Finite Group Databasebeta

← #27280984

Subgroups — #27280984

Order 13568.

612 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284810C211yesmaximal, characteristic
2#20277077C211yesmaximal, characteristic
3#20275660C211yesmaximal, characteristic, complemented
4#20273157C211yesmaximal, characteristic
5#20272608C211yesmaximal, characteristic, complemented
6#20272570C211yesmaximal, characteristic, complemented
7C53×#1506C211yesmaximal, characteristic, complemented
8C2×C4×D212—21no—
9C2×C4×D212C2211yescharacteristic
10#16810956—21no—
11#16810900C412yescomplemented
12#16810825C412yescomplemented
13#16810773—21no—
14#16810721—22no—
15#16810680C2211yescharacteristic
16C4×#11719917C2211yescharacteristic
17C4×C4⋊Dic53—21no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810470—22no—
20#16810470—21no—
21#95×D53—21no—
22#16810382—22no—
23#16810350—22no—
24C53×#533C2211yescharacteristic
25C53×#432C2211yescharacteristic
26C53×#376C2211yescharacteristic
27C53×#328—21no—
28C53×#328—21no—
29C22×D212D411yescharacteristic
30C22×C4×D53—22no—
31C22×C4×D53—21no—
32C2×#11719924—22no—
33C2×#11719917—21no—
34C4×D212—28no—
35C4×D212—21no—
36C4×D212—21no—
37C4×D212C2×C412yes—
38C4×D212D412yes—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53D411yescharacteristic
41C2×#11719921—22no—
42C2×#11719921—21no—
43C2×C4×Dic53—22no—
44C2×C4×Dic53D412yes—
45#13133150—22no—
46C4⋊#88322C2×C412yescomplemented
47#13133141—22no—
48#13133137—22no—
49C22⋊#88322—21no—
50#13133128—21no—
51#13133128—21no—
52C4×#88322—21no—
53#13133124—22no—
54#13133124—21no—
55C4⋊Q8×C53C2×C411yescharacteristic, complemented
56C4⋊Q8×C53C2×C411yescharacteristic, complemented
57#125×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62#117×C53—21no—
63C2×C42×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67#97×C53—21no—
68#95×C53—21no—
69#95×C53D411yescharacteristic
70#95×C53D411yescharacteristic
71C23×D53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—21no—
80C22×Dic53—22no—
81C2×D212—21no—
82C2×D212—24no—
83C2×D212C22⋊C412yes—
84C2×C4×D53—28no—
85C2×C4×D53—22no—
86C2×C4×D53—22no—
87C2×C4×D53—82no—
88C2×C4×D53—82no—
89C2×C4×D53—82no—
90C2×C8×C53—41no—
91C4⋊C4×C53—21no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C42×C53—24no—
99C42×C53—41no—
100C42×C53C22×C411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53#4111yescharacteristic
103C42×C53#4111yescharacteristic
104#11719924—44no—
105#11719921—28no—
106#11719921—22no—
107#11719921—22no—
108C4⋊Dic53—24no—
109C4⋊Dic53—21no—
110C4⋊Dic53C22⋊C412yes—
111C4×Dic53—42no—
112C4×Dic53—82no—
113#11719917—22no—
114#11719917—41no—
115C2×#88322—22no—
116C2×#88322—41no—
117C23×C53#12011yescharacteristic
118C22×D53—24no—
119C22×D53—22no—
120C22×D53—42no—
121C22×D53—41no—
122C22×D53—41no—
123Q8×C53—42no—
124Q8×C53—81no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
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×Dic53—24no—
144C2×Dic53—22no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148D212—24no—
149D212—22no—
150D212#10414yes—
151C4×D53—82no—
152C4×D53—82no—
153C4×D53—88no—
154C8×C53—81no—
155#88322—42no—
156#88322—42no—
157#9323—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53#38611yescharacteristic
161C22×C53#45511yescharacteristic
162C22×C53#45611yescharacteristic
163C2×D53—24no—
164C2×D53—44no—
165C2×D53—44no—
166C2×D53—42no—
167C2×D53—42no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—42no—
174C4×C53—42no—
175C4×C53—81no—
176C4×C53—81no—
177C4×C53C2×#10412yes—
178Dic53—44no—
179Dic53—42no—
180#3110—531no—
181#2275—531no—
182#1986—531no—
183#1506D5311yescharacteristic, complemented
184#1490—531no—
185#1300—531no—
186#1285—531no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53#188211yescharacteristic
190C2×C53#189611yescharacteristic
191C2×C53#190411yescharacteristic
192D53—44no—
193#533C2×D5311yescharacteristic
