Finite Group Databasebeta

← #27351827

Subgroups — #27351827

Order 13568.

918 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288582C211yesmaximal, characteristic
2#20283795C211yesmaximal, characteristic, complemented
3#20277682C211yesmaximal, characteristic, complemented
4#20273291C214yesmaximal, complemented
5#20273274C211yesmaximal, characteristic
6#20273214C211yesmaximal, characteristic, complemented
7#20273211C211yesmaximal, characteristic
8#20273200C214yesmaximal, complemented
9C53×#1518C211yesmaximal, characteristic, complemented
10#16811358C412yescomplemented
11#16811352C412yescomplemented
12#16811347C412yescomplemented
13#16811346—22no—
14#16811334C412yescomplemented
15#16811311—22no—
16#16811222—22no—
17#16811199—21no—
18#16811173C2211yescharacteristic
19#16811159C2214yes—
20#16811103—21no—
21C2×C4×Dic106C2211yescharacteristic
22C8×Dic106—22no—
23#16810747—22no—
24#16810676C2211yescharacteristic
25#16810673C2214yes—
26#16810528—21no—
27#16810510—22no—
28#16810490C2212yes—
29C4⋊C4×Dic53—21no—
30C4×C4⋊Dic53C2211yescharacteristic
31#16810468C2212yes—
32#16810360C2212yes—
33#16810360C2212yes—
34#16810358C2212yes—
35#16810358C2212yes—
36#16810356C2214yes—
37C53×#533C2211yescharacteristic
38C53×#431C2211yescharacteristic
39C53×#382C2211yescharacteristic
40C53×#328C2214yes—
41C2×#11719945—21no—
42C22×Dic106D411yescharacteristic
43#13133217—22no—
44#13133217C2×C411yescharacteristic
45#13133214C2×C411yescharacteristic
46#13133212—24no—
47#13133212—24no—
48#13133208—24no—
49#13133208C2×C414yes—
50#13133206C2×C414yes—
51#13133205—21no—
52#13133205—22no—
53#13133202—24no—
54#13133201—24no—
55#13133197—22no—
56#13133197—22no—
57#13133193—24no—
58#13133191—21no—
59#13133191C2312yes—
60Q8×Dic53C2×C414yes—
61C4×Dic106—22no—
62C4×Dic106C2×C412yes—
63C4×Dic106C2×C412yes—
64C4×Dic106D412yes—
65C4×Dic106D412yes—
66C4×D212—21no—
67C4×D212—22no—
68C4×D212C2×C412yes—
69#13133179C2×C414yes—
70#13133177—24no—
71#13133175—22no—
72#13133175—24no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53D411yescharacteristic
75C2×#11719919D412yes—
76C4⋊C4×D53—22no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—22no—
79C2×C4×Dic53C2312yes—
80C4⋊#88322—21no—
81#13133146—24no—
82C22⋊#88322C2312yes—
83C4×#88322—21no—
84C8×Dic53—24no—
85#13133124—22no—
86#13133124—22no—
87#13133124—22no—
88#13133124C2311yescharacteristic
89#13133124C2311yescharacteristic
90C4⋊Q8×C53C2×C411yescharacteristic
91C4⋊Q8×C53C2×C411yescharacteristic
92#125×C53—21no—
93C22⋊Q8×C53—21no—
94C22⋊Q8×C53C2312yes—
95C4×Q8×C53—21no—
96C4×Q8×C53C2×C412yes—
97#117×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C42×C53C2311yescharacteristic
100C4⋊C8×C53—21no—
101C22⋊C8×C53C2312yes—
102C4×C8×C53—21no—
103#95×C53—21no—
104#95×C53C2312yes—
105C2×Q8×C53—21no—
106C2×Q8×C53C22×C412yes—
107C22×C4×C53—22no—
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2411yescharacteristic
111C2×C22⋊D53—22no—
112C22×Dic53—21no—
113C22×Dic53—21no—
114C22×Dic53C2×D412yes—
115C22×Dic53C2×D412yes—
116#11719945—42no—
117#11719945—42no—
118C2×D212—21no—
119C2×D212—21no—
120C2×D212C22×C412yes—
121C2×C4×D53—22no—
122C2×C4×D53—24no—
123C2×Dic106—22no—
124C2×Dic106—22no—
125C2×Dic106—21no—
126C2×Dic106C22⋊C412yes—
127C2×Dic106C22⋊C412yes—
128C2×C8×C53—22no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—42no—
135C4⋊C4×C53C22×C412yes—
136C22⋊C4×C53—22no—
137C42×C53—22no—
138C42×C53C22×C411yescharacteristic
139C42×C53C22×C411yescharacteristic
