Finite Group Databasebeta

← #27287908

Subgroups — #27287908

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×#16810825C211yesmaximal, characteristic
2#20275677C211yesmaximal, characteristic, complemented
3#20275508C211yesmaximal, characteristic
4#20274716C211yesmaximal, characteristic
5C2×#16810375C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×#332C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×C4×D212C2211yescharacteristic
10#16810825—22no—
11#16810825—22no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133141—21no—
14C2×#13133139—21no—
15C2×#13133129—21no—
16C2×#13133129—21no—
17C2×C4×#88322C2211yescharacteristic
18#16810493C2211yescharacteristic
19#16810386—21no—
20#16810385—21no—
21#16810385—21no—
22#16810375—22no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C53×#332—22no—
27C22×D212D411yescharacteristic
28C22×C4×D53—21no—
29C2×#11719924—24no—
30C2×#11719924—42no—
31C2×#11719924—42no—
32C22×#88322—21no—
33C22×#88322D411yescharacteristic
34C22×#88322D411yescharacteristic
35C4×D212—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53D411yescharacteristic
43C2×#11719921—21no—
44C4⋊#88322—21no—
45C4⋊#88322D412yes—
46#13133141—24no—
47#13133139—42no—
48#13133129—42no—
49#13133129—42no—
50C4×#88322—21no—
51C4×#88322—21no—
52C22×Q8×C53—21no—
53C22×C8×C53—21no—
54C22×C8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53D412yes—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C23×D53—21no—
67C2×Q8×C53—22no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53#4111yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—41no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212D812yes—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53#4111yescharacteristic
104C42×C53#4111yescharacteristic
105#11719924—48no—
106#11719921—41no—
107#11719921—41no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C4⋊Dic53—41no—
111C4⋊Dic53—41no—
112C4⋊Dic53—41no—
113C4⋊Dic53—41no—
114C4⋊Dic53—81no—
115C2×#88322—21no—
116C2×#88322—21no—
117C2×#88322—22no—
118C2×#88322—21no—
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—41no—
122C2×#88322—41no—
123C2×#88322SD1612yescomplemented
124C23×C53#12111yescharacteristic
125C22×D53—22no—
126C22×D53—41no—
127C22×D53—41no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—41no—
131C22×D53—41no—
132Q8×C53—42no—
133Q8×C53—42no—
134Q8×C53—81no—
135Q8×C53—81no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—42no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12311yescharacteristic
145C2×C4×C53#12311yescharacteristic
146C2×C4×C53#12411yescharacteristic
147C2×C4×C53C2×D811yescharacteristic
148C2×C4×C53C2×SD1611yescharacteristic
149C2×C4×C53#13511yescharacteristic
150C2×C4×C53#13511yescharacteristic
151C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C53#13711yescharacteristic
154C2×C4×C53#13711yescharacteristic
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—81no—
159C2×Dic53—81no—
160C2×Dic53—81no—
161D212—22no—
162D212—22no—
163D212—41no—
164D212—41no—
165C4×D53—81no—
166C4×D53—81no—
167C4×D53—81no—
168C4×D53—81no—
169C8×C53—81no—
170C8×C53—81no—
171#88322—22no—
172#88322—42no—
173#88322—41no—
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—82no—
192C4×C53#44711yescharacteristic
193C4×C53#45211yescharacteristic
194C4×C53#46011yescharacteristic
195C4×C53#46411yescharacteristic
196C4×C53#46511yescharacteristic
197C4×C53#46811yescharacteristic
198C4×C53#48011yescharacteristic
199C4×C53#48211yescharacteristic
200Dic53—81no—
201Dic53—161no—
202C2×#461—531no—
203#1990—531no—
204#1954—531no—
205#1802—531no—
206C2×#335—531no—
207C2×#332D5311yescharacteristic, complemented
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⋊Q8C2×D5311yescharacteristic
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×C8—531no—
230#384—531no—
231#340—1061no—
232#340—1061no—
233#340—1061no—
234#335—1062no—
235#332—22no—
236C53#1047611yescharacteristic, complemented, Sylow
237C22×Q8—21no—
238C22×D4—531no—
239C23×C4—1061no—
240C2×SD16—1064no—
241C2×SD16—2122no—
