Finite Group Databasebeta

← #27285155

Subgroups — #27285155

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810999C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic, complemented
3#20275642C211yesmaximal, characteristic, complemented
4C2×#16810367C211yesmaximal, characteristic, complemented
5C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2548C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C22×#11719922—22no—
10C2×#13133214C2211yescharacteristic
11#16810999—22no—
12#16810999—22no—
13#16810823—22no—
14#16810823—22no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133137—21no—
18C2×#13133137—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810367—22no—
21Q8⋊#88322—22no—
22C2×#127×C53C2211yescharacteristic
23C2×#124×C53—21no—
24C2×C4×Q8×C53C2211yescharacteristic
25C53×#394—22no—
26C53×#390—22no—
27C53×#390—21no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719924—24no—
33C2×#11719924—42no—
34C2×#11719924—42no—
35C2×#11719922—28no—
36C2×#11719922—48no—
37C22×#88322—21no—
38C22×#88322—21no—
39C22×#88322—21no—
40#13133214—21no—
41#13133214—21no—
42#13133214—21no—
43#13133214D412yes—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322D412yes—
48#13133137—24no—
49#13133137—44no—
50C4×#88322—21no—
51C4×#88322D412yes—
52C22×Q8×C53D411yescharacteristic
53C22×D4×C53—21no—
54C22×D4×C53—21no—
55C22×D4×C53D412yes—
56#127×C53—21no—
57#127×C53—21no—
58#127×C53—21no—
59#127×C53—21no—
60#124×C53—42no—
61#124×C53—42no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C4×Q8×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C4⋊C4×C53D412yes—
69C2×C22⋊C4×C53—24no—
70C2×C22⋊C4×C53—24no—
71C2×C42×C53—21no—
72C2×C42×C53—21no—
73C2×C42×C53C2311yescharacteristic
74#95×C53—21no—
75C24×C53—22no—
76C23×D53—41no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53D812yescomplemented
83C2×D4×C53—22no—
84C2×D4×C53—22no—
85C2×D4×C53—44no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C2×D4×C53—42no—
93C2×D4×C53—41no—
94C2×D4×C53—41no—
95C2×D4×C53—42no—
96C22×C4×C53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—21no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C2×D212—22no—
103C2×D212—22no—
104C2×D212—22no—
105C2×D212—44no—
106C2×D212—41no—
107C2×D212—41no—
108C2×D212—42no—
109C2×D212—42no—
110C2×D212—41no—
111C2×D212—41no—
112C2×D212—41no—
113C2×D212—41no—
114C2×D212—41no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—41no—
120C4⋊C4×C53—41no—
121C22⋊C4×C53—48no—
122C22⋊C4×C53—84no—
123C42×C53—41no—
124C42×C53—41no—
125C42×C53—41no—
126C42×C53—41no—
127C42×C53C2×D411yescharacteristic
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131#11719924—48no—
132#11719922—416no—
133C2×#88322—24no—
134C2×#88322—22no—
135C2×#88322—22no—
136C2×#88322—41no—
137C2×#88322—41no—
138C2×#88322—41no—
139C2×#88322—42no—
140C23×C53—42no—
141C23×C53—42no—
142C23×C53—42no—
143C23×C53—44no—
144C23×C53—44no—
145C23×C53#12011yescharacteristic
146C22×D53—42no—
147C22×D53—81no—
148C22×D53—81no—
149C22×D53—82no—
150C22×D53—82no—
151Q8×C53—22no—
152Q8×C53—22no—
153Q8×C53—41no—
154Q8×C53—41no—
155D4×C53—44no—
156D4×C53—82no—
157D4×C53—81no—
158D4×C53—81no—
159D4×C53—82no—
160D4×C53—81no—
161D4×C53—81no—
