Finite Group Databasebeta

← #27369330

Subgroups — #27369330

Order 13568.

728 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285833C211yesmaximal, characteristic, complemented
2C2×#16810828C211yesmaximal, characteristic, complemented
3#20283739C211yesmaximal, characteristic, complemented
4#20274150C214yesmaximal, complemented
5#20274102C214yesmaximal, complemented
6#20273491C211yesmaximal, characteristic, complemented
7#20273388C211yesmaximal, characteristic, complemented
8#20273364C211yesmaximal, characteristic, complemented
9C53×#3036C211yesmaximal, characteristic
10C22×#11719923—21no—
11C2×#13133217C2211yescharacteristic
12#16811026C2212yescomplemented
13#16810969C2212yescomplemented
14#16810934—22no—
15#16810903—22no—
16#16810828—22no—
17#16810828C2214yes—
18D4×#88322—22no—
19D4×#88322C2214yes—
20#16810676C2211yescharacteristic
21#16810676C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133140—21no—
24C2×C22⋊#88322—22no—
25#16810375C2214yescomplemented
26#16810370C2214yescomplemented
27#16810369C2212yescomplemented
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—
39C2×#11719923—44no—
40C22×#88322—24no—
41C22×#88322—21no—
42#13133217—21no—
43#13133217—21no—
44#13133217—21no—
45#13133217C2312yes—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48#13133150—24no—
49C4⋊#88322—21no—
50C4⋊#88322D412yes—
51C4⋊#88322C2312yes—
52C4⋊#88322C2312yes—
53#13133140—24no—
54#13133138—24no—
55C22⋊#88322—24no—
56C22⋊#88322—21no—
57C22⋊#88322D412yes—
58C4×#88322—21no—
59C4×#88322C2312yes—
60C2×#41×C53—21no—
61C22×D4×C53D411yescharacteristic
62C23×C4×C53—21no—
63AGL(1,ℤ/8)×C53—44no—
64C2×SD16×C53—22no—
65C2×D8×C53—22no—
66C2×M16×C53—21no—
67C4⋊Q8×C53C2311yescharacteristic
68#127×C53C2311yescharacteristic
69#124×C53—21no—
70#121×C53—21no—
71#121×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53—21no—
76C4×D4×C53D412yes—
77C4×D4×C53C2312yes—
78C2×C4⋊C4×C53D411yescharacteristic
79C2×C22⋊C4×C53—21no—
80C2×C42×C53C2311yescharacteristic
81C4⋊C8×C53—21no—
82C4⋊C8×C53C2312yes—
83#103×C53—22no—
84#102×C53—22no—
85C24×C53—21no—
86#41×C53—42no—
87#41×C53—42no—
88C2×Q8×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—22no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—41no—
95C2×D4×C53D814yes—
96C2×D4×C53C2×D412yes—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—41no—
101C22×C4×C53—82no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×Dic53—41no—
106C2×Dic106—22no—
107C2×Dic106—42no—
108C2×Dic106—42no—
109C2×Dic106—41no—
110SD16×C53—44no—
111D8×C53—44no—
112M16×C53—42no—
113C2×C8×C53—22no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53C2×D412yes—
118C22⋊C4×C53—22no—
119C22⋊C4×C53—22no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D411yescharacteristic
124C42×C53#4111yescharacteristic
125C42×C53C2411yescharacteristic
126#11719923—48no—
127C4⋊Dic53—22no—
128C4⋊Dic53—22no—
129C4⋊Dic53—41no—
130C4⋊Dic53—41no—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322—42no—
134C2×#88322—84no—
135C2×#88322C2×D414yes—
136C23×C53—22no—
137C23×C53—24no—
138C23×C53—41no—
139C23×C53—81no—
140C23×C53—81no—
141C23×C53#12111yescharacteristic
142Q8×C53—41no—
143Q8×C53—41no—
144D4×C53—22no—
145D4×C53—24no—
146D4×C53—41no—
147D4×C53—41no—
148D4×C53—42no—
149D4×C53—81no—
150D4×C53—82no—
151D4×C53C2×D814yes—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—41no—
155C2×C4×C53—44no—
156C2×C4×C53—41no—
157C2×C4×C53—82no—
158C2×C4×C53—81no—
159C2×C4×C53—82no—
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53C2×D812yes—
164C2×C4×C53#13512yes—
165C2×C4×C53C22×D411yescharacteristic
166C2×C4×C53C22×D411yescharacteristic
167C2×C4×C53C2×#4111yescharacteristic
168C2×Dic53—42no—
169C2×Dic53—82no—
170Dic106—42no—
171Dic106—42no—
172Dic106—82no—
173C8×C53—42no—
174#88322—24no—
175#88322—42no—
176#88322—42no—
177#19392—531nomaximal, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—24no—
181C22×C53—42no—
182C22×C53—42no—
183C22×C53—84no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53#46612yes—
188C22×C53C2×#12111yescharacteristic
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—42no—
193C4×C53—81no—
194C4×C53#46612yes—
195C4×C53C22×D811yescharacteristic
196C4×C53C2×#13511yescharacteristic
197C4×C53#57712yes—
198Dic53—82no—
199#3226—531no—
200#3036D5311yescharacteristic
201C2×#461—531no—
202#2972—531no—
203#1681—534no—
204#1673—534no—
205#1571—531no—
206#1546—531no—
207#1540—531no—
208C2×C53—22no—
