yescharacteristic
71C2×C22⋊D53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74#11719945—21no—
75#11719945—21no—
76#11719945—21no—
77#11719945—21no—
78C2×D212C4⋊C411yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×Dic106C4⋊C411yescharacteristic
83M16×C53—21no—
84M16×C53—21no—
85M16×C53—21no—
86M16×C53M1612yescomplemented
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—22no—
90C2×C8×C53—42no—
91C2×C8×C53—41no—
92C42×C53C2×D411yescharacteristic
93C42×C53C2×D411yescharacteristic
94C42×C53C2×Q811yescharacteristic
95C42×C53#4111yescharacteristic
96C22⋊Dic53—21no—
97#11719921—21no—
98#11719921—21no—
99C4⋊Dic53C4⋊C411yescharacteristic
100C4⋊Dic53C4⋊C411yescharacteristic
101#11719919—21no—
102#11719919—21no—
103C4×Dic53—22no—
104C4×Dic53—22no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C4×Dic53—21no—
109C4×Dic53—21no—
110C4×Dic53—21no—
111C4×Dic53—42no—
112C4×Dic53M1612yescomplemented
113C2×#88322—21no—
114C2×#88322—21no—
115C2×#88322—42no—
116C2×#88322—41no—
117#11719912—82no—
118C23×C53C4×D411yescharacteristic
119C22×D53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—41no—
123C2×C4×C53C2×C4⋊C411yescharacteristic
124C2×C4×C53C4×D411yescharacteristic
125C2×C4×C53#12011yescharacteristic
126C2×C4×C53C22⋊Q811yescharacteristic
127C2×C4×C53C22⋊Q811yescharacteristic
128C2×C4×C53#12311yescharacteristic
129C2×C4×C53C2×M1611yescharacteristic
130C2×C4×C53C2×M1611yescharacteristic
131C2×C4×C53#13111yescharacteristic
132C2×C4×C53#13111yescharacteristic
133C22⋊D53—41no—
134C22⋊D53—41no—
135C2×Dic53—22no—
136C2×Dic53—22no—
137C2×Dic53—22no—
138C2×Dic53—22no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144D212—21no—
145D212—21no—
146C4×D53—41no—
147C4×D53—41no—
148C4×D53—41no—
149C4×D53—41no—
150C4×D53—81no—
151Dic106—21no—
152Dic106—21no—
153C8×C53—22no—
154C8×C53—41no—
155C8×C53—42no—
156C8×C53—81no—
157#88322—41no—
158#88322—41no—
159#88322—81no—
160#8841—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53#38511yescharacteristic
164C22×C53#43511yescharacteristic
165C22×C53#43611yescharacteristic
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53#42311yescharacteristic
173C4×C53#42411yescharacteristic
174C4×C53#43511yescharacteristic
175C4×C53#43611yescharacteristic
176C4×C53#45311yescharacteristic
177C4×C53#45611yescharacteristic
178Dic53—24no—
179Dic53—42no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183#2989—531no—
184#1842—531no—
185C4×#104—531no—
186#1565—531no—
187#1477—531no—
188#1283D5311yescharacteristic, complemented
189#1276—531no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53#185111yescharacteristic
193C2×C53#185911yescharacteristic
194C2×C53#186111yescharacteristic
195D53—81no—
196C4×#41—531no—
197#446—1061no—
198#436—1061no—
199#435—1062no—
200#432—1061no—
201#431—531no—
202#431C2×D5311yescharacteristic
203#423—1061no—
204C2×C4⋊C8—531no—
205C2×C4⋊C8—1062no—
206C2×C4⋊C8C2×D5311yescharacteristic
207C2×#104—531no—
208C2×#104—531no—
209C4×M16C2×D5311yescharacteristic
210C43—531no—
211#339—531no—
212#339—531no—
213#335—1061no—
214#334—1061no—
215Q8⋊C8—1061no—
216D4⋊C8—1061no—
217C53#884111yescharacteristic, complemented, Sylow
218C2×#41—531no—
219C2×M16—1062no—
220C2×M16C4×D5311yescharacteristic
221C2×M16C4×D5311yescharacteristic
222C22×C8—1061no—
223C22×C8D21212yes—
224C4×Q8—531no—
225C4×Q8—531no—
226C4×D4—531no—
227C4×D4—531no—
228C4×D4—1061no—
229C4×D4—1061no—
230#117—531no—
