Finite Group Databasebeta

← #27285169

Subgroups — #27285169

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×#16810997C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic
3#20275642C211yesmaximal, characteristic
4C2×#16810372C211yesmaximal, characteristic, complemented
5C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#2567C211yesmaximal, 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×#13133137—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810372—22no—
21Q8⋊#88322—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#395—22no—
25C53×#391—21no—
26C53×#389—22no—
27C53×#384—21no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719924—24no—
33C2×#11719924—28no—
34C2×#11719924—42no—
35C2×#11719924—48no—
36C2×#11719924—42no—
37C22×#88322—21no—
38C22×#88322—21no—
39C22×#88322—21no—
40#13133214—21no—
41#13133214—21no—
42#13133214—21no—
43#13133214D412yes—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322D412yes—
48#13133137—24no—
49#13133137—44no—
50C4×#88322—21no—
51C4×#88322D412yes—
52C22×Q8×C53D411yescharacteristic
53C22×Q8×C53D412yes—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4×Q8×C53—21no—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61C4×Q8×C53—21no—
62C2×C4⋊C4×C53—24no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C4⋊C4×C53D412yes—
68C2×C42×C53—21no—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71#95×C53—24no—
72#95×C53—21no—
73C23×D53—41no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—22no—
77C2×Q8×C53—21no—
78C2×Q8×C53—22no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—44no—
82C2×Q8×C53SD1612yescomplemented
83C22×C4×C53—21no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—21no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C2×D212—22no—
91C2×D212—22no—
92C2×D212—22no—
93C2×D212—44no—
94C2×D212—42no—
95C2×D212—41no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—42no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—84no—
113C4⋊C4×C53SD1614yescomplemented
114C42×C53—41no—
115C42×C53—41no—
116C42×C53—41no—
117C42×C53—41no—
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53#4111yescharacteristic
122#11719924—416no—
123#11719924—48no—
124C2×#88322—24no—
125C2×#88322—22no—
126C2×#88322—22no—
127C2×#88322—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—42no—
131C23×C53#12011yescharacteristic
132C22×D53—42no—
133C22×D53—81no—
134C22×D53—82no—
135C22×D53—81no—
136C22×D53—82no—
137Q8×C53—22no—
138Q8×C53—22no—
139Q8×C53—44no—
140Q8×C53—41no—
141Q8×C53—41no—
142Q8×C53—82no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—24no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—41no—
152C2×C4×C53—44no—
153C2×C4×C53—42no—
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12011yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12711yescharacteristic
160C2×C4×C53C2×SD1611yescharacteristic
161C2×C4×C53C2×SD1612yes—
162C2×C4×C53#13511yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53#13711yescharacteristic
165C2×C4×C53#13712yes—
166D212—44no—
167D212—42no—
168D212—42no—
169D212—44no—
170D212—82no—
171D212—81no—
172D212—81no—
173D212—81no—
174D212—81no—
175#88322—24no—
176#88322—42no—
177#88322—81no—
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—44no—
192C4×C53—42no—
193C4×C53—81no—
194C4×C53—82no—
195C4×C53#45011yescharacteristic
196C4×C53#45211yescharacteristic
197C4×C53#46112yes—
198C4×C53#46511yescharacteristic
199C4×C53#46811yescharacteristic
200C4×C53#49211yescharacteristic
201C4×C53#49711yescharacteristic
202C2×#492—531no—
203C2×#460—531no—
204#2567D5311yescharacteristic, complemented
205#1982—531no—
206C2×#333—531no—
207C2×Q8⋊C8—531no—
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⋊Q8C2×D5311yescharacteristic
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⋊C8—531no—
226C2×#102—1061no—
227C2×#102—1062no—
228C2×C4×C8—531no—
229#395—22no—
230#391—21no—
