Finite Group Databasebeta

← #27285299

Subgroups — #27285299

Order 13568.

670 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×#16810825C211yesmaximal, characteristic
2#20279669C211yesmaximal, characteristic
3#20275668C211yesmaximal, characteristic, complemented
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×#498C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×#13133193—22no—
10C2×C4×D212C2211yescharacteristic
11#16810825—22no—
12#16810825—22no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133141—21no—
16#16810558—22no—
17#16810524—21no—
18#16810492—21no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810385—22no—
21#16810369—22no—
22#16810352—22no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4⋊Q8×C53C2211yescharacteristic
25C53×#498—22no—
26C53×#498—22no—
27C2×#106×C53—22no—
28C2×C4×C8×C53C2211yescharacteristic
29C22×D212D411yescharacteristic
30C22×C4×D53—21no—
31C2×#11719924—24no—
32C2×#11719924—42no—
33C2×#11719924—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133193—44no—
37#13133193—44no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212—21no—
42C2×C4⋊Dic53D411yescharacteristic
43C2×C4⋊Dic53D412yes—
44C2×#11719919—24no—
45C2×#11719921—24no—
46C2×#11719921—21no—
47C2×C4×Dic53—21no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322D412yes—
52#13133141—24no—
53#13133124—21no—
54C22×Q8×C53—21no—
55C22×Q8×C53—21no—
56C22×C8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53D412yes—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53D412yes—
69C2×C42×C53C2311yescharacteristic
70#106×C53—28no—
71C4×C8×C53—21no—
72C4×C8×C53D412yes—
73C23×D53—21no—
74C2×Q8×C53—22no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—42no—
79C2×Q8×C53—42no—
80C2×Q8×C53—41no—
81C2×Q8×C53—41no—
82C2×Q8×C53—41no—
83C2×Q8×C53—41no—
84C22×C4×C53—22no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×Dic53—22no—
90C22×Dic53—21no—
91C2×D212—21no—
92C2×D212—21no—
93C2×D212—21no—
94C2×D212—21no—
95C2×D212—21no—
96C2×D212SD1612yes—
97C2×C4×D53—41no—
98C2×C4×D53—41no—
99C2×C4×D53—42no—
100C2×C4×D53—42no—
101C2×C8×C53—24no—
102C2×C8×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53#4111yescharacteristic
115#11719924—48no—
116#11719921—48no—
117#11719921—41no—
118#11719921—41no—
119C4⋊Dic53—22no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122C4⋊Dic53SD1614yescomplemented
123#11719919—48no—
124C4×Dic53—41no—
125C4×Dic53—41no—
126C2×#88322—22no—
127C2×#88322—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—42no—
131C23×C53#12011yescharacteristic
132C22×D53—22no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—42no—
136C22×D53—42no—
137Q8×C53—42no—
138Q8×C53—42no—
139Q8×C53—82no—
140Q8×C53—81no—
141Q8×C53—81no—
142Q8×C53—81no—
143Q8×C53—81no—
144C2×C4×C53—22no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—44no—
148C2×C4×C53—81no—
149C2×C4×C53—81no—
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12711yescharacteristic
156C2×C4×C53C2×SD1611yescharacteristic
157C2×C4×C53C2×SD1612yes—
158C2×C4×C53#13511yescharacteristic
159C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C53AGL(1,ℤ/8)12yes—
161C2×C4×C53#13711yescharacteristic
162C2×Dic53—24no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165C2×Dic53—41no—
166C2×Dic53—42no—
167D212—22no—
168D212—22no—
169D212—41no—
170D212—41no—
171C4×D53—82no—
172C4×D53—81no—
173C4×D53—81no—
174C8×C53—24no—
175#88322—42no—
176#88322—81no—
177#10068—531nomaximal, Sylow
178C22×C53#39211yescharacteristic
179C22×C53#44811yescharacteristic
180C22×C53#44911yescharacteristic
181C22×C53#45012yes—
182C22×C53#45612yes—
183C2×D53—44no—
184C2×D53—42no—
185C2×D53—82no—
186C2×D53—81no—
187C2×D53—81no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—82no—
191C4×C53—82no—
192C4×C53#45011yescharacteristic
193C4×C53#45211yescharacteristic
194C4×C53#46012yes—
195C4×C53#46511yescharacteristic
196C4×C53#46811yescharacteristic
197C4×C53#49211yescharacteristic
198C4×C53#49611yescharacteristic
199Dic53—44no—
200Dic53—81no—
201C2×#498D5311yescharacteristic, complemented
202C2×#461—531no—
203#2542—531no—
204#1988—531no—
205C2×#332—531no—
206C2×D4⋊C8—531no—
207#1279—531no—
208C2×C53#165112yes—
209C2×C53#165412yes—
210C2×C53#200011yescharacteristic
211C2×C53#200511yescharacteristic
212C2×C53#201411yescharacteristic