162D4×C53—81no—
163D4×C53—81no—
164C2×C4×C53—22no—
165C2×C4×C53—22no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53#12011yescharacteristic
174C2×C4×C53#12011yescharacteristic
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12711yescharacteristic
179C2×C4×C53C2×D811yescharacteristic
180C2×C4×C53#13511yescharacteristic
181C2×C4×C53#13512yes—
182C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C53AGL(1,ℤ/8)12yes—
184C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
185D212—44no—
186D212—42no—
187D212—42no—
188D212—44no—
189D212—82no—
190D212—81no—
191D212—81no—
192D212—81no—
193D212—81no—
194#88322—24no—
195#88322—42no—
196#88322—81no—
197#10047—531nomaximal, Sylow
198C22×C53—84no—
199C22×C53—84no—
200C22×C53—82no—
201C22×C53—82no—
202C22×C53—82no—
203C22×C53#39211yescharacteristic
204C22×C53#44711yescharacteristic
205C22×C53#44911yescharacteristic
206C22×C53#44912yes—
207C22×C53#45312yes—
208C2×D53—82no—
209C2×D53—82no—
210C2×D53—82no—
211C2×D53—162no—
212C2×D53—162no—
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—42no—
216C4×C53—82no—
217C4×C53—81no—
218C4×C53#44711yescharacteristic
219C4×C53#44911yescharacteristic
220C4×C53#46312yes—
221C4×C53#46611yescharacteristic
222C4×C53#46911yescharacteristic
223C4×C53#49511yescharacteristic
224C4×C53#49611yescharacteristic
225C2×#493—531no—
226C2×#460—531no—
227#2548D5311yescharacteristic, complemented
228#1982—531no—
229C2×#331—531no—
230C2×Q8⋊C8—531no—
231#1279—531no—
232C2×C53—162no—
233C2×C53#162012yes—
234C2×C53#162112yes—
235C2×C53#199911yescharacteristic
236C2×C53#200211yescharacteristic
237C2×C53#201211yescharacteristic
238D53—162no—
239C22×SD16—1061no—
240C22×D8—1062no—
241C2×#127—531no—
242C2×#127C2×D5311yescharacteristic
243C2×#124—21no—
244C2×C4×Q8C2×D5311yescharacteristic
245#493—1062no—
246#493—1062no—
247#460—1062no—
248#460—1062no—
249C2×C4⋊C8—531no—
250C2×C4⋊C8—531no—
251C2×#102—1061no—
252C2×#102—1062no—
253C2×C4×C8—531no—
254#394—22no—
255#390—22no—
256#390—21no—
257C4×C4⋊C4C2×D5311yescharacteristic
258#331—1062no—
259Q8⋊C8—1062no—
260C53#1004711yescharacteristic, complemented, Sylow
261C22×Q8C22⋊D5311yescharacteristic
262C22×D4—21no—
263C22×D4—21no—
264C22×D4—1061no—
265C22×D4—1061no—
266C22×D4—1061no—
267C22×D4C22⋊D5312yes—
268C2×SD16—1064no—
269C2×SD16—2122no—
270C2×SD16—2122no—
271C2×D8—1068no—
272C2×D8—2128no—
273C22×C8—1061no—
274C22×C8—1061no—
275C22×C8—1061no—
276#127—21no—
277#127—21no—
278#127—21no—
279#127—21no—
280#127—532no—
281#127—1061no—
282#127—1061no—
283#127—1061no—
284#124—42no—
285#124—42no—
286C4×Q8—21no—
287C4×Q8—21no—
288C4×Q8—21no—
289C4×Q8—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C22⋊D5311yescharacteristic
292C2×C4⋊C4C22⋊D5312yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42C22×D5311yescharacteristic
298C4⋊C8—532no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301C4⋊C8—1061no—
302#102—1064no—
303#102—2124no—
304C4×C8—532no—
305C4×C8—1061no—
306#95—21no—
307C24—22no—
308C24—2121no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8#1171992212yescomplemented
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—44no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—41no—
