Finite Group Databasebeta

← #27350186

Subgroups — #27350186

Order 13568.

1034 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133210C211yesmaximal, characteristic
2#20283344C212yesmaximal, complemented
3#20273081C218yesmaximal, complemented
4#20273044C212yesmaximal, complemented
5#20273042C211yesmaximal, characteristic
6C53×#1462C211yesmaximal, characteristic
7C2×#13133210—22no—
8C2×#13133210—22no—
9C2×#13133210—22no—
10C2×#13133210—22no—
11C2×#13133210C414yes—
12C2×#13133210C2214yes—
13C22×C4⋊Dic53C411yescharacteristic
14C22×#11719919C411yescharacteristic
15C22×#11719919C2211yescharacteristic
16C22×C22⋊Dic53C411yescharacteristic
17C22×C22⋊Dic53C411yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19#16810677—22no—
20#16810677C2214yescomplemented
21#16810673—22no—
22#16810673C2214yescomplemented
23#16810646—22no—
24#16810587C2212yes—
25M16×Dic53—22no—
26#16810348C2218yescomplemented
27#16810347C2214yes—
28C22×C22⋊C4×C53C2211yescharacteristic
29C53×#407C2212yes—
30C53×#323C2214yes—
31C23×Dic53C2×C411yescharacteristic
32C23×Dic53C2×C411yescharacteristic
33C23×Dic53C2×C411yescharacteristic
34C23×Dic53C2×C411yescharacteristic
35C2×#11719917—22no—
36#13133210—24no—
37#13133210—24no—
38#13133210—24no—
39#13133210—24no—
40#13133210—24no—
41#13133210—24no—
42#13133210—24no—
43#13133210—24no—
44C2×C4⋊Dic53—22no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53C2×C414yes—
48C2×#11719919—22no—
49C2×#11719919—22no—
50C2×#11719919—22no—
51C2×#11719919—21no—
52C2×#11719919—21no—
53C2×#11719919C2×C414yes—
54C2×#11719919C2×C412yes—
55C2×#11719919C2312yes—
56C2×C22⋊Dic53—21no—
57C2×C22⋊Dic53—21no—
58C2×C22⋊Dic53—22no—
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—21no—
61C2×C22⋊Dic53—21no—
62C2×C22⋊Dic53C2×C414yes—
63C2×C22⋊Dic53C2×C414yes—
64C2×C4×Dic53C2×C412yes—
65C2×C4×Dic53D412yes—
66C2×C4×Dic53D412yes—
67C2×C4×Dic53D412yes—
68C2×C4×Dic53D412yes—
69C2×C4×Dic53C2312yes—
70#13133146—24no—
71#13133146—24no—
72C22⋊#88322—22no—
73C22⋊#88322C2314yes—
74#13133127—24no—
75C8×Dic53—24no—
76C25×C53C2×C411yescharacteristic
77C23×C4×C53C2×C411yescharacteristic
78C23×C4×C53C2311yescharacteristic
79C2×M16×C53—22no—
80C2×C22⋊C4×C53—22no—
81C2×C22⋊C4×C53—22no—
82C2×C22⋊C4×C53C2×C412yes—
83C2×C22⋊C4×C53C2312yes—
84C22⋊C8×C53—22no—
85C22⋊C8×C53C2314yes—
86C24×C53—21no—
87C24×C53—21no—
88C24×C53—42no—
89C24×C53—42no—
90C24×C53—42no—
91C24×C53C22×C411yescharacteristic
92C24×C53C22×C412yes—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—42no—
96C22×C4×C53—42no—
97C22×C4×C53C22×C412yes—
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2×D412yes—
101C22×C4×C53C2411yescharacteristic
102C22×Dic53—21no—
103C22×Dic53—21no—
104C22×Dic53—21no—
105C22×Dic53—21no—
106C22×Dic53—42no—
107C22×Dic53—42no—
108C22×Dic53—42no—
109C22×Dic53—42no—
110C22×Dic53C22⋊C412yes—
111C22×Dic53C22⋊C411yescharacteristic
112C22×Dic53C22⋊C411yescharacteristic
113C22×Dic53C22⋊C412yes—
114C22×Dic53C22⋊C411yescharacteristic
115C22×Dic53C22⋊C411yescharacteristic
116C22×Dic53C22×C412yes—
117C22×Dic53C22×C412yes—
118C22×Dic53C22×C412yes—
119C22×Dic53C22×C412yes—
120M16×C53—44no—
121C2×C8×C53—24no—
122C22⋊C4×C53—24no—
123C22⋊C4×C53—24no—