231#389—22no—
232#384—21no—
233C4×C4⋊C4C2×D5311yescharacteristic
234#333—1062no—
235Q8⋊C8—1062no—
236C53#1004911yescharacteristic, complemented, Sylow
237C22×Q8C22⋊D5311yescharacteristic
238C22×Q8C22⋊D5312yes—
239C22×D4—1061no—
240C22×D4—1061no—
241C22×D4—1061no—
242C2×SD16—1064no—
243C2×SD16—1068no—
244C2×SD16—2128no—
245C2×SD16—2122no—
246C2×SD16—2122no—
247C22×C8—1061no—
248C22×C8—1061no—
249C22×C8—1061no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
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—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4C22⋊D5311yescharacteristic
267C2×C4⋊C4C22⋊D5312yes—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42C22×D5311yescharacteristic
271C4⋊C8—532no—
272C4⋊C8—1061no—
273C4⋊C8—1061no—
274C4⋊C8—1061no—
275#102—1064no—
276#102—2124no—
277C4×C8—532no—
278C4×C8—1061no—
279#95—24no—
280#95—21no—
281C24—2121no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—21no—
286C2×Q8—22no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—44no—
290C2×Q8#1171992412yescomplemented
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—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4C2×C22⋊D5311yescharacteristic
309C22×C4C2×C22⋊D5311yescharacteristic
310C22×C4C2×C22⋊D5311yescharacteristic
311SD16—21216no—
312SD16—2128no—
313C2×C8—1064no—
314C2×C8—1062no—
315C2×C8—1062no—
316C2×C8—2121no—
317C2×C8—2122no—
318C2×C8—2121no—
319C2×C8—2121no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—84no—
330C4⋊C4#1171992414yescomplemented
331C42—41no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42D4×D5311yescharacteristic
336C42D4×D5311yescharacteristic
337C42D4×D5311yescharacteristic
338C42#1171994711yescharacteristic
339C23—4242no—
340C23—4241no—
341C23—4241no—
342C23—4242no—
343C23—2122no—
344C23#1313318611yescharacteristic
345Q8—22no—
346Q8—22no—
347Q8—44no—
348Q8—41no—
349Q8—41no—
350Q8—82no—
351D4—4242no—
352D4—4241no—
353D4—4241no—
354D4—4241no—
355D4—4241no—
356D4—2124no—
357D4—2122no—
358D4—2122no—
359D4—2124no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4#1313318511yescharacteristic
372C2×C4#1313318511yescharacteristic
373C2×C4#1313318811yescharacteristic
374C2×C4#1313318911yescharacteristic
375C2×C4#1313318911yescharacteristic
376C2×C4#1313321311yescharacteristic
377C2×C4C2×#1171992411yescharacteristic
378C2×C4C2×#1171992412yes—
379C2×C4#1313323711yescharacteristic
380C2×C4#1313324211yescharacteristic
381C2×C4#1313324812yes—
382C2×C4#1313324811yescharacteristic
383C8—1064no—
384C8—4241no—
385C8—2122no—
386C22—8482no—
387C22—8482no—
388C22—4242no—
389C22—4242no—
390C22—4242no—
391C22#1681053111yescharacteristic
392C22#1681076211yescharacteristic
393C22#1681076212yes—
394C22#1681076811yescharacteristic
395C22#1681079112yes—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—44no—
400C4—81no—
401C4—82no—
402C4#1681077311yescharacteristic
403C4#1681078311yescharacteristic
404C4#1681082512yes—
405C4#1681085411yescharacteristic
406C4#1681087111yescharacteristic
407C4#1681099711yescharacteristic
408C4#1681101511yescharacteristic
409C2—8482no—
410C2#2027378712yes—
411C2#2027379312yes—
412C2#2027571611yescharacteristic
413C2#2027572811yescharacteristic
414C2#2027579811yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2567C2
socleC23×C53#120
prime radicalC2×#13133214C22
hypercenter#2567D53
hypocenterC53#10049
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2567

Complements

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

NH ≅ G/Ntype
4. C2×#16810372C2semidirect
5. C2×Q8⋊#88322C2semidirect
6. #20272556C2semidirect
7. C53×#2567C2semidirect
82. C2×Q8×C53SD16semidirect
113. C4⋊C4×C53SD16semidirect
204. #2567D53semidirect
236. C53#10049semidirect
290. C2×Q8#11719924semidirect
330. C4⋊C4#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
#202756422
C2×#168103722
C2×Q8⋊#883222
#202725562
C53×#25672
#1004953