Finite Group Databasebeta

← #27279842

Subgroups — #27279842

Order 13568.

1088 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 147 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#20282997C211yesmaximal, characteristic
2#20274625C211yesmaximal, characteristic, complemented
3C2×#16810356C212yesmaximal, complemented
4#20272594C212yesmaximal, complemented
5C2×C4×C22⋊C4×C53C211yesmaximal, characteristic, complemented
6C2×#13133191C412yescomplemented
7C2×#13133191C2211yescharacteristic
8C2×#13133186—21no—
9C2×C22⋊#88322—22no—
10C2×C22⋊#88322C2212yes—
11#16810502—21no—
12#16810502—22no—
13#16810483C412yescomplemented
14#16810483C2211yescharacteristic
15#16810356—24no—
16C24×C4×C53C2211yescharacteristic
17C22×C22⋊C4×C53—21no—
18C22×C42×C53—21no—
19C4×C22⋊C4×C53—24no—
20C2×#95×C53C2212yes—
21C22×D212C2×C411yescharacteristic
22C22×C22⋊D53—22no—
23C22×C22⋊D53—21no—
24C22×#88322—22no—
25#13133191—22no—
26#13133191—22no—
27#13133191—44no—
28#13133186—44no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4⋊Dic53C2×C412yes—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—21no—
33C2×#11719921—22no—
34C2×#11719921—21no—
35C22⋊#88322—22no—
36C22⋊#88322—22no—
37C22⋊#88322—44no—
38#13133124—22no—
39#13133124—21no—
40C25×C53D411yescharacteristic
41C23×C4×C53—22no—
42C23×C4×C53—22no—
43C23×C4×C53—22no—
44C23×C4×C53—44no—
45C23×C4×C53C2×C412yes—
46C23×C4×C53D411yescharacteristic
47C23×C4×C53D412yes—
48C23×C4×C53C2311yescharacteristic
49C2×C22⋊C4×C53—28no—
50C2×C22⋊C4×C53—24no—
51C2×C42×C53—24no—
52C2×C42×C53—44no—
53#95×C53—22no—
54#95×C53D414yescomplemented
55C24×C53—24no—
56C24×C53—21no—
57C24×C53—21no—
58C24×C53—22no—
59C24×C53—22no—
60C24×C53—42no—
61C24×C53C22⋊C412yes—
62C24×C53C2×D411yescharacteristic
63C23×D53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—24no—
67C22×C4×C53—28no—
68C22×C4×C53—24no—
69C22×C4×C53—24no—
70C22×C4×C53—24no—
71C22×C4×C53—24no—
72C22×C4×C53—24no—
73C22×C4×C53—42no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53—48no—
77C22×C4×C53—48no—
78C22×C4×C53—44no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53C22⋊C412yes—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D412yes—
85C22×C4×C53#4112yes—
86C2×C22⋊D53—44no—
87C2×C22⋊D53—44no—
88C2×C22⋊D53—44no—
89C2×C22⋊D53—41no—
90C2×C22⋊D53—41no—
91C2×C22⋊D53—42no—
92C2×C22⋊D53—42no—
93C22×Dic53—21no—
94C22×Dic53—22no—
95C2×D212—22no—
96C2×D212—42no—
97C22⋊C4×C53—216no—
98C42×C53—48no—
99C22⋊Dic53—44no—
100C22⋊Dic53—42no—
101#11719921—44no—
102#11719921—42no—
103C4⋊Dic53—22no—
104C4⋊Dic53—42no—
105C2×#88322—44no—
106C2×#88322—42no—
107C23×C53—22no—
108C23×C53—22no—
109C23×C53—24no—
110C23×C53—24no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—22no—
117C23×C53—48no—
118C23×C53—48no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53#10414yes—
124C23×C53C2×C22⋊C411yescharacteristic
125C23×C53#12011yescharacteristic
126C23×C53#12311yescharacteristic
127C23×C53#13512yes—
128C23×C53AGL(1,ℤ/8)12yes—
129C22×D53—41no—
130C22×D53—42no—
131C22×D53—41no—
132C22×D53—41no—
133C22×D53—42no—
134C2×C4×C53—24no—
135C2×C4×C53—24no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—28no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—48no—
144C2×C4×C53—48no—
145C2×C4×C53—48no—
146C2×C4×C53—48no—
147C2×C4×C53—44no—
148C2×C4×C53—44no—
149C2×C4×C53#9914yes—
150C2×C4×C53C4×D412yes—
151C2×C4×C53#12112yes—
152C22⋊D53—88no—
153C22⋊D53—82no—
154C22⋊D53—82no—
155C2×Dic53—41no—
156C2×Dic53—42no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161D212—82no—
162#88322—82no—
163#9150—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—24no—
169C22×C53—24no—
170C22×C53—24no—
171C22×C53—24no—
172C22×C53—24no—
173C22×C53—48no—
174C22×C53—48no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—42no—
178C22×C53—42no—
179C22×C53#38611yescharacteristic
180C22×C53C2×#9912yes—
181C22×C53#41611yescharacteristic
182C22×C53#41811yescharacteristic
183C22×C53C2×#10412yes—
184C22×C53#44912yes—
185C22×C53#48212yes—
186C2×D53—81no—
187C2×D53—81no—
188C2×D53—81no—
189C2×D53—82no—
190C2×D53—82no—
191C4×C53—22no—
192C4×C53—22no—
193C4×C53—24no—
194C4×C53—48no—
