Finite Group Databasebeta

← #27280065

Subgroups — #27280065

Order 13568.

582 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 103 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#20280599C211yesmaximal, characteristic
2C4×#13133132C211yesmaximal, characteristic, complemented
3#20273232C212yesmaximal, complemented
4#20272569C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic, complemented
6C2×#13133191C2211yescharacteristic
7C2×#13133132—21no—
8#16810562C411yescharacteristic, complemented
9#16810559C411yescharacteristic, complemented
10#16810540C411yescharacteristic, complemented
11#16810538—21no—
12#16810532C411yescharacteristic, complemented
13#16810531—21no—
14#16810514—21no—
15#16810502—21no—
16#16810486C2211yescharacteristic
17C4×C22⋊Dic53C2212yes—
18#16810390—21no—
19#16810358—24no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#435—24no—
22C53×#432—21no—
23C53×#407C2212yes—
24C2×C22⋊C8×C53—21no—
25C22×D212C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C22×C22⋊D53—21no—
28#13133191—22no—
29#13133191—22no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—22no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53—22no—
40#13133132—24no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—22no—
45#13133124C2×C411yescharacteristic
46#13133124C2×C411yescharacteristic
47C22×D4×C53D411yescharacteristic
48C23×C4×C53D411yescharacteristic
49C23×C4×C53C2311yescharacteristic
50C2×M16×C53—22no—
51C22×C8×C53—22no—
52C4×D4×C53—22no—
53C4×D4×C53—22no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53D411yescharacteristic
57C2×C42×C53D411yescharacteristic
58C4⋊C8×C53—22no—
59C22⋊C8×C53—22no—
60C22⋊C8×C53—22no—
61C22⋊C8×C53D414yescomplemented
62#97×C53—22no—
63C24×C53C22⋊C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C23×D53—21no—
66C2×D4×C53—22no—
67C2×D4×C53—22no—
68C2×D4×C53—41no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—81no—
72C22×C4×C53—81no—
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D412yes—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53#4112yes—
80C2×C22⋊D53—42no—
81C2×C22⋊D53—42no—
82C2×C22⋊D53—81no—
83C2×C22⋊D53—81no—
84C2×C22⋊D53—81no—
85C2×C22⋊D53—81no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—42no—
90C2×D212—22no—
91C2×D212—81no—
92M16×C53—82no—
93C2×C8×C53—24no—
94C2×C8×C53—82no—
95C4⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C22⋊C4×C53—22no—
98C42×C53—21no—
99C42×C53—21no—
100C22⋊Dic53—24no—
101C22⋊Dic53—42no—
102C22⋊Dic53—81no—
103C22⋊Dic53—81no—
104#11719921—42no—
105#11719921—41no—
106#11719921—41no—
107#11719921—81no—
108#11719921—81no—
109C4⋊Dic53—22no—
110#11719919—81no—
111C4×Dic53—82no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53#10412yes—
121C23×C53C2×C22⋊C411yescharacteristic
122C23×C53#12112yes—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—41no—
126C22×D53—41no—
127C22×D53—41no—
128C22×D53—41no—
129C22×D53—41no—
130D4×C53—44no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—81no—
144C2×C4×C53C4×D412yes—
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12311yescharacteristic
147C22⋊D53—84no—
148C22⋊D53—84no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—44no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—82no—
160C2×Dic53—82no—
161D212—82no—
162C8×C53—44no—
163#9185—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—42no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53#38611yescharacteristic
177C22×C53C2×#10411yescharacteristic
178C22×C53#42111yescharacteristic
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—81no—
182C2×D53—81no—
183C2×D53—81no—
184C2×D53—81no—
185C2×D53—81no—
186C4×C53—21no—
187C4×C53—22no—
188C4×C53—21no—
189C4×C53—41no—
190C4×C53—41no—
191C4×C53—41no—
192Dic53—81no—
193Dic53—81no—
194Dic53—84no—
195Dic53—81no—
196#2973D5311yescharacteristic, complemented
197#2635—531no—
198C4×#99—531no—
199#1507—532no—
200#1284—532no—
201C2×C53—21no—
202C2×C53—21no—
203C2×C53—42no—
