Finite Group Databasebeta

← #27279889

Subgroups — #27279889

Order 13568.

452 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#20283729C211yesmaximal, characteristic
2#20274525C211yesmaximal, characteristic
3#20273310C211yesmaximal, characteristic, complemented
4#20273227C211yesmaximal, characteristic, complemented
5#20272607C211yesmaximal, characteristic, complemented
6#20272586C211yesmaximal, characteristic, complemented
7C53×#3116C211yesmaximal, characteristic
8#16810754—21no—
9#16810695—22no—
10D4×#88322—21no—
11#16810680—21no—
12#16810680C2211yescharacteristic
13C2×#13133144—21no—
14C4×#11719917C2211yescharacteristic
15C4×#11719917C2211yescharacteristic
16C2×#13133128—21no—
17#16810469C2211yescharacteristic
18#16810382—21no—
19#16810365—21no—
20#16810365—21no—
21#16810360—22no—
22C53×#533C2211yescharacteristic
23C4×#41×C53C2211yescharacteristic
24C53×#485—21no—
25C53×#477C411yescharacteristic, complemented
26C53×#476C411yescharacteristic, complemented
27C53×#462C411yescharacteristic, complemented
28C53×#461C411yescharacteristic, complemented
29C53×#452—21no—
30C53×#441—21no—
31C53×#440—21no—
32C53×#432C2211yescharacteristic
33C2×#11719917—21no—
34C2×#11719917—21no—
35C22×#88322—22no—
36C2×C4×Dic53—21no—
37C4⋊#88322—22no—
38C4⋊#88322—21no—
39#13133144—24no—
40C22⋊#88322—22no—
41C22⋊#88322D412yes—
42#13133128—22no—
43#13133128—21no—
44#13133128—21no—
45#13133128D411yescharacteristic
46#13133128D411yescharacteristic
47C4×#88322—21no—
48C4×#88322—21no—
49#13133124—21no—
50#13133124—21no—
51C2×#41×C53D411yescharacteristic
52C2×Q16×C53—21no—
53C2×SD16×C53—21no—
54C4⋊Q8×C53C2×C411yescharacteristic
55C4⋊Q8×C53C2×C411yescharacteristic
56#125×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53C2×C411yescharacteristic
62C4×Q8×C53D411yescharacteristic
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53C2×C411yescharacteristic
66C4×D4×C53D411yescharacteristic
67#117×C53—21no—
68#117×C53—21no—
69#117×C53D411yescharacteristic
70C2×C42×C53—21no—
71C2×C42×C53C2311yescharacteristic
72#107×C53—21no—
73#106×C53—21no—
74C4⋊C8×C53C2×C411yescharacteristic
75C4⋊C8×C53C2×C411yescharacteristic
76#103×C53—21no—
77#103×C53—21no—
78#103×C53—21no—
79#102×C53—21no—
80C22⋊C8×C53—21no—
81#97×C53—21no—
82#41×C53—22no—
83#41×C53—41no—
84C2×Q8×C53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53C22⋊C411yescharacteristic
87C2×D4×C53—21no—
88C2×D4×C53C22⋊C411yescharacteristic
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C22×Dic53—41no—
96Q16×C53—42no—
97SD16×C53—42no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53C22⋊C411yescharacteristic
109C4⋊C4×C53C22⋊C411yescharacteristic
110C22⋊C4×C53—21no—
111C22⋊C4×C53—21no—
112C22⋊C4×C53—21no—
113C22⋊C4×C53—41no—
114C42×C53—21no—
115C42×C53—21no—
116C42×C53—21no—
117C42×C53—41no—
118C42×C53C22×C411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122C4×Dic53—22no—
123C4×Dic53—41no—
124#11719917—22no—
125#11719917—42no—
126#11719917—41no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—22no—
130C2×#88322—44no—
131C2×#88322—41no—
132C23×C53—21no—
133C23×C53#12111yescharacteristic
134Q8×C53—21no—
135Q8×C53—21no—
136Q8×C53—41no—
137Q8×C53—41no—
138Q8×C53—81no—
139D4×C53—21no—
140D4×C53—42no—
141D4×C53#10412yes—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—81no—
154C2×C4×C53C2×C22⋊C411yescharacteristic
155C2×C4×C53C4×D411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53#12011yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12311yescharacteristic
160C2×Dic53—42no—
161C2×Dic53—81no—
162C2×Dic53—81no—
163C8×C53—41no—
164C8×C53—41no—
165#88322—22no—
166#88322—42no—
167#88322—41no—
168#88322—42no—
169#9159—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—22no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53#38611yescharacteristic
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—41no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—81no—
186C4×C53—81no—
187C4×C53C2×#10411yescharacteristic
188C4×C53#42111yescharacteristic
189Dic53—82no—
190#3116D5311yescharacteristic
191#2970—531no—
192#1759—531no—
193#1526—531no—
194#1506—531no—
195#1300—531no—
196#1291—531no—
197C2×C53—41no—
198C2×C53—42no—
199C2×C53#187111yescharacteristic
