Finite Group Databasebeta

← #27287912

Subgroups — #27287912

Order 13568.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810829C211yesmaximal, characteristic
2#20275676C211yesmaximal, characteristic, complemented
3#20275509C211yesmaximal, characteristic
4#20274716C211yesmaximal, characteristic
5C2×#16810370C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×#335C211yesmaximal, characteristic, complemented
8C22×#11719913—21no—
9C2×#13133217C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11#16810829—22no—
12#16810829—22no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133142—21no—
15C2×#13133139—21no—
16C2×#13133129—21no—
17C2×#13133129—21no—
18#16810493C2211yescharacteristic
19#16810386—21no—
20#16810386—21no—
21#16810386—21no—
22#16810370—22no—
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4×C8×C53C2211yescharacteristic
26C53×#335—22no—
27C22×Dic106—21no—
28C22×D212D411yescharacteristic
29C22×C4×D53—21no—
30C2×#11719913—24no—
31C2×#11719913—42no—
32C2×#11719913—42no—
33C22×#88322—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217D412yes—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212—21no—
42C2×C4⋊Dic53—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53D411yescharacteristic
48C2×#11719921—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51#13133142—24no—
52#13133139—42no—
53#13133129—42no—
54#13133129—42no—
55C22×C8×C53—21no—
56C22×C8×C53—21no—
57C22×C8×C53D411yescharacteristic
58C22×C8×C53D411yescharacteristic
59C2×C42×C53C2311yescharacteristic
60C4⋊C8×C53—21no—
61C4⋊C8×C53—21no—
62C4⋊C8×C53—21no—
63C4⋊C8×C53D412yes—
64C4×C8×C53—21no—
65C4×C8×C53—21no—
66C23×D53—21no—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4111yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C2×D212—21no—
74C2×D212—21no—
75C2×D212—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212D812yes—
79C2×C4×D53—41no—
80C2×C4×D53—41no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×Dic106—22no—
86C2×Dic106—41no—
87C2×Dic106—41no—
88C2×Dic106—41no—
89C2×Dic106—41no—
90C2×Dic106—41no—
91C2×C8×C53—21no—
92C2×C8×C53—21no—
93C2×C8×C53—22no—
94C2×C8×C53—21no—
95C2×C8×C53—21no—
96C2×C8×C53—21no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53SD1612yescomplemented
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53#4111yescharacteristic
106C42×C53#4111yescharacteristic
107#11719921—41no—
108#11719921—41no—
109C4⋊Dic53—21no—
110C4⋊Dic53—22no—
111C4⋊Dic53—22no—
112C4⋊Dic53—21no—
113C4⋊Dic53—41no—
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116C4⋊Dic53—41no—
117C4⋊Dic53—41no—
118C4⋊Dic53—41no—
119C4⋊Dic53—81no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322—41no—
123#11719913—48no—
124C23×C53#12111yescharacteristic
125C22×D53—22no—
126C22×D53—41no—
127C22×D53—41no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—41no—
131C22×D53—41no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53#12011yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12311yescharacteristic
138C2×C4×C53#12311yescharacteristic
139C2×C4×C53#12411yescharacteristic
140C2×C4×C53C2×D811yescharacteristic
141C2×C4×C53C2×SD1611yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53#13511yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53#13711yescharacteristic