195Dic53—81no—
196Dic53—82no—
197#2847—531no—
198C2×C4×C22⋊C4D5311yescharacteristic, complemented
199#1779—531no—
200C2×#327—532no—
201#1294—532no—
202C2×C53—22no—
203C2×C53—22no—
204C2×C53—44no—
205C2×C53#150112yes—
206C2×C53#150712yes—
207C2×C53#187511yescharacteristic
208C2×C53#189611yescharacteristic
209C2×C53#189811yescharacteristic
210D53—161no—
211C24×C4C2×D5311yescharacteristic
212C2×#121—531no—
213C2×#121—532no—
214C2×#120—1061no—
215C22×C22⋊C4—21no—
216C22×C42—21no—
217C2×C22⋊C8—532no—
218C2×C22⋊C8—1062no—
219#386—1062no—
220#386—1061no—
221#380—532no—
222#380—531no—
223C4×C22⋊C4—24no—
224C2×#95C2×D5312yes—
225#327—1064no—
226C53#915011yescharacteristic, complemented, Sylow
227C25C22⋊D5311yescharacteristic
228C22×D4—531no—
229C22×D4—1061no—
230C22×D4—1062no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—44no—
235C23×C4C4×D5312yes—
236C23×C4D21212yes—
237C23×C4C22⋊D5311yescharacteristic
238C23×C4C22×D5311yescharacteristic
239C22×C8—1062no—
240#121—1062no—
241#121—1062no—
242#121—2124no—
243#120—2124no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—532no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—1061no—
249C2×C22⋊C4—1062no—
250C2×C22⋊C4—1061no—
251C2×C22⋊C4—1062no—
252C2×C42—24no—
253C2×C42—44no—
254C22⋊C8—1062no—
255C22⋊C8—1062no—
256C22⋊C8—2124no—
257#95—22no—
258#95—1061no—
259#95—1062no—
260#95C22⋊D5314yescomplemented
261C24—24no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—42no—
267C24—1061no—
268C24#1171992112yes—
269C24C2×C22⋊D5311yescharacteristic
270C2×D4—1062no—
271C2×D4—2122no—
272C2×D4—2124no—
273C2×D4—2124no—
274C2×D4—2124no—
275C2×D4—2121no—
276C2×D4—2121no—
277C2×D4—2122no—
278C2×D4—2122no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—1061no—
297C22×C4—1062no—
298C22×C4#1171992112yes—
299C22×C4C2×C4×D5311yescharacteristic
300C22×C4C2×D21211yescharacteristic
301C22×C4#1171994512yes—
302C22×C4C2×C22⋊D5312yes—
303C2×C8—2124no—
304C2×C8—2122no—
305C4⋊C4—1062no—
306C4⋊C4—2122no—
307C22⋊C4—216no—
308C22⋊C4—2124no—
309C22⋊C4—2122no—
310C22⋊C4—2124no—
311C22⋊C4—2122no—
312C42—48no—
313C23—22no—
314C23—22no—
315C23—24no—
316C23—24no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—48no—
324C23—48no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—2121no—
330C23—2122no—
331C23—2121no—
332C23—2121no—
333C23—2122no—
334C23#1313314514yes—
335C23C2×#1171992111yescharacteristic
336C23#1313318611yescharacteristic
337C23#1313320011yescharacteristic
338C23#1313323712yes—
339C23#1313324212yes—
340D4—4242no—
341D4—4248no—
342D4—4242no—
343D4—4242no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—28no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—2121no—
360C2×C4—2122no—
361C2×C4—2121no—
362C2×C4—2122no—
363C2×C4—2121no—
364C2×C4—2122no—
365C2×C4#1313313114yes—
366C2×C4C4×C22⋊D5312yes—
367C2×C4#1313319112yes—
368C8—4242no—
369C22—24no—
370C22—24no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—48no—
379C22—48no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—4242no—
385C22—4242no—
386C22—4241no—
387C22—4241no—
388C22—4241no—
389C22#1681050211yescharacteristic
390C22C2×#1313313112yes—
391C22#1681061711yescharacteristic
392C22#1681062711yescharacteristic
393C22C2×#1313314512yes—
394C22#1681076812yes—
395C22#1681094712yes—
396C4—22no—
397C4—24no—
398C4—22no—
399C4—48no—
400C4—4242no—
401C4—4241no—
402C2—22no—
403C2—22no—
404C2—44no—
405C2—8481no—
406C2#2027320412yes—
407C2#2027323012yes—
408C2#2027503711yescharacteristic
409C2#2027515811yescharacteristic
410C2#2027516811yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C4×C22⋊C4×C53C2
socleC23×C53C2×C22⋊C4
prime radical#20282997C2
cosocleC23×C4×C53C23
hypercenterC2×C4×C22⋊C4D53
hypocenterC53#9150
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20274625C2semidirect
3. C2×#16810356C2semidirect
4. #20272594C2semidirect
5. C2×C4×C22⋊C4×C53C2semidirect
6. C2×#13133191C4semidirect
13. #16810483C4semidirect
54. #95×C53D4semidirect
198. C2×C4×C22⋊C4D53semidirect
226. C53#9150semidirect
260. #95C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202829972
#202746252
C2×#168103562
#202725942
C2×C4×C22⋊C4×C532
#915053