124C22⋊Dic53—24no—
125C22⋊Dic53—24no—
126C22⋊Dic53—24no—
127C22⋊Dic53—24no—
128C4⋊Dic53—24no—
129C4⋊Dic53—24no—
130#11719919—24no—
131#11719919—24no—
132#11719919—24no—
133#11719919—24no—
134C4×Dic53C22⋊C412yes—
135C4×Dic53C22⋊C412yes—
136C4×Dic53C22⋊C412yes—
137C4×Dic53C22⋊C412yes—
138C4×Dic53C2×D412yes—
139C4×Dic53C2×D412yes—
140C4×Dic53C2×D412yes—
141C4×Dic53C2×D412yes—
142#11719917—44no—
143C2×#88322—24no—
144C23×C53—22no—
145C23×C53—22no—
146C23×C53—22no—
147C23×C53—22no—
148C23×C53—24no—
149C23×C53—42no—
150C23×C53—42no—
151C23×C53—42no—
152C23×C53—42no—
153C23×C53—42no—
154C23×C53—42no—
155C23×C53—42no—
156C23×C53—42no—
157C23×C53—88no—
158C23×C53C2×C22⋊C411yescharacteristic
159C23×C53C2×C22⋊C411yescharacteristic
160C23×C53C23×C411yescharacteristic
161C2×C4×C53—24no—
162C2×C4×C53—22no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53C2×C22⋊C412yes—
170C2×C4×C53C22×D412yes—
171C2×Dic53—24no—
172C2×Dic53—24no—
173C2×Dic53—22no—
174C2×Dic53—22no—
175C2×Dic53—24no—
176C2×Dic53—42no—
177C2×Dic53—41no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—42no—
182C2×Dic53—42no—
183C2×Dic53—42no—
184C2×Dic53—42no—
185C2×Dic53—42no—
186C2×Dic53—42no—
187C2×Dic53C2×C22⋊C412yes—
188C2×Dic53C2×C22⋊C412yes—
189C2×Dic53C2×C22⋊C412yes—
190C2×Dic53C2×C22⋊C412yes—
191C8×C53—44no—
192#88322—44no—
193#17398—531nomaximal, Sylow
194C22×C53—28no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—42no—
202C22×C53—42no—
203C22×C53—42no—
204C22×C53—84no—
205C22×C53—84no—
206C22×C53—84no—
207C22×C53#41012yes—
208C22×C53#41312yes—
209C22×C53C22×C22⋊C411yescharacteristic
210C22×C53#56812yes—
211C4×C53—22no—
212C4×C53—22no—
213C4×C53—42no—
214Dic53—22no—
215Dic53—22no—
216Dic53—22no—
217Dic53—22no—
218Dic53—42no—
219Dic53—42no—
220C22×#126—531no—
221#2920—532no—
222#1473—538no—
223#1463—532no—
224#1462—531no—
225#1462D5311yescharacteristic
226C2×C53—22no—
227C2×C53—22no—
228C2×C53—22no—
229C2×C53—82no—
230C2×C53#287911yescharacteristic
231C2×C53C2×#41011yescharacteristic
232C2×C53C2×#41311yescharacteristic
233C2×#126—534no—
234C2×#126—534no—
235C2×#126—1062no—
236C2×#126—1062no—
237C2×#126—1062no—
238C2×#126—1062no—
239C22×C4⋊C4—531no—
240C22×C4⋊C4—531no—
241C22×C4⋊C4—531no—
242C22×C22⋊C4—531no—
243C22×C22⋊C4—531no—
244C22×C22⋊C4C2×D5311yescharacteristic
245C22×C42—531no—
246#432—534no—
247#432—1062no—
248#431—534no—
249#431—1062no—
250#423—1062no—
251#407—532no—
252#407C2×D5312yes—
253C4×M16—1062no—
254#324—538no—
255#323—534no—
256#323C2×D5314yes—
257C53#1739811yescharacteristic, complemented, Sylow
258C25C4×D5311yescharacteristic
259C23×C4—531no—
260C23×C4—531no—
261C23×C4—531no—
262C23×C4—531no—
263C23×C4C4×D5311yescharacteristic
264C23×C4C22×D5311yescharacteristic
265C2×M16—22no—
266C2×M16—1062no—
267#126—1064no—
268#126—1064no—
269#126—1064no—
270#126—1064no—
271#126—1064no—
272#126—1064no—
273#126—1064no—
274#126—1064no—
275C2×C4⋊C4—534no—
276C2×C4⋊C4—532no—
277C2×C4⋊C4—534no—
278C2×C4⋊C4—532no—
279C2×C4⋊C4—1062no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—1061no—
282C2×C4⋊C4—1062no—
283C2×C4⋊C4—1061no—
284C2×C4⋊C4—1061no—
285C2×C4⋊C4—1062no—
286C2×C4⋊C4—1062no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—534no—
