Finite Group Databasebeta

← #27367951

Subgroups — #27367951

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#20285274C211yesmaximal, characteristic
2#20284845C211yesmaximal, characteristic, complemented
3#20284837C211yesmaximal, characteristic
4#20274173C212yesmaximal, complemented
5#20274159C212yesmaximal, complemented
6#20274097C212yesmaximal, complemented
7#20274063C212yesmaximal, complemented
8#20273478C211yesmaximal, characteristic, complemented
9#20273413C211yesmaximal, characteristic, complemented
10#20273408C211yesmaximal, characteristic
11C2×C8×Q8×C53C211yesmaximal, characteristic, complemented
12#16811180C2211yescharacteristic
13#16811173C2211yescharacteristic
14#16811021C2211yescharacteristic
15#16810996C2211yescharacteristic
16#16810974C2211yescharacteristic
17#16810970C2211yescharacteristic
18#16810953—21no—
19#16810936—21no—
20#16810918C2212yes—
21#16810908C2212yes—
22#16810852—21no—
23#16810852—21no—
24#16810833C2212yes—
25#16810823C2212yes—
26#16810779—21no—
27#16810762—21no—
28#16810728—22no—
29#16810721—22no—
30C4×#11719913—21no—
31C4×#11719913—21no—
32#16810680C2211yescharacteristic
33#16810680C2211yescharacteristic
34#16810376C2212yes—
35#16810374C2212yes—
36#16810371C2212yes—
37#16810370C2212yes—
38#16810368C2212yes—
39#16810366C2212yes—
40C2×C4×Q8×C53C2211yescharacteristic
41C8×Q8×C53—21no—
42C8×Q8×C53—21no—
43C8×Q8×C53C2212yes—
44C8×Q8×C53C2212yes—
45C2×C4⋊C8×C53—21no—
46C2×C4⋊C8×C53C2211yescharacteristic
47C2×C4×C8×C53—21no—
48C2×C4×C8×C53C2211yescharacteristic
49C2×#11719945—21no—
50C2×#11719925—22no—
51C2×#11719913—21no—
52C2×#11719913—21no—
53C2×#11719924—22no—
54#13133217C2311yescharacteristic
55#13133217C2311yescharacteristic
56#13133217C2311yescharacteristic
57#13133214C2311yescharacteristic
58#13133209—21no—
59#13133205—21no—
60#13133196—21no—
61#13133196—21no—
62#13133191—21no—
63#13133191—21no—
64C4×Dic106—21no—
65C4×Dic106—21no—
66C4×D212—21no—
67C4×D212—21no—
68#13133176—21no—
69#13133150—22no—
70#13133149—21no—
71#13133149—21no—
72#13133148—22no—
73C4⋊#88322C2312yes—
74C4⋊#88322C2312yes—
75#13133142—21no—
76#13133142—21no—
77#13133141—22no—
78#13133141—22no—
79#13133140—22no—
80#13133139—21no—
81#13133139—21no—
82#13133137—22no—
83C22⋊#88322—21no—
84C22⋊#88322—21no—
85#13133128—21no—
86#13133128—21no—
87C22×Q8×C53D411yescharacteristic
88C22×C8×C53—21no—
89C22×C8×C53D411yescharacteristic
90C22×C8×C53D411yescharacteristic
91C4×Q8×C53—21no—
92C4×Q8×C53—21no—
93C4×Q8×C53D412yes—
94C4×Q8×C53C2312yes—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53D411yescharacteristic
97C2×C42×C53—21no—
98C2×C42×C53C2311yescharacteristic
99C4⋊C8×C53—22no—
100C4⋊C8×C53—22no—
101C4⋊C8×C53—21no—
102C4⋊C8×C53C2312yes—
103C4×C8×C53—22no—
104C4×C8×C53—22no—
105C4×C8×C53D411yescharacteristic
106C4×C8×C53D411yescharacteristic
107C4×C8×C53C2311yescharacteristic
108C4×C8×C53C2311yescharacteristic
109C2×Q8×C53—24no—
110C2×Q8×C53—21no—
111C2×Q8×C53C2×D412yes—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C2×C22⋊D53—41no—
118#11719945—82no—
119C2×D212—21no—
120C2×D212—21no—
121C2×D212—21no—
122C2×C4×D53—41no—
123C2×Dic106—21no—
124C2×Dic106—21no—
125C2×Dic106—21no—
126C2×C8×C53—22no—
127C2×C8×C53—22no—
128C2×C8×C53—41no—
