Finite Group Databasebeta

← #27369328

Subgroups — #27369328

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#20285838C211yesmaximal, characteristic, complemented
2#20284179C211yesmaximal, characteristic
3C2×#16810828C211yesmaximal, characteristic, complemented
4#20274152C214yesmaximal, complemented
5#20274104C214yesmaximal, complemented
6#20273490C211yesmaximal, characteristic, complemented
7#20273386C211yesmaximal, characteristic, complemented
8#20273362C211yesmaximal, characteristic, complemented
9C53×#2972C211yesmaximal, characteristic, complemented
10C2×#13133243—21no—
11C22×#11719923—21no—
12#16811173C2211yescharacteristic
13C2×#13133217C2211yescharacteristic
14#16811028C2212yescomplemented
15#16810970C2212yescomplemented
16#16810937—22no—
17#16810904—22no—
18#16810838—22no—
19#16810828—22no—
20#16810828C2212yes—
21#16810828C2214yes—
22#16810818C2212yes—
23#16810647C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25#16810622—21no—
26C2×#13133140—21no—
27#16810376C2214yescomplemented
28#16810368C2212yescomplemented
29#16810368C2214yescomplemented
30#16810366C2212yescomplemented
31C2×C4×D4×C53C2211yescharacteristic
32C8×D4×C53—22no—
33C8×D4×C53C2214yes—
34C53×#431C2211yescharacteristic
35C53×#431C2211yescharacteristic
36C2×C22⋊C8×C53—22no—
37C2×#11719945—21no—
38C22×Dic106—21no—
39#13133243—44no—
40C2×#11719923—24no—
41C2×#11719923—22no—
42C2×#11719923—44no—
43C2×#11719922—22no—
44C2×#11719917—21no—
45C22×#88322—21no—
46#13133217—21no—
47#13133217—21no—
48#13133217—21no—
49#13133217C2311yescharacteristic
50#13133217C2312yes—
51#13133214C2311yescharacteristic
52#13133205—21no—
53#13133191—21no—
54#13133191—21no—
55C4×D212—21no—
56C4×D212—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—21no—
59#13133151—24no—
60C4⋊#88322—21no—
61C4⋊#88322—21no—
62C4⋊#88322C2312yes—
63C4⋊#88322C2312yes—
64#13133140—24no—
65#13133140—22no—
66#13133139—24no—
67#13133137—22no—
68C22×D4×C53D411yescharacteristic
69C23×C4×C53—21no—
70C22×C8×C53—24no—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53D412yes—
74C4×D4×C53C2312yes—
75C2×C4⋊C4×C53D411yescharacteristic
76C2×C22⋊C4×C53—21no—
77C2×C42×C53C2311yescharacteristic
78C4⋊C8×C53D412yes—
79C4⋊C8×C53C2312yes—
80C22⋊C8×C53—24no—
81C22⋊C8×C53—21no—
82C22⋊C8×C53D412yes—
83C4×C8×C53—21no—
84C4×C8×C53C2312yes—
85C24×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—22no—
88C2×D4×C53D814yes—
89C2×D4×C53C2×D412yes—
90C22×C4×C53—21no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—82no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53#4111yescharacteristic
97C2×C22⋊D53—41no—
98C22×Dic53—41no—
99#11719945—42no—
100#11719945—42no—
101C2×D212—21no—
102C2×D212—21no—
103C2×D212—21no—
104C2×C4×D53—41no—
105C2×Dic106—21no—
106C2×Dic106—22no—
107C2×Dic106—42no—
108C2×Dic106—42no—
109C2×Dic106—41no—
110C2×C8×C53—22no—
111C2×C8×C53—84no—
112C2×C8×C53C2×D414yes—
113C4⋊C4×C53—21no—
114C4⋊C4×C53C2×D412yes—
115C22⋊C4×C53—22no—
116C22⋊C4×C53—22no—
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53#4111yescharacteristic
120C42×C53C2411yescharacteristic