147C2×C4×C53#13711yescharacteristic
148C2×Dic53—42no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—81no—
153C2×Dic53—81no—
154C2×Dic53—81no—
155C2×Dic53—81no—
156C2×Dic53—81no—
157D212—22no—
158D212—22no—
159D212—41no—
160D212—41no—
161C4×D53—81no—
162C4×D53—81no—
163C4×D53—81no—
164C4×D53—81no—
165Dic106—42no—
166Dic106—42no—
167Dic106—81no—
168Dic106—81no—
169C8×C53—22no—
170C8×C53—42no—
171C8×C53—41no—
172C8×C53—81no—
173#88322—81no—
174#10476—531nomaximal, Sylow
175C22×C53#39411yescharacteristic
176C22×C53#46111yescharacteristic
177C22×C53#46311yescharacteristic
178C22×C53#46611yescharacteristic
179C22×C53#46811yescharacteristic
180C22×C53#47111yescharacteristic
181C22×C53#47311yescharacteristic
182C2×D53—42no—
183C2×D53—42no—
184C2×D53—42no—
185C2×D53—81no—
186C2×D53—81no—
187C2×D53—81no—
188C2×D53—81no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53#44711yescharacteristic
192C4×C53#45211yescharacteristic
193C4×C53#46011yescharacteristic
194C4×C53#46411yescharacteristic
195C4×C53#46511yescharacteristic
196C4×C53#46811yescharacteristic
197C4×C53#48011yescharacteristic
198C4×C53#48211yescharacteristic
199Dic53—81no—
200Dic53—82no—
201Dic53—161no—
202C2×#461—531no—
203#1990—531no—
204#1954—531no—
205#1802—531no—
206C2×#335D5311yescharacteristic, complemented
207C2×#332—531no—
208#1296—531no—
209C2×C53#167311yescharacteristic
210C2×C53#168111yescharacteristic
211C2×C53#169111yescharacteristic
212C2×C53#169911yescharacteristic
213C2×C53#203311yescharacteristic
214C2×C53#205211yescharacteristic
215C2×C53#205511yescharacteristic
216D53—82no—
217C22×SD16—1061no—
218C2×C4⋊Q8—531no—
219C2×C4×D4—531no—
220#461—1062no—
221#461—1062no—
222C2×C4⋊C8—531no—
223C2×C4⋊C8C2×D5311yescharacteristic
224C2×C4⋊C8C2×D5311yescharacteristic
225C2×#103—1061no—
226C2×#102—1061no—
227C2×C22⋊C8—1061no—
228C2×C22⋊C8—1061no—
229C2×C4×C8C2×D5311yescharacteristic
230#384—531no—
231#340—1061no—
232#340—1061no—
233#340—1061no—
234#335—22no—
235#332—1062no—
236C53#1047611yescharacteristic, complemented, Sylow
237C22×Q8—1061no—
238C22×D4—531no—
239C23×C4—1061no—
240C2×SD16—1064no—
241C2×SD16—2122no—
242C2×SD16—2122no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—1061no—
246C22×C8D21211yescharacteristic
247C22×C8D21211yescharacteristic
248C4⋊Q8—532no—
249C4⋊Q8—1061no—
250C4⋊Q8—1061no—
251C4⋊Q8—1061no—
252C4×D4—1061no—
253C4×D4—1061no—
254C4×D4—1061no—
255C4×D4—1061no—
256C2×C4⋊C4—531no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4—1061no—
260C2×C4⋊C4—1061no—
261C2×C4⋊C4—1061no—
262C2×C22⋊C4—1061no—
263C2×C42C22×D5311yescharacteristic
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269C4⋊C8C22⋊D5312yes—
270#103—1064no—
271#102—2122no—
272C22⋊C8—2122no—
273C22⋊C8—2122no—
274C4×C8—21no—
275C4×C8—21no—
276C24—1061no—
277C2×Q8—1062no—
278C2×Q8—2121no—
279C2×Q8—2121no—
280C2×Q8—2121no—
281C2×Q8—2121no—
282C2×Q8—2121no—
283C2×D4—532no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C22×C4—1061no—
290C22×C4—2121no—
291C22×C4—2121no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4—2121no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4—2121no—
298C22×C4C2×D21211yescharacteristic
299C22×C4D4×D5311yescharacteristic