200C2×C53#190211yescharacteristic
201C2×C53#190611yescharacteristic
202#533C2×D5311yescharacteristic
203C4×#41C2×D5311yescharacteristic
204#485—21no—
205#477Dic5311yescharacteristic, complemented
206#476Dic5311yescharacteristic, complemented
207#462Dic5311yescharacteristic, complemented
208#461Dic5311yescharacteristic, complemented
209#452—21no—
210#446—1061no—
211#441—21no—
212#440—21no—
213#435—1062no—
214C8×D4—1061no—
215#432—531no—
216#432—1061no—
217#432C2×D5311yescharacteristic
218C2×#104—1061no—
219C4×M16—531no—
220C4×M16—531no—
221C2×#97—1061no—
222#376—531no—
223#339—1061no—
224#330—1061no—
225#330—1061no—
226#328—1062no—
227C53#915911yescharacteristic, complemented, Sylow
228C2×#41C22⋊D5311yescharacteristic
229C2×Q16—21no—
230C2×SD16—21no—
231C2×M16—1061no—
232C2×M16—1061no—
233C22×C8—1062no—
234C4⋊Q8C2×Dic5311yescharacteristic
235C4⋊Q8C2×Dic5311yescharacteristic
236#125—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8C2×Dic5311yescharacteristic
242C4×Q8C22⋊D5311yescharacteristic
243C4×D4—21no—
244C4×D4—21no—
245C4×D4C2×Dic5311yescharacteristic
246C4×D4C22⋊D5311yescharacteristic
247#117—21no—
248#117—21no—
249#117C22⋊D5311yescharacteristic
250C2×C42—21no—
251C2×C42—1061no—
252C2×C42C22×D5311yescharacteristic
253#107—21no—
254#106—21no—
255C4⋊C8—1062no—
256C4⋊C8—1061no—
257C4⋊C8C2×Dic5311yescharacteristic
258C4⋊C8C2×Dic5311yescharacteristic
259#104—1064no—
260#103—21no—
261#103—21no—
262#103—21no—
263#102—21no—
264C22⋊C8—21no—
265C22⋊C8—532no—
266C22⋊C8—1062no—
267#97—21no—
268#97—531no—
269#97—531no—
270#97—1062no—
271#97—1061no—
272#97—1061no—
273C4×C8—1061no—
274C4×C8—1061no—
275#95—1061no—
276#95—1061no—
277#41—22no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8C22⋊Dic5311yescharacteristic
282C2×D4—21no—
283C2×D4C22⋊Dic5311yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—2121no—
288C22×C4D4×D5311yescharacteristic
289C22×C4#1171994711yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291Q16—42no—
292SD16—42no—
293M16—1062no—
294M16—2122no—
295M16—2121no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8—1062no—
301C2×C8—2124no—
302C2×C8—2121no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4C22⋊Dic5311yescharacteristic
312C4⋊C4C22⋊Dic5311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—41no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—41no—
321C42—1062no—
322C42—2121no—
323C42D4×D5311yescharacteristic
324C42#1171994711yescharacteristic
325C42C22×Dic5311yescharacteristic
326C42C2×C22⋊D5311yescharacteristic
327C23—21no—
328C23#1313318911yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—41no—
332Q8—41no—
333Q8—81no—
334D4—21no—
335D4—42no—
336D4#1313314412yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—4241no—
350C2×C4—4241no—
351C2×C4—2122no—
352C2×C4C2×C22⋊Dic5311yescharacteristic
353C2×C4D4×Dic5311yescharacteristic
354C2×C4D4×Dic5311yescharacteristic
355C2×C4#1313318511yescharacteristic
356C2×C4#1313318911yescharacteristic
357C2×C4#1313319811yescharacteristic
358C8—41no—
359C8—41no—
360C8—1062no—
361C8—2122no—
362C8—2121no—
363C8—2122no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22#1681050311yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—4242no—
382C4C2×#1313314411yescharacteristic
383C4#1681064011yescharacteristic
384C2—41no—
385C2—42no—
386C2#2027501811yescharacteristic
387C2#2027519011yescharacteristic
388C2#2027521811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3116C2
socleC22×C53#386
prime radicalC2×#41×C53D4
cosocleC2×C42×C53C23
hypercenter#3116D53
hypocenterC53#9159
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3116

Complements

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

NH ≅ G/Ntype
3. #20273310C2semidirect
4. #20273227C2semidirect
5. #20272607C2semidirect
6. #20272586C2semidirect
25. C53×#477C4semidirect
26. C53×#476C4semidirect
27. C53×#462C4semidirect
28. C53×#461C4semidirect
205. #477Dic53semidirect
206. #476Dic53semidirect
207. #462Dic53semidirect
208. #461Dic53semidirect
227. C53#9159semidirect

Maximal subgroups

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

subgroupindex
#202837292
#202745252
#202733102
#202732272
#202726072
#202725862
C53×#31162
#915953