121#11719923—48no—
122#11719923—44no—
123#11719922—44no—
124#11719921—41no—
125#11719921—41no—
126C4⋊Dic53—22no—
127C4⋊Dic53—21no—
128C4⋊Dic53—22no—
129C4⋊Dic53—21no—
130C4⋊Dic53—41no—
131C4⋊Dic53—41no—
132#11719917—42no—
133C2×#88322—22no—
134C2×#88322—22no—
135C2×#88322—42no—
136C23×C53—22no—
137C23×C53—24no—
138C23×C53—81no—
139C23×C53—81no—
140C23×C53#12111yescharacteristic
141C22×D53—41no—
142D4×C53—22no—
143D4×C53—24no—
144D4×C53C2×D814yes—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53—44no—
149C2×C4×C53—82no—
150C2×C4×C53—81no—
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53C2×D812yes—
155C2×C4×C53#13512yes—
156C2×C4×C53C22×D411yescharacteristic
157C2×C4×C53C22×D411yescharacteristic
158C2×C4×C53C2×#4111yescharacteristic
159C22⋊D53—82no—
160C2×Dic53—42no—
161C2×Dic53—41no—
162C2×Dic53—82no—
163D212—41no—
164D212—41no—
165D212—42no—
166D212—81no—
167C4×D53—82no—
168Dic106—42no—
169Dic106—41no—
170Dic106—42no—
171Dic106—41no—
172Dic106—82no—
173C8×C53—24no—
174C8×C53—42no—
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#46612yes—
185C22×C53C2×#12111yescharacteristic
186C2×D53—81no—
187C2×D53—81no—
188C2×D53—81no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—42no—
193C4×C53#46612yes—
194C4×C53C22×D811yescharacteristic
195C4×C53C2×#13511yescharacteristic
196C4×C53#57712yes—
197Dic53—82no—
198Dic53—81no—
199#3226—531no—
200#3036—531no—
201C2×#461—531no—
202#2972D5311yescharacteristic, complemented
203#1681—534no—
204#1673—534no—
205#1571—531no—
206#1546—531no—
207#1540—531no—
208C2×C53—22no—
209C2×C53—44no—
210C2×C53#304211yescharacteristic
211C2×C53C2×#46611yescharacteristic
212C2×C53#306911yescharacteristic
213D53—161no—
214C2×AGL(1,ℤ/8)—1061no—
215C22×SD16—1061no—
216#532—531no—
217C2×C4⋊Q8—531no—
218C2×C4×D4C2×D5311yescharacteristic
219#500—532no—
220#486—532no—
221#480—1062no—
222#474—1062no—
223#463—1062no—
224#461—532no—
225#461—534no—
226#461—1062no—
227#459—532no—
228C8×D4—22no—
229C8×D4C2×D5314yes—
230#431C2×D5311yescharacteristic
231#431C2×D5311yescharacteristic
232#423—531no—
233C2×C4⋊C8—531no—
234#417—1061no—
235C2×#103—1061no—
236C2×C22⋊C8—22no—
237#335—534no—
238#332—532no—
239#332—534no—
240#331—532no—
241C53#1939211yescharacteristic, complemented, Sylow
242C2×#41—1061no—
243C22×Q8—1061no—
244C22×D4D21211yescharacteristic
245C23×C4—21no—
246AGL(1,ℤ/8)—2124no—
247C2×SD16—1064no—
248C2×SD16—1062no—
249C2×SD16—2124no—
250C2×D8—1062no—
251C2×M16—1061no—
252C22×C8—24no—
253C22×C8—1061no—
254C4⋊Q8—532no—
255C4⋊Q8—531no—
256C4⋊Q8—1061no—
257C4⋊Q8—1061no—
258C4⋊Q8—1061no—
259#127—531no—
260#124—1061no—
261#121—1061no—
262#121—1061no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—1061no—
266C4×D4—1061no—
267C4×D4D21212yes—
268C4×D4C22×D5312yes—
269C2×C4⋊C4—1061no—
270C2×C4⋊C4—1061no—
271C2×C4⋊C4D21211yescharacteristic
272C2×C22⋊C4—21no—
273C2×C42C22×D5311yescharacteristic
274#107—1064no—
275C4⋊C8—532no—
276C4⋊C8—532no—
277C4⋊C8—1061no—
278C4⋊C8—1061no—
279C4⋊C8C22⋊D5312yes—
280C4⋊C8C22×D5312yes—
281#103—1064no—
282#103—1062no—
283#102—1064no—
284#102—1062no—
