Finite Group Databasebeta

← #27283239

Subgroups — #27283239

Order 13568.

468 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285274C211yesmaximal, characteristic
2#20275658C211yesmaximal, characteristic, complemented
3#20275633C211yesmaximal, characteristic
4#20273213C211yesmaximal, characteristic, complemented
5#20273209C211yesmaximal, characteristic
6C8×#11719917C211yesmaximal, characteristic, complemented
7C4×#106×C53C211yesmaximal, characteristic, complemented
8#16811180C2211yescharacteristic
9#16811173C2211yescharacteristic
10#16811021C411yescharacteristic, complemented
11#16810996C411yescharacteristic, complemented
12#16810974C411yescharacteristic, complemented
13#16810970C411yescharacteristic, complemented
14#16810852—21no—
15#16810852—21no—
16C4×#11719913—21no—
17C4×#11719913—21no—
18#16810676C2211yescharacteristic
19C2×#13133145—22no—
20C4×#11719917C2211yescharacteristic
21#16810383—22no—
22#16810359—22no—
23#16810356—22no—
24#16810345—21no—
25C8×#88322—21no—
26C2×#106×C53—21no—
27C2×C4×C8×C53C2211yescharacteristic
28C4×C4⋊C4×C53C2211yescharacteristic
29C4×C4⋊C4×C53C2211yescharacteristic
30C53×#340—21no—
31C2×#11719945—21no—
32C2×#11719913—21no—
33C2×#11719913—21no—
34C2×#11719917D412yes—
35#13133217C2×C411yescharacteristic
36#13133217C2×C411yescharacteristic
37#13133217C2×C411yescharacteristic
38#13133214C2×C411yescharacteristic
39#13133209—21no—
40#13133205—21no—
41#13133196—21no—
42#13133196—21no—
43#13133191—21no—
44#13133191—21no—
45C4×Dic106—21no—
46C4×Dic106—21no—
47C4×D212—21no—
48C4×D212—21no—
49#13133176—21no—
50#13133149—21no—
51#13133149—21no—
52C4⋊#88322—21no—
53#13133145—44no—
54#13133142—21no—
55#13133142—21no—
56#13133139—21no—
57#13133139—21no—
58C22⋊#88322D412yes—
59#13133128—21no—
60C4×#88322—21no—
61C4×#88322—21no—
62C22×C8×C53D411yescharacteristic
63C22×C8×C53D411yescharacteristic
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C42×C53—21no—
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69#106×C53—24no—
70C4×C8×C53C2×C411yescharacteristic
71C4×C8×C53C2×C411yescharacteristic
72C4×C8×C53D411yescharacteristic
73C4×C8×C53D411yescharacteristic
74#95×C53—21no—
75#95×C53—21no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C2×C22⋊D53—41no—
82#11719945—42no—
83#11719945—42no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×C4×D53—41no—
88C2×Dic106—21no—
89C2×Dic106—21no—
90C2×Dic106—21no—
91C2×C8×C53—22no—
92C2×C8×C53—41no—
93C2×C8×C53C22⋊C411yescharacteristic
94C2×C8×C53C22⋊C411yescharacteristic
95C2×C8×C53C22⋊C411yescharacteristic
96C2×C8×C53C22⋊C411yescharacteristic
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C42×C53—42no—
102C42×C53—42no—
103C42×C53C22×C411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53#4111yescharacteristic
106C42×C53#4111yescharacteristic
107C22⋊Dic53—41no—
108#11719921—41no—
109#11719921—41no—
110C4⋊Dic53—21no—
111C4⋊Dic53—21no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C4⋊Dic53—21no—
115C4⋊Dic53—21no—
116#11719919—41no—
117#11719919—41no—
118C4×Dic53—41no—
119#11719917—22no—
120#11719917SD1614yescomplemented
121C2×#88322—22no—
122C2×#88322—22no—
123#11719913—42no—
124#11719913—42no—
125C23×C53#12711yescharacteristic
126C22×D53—41no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53—81no—
135C2×C4×C53—81no—
136C2×C4×C53C2×C22⋊C411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12411yescharacteristic
142C2×C4×C53C2×SD1612yes—
143C2×C4×C53#13512yes—
144C22⋊D53—81no—
145C22⋊D53—81no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149D212—41no—
150D212—41no—
151D212—42no—
152D212—81no—
153C4×D53—82no—
154Dic106—41no—
155Dic106—41no—
156Dic106—42no—
157Dic106—81no—
158C8×C53—21no—
159C8×C53—21no—
160C8×C53—21no—
161C8×C53#10412yescomplemented
162#88322—24no—
163#88322—42no—
164#9717—531nomaximal, Sylow
165C22×C53—21no—
166C22×C53—21no—
167C22×C53#39011yescharacteristic
168C22×C53#49211yescharacteristic
169C22×C53#49511yescharacteristic
170C2×D53—81no—
171C2×D53—81no—
172C2×D53—81no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—82no—
177C4×C53—82no—
178C4×C53C2×#10411yescharacteristic
179C4×C53#42111yescharacteristic
180C4×C53C4×SD1612yes—
181C4×C53#46512yes—
182Dic53—81no—
183Dic53—81no—
184Dic53—81no—
185#3167—531no—
186#1986—531no—
187#1980—531no—
188C4×#106D5311yescharacteristic, complemented
189#1503—531no—
190#1502—531no—
191C8×M16—531no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53#196811yescharacteristic
