Finite Group Databasebeta

← #27285260

Subgroups — #27285260

Order 13568.

806 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810969C211yesmaximal, characteristic, complemented
2C2×#16810828C211yesmaximal, characteristic, complemented
3#20275667C211yesmaximal, characteristic, complemented
4C2×#16810367C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2541C211yesmaximal, characteristic
8C22×#11719923—21no—
9C2×#13133217C2211yescharacteristic
10#16810969—22no—
11#16810969—22no—
12#16810828—22no—
13#16810828—22no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133140—21no—
17C2×#13133138—22no—
18C2×C4×#88322C2211yescharacteristic
19#16810385—22no—
20#16810367—22no—
21D4⋊#88322—22no—
22C2×#127×C53—21no—
23C2×#127×C53C2211yescharacteristic
24C2×#121×C53—22no—
25C2×C4×D4×C53C2211yescharacteristic
26C53×#392—22no—
27C53×#390—21no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×Dic106—21no—
30C2×#11719923—24no—
31C2×#11719923—42no—
32C2×#11719923—42no—
33C22×#88322—21no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133217—21no—
37#13133217—21no—
38#13133217—21no—
39#13133217D412yescomplemented
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D412yescomplemented
46#13133140—24no—
47#13133138—28no—
48C4×#88322—21no—
49C4×#88322D412yescomplemented
50C22×D4×C53—24no—
51C22×D4×C53—21no—
52C22×D4×C53—21no—
53C22×D4×C53D411yescharacteristic
54C22×D4×C53D412yes—
55C23×C4×C53—21no—
56#127×C53—21no—
57#127×C53—21no—
58#127×C53—21no—
59#127×C53—21no—
60#127×C53—42no—
61#127×C53—42no—
62#121×C53—44no—
63#121×C53—44no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C2×C4⋊C4×C53D411yescharacteristic
69C2×C4⋊C4×C53D412yes—
70C2×C22⋊C4×C53—24no—
71C2×C22⋊C4×C53—21no—
72C2×C42×C53—21no—
73C2×C42×C53C2311yescharacteristic
74#95×C53—21no—
75C24×C53—21no—
76C24×C53—22no—
77C2×D4×C53—21no—
78C2×D4×C53—22no—
79C2×D4×C53—21no—
80C2×D4×C53—22no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—44no—
85C2×D4×C53—48no—
86C2×D4×C53—48no—
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—42no—
95C2×D4×C53—41no—
96C2×D4×C53—41no—
97C2×D4×C53—84no—
98C2×D4×C53D812yes—
99C22×C4×C53—22no—
100C22×C4×C53—21no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—42no—
104C22×C4×C53—42no—
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×Dic53—41no—
109C2×Dic106—22no—
110C2×Dic106—41no—
111C2×Dic106—41no—
112C2×Dic106—41no—
113C2×Dic106—41no—
114C2×Dic106—41no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53SD1614yescomplemented
119C22⋊C4×C53—48no—
120C22⋊C4×C53—41no—
121C22⋊C4×C53—41no—
122C42×C53—41no—
123C42×C53—41no—
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128#11719923—48no—
129C4⋊Dic53—22no—
130C4⋊Dic53—22no—
131C4⋊Dic53—41no—
132C4⋊Dic53—41no—
133C2×#88322—22no—
134C2×#88322—22no—
135C2×#88322—42no—
136C2×#88322—41no—
137C2×#88322—41no—
138C2×#88322—41no—
139C2×#88322—42no—
140C23×C53—22no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—42no—
144C23×C53—42no—
145C23×C53—42no—
146C23×C53—42no—
147C23×C53—42no—
148C23×C53—44no—
149C23×C53—44no—
150C23×C53#12011yescharacteristic
151D4×C53—22no—
152D4×C53—22no—
153D4×C53—44no—
154D4×C53—41no—
155D4×C53—41no—
156D4×C53—82no—
157D4×C53—816no—
158D4×C53—81no—
159D4×C53—81no—
160D4×C53—82no—
