Finite Group Databasebeta

← #27280135

Subgroups — #27280135

Order 13568.

684 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 131 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#20280520C211yesmaximal, characteristic
2#20273286C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic
4#20272582C212yesmaximal, complemented
5C53×#2315C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7C2×C22⋊#88322C2212yes—
8#16810559C412yescomplemented
9#16810546—22no—
10#16810535C412yescomplemented
11#16810496—21no—
12#16810483C2211yescharacteristic
13#16810479—21no—
14#16810359—24no—
15#16810347—22no—
16C22×C22⋊C4×C53—21no—
17C22×C22⋊C4×C53C2211yescharacteristic
18C53×#385—24no—
19C53×#380—21no—
20C53×#380C2212yes—
21C22×Dic106C2×C411yescharacteristic, complemented
22C22×#88322—22no—
23#13133196—22no—
24#13133196—22no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C411yescharacteristic, complemented
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C22⋊#88322—22no—
34C22⋊#88322—22no—
35#13133124—22no—
36#13133124C2×C412yescomplemented
37C25×C53D411yescharacteristic
38C23×C4×C53—22no—
39C23×C4×C53D411yescharacteristic
40C23×C4×C53C2311yescharacteristic
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—42no—
48C2×C22⋊C4×C53—42no—
49C2×C22⋊C4×C53C2×C412yes—
50C2×C22⋊C4×C53D412yes—
51#95×C53—22no—
52#95×C53D414yes—
53C24×C53—21no—
54C24×C53—21no—
55C24×C53—21no—
56C24×C53—21no—
57C24×C53—81no—
58C24×C53—81no—
59C24×C53C22⋊C412yes—
60C24×C53SD1612yes—
61C24×C53Q1612yes—
62C24×C53C2×D411yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—24no—
66C22×C4×C53—44no—
67C22×C4×C53—81no—
68C22×C4×C53—81no—
69C22×C4×C53—82no—
70C22×C4×C53—82no—
71C22×C4×C53C22⋊C412yes—
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D412yes—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53#4112yes—
76C22×Dic53—21no—
77C22×Dic53—22no—
78C22×Dic53—21no—
79C2×Dic106—22no—
80C2×Dic106—81no—
81C4⋊C4×C53—82no—
82C4⋊C4×C53—82no—
83C22⋊C4×C53—24no—
84C22⋊C4×C53—24no—
85C22⋊C4×C53—44no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C22⋊Dic53—41no—
89C22⋊Dic53—41no—
90C22⋊Dic53—41no—
91C22⋊Dic53—41no—
92C22⋊Dic53—82no—
93C22⋊Dic53—82no—
94C4⋊Dic53—22no—
95C4⋊Dic53—81no—
96#11719919—42no—
97#11719919—82no—
98C2×#88322—42no—
99C2×#88322—82no—
100C23×C53—22no—
101C23×C53—22no—
102C23×C53—22no—
103C23×C53—22no—
104C23×C53—22no—
105C23×C53—22no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—42no—
109C23×C53—42no—
110C23×C53—42no—
111C23×C53—42no—
112C23×C53—81no—
113C23×C53—81no—
114C23×C53—82no—
115C23×C53—82no—
116C23×C53—82no—
117C23×C53—82no—
118C23×C53#10314yes—
119C23×C53C2×C22⋊C411yescharacteristic
120C23×C53#12011yescharacteristic
121C23×C53#12311yescharacteristic
122C23×C53C2×SD1611yescharacteristic
123C23×C53C2×Q1611yescharacteristic
124C23×C53AGL(1,ℤ/8)11yescharacteristic
125C23×C53#13711yescharacteristic
126C2×C4×C53—22no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—44no—
134C2×C4×C53—44no—
135C2×C4×C53—44no—
136C2×C4×C53—81no—
137C2×C4×C53—88no—
138C2×C4×C53C4×D412yes—
139C2×C4×C53#12112yes—
140C2×Dic53—42no—
141C2×Dic53—42no—
142C2×Dic53—42no—
143C2×Dic53—42no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—42no—
147Dic106—82no—
148#88322—82no—
149#9199—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—21no—
156C22×C53—24no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—44no—
162C22×C53—44no—
163C22×C53—44no—
164C22×C53—42no—
165C22×C53—42no—
166C22×C53—82no—
167C22×C53—82no—
168C22×C53—82no—
169C22×C53#38611yescharacteristic
170C22×C53C2×#10311yescharacteristic
171C22×C53#41711yescharacteristic
172C22×C53#45011yescharacteristic
173C22×C53#45111yescharacteristic
174C22×C53#48311yescharacteristic
175C22×C53#48411yescharacteristic
176C4×C53—22no—
