Finite Group Databasebeta

← #27367954

Subgroups — #27367954

Order 13568.

628 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285275C211yesmaximal, characteristic
2#20284837C211yesmaximal, characteristic
3C2×Q8×#88322C211yesmaximal, characteristic, complemented
4#20274171C212yesmaximal, complemented
5#20274161C212yesmaximal, complemented
6#20274095C212yesmaximal, complemented
7#20274065C212yesmaximal, complemented
8#20273476C211yesmaximal, characteristic, complemented
9#20273415C211yesmaximal, characteristic, complemented
10#20273411C211yesmaximal, characteristic
11C53×#3113C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16811019C2211yescharacteristic
14#16810997C2211yescharacteristic
15#16810973C2211yescharacteristic
16#16810971C2211yescharacteristic
17#16810936—21no—
18#16810918C2212yes—
19#16810854—21no—
20#16810854—21no—
21#16810823C2212yes—
22#16810762—21no—
23Q8×#88322—21no—
24Q8×#88322—21no—
25Q8×#88322C2212yes—
26Q8×#88322C2212yes—
27#16810721—22no—
28C4×#11719924—21no—
29C4×#11719924—21no—
30#16810680C2211yescharacteristic
31C2×C4⋊#88322—21no—
32C2×C4⋊#88322C2211yescharacteristic
33C2×C4×#88322—21no—
34C2×C4×#88322C2211yescharacteristic
35#16810375C2212yes—
36#16810373C2212yes—
37#16810371C2212yes—
38#16810369C2212yes—
39#16810369C2212yes—
40#16810366C2212yes—
41C53×#533C2211yescharacteristic
42C2×C4×Q8×C53C2211yescharacteristic
43C53×#483—21no—
44C53×#475C2212yes—
45C53×#462C2212yes—
46C53×#451—21no—
47C53×#441—22no—
48C53×#432C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719924—21no—
51C2×#11719924—21no—
52C2×#11719924—22no—
53C22×#88322—21no—
54C22×#88322D411yescharacteristic
55C22×#88322D411yescharacteristic
56#13133217C2311yescharacteristic
57#13133214C2311yescharacteristic
58#13133205—21no—
59#13133191—21no—
60#13133191—21no—
61C4×D212—21no—
62C4×D212—21no—
63#13133150—22no—
64#13133148—21no—
65#13133148—21no—
66C4⋊#88322—21no—
67C4⋊#88322—22no—
68C4⋊#88322—22no—
69C4⋊#88322C2312yes—
70C4⋊#88322C2312yes—
71#13133141—21no—
72#13133141—21no—
73#13133141—22no—
74#13133137—22no—
75#13133137—21no—
76#13133137—21no—
77C22⋊#88322—21no—
78#13133128—21no—
79C4×#88322—22no—
80C4×#88322—22no—
81C4×#88322D411yescharacteristic
82C4×#88322D411yescharacteristic
83C4×#88322C2311yescharacteristic
84C4×#88322C2311yescharacteristic
85C22×Q8×C53D411yescharacteristic
86C2×Q16×C53—22no—
87C4⋊Q8×C53C2311yescharacteristic
88C4⋊Q8×C53C2311yescharacteristic
89#125×C53—21no—
90C22⋊Q8×C53—21no—
91C22⋊Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×Q8×C53D412yes—
97C4×Q8×C53C2312yes—
98#117×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C42×C53—21no—
102C2×C42×C53C2311yescharacteristic
103#106×C53—22no—
104C4⋊C8×C53C2312yes—
105#103×C53—22no—
106#103×C53—22no—
107C22⋊C8×C53—21no—
108#97×C53—21no—
109C2×Q8×C53—24no—
110C2×Q8×C53—21no—
111C2×Q8×C53—21no—
112C2×Q8×C53—21no—
113C2×Q8×C53C2×D412yes—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D411yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C2×C22⋊D53—41no—
120#11719945—82no—
121C2×D212—21no—
122C2×D212—21no—
123C2×D212—21no—
124C2×C4×D53—41no—
125C2×Dic106—21no—
126Q16×C53—44no—
127C2×C8×C53—22no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53C2×D412yes—
138C22⋊C4×C53—41no—
139C42×C53—22no—
140C42×C53—22no—
