Finite Group Databasebeta

← #27279847

Subgroups — #27279847

Order 13568.

946 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283002C211yesmaximal, characteristic
2#20274625C211yesmaximal, characteristic
3C2×#16810359C212yesmaximal, complemented
4#20272594C212yesmaximal, complemented
5C2×C53×#380C211yesmaximal, characteristic
6C2×#13133196C412yescomplemented
7C2×#13133196C2211yescharacteristic
8C2×C22⋊#88322—22no—
9C2×C22⋊#88322C2212yes—
10#16810496—21no—
11#16810496—22no—
12#16810483C412yescomplemented
13#16810483C2211yescharacteristic
14#16810479—21no—
15#16810359—24no—
16C24×C4×C53C2211yescharacteristic
17C22×C4⋊C4×C53—21no—
18C22×C22⋊C4×C53—21no—
19C53×#380—24no—
20C2×#95×C53C2212yes—
21C22×Dic106C2×C411yescharacteristic
22C22×#88322—22no—
23#13133196—22no—
24#13133196—22no—
25#13133196—44no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×C4⋊Dic53C2×C412yes—
28C2×#11719919—22no—
29C2×#11719919—21no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C22⋊#88322—22no—
35C22⋊#88322—22no—
36C22⋊#88322—44no—
37#13133124—22no—
38#13133124—21no—
39C25×C53D411yescharacteristic
40C23×C4×C53—22no—
41C23×C4×C53—22no—
42C23×C4×C53—22no—
43C23×C4×C53—44no—
44C23×C4×C53C2×C412yes—
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D412yes—
47C23×C4×C53C2311yescharacteristic
48C2×C4⋊C4×C53—24no—
49C2×C4⋊C4×C53—44no—
50C2×C22⋊C4×C53—24no—
51C2×C22⋊C4×C53—42no—
52C2×C22⋊C4×C53—42no—
53#95×C53—22no—
54#95×C53D414yes—
55C24×C53—21no—
56C24×C53—21no—
57C24×C53—21no—
58C24×C53—21no—
59C24×C53—42no—
60C24×C53—42no—
61C24×C53C22⋊C412yes—
62C24×C53SD1612yes—
63C24×C53Q1612yes—
64C24×C53C2×D411yescharacteristic
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—24no—
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—44no—
80C22×C4×C53—42no—
81C22×C4×C53—42no—
82C22×C4×C53C22⋊C412yes—
83C22×C4×C53C22×C411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D412yes—
86C22×C4×C53#4112yes—
87C22×Dic53—21no—
88C22×Dic53—22no—
89C22×Dic53—21no—
90C2×Dic106—22no—
91C2×Dic106—42no—
92C4⋊C4×C53—48no—
93C22⋊C4×C53—48no—
94C22⋊Dic53—44no—
95C22⋊Dic53—44no—
96C22⋊Dic53—41no—
97C22⋊Dic53—41no—
98C22⋊Dic53—41no—
99C22⋊Dic53—41no—
100C4⋊Dic53—22no—
101C4⋊Dic53—42no—
102#11719919—44no—
103#11719919—42no—
104C2×#88322—44no—
105C2×#88322—42no—
106C23×C53—22no—
107C23×C53—22no—
108C23×C53—22no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53—22no—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—22no—
117C23×C53—22no—
118C23×C53—44no—
119C23×C53—44no—
120C23×C53—42no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—44no—
125C23×C53—44no—
126C23×C53#10314yes—
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53#12011yescharacteristic
129C23×C53#12311yescharacteristic
130C23×C53C2×SD1611yescharacteristic
131C23×C53C2×Q1611yescharacteristic
132C23×C53AGL(1,ℤ/8)11yescharacteristic
133C23×C53#13711yescharacteristic
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—42no—
141C2×C4×C53—42no—
142C2×C4×C53—48no—
143C2×C4×C53—48no—
144C2×C4×C53—44no—
145C2×C4×C53—44no—
146C2×C4×C53—44no—
147C2×C4×C53—84no—
148C2×C4×C53—84no—
149C2×C4×C53#9914yes—
150C2×C4×C53C4×D412yes—
151C2×C4×C53#12112yes—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159Dic106—82no—
160#88322—82no—
161#9151—531nomaximal, Sylow
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—22no—
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—24no—
171C22×C53—24no—
172C22×C53—24no—
173C22×C53—44no—
174C22×C53—44no—
175C22×C53—44no—
176C22×C53—44no—
177C22×C53—42no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53#38611yescharacteristic
182C22×C53C2×#9912yes—
183C22×C53C2×#10311yescharacteristic
184C22×C53#41711yescharacteristic
185C22×C53#42112yes—
186C22×C53#45011yescharacteristic
187C22×C53#45111yescharacteristic
188C22×C53#48311yescharacteristic
189C22×C53#48411yescharacteristic
190C4×C53—22no—
