Finite Group Databasebeta

← #27286643

Subgroups — #27286643

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810997C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic
3#20275652C211yesmaximal, characteristic, complemented
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#475C211yesmaximal, characteristic, complemented
8C22×#11719924—22no—
9C22×#11719924—21no—
10C2×#13133214C2211yescharacteristic
11#16810997—22no—
12#16810997—22no—
13#16810823—22no—
14#16810823—22no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133137—21no—
18C2×C4×#88322C2211yescharacteristic
19#16810378—22no—
20#16810369—22no—
21#16810366—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#475—22no—
25C53×#475—22no—
26C2×#106×C53—21no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×#103×C53—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719924—24no—
33C2×#11719924—28no—
34C2×#11719924—42no—
35C2×#11719924—44no—
36C2×#11719924—44no—
37C2×#11719924—42no—
38C22×#88322—21no—
39C22×#88322—21no—
40C22×#88322D412yes—
41#13133214—21no—
42#13133214—21no—
43#13133214—21no—
44#13133214D412yes—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322D412yes—
49#13133137—24no—
50C4×#88322—21no—
51C4×#88322—21no—
52C22×Q8×C53—21no—
53C22×Q8×C53D411yescharacteristic
54C22×C8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53D412yes—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69#106×C53—24no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53D412yes—
72#103×C53—24no—
73C23×D53—41no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—22no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—41no—
81C2×Q8×C53—42no—
82C2×Q8×C53—42no—
83C2×Q8×C53SD1612yescomplemented
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—41no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C2×D212—22no—
91C2×D212—22no—
92C2×D212—22no—
93C2×D212—41no—
94C2×D212—42no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—42no—
98C2×D212—41no—
99C2×D212—41no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—82no—
103C2×C8×C53—22no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C42×C53—41no—
115C42×C53—41no—
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120#11719924—416no—
121#11719924—48no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—42no—
129C2×#88322SD1614yescomplemented
130C23×C53#12111yescharacteristic
131C22×D53—42no—
132C22×D53—82no—
133C22×D53—81no—
134C22×D53—81no—
135C22×D53—82no—
136Q8×C53—22no—
137Q8×C53—22no—
138Q8×C53—41no—
139Q8×C53—41no—
140Q8×C53—44no—
141Q8×C53—81no—
142Q8×C53—81no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—82no—
151C2×C4×C53#12011yescharacteristic
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12711yescharacteristic
157C2×C4×C53C2×SD1612yes—
158C2×C4×C53C2×SD1611yescharacteristic
159C2×C4×C53#13511yescharacteristic
160C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C53AGL(1,ℤ/8)12yes—
162C2×C4×C53#13711yescharacteristic
163D212—44no—
164D212—42no—
165D212—42no—
166D212—82no—
167D212—82no—
168D212—81no—
169D212—81no—
170D212—81no—
171D212—81no—
172C8×C53—42no—
173#88322—24no—
174#88322—42no—
175#88322—81no—
176#10268—531nomaximal, Sylow
177C22×C53#39211yescharacteristic
178C22×C53#46012yes—
179C22×C53#46511yescharacteristic
180C22×C53#46811yescharacteristic
181C22×C53#47112yes—
182C2×D53—82no—
183C2×D53—82no—
184C2×D53—82no—
185C2×D53—162no—
186C2×D53—162no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—42no—
190C4×C53—82no—
191C4×C53—81no—
192C4×C53#44811yescharacteristic
193C4×C53#44911yescharacteristic
194C4×C53#45012yes—
195C4×C53#45011yescharacteristic
196C4×C53#45211yescharacteristic
197C4×C53#49211yescharacteristic
198C4×C53#49611yescharacteristic
199C2×#492—531no—
200C2×#475D5311yescharacteristic, complemented
201C2×#460—531no—
202#1984—531no—
203C2×#332—531no—
204C2×#331—531no—
205#1277—531no—
206C2×C53#166612yes—
207C2×C53#168612yes—
208C2×C53#200011yescharacteristic
209C2×C53#201411yescharacteristic
210C2×C53#201911yescharacteristic
211D53—162no—
212C22×SD16—1062no—
213C22×SD16—1061no—
214C2×C4⋊Q8C2×D5311yescharacteristic
215C2×#127—531no—