141C42×C53—41no—
142C42×C53C2×D411yescharacteristic
143C42×C53C2×D411yescharacteristic
144C42×C53#4111yescharacteristic
145C42×C53C2411yescharacteristic
146#11719924—42no—
147#11719924—44no—
148#11719924—42no—
149#11719921—41no—
150#11719921—41no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153C2×#88322—22no—
154C2×#88322—22no—
155C2×#88322—22no—
156C2×#88322—41no—
157C2×#88322—42no—
158C2×#88322C2×D411yescharacteristic
159C2×#88322C2×D411yescharacteristic
160C2×#88322C2×D411yescharacteristic
161C2×#88322C2×D411yescharacteristic
162C23×C53#12111yescharacteristic
163C22×D53—41no—
164Q8×C53—24no—
165Q8×C53—22no—
166Q8×C53—42no—
167Q8×C53—41no—
168Q8×C53—41no—
169Q8×C53#13514yes—
170C2×C4×C53—24no—
171C2×C4×C53—22no—
172C2×C4×C53—22no—
173C2×C4×C53—22no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—22no—
177C2×C4×C53—42no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53#12111yescharacteristic
181C2×C4×C53#12111yescharacteristic
182C2×C4×C53#12111yescharacteristic
183C2×C4×C53C22×D411yescharacteristic
184C2×C4×C53C22×D411yescharacteristic
185C2×C4×C53C2×#4111yescharacteristic
186C22⋊D53—82no—
187C2×Dic53—41no—
188D212—42no—
189D212—41no—
190D212—41no—
191D212—81no—
192C4×D53—82no—
193Dic106—81no—
194C8×C53—42no—
195#88322—21no—
196#88322—21no—
197#88322—21no—
198#88322—42no—
199#88322—42no—
200#88322#13712yes—
201#19263—531nomaximal, Sylow
202C22×C53—21no—
203C22×C53—21no—
204C22×C53#47112yes—
205C22×C53C2×#12111yescharacteristic
206C2×D53—81no—
207C2×D53—81no—
208C2×D53—81no—
209C4×C53—21no—
210C4×C53—24no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—42no—
215C4×C53—81no—
216C4×C53—81no—
217C4×C53C2×#13512yes—
218C4×C53C2×#13711yescharacteristic
219C4×C53#57511yescharacteristic
220Dic53—81no—
221#3167—531no—
222#3113D5311yescharacteristic, complemented
223#3112—531no—
224C2×C8×Q8—531no—
225#1684—532no—
226#1682—532no—
227#1672—532no—
228#1666—532no—
229#1568—531no—
230#1552—531no—
231#1551—531no—
232C2×C53—22no—
233C2×C53#303711yescharacteristic
234C2×C53#305011yescharacteristic
235C2×C53C2×#47111yescharacteristic
236D53—161no—
237#533C2×D5311yescharacteristic
238#532—531no—
239C2×C4×Q8C2×D5311yescharacteristic
240#498—531no—
241#492—531no—
242#487—531no—
243#486—531no—
244#483—21no—
245#480—1061no—
246#477—532no—
247#475C2×D5312yes—
248#465—1061no—
249#465—1061no—
250#462C2×D5312yes—
251#460—532no—
252#451—21no—
253#448—1061no—
254C8×Q8—532no—
255C8×Q8—532no—
256C8×Q8—1061no—
257C8×Q8—1061no—
258#441—22no—
259#440—1062no—
260C4×SD16—1061no—
261C4×SD16—1061no—
262#432—531no—
263#432C2×D5311yescharacteristic
264C2×C4⋊C8—531no—
265C2×C4⋊C8—1061no—
266C2×C4×C8—531no—
267C2×C4×C8—1061no—
268#335—532no—
269#334—532no—
270#333—532no—
271#332—532no—
272#332—532no—
273#331—532no—
274C53#1926311yescharacteristic, complemented, Sylow
275C2×#41—1061no—
276C22×Q8C22⋊D5311yescharacteristic
277C2×Q16—22no—
278C2×SD16—1061no—
279C2×SD16—1061no—
280C2×SD16—1062no—
281C22×C8—531no—
282C22×C8—531no—
283C22×C8—1061no—
284C4⋊Q8—531no—
285C4⋊Q8C22×D5311yescharacteristic
286C4⋊Q8C22×D5311yescharacteristic
287#127—531no—
288#125—21no—
289#124—1061no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292#121—1061no—
293#121—1061no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8C22⋊D5312yes—
299C4×Q8C22×D5312yes—
