Finite Group Databasebeta

← #27285171

Subgroups — #27285171

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810996C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic
3#20279920C211yesmaximal, characteristic, complemented
4#20275641C211yesmaximal, characteristic
5C2×#16810371C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#11719913—22no—
9C22×#11719924—21no—
10C2×#13133217C2211yescharacteristic
11C2×#13133214C2211yescharacteristic
12#16810996—22no—
13#16810996—22no—
14#16810823—22no—
15#16810823—22no—
16C2×C4⋊#88322C2211yescharacteristic
17C2×#13133139—22no—
18C2×#13133137—21no—
19#16810543—22no—
20#16810530—21no—
21#16810521—22no—
22#16810492—21no—
23C4×C4⋊Dic53C2211yescharacteristic
24#16810371—22no—
25C2×C4×Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4×C8×C53C2211yescharacteristic
28C53×Q8⋊C8—22no—
29C22×Dic106D412yes—
30C22×D212—21no—
31C22×D212—21no—
32C22×D212—21no—
33C2×#11719913—28no—
34C2×#11719913—48no—
35C2×#11719924—24no—
36C2×#11719924—42no—
37C2×#11719924—42no—
38C22×#88322—21no—
39#13133217—21no—
40#13133217—21no—
41#13133217—21no—
42#13133217—21no—
43#13133214—21no—
44#13133214—21no—
45#13133214—21no—
46#13133214D412yes—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53D412yes—
50C2×#11719919—24no—
51C2×C4×Dic53—21no—
52C4⋊#88322—21no—
53C4⋊#88322D412yes—
54#13133139—44no—
55#13133137—24no—
56#13133124—24no—
57#13133124—21no—
58C22×Q8×C53D411yescharacteristic
59C22×C8×C53—21no—
60C22×C8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4×C8×C53—21no—
72C4×C8×C53D412yes—
73C23×D53—41no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53SD1612yescomplemented
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—22no—
86C22×Dic53—22no—
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—22no—
90C2×D212—44no—
91C2×D212—42no—
92C2×D212—41no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—41no—
96C2×D212—42no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—41no—
100C2×Dic106—22no—
101C2×Dic106—22no—
102C2×Dic106—44no—
103C2×C8×C53—24no—
104C2×C8×C53—22no—
105C2×C8×C53—41no—
106C2×C8×C53—42no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C42×C53—41no—
112C42×C53—41no—
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117#11719924—48no—
118C4⋊Dic53—22no—
119C4⋊Dic53—41no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122C4⋊Dic53—41no—
123C4⋊Dic53SD1614yescomplemented
124#11719919—84no—
125C4×Dic53—41no—
126C4×Dic53—41no—
127C2×#88322—22no—
128C2×#88322—41no—
129C2×#88322—41no—
130#11719913—416no—
131C23×C53#12011yescharacteristic
132C22×D53—42no—
133C22×D53—82no—
134C22×D53—81no—
135C22×D53—81no—
136C22×D53—82no—
137Q8×C53—22no—
138Q8×C53—22no—
139Q8×C53—41no—
140Q8×C53—41no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53#12011yescharacteristic
148C2×C4×C53#12011yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12711yescharacteristic
153C2×C4×C53C2×SD1612yes—
154C2×C4×C53C2×SD1611yescharacteristic
155C2×C4×C53#13511yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C53#13712yes—
158C2×C4×C53#13711yescharacteristic
159C2×Dic53—24no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53—44no—
164D212—44no—
165D212—42no—
166D212—42no—
167D212—44no—
168D212—82no—
169D212—81no—
170D212—81no—
171D212—81no—
172D212—81no—
173Dic106—44no—
174Dic106—82no—
175C8×C53—24no—
176C8×C53—81no—
177#88322—42no—
178#10049—531nomaximal, Sylow
179C22×C53#39211yescharacteristic
180C22×C53#44812yes—
181C22×C53#44811yescharacteristic
182C22×C53#44911yescharacteristic
183C22×C53#45312yes—
184C2×D53—82no—
185C2×D53—82no—
186C2×D53—82no—
187C2×D53—162no—
188C2×D53—162no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—42no—
192C4×C53—81no—
193C4×C53#45011yescharacteristic
194C4×C53#45211yescharacteristic
195C4×C53#46112yes—
196C4×C53#46511yescharacteristic
197C4×C53#46811yescharacteristic
198C4×C53#49211yescharacteristic
199C4×C53#49711yescharacteristic
200Dic53—44no—
201Dic53—82no—
202C2×#492—531no—
203C2×#460—531no—
204#2567—531no—
205#1982—531no—
206C2×#333—531no—
207C2×Q8⋊C8D5311yescharacteristic, complemented
208#1279—531no—
209C2×C53#162212yes—
210C2×C53#162312yes—
211C2×C53#200011yescharacteristic
212C2×C53#200211yescharacteristic
213C2×C53#201311yescharacteristic
214D53—162no—
215C22×SD16—1062no—
216C22×SD16—1061no—
217C2×C4⋊Q8—531no—
218C2×#127—531no—
219C2×C4×Q8C2×D5311yescharacteristic
220#492—1062no—
