Finite Group Databasebeta

← #27280364

Subgroups — #27280364

Order 13568.

604 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284746C211yesmaximal, characteristic
2#20275543C211yesmaximal, characteristic
3#20273209C211yesmaximal, characteristic
4#20273144C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272548C211yesmaximal, characteristic, complemented
7C42×Q8×C53C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16810941—21no—
10#16810908C412yescomplemented
11#16810823C412yescomplemented
12#16810762—21no—
13C4×#11719924—22no—
14#16810676C2211yescharacteristic
15#16810676C2211yescharacteristic
16#16810647C2211yescharacteristic
17C2×#13133145—21no—
18C2×#13133145—21no—
19#16810356—22no—
20Q8⋊#88322—22no—
21C2×C4×Q8×C53—21no—
22C2×C4×Q8×C53C2211yescharacteristic
23C4×C4⋊C4×C53—21no—
24C4×C4⋊C4×C53—21no—
25C4×C4⋊C4×C53—22no—
26C4×C4⋊C4×C53C2211yescharacteristic
27C43×C53—21no—
28C43×C53C2211yescharacteristic
29C2×#11719945—21no—
30C2×#11719924—22no—
31C2×#11719917—21no—
32#13133217C2×C411yescharacteristic
33#13133214C2×C411yescharacteristic
34#13133205—21no—
35#13133191—21no—
36#13133191—21no—
37C4×D212—21no—
38C4×D212—21no—
39#13133148—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322C2×C412yes—
44#13133145—42no—
45#13133145—42no—
46#13133141—22no—
47#13133137—22no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52C22×Q8×C53D411yescharacteristic
53C4×Q8×C53—28no—
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53C2×C412yes—
57C4×Q8×C53D412yes—
58C2×C4⋊C4×C53—22no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C42×C53—22no—
63C2×C42×C53—22no—
64C2×C42×C53—21no—
65C2×C42×C53—21no—
66C2×C42×C53—21no—
67C2×C42×C53—21no—
68C2×C42×C53D411yescharacteristic
69C2×C42×C53D411yescharacteristic
70C2×C42×C53C2311yescharacteristic
71#95×C53—22no—
72#95×C53D412yes—
73C2×Q8×C53—24no—
74C2×Q8×C53—21no—
75C2×Q8×C53C22⋊C412yes—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—22no—
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C2×C22⋊D53—41no—
85#11719945—41no—
86#11719945—41no—
87#11719945—41no—
88#11719945—41no—
89C2×D212—21no—
90C2×D212—21no—
91C2×D212—21no—
92C2×C4×D53—41no—
93C2×Dic106—21no—
94C4⋊C4×C53—28no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53C22⋊C412yes—
100C42×C53—28no—
101C42×C53—24no—
102C42×C53—22no—
103C42×C53—22no—
104C42×C53—21no—
105C42×C53—21no—
106C42×C53—21no—
107C42×C53—42no—
108C42×C53—44no—
109C42×C53—41no—
110C42×C53—41no—
111C42×C53SD1612yescomplemented
112C42×C53C22×C411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53#4111yescharacteristic
115C42×C53#4111yescharacteristic
116#11719924—44no—
117#11719921—41no—
118#11719921—41no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121#11719917—41no—
122#11719917—41no—
123C2×#88322—22no—
124C2×#88322—41no—
125C2×#88322—41no—
126C23×C53#12011yescharacteristic
127C22×D53—41no—
128Q8×C53—24no—
129Q8×C53—22no—
130Q8×C53#10414yescomplemented
131C2×C4×C53—24no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—24no—
139C2×C4×C53—22no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—41no—
145C2×C4×C53—42no—
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
156C22⋊D53—81no—
157C22⋊D53—81no—
158C2×Dic53—41no—
159D212—42no—
160D212—41no—
161D212—41no—
162D212—81no—
163C4×D53—81no—
164C4×D53—81no—
165Dic106—41no—
166Dic106—41no—
167#88322—42no—
168#88322—81no—
169#88322—81no—
170#9234—531nomaximal, Sylow
171C22×C53—21no—
172C22×C53—21no—
173C22×C53#38611yescharacteristic
174C22×C53#44811yescharacteristic
175C22×C53#44911yescharacteristic
176C2×D53—81no—
177C2×D53—81no—
178C2×D53—81no—
179C4×C53—24no—
180C4×C53—24no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—42no—
185C4×C53—44no—
186C4×C53—42no—
187C4×C53C2×#10412yes—
188C4×C53C4×SD1611yescharacteristic
189C4×C53#44011yescharacteristic
190C4×C53#46011yescharacteristic
191C4×C53#46411yescharacteristic
192Dic53—81no—
193#3103—531no—
194C42×Q8D5311yescharacteristic, complemented
195#1962—531no—
196#1502—531no—
197#1487—531no—
198#1303—531no—
199#1276—531no—
200C2×C53—21no—
201C2×C53—21no—
202C2×C53#188011yescharacteristic
203C2×C53#189611yescharacteristic