300C4×D4—1061no—
301C4×D4—1061no—
302#117—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C22⋊D5311yescharacteristic
305C2×C42—21no—
306C2×C42C22×D5311yescharacteristic
307#107—1062no—
308#106—22no—
309#106—1061no—
310#106—1061no—
311C4⋊C8—532no—
312C4⋊C8—532no—
313C4⋊C8—1062no—
314C4⋊C8—1062no—
315C4⋊C8—1061no—
316C4⋊C8C22×D5312yes—
317#103—22no—
318#103—22no—
319#103—1061no—
320#103—1061no—
321#103—1062no—
322#102—1061no—
323#102—1061no—
324#102—1062no—
325C22⋊C8—21no—
326C22⋊C8—1061no—
327#97—21no—
328#97—1061no—
329C4×C8—531no—
330C4×C8—531no—
331C4×C8—531no—
332C4×C8—531no—
333C4×C8—1062no—
334C4×C8—1062no—
335#41—4242no—
336C2×Q8—24no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—1061no—
341C2×Q8C2×C22⋊D5312yes—
342C2×D4—1061no—
343C2×D4—1061no—
344C2×D4—1061no—
345C2×D4—2121no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—2121no—
349C22×C4D4×D5311yescharacteristic
350C22×C4#1171994711yescharacteristic
351C22×C4C2×C22⋊D5311yescharacteristic
352Q16—44no—
353SD16—2122no—
354SD16—2122no—
355SD16—2124no—
356C2×C8—22no—
357C2×C8—531no—
358C2×C8—531no—
359C2×C8—531no—
360C2×C8—531no—
361C2×C8—1062no—
362C2×C8—1062no—
363C2×C8—1062no—
364C2×C8—2121no—
365C2×C8—2122no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—1061no—
376C4⋊C4—1061no—
377C4⋊C4C2×C22⋊D5312yes—
378C22⋊C4—41no—
379C22⋊C4—2121no—
380C22⋊C4—2121no—
381C42—22no—
382C42—22no—
383C42—41no—
384C42D4×D5311yescharacteristic
385C42#1171994711yescharacteristic
386C42C2×C22⋊D5311yescharacteristic
387C42C23×D5311yescharacteristic
388C23—2121no—
389C23#1313318911yescharacteristic
390Q8—24no—
391Q8—22no—
392Q8—42no—
393Q8—41no—
394Q8—41no—
395Q8—4241no—
396Q8#1313323714yes—
397D4—4241no—
398D4—4242no—
399D4—2122no—
400D4—2121no—
401D4—2121no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—4242no—
413C2×C4—2121no—
414C2×C4#1313318911yescharacteristic
415C2×C4#1313318911yescharacteristic
416C2×C4#1313318911yescharacteristic
417C2×C4C2×D4×D5311yescharacteristic
418C2×C4C22×C22⋊D5311yescharacteristic
419C2×C4C2×#1171994711yescharacteristic
420C8—42no—
421C8—532no—
422C8—1061no—
423C8—1061no—
424C8—1061no—
425C8—2122no—
426C8—2122no—
427C22—21no—
428C22—21no—
429C22—4241no—
430C22—4241no—
431C22—4241no—
432C22#1681088812yes—
433C22C2×#1313318911yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—24no—
438C4—21no—
439C4—42no—
440C4—81no—
441C4—81no—
442C4—4241no—
443C4C2×#1313323712yes—
444C4#137×D5311yescharacteristic
445C4#1681142211yescharacteristic
446C2—22no—
447C2—8481no—
448C2#2028418411yescharacteristic
449C2#2028429311yescharacteristic
450C2C2×#1681088811yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3113C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#3113D53
hypocenterC53#19263
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
3. C2×Q8×#88322C2semidirect
4. #20274171C2semidirect
5. #20274161C2semidirect
6. #20274095C2semidirect
7. #20274065C2semidirect
8. #20273476C2semidirect
9. #20273415C2semidirect
11. C53×#3113C2semidirect
222. #3113D53semidirect
274. C53#19263semidirect

Maximal subgroups

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

subgroupindex
#202852752
#202848372
C2×Q8×#883222
#202741712
#202741612
#202740952
#202740652
#202734762
#202734152
#202734112
C53×#31132
#1926353