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142#11719921—22no—
143#11719921—22no—
144#11719921—24no—
145#11719921—24no—
146#11719921—24no—
147C4⋊Dic53—21no—
148C4⋊Dic53—24no—
149C4⋊Dic53—24no—
150C4⋊Dic53—42no—
151C4⋊Dic53C22⋊C412yes—
152C4⋊Dic53C22⋊C412yes—
153C4⋊Dic53C22×C412yes—
154#11719919—24no—
155#11719919C22⋊C414yes—
156#11719919C22⋊C414yes—
157C4×Dic53—24no—
158C4×Dic53—44no—
159C4×Dic53—42no—
160C4×Dic53C22×C414yes—
161C4×Dic53C2×D414yes—
162C2×#88322—22no—
163C23×C53C22×D411yescharacteristic
164C22×D53—22no—
165Q8×C53—42no—
166Q8×C53—81no—
167C2×C4×C53—22no—
168C2×C4×C53—22no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—42no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53C2×C22⋊C411yescharacteristic
175C2×C4×C53C2×C22⋊C411yescharacteristic
176C2×C4×C53C2×C22⋊C411yescharacteristic
177C2×C4×C53C2×C22⋊C411yescharacteristic
178C2×C4×C53#13512yes—
179C2×C4×C53#13512yes—
180C2×C4×C53C23×C411yescharacteristic
181C2×C4×C53C22×D411yescharacteristic
182C22⋊D53—42no—
183C22⋊D53—42no—
184C2×Dic53—22no—
185C2×Dic53—22no—
186C2×Dic53—22no—
187C2×Dic53—22no—
188C2×Dic53—42no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53C2×C22⋊C414yes—
195C2×Dic53C2×C22⋊C414yes—
196D212—42no—
197D212—42no—
198D212—81no—
199C4×D53—44no—
200C4×D53—84no—
201Dic106—24no—
202Dic106—24no—
203Dic106—42no—
204C8×C53—42no—
205#88322—42no—
206#17616—531nomaximal, Sylow
207C22×C53—21no—
208C22×C53—21no—
209C22×C53C22×C22⋊C411yescharacteristic
210C22×C53C2×#13511yescharacteristic
211C22×C53C2×#13511yescharacteristic
212C2×D53—42no—
213C2×D53—42no—
214C2×D53—42no—
215C4×C53—21no—
216C4×C53—21no—
217C4×C53—21no—
218C4×C53—21no—
219C4×C53—42no—
220C4×C53—42no—
221C4×C53—82no—
222C4×C53#41612yes—
223C4×C53#41612yes—
224Dic53—24no—
225Dic53—24no—
226Dic53—42no—
227Dic53—42no—
228Dic53—81no—
229#3507—531no—
230#2982—531no—
231#2356—531no—
232#1521—534no—
233#1518D5311yescharacteristic, complemented
234#1517—531no—
235#1503—531no—
236#1502—531no—
237#1500—534no—
238C2×C53—21no—
239C2×C53—21no—
240C2×C53#288411yescharacteristic
241C2×C53C2×#41611yescharacteristic
242C2×C53#290111yescharacteristic
243D53—82no—
244#559—532no—
245#557—532no—
246#556—532no—
247#556—1062no—
248#554—532no—
249#552—1062no—
250#539—1062no—
251#536—1061no—
252#533C2×D5311yescharacteristic
253#532—531no—
254#529—534no—
255#520—1061no—
256C2×C4×Q8—531no—
257C8×Q8—1062no—
258C8×Q8—1062no—
259#431—531no—
260#431—534no—
261#431C2×D5311yescharacteristic
262#391—1061no—
263#387—1062no—
264#383—532no—
265#382C2×D5311yescharacteristic
266C4×C4⋊C4—531no—
267C4×C4⋊C4—1061no—
268#376—532no—
269#328—532no—
270#328—532no—
271#328C2×D5314yes—
272#327—532no—
273#327—532no—
274#327—534no—
275C53#1761611yescharacteristic, complemented, Sylow
276C2×#41—1061no—
277C22×Q8—531no—
278C4⋊Q8—531no—
279C4⋊Q8—1062no—
280C4⋊Q8C4×D5311yescharacteristic
281C4⋊Q8C4×D5311yescharacteristic
282#127—531no—
283#126—1064no—
284#126—1064no—
285#125—21no—
286#125—534no—
287#125—1064no—
288#124—534no—
289#124—1062no—
290#124—1061no—
291#123—1064no—
292#123—1064no—
293C22⋊Q8—21no—
294C22⋊Q8—1062no—
295C22⋊Q8—1062no—
296C22⋊Q8—1064no—
297C22⋊Q8C22×D5312yes—
298#121—532no—
299#121—1061no—
300C4×Q8—21no—
301C4×Q8—532no—
302C4×Q8—532no—
303C4×Q8—532no—
304C4×Q8—532no—
305C4×Q8—534no—
306C4×Q8—1062no—
307C4×Q8C4×D5312yes—
308C4×D4—534no—
309C4×D4—532no—
310C4×D4—1062no—
311C4×D4—1064no—
312C4×D4—1061no—
313#117—21no—
314#117—1064no—
315#117—1062no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—531no—