231#117—1061no—
232C2×C42—531no—
233C2×C42—531no—
234C2×C42—1061no—
235C2×C42—1061no—
236C2×C42—1061no—
237C2×C42C22×D5311yescharacteristic
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—531no—
242C4⋊C8—531no—
243C4⋊C8—1062no—
244C4⋊C8—1061no—
245C4⋊C8—1061no—
246C4⋊C8—2124no—
247#104—1062no—
248#104—1062no—
249C22⋊C8—21no—
250C22⋊C8—1062no—
251C22⋊C8—1061no—
252#97—21no—
253#97—1062no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—1061no—
257#41—1061no—
258#41—1061no—
259#41—1061no—
260#41—1061no—
261C2×Q8—531no—
262C2×D4—531no—
263C2×D4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4C2×C4×D5311yescharacteristic
270C22×C4C2×D21211yescharacteristic
271C22×C4#1171994511yescharacteristic
272M16—21no—
273M16—21no—
274M16—21no—
275M16—4242no—
276M16#1171991212yescomplemented
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—42no—
281C2×C8—41no—
282C2×C8—1061no—
283C2×C8—1061no—
284C2×C8—2122no—
285C2×C8—2121no—
286C4⋊C4—531no—
287C4⋊C4—531no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C42—532no—
294C42—1062no—
295C42—1062no—
296C42—1061no—
297C42—1061no—
298C42—1061no—
299C42—1061no—
300C42—1061no—
301C42—1061no—
302C42—2122no—
303C42D4×D5311yescharacteristic
304C42D4×D5311yescharacteristic
305C42#1171994711yescharacteristic
306C42Q8×D5311yescharacteristic
307C23—1061no—
308C23C4×D21211yescharacteristic
309Q8—1061no—
310Q8—1061no—
311D4—1061no—
312D4—1061no—
313D4—2121no—
314D4—2121no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—1062no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—4241no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4C4⋊C4×D5311yescharacteristic
333C2×C4#1313317711yescharacteristic
334C2×C4#1313318711yescharacteristic
335C2×C4#1313319511yescharacteristic
336C2×C4#1313319711yescharacteristic
337C2×C4#1313319911yescharacteristic
338C2×C4M16×D5311yescharacteristic
339C2×C4C2×#1171991211yescharacteristic
340C2×C4#1313322311yescharacteristic
341C2×C4#1313322511yescharacteristic
342C8—22no—
343C8—42no—
344C8—41no—
345C8—81no—
346C8—4241no—
347C8—2121no—
348C8—2121no—
349C22—21no—
350C22—21no—
351C22—2121no—
352C22—2121no—
353C22—2121no—
354C22#1681049711yescharacteristic
355C22#1681069611yescharacteristic
356C22#1681070211yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—1064no—
361C4—2122no—
362C4—2121no—
363C4—2121no—
364C4—2121no—
365C4#1681064511yescharacteristic
366C4#1681064811yescharacteristic
367C4#1681069111yescharacteristic
368C4#1681069811yescharacteristic
369C4#1681079311yescharacteristic
370C4#1681080811yescharacteristic
371C2—21no—
372C2—21no—
373C2—4241no—
374C2#2027489911yescharacteristic
375C2#2027494411yescharacteristic
376C2#2027495811yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1283C2
socleC22×C53#385
prime radicalC2×#11719945C2×C4
cosocleC2×C42×C53C23
hypercenter#1283D53
hypocenterC53#8841
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1283

Complements

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

NH ≅ G/Ntype
2. #20274866C2semidirect
4. #20273467C2semidirect
6. #20272547C2semidirect
7. C53×#1283C2semidirect
86. M16×C53M16semidirect
112. C4×Dic53M16semidirect
188. #1283D53semidirect
217. C53#8841semidirect
276. M16#11719912semidirect

Maximal subgroups

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

subgroupindex
#202838332
#202748662
C4×#131331432
#202734672
#202731032
#202725472
C53×#12832
#884153