326C2×D4—42no—
327C2×D4—41no—
328C2×D4—1062no—
329C2×D4—1062no—
330C2×D4—1062no—
331C2×D4—2124no—
332C2×D4—2121no—
333C2×D4—2121no—
334C2×D4—2121no—
335C2×D4—2121no—
336C2×D4—2121no—
337C2×D4—2122no—
338C2×D4—2122no—
339C2×D4—2121no—
340C2×D4—2121no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4C2×C22⋊D5311yescharacteristic
345C22×C4C2×C22⋊D5311yescharacteristic
346C22×C4C2×C22⋊D5311yescharacteristic
347SD16—2128no—
348D8—21216no—
349C2×C8—1064no—
350C2×C8—1062no—
351C2×C8—1062no—
352C2×C8—2121no—
353C2×C8—2122no—
354C2×C8—2121no—
355C2×C8—2121no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C22⋊C4—48no—
363C22⋊C4—84no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42D4×D5311yescharacteristic
369C42D4×D5311yescharacteristic
370C42D4×D5311yescharacteristic
371C42#1171994711yescharacteristic
372C23—42no—
373C23—44no—
374C23—42no—
375C23—42no—
376C23—44no—
377C23—4241no—
378C23—4241no—
379C23—4242no—
380C23—4242no—
381C23—2122no—
382C23#1313318611yescharacteristic
383Q8—22no—
384Q8—22no—
385Q8—41no—
386Q8—41no—
387D4—44no—
388D4—82no—
389D4—81no—
390D4—81no—
391D4—81no—
392D4—82no—
393D4—81no—
394D4—81no—
395D4—81no—
396D4—4242no—
397D4—4241no—
398D4—4241no—
399D4—4241no—
400D4—4241no—
401D4—2124no—
402D4—2122no—
403D4—2122no—
404D4—2124no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4#1313318511yescharacteristic
415C2×C4#1313318511yescharacteristic
416C2×C4#1313318811yescharacteristic
417C2×C4#1313318911yescharacteristic
418C2×C4#1313318911yescharacteristic
419C2×C4#1313321311yescharacteristic
420C2×C4C2×#1171992211yescharacteristic
421C2×C4#1313323711yescharacteristic
422C2×C4#1313323712yes—
423C2×C4#1313324211yescharacteristic
424C2×C4#1313324212yes—
425C2×C4#1313324311yescharacteristic
426C8—1064no—
427C8—4241no—
428C8—2122no—
429C22—82no—
430C22—82no—
431C22—84no—
432C22—84no—
433C22—82no—
434C22—8482no—
435C22—8482no—
436C22—4242no—
437C22—4242no—
438C22—4242no—
439C22#1681053111yescharacteristic
440C22#1681075711yescharacteristic
441C22#1681076812yes—
442C22#1681076811yescharacteristic
443C22#1681079112yes—
444C4—21no—
445C4—21no—
446C4—42no—
447C4—81no—
448C4—82no—
449C4#1681075611yescharacteristic
450C4#1681076611yescharacteristic
451C4#1681083512yes—
452C4#1681085711yescharacteristic
453C4#1681087411yescharacteristic
454C4#1681100511yescharacteristic
455C4#1681101311yescharacteristic
456C2—162no—
457C2—8482no—
458C2#2027377412yes—
459C2#2027378012yes—
460C2#2027571111yescharacteristic
461C2#2027572811yescharacteristic
462C2#2027579211yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2548C2
socleC23×C53#120
prime radicalC2×#16810999C2
hypercenter#2548D53
hypocenterC53#10047
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
2. C2×#16810823C2semidirect
3. #20275642C2semidirect
4. C2×#16810367C2semidirect
5. C2×Q8⋊#88322C2semidirect
6. #20272556C2semidirect
7. C53×#2548C2semidirect
82. C2×Q8×C53D8semidirect
227. #2548D53semidirect
260. C53#10047semidirect
314. C2×Q8#11719922semidirect

Maximal subgroups

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

subgroupindex
C2×#168109992
C2×#168108232
#202756422
C2×#168103672
C2×Q8⋊#883222
#202725562
C53×#25482
#1004753