129C2×C8×C53—42no—
130C2×C8×C53C2×D411yescharacteristic
131C2×C8×C53C2×D411yescharacteristic
132C2×C8×C53C2×D411yescharacteristic
133C2×C8×C53C2×D411yescharacteristic
134C4⋊C4×C53—22no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53C2×D412yes—
138C42×C53—22no—
139C42×C53—22no—
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142C42×C53#4111yescharacteristic
143C42×C53C2411yescharacteristic
144C22⋊Dic53—41no—
145#11719925—44no—
146#11719924—44no—
147#11719921—41no—
148#11719921—41no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155#11719919—41no—
156#11719919—41no—
157C4×Dic53—41no—
158C2×#88322—22no—
159C2×#88322—22no—
160#11719913—42no—
161#11719913—42no—
162C23×C53#12111yescharacteristic
163C22×D53—41no—
164Q8×C53—24no—
165Q8×C53—22no—
166Q8×C53#13514yes—
167C2×C4×C53—24no—
168C2×C4×C53—22no—
169C2×C4×C53—22no—
170C2×C4×C53—22no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—22no—
174C2×C4×C53—42no—
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53C22×D411yescharacteristic
179C2×C4×C53C22×D411yescharacteristic
180C2×C4×C53C2×#4111yescharacteristic
181C22⋊D53—82no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—41no—
185D212—42no—
186D212—41no—
187D212—41no—
188D212—81no—
189C4×D53—82no—
190Dic106—42no—
191Dic106—41no—
192Dic106—41no—
193Dic106—81no—
194C8×C53—21no—
195C8×C53—21no—
196C8×C53—21no—
197C8×C53—42no—
198C8×C53#13712yes—
199#88322—42no—
200#88322—42no—
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×C53C2×#13512yes—
216C4×C53C2×#13711yescharacteristic
217C4×C53#57511yescharacteristic
218Dic53—81no—
219Dic53—81no—
220Dic53—81no—
221#3167—531no—
222#3113—531no—
223#3112—531no—
224C2×C8×Q8D5311yescharacteristic, complemented
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#533—531no—
238#532—531no—
239C2×C4×Q8C2×D5311yescharacteristic
240#498—531no—
241#492—531no—
242#487—531no—
243#486—531no—
244#483—1061no—
245#480—1061no—
246#477—532no—
247#475—532no—
248#465—1061no—
249#465—1061no—
250#462—532no—
251#460—532no—
252#451—1061no—
253#448—1061no—
254C8×Q8—21no—
255C8×Q8—21no—
256C8×Q8C2×D5312yes—
257C8×Q8C2×D5312yes—
258#441—1062no—
259#440—1062no—
260C4×SD16—1061no—
261C4×SD16—1061no—
262#432—531no—
263#432—531no—
264C2×C4⋊C8—21no—
265C2×C4⋊C8C2×D5311yescharacteristic
266C2×C4×C8—21no—
267C2×C4×C8C2×D5311yescharacteristic
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×Q8D21211yescharacteristic
277C2×Q16—1062no—
278C2×SD16—1062no—
279C2×SD16—1061no—
280C2×SD16—1061no—
281C22×C8—21no—
282C22×C8C22⋊D5311yescharacteristic
283C22×C8C22⋊D5311yescharacteristic
284C4⋊Q8—531no—
285C4⋊Q8—531no—
286C4⋊Q8—531no—
287#127—531no—
288#125—1061no—
289#124—1061no—
290C22⋊Q8—1061no—
291C22⋊Q8—1061no—
292#121—1061no—
293#121—1061no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8—1061no—
297C4×Q8—1061no—
298C4×Q8D21212yes—
299C4×Q8C22×D5312yes—
300C4×D4—1061no—
301C4×D4—1061no—
302#117—1061no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4D21211yescharacteristic
305C2×C42—21no—
306C2×C42C22×D5311yescharacteristic
307#107—1062no—
308#106—1061no—
309#106—1061no—
310#106—1062no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—21no—
314C4⋊C8—532no—
315C4⋊C8—532no—
316C4⋊C8C22×D5312yes—