242C2×SD16—2122no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—531no—
246C22×C8—531no—
247C22×C8—1061no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C22⋊D5312yes—
252C4×D4—1061no—
253C4×D4—1061no—
254C4×D4—1061no—
255C4×D4—1061no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—531no—
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—21no—
268C4⋊C8—532no—
269C4⋊C8—1061no—
270#103—1064no—
271#102—2122no—
272C22⋊C8—2122no—
273C22⋊C8—2122no—
274C4×C8—1061no—
275C4×C8—1061no—
276C24—1061no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×D4—532no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C22×C4—41no—
290C22×C4—1061no—
291C22×C4—2121no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4—2121no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4—2121no—
298C22×C4#1171994511yescharacteristic
299C22×C4D4×D5311yescharacteristic
300C22×C4D4×D5311yescharacteristic
301SD16—2128no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—532no—
309C2×C8—1061no—
310C2×C8—1061no—
311C2×C8—1062no—
312C2×C8—1061no—
313C2×C8—1061no—
314C2×C8—1061no—
315C2×C8—2121no—
316C2×C8—2121no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—1061no—
322C4⋊C4—1061no—
323C4⋊C4—4241no—
324C4⋊C4—2121no—
325C4⋊C4—2121no—
326C4⋊C4—2121no—
327C4⋊C4—2121no—
328C22⋊C4—2121no—
329C22⋊C4—2121no—
330C42#1171994511yescharacteristic
331C42D4×D5311yescharacteristic
332C42#1171994711yescharacteristic
333C42C2×C22⋊D5311yescharacteristic
334C23—1062no—
335C23—2121no—
336C23—2121no—
337C23—2121no—
338C23—2121no—
339C23—2121no—
340C23—2121no—
341C23#1313319011yescharacteristic
342Q8—42no—
343Q8—42no—
344Q8—81no—
345Q8—81no—
346D4—1062no—
347D4—1062no—
348D4—2121no—
349D4—2121no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—4241no—
356C2×C4—4241no—
357C2×C4—4241no—
358C2×C4—4241no—
359C2×C4—4241no—
360C2×C4—4241no—
361C2×C4—4241no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4#1313318511yescharacteristic
366C2×C4#1313318911yescharacteristic
367C2×C4#1313319011yescharacteristic
368C2×C4#1313319911yescharacteristic
369C2×C4#1313320011yescharacteristic
370C2×C4#1313320311yescharacteristic
371C2×C4D8×D5311yescharacteristic
372C2×C4SD16×D5311yescharacteristic
373C2×C4#1313323811yescharacteristic
374C2×C4#1313324011yescharacteristic
375C2×C4#1313324111yescharacteristic
376C2×C4#1313324411yescharacteristic
377C2×C4#1313324611yescharacteristic
378C2×C4#1313324911yescharacteristic
379C8—81no—
380C8—81no—
381C8—1062no—
382C8—2122no—
383C8—2121no—
384C22—4241no—
385C22—4241no—
386C22—4241no—
387C22—4241no—
388C22—2122no—
389C22—2122no—
390C22—2122no—
391C22#1681053811yescharacteristic
392C22#1681082711yescharacteristic
393C22#1681084111yescharacteristic
394C22#1681085811yescharacteristic
395C22#1681086811yescharacteristic
396C22#1681088911yescharacteristic
397C22#1681089711yescharacteristic
398C4—21no—
399C4—21no—
400C4—82no—
401C4—8481no—
402C4—4241no—
403C4#1681075511yescharacteristic
404C4#1681078211yescharacteristic
405C4#1681082111yescharacteristic
406C4#1681084711yescharacteristic
407C4#1681085511yescharacteristic
408C4#1681087211yescharacteristic
409C4#1681093511yescharacteristic
410C4#1681094611yescharacteristic
411C2—4242no—
412C2#2027410311yescharacteristic
413C2#2027415511yescharacteristic
414C2#2027421311yescharacteristic
415C2#2027426511yescharacteristic
416C2#2027589511yescharacteristic
417C2#2027601411yescharacteristic
418C2#2027602911yescharacteristic
subgroupG/H
centerC23#13133190
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#332C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenterC2×#332D53
hypocenterC53#10476
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #20275677C2semidirect
5. C2×#16810375C2semidirect
6. #20272599C2semidirect
7. C2×C53×#332C2semidirect
123. C2×#88322SD16semidirect
207. C2×#332D53semidirect
236. C53#10476semidirect

Maximal subgroups

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

subgroupindex
C2×#168108252
#202756772
#202755082
#202747162
C2×#168103752
#202725992
C2×C53×#3322
#1047653