161D4×C53—81no—
162D4×C53—81no—
163D4×C53—81no—
164D4×C53—81no—
165C2×C4×C53—22no—
166C2×C4×C53—24no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53—41no—
170C2×C4×C53—42no—
171C2×C4×C53—82no—
172C2×C4×C53—81no—
173C2×C4×C53—81no—
174C2×C4×C53#12011yescharacteristic
175C2×C4×C53#12011yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12711yescharacteristic
180C2×C4×C53C2×D811yescharacteristic
181C2×C4×C53C2×SD1612yes—
182C2×C4×C53#13511yescharacteristic
183C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C53AGL(1,ℤ/8)12yes—
186C2×Dic53—42no—
187C2×Dic53—81no—
188C2×Dic53—81no—
189Dic106—42no—
190Dic106—42no—
191Dic106—81no—
192Dic106—81no—
193#88322—42no—
194#88322—42no—
195#88322—81no—
196#10062—531nomaximal, Sylow
197C22×C53—44no—
198C22×C53—42no—
199C22×C53—82no—
200C22×C53—81no—
201C22×C53—81no—
202C22×C53—84no—
203C22×C53—84no—
204C22×C53—82no—
205C22×C53—82no—
206C22×C53—82no—
207C22×C53#39211yescharacteristic
208C22×C53#44711yescharacteristic
209C22×C53#44911yescharacteristic
210C22×C53#45012yes—
211C22×C53#45512yes—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—44no—
215C4×C53—81no—
216C4×C53#44711yescharacteristic
217C4×C53#44911yescharacteristic
218C4×C53#46012yes—
219C4×C53#46611yescharacteristic
220C4×C53#46911yescharacteristic
221C4×C53#49211yescharacteristic
222C4×C53#49611yescharacteristic
223Dic53—82no—
224C2×#486—531no—
225C2×#461—531no—
226#2541D5311yescharacteristic
227#1988—531no—
228C2×#331—531no—
229C2×D4⋊C8—531no—
230#1279—531no—
231C2×C53—82no—
232C2×C53—162no—
233C2×C53#163912yes—
234C2×C53#164212yes—
235C2×C53#199911yescharacteristic
236C2×C53#200411yescharacteristic
237C2×C53#201411yescharacteristic
238C22×SD16—1061no—
239C2×C4⋊Q8—531no—
240C2×#127—21no—
241C2×#127C2×D5311yescharacteristic
242C2×#121—22no—
243C2×C4×D4C2×D5311yescharacteristic
244#486—1062no—
245#486—1062no—
246#461—1062no—
247#461—1062no—
248C2×C4⋊C8—531no—
249C2×C4⋊C8—531no—
250C2×#103—1061no—
251C2×#102—1062no—
252C2×C4×C8—531no—
253#392—22no—
254#390—21no—
255C4×C4⋊C4C2×D5311yescharacteristic
256#340—1062no—
257#331—1062no—
258D4⋊C8—1062no—
259C53#1006211yescharacteristic, complemented, Sylow
260C22×Q8—1061no—
261C22×D4—24no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4C22⋊D5311yescharacteristic
265C22×D4C22⋊D5312yes—
266C23×C4—21no—
267C2×SD16—1064no—
268C2×SD16—2122no—
269C2×SD16—2122no—
270C22×C8—1061no—
271C22×C8—1061no—
272C22×C8—1061no—
273C4⋊Q8—532no—
274C4⋊Q8—1061no—
275C4⋊Q8—1061no—
276C4⋊Q8—1061no—
277#127—21no—
278#127—21no—
279#127—21no—
280#127—21no—
281#127—42no—
282#127—42no—
283#121—44no—
284#121—44no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C2×C4⋊C4—1061no—
290C2×C4⋊C4—1061no—
291C2×C4⋊C4C22⋊D5311yescharacteristic
292C2×C4⋊C4C22⋊D5312yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—21no—
295C2×C42—21no—
296C2×C42C22×D5311yescharacteristic
297C4⋊C8—532no—
298C4⋊C8—1061no—
299C4⋊C8—1061no—
300C4⋊C8—1061no—
301#103—1064no—
302#102—1068no—
303C4×C8—532no—
304C4×C8—1061no—
305#95—21no—
306C24—21no—
307C24—22no—
308C2×Q8—1062no—
309C2×Q8—2121no—
310C2×Q8—2121no—
311C2×Q8—2121no—
312C2×Q8—2121no—
313C2×Q8—2121no—
314C2×D4—21no—
315C2×D4—22no—