290C2×C22⋊C4—534no—
291C2×C22⋊C4—1062no—
292C2×C22⋊C4—1061no—
293C2×C22⋊C4—1061no—
294C2×C22⋊C4—1062no—
295C2×C22⋊C4—1061no—
296C2×C22⋊C4—1061no—
297C2×C22⋊C4C4×D5312yes—
298C2×C22⋊C4C22×D5312yes—
299C2×C42—532no—
300C2×C42—532no—
301C2×C42—532no—
302C2×C42—532no—
303C2×C42—532no—
304C2×C42—532no—
305C4⋊C8—1064no—
306C4⋊C8—1064no—
307C22⋊C8—22no—
308C22⋊C8—534no—
309C22⋊C8—1062no—
310C22⋊C8C22×D5314yes—
311#97—1064no—
312C4×C8—1064no—
313C24—21no—
314C24—21no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24C2×C4×D5311yescharacteristic
319C24C2×C4×D5312yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—532no—
325C22×C4—532no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—532no—
329C22×C4—531no—
330C22×C4—531no—
331C22×C4—532no—
332C22×C4—532no—
333C22×C4—532no—
334C22×C4—1061no—
335C22×C4—1061no—
336C22×C4—1061no—
337C22×C4—1061no—
338C22×C4—2122no—
339C22×C4—2122no—
340C22×C4—2122no—
341C22×C4—2122no—
342C22×C4C2×C4×D5312yes—
343C22×C4C2×C4×D5311yescharacteristic
344C22×C4D4×D5312yes—
345C22×C4D4×D5312yes—
346C22×C4C23×D5311yescharacteristic
347M16—44no—
348M16—2124no—
349C2×C8—24no—
350C2×C8—1064no—
351C4⋊C4—1064no—
352C4⋊C4—1064no—
353C4⋊C4—1064no—
354C4⋊C4—1064no—
355C4⋊C4—1064no—
356C4⋊C4—1064no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—1064no—
360C22⋊C4—1064no—
361C22⋊C4—1064no—
362C22⋊C4—1064no—
363C42—532no—
364C42—532no—
365C42—532no—
366C42—532no—
367C42—532no—
368C42—532no—
369C42—532no—
370C42—532no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—88no—
385C23C22⋊C4×D5311yescharacteristic
386C23C22⋊C4×D5311yescharacteristic
387C23C22×C4×D5311yescharacteristic
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—532no—
397C2×C4—532no—
398C2×C4—532no—
399C2×C4—532no—
400C2×C4—1064no—
401C2×C4—1064no—
402C2×C4—1062no—
403C2×C4—1062no—
404C2×C4—1064no—
405C2×C4—2122no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4—2121no—
410C2×C4—2122no—
411C2×C4—2122no—
412C2×C4—2122no—
413C2×C4—2122no—
414C2×C4—2122no—
415C2×C4—2122no—
416C2×C4C22⋊C4×D5312yes—
417C2×C4C2×D4×D5312yes—
418C8—44no—
419C8—2124no—
420C22—28no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—84no—
431C22—84no—
432C22—84no—
433C22#1681059512yes—
434C22#1681060412yes—
435C22C2×C22⋊C4×D5311yescharacteristic
436C22#1681138912yes—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—1062no—
441C4—1062no—
442C4—1062no—
443C4—1062no—
444C4—2122no—
445C4—2122no—
446C2—22no—
447C2—22no—
448C2—22no—
449C2—82no—
450C2#2028315811yescharacteristic
451C2C2×#1681059511yescharacteristic
452C2C2×#1681060411yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1462C2
socleC22×C53C22×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC22×C4×C53C24
hypercenter#1462D53
hypocenterC53#17398
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #20283344C2semidirect
3. #20273081C2semidirect
4. #20273044C2semidirect
20. #16810677C22semidirect
22. #16810673C22semidirect
26. #16810348C22semidirect
257. C53#17398semidirect

Maximal subgroups

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

subgroupindex
C22×#131332102
#202833442
#202730812
#202730442
#202730422
C53×#14622
#1739853