Finite Group Databasebeta

← #27280362

Subgroups — #27280362

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
2C4×Q8×Dic53C211yesmaximal, characteristic, complemented
3#20275542C211yesmaximal, characteristic
4#20273211C211yesmaximal, characteristic
5#20272616C211yesmaximal, characteristic, complemented
6#20272547C211yesmaximal, characteristic, complemented
7C53×#1487C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9C2×Q8×Dic53—21no—
10#16810941—21no—
11#16810908C412yescomplemented
12#16810823C412yescomplemented
13#16810762—21no—
14C4×#11719924—22no—
15#16810676C2211yescharacteristic
16C2×#13133143—21no—
17C2×#13133143—21no—
18C4⋊C4×Dic53—22no—
19C4⋊C4×Dic53—21no—
20C4×C4⋊Dic53—21no—
21C4×C4⋊Dic53C2211yescharacteristic
22C42×Dic53—21no—
23C42×Dic53C2211yescharacteristic
24#16810358—22no—
25C2×C4×Q8×C53C2211yescharacteristic
26C53×#431C2211yescharacteristic
27C53×#423C2211yescharacteristic
28C53×Q8⋊C8—22no—
29C2×#11719945—21no—
30C2×#11719924—22no—
31#13133217C2×C411yescharacteristic
32#13133214C2×C411yescharacteristic
33#13133205—21no—
34#13133191—21no—
35#13133191—21no—
36Q8×Dic53—28no—
37C4×D212—21no—
38C4×D212—21no—
39C2×C4⋊Dic53—22no—
40C2×C4⋊Dic53—21no—
41C2×C4×Dic53—22no—
42C2×C4×Dic53—22no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53—21no—
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48#13133148—22no—
49C4⋊#88322C2×C412yes—
50#13133143—42no—
51#13133143—42no—
52#13133141—22no—
53#13133137—22no—
54C22⋊#88322—21no—
55C4×#88322—21no—
56#13133124—22no—
57#13133124D412yes—
58C22×Q8×C53D411yescharacteristic
59C2×M16×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53C2×C412yes—
63C4×Q8×C53D412yes—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C22⋊C8×C53—21no—
72C4×C8×C53—21no—
73C2×Q8×C53—24no—
74C2×Q8×C53—21no—
75C2×Q8×C53C22⋊C412yes—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C2×C22⋊D53—41no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—22no—
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—
94M16×C53—41no—
95M16×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53—41no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53C22⋊C412yes—
102C42×C53—22no—
103C42×C53—22no—
104C42×C53C22×C411yescharacteristic
105C42×C53C2×D411yescharacteristic
106C42×C53#4111yescharacteristic
107C42×C53#4111yescharacteristic
108#11719924—44no—
109#11719921—41no—
110#11719921—41no—
111C4⋊Dic53—28no—
112C4⋊Dic53—24no—
113C4⋊Dic53—21no—
114C4⋊Dic53—21no—
115C4×Dic53—28no—
116C4×Dic53—24no—
117C4×Dic53—21no—
118C4×Dic53—21no—
119C4×Dic53—21no—
120C4×Dic53—42no—
121C4×Dic53—44no—
122C4×Dic53—41no—
123C4×Dic53—41no—
124C4×Dic53SD1612yescomplemented
125C2×#88322—22no—
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—41no—
136C2×C4×C53—42no—
137C2×C4×C53C2×C22⋊C411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12311yescharacteristic
143C2×C4×C53C2×SD1611yescharacteristic
144C2×C4×C53#13511yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53#13711yescharacteristic
147C22⋊D53—81no—
148C22⋊D53—81no—
149C2×Dic53—22no—
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—22no—
153C2×Dic53—24no—
154C2×Dic53—41no—
155C2×Dic53—42no—
156C2×Dic53—41no—
157C2×Dic53—42no—
158C2×Dic53—41no—
159D212—42no—
160D212—41no—
161D212—41no—
162D212—81no—
163C4×D53—81no—
164C4×D53—81no—
165Dic106—41no—
166Dic106—41no—
167C8×C53—81no—
168C8×C53—81no—
169#88322—42no—
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—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—42no—
184C4×C53C2×#10412yes—
185C4×C53C4×SD1611yescharacteristic
186C4×C53#44011yescharacteristic
187C4×C53#46011yescharacteristic
188C4×C53#46411yescharacteristic
189Dic53—24no—
190Dic53—42no—
191Dic53—44no—
192Dic53—81no—
193#3103—531no—
194C42×Q8—531no—
195#1962—531no—
196#1502—531no—
197#1487D5311yescharacteristic, complemented
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—1061no—
208C2×C4×Q8C2×D5311yescharacteristic
209#481—1061no—
210#475—532no—
211#460—532no—
212#448—1061no—
213C4×SD16—1062no—
214#431—531no—
215#431C2×D5311yescharacteristic
