Finite Group Databasebeta

← #27367644

Subgroups — #27367644

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286022C211yesmaximal, characteristic
2#20285111C211yesmaximal, characteristic, complemented
3#20283459C211yesmaximal, characteristic, complemented
4#20274292C212yesmaximal, complemented
5#20274250C212yesmaximal, complemented
6#20274168C212yesmaximal, complemented
7#20274126C212yesmaximal, complemented
8#20273468C211yesmaximal, characteristic, complemented
9C2×#16810372C211yesmaximal, characteristic, complemented
10#20273362C211yesmaximal, characteristic, complemented
11C53×#3227C211yesmaximal, characteristic, complemented
12C2×#13133224—21no—
13#16811173C2211yescharacteristic
14#16810985—22no—
15#16810975C2212yes—
16#16810971C2212yes—
17#16810903C2212yes—
18#16810900—21no—
19#16810857—22no—
20#16810818C2212yes—
21#16810815—21no—
22#16810701C2212yes—
23#16810695—22no—
24D4×#88322C2212yes—
25C2×C4⋊#88322C2211yescharacteristic
26C2×C4⋊#88322C2211yescharacteristic
27#16810622—21no—
28#16810622—21no—
29#16810590—21no—
30C4×#11719917C2211yescharacteristic
31#16810375C2212yes—
32#16810373C2212yes—
33#16810372C2212yes—
34#16810372C2212yes—
35#16810368C2211yescharacteristic
36#16810368C2211yescharacteristic
37#16810366C2211yescharacteristic
38#16810366C2211yescharacteristic
39C53×#532C2211yescharacteristic
40C2×C4⋊Q8×C53C2211yescharacteristic
41C53×#498—21no—
42C53×#486—21no—
43C53×#481—22no—
44C53×#476C2212yes—
45C53×#476C2212yes—
46C53×#431C2211yescharacteristic
47C2×#11719945—21no—
48C2×#11719922—22no—
49#13133224—44no—
50C2×#11719917—22no—
51C2×#11719917D411yescharacteristic
52C2×#11719917D411yescharacteristic
53C22×#88322—22no—
54C22×#88322—21no—
55C22×#88322—21no—
56#13133217C2311yescharacteristic
57#13133214C2311yescharacteristic
58#13133205—21no—
59#13133191—21no—
60#13133191—21no—
61C4×D212—21no—
62C4×D212—21no—
63#13133150—22no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322C2312yes—
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2311yescharacteristic
69#13133140—22no—
70#13133140—22no—
71#13133138—22no—
72#13133137—22no—
73#13133137—22no—
74#13133137—22no—
75C22⋊#88322—22no—
76C22⋊#88322—22no—
77#13133128D412yes—
78C4×#88322C2312yes—
79C2×#41×C53—21no—
80C22×Q8×C53—21no—
81C4⋊Q8×C53—21no—
82C4⋊Q8×C53—21no—
83C4⋊Q8×C53—21no—
84C4⋊Q8×C53D411yescharacteristic
85C4⋊Q8×C53C2312yes—
86#127×C53D411yescharacteristic
87#124×C53—21no—
88#121×C53—21no—
89#121×C53D412yes—
90C4×D4×C53—21no—
91C4×D4×C53C2312yes—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C42×C53C2311yescharacteristic
95#106×C53—22no—
96#106×C53—22no—
97C4⋊C8×C53C2311yescharacteristic
98C4⋊C8×C53C2311yescharacteristic
99#102×C53—22no—
100#102×C53—22no—
101C22⋊C8×C53—21no—
102C4×C8×C53—21no—
103#41×C53—42no—
104#41×C53—42no—
105C2×Q8×C53—21no—
106C2×Q8×C53—22no—
107C2×Q8×C53—41no—
108C2×Q8×C53—82no—
109C2×D4×C53—22no—
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53C2×D412yes—
113C22×C4×C53—22no—
114C22×C4×C53—41no—
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53#4111yescharacteristic
118C2×C22⋊D53—41no—
119#11719945—42no—
120#11719945—42no—
121C2×D212—21no—
122C2×D212—21no—
123C2×D212—21no—
124C2×C4×D53—41no—
125C2×Dic106—21no—
126C2×C8×C53—21no—
127C2×C8×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53C2×D412yes—
134C22⋊C4×C53—22no—
135C22⋊C4×C53—22no—
136C42×C53C2×D411yescharacteristic