316C2×D4—21no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—44no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—42no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—41no—
334C2×D4—84no—
335C2×D4#1171992212yes—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—2121no—
343C22×C4C2×C22⋊D5311yescharacteristic
344C22×C4C2×C22⋊D5311yescharacteristic
345C22×C4C2×C22⋊D5311yescharacteristic
346SD16—2128no—
347C2×C8—1062no—
348C2×C8—1062no—
349C2×C8—2121no—
350C2×C8—2122no—
351C2×C8—2122no—
352C2×C8—2121no—
353C2×C8—2121no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—1062no—
358C4⋊C4—1062no—
359C4⋊C4—2121no—
360C4⋊C4—2121no—
361C4⋊C4#1171992314yescomplemented
362C22⋊C4—48no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C42—41no—
366C42—41no—
367C42D4×D5311yescharacteristic
368C42D4×D5311yescharacteristic
369C42D4×D5311yescharacteristic
370C42#1171994711yescharacteristic
371C23—22no—
372C23—41no—
373C23—42no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—44no—
379C23—42no—
380C23—44no—
381C23#1313318611yescharacteristic
382Q8—4241no—
383Q8—4241no—
384Q8—2122no—
385Q8—2122no—
386D4—22no—
387D4—22no—
388D4—44no—
389D4—41no—
390D4—41no—
391D4—82no—
392D4—816no—
393D4—81no—
394D4—81no—
395D4—81no—
396D4—82no—
397D4—81no—
398D4—81no—
399D4—81no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—81no—
407C2×C4—82no—
408C2×C4—81no—
409C2×C4—4241no—
410C2×C4—4241no—
411C2×C4—2122no—
412C2×C4#1313318511yescharacteristic
413C2×C4#1313318511yescharacteristic
414C2×C4#1313318811yescharacteristic
415C2×C4#1313318911yescharacteristic
416C2×C4#1313318911yescharacteristic
417C2×C4#1313321311yescharacteristic
418C2×C4C2×#1171992211yescharacteristic
419C2×C4C2×#1171992312yes—
420C2×C4#1313323711yescharacteristic
421C2×C4#1313324211yescharacteristic
422C2×C4#1313324312yes—
423C2×C4#1313324311yescharacteristic
424C8—4241no—
425C8—2122no—
426C8—2122no—
427C22—44no—
428C22—42no—
429C22—81no—
430C22—81no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—84no—
435C22—84no—
436C22—82no—
437C22#1681053111yescharacteristic
438C22#1681075711yescharacteristic
439C22#1681076811yescharacteristic
440C22#1681077412yes—
441C22#1681080112yes—
442C4—21no—
443C4—21no—
444C4—44no—
445C4—81no—
446C4—4242no—
447C4#1681075611yescharacteristic
448C4#1681076611yescharacteristic
449C4#1681082012yes—
450C4#1681085711yescharacteristic
451C4#1681087411yescharacteristic
452C4#1681099511yescharacteristic
453C4#1681101011yescharacteristic
454C2—82no—
455C2—162no—
456C2#2027389412yes—
457C2#2027391512yes—
458C2#2027571111yescharacteristic
459C2#2027574211yescharacteristic
460C2#2027580311yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#2541C2
socleC23×C53#120
hypercenter#2541D53
hypocenterC53#10062
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C2×#16810969C2semidirect
2. C2×#16810828C2semidirect
3. #20275667C2semidirect
4. C2×#16810367C2semidirect
5. C2×D4⋊#88322C2semidirect
6. #20272556C2semidirect
39. #13133217D4semidirect
45. C4⋊#88322D4semidirect
49. C4×#88322D4semidirect
118. C4⋊C4×C53SD16semidirect
259. C53#10062semidirect
361. C4⋊C4#11719923semidirect

Maximal subgroups

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

subgroupindex
C2×#168109692
C2×#168108282
#202756672
C2×#168103672
C2×D4⋊#883222
#202725562
C53×#25412
#1006253