Finite Group Databasebeta

← #27280290

Subgroups — #27280290

Order 13568.

546 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810910C211yesmaximal, characteristic
2C2×#16810702C211yesmaximal, characteristic
3#20274712C211yesmaximal, characteristic
4C2×#16810372C211yesmaximal, characteristic, complemented
5#20272598C211yesmaximal, characteristic, complemented
6#20272550C211yesmaximal, characteristic, complemented
7C53×#1990C211yesmaximal, characteristic, complemented
8C22×#11719912—21no—
9C2×C4×D212C2211yescharacteristic
10#16810910—22no—
11#16810910C414yescomplemented
12#16810702—22no—
13#16810702—22no—
14C2×#13133148—21no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133137—21no—
19C2×#13133137—21no—
20C2×#13133129—21no—
21#16810385—21no—
22#16810372—22no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4×C8×C53C2211yescharacteristic
25C53×#384C2211yescharacteristic
26C53×#340—21no—
27C53×#340—21no—
28C22×D212D411yescharacteristic
29C22×C4×D53—21no—
30C2×#11719912—24no—
31C2×#11719912—42no—
32C2×#11719912—42no—
33C22×#88322—21no—
34C22×#88322—21no—
35C22×#88322—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212C2×C412yes—
39C4×D212D412yes—
40C2×C4⋊Dic53D411yescharacteristic
41C2×#11719921—21no—
42#13133148—24no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322C2×C412yes—
48C4⋊#88322D412yes—
49#13133137—24no—
50#13133137—42no—
51#13133129—24no—
52C22×Q8×C53—21no—
53C22×C8×C53D411yescharacteristic
54C22×C8×C53D411yescharacteristic
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53C2×C412yes—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4×C8×C53—21no—
66C4×C8×C53—21no—
67C23×D53—21no—
68C2×Q8×C53—22no—
69C2×Q8×C53—41no—
70C2×Q8×C53—81no—
71C2×Q8×C53—81no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53#4111yescharacteristic
77C22×Dic53—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212C22⋊C412yes—
84C2×C4×D53—22no—
85C2×C4×D53—22no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C8×C53—21no—
91C2×C8×C53—21no—
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53—21no—
95C2×C8×C53SD1612yes—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—81no—
105C42×C53C22×C411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109#11719921—22no—
110#11719921—22no—
111C4⋊Dic53—21no—
112C4⋊Dic53C22⋊C412yes—
113C2×#88322—22no—
114C2×#88322—22no—
115C2×#88322—41no—
116C2×#88322—41no—
117C2×#88322—41no—
118C2×#88322—41no—
119C2×#88322—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122#11719912—48no—
123C23×C53#12111yescharacteristic
124C22×D53—22no—
125C22×D53—41no—
126C22×D53—41no—
127C22×D53—41no—
128C22×D53—41no—
129C22×D53—41no—
130C22×D53—41no—
131Q8×C53—82no—
132Q8×C53—82no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—42no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53C2×C22⋊C411yescharacteristic
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12311yescharacteristic
147C2×C4×C53C2×SD1611yescharacteristic
148C2×C4×C53#13511yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C53#13711yescharacteristic
152C2×C4×C53#13711yescharacteristic
153C2×C4×C53#13711yescharacteristic
154C2×C4×C53#13711yescharacteristic
155C2×Dic53—22no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158D212—22no—
159D212—22no—
160D212—41no—
161D212—41no—
162C4×D53—42no—
163C4×D53—42no—
164C4×D53—42no—
165C4×D53—42no—
166C8×C53—22no—
167C8×C53—41no—
168#88322—42no—
169#88322—42no—
170#88322—81no—
171#9223—531nomaximal, Sylow
172C22×C53#38611yescharacteristic
173C22×C53#46111yescharacteristic
174C22×C53#46311yescharacteristic
175C22×C53#46811yescharacteristic
176C22×C53#47011yescharacteristic
177C22×C53#47111yescharacteristic
178C22×C53#47211yescharacteristic
179C2×D53—42no—
180C2×D53—42no—
181C2×D53—42no—
182C2×D53—81no—
183C2×D53—81no—
184C2×D53—81no—
185C2×D53—81no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—82no—
189C4×C53—161no—
190C4×C53#41711yescharacteristic
191C4×C53#41911yescharacteristic
192C4×C53C4×SD1611yescharacteristic
193C4×C53#44011yescharacteristic
194C4×C53#45011yescharacteristic
195C4×C53#45211yescharacteristic
196C4×C53#46811yescharacteristic
197C4×C53#47011yescharacteristic
198Dic53—42no—
199C2×#476—531no—
200C2×#436—531no—
201#1990D5311yescharacteristic, complemented
202#1802—531no—
203C2×#333—531no—
204#1296—531no—
205#1277—531no—
206C2×C53#169111yescharacteristic
207C2×C53#169511yescharacteristic
208C2×C53#169711yescharacteristic
209C2×C53#170311yescharacteristic
210C2×C53#187711yescharacteristic