216C2×C4×Q8C2×D5311yescharacteristic
217#492—1062no—
218#492—1062no—
219#475—22no—
220#475—22no—
221#460—1062no—
222#460—1062no—
223C2×#106—21no—
224C2×C4⋊C8—531no—
225C2×C4⋊C8—531no—
226C2×C4⋊C8C2×D5311yescharacteristic
227C2×#103—21no—
228C2×#102—1061no—
229C2×C4×C8—531no—
230#336—1062no—
231#332—1062no—
232#331—1062no—
233C53#1026811yescharacteristic, complemented, Sylow
234C22×Q8—21no—
235C22×Q8C22⋊D5311yescharacteristic
236C22×D4—1061no—
237C22×D4—1061no—
238C22×D4—1061no—
239C2×SD16—1064no—
240C2×SD16—1068no—
241C2×SD16—2124no—
242C2×SD16—2124no—
243C2×SD16—2122no—
244C2×SD16—2122no—
245C22×C8—21no—
246C22×C8—532no—
247C22×C8—1061no—
248C22×C8—1061no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8C22⋊D5312yes—
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—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C22⋊D5311yescharacteristic
265C2×C42—21no—
266C2×C42C22×D5311yescharacteristic
267#106—24no—
268C4⋊C8—21no—
269C4⋊C8—532no—
270C4⋊C8—1061no—
271C4⋊C8—1061no—
272C4⋊C8—1061no—
273C4⋊C8C22⋊D5312yes—
274#103—24no—
275#102—1064no—
276C4×C8—1061no—
277C4×C8—1061no—
278C24—2121no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8#1171992412yescomplemented
289C2×D4—1062no—
290C2×D4—1062no—
291C2×D4—1062no—
292C2×D4—4242no—
293C2×D4—2121no—
294C2×D4—2121no—
295C2×D4—2121no—
296C2×D4—2122no—
297C2×D4—2121no—
298C2×D4—2121no—
299C2×D4—2121no—
300C2×D4—2122no—
301C2×D4—2121no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4D4×D5311yescharacteristic
306C22×C4#1171994711yescharacteristic
307C22×C4C2×C22⋊D5311yescharacteristic
308SD16—21216no—
309SD16—2128no—
310C2×C8—22no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—534no—
314C2×C8—1062no—
315C2×C8—1062no—
316C2×C8—1062no—
317C2×C8—2121no—
318C2×C8—2122no—
319C2×C8—2121no—
320C2×C8—2121no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C42—41no—
330C42—41no—
331C42D4×D5311yescharacteristic
332C42D4×D5311yescharacteristic
333C42C2×C22⋊D5311yescharacteristic
334C42C2×C22⋊D5311yescharacteristic
335C23—4242no—
336C23—4241no—
337C23—4241no—
338C23—4242no—
339C23—2122no—
340C23#1313318911yescharacteristic
341Q8—22no—
342Q8—22no—
343Q8—41no—
344Q8—41no—
345Q8—44no—
346Q8—81no—
347Q8—81no—
348D4—4242no—
349D4—4242no—
350D4—4241no—
351D4—4241no—
352D4—4241no—
353D4—4241no—
354D4—2124no—
355D4—2122no—
356D4—2122no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4#1313318511yescharacteristic
366C2×C4#1313318511yescharacteristic
367C2×C4#1313318611yescharacteristic
368C2×C4#1313318811yescharacteristic
369C2×C4#1313318911yescharacteristic
370C2×C4#1313321311yescharacteristic
371C2×C4SD16×D5312yes—
372C2×C4C2×#1171992411yescharacteristic
373C2×C4#1313323711yescharacteristic
374C2×C4#1313324211yescharacteristic
375C2×C4#1313324412yes—
376C2×C4#1313324811yescharacteristic
377C8—42no—
378C8—1064no—
379C8—4241no—
380C8—2122no—
381C22—8482no—
382C22—8482no—
383C22—4242no—
384C22—4242no—
385C22—4242no—
386C22#1681053111yescharacteristic
387C22#1681082112yes—
388C22#1681085411yescharacteristic
389C22#1681087111yescharacteristic
390C22#1681088812yes—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—82no—
395C4—81no—
396C4#1681076211yescharacteristic
397C4#1681076811yescharacteristic
398C4#1681077212yes—
399C4#1681077311yescharacteristic
400C4#1681078311yescharacteristic
401C4#1681099411yescharacteristic
402C4#1681100911yescharacteristic
403C2—8482no—
404C2#2027406512yes—
405C2#2027418912yes—
406C2#2027571611yescharacteristic
407C2#2027580111yescharacteristic
408C2#2027582611yescharacteristic
subgroupG/H
centerC23#13133189
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#475C2
socleC23×C53#121
prime radicalC2×#13133214C22
hypercenterC2×#475D53
hypocenterC53#10268
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. #20275652C2semidirect
4. C2×#16810369C2semidirect
6. #20272549C2semidirect
7. C2×C53×#475C2semidirect
83. C2×Q8×C53SD16semidirect
129. C2×#88322SD16semidirect
200. C2×#475D53semidirect
233. C53#10268semidirect
288. C2×Q8#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109972
C2×#168108232
#202756522
C2×#168103692
C2×#168103662
#202725492
C2×C53×#4752
#1026853