Finite Group Databasebeta

← #27282371

Subgroups — #27282371

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275508C211yesmaximal, characteristic
2#20275434C211yesmaximal, characteristic, complemented
3C4×#13133137C211yesmaximal, characteristic
4C2×#16810352C211yesmaximal, characteristic
5C2×#16810345C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9C2×C4⋊#88322—21no—
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133137C411yescharacteristic
12C2×#13133137C411yescharacteristic
13C2×#13133129—21no—
14C2×#13133129—21no—
15C2×C4×#88322C2211yescharacteristic
16C2×C4×#88322C2211yescharacteristic
17#16810385C411yescharacteristic
18#16810385C411yescharacteristic
19#16810378—21no—
20#16810352—22no—
21#16810345—22no—
22C2×C4⋊C8×C53C2211yescharacteristic
23C2×C4×C8×C53C2211yescharacteristic
24C4×C4⋊C4×C53C2211yescharacteristic
25C22×D212C2×C411yescharacteristic
26C22×C4×D53—21no—
27C22×#88322—21no—
28C22×#88322C2×C411yescharacteristic
29C22×#88322C2×C411yescharacteristic
30C22×#88322D411yescharacteristic
31C22×#88322D411yescharacteristic
32C4×D212—21no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—21no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719921—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—42no—
41#13133137—22no—
42#13133137C814yescomplemented
43#13133129—42no—
44#13133129—42no—
45C4×#88322—21no—
46C4×#88322—21no—
47C4×#88322—21no—
48C4×#88322—21no—
49C22×C8×C53—21no—
50C22×C8×C53—21no—
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C42×C53—21no—
54C2×C42×C53C2311yescharacteristic
55C4⋊C8×C53—21no—
56C4⋊C8×C53—21no—
57C4×C8×C53—21no—
58C4×C8×C53D412yes—
59#95×C53—21no—
60C23×D53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53C22×C411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×C4×C53#4111yescharacteristic
66C22×Dic53—21no—
67C2×D212—21no—
68C2×D212—21no—
69C2×D212—21no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212C2×C812yes—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×C4×D53—41no—
78C2×C4×D53—41no—
79C2×C8×C53—22no—
80C2×C8×C53—41no—
81C2×C8×C53—41no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C2×C8×C53—41no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—21no—
87C4⋊C4×C53—21no—
88C4⋊C4×C53C2×C812yes—
89C42×C53—41no—
90C42×C53—41no—
91C42×C53C2×D411yescharacteristic
92C42×C53#4111yescharacteristic
93C42×C53#4111yescharacteristic
94C42×C53#4111yescharacteristic
95#11719921—41no—
96#11719921—41no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99C2×#88322—21no—
100C2×#88322—21no—
101C2×#88322—21no—
102C2×#88322—21no—
103C2×#88322—21no—
104C2×#88322—21no—
105C2×#88322—21no—
106C2×#88322—21no—
107C2×#88322—21no—
108C2×#88322—21no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322C2×C812yes—
113C2×#88322M1612yes—
114C23×C53C4×D411yescharacteristic
115C22×D53—22no—
116C22×D53—41no—
117C22×D53—41no—
118C22×D53—41no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53—41no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—22no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53#11711yescharacteristic
131C2×C4×C53C4×D411yescharacteristic
132C2×C4×C53#12111yescharacteristic
133C2×C4×C53#12311yescharacteristic
134C2×C4×C53#12411yescharacteristic
135C2×C4×C53#12611yescharacteristic
136C2×C4×C53C22×C811yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53#13111yescharacteristic
139C2×C4×C53#13111yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C53#13711yescharacteristic
144C2×Dic53—41no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147D212—22no—
148D212—22no—
149D212—41no—
150D212—41no—
151C4×D53—81no—
152C4×D53—81no—
153C4×D53—81no—
154C4×D53—81no—
155C8×C53—42no—
156C8×C53—81no—
157#88322—22no—
158#88322—22no—
159#88322—41no—
160#88322—41no—
161#88322—81no—
162#9556—531nomaximal, Sylow
163C22×C53#38811yescharacteristic
164C22×C53C8×D411yescharacteristic
165C22×C53#43611yescharacteristic
166C22×C53#44011yescharacteristic
167C22×C53#44211yescharacteristic
168C22×C53#44411yescharacteristic
169C22×C53#44411yescharacteristic
170C2×D53—42no—
171C2×D53—42no—
172C2×D53—42no—
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C2×D53—81no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—42no—
180C4×C53—81no—
181C4×C53#43111yescharacteristic
182C4×C53#43311yescharacteristic
183C4×C53C8×D411yescharacteristic
184C4×C53#43511yescharacteristic
185C4×C53#46811yescharacteristic
186C4×C53#46911yescharacteristic
187C4×C53#48911yescharacteristic
188C4×C53#49011yescharacteristic
189Dic53—81no—
190#1954—531no—
191#1939—531no—
192C4×#102—531no—
193C2×D4⋊C8—531no—
194C2×#322—531no—
195#1279D5311yescharacteristic, complemented
196#1277—531no—
197C2×C53SD16⋊C811yescharacteristic
198C2×C53#158011yescharacteristic