195C2×C53#197611yescharacteristic
196C2×C53#198011yescharacteristic
197D53—161no—
198#533—531no—
199#532—531no—
200#498—531no—
201#492—531no—
202#487—531no—
203#486—531no—
204#465—1061no—
205#465—1061no—
206C4×SD16—1061no—
207C4×SD16—1061no—
208#431—531no—
209C2×#106—21no—
210C2×#104—1062no—
211C4×M16—531no—
212C2×C4×C8C2×D5311yescharacteristic
213C4×C4⋊C4C2×D5311yescharacteristic
214C4×C4⋊C4C2×D5311yescharacteristic
215#340—21no—
216#339—1062no—
217#328—1062no—
218#327—1062no—
219#322—1061no—
220C82—1061no—
221C53#971711yescharacteristic, complemented, Sylow
222C2×#41—1061no—
223C2×SD16—1061no—
224C2×SD16—1061no—
225C2×M16—532no—
226C22×C8C22⋊D5311yescharacteristic
227C22×C8C22⋊D5311yescharacteristic
228C4⋊Q8—531no—
229C4⋊Q8—531no—
230C4⋊Q8—531no—
231#127—531no—
232#125—1061no—
233#124—1061no—
234C22⋊Q8—1061no—
235C22⋊Q8—1061no—
236#121—1061no—
237#121—1061no—
238C4×Q8—1061no—
239C4×Q8—1061no—
240C4×D4—1061no—
241C4×D4—1061no—
242#117—1061no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42C22×D5311yescharacteristic
248#106—24no—
249#106—1061no—
250#106—1061no—
251C4⋊C8—1061no—
252#104—2124no—
253#103—1061no—
254#103—1061no—
255#102—1061no—
256#102—1061no—
257C22⋊C8—532no—
258#97—1061no—
259C4×C8—1061no—
260C4×C8—1061no—
261C4×C8C4×D5311yescharacteristic
262C4×C8C4×D5311yescharacteristic
263C4×C8D21211yescharacteristic
264C4×C8D21211yescharacteristic
265#95—21no—
266#95—21no—
267#41—2122no—
268#41—2122no—
269C2×Q8—1061no—
270C2×Q8—1061no—
271C2×Q8—1061no—
272C2×D4—1061no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—2121no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—2121no—
279C22×C4D4×D5311yescharacteristic
280C22×C4D4×D5311yescharacteristic
281C22×C4C2×C22⋊D5311yescharacteristic
282SD16—2122no—
283SD16—2122no—
284M16—534no—
285M16—1062no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—1062no—
289C2×C8—1062no—
290C2×C8#1171992111yescharacteristic
291C2×C8#1171992111yescharacteristic
292C2×C8#1171992111yescharacteristic
293C2×C8#1171992111yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—1061no—
299C4⋊C4—1061no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4—1061no—
304C4⋊C4—2121no—
305C4⋊C4—2121no—
306C22⋊C4—2121no—
307C22⋊C4—2121no—
308C22⋊C4—2121no—
309C42—42no—
310C42—42no—
311C42—2121no—
312C42C2×C4×D5311yescharacteristic
313C42C2×D21211yescharacteristic
314C42#1171994911yescharacteristic
315C42#1171994911yescharacteristic
316C23—2121no—
317C23#1313321311yescharacteristic
318Q8—4241no—
319Q8—2121no—
320Q8—2121no—
321Q8—2122no—
322D4—4241no—
323D4—4241no—
324D4—4241no—
325D4—2121no—
326D4—2121no—
327D4—2122no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—4242no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4C2×#1171992111yescharacteristic
342C2×C4#1313317911yescharacteristic
343C2×C4#1313317911yescharacteristic
344C2×C4#1313319211yescharacteristic
345C2×C4#1313319211yescharacteristic
346C2×C4#1313320611yescharacteristic
347C2×C4SD16×D5312yes—
348C2×C4#1313323812yes—
349C8—21no—
350C8—21no—
351C8—21no—
352C8—1064no—
353C8—2122no—
354C8#1313314512yescomplemented
355C22—21no—
356C22—21no—
357C22—4241no—
358C22—4241no—
359C22—4241no—
360C22#1681052411yescharacteristic
361C22#1681099411yescharacteristic
362C22#1681100411yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—82no—
368C4—4241no—
369C4—4241no—
370C4—4241no—
371C4C2×#1313314511yescharacteristic
372C4#1681064111yescharacteristic
373C4#1681071012yes—
374C4#1681085312yes—
375C2—21no—
376C2—21no—
377C2—8481no—
378C2#2027556711yescharacteristic
379C2#2027561111yescharacteristic
380C2#2027563311yescharacteristic
subgroupG/H
centerC2×C4#13133192
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×#106×C53C2
socleC22×C53#390
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenterC4×#106D53
hypocenterC53#9717
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
2. #20275658C2semidirect
4. #20273213C2semidirect
6. C8×#11719917C2semidirect
7. C4×#106×C53C2semidirect
10. #16811021C4semidirect
11. #16810996C4semidirect
12. #16810974C4semidirect
13. #16810970C4semidirect
120. #11719917SD16semidirect
161. C8×C53#104semidirect
188. C4×#106D53semidirect
221. C53#9717semidirect
354. C8#13133145semidirect

Maximal subgroups

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

subgroupindex
#202852742
#202756582
#202756332
#202732132
#202732092
C8×#117199172
C4×#106×C532
#971753