317#103—1062no—
318#103—1062no—
319#103—1061no—
320#103—1061no—
321#103—1062no—
322#102—1061no—
323#102—1061no—
324#102—1062no—
325C22⋊C8—1061no—
326C22⋊C8—1061no—
327#97—1061no—
328#97—1061no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8C22⋊D5311yescharacteristic
332C4×C8C22⋊D5311yescharacteristic
333C4×C8C22×D5311yescharacteristic
334C4×C8C22×D5311yescharacteristic
335#41—4242no—
336C2×Q8—24no—
337C2×Q8—21no—
338C2×Q8—1061no—
339C2×Q8—1061no—
340C2×Q8—1061no—
341C2×Q8C2×D21212yes—
342C2×D4—1061no—
343C2×D4—1061no—
344C2×D4—1061no—
345C2×D4—2121no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—2121no—
349C22×C4C2×D21211yescharacteristic
350C22×C4#1171994511yescharacteristic
351C22×C4C2×C22⋊D5311yescharacteristic
352Q16—2124no—
353SD16—2122no—
354SD16—2124no—
355SD16—2122no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—1062no—
361C2×C8—1062no—
362C2×C8C2×C22⋊D5311yescharacteristic
363C2×C8C2×C22⋊D5311yescharacteristic
364C2×C8C2×C22⋊D5311yescharacteristic
365C2×C8C2×C22⋊D5311yescharacteristic
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—1061no—
370C4⋊C4—1061no—
371C4⋊C4—1061no—
372C4⋊C4—1061no—
373C4⋊C4—1061no—
374C4⋊C4—1061no—
375C4⋊C4—2121no—
376C4⋊C4—2121no—
377C4⋊C4C2×D21212yes—
378C22⋊C4—2121no—
379C22⋊C4—2121no—
380C22⋊C4—2121no—
381C42—22no—
382C42—22no—
383C42—2121no—
384C42C2×D21211yescharacteristic
385C42#1171994511yescharacteristic
386C42C2×C22⋊D5311yescharacteristic
387C42C23×D5311yescharacteristic
388C23—2121no—
389C23#1313319111yescharacteristic
390Q8—24no—
391Q8—22no—
392Q8—4241no—
393Q8—2122no—
394Q8—2121no—
395Q8—2121no—
396Q8#1313323914yes—
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—4242no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4—2121no—
414C2×C4#1313319111yescharacteristic
415C2×C4#1313319111yescharacteristic
416C2×C4#1313319111yescharacteristic
417C2×C4C22×C22⋊D5311yescharacteristic
418C2×C4C22×D21211yescharacteristic
419C2×C4C2×#1171994511yescharacteristic
420C8—21no—
421C8—21no—
422C8—21no—
423C8—42no—
424C8—2122no—
425C8—2122no—
426C8#1313324812yes—
427C22—21no—
428C22—21no—
429C22—4241no—
430C22—4241no—
431C22—4241no—
432C22#1681088612yes—
433C22C2×#1313319111yescharacteristic
434C4—21no—
435C4—24no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—42no—
440C4—4241no—
441C4—4241no—
442C4—4241no—
443C4C2×#1313323912yes—
444C4C2×#1313324811yescharacteristic
445C4#1681142111yescharacteristic
446C2—22no—
447C2—8481no—
448C2#2028418711yescharacteristic
449C2#2028429611yescharacteristic
450C2C2×#1681088611yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C8×Q8×C53C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenterC2×C8×Q8D53
hypocenterC53#19263
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #20284845C2semidirect
4. #20274173C2semidirect
5. #20274159C2semidirect
6. #20274097C2semidirect
7. #20274063C2semidirect
8. #20273478C2semidirect
9. #20273413C2semidirect
11. C2×C8×Q8×C53C2semidirect
224. C2×C8×Q8D53semidirect
274. C53#19263semidirect

Maximal subgroups

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

subgroupindex
#202852742
#202848452
#202848372
#202741732
#202741592
#202740972
#202740632
#202734782
#202734132
#202734082
C2×C8×Q8×C532
#1926353