209C2×C53—44no—
210C2×C53—161no—
211C2×C53#304211yescharacteristic
212C2×C53C2×#46611yescharacteristic
213C2×C53#306911yescharacteristic
214C2×AGL(1,ℤ/8)—21no—
215C22×SD16—1061no—
216#532C2×D5311yescharacteristic
217C2×C4⋊Q8—531no—
218C2×C4×D4C2×D5311yescharacteristic
219#500—532no—
220#486—532no—
221#480—1062no—
222#474—1062no—
223#463—22no—
224#461—534no—
225#461—1062no—
226#461C2×D5312yes—
227#459C2×D5312yes—
228C8×D4—534no—
229C8×D4—1062no—
230#431—531no—
231#431—531no—
232#423C2×D5311yescharacteristic
233C2×C4⋊C8—531no—
234#417—21no—
235C2×#103—1061no—
236C2×C22⋊C8—1062no—
237#335—534no—
238#332—532no—
239#332—534no—
240#331—532no—
241C53#1939211yescharacteristic, complemented, Sylow
242C2×#41—21no—
243C22×Q8—1061no—
244C22×D4C22⋊D5311yescharacteristic
245C23×C4—21no—
246AGL(1,ℤ/8)—44no—
247C2×SD16—22no—
248C2×SD16—1064no—
249C2×SD16—2124no—
250C2×D8—22no—
251C2×M16—21no—
252C22×C8—1064no—
253C22×C8—1061no—
254C4⋊Q8—532no—
255C4⋊Q8—1061no—
256C4⋊Q8—1061no—
257C4⋊Q8—1061no—
258C4⋊Q8C22×D5311yescharacteristic
259#127C22×D5311yescharacteristic
260#124—21no—
261#121—21no—
262#121—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4C22⋊D5312yes—
268C4×D4C22×D5312yes—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4C22⋊D5311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C42C22×D5311yescharacteristic
274#107—1064no—
275C4⋊C8—21no—
276C4⋊C8—532no—
277C4⋊C8—532no—
278C4⋊C8—532no—
279C4⋊C8—1061no—
280C4⋊C8C22×D5312yes—
281#103—22no—
282#103—1064no—
283#102—22no—
284#102—1064no—
285C22⋊C8—532no—
286C22⋊C8—1064no—
287C22⋊C8—1061no—
288C4×C8—532no—
289C4×C8—1061no—
290C24—21no—
291#41—42no—
292#41—42no—
293C2×Q8—21no—
294C2×Q8—1062no—
295C2×Q8—2122no—
296C2×Q8—2122no—
297C2×Q8—2121no—
298C2×D4—21no—
299C2×D4—22no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—41no—
304C2×D4#1171992214yes—
305C2×D4C2×C22⋊D5312yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—82no—
311C22×C4—2121no—
312C22×C4D4×D5311yescharacteristic
313C22×C4#1171994711yescharacteristic
314C22×C4C2×C22⋊D5311yescharacteristic
315SD16—44no—
316SD16—2128no—
317D8—44no—
318M16—42no—
319C2×C8—22no—
320C2×C8—534no—
321C2×C8—1062no—
322C2×C8—1062no—
323C2×C8—4244no—
324C2×C8—2122no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—1062no—
329C4⋊C4—1062no—
330C4⋊C4—2121no—
331C4⋊C4—2121no—
332C4⋊C4C2×C22⋊D5312yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C42D4×D5311yescharacteristic
338C42#1171994711yescharacteristic
339C42C2×C22⋊D5311yescharacteristic
340C42C23×D5311yescharacteristic
341C23—22no—
342C23—24no—
343C23—41no—
344C23—81no—
345C23—81no—
346C23#1313318911yescharacteristic
347Q8—41no—
348Q8—41no—
349Q8—4242no—
350Q8—2122no—
351Q8—2122no—
352D4—24no—
353D4—22no—
354D4—41no—
355D4—41no—
356D4—42no—
357D4—81no—
358D4—82no—
359D4C2×#1171992214yes—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—4242no—
369C2×C4—2122no—
370C2×C4#1313318911yescharacteristic
371C2×C4#1313318911yescharacteristic
372C2×C4#1313318911yescharacteristic
373C2×C4C2×#1171992212yes—
374C2×C4#1313323712yes—
375C2×C4C2×D4×D5311yescharacteristic
376C2×C4C22×C22⋊D5311yescharacteristic
377C2×C4C2×#1171994711yescharacteristic
378C8—42no—
379C8—1064no—
380C8—2122no—
381C8—2122no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—84no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22#1681085712yes—
392C22C2×#1313318911yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—4242no—
399C4#1681085712yes—
400C4C22×#1171992211yescharacteristic
401C4C2×#1313323711yescharacteristic
402C4#1681143312yes—
403C2—22no—
404C2—44no—
405C2—161no—
406C2#2028422911yescharacteristic
407C2C2×#1681085711yescharacteristic
408C2#2028445411yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3036C2
socleC22×C53C2×#121
hypercenter#3036D53
hypocenterC53#19392
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. #20285833C2semidirect
2. C2×#16810828C2semidirect
3. #20283739C2semidirect
4. #20274150C2semidirect
5. #20274102C2semidirect
6. #20273491C2semidirect
7. #20273388C2semidirect
8. #20273364C2semidirect
12. #16811026C22semidirect
13. #16810969C22semidirect
25. #16810375C22semidirect
26. #16810370C22semidirect
27. #16810369C22semidirect
28. #16810367C22semidirect
241. C53#19392semidirect

Maximal subgroups

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

subgroupindex
#202858332
C2×#168108282
#202837392
#202741502
#202741022
#202734912
#202733882
#202733642
C53×#30362
#1939253