199C2×C53#159211yescharacteristic
200C2×C53#159411yescharacteristic
201C2×C53#192311yescharacteristic
202C2×C53#193611yescharacteristic
203C2×C53#193811yescharacteristic
204D53—82no—
205C2×C4×D4—531no—
206C2×C4⋊C8—531no—
207C2×C4⋊C8—1061no—
208C2×C4⋊C8C2×D5311yescharacteristic
209C2×#102—531no—
210C2×#102—531no—
211C2×C22⋊C8—1061no—
212C2×C22⋊C8—1061no—
213C2×C4×C8—531no—
214C2×C4×C8—531no—
215C2×C4×C8C2×D5311yescharacteristic
216C4×C4⋊C4C2×D5311yescharacteristic
217#340—531no—
218#340—531no—
219#336—1061no—
220D4⋊C8—1062no—
221#322—1062no—
222C53#955611yescharacteristic, complemented, Sylow
223C22×D4—531no—
224C23×C4—1061no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—531no—
228C22×C8—531no—
229C22×C8—531no—
230C22×C8—531no—
231C22×C8—1061no—
232C4×D4—1061no—
233C4×D4—1061no—
234C4×D4—1061no—
235C4×D4—1061no—
236C2×C4⋊C4—531no—
237C2×C4⋊C4C4×D5311yescharacteristic
238C2×C4⋊C4C4×D5311yescharacteristic
239C2×C22⋊C4—1061no—
240C2×C42—21no—
241C2×C42C22×D5311yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—1061no—
245C4⋊C8—1061no—
246C4⋊C8—2122no—
247#102—534no—
248#102—1062no—
249C22⋊C8—2122no—
250C22⋊C8—2122no—
251C4×C8—21no—
252C4×C8—1061no—
253C4×C8—1061no—
254C4×C8—1061no—
255C4×C8—1061no—
256C4×C8D21212yes—
257#95—21no—
258C24—1061no—
259C2×D4—532no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C2×D4—1061no—
264C2×D4—1061no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—1061no—
268C22×C4—2121no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4—2121no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4C2×C4×D5311yescharacteristic
275C22×C4D4×D5311yescharacteristic
276C22×C4#1171994711yescharacteristic
277C2×C8—22no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—532no—
284C2×C8—532no—
285C2×C8—1061no—
286C2×C8—1061no—
287C2×C8—1061no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8—1061no—
292C2×C8—1061no—
293C2×C8—1061no—
294C2×C8—1061no—
295C2×C8—2121no—
296C2×C8—2121no—
297C2×C8—2121no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—1061no—
302C4⋊C4—1061no—
303C4⋊C4C8×D5312yes—
304C22⋊C4—2121no—
305C22⋊C4—2121no—
306C42—41no—
307C42—41no—
308C42C2×D21211yescharacteristic
309C42#1171994511yescharacteristic
310C42#1171994711yescharacteristic
311C42#1171994911yescharacteristic
312C23—1062no—
313C23—2121no—
314C23—2121no—
315C23—2121no—
316C23—2121no—
317C23—2121no—
318C23—2121no—
319C23#1313317711yescharacteristic
320D4—1062no—
321D4—1062no—
322D4—2121no—
323D4—2121no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—4241no—
333C2×C4—4241no—
334C2×C4—4241no—
335C2×C4—4241no—
336C2×C4—2121no—
337C2×C4—2121no—
338C2×C4—2121no—
339C2×C4#1313317511yescharacteristic
340C2×C4C4×D21211yescharacteristic
341C2×C4#1313319211yescharacteristic
342C2×C4#1313320111yescharacteristic
343C2×C4#1313320311yescharacteristic
344C2×C4#1313321211yescharacteristic
345C2×C4C2×C8×D5311yescharacteristic
346C2×C4M16×D5311yescharacteristic
347C2×C4#1313322311yescharacteristic
348C2×C4#1313322511yescharacteristic
349C2×C4#1313324111yescharacteristic
350C2×C4#1313324411yescharacteristic
351C2×C4#1313324411yescharacteristic
352C2×C4#1313324911yescharacteristic
353C8—42no—
354C8—81no—
355C8—1062no—
356C8—1062no—
357C8—4241no—
358C8—2121no—
359C8—2121no—
360C22—4241no—
361C22—4241no—
362C22—4241no—
363C22—4241no—
364C22—2122no—
365C22—2122no—
366C22—2122no—
367C22#1681051511yescharacteristic
368C22#1681068611yescharacteristic
369C22#1681069811yescharacteristic
370C22#1681072611yescharacteristic
371C22#1681073211yescharacteristic
372C22#1681074211yescharacteristic
373C22#1681074211yescharacteristic
374C4—21no—
375C4—21no—
376C4—42no—
377C4—81no—
378C4—4241no—
379C4#1681067511yescharacteristic
380C4#1681068311yescharacteristic
381C4C8×D21211yescharacteristic
382C4#1681069611yescharacteristic
383C4#1681087011yescharacteristic
384C4#1681087711yescharacteristic
385C4#1681098411yescharacteristic
386C4#1681098611yescharacteristic
387C2—4242no—
388C2#2027353011yescharacteristic
389C2#2027353611yescharacteristic
390C2#2027362011yescharacteristic
391C2#2027362611yescharacteristic
392C2#2027532211yescharacteristic
393C2#2027541311yescharacteristic
394C2#2027542711yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53C4×D4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1279D53
hypocenterC53#9556
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #20275434C2semidirect
5. C2×#16810345C2semidirect
6. #20272549C2semidirect
7. C53×#1279C2semidirect
42. #13133137C8semidirect
195. #1279D53semidirect
222. C53#9556semidirect

Maximal subgroups

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

subgroupindex
#202755082
#202754342
C4×#131331372
C2×#168103522
C2×#168103452
#202725492
C53×#12792
#955653