137C42×C53C2×D411yescharacteristic
138C42×C53#4111yescharacteristic
139C42×C53C2411yescharacteristic
140#11719922—82no—
141#11719921—41no—
142#11719921—41no—
143C4⋊Dic53—21no—
144C4⋊Dic53—21no—
145#11719917—22no—
146#11719917—22no—
147#11719917—44no—
148C2×#88322—21no—
149C2×#88322—21no—
150C2×#88322—22no—
151C2×#88322—44no—
152C2×#88322—42no—
153C2×#88322—81no—
154C2×#88322C2×D412yes—
155C2×#88322C2×D412yes—
156C23×C53—22no—
157C23×C53#12111yescharacteristic
158C22×D53—41no—
159Q8×C53—42no—
160Q8×C53—82no—
161Q8×C53—82no—
162D4×C53—22no—
163D4×C53—22no—
164D4×C53—44no—
165D4×C53—42no—
166D4×C53—81no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—22no—
170C2×C4×C53—41no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53—82no—
175C2×C4×C53#12111yescharacteristic
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#13512yes—
179C2×C4×C53AGL(1,ℤ/8)12yes—
180C2×C4×C53C22×D411yescharacteristic
181C2×C4×C53C22×D411yescharacteristic
182C2×C4×C53C2×#4111yescharacteristic
183C22⋊D53—82no—
184C2×Dic53—41no—
185D212—41no—
186D212—41no—
187D212—41no—
188D212—41no—
189D212—81no—
190C4×D53—82no—
191Dic106—41no—
192Dic106—41no—
193C8×C53—41no—
194C8×C53—41no—
195#88322—22no—
196#88322—22no—
197#88322—41no—
198#88322—41no—
199#88322—42no—
200#19236—531nomaximal, Sylow
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—42no—
204C22×C53—42no—
205C22×C53—42no—
206C22×C53#46312yes—
207C22×C53C2×#12111yescharacteristic
208C2×D53—81no—
209C2×D53—81no—
210C2×D53—81no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—42no—
215C4×C53—82no—
216C4×C53#46312yes—
217C4×C53C2×#13511yescharacteristic
218C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
219C4×C53#57711yescharacteristic
220C4×C53#57811yescharacteristic
221Dic53—81no—
222#3245—531no—
223#3227D5311yescharacteristic, complemented
224#3141—531no—
225#2935—531no—
226#1703—532no—
227#1697—532no—
228#1683—532no—
229#1677—532no—
230#1565—531no—
231C2×#333—531no—
232#1540—531no—
233C2×C53—22no—
234C2×C53—82no—
235C2×C53C2×#46311yescharacteristic
236C2×C53#306311yescharacteristic
237C2×C53#306811yescharacteristic
238D53—161no—
239C2×#131—1061no—
240#532—531no—
241#532C2×D5311yescharacteristic
242C2×C4⋊Q8C2×D5311yescharacteristic
243#498—21no—
244#489—1062no—
245#487—532no—
246#486—21no—
247#486—532no—
248#481—22no—
249#476C2×D5312yes—
250#476C2×D5312yes—
251#474—532no—
252#474—1061no—
253#466—1062no—
254#459—532no—
255#459—1061no—
256#436—532no—
257#435—1062no—
258C8×D4—532no—
259#431C2×D5311yescharacteristic
260C2×C4⋊C8—531no—
261C2×C4⋊C8—531no—
262#417—1061no—
263#417—1061no—
264#408—1061no—
265C4×M16—531no—
266#335—532no—
267#334—532no—
268#333—532no—
269#333—532no—
270#332—531no—
271#332—531no—
272#331—531no—
273#331—531no—
274C53#1923611yescharacteristic, complemented, Sylow
275C2×#41—21no—
276C2×#41—1061no—
277C22×Q8—21no—
278C2×D8—1062no—
279#131—2124no—
280C2×M16—531no—
281C2×M16—531no—
282C2×M16—1062no—
283C22×C8—1062no—
284C22×C8—1061no—
285C22×C8—1061no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—531no—
290C4⋊Q8C22⋊D5311yescharacteristic
291C4⋊Q8C22×D5312yes—
292#127—531no—
293#127C22⋊D5311yescharacteristic
294#124—21no—
295#124—1061no—
296#121—21no—
297#121—1061no—
298#121—1061no—
299#121C22⋊D5312yes—
300C4×D4—21no—
301C4×D4—1061no—
302C4×D4—1061no—
303C4×D4C22×D5312yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C42C22×D5311yescharacteristic
