Finite Group Databasebeta

← #27369353

Subgroups — #27369353

Order 13568.

628 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284810C211yesmaximal, characteristic
2#20284719C211yesmaximal, characteristic
3#20283738C211yesmaximal, characteristic
4#20274163C212yesmaximal, complemented
5#20274135C212yesmaximal, complemented
6#20274101C212yesmaximal, complemented
7#20274087C212yesmaximal, complemented
8#20273495C211yesmaximal, characteristic, complemented
9#20273425C211yesmaximal, characteristic, complemented
10#20273380C211yesmaximal, characteristic, complemented
11C53×#3231C211yesmaximal, characteristic, complemented
12C2×C4×D212C2211yescharacteristic
13#16810956—21no—
14#16810951—21no—
15#16810910C2212yes—
16#16810900C2212yes—
17#16810825C2212yes—
18#16810825C2212yes—
19#16810773—21no—
20#16810773—21no—
21#16810721—22no—
22C4×#11719924—22no—
23C8×D212C2212yes—
24C8×D212C2212yes—
25#16810688—22no—
26#16810680C2211yescharacteristic
27#16810676C2211yescharacteristic
28#16810676C2211yescharacteristic
29C22⋊C8×D53—22no—
30#16810376C2212yescomplemented
31#16810374C2212yescomplemented
32#16810372C2212yescomplemented
33#16810369C2212yescomplemented
34#16810369C2212yescomplemented
35#16810369C2212yescomplemented
36C53×#533C2211yescharacteristic
37C53×#533C2211yescharacteristic
38C53×#500C2211yescharacteristic, complemented
39C53×#498C2211yescharacteristic, complemented
40C53×#487C2211yescharacteristic, complemented
41C53×#487C2211yescharacteristic, complemented
42C53×#485—22no—
43C53×#479—22no—
44C53×#431C2211yescharacteristic
45C22×D212D411yescharacteristic
46C22×C4×D53—21no—
47C2×#11719924—22no—
48C2×#11719924—22no—
49C2×C8×D53—22no—
50C2×C8×D53—22no—
51C4×D212—21no—
52C4×D212—21no—
53C4×D212D412yes—
54C4×D212C2312yes—
55C2×C4⋊Dic53D411yescharacteristic
56C2×#11719921—21no—
57#13133150—22no—
58#13133148—22no—
59C4⋊#88322C2312yes—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133141—22no—
63#13133141—22no—
64#13133137—22no—
65#13133137—22no—
66C22⋊#88322—21no—
67C22⋊#88322—21no—
68C22⋊#88322—21no—
69#13133129—22no—
70#13133129—22no—
71#13133128—21no—
72C4×#88322—21no—
73C4×#88322—21no—
74C4⋊Q8×C53C2311yescharacteristic
75C4⋊Q8×C53C2311yescharacteristic
76C4⋊Q8×C53C2311yescharacteristic
77C4⋊Q8×C53C2311yescharacteristic
78#125×C53—21no—
79#125×C53—21no—
80C22⋊Q8×C53—21no—
81C22⋊Q8×C53—21no—
82C22⋊Q8×C53—21no—
83C22⋊Q8×C53—21no—
84C4×Q8×C53—21no—
85C4×Q8×C53—21no—
86C4×Q8×C53—21no—
87C4×Q8×C53—21no—
88#117×C53—21no—
89#117×C53—21no—
90C2×C42×C53C2311yescharacteristic
91#107×C53—22no—
92#106×C53—22no—
93C4⋊C8×C53D412yes—
94#103×C53—22no—
95#103×C53—22no—
96C22⋊C8×C53D412yes—
97C4×C8×C53C2311yescharacteristic
98C4×C8×C53C2311yescharacteristic
99C23×D53—21no—
100C2×Q8×C53—21no—
101C2×Q8×C53—21no—
102C2×Q8×C53—21no—
103C2×Q8×C53—21no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×Dic53—21no—
108C2×D212—21no—