191C4×C53—22no—
192C4×C53—24no—
193C4×C53—84no—
194Dic53—81no—
195Dic53—81no—
196Dic53—82no—
197#2848—531no—
198C2×#380D5311yescharacteristic
199#1779—531no—
200C2×#328—532no—
201#1294—532no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53—44no—
207C2×C53#151312yes—
208C2×C53#152012yes—
209C2×C53#187211yescharacteristic
210C2×C53#189711yescharacteristic
211C2×C53#189811yescharacteristic
212C24×C4C2×D5311yescharacteristic
213C2×C22⋊Q8—531no—
214C2×C22⋊Q8—532no—
215C22×C4⋊C4—21no—
216C22×C22⋊C4—21no—
217C2×C22⋊C8—532no—
218C2×C22⋊C8—1062no—
219#385—1062no—
220#385—1061no—
221#380—24no—
222#380—532no—
223#380—531no—
224#379—1061no—
225C2×#95C2×D5312yes—
226#328—1064no—
227C53#915111yescharacteristic, complemented, Sylow
228C25C22⋊D5311yescharacteristic
229C22×Q8—531no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—44no—
234C23×C4C4×D5312yes—
235C23×C4D21212yes—
236C23×C4C22⋊D5311yescharacteristic
237C23×C4C22×D5311yescharacteristic
238C22×C8—1062no—
239C22⋊Q8—1062no—
240C22⋊Q8—1062no—
241C22⋊Q8—2124no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—531no—
245C2×C4⋊C4—532no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4—1062no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—1061no—
252C2×C22⋊C4—1061no—
253C2×C22⋊C4—1062no—
254C2×C22⋊C4—1062no—
255C22⋊C8—1062no—
256C22⋊C8—1062no—
257C22⋊C8—2124no—
258#95—22no—
259#95—1061no—
260#95—1062no—
261#95C22⋊D5314yes—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—42no—
267C24—42no—
268C24#1171992112yes—
269C24#1171992312yes—
270C24#1171992512yes—
271C24C2×C22⋊D5311yescharacteristic
272C2×Q8—1062no—
273C2×Q8—2122no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—1061no—
292C22×C4—1062no—
293C22×C4—1061no—
294C22×C4#1171992112yes—
295C22×C4C2×C4×D5311yescharacteristic
296C22×C4C2×D21211yescharacteristic
297C22×C4#1171994512yes—
298C22×C4C2×C22⋊D5312yes—
299C2×C8—2124no—
300C2×C8—2122no—
301C4⋊C4—48no—
302C4⋊C4—1062no—
303C4⋊C4—2122no—
304C4⋊C4—2124no—
305C4⋊C4—2122no—
306C22⋊C4—48no—
307C22⋊C4—2124no—
308C22⋊C4—2124no—
309C22⋊C4—2121no—
310C22⋊C4—2121no—
311C22⋊C4—2121no—
312C22⋊C4—2121no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—44no—
326C23—44no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23#1313314014yes—
334C23C2×#1171992111yescharacteristic
335C23#1313318611yescharacteristic
336C23#1313320011yescharacteristic
337C23C2×#1171992311yescharacteristic
338C23C2×#1171992511yescharacteristic
339C23#1313324311yescharacteristic
340C23#1313324811yescharacteristic
341Q8—4242no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4#1313313114yes—
365C2×C4C4×C22⋊D5312yes—
366C2×C4#1313319112yes—
367C8—4242no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—44no—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—42no—
387C22#1681050211yescharacteristic
388C22C2×#1313313112yes—
389C22C2×#1313314011yescharacteristic
390C22#1681062211yescharacteristic
391C22#1681064112yes—
392C22#1681077411yescharacteristic
393C22#1681077911yescharacteristic
394C22#1681095311yescharacteristic
395C22#1681095811yescharacteristic
396C4—22no—
397C4—24no—
398C4—22no—
399C4—84no—
400C4—4242no—
401C4—4241no—
402C4—4241no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—44no—
408C2#2027325612yes—
409C2#2027328612yes—
410C2#2027502211yescharacteristic
411C2#2027516311yescharacteristic
412C2#2027516811yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#380C2
socleC23×C53C2×C22⋊C4
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenterC2×#380D53
hypocenterC53#9151
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
3. C2×#16810359C2semidirect
4. #20272594C2semidirect
6. C2×#13133196C4semidirect
12. #16810483C4semidirect
227. C53#9151semidirect

Maximal subgroups

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

subgroupindex
#202830022
#202746252
C2×#168103592
#202725942
C2×C53×#3802
#915153