Finite Group Databasebeta

← #27369263

Subgroups — #27369263

Order 13568.

708 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285637C211yesmaximal, characteristic, complemented
2C2×#16810903C211yesmaximal, characteristic, complemented
3#20283739C211yesmaximal, characteristic, complemented
4#20274140C214yesmaximal, complemented
5#20274054C214yesmaximal, complemented
6#20273487C211yesmaximal, characteristic, complemented
7#20273401C211yesmaximal, characteristic, complemented
8#20273364C211yesmaximal, characteristic, complemented
9C53×#3036C211yesmaximal, characteristic
10C2×#13133217C2211yescharacteristic
11#16810995C2212yescomplemented
12#16810975C2212yescomplemented
13#16810903—22no—
14#16810903C2214yes—
15#16810828—22no—
16#16810761—22no—
17D4×#88322—22no—
18D4×#88322C2214yes—
19#16810676C2211yescharacteristic
20#16810676C2211yescharacteristic
21C2×#13133150—21no—
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133138—21no—
24C2×C22⋊#88322—22no—
25#16810373C2214yescomplemented
26#16810369C2212yescomplemented
27#16810368C2214yescomplemented
28#16810367C2212yescomplemented
29C2×AGL(1,ℤ/8)×C53—21no—
30C53×#532C2211yescharacteristic
31C2×C4×D4×C53C2211yescharacteristic
32C53×#463—22no—
33C53×#461C2212yes—
34C53×#459C2212yes—
35C53×#423C2211yescharacteristic
36C53×#417—21no—
37C22×Dic106—21no—
38C2×#11719923—24no—
39C22×#88322—24no—
40C22×#88322—21no—
41#13133217—21no—
42#13133217—21no—
43#13133217—21no—
44#13133217C2312yes—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47#13133150—24no—
48C4⋊#88322—21no—
49C4⋊#88322D412yes—
50C4⋊#88322C2312yes—
51C4⋊#88322C2312yes—
52#13133140—24no—
53#13133138—24no—
54C22⋊#88322—24no—
55C22⋊#88322—21no—
56C22⋊#88322D412yes—
57C4×#88322—21no—
58C4×#88322C2312yes—
59C2×#41×C53—21no—
60C22×D4×C53D411yescharacteristic
61C23×C4×C53—21no—
62AGL(1,ℤ/8)×C53—44no—
63C2×SD16×C53—22no—
64C2×D8×C53—22no—
65C2×M16×C53—21no—
66C4⋊Q8×C53C2311yescharacteristic
67#127×C53C2311yescharacteristic
68#124×C53—21no—
69#121×C53—21no—
70#121×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53D412yes—
76C4×D4×C53C2312yes—
77C2×C4⋊C4×C53D411yescharacteristic
78C2×C22⋊C4×C53—21no—
79C2×C42×C53C2311yescharacteristic
80C4⋊C8×C53—21no—
81C4⋊C8×C53C2312yes—
82#103×C53—22no—
83#102×C53—22no—
84C24×C53—21no—
85#41×C53—42no—
86#41×C53—42no—
87C2×Q8×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—22no—
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—41no—
94C2×D4×C53SD1614yescomplemented
95C2×D4×C53C2×D412yes—
96C22×C4×C53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—41no—
100C22×C4×C53—82no—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C22×Dic53—41no—
105C2×Dic106—22no—
106C2×Dic106—41no—
107C2×Dic106—82no—
108SD16×C53—44no—
109D8×C53—44no—
110M16×C53—42no—
111C2×C8×C53—22no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53C2×D412yes—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53#4111yescharacteristic
123C42×C53C2411yescharacteristic
124#11719923—84no—
125C4⋊Dic53—22no—
126C4⋊Dic53—22no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129C2×#88322—22no—
130C2×#88322—22no—
131C2×#88322—42no—
132C2×#88322—84no—
133C2×#88322C2×D414yes—
134C23×C53—22no—
135C23×C53—24no—
136C23×C53—41no—
137C23×C53—81no—
138C23×C53—81no—
139C23×C53#12111yescharacteristic
140Q8×C53—41no—
141Q8×C53—41no—
142D4×C53—22no—
143D4×C53—24no—
144D4×C53—41no—
145D4×C53—41no—
146D4×C53—42no—
147D4×C53—81no—
148D4×C53—82no—
149D4×C53C2×SD1614yes—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—41no—
153C2×C4×C53—44no—
154C2×C4×C53—41no—
155C2×C4×C53—82no—
156C2×C4×C53—81no—
157C2×C4×C53—82no—
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53C2×SD1612yes—
162C2×C4×C53#13512yes—
163C2×C4×C53C22×D411yescharacteristic
164C2×C4×C53C22×D411yescharacteristic
165C2×C4×C53C2×#4111yescharacteristic
166C2×Dic53—42no—
167C2×Dic53—82no—
168Dic106—44no—
169Dic106—82no—
170C8×C53—42no—
171#88322—24no—
172#88322—42no—
173#88322—42no—
174#19385—531nomaximal, Sylow
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—24no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—84no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53—81no—
184C22×C53#46512yes—
185C22×C53C2×#12111yescharacteristic
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—42no—
190C4×C53—81no—
191C4×C53#46512yes—
192C4×C53C22×SD1611yescharacteristic
193C4×C53C2×#13511yescharacteristic
194C4×C53#57612yes—
195Dic53—82no—
196#3206—531no—
197C2×#474—531no—
198#3036D5311yescharacteristic
199#2972—531no—
200#1679—534no—