177C4×C53—22no—
178C4×C53—42no—
179C4×C53—84no—
180Dic53—82no—
181Dic53—81no—
182Dic53—81no—
183#2626—531no—
184#2315D5311yescharacteristic
185#1520—532no—
186C2×#323—531no—
187#1289—532no—
188C2×C53—21no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—44no—
195C2×C53#187211yescharacteristic
196C2×C53#191111yescharacteristic
197C2×C53#191311yescharacteristic
198C2×C22⋊Q8—531no—
199C22×C22⋊C4—21no—
200C22×C22⋊C4C2×D5311yescharacteristic
201C2×C22⋊C8—532no—
202#399—532no—
203#396—1062no—
204#393—532no—
205#385—24no—
206#385—1061no—
207#380—21no—
208#380—531no—
209#380C2×D5312yes—
210#379—1061no—
211#328—1064no—
212#323—1062no—
213C53#919911yescharacteristic, complemented, Sylow
214C25C22⋊D5311yescharacteristic
215C22×Q8—531no—
216C23×C4—22no—
217C23×C4C22⋊D5311yescharacteristic
218C23×C4C22×D5311yescharacteristic
219C22×C8—1062no—
220C22⋊Q8—1062no—
221C22⋊Q8—1062no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1062no—
227C2×C4⋊C4—1061no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—1062no—
235C2×C22⋊C4—1062no—
236C2×C22⋊C4—1061no—
237C2×C22⋊C4—1061no—
238C2×C22⋊C4C4×D5312yes—
239C2×C22⋊C4D21212yes—
240C22⋊C8—1062no—
241C22⋊C8—1062no—
242#95—22no—
243#95—532no—
244#95—1062no—
245#95C22⋊D5314yes—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—81no—
251C24—81no—
252C24#1171992112yes—
253C24#1171992312yes—
254C24#1171992512yes—
255C24C2×C22⋊D5311yescharacteristic
256C2×Q8—1062no—
257C2×Q8—4241no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—44no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—1062no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4#1171992112yes—
270C22×C4C2×C4×D5311yescharacteristic
271C22×C4C2×D21211yescharacteristic
272C22×C4#1171994512yes—
273C22×C4C2×C22⋊D5312yes—
274C2×C8—4242no—
275C2×C8—2122no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—1062no—
279C4⋊C4—4242no—
280C4⋊C4—4241no—
281C4⋊C4—2122no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—4242no—
288C22⋊C4—4242no—
289C22⋊C4—2121no—
290C22⋊C4—2121no—
291C22⋊C4—2121no—
292C22⋊C4—2121no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—81no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23#1313314014yes—
312C23C2×#1171992111yescharacteristic
313C23#1313318611yescharacteristic
314C23#1313320011yescharacteristic
315C23C2×#1171992311yescharacteristic
316C23C2×#1171992511yescharacteristic
317C23#1313324311yescharacteristic
318C23#1313324811yescharacteristic
319Q8—4242no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—81no—
331C2×C4—88no—
332C2×C4—2122no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2122no—
339C2×C4C4×C22⋊D5312yes—
340C2×C4#1313319112yes—
341C8—4242no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#1681050211yescharacteristic
362C22C2×#1313314011yescharacteristic
363C22#1681062211yescharacteristic
364C22#1681077411yescharacteristic
365C22#1681077911yescharacteristic
366C22#1681095311yescharacteristic
367C22#1681095811yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—84no—
372C4—4242no—
373C4—4241no—
374C4—4241no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2027502211yescharacteristic
383C2#2027524711yescharacteristic
384C2#2027525711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2315C2
socleC22×C53#386
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenter#2315D53
hypocenterC53#9199
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
2. #20273286C2semidirect
4. #20272582C2semidirect
8. #16810559C4semidirect
10. #16810535C4semidirect
21. C22×Dic106C2×C4semidirect
26. C2×C4⋊Dic53C2×C4semidirect
36. #13133124C2×C4semidirect
213. C53#9199semidirect

Maximal subgroups

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

subgroupindex
#202805202
#202732862
C2×#168103472
#202725822
C53×#23152
#919953