204C2×C53#190611yescharacteristic
205D53—161no—
206#532—531no—
207C2×C4×Q8—21no—
208C2×C4×Q8C2×D5311yescharacteristic
209#481—1061no—
210#475—532no—
211#460—532no—
212#448—1061no—
213C4×SD16—1062no—
214#431—531no—
215#431—531no—
216#423—531no—
217C2×#104—1061no—
218C2×#104—1061no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2×D5311yescharacteristic
223C43—21no—
224C43C2×D5311yescharacteristic
225#327—1062no—
226Q8⋊C8—1062no—
227C53#923411yescharacteristic, complemented, Sylow
228C2×#41—1061no—
229C22×Q8C22⋊D5311yescharacteristic
230C2×SD16—1062no—
231C2×M16—1061no—
232C4⋊Q8—531no—
233#127—531no—
234#124—1061no—
235#121—1061no—
236#121—1061no—
237C4×Q8—28no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C4×D5312yes—
241C4×Q8D21212yes—
242C4×D4—1061no—
243C4×D4—1061no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22⋊D5311yescharacteristic
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C22⋊D5311yescharacteristic
255C2×C42C22⋊D5311yescharacteristic
256C2×C42C22×D5311yescharacteristic
257#106—1062no—
258C4⋊C8—532no—
259C4⋊C8—1061no—
260C4⋊C8—1061no—
261C4⋊C8—1061no—
262#104—2122no—
263#104—2122no—
264#103—1062no—
265#102—1062no—
266C22⋊C8—1061no—
267C22⋊C8—1061no—
268C4×C8—1061no—
269C4×C8—1061no—
270#95—22no—
271#95C22⋊D5312yes—
272#41—2121no—
273#41—2121no—
274#41—2121no—
275#41—2121no—
276C2×Q8—24no—
277C2×Q8—21no—
278C2×Q8—1061no—
279C2×Q8#1171992112yes—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—2121no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—2121no—
290C22×C4C2×C22⋊D5311yescharacteristic
291C22×C4C2×C22⋊D5311yescharacteristic
292C22×C4C2×C22⋊D5311yescharacteristic
293SD16—2124no—
294M16—2121no—
295M16—2121no—
296C2×C8—1062no—
297C2×C8—2121no—
298C2×C8—2121no—
299C4⋊C4—28no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4#1171992112yes—
307C22⋊C4—2121no—
308C22⋊C4—2121no—
309C42—28no—
310C42—24no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—42no—
317C42—44no—
318C42—41no—
319C42—41no—
320C42#1171992412yescomplemented
321C42C2×C4×D5311yescharacteristic
322C42C2×D21211yescharacteristic
323C42#1171994511yescharacteristic
324C42#1171994511yescharacteristic
325C23—2121no—
326C23#1313318611yescharacteristic
327Q8—24no—
328Q8—22no—
329Q8—2121no—
330Q8—2121no—
331Q8#1313314514yescomplemented
332D4—4241no—
333D4—4241no—
334D4—4241no—
335D4—2122no—
336D4—2121no—
337D4—2121no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—4241no—
354C2×C4—4241no—
355C2×C4—2121no—
356C2×C4C2×#1171992111yescharacteristic
357C2×C4C4×C22⋊D5311yescharacteristic
358C2×C4C4×C22⋊D5311yescharacteristic
359C2×C4#1313319111yescharacteristic
360C2×C4#1313319111yescharacteristic
361C2×C4#1313320011yescharacteristic
362C2×C4C2×#1171992411yescharacteristic
363C2×C4#1313323711yescharacteristic
364C2×C4#1313324211yescharacteristic
365C2×C4#1313324811yescharacteristic
366C8—4241no—
367C8—4241no—
368C8—2122no—
369C22—21no—
370C22—21no—
371C22—4241no—
372C22—4241no—
373C22—4241no—
374C22#1681050211yescharacteristic
375C22#1681076211yescharacteristic
376C22#1681076811yescharacteristic
377C4—24no—
378C4—21no—
379C4—21no—
380C4—24no—
381C4—21no—
382C4—42no—
383C4—44no—
384C4—42no—
385C4—4241no—
386C4C2×#1313314512yes—
387C4C4×#1171992411yescharacteristic
388C4#1681072111yescharacteristic
389C4#1681082311yescharacteristic
390C4#1681084511yescharacteristic
391C2—21no—
392C2—21no—
393C2—8481no—
394C2#2027507011yescharacteristic
395C2#2027515811yescharacteristic
396C2#2027521611yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC42×Q8×C53C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenterC42×Q8D53
hypocenterC53#9234
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
4. #20273144C2semidirect
5. #20272618C2semidirect
6. #20272548C2semidirect
7. C42×Q8×C53C2semidirect
10. #16810908C4semidirect
11. #16810823C4semidirect
111. C42×C53SD16semidirect
130. Q8×C53#104semidirect
194. C42×Q8D53semidirect
227. C53#9234semidirect
320. C42#11719924semidirect
331. Q8#13133145semidirect

Maximal subgroups

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

subgroupindex
#202847462
#202755432
#202732092
#202731442
#202726182
#202725482
C42×Q8×C532
#923453