201#1665—534no—
202#1570—531no—
203#1549—531no—
204#1540—531no—
205C2×C53—22no—
206C2×C53—44no—
207C2×C53—161no—
208C2×C53#304111yescharacteristic
209C2×C53C2×#46511yescharacteristic
210C2×C53#306911yescharacteristic
211C2×AGL(1,ℤ/8)—21no—
212#532C2×D5311yescharacteristic
213C2×C4⋊Q8—531no—
214C2×C4×D4C2×D5311yescharacteristic
215#492—532no—
216#487—532no—
217#474—534no—
218#474—1062no—
219#463—22no—
220#461—1062no—
221#461C2×D5312yes—
222#459C2×D5312yes—
223#448—1062no—
224C8×D4—534no—
225C8×D4—1062no—
226#431—531no—
227#431—531no—
228C2×#107—1061no—
229#423C2×D5311yescharacteristic
230C2×C4⋊C8—531no—
231#417—21no—
232C2×#102—1061no—
233C2×C22⋊C8—1062no—
234#334—534no—
235#332—532no—
236#332—534no—
237#331—532no—
238C53#1938511yescharacteristic, complemented, Sylow
239C2×#41—21no—
240C22×Q8—1061no—
241C22×D4C22⋊D5311yescharacteristic
242C23×C4—21no—
243AGL(1,ℤ/8)—44no—
244C2×SD16—22no—
245C2×SD16—1064no—
246C2×D8—22no—
247C2×M16—21no—
248C22×C8—1064no—
249C22×C8—1061no—
250C4⋊Q8—532no—
251C4⋊Q8—1061no—
252C4⋊Q8—1061no—
253C4⋊Q8—1061no—
254C4⋊Q8C22×D5311yescharacteristic
255#127C22×D5311yescharacteristic
256#124—21no—
257#121—21no—
258#121—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4C22⋊D5312yes—
264C4×D4C22×D5312yes—
265C2×C4⋊C4—1061no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4C22⋊D5311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C42C22×D5311yescharacteristic
270#107—1064no—
271C4⋊C8—21no—
272C4⋊C8—532no—
273C4⋊C8—532no—
274C4⋊C8—532no—
275C4⋊C8—1061no—
276C4⋊C8C22×D5312yes—
277#103—22no—
278#103—1064no—
279#102—22no—
280#102—1064no—
281C22⋊C8—532no—
282C22⋊C8—1064no—
283C22⋊C8—1061no—
284C4×C8—532no—
285C4×C8—1061no—
286C24—21no—
287#41—42no—
288#41—42no—
289C2×Q8—21no—
290C2×Q8—1062no—
291C2×Q8—4242no—
292C2×Q8—2121no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—41no—
299C2×D4#1171992314yescomplemented
300C2×D4C2×C22⋊D5312yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—2121no—
307C22×C4D4×D5311yescharacteristic
308C22×C4#1171994711yescharacteristic
309C22×C4C2×C22⋊D5311yescharacteristic
310SD16—44no—
311SD16—4244no—
312D8—44no—
313M16—42no—
314C2×C8—22no—
315C2×C8—534no—
316C2×C8—1062no—
317C2×C8—1062no—
318C2×C8—4244no—
319C2×C8—2122no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—1062no—
324C4⋊C4—1062no—
325C4⋊C4—2121no—
326C4⋊C4—2121no—
327C4⋊C4C2×C22⋊D5312yes—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C42D4×D5311yescharacteristic
333C42#1171994711yescharacteristic
334C42C2×C22⋊D5311yescharacteristic
335C42C23×D5311yescharacteristic
336C23—22no—
337C23—24no—
338C23—41no—
339C23—81no—
340C23—81no—
341C23#1313318911yescharacteristic
342Q8—41no—
343Q8—41no—
344Q8—4242no—
345Q8—2124no—
346D4—24no—
347D4—22no—
348D4—41no—
349D4—41no—
350D4—42no—
351D4—81no—
352D4—82no—
353D4C2×#1171992314yes—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—41no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—4242no—
363C2×C4—2122no—
364C2×C4#1313318911yescharacteristic
365C2×C4#1313318911yescharacteristic
366C2×C4#1313318911yescharacteristic
367C2×C4C2×#1171992312yes—
368C2×C4#1313323712yes—
369C2×C4C2×D4×D5311yescharacteristic
370C2×C4C22×C22⋊D5311yescharacteristic
371C2×C4C2×#1171994711yescharacteristic
372C8—42no—
373C8—1064no—
374C8—2122no—
375C8—2122no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#1681085012yes—
386C22C2×#1313318911yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—81no—
392C4—4242no—
393C4#1681085012yes—
394C4C22×#1171992311yescharacteristic
395C4C2×#1313323711yescharacteristic
396C4#1681142712yes—
397C2—22no—
398C2—44no—
399C2—161no—
400C2#2028422011yescharacteristic
401C2C2×#1681085011yescharacteristic
402C2#2028445411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3036C2
socleC22×C53C2×#121
hypercenter#3036D53
hypocenterC53#19385
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #20285637C2semidirect
2. C2×#16810903C2semidirect
3. #20283739C2semidirect
4. #20274140C2semidirect
5. #20274054C2semidirect
6. #20273487C2semidirect
7. #20273401C2semidirect
8. #20273364C2semidirect
11. #16810995C22semidirect
12. #16810975C22semidirect
25. #16810373C22semidirect
26. #16810369C22semidirect
27. #16810368C22semidirect
28. #16810367C22semidirect
94. C2×D4×C53SD16semidirect
238. C53#19385semidirect
299. C2×D4#11719923semidirect

Maximal subgroups

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

subgroupindex
#202856372
C2×#168109032
#202837392
#202741402
#202740542
#202734872
#202734012
#202733642
C53×#30362
#1938553