318C2×C4⋊C4—532no—
319C2×C4⋊C4—1062no—
320C2×C4⋊C4—1061no—
321C2×C42—532no—
322C2×C42—1062no—
323C2×C42—1061no—
324C2×C42C22×D5311yescharacteristic
325C4⋊C8—21no—
326C4⋊C8—1064no—
327C4⋊C8—1061no—
328C22⋊C8—532no—
329C22⋊C8C22×D5312yes—
330C4×C8—21no—
331C4×C8—1064no—
332C4×C8—1061no—
333#95—21no—
334#95—531no—
335#95—531no—
336#95—1062no—
337#95—1062no—
338#95—1062no—
339#95C22×D5312yes—
340#41—2122no—
341#41—2122no—
342C2×Q8—21no—
343C2×Q8—532no—
344C2×Q8—532no—
345C2×Q8—1061no—
346C2×Q8—1062no—
347C2×Q8—1062no—
348C2×Q8C2×C4×D5312yes—
349C2×D4—532no—
350C2×D4—1062no—
351C2×D4—1061no—
352C2×D4—1061no—
353C22×C4—22no—
354C22×C4—532no—
355C22×C4—532no—
356C22×C4—1064no—
357C22×C4—1061no—
358C22×C4—1062no—
359C22×C4—1061no—
360C22×C4D4×D5311yescharacteristic
361C22×C4D4×D5311yescharacteristic
362C22×C4C23×D5311yescharacteristic
363C2×C8—22no—
364C2×C8—1062no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—42no—
371C4⋊C4—532no—
372C4⋊C4—532no—
373C4⋊C4—534no—
374C4⋊C4—534no—
375C4⋊C4—532no—
376C4⋊C4—1064no—
377C4⋊C4—1064no—
378C4⋊C4—1064no—
379C4⋊C4—1061no—
380C4⋊C4—2122no—
381C4⋊C4C2×C4×D5312yes—
382C22⋊C4—22no—
383C22⋊C4—1064no—
384C22⋊C4—1064no—
385C22⋊C4—1064no—
386C22⋊C4—1062no—
387C22⋊C4—1062no—
388C42—22no—
389C42—534no—
390C42—534no—
391C42—1064no—
392C42—2124no—
393C42—2122no—
394C42C2×C4×D5311yescharacteristic
395C42C2×C4×D5311yescharacteristic
396C42D4×D5311yescharacteristic
397C42D4×D5311yescharacteristic
398C23—1062no—
399C23C2×D4×D5311yescharacteristic
400Q8—42no—
401Q8—81no—
402Q8—1064no—
403Q8—1064no—
404Q8—2122no—
405D4—4241no—
406D4—2122no—
407D4—2122no—
408D4—2122no—
409D4—2122no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—534no—
418C2×C4—534no—
419C2×C4—1062no—
420C2×C4—1062no—
421C2×C4—1062no—
422C2×C4—1062no—
423C2×C4—4244no—
424C2×C4—2122no—
425C2×C4—2121no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2121no—
429C2×C4—2121no—
430C2×C4—2124no—
431C2×C4C22⋊C4×D5311yescharacteristic
432C2×C4C22⋊C4×D5311yescharacteristic
433C2×C4C22⋊C4×D5311yescharacteristic
434C2×C4C22⋊C4×D5311yescharacteristic
435C2×C4#1313323812yes—
436C2×C4#1313324012yes—
437C2×C4C22×C4×D5311yescharacteristic
438C2×C4C2×D4×D5311yescharacteristic
439C8—42no—
440C8—2122no—
441C22—21no—
442C22—21no—
443C22—2122no—
444C22—2122no—
445C22—2122no—
446C22C2×C22⋊C4×D5311yescharacteristic
447C22C2×#1313323811yescharacteristic
448C22C2×#1313324011yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—21no—
453C4—42no—
454C4—42no—
455C4—82no—
456C4—1064no—
457C4—1064no—
458C4—4241no—
459C4—2122no—
460C4—2122no—
461C4#1681061812yes—
462C4#1681061812yes—
463C2—21no—
464C2—21no—
465C2—4242no—
466C2#2028317711yescharacteristic
467C2C2×#1681061811yescharacteristic
468C2#2028326111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1518C2
socleC22×C53C22×C22⋊C4
prime radical#16811173C22
cosocleC22×C4×C53C24
hypercenter#1518D53
hypocenterC53#17616
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
2. #20283795C2semidirect
3. #20277682C2semidirect
4. #20273291C2semidirect
6. #20273214C2semidirect
8. #20273200C2semidirect
9. C53×#1518C2semidirect
10. #16811358C4semidirect
11. #16811352C4semidirect
12. #16811347C4semidirect
14. #16811334C4semidirect
233. #1518D53semidirect
275. C53#17616semidirect

Maximal subgroups

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

subgroupindex
#202885822
#202837952
#202776822
#202732912
#202732742
#202732142
#202732112
#202732002
C53×#15182
#1761653