Finite Group Databasebeta

← #27294879

Subgroups — #27294879

Order 13568.

476 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 79 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#20280567C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic, complemented
3#20272737C212yesmaximal, complemented
4#20272713C212yesmaximal, complemented
5C53×#1465C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic, complemented
7C2×#13133132—21no—
8#16810549—21no—
9#16810535C411yescharacteristic, complemented
10#16810535C2211yescharacteristic
11#16810532C411yescharacteristic, complemented
12#16810532C2211yescharacteristic
13#16810531—21no—
14#16810502—21no—
15#16810502—21no—
16#16810491C411yescharacteristic, complemented
17#16810483—21no—
18#16810390C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#324—21no—
22C53×#323C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×D212C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26C22×C22⋊D53—21no—
27#13133205—41no—
28#13133205—41no—
29C2×C4⋊Dic53—41no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—22no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×#11719921—21no—
37#13133132—42no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—41no—
43#13133124C2×C411yescharacteristic
44#13133124C2×C411yescharacteristic
45C22×D4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C2×M16×C53—41no—
49C4×D4×C53—41no—
50C4×D4×C53—41no—
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54C2×C42×C53C2×C411yescharacteristic
55C22⋊C8×C53—22no—
56C22⋊C8×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C23×D53—21no—
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—81no—
69C22×C4×C53—81no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C2×C22⋊D53—42no—
76C2×C22⋊D53—41no—
77C2×C22⋊D53—41no—
78C2×C22⋊D53—81no—
79C2×C22⋊D53—81no—
80C2×C22⋊D53—81no—
81C2×C22⋊D53—81no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—42no—
86C2×D212—41no—
87C2×D212—81no—
88C2×Dic106—41no—
89C2×Dic106—81no—
90M16×C53—82no—
91C2×C8×C53—42no—
92C4⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C42×C53—21no—
96C42×C53—21no—
97C22⋊Dic53—42no—
98C22⋊Dic53—42no—
99C22⋊Dic53—81no—
100C22⋊Dic53—81no—
101#11719921—42no—
102#11719921—41no—
103#11719921—41no—
104#11719921—81no—
105#11719921—81no—
106C4⋊Dic53—82no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53C2×C22⋊C411yescharacteristic
117C23×C53AGL(1,ℤ/8)11yescharacteristic
118C23×C53#13711yescharacteristic
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53—41no—
124C22×D53—41no—
125C22×D53—41no—
126D4×C53—42no—
127D4×C53—42no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—81no—
141C2×C4×C53—81no—
142C2×C4×C53#9911yescharacteristic
143C2×C4×C53#9911yescharacteristic
144C22⋊D53—82no—
145C22⋊D53—82no—
146C22⋊D53—84no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—82no—
157C2×Dic53—82no—
158C2×Dic53—82no—
159D212—82no—
160Dic106—82no—
161C8×C53—82no—
162#11825—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—42no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53C2×#9911yescharacteristic
178C22×C53#41711yescharacteristic
179C22×C53#42111yescharacteristic
180C2×D53—81no—
181C2×D53—81no—
182C2×D53—81no—
183C2×D53—81no—
184C2×D53—81no—
185C2×D53—81no—
186C2×D53—81no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—41no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—41no—
193Dic53—81no—
194Dic53—81no—
195Dic53—81no—
196Dic53—162no—
197#2631—531no—
198#1799—531no—
199#1465D5311yescharacteristic, complemented
200#1351—532no—
201#1343—532no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—41no—
205C2×C53—41no—
206C2×C53—81no—
207C2×C53#151311yescharacteristic
208C2×C53#212211yescharacteristic