285C22⋊C8—21no—
286C22⋊C8—24no—
287C22⋊C8C22⋊D5312yes—
288C4×C8—21no—
289C4×C8C22×D5312yes—
290C24—21no—
291#41—2122no—
292#41—2122no—
293C2×Q8—1061no—
294C2×Q8—1062no—
295C2×Q8—2122no—
296C2×Q8—2122no—
297C2×Q8—2121no—
298C2×D4—21no—
299C2×D4—22no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—2121no—
304C2×D4D42414yes—
305C2×D4C2×D21212yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—82no—
310C22×C4—2121no—
311C22×C4—2121no—
312C22×C4C2×D21211yescharacteristic
313C22×C4#1171994511yescharacteristic
314C22×C4C2×C22⋊D5311yescharacteristic
315SD16—2128no—
316SD16—2124no—
317D8—2124no—
318M16—2122no—
319C2×C8—22no—
320C2×C8—84no—
321C2×C8—1062no—
322C2×C8—1062no—
323C2×C8—2122no—
324C2×C8C2×C22⋊D5314yes—
325C4⋊C4—21no—
326C4⋊C4—1062no—
327C4⋊C4—1061no—
328C4⋊C4—1062no—
329C4⋊C4—1061no—
330C4⋊C4—2121no—
331C4⋊C4—2121no—
332C4⋊C4C2×D21212yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—2121no—
336C22⋊C4—2121no—
337C42C2×D21211yescharacteristic
338C42#1171994511yescharacteristic
339C42C2×C22⋊D5311yescharacteristic
340C42C23×D5311yescharacteristic
341C23—22no—
342C23—24no—
343C23—81no—
344C23—81no—
345C23—2121no—
346C23#1313319111yescharacteristic
347Q8—4242no—
348Q8—2122no—
349Q8—2121no—
350Q8—2122no—
351Q8—2121no—
352D4—24no—
353D4—22no—
354D4—4241no—
355D4—4242no—
356D4—2121no—
357D4—2121no—
358D4—2122no—
359D4C2×D42414yes—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—4242no—
367C2×C4—4242no—
368C2×C4—2121no—
369C2×C4—2122no—
370C2×C4#1313319111yescharacteristic
371C2×C4#1313319111yescharacteristic
372C2×C4#1313319111yescharacteristic
373C2×C4C2×D42412yes—
374C2×C4#1313323912yes—
375C2×C4C22×C22⋊D5311yescharacteristic
376C2×C4C22×D21211yescharacteristic
377C2×C4C2×#1171994511yescharacteristic
378C8—24no—
379C8—42no—
380C8—2122no—
381C8—2122no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—84no—
388C22—4241no—
389C22—4241no—
390C22—4241no—
391C22#1681085912yes—
392C22C2×#1313319111yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—4241no—
398C4—4242no—
399C4#1681085912yes—
400C4C22×D42411yescharacteristic
401C4C2×#1313323911yescharacteristic
402C4#1681143412yes—
403C2—22no—
404C2—44no—
405C2—8481no—
406C2#2028422711yescharacteristic
407C2C2×#1681085911yescharacteristic
408C2#2028445211yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2972C2
socleC22×C53C2×#121
prime radical#20284179C2
hypercenter#2972D53
hypocenterC53#19392
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #20285838C2semidirect
3. C2×#16810828C2semidirect
4. #20274152C2semidirect
5. #20274104C2semidirect
6. #20273490C2semidirect
7. #20273386C2semidirect
8. #20273362C2semidirect
9. C53×#2972C2semidirect
14. #16811028C22semidirect
15. #16810970C22semidirect
27. #16810376C22semidirect
28. #16810368C22semidirect
29. #16810368C22semidirect
30. #16810366C22semidirect
202. #2972D53semidirect
241. C53#19392semidirect

Maximal subgroups

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

subgroupindex
#202858382
#202841792
C2×#168108282
#202741522
#202741042
#202734902
#202733862
#202733622
C53×#29722
#1939253