Finite Group Databasebeta

← #27287897

Subgroups — #27287897

Order 13568.

522 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810823C211yesmaximal, characteristic
2#20275521C211yesmaximal, characteristic
3#20274720C211yesmaximal, characteristic, complemented
4C2×#16810376C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6#20272599C211yesmaximal, characteristic, complemented
7C53×#1984C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×#13133214C2211yescharacteristic
10#16810823—22no—
11#16810823—22no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133141—21no—
15C2×#13133139—21no—
16C2×#13133139—21no—
17C2×#13133137—21no—
18#16810493C2211yescharacteristic
19#16810385—21no—
20#16810376—22no—
21#16810366—22no—
22C2×C4×Q8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×C4×C8×C53C2211yescharacteristic
25C53×#336—21no—
26C53×#336—21no—
27C22×D212—21no—
28C22×D212—21no—
29C22×D212—21no—
30C2×#11719924—24no—
31C2×#11719924—42no—
32C2×#11719924—42no—
33C22×#88322—21no—
34C22×#88322—21no—
35#13133214—21no—
36#13133214—21no—
37#13133214—21no—
38#13133214D412yes—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D412yes—
46#13133141—42no—
47#13133139—42no—
48#13133139—42no—
49#13133137—24no—
50C22×Q8×C53D411yescharacteristic
51C22×C8×C53—21no—
52C22×C8×C53D411yescharacteristic
53C22×C8×C53D411yescharacteristic
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53D411yescharacteristic
60C2×C42×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4×C8×C53—21no—
65C4×C8×C53—21no—
66C23×D53—41no—
67C2×Q8×C53—21no—
68C2×Q8×C53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53D812yes—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53#4111yescharacteristic
78C22×Dic53—41no—
79C2×D212—22no—
80C2×D212—22no—
81C2×D212—22no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—81no—
94C2×D212—81no—
95C2×C8×C53—21no—
96C2×C8×C53—21no—
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C2×C8×C53SD1612yescomplemented
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C42×C53—41no—
109C42×C53—41no—
110C42×C53C2×D411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53#4111yescharacteristic
113C42×C53#4111yescharacteristic
114#11719924—48no—
115C4⋊Dic53—41no—
116C4⋊Dic53—41no—
117C4⋊Dic53—41no—
118C4⋊Dic53—41no—
119C4⋊Dic53—81no—
120C2×#88322—22no—
121C2×#88322—41no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322—41no—
126C23×C53#12111yescharacteristic
127C22×D53—42no—
128C22×D53—81no—
129C22×D53—81no—
130C22×D53—81no—
131C22×D53—81no—
132C22×D53—81no—
133C22×D53—81no—
134Q8×C53—22no—
135Q8×C53—22no—
136Q8×C53—41no—
137Q8×C53—41no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—22no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12311yescharacteristic
150C2×C4×C53#12311yescharacteristic
151C2×C4×C53#12411yescharacteristic
152C2×C4×C53C2×D811yescharacteristic
153C2×C4×C53C2×SD1611yescharacteristic
154C2×C4×C53#13511yescharacteristic
155C2×C4×C53#13511yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53#13711yescharacteristic
159C2×C4×C53#13711yescharacteristic
160C2×Dic53—81no—
161C2×Dic53—81no—
162C2×Dic53—81no—
163D212—42no—
164D212—42no—
165D212—42no—
166D212—42no—
167D212—82no—
168D212—82no—
169D212—81no—
170D212—81no—
171D212—81no—
172D212—81no—
173C8×C53—22no—
174C8×C53—41no—
175C8×C53—81no—
176#88322—42no—
177#88322—81no—
178#10474—531nomaximal, Sylow
179C22×C53#39411yescharacteristic
180C22×C53#46011yescharacteristic
181C22×C53#46411yescharacteristic
182C22×C53#46611yescharacteristic
183C22×C53#46811yescharacteristic
184C22×C53#47111yescharacteristic
185C22×C53#47211yescharacteristic
186C2×D53—82no—
187C2×D53—82no—
188C2×D53—82no—
189C2×D53—161no—
190C2×D53—161no—
191C2×D53—161no—
192C2×D53—161no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—42no—
196C4×C53—81no—
197C4×C53#44711yescharacteristic
198C4×C53#45211yescharacteristic
199C4×C53#46111yescharacteristic
200C4×C53#46311yescharacteristic
201C4×C53#46511yescharacteristic
202C4×C53#46811yescharacteristic
203C4×C53#48011yescharacteristic
204C4×C53#48211yescharacteristic
205Dic53—161no—
206C2×#460—531no—
207#1984D5311yescharacteristic, complemented
208#1957—531no—
209#1803—531no—
210C2×#335—531no—
211C2×#331—531no—
212#1296—531no—
213C2×C53#166811yescharacteristic
214C2×C53#168211yescharacteristic
215C2×C53#168611yescharacteristic
216C2×C53#169811yescharacteristic
217C2×C53#203311yescharacteristic
218C2×C53#205111yescharacteristic
219C2×C53#205511yescharacteristic
220D53—162no—
221C22×SD16—1061no—
222C2×#127—531no—
223C2×C4×Q8C2×D5311yescharacteristic
224#460—1062no—
225#460—1062no—