307#107—1062no—
308#106—22no—
309#106—22no—
310C4⋊C8—531no—
311C4⋊C8—531no—
312C4⋊C8—532no—
313C4⋊C8—1061no—
314C4⋊C8—1061no—
315C4⋊C8C22×D5311yescharacteristic
316C4⋊C8C22×D5311yescharacteristic
317#103—1062no—
318#103—1062no—
319#102—22no—
320#102—22no—
321#102—1062no—
322#102—1062no—
323#102—1062no—
324#102—1062no—
325C22⋊C8—21no—
326C22⋊C8—1062no—
327C22⋊C8—1062no—
328#97—532no—
329C4×C8—21no—
330C4×C8—532no—
331#41—42no—
332#41—42no—
333#41—2122no—
334#41—2122no—
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—41no—
338C2×Q8—82no—
339C2×Q8—1061no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—1061no—
344C2×D4—1061no—
345C2×D4—1061no—
346C2×D4—2121no—
347C2×D4C2×C22⋊D5312yes—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—2121no—
351C22×C4D4×D5311yescharacteristic
352C22×C4#1171994711yescharacteristic
353C22×C4C2×C22⋊D5311yescharacteristic
354D8—4242no—
355M16—1062no—
356M16—1062no—
357M16—2124no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—532no—
361C2×C8—532no—
362C2×C8—1061no—
363C2×C8—1061no—
364C2×C8—1062no—
365C2×C8—4241no—
366C2×C8—2124no—
367C2×C8—2122no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—1061no—
374C4⋊C4—1061no—
375C4⋊C4C2×C22⋊D5312yes—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—2121no—
379C22⋊C4—2121no—
380C42D4×D5311yescharacteristic
381C42#1171994711yescharacteristic
382C42C2×C22⋊D5311yescharacteristic
383C42C23×D5311yescharacteristic
384C23—22no—
385C23—2121no—
386C23#1313318911yescharacteristic
387Q8—42no—
388Q8—82no—
389Q8—82no—
390Q8—2121no—
391Q8—2121no—
392D4—22no—
393D4—22no—
394D4—44no—
395D4—42no—
396D4—81no—
397D4—4241no—
398D4—4242no—
399D4—2121no—
400D4—2121no—
401D4—2121no—
402D4—2121no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—82no—
411C2×C4—4242no—
412C2×C4—2121no—
413C2×C4#1313318911yescharacteristic
414C2×C4#1313318911yescharacteristic
415C2×C4#1313318911yescharacteristic
416C2×C4#1313323812yes—
417C2×C4#1313324412yes—
418C2×C4C2×D4×D5311yescharacteristic
419C2×C4C22×C22⋊D5311yescharacteristic
420C2×C4C2×#1171994711yescharacteristic
421C8—41no—
422C8—41no—
423C8—1062no—
424C8—1062no—
425C8—2121no—
426C8—2121no—
427C8—2122no—
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—4241no—
434C22—4241no—
435C22—4241no—
436C22#1681084012yes—
437C22C2×#1313318911yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—82no—
443C4—4241no—
444C4#1681084012yes—
445C4C2×#1313323811yescharacteristic
446C4C2×#1313324411yescharacteristic
447C4#1681143111yescharacteristic
448C4#1681143811yescharacteristic
449C2—22no—
450C2—82no—
451C2—8481no—
452C2C2×#1681084011yescharacteristic
453C2#2028440211yescharacteristic
454C2#2028444011yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3227C2
socleC22×C53C2×#121
prime radical#20286022C2
hypercenter#3227D53
hypocenterC53#19236
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
2. #20285111C2semidirect
3. #20283459C2semidirect
4. #20274292C2semidirect
5. #20274250C2semidirect
6. #20274168C2semidirect
7. #20274126C2semidirect
8. #20273468C2semidirect
9. C2×#16810372C2semidirect
10. #20273362C2semidirect
11. C53×#3227C2semidirect
223. #3227D53semidirect
274. C53#19236semidirect

Maximal subgroups

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

subgroupindex
#202860222
#202851112
#202834592
#202742922
#202742502
#202741682
#202741262
#202734682
C2×#168103722
#202733622
C53×#32272
#1923653