216#423C2×D5311yescharacteristic
217C2×#104—1061no—
218C2×#104—1061no—
219C4×C4⋊C4—531no—
220C4×C4⋊C4—1062no—
221C4×C4⋊C4—1061no—
222C4×C4⋊C4—1061no—
223C43—531no—
224C43—1061no—
225#327—1062no—
226Q8⋊C8—22no—
227C53#923411yescharacteristic, complemented, Sylow
228C2×#41—1061no—
229C22×Q8D21211yescharacteristic
230C2×SD16—1062no—
231C2×M16—21no—
232C4⋊Q8—531no—
233#127—531no—
234#124—1061no—
235#121—1061no—
236#121—1061no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—1068no—
240C4×Q8C4×D5312yes—
241C4×Q8C22⋊D5312yes—
242C4×D4—1061no—
243C4×D4—1061no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—1062no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4D21211yescharacteristic
248C2×C42—21no—
249C2×C42—531no—
250C2×C42—531no—
251C2×C42—1062no—
252C2×C42—1061no—
253C2×C42—1062no—
254C2×C42—1061no—
255C2×C42—1061no—
256C2×C42C22×D5311yescharacteristic
257#106—1062no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—532no—
262#104—2122no—
263#104—2122no—
264#103—1062no—
265#102—1062no—
266C22⋊C8—21no—
267C22⋊C8—1061no—
268C4×C8—21no—
269C4×C8—1061no—
270#95—532no—
271#95—1062no—
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—1061no—
287C22×C4—1062no—
288C22×C4—1061no—
289C22×C4—2121no—
290C22×C4C2×D21211yescharacteristic
291C22×C4D4×D5311yescharacteristic
292C22×C4D4×D5311yescharacteristic
293SD16—2124no—
294M16—41no—
295M16—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—1062no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—1068no—
303C4⋊C4—1064no—
304C4⋊C4—1061no—
305C4⋊C4—1061no—
306C4⋊C4#1171992112yes—
307C22⋊C4—2121no—
308C22⋊C4—2121no—
309C42—22no—
310C42—22no—
311C42—532no—
312C42—1068no—
313C42—1064no—
314C42—1061no—
315C42—1061no—
316C42—1061no—
317C42—2122no—
318C42—2124no—
319C42—2121no—
320C42—2121no—
321C42C2×C4×D5311yescharacteristic
322C42#1171994711yescharacteristic
323C42#1171994711yescharacteristic
324C42C2×C22⋊D5311yescharacteristic
325C23—2121no—
326C23#1313318711yescharacteristic
327Q8—24no—
328Q8—22no—
329Q8—2121no—
330Q8—2121no—
331Q8#1313314314yescomplemented
332D4—4241no—
333D4—4241no—
334D4—4241no—
335D4—2122no—
336D4—2121no—
337D4—2121no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—1062no—
345C2×C4—1062no—
346C2×C4—1062no—
347C2×C4—1064no—
348C2×C4—1062no—
349C2×C4—4241no—
350C2×C4—4241no—
351C2×C4—2121no—
352C2×C4—2122no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2122no—
356C2×C4C2×#1171992111yescharacteristic
357C2×C4#1313317711yescharacteristic
358C2×C4#1313317711yescharacteristic
359C2×C4#1313318911yescharacteristic
360C2×C4#1313318911yescharacteristic
361C2×C4#1313320111yescharacteristic
362C2×C4SD16×D5311yescharacteristic
363C2×C4#1313324011yescharacteristic
364C2×C4#1313324411yescharacteristic
365C2×C4#1313324911yescharacteristic
366C8—81no—
367C8—81no—
368C8—2122no—
369C22—21no—
370C22—21no—
371C22—4241no—
372C22—4241no—
373C22—4241no—
374C22#1681050011yescharacteristic
375C22#1681076311yescharacteristic
376C22#1681077111yescharacteristic
377C4—21no—
378C4—21no—
379C4—24no—
380C4—21no—
381C4—42no—
382C4—1064no—
383C4—4241no—
384C4—2122no—
385C4—2124no—
386C4C2×#1313314312yes—
387C4#1681070811yescharacteristic
388C4#1681072611yescharacteristic
389C4#1681082111yescharacteristic
390C4#1681084711yescharacteristic
391C2—21no—
392C2—21no—
393C2—8481no—
394C2#2027506811yescharacteristic
395C2#2027515611yescharacteristic
396C2#2027521411yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1487C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#1487D53
hypocenterC53#9234
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
2. C4×Q8×Dic53C2semidirect
5. #20272616C2semidirect
6. #20272547C2semidirect
7. C53×#1487C2semidirect
11. #16810908C4semidirect
12. #16810823C4semidirect
124. C4×Dic53SD16semidirect
130. Q8×C53#104semidirect
197. #1487D53semidirect
227. C53#9234semidirect
331. Q8#13133143semidirect

Maximal subgroups

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

subgroupindex
#202847462
C4×Q8×Dic532
#202755422
#202732112
#202726162
#202725472
C53×#14872
#923453