213D53—82no—
214C22×SD16—1061no—
215C2×C4⋊Q8C2×D5311yescharacteristic
216C2×C4⋊Q8C2×D5311yescharacteristic
217C2×C22⋊Q8—1062no—
218C2×C4×D4—531no—
219#498—22no—
220#498—22no—
221#461—1062no—
222#461—1062no—
223C2×#106—22no—
224C2×C4⋊C8—531no—
225C2×C4⋊C8—531no—
226C2×#103—1061no—
227C2×C4×C8C2×D5311yescharacteristic
228#399—1062no—
229#390—1061no—
230#384—1061no—
231C4×C4⋊C4—531no—
232#340—1062no—
233#332—1062no—
234D4⋊C8—1062no—
235C53#1006811yescharacteristic, complemented, Sylow
236C22×Q8—21no—
237C22×Q8—21no—
238C22×D4—531no—
239C23×C4—1061no—
240C2×SD16—1064no—
241C2×SD16—2122no—
242C2×SD16—2122no—
243C22×C8—21no—
244C22×C8—1061no—
245C22×C8—1061no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8C22⋊D5312yes—
254C22⋊Q8—2124no—
255C22⋊Q8—2124no—
256C4×D4—1061no—
257C4×D4—1061no—
258C4×D4—1061no—
259C4×D4—1061no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—532no—
264C2×C4⋊C4—531no—
265C2×C4⋊C4—1064no—
266C2×C4⋊C4C22⋊D5312yes—
267C2×C22⋊C4—1064no—
268C2×C22⋊C4—1061no—
269C2×C42—1061no—
270C2×C42C22×D5311yescharacteristic
271#106—28no—
272C4⋊C8—532no—
273C4⋊C8—1061no—
274C4⋊C8—1061no—
275C4⋊C8—1061no—
276#103—1064no—
277C4×C8—21no—
278C4×C8C22⋊D5312yes—
279#95—1061no—
280C24—1061no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×D4—532no—
292C2×D4—1061no—
293C2×D4—1061no—
294C2×D4—1061no—
295C2×D4—1061no—
296C2×D4—1061no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—1062no—
300C22×C4—1061no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4—2122no—
304C22×C4—2122no—
305C22×C4D4×D5311yescharacteristic
306C22×C4D4×D5311yescharacteristic
307C22×C4C2×C22⋊D5311yescharacteristic
308SD16—2128no—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—1062no—
312C2×C8—2121no—
313C2×C8—2122no—
314C2×C8—2121no—
315C2×C8—2121no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—534no—
325C4⋊C4—1062no—
326C4⋊C4—1061no—
327C4⋊C4—1061no—
328C4⋊C4—2128no—
329C22⋊C4—2128no—
330C22⋊C4—2121no—
331C22⋊C4—2121no—
332C42—2121no—
333C42—2121no—
334C42D4×D5311yescharacteristic
335C42#1171994711yescharacteristic
336C42C2×C22⋊D5311yescharacteristic
337C42C2×C22⋊D5311yescharacteristic
338C23—1062no—
339C23—2121no—
340C23—2122no—
341C23—2121no—
342C23—2122no—
343C23#1313318511yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—81no—
347Q8—82no—
348Q8—81no—
349Q8—81no—
350Q8—81no—
351D4—1062no—
352D4—1062no—
353D4—2121no—
354D4—2121no—
355C2×C4—22no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—1064no—
362C2×C4—4241no—
363C2×C4—4242no—
364C2×C4—4241no—
365C2×C4—2122no—
366C2×C4—2121no—
367C2×C4—2122no—
368C2×C4—2122no—
369C2×C4#1313318511yescharacteristic
370C2×C4#1313318611yescharacteristic
371C2×C4#1313318811yescharacteristic
372C2×C4#1313318911yescharacteristic
373C2×C4#1313318911yescharacteristic
374C2×C4#1313321311yescharacteristic
375C2×C4SD16×D5311yescharacteristic
376C2×C4SD16×D5312yes—
377C2×C4#1313323811yescharacteristic
378C2×C4#1313324412yes—
379C2×C4#1313324411yescharacteristic
380C2×C4#1313324611yescharacteristic
381C8—24no—
382C8—4241no—
383C8—2122no—
384C22—4241no—
385C22—4242no—
386C22—4241no—
387C22—2122no—
388C22—2124no—
389C22#1681053111yescharacteristic
390C22#1681076011yescharacteristic
391C22#1681076711yescharacteristic
392C22#1681077212yes—
393C22#1681080512yes—
394C4—21no—
395C4—21no—
396C4—82no—
397C4—82no—
398C4—4241no—
399C4—2124no—
400C4#1681077211yescharacteristic
401C4#1681078211yescharacteristic
402C4#1681082112yes—
403C4#1681084911yescharacteristic
404C4#1681086611yescharacteristic
405C4#1681099411yescharacteristic
406C4#1681100911yescharacteristic
407C2—4242no—
408C2#2027397512yes—
409C2#2027399612yes—
410C2#2027571511yescharacteristic
411C2#2027575111yescharacteristic
412C2#2027580111yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#498C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenterC2×#498D53
hypocenterC53#10068
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
3. #20275668C2semidirect
4. C2×#16810369C2semidirect
6. #20272555C2semidirect
7. C2×C53×#498C2semidirect
122. C4⋊Dic53SD16semidirect
201. C2×#498D53semidirect
235. C53#10068semidirect

Maximal subgroups

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

subgroupindex
C2×#168108252
#202796692
#202756682
C2×#168103692
C2×#168103522
#202725552
C2×C53×#4982
#1006853