226C2×C4⋊C8—531no—
227C2×C4⋊C8—531no—
228C2×C4⋊C8C2×D5311yescharacteristic
229C2×#103—1061no—
230C2×#102—1061no—
231C2×#102—1061no—
232C2×#102—1061no—
233C2×C4×C8C2×D5311yescharacteristic
234#384—531no—
235#340—1061no—
236#336—21no—
237#336—21no—
238#335—1062no—
239#331—1062no—
240C53#1047411yescharacteristic, complemented, Sylow
241C22×Q8D21211yescharacteristic
242C22×D4—1061no—
243C22×D4—1061no—
244C22×D4—1061no—
245C2×SD16—1064no—
246C2×SD16—2122no—
247C2×SD16—2122no—
248C22×C8—21no—
249C22×C8—1061no—
250C22×C8—1061no—
251C22×C8C22⋊D5311yescharacteristic
252C22×C8C22⋊D5311yescharacteristic
253#127—532no—
254#127—1061no—
255#127—1061no—
256#127—1061no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—1061no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4D21211yescharacteristic
266C2×C42—21no—
267C2×C42C22×D5311yescharacteristic
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—532no—
271C4⋊C8—1061no—
272C4⋊C8—1061no—
273C4⋊C8—1061no—
274#103—2122no—
275#102—1064no—
276#102—2122no—
277#102—2122no—
278C4×C8—21no—
279C4×C8—21no—
280C24—2121no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8D42412yes—
287C2×D4—1062no—
288C2×D4—1062no—
289C2×D4—1062no—
290C2×D4—4241no—
291C2×D4—4241no—
292C2×D4—2121no—
293C2×D4—2121no—
294C2×D4—2121no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2121no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—2121no—
306C22×C4C2×D21211yescharacteristic
307C22×C4#1171994511yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309SD16—2128no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—1062no—
319C2×C8—2121no—
320C2×C8—2121no—
321C2×C8—2121no—
322C2×C8—2121no—
323C2×C8—2121no—
324C2×C8#1171992412yescomplemented
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—4241no—
330C4⋊C4—2121no—
331C4⋊C4—2121no—
332C4⋊C4—2121no—
333C4⋊C4—2121no—
334C42—41no—
335C42—41no—
336C42D4×D5311yescharacteristic
337C42D4×D5311yescharacteristic
338C42#1171994711yescharacteristic
339C42#1171994711yescharacteristic
340C23—4241no—
341C23—4241no—
342C23—4241no—
343C23—4241no—
344C23—4241no—
345C23—4241no—
346C23—2122no—
347C23#1313319111yescharacteristic
348Q8—22no—
349Q8—22no—
350Q8—41no—
351Q8—41no—
352D4—4242no—
353D4—4242no—
354D4—4241no—
355D4—4241no—
356D4—4241no—
357D4—4241no—
358D4—2122no—
359D4—2122no—
360D4—2122no—
361D4—2122no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—4241no—
371C2×C4—4241no—
372C2×C4—4241no—
373C2×C4#1313318711yescharacteristic
374C2×C4#1313318811yescharacteristic
375C2×C4#1313318911yescharacteristic
376C2×C4#1313319911yescharacteristic
377C2×C4#1313320111yescharacteristic
378C2×C4#1313320311yescharacteristic
379C2×C4C2×D42411yescharacteristic
380C2×C4C2×#1171992411yescharacteristic
381C2×C4#1313323711yescharacteristic
382C2×C4#1313323911yescharacteristic
383C2×C4#1313324211yescharacteristic
384C2×C4#1313324511yescharacteristic
385C2×C4#1313324811yescharacteristic
386C2×C4#1313325011yescharacteristic
387C8—22no—
388C8—41no—
389C8—81no—
390C8—4241no—
391C8—2122no—
392C22—8481no—
393C22—8481no—
394C22—8481no—
395C22—8481no—
396C22—4242no—
397C22—4242no—
398C22—4242no—
399C22#1681053911yescharacteristic
400C22#1681082311yescharacteristic
401C22#1681084511yescharacteristic
402C22#1681085911yescharacteristic
403C22#1681086911yescharacteristic
404C22#1681088611yescharacteristic
405C22#1681089311yescharacteristic
406C4—21no—
407C4—21no—
408C4—42no—
409C4—81no—
410C4—8481no—
411C4#1681075911yescharacteristic
412C4#1681078711yescharacteristic
413C4#1681082511yescharacteristic
414C4#1681083511yescharacteristic
415C4#1681085411yescharacteristic
416C4#1681087111yescharacteristic
417C4#1681093711yescharacteristic
418C4#1681094811yescharacteristic
419C2—8482no—
420C2#2027407511yescharacteristic
421C2#2027415911yescharacteristic
422C2#2027418711yescharacteristic
423C2#2027425511yescharacteristic
424C2#2027589811yescharacteristic
425C2#2027600611yescharacteristic
426C2#2027603011yescharacteristic
subgroupG/H
centerC23#13133191
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1984C2
socleC23×C53#121
prime radicalC2×#13133214C22
hypercenter#1984D53
hypocenterC53#10474
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984

Complements

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

NH ≅ G/Ntype
3. #20274720C2semidirect
4. C2×#16810376C2semidirect
6. #20272599C2semidirect
7. C53×#1984C2semidirect
103. C2×C8×C53SD16semidirect
207. #1984D53semidirect
240. C53#10474semidirect
324. C2×C8#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168108232
#202755212
#202747202
C2×#168103762
C2×#168103662
#202725992
C53×#19842
#1047453