209C2×C53#212711yescharacteristic
210D53—161no—
211C2×#124—531no—
212C2×C4×D4C2×D5311yescharacteristic
213C2×#99—1061no—
214#407—21no—
215#397—1061no—
216#393—531no—
217#393—531no—
218#392—531no—
219#392—531no—
220#392—1061no—
221#386—1061no—
222#386—1061no—
223#383—531no—
224#380—1061no—
225#342—532no—
226#324—21no—
227#323C2×D5312yes—
228C53#1182511yescharacteristic, complemented, Sylow
229C22×Q8—531no—
230C22×D4—531no—
231C22×D4—1061no—
232C22×D4—1061no—
233C22×D4D21211yescharacteristic
234C23×C4D21211yescharacteristic
235C23×C4C22⋊D5311yescharacteristic
236C2×M16—41no—
237#124—2121no—
238#124—2121no—
239C4×D4—41no—
240C4×D4—41no—
241C2×C4⋊C4—2121no—
242C2×C4⋊C4C22⋊D5311yescharacteristic
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C22⋊C4—1061no—
247C2×C22⋊C4—1061no—
248C2×C22⋊C4—1061no—
249C2×C22⋊C4—1062no—
250C2×C22⋊C4C4×D5311yescharacteristic
251C2×C22⋊C4C22×D5311yescharacteristic
252C2×C42C4×D5311yescharacteristic
253#99—2122no—
254C22⋊C8—22no—
255C22⋊C8—41no—
256#95—531no—
257#95—531no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262#95—2121no—
263C24—1061no—
264C24#1171992111yescharacteristic
265C24C2×D21211yescharacteristic
266C2×Q8—4241no—
267C2×Q8—2121no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—4241no—
273C2×D4—4241no—
274C2×D4—4241no—
275C2×D4—4241no—
276C2×D4—4241no—
277C2×D4—2121no—
278C2×D4—2122no—
279C2×D4—2121no—
280C2×D4—2121no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—2122no—
291C22×C4#1171992111yescharacteristic
292C22×C4#1171992111yescharacteristic
293C22×C4#1171992111yescharacteristic
294C22×C4C2×C4×D5311yescharacteristic
295C22×C4C2×C22⋊D5311yescharacteristic
296M16—82no—
297C2×C8—42no—
298C4⋊C4—41no—
299C4⋊C4—4242no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—4241no—
303C22⋊C4—4241no—
304C22⋊C4—4241no—
305C22⋊C4—4241no—
306C22⋊C4—2122no—
307C22⋊C4—2122no—
308C22⋊C4—2122no—
309C22⋊C4—2121no—
310C22⋊C4—2121no—
311C42—21no—
312C42—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—2121no—
323C23—2121no—
324C23—2121no—
325C23—2121no—
326C23—2121no—
327C23—2121no—
328C23—2121no—
329C23C2×#1171992111yescharacteristic
330C23#1313324511yescharacteristic
331C23#1313325011yescharacteristic
332Q8—4242no—
333D4—42no—
334D4—42no—
335D4—4242no—
336D4—4242no—
337D4—4242no—
338D4—4244no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—4242no—
354C2×C4—4242no—
355C2×C4—4242no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4#1313313111yescharacteristic
366C2×C4#1313313111yescharacteristic
367C8—82no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—4241no—
383C22—4241no—
384C22—4241no—
385C22—4241no—
386C22—4241no—
387C22—4241no—
388C22—4241no—
389C22C2×#1313313111yescharacteristic
390C22#1681062511yescharacteristic
391C22#1681063911yescharacteristic
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—8482no—
399C4—4241no—
400C4—4241no—
401C4—4241no—
402C2—21no—
403C2—21no—
404C2—41no—
405C2—41no—
406C2—81no—
407C2—8481no—
408C2#2027325911yescharacteristic
409C2#2027636611yescharacteristic
410C2#2027638211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1465C2
socleC22×C53C2×#99
prime radical#20280567C2
cosocleC2×C22⋊C4×C53C23
hypercenter#1465D53
hypocenterC53#11825
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #20274703C2semidirect
3. #20272737C2semidirect
4. #20272713C2semidirect
5. C53×#1465C2semidirect
6. C2×#13133205C4semidirect
9. #16810535C4semidirect
11. #16810532C4semidirect
16. #16810491C4semidirect
199. #1465D53semidirect
228. C53#11825semidirect

Maximal subgroups

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

subgroupindex
#202805672
#202747032
#202727372
#202727132
C53×#14652
#1182553