221#492—1062no—
222#460—1062no—
223#460—1062no—
224C2×C4⋊C8—531no—
225C2×C4⋊C8C2×D5311yescharacteristic
226C2×#102—1061no—
227C2×#102—1062no—
228C2×C4×C8C2×D5311yescharacteristic
229#395—1062no—
230#391—1061no—
231#389—1062no—
232#384—1061no—
233C4×C4⋊C4—531no—
234#333—1062no—
235Q8⋊C8—22no—
236C53#1004911yescharacteristic, complemented, Sylow
237C22×Q8—532no—
238C22×Q8D21211yescharacteristic
239C22×D4—1061no—
240C22×D4—1061no—
241C22×D4—1061no—
242C2×SD16—1068no—
243C2×SD16—1064no—
244C2×SD16—2128no—
245C2×SD16—2122no—
246C2×SD16—2122no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—1061no—
250C4⋊Q8—1061no—
251C4⋊Q8—1061no—
252C4⋊Q8—1061no—
253C4⋊Q8—1061no—
254#127—532no—
255#127—1061no—
256#127—1061no—
257#127—1061no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—532no—
264C2×C4⋊C4—1064no—
265C2×C4⋊C4—1061no—
266C2×C4⋊C4—1061no—
267C2×C4⋊C4D21211yescharacteristic
268C2×C42—21no—
269C2×C42—1061no—
270C2×C42C22×D5311yescharacteristic
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—532no—
274C4⋊C8—1061no—
275#102—1064no—
276#102—2124no—
277C4×C8—21no—
278C4×C8C22⋊D5312yes—
279#95—1064no—
280#95—1061no—
281C24—2121no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—1062no—
288C2×Q8—1062no—
289C2×Q8—2124no—
290C2×Q8#1171991312yescomplemented
291C2×D4—1062no—
292C2×D4—1062no—
293C2×D4—1062no—
294C2×D4—2124no—
295C2×D4—2121no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4—2122no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2122no—
303C2×D4—2121no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—1062no—
307C22×C4—1062no—
308C22×C4C2×D21211yescharacteristic
309C22×C4D4×D5311yescharacteristic
310C22×C4D4×D5311yescharacteristic
311SD16—21216no—
312SD16—2128no—
313C2×C8—24no—
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—1062no—
318C2×C8—2121no—
319C2×C8—2121no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—534no—
325C4⋊C4—1062no—
326C4⋊C4—4244no—
327C4⋊C4—2121no—
328C4⋊C4—2121no—
329C4⋊C4—2121no—
330C4⋊C4—2121no—
331C42—41no—
332C42—41no—
333C42—2121no—
334C42—2121no—
335C42#1171994511yescharacteristic
336C42D4×D5311yescharacteristic
337C42D4×D5311yescharacteristic
338C42C2×C22⋊D5311yescharacteristic
339C23—4242no—
340C23—4241no—
341C23—4241no—
342C23—4242no—
343C23—2122no—
344C23#1313318711yescharacteristic
345Q8—22no—
346Q8—22no—
347Q8—41no—
348Q8—41no—
349Q8—4242no—
350Q8—2124no—
351D4—4242no—
352D4—4241no—
353D4—4241no—
354D4—4241no—
355D4—4241no—
356D4—2124no—
357D4—2122no—
358D4—2122no—
359D4—2124no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—1064no—
367C2×C4—2122no—
368C2×C4—2124no—
369C2×C4—2122no—
370C2×C4—2122no—
371C2×C4#1313318511yescharacteristic
372C2×C4#1313318711yescharacteristic
373C2×C4#1313319011yescharacteristic
374C2×C4#1313319011yescharacteristic
375C2×C4#1313319111yescharacteristic
376C2×C4#1313321311yescharacteristic
377C2×C4SD16×D5312yes—
378C2×C4C2×#1171991311yescharacteristic
379C2×C4#1313323911yescharacteristic
380C2×C4#1313324511yescharacteristic
381C2×C4#1313324912yes—
382C2×C4#1313325011yescharacteristic
383C8—24no—
384C8—81no—
385C8—2122no—
386C22—8482no—
387C22—8482no—
388C22—4242no—
389C22—4242no—
390C22—4242no—
391C22#1681053211yescharacteristic
392C22#1681076312yes—
393C22#1681076411yescharacteristic
394C22#1681077011yescharacteristic
395C22#1681079212yes—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—81no—
400C4—4242no—
401C4—2124no—
402C4#1681077611yescharacteristic
403C4#1681078711yescharacteristic
404C4#1681082712yes—
405C4#1681085211yescharacteristic
406C4#1681086911yescharacteristic
407C4#1681099411yescharacteristic
408C4#1681101611yescharacteristic
409C2—8482no—
410C2#2027379012yes—
411C2#2027379512yes—
412C2#2027572011yescharacteristic
413C2#2027572911yescharacteristic
414C2#2027579911yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenterC2×Q8⋊C8D53
hypocenterC53#10049
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #20279920C2semidirect
5. C2×#16810371C2semidirect
6. #20272555C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
79. C2×Q8×C53SD16semidirect
123. C4⋊Dic53SD16semidirect
207. C2×Q8⋊C8D53semidirect
236. C53#10049semidirect
290. C2×Q8#11719913semidirect

Maximal subgroups

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

subgroupindex
C2×#168109962
C2×#168108232
#202799202
#202756412
C2×#168103712
#202725552
C2×C53×Q8⋊C82
#1004953