300C22×C4#1171994911yescharacteristic
301SD16—2128no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—2121no—
314C2×C8—2121no—
315C2×C8—2121no—
316C2×C8#1171991312yescomplemented
317C4⋊C4—1061no—
318C4⋊C4—1062no—
319C4⋊C4—1061no—
320C4⋊C4—1062no—
321C4⋊C4—4241no—
322C4⋊C4—2121no—
323C4⋊C4—2121no—
324C4⋊C4—2121no—
325C4⋊C4—2121no—
326C4⋊C4—2121no—
327C4⋊C4—2121no—
328C22⋊C4—2121no—
329C22⋊C4—2121no—
330C42#1171994511yescharacteristic
331C42D4×D5311yescharacteristic
332C42#1171994911yescharacteristic
333C42C2×C22⋊D5311yescharacteristic
334C23—1062no—
335C23—2121no—
336C23—2121no—
337C23—2121no—
338C23—2121no—
339C23—2121no—
340C23—2121no—
341C23#1313319211yescharacteristic
342Q8—4241no—
343Q8—4241no—
344Q8—2122no—
345Q8—2122no—
346D4—1062no—
347D4—1062no—
348D4—2121no—
349D4—2121no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—4241no—
353C2×C4—4241no—
354C2×C4—4241no—
355C2×C4—4241no—
356C2×C4—4241no—
357C2×C4—4241no—
358C2×C4—4241no—
359C2×C4—4241no—
360C2×C4—4241no—
361C2×C4—2121no—
362C2×C4—2122no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4#1313318511yescharacteristic
366C2×C4#1313319111yescharacteristic
367C2×C4#1313319211yescharacteristic
368C2×C4#1313320211yescharacteristic
369C2×C4#1313320211yescharacteristic
370C2×C4#1313320611yescharacteristic
371C2×C4D8×D5311yescharacteristic
372C2×C4C2×#1171991311yescharacteristic
373C2×C4#1313323911yescharacteristic
374C2×C4#1313324011yescharacteristic
375C2×C4#1313324111yescharacteristic
376C2×C4#1313324511yescharacteristic
377C2×C4#1313324911yescharacteristic
378C2×C4#1313325011yescharacteristic
379C8—22no—
380C8—42no—
381C8—41no—
382C8—81no—
383C8—4241no—
384C22—4241no—
385C22—4241no—
386C22—4241no—
387C22—4241no—
388C22—2122no—
389C22—2122no—
390C22—2122no—
391C22#1681054011yescharacteristic
392C22#1681082911yescharacteristic
393C22#1681083911yescharacteristic
394C22#1681086011yescharacteristic
395C22#1681087011yescharacteristic
396C22#1681088711yescharacteristic
397C22#1681089911yescharacteristic
398C4—21no—
399C4—21no—
400C4—8481no—
401C4—4241no—
402C4—4242no—
403C4#1681075511yescharacteristic
404C4#1681078411yescharacteristic
405C4#1681082411yescharacteristic
406C4#1681084611yescharacteristic
407C4#1681085211yescharacteristic
408C4#1681086911yescharacteristic
409C4#1681093811yescharacteristic
410C4#1681095011yescharacteristic
411C2—4242no—
412C2#2027410511yescharacteristic
413C2#2027415311yescharacteristic
414C2#2027421511yescharacteristic
415C2#2027426311yescharacteristic
416C2#2027589911yescharacteristic
417C2#2027601611yescharacteristic
418C2#2027603311yescharacteristic
subgroupG/H
centerC23#13133192
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#335C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenterC2×#335D53
hypocenterC53#10476
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
2. #20275676C2semidirect
5. C2×#16810370C2semidirect
6. #20272599C2semidirect
7. C2×C53×#335C2semidirect
102. C2×C8×C53SD16semidirect
206. C2×#335D53semidirect
236. C53#10476semidirect
316. C2×C8#11719913semidirect

Maximal subgroups

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

subgroupindex
C2×#168108292
#202756762
#202755092
#202747162
C2×#168103702
#202725992
C2×C53×#3352
#1047653