204C2×C53—42no—
205C2×C53#187111yescharacteristic
206C2×C53#191011yescharacteristic
207C2×C53#191311yescharacteristic
208D53—161no—
209C2×#121—531no—
210C2×C4×D4C2×D5311yescharacteristic
211#435—24no—
212#432—21no—
213C2×#99—1061no—
214#407C2×D5312yes—
215C2×C22⋊C8—21no—
216#400—531no—
217#399—531no—
218#394—531no—
219#394—1061no—
220#392—531no—
221#392—1061no—
222#388—1061no—
223#386—1061no—
224#381—531no—
225C4×C22⋊C4—532no—
226#342—1061no—
227#327—1064no—
228C53#918511yescharacteristic, complemented, Sylow
229C22×D4—531no—
230C22×D4—1061no—
231C22×D4—1061no—
232C22×D4D21211yescharacteristic
233C23×C4C22⋊D5311yescharacteristic
234C23×C4C22×D5311yescharacteristic
235C2×M16—22no—
236C22×C8—22no—
237#121—1062no—
238#121—1062no—
239C4×D4—22no—
240C4×D4—22no—
241C2×C4⋊C4—531no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4C4×D5311yescharacteristic
244C2×C22⋊C4—1062no—
245C2×C22⋊C4—1061no—
246C2×C22⋊C4—1061no—
247C2×C22⋊C4—1061no—
248C2×C22⋊C4—1061no—
249C2×C22⋊C4—1061no—
250C2×C22⋊C4—1061no—
251C2×C22⋊C4C4×D5311yescharacteristic
252C2×C22⋊C4D21211yescharacteristic
253C2×C42—1062no—
254C2×C42C22⋊D5311yescharacteristic
255C4⋊C8—22no—
256#99—1064no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8C22⋊D5314yescomplemented
260#97—22no—
261#95—531no—
262#95—531no—
263#95—1061no—
264#95—1061no—
265#95—1061no—
266#95—1062no—
267C24—1061no—
268C24#1171992111yescharacteristic
269C24C2×D21211yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—1062no—
274C2×D4—4241no—
275C2×D4—4241no—
276C2×D4—4241no—
277C2×D4—4241no—
278C2×D4—4241no—
279C2×D4—2122no—
280C2×D4—2122no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—2122no—
289C22×C4#1171992111yescharacteristic
290C22×C4#1171992111yescharacteristic
291C22×C4#1171992111yescharacteristic
292C22×C4C2×C4×D5311yescharacteristic
293C22×C4#1171994512yes—
294C22×C4C2×C22⋊D5312yes—
295C22×C4C2×C22⋊D5311yescharacteristic
296M16—82no—
297C2×C8—24no—
298C2×C8—82no—
299C4⋊C4—22no—
300C4⋊C4—1062no—
301C4⋊C4—4241no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—1064no—
305C22⋊C4—4241no—
306C22⋊C4—4241no—
307C22⋊C4—4241no—
308C22⋊C4—4241no—
309C22⋊C4—2122no—
310C22⋊C4—2122no—
311C22⋊C4—2121no—
312C22⋊C4—2121no—
313C42—21no—
314C42—21no—
315C42—4242no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—2121no—
325C23—2121no—
326C23—2121no—
327C23—2121no—
328C23—2121no—
329C23—2121no—
330C23—2121no—
331C23#1313314312yes—
332C23C2×#1171992111yescharacteristic
333C23#1313319112yes—
334D4—44no—
335D4—4242no—
336D4—4244no—
337D4—4244no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—4242no—
352C2×C4—4242no—
353C2×C4—2124no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4C4×C22⋊D5312yes—
364C2×C4#1313318611yescharacteristic
365C2×C4#1313320011yescharacteristic
366C8—44no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—42no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—4241no—
380C22—4241no—
381C22—4241no—
382C22—4241no—
383C22—4241no—
384C22—4241no—
385C22—4241no—
386C22#1681050211yescharacteristic
387C22C2×#1313314311yescharacteristic
388C22#1681063911yescharacteristic
389C4—21no—
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—4241no—
396C4—4241no—
397C4—4244no—
398C4—4241no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2—8481no—
404C2#2027501711yescharacteristic
405C2#2027524211yescharacteristic
406C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radical#20280599C2
cosocleC23×C4×C53C23
hypercenter#2973D53
hypocenterC53#9185
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. C4×#13133132C2semidirect
3. #20273232C2semidirect
4. #20272569C2semidirect
5. C53×#2973C2semidirect
8. #16810562C4semidirect
9. #16810559C4semidirect
10. #16810540C4semidirect
12. #16810532C4semidirect
61. C22⋊C8×C53D4semidirect
196. #2973D53semidirect
228. C53#9185semidirect
259. C22⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202805992
C4×#131331322
#202732322
#202725692
C53×#29732
#918553