194C2×C4×D4—531no—
195C2×C4×D4—1061no—
196#484—1061no—
197#474—532no—
198#461—532no—
199#450—1061no—
200#440—1062no—
201#432—531no—
202#432C2×D5311yescharacteristic
203C4×M16—531no—
204C4×C4⋊C4—531no—
205C4×C4⋊C4—1061no—
206C4×C22⋊C4—1062no—
207C4×C22⋊C4—1061no—
208#376C2×D5311yescharacteristic
209C2×#95—1061no—
210#339—1062no—
211#328—21no—
212#328—21no—
213D4⋊C8—1062no—
214C53#932311yescharacteristic, complemented, Sylow
215C22×D4—531no—
216C23×C4—1062no—
217C23×C4—1061no—
218C2×SD16—1062no—
219C2×M16—1061no—
220C4⋊Q8C4×D5311yescharacteristic, complemented
221C4⋊Q8C4×D5311yescharacteristic, complemented
222#125—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×D4—532no—
228C4×D4—532no—
229C4×D4—1068no—
230C4×D4—1061no—
231C4×D4—1061no—
232#117—21no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—1061no—
235C2×C22⋊C4—1062no—
236C2×C22⋊C4—1061no—
237C2×C42—21no—
238C2×C42—532no—
239C2×C42—1062no—
240C2×C42C22×D5311yescharacteristic
241#107—1062no—
242C4⋊C8—21no—
243C4⋊C8—532no—
244#103—1062no—
245#102—1062no—
246C22⋊C8—21no—
247C22⋊C8—1061no—
248#97—21no—
249#97—1061no—
250#97—1061no—
251C4×C8—1061no—
252#95—21no—
253#95—1062no—
254#95—1061no—
255#95C22⋊D5311yescharacteristic
256#95C22⋊D5311yescharacteristic
257C24—1061no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×D4—532no—
261C2×D4—1064no—
262C2×D4—1061no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—1068no—
266C22×C4—1062no—
267C22×C4—1061no—
268C22×C4—1062no—
269C22×C4—1062no—
270C22×C4—4242no—
271C22×C4—4242no—
272C22×C4—4242no—
273C22×C4D4×D5311yescharacteristic
274C22×C4D4×D5311yescharacteristic
275C22×C4C2×C22⋊D5311yescharacteristic
276SD16—2124no—
277M16—1062no—
278M16—2121no—
279C2×C8—41no—
280C2×C8—1062no—
281C2×C8—2121no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—532no—
289C4⋊C4—1064no—
290C4⋊C4—1061no—
291C22⋊C4—41no—
292C22⋊C4—1068no—
293C22⋊C4—1062no—
294C22⋊C4—1062no—
295C42—24no—
296C42—41no—
297C42—4242no—
298C42—2122no—
299C42C2×C4×D5311yescharacteristic
300C42#1171994511yescharacteristic
301C42D4×D5311yescharacteristic
302C42#1171994711yescharacteristic
303C23—1064no—
304C23—1062no—
305C23—2122no—
306C23—2121no—
307C23—2121no—
308C23#1313318511yescharacteristic
309Q8—42no—
310Q8—81no—
311D4—534no—
312D4—1064no—
313D4—1062no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—1064no—
327C2×C4—1062no—
328C2×C4—4242no—
329C2×C4—4242no—
330C2×C4—4248no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4C22⋊C4×D5311yescharacteristic
335C2×C4#1313317711yescharacteristic
336C2×C4C4×C22⋊D5311yescharacteristic
337C2×C4#1313318811yescharacteristic
338C2×C4#1313319011yescharacteristic
339C2×C4#1313319911yescharacteristic
340C8—81no—
341C8—2122no—
342C8—2122no—
343C22—21no—
344C22—21no—
345C22—1064no—
346C22—2124no—
347C22—2124no—
348C22—2122no—
349C22—2122no—
350C22#1681049911yescharacteristic
351C22#1681079911yescharacteristic
352C22#1681080411yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—42no—
360C4—81no—
361C4—81no—
362C4—2124no—
363C4—2122no—
364C4#104×D5312yes—
365C2—21no—
366C2—21no—
367C2—2124no—
368C2#2027508111yescharacteristic
369C2#2027515511yescharacteristic
370C2#2027520511yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1506C2
socleC22×C53#386
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1506D53
hypocenterC53#9323
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
3. #20275660C2semidirect
5. #20272608C2semidirect
6. #20272570C2semidirect
7. C53×#1506C2semidirect
11. #16810900C4semidirect
12. #16810825C4semidirect
46. C4⋊#88322C2×C4semidirect
55. C4⋊Q8×C53C2×C4semidirect
56. C4⋊Q8×C53C2×C4semidirect
183. #1506D53semidirect
214. C53#9323semidirect
220. C4⋊Q8C4×D53semidirect
221. C4⋊Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202848102
#202770772
#202756602
#202731572
#202726082
#202725702
C53×#15062
#932353