211C2×C53#189411yescharacteristic
212C2×C53#190711yescharacteristic
213D53—82no—
214C22×M16—1061no—
215C2×C4⋊Q8C2×D5311yescharacteristic
216C2×C4×D4—531no—
217#476—534no—
218#476—1062no—
219#436—1062no—
220#436—1062no—
221C2×#106—1061no—
222C2×C4⋊C8—531no—
223C2×C4⋊C8—531no—
224C2×C4⋊C8—531no—
225C2×#102—1061no—
226C2×#102—1061no—
227C2×C22⋊C8—1061no—
228C2×C4×C8C2×D5311yescharacteristic
229#384C2×D5311yescharacteristic
230#340—21no—
231#340—21no—
232#340—1061no—
233#333—1062no—
234C53#922311yescharacteristic, complemented, Sylow
235C22×Q8—21no—
236C22×D4—531no—
237C23×C4—1061no—
238C2×M16—1064no—
239C2×M16—2122no—
240C2×M16—2122no—
241C22×C8—1061no—
242C22×C8—1061no—
243C22×C8—1061no—
244C22×C8C22⋊D5311yescharacteristic
245C22×C8C22⋊D5311yescharacteristic
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8C4×D5312yes—
250C4×D4—532no—
251C4×D4—532no—
252C4×D4—1061no—
253C4×D4—1061no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—531no—
260C2×C22⋊C4—1061no—
261C2×C42C22×D5311yescharacteristic
262#106—1064no—
263C4⋊C8—532no—
264C4⋊C8—532no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267C4⋊C8—1061no—
268C4⋊C8—1061no—
269#102—1064no—
270#102—2122no—
271C22⋊C8—1064no—
272C4×C8—21no—
273C4×C8—21no—
274C24—1061no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×D4—532no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—1061no—
288C22×C4—1062no—
289C22×C4—1062no—
290C22×C4—2121no—
291C22×C4—2121no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4D4×D5311yescharacteristic
295C22×C4#1171994711yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297M16—2128no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—1062no—
304C2×C8—1062no—
305C2×C8—2121no—
306C2×C8—2121no—
307C2×C8—2121no—
308C2×C8—2121no—
309C2×C8—2121no—
310C2×C8—2121no—
311C2×C8—2121no—
312C2×C8#1171992412yes—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—81no—
322C4⋊C4—532no—
323C4⋊C4—1061no—
324C22⋊C4—1062no—
325C22⋊C4—1062no—
326C42C2×C4×D5311yescharacteristic
327C42#1171994511yescharacteristic
328C42D4×D5311yescharacteristic
329C42D4×D5311yescharacteristic
330C23—1062no—
331C23—2121no—
332C23—2121no—
333C23—2121no—
334C23—2121no—
335C23—2121no—
336C23—2121no—
337C23#1313318811yescharacteristic
338Q8—82no—
339Q8—82no—
340D4—1062no—
341D4—1062no—
342D4—2121no—
343D4—2121no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—1062no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4C22⋊C4×D5311yescharacteristic
360C2×C4#1313317711yescharacteristic
361C2×C4C4×C22⋊D5311yescharacteristic
362C2×C4#1313318511yescharacteristic
363C2×C4#1313319011yescharacteristic
364C2×C4#1313319911yescharacteristic
365C2×C4C2×#1171992411yescharacteristic
366C2×C4#1313323711yescharacteristic
367C2×C4#1313324211yescharacteristic
368C2×C4#1313324411yescharacteristic
369C2×C4#1313324611yescharacteristic
370C2×C4#1313324811yescharacteristic
371C2×C4#1313324911yescharacteristic
372C2×C4#1313324911yescharacteristic
373C8—22no—
374C8—41no—
375C8—4241no—
376C8—2122no—
377C8—2122no—
378C22—4241no—
379C22—4241no—
380C22—4241no—
381C22—4241no—
382C22—2122no—
383C22—2122no—
384C22—2122no—
385C22#1681049911yescharacteristic
386C22#1681082511yescharacteristic
387C22#1681083511yescharacteristic
388C22#1681086611yescharacteristic
389C22#1681087811yescharacteristic
390C22#1681088311yescharacteristic
391C22#1681089011yescharacteristic
392C4—21no—
393C4—21no—
394C4—82no—
395C4—161no—
396C4—2122no—
397C4#1681062311yescharacteristic
398C4#1681063011yescharacteristic
399C4C4×#1171992411yescharacteristic
400C4#1681072111yescharacteristic
401C4#1681077311yescharacteristic
402C4#1681078311yescharacteristic
403C4#1681087211yescharacteristic
404C4#1681088011yescharacteristic
405C2—4242no—
406C2#2027421111yescharacteristic
407C2#2027423511yescharacteristic
408C2#2027424511yescharacteristic
409C2#2027428711yescharacteristic
410C2#2027505211yescharacteristic
411C2#2027514511yescharacteristic
412C2#2027522611yescharacteristic
subgroupG/H
centerC23#13133188
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1990C2
socleC23×C53#121
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1990D53
hypocenterC53#9223
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
4. C2×#16810372C2semidirect
5. #20272598C2semidirect
6. #20272550C2semidirect
7. C53×#1990C2semidirect
11. #16810910C4semidirect
201. #1990D53semidirect
234. C53#9223semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
C2×#168107022
#202747122
C2×#168103722
#202725982
#202725502
C53×#19902
#922353