109C2×D212—24no—
110C2×D212C2×D412yes—
111C2×C4×D53—22no—
112C2×C4×D53—22no—
113C2×C4×D53—82no—
114C2×C8×C53C2×D412yes—
115C2×C8×C53C2×D412yes—
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—41no—
125C4⋊C4×C53—41no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—41no—
128C22⋊C4×C53—41no—
129C22⋊C4×C53—41no—
130C42×C53—41no—
131C42×C53—41no—
132C42×C53C2×D411yescharacteristic
133C42×C53C2×D411yescharacteristic
134C42×C53#4111yescharacteristic
135C42×C53C2411yescharacteristic
136#11719924—44no—
137#11719924—44no—
138#11719921—22no—
139#11719921—22no—
140C4⋊Dic53—21no—
141C4⋊Dic53C2×D412yes—
142C2×#88322—22no—
143C2×#88322—22no—
144C2×#88322—22no—
145C8×D53—82no—
146C8×D53—82no—
147C23×C53#12111yescharacteristic
148C22×D53—22no—
149C22×D53—24no—
150C22×D53—81no—
151C22×D53—81no—
152Q8×C53—42no—
153Q8×C53—42no—
154Q8×C53—42no—
155Q8×C53—81no—
156C2×C4×C53—22no—
157C2×C4×C53—22no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C2×C4×C53C22×D411yescharacteristic
167C2×C4×C53C22×D411yescharacteristic
168C2×C4×C53C2×#4111yescharacteristic
169C2×Dic53—22no—
170C2×Dic53—81no—
171D212—24no—
172D212—22no—
173D212#13514yes—
174C4×D53—42no—
175C4×D53—42no—
176C4×D53—82no—
177C8×C53—22no—
178C8×C53—22no—
179#88322—42no—
180#88322—42no—
181#88322—42no—
182#19395—531nomaximal, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53C2×#12111yescharacteristic
187C2×D53—24no—
188C2×D53—42no—
189C2×D53—42no—
190C2×D53—84no—
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—81no—
196C4×C53—81no—
197C4×C53—81no—
198C4×C53—81no—
199C4×C53C2×#13512yes—
200C4×C53#57712yes—
201Dic53—42no—
202#3231D5311yescharacteristic, complemented
203#3110—531no—
204#3101—531no—
205#2972—531no—
206#1683—532no—
207#1679—532no—
208#1673—532no—
209#1671—532no—
210#1572—531no—
211#1554—531no—
212#1544—531no—
213C2×C53—41no—
214C2×C53#304211yescharacteristic
215C2×C53#305011yescharacteristic
216C2×C53#306911yescharacteristic
217D53—44no—
218#533C2×D5311yescharacteristic
219#533C2×D5311yescharacteristic
220C2×C4×D4—531no—
221#500C2×D5311yescharacteristic, complemented
222#498C2×D5311yescharacteristic, complemented
223#487C2×D5311yescharacteristic, complemented
224#487C2×D5311yescharacteristic, complemented
225#485—22no—
226#484—1061no—
227#483—1061no—
228#479—22no—
229#476—532no—
230#474—532no—
231#461—532no—
232#461—532no—
233#450—1061no—
234#450—1061no—
235#440—1062no—
236C4×SD16—1062no—
237C8×D4—532no—
238C8×D4—532no—
239C8×D4—1062no—
240#432—531no—
241#431—531no—
242#431—531no—
243#431C2×D5311yescharacteristic
244C2×C22⋊C8—1062no—
245#335—532no—
246#334—532no—
247#333—532no—
248#332—532no—
249#332—532no—
250#332—532no—
251C53#1939511yescharacteristic, complemented, Sylow
252C22×D4—531no—
253C23×C4—1061no—
254C2×SD16—1062no—
255C2×SD16—1062no—
256C22×C8—1062no—
257C22×C8—1062no—
258C4⋊Q8C22×D5311yescharacteristic
259C4⋊Q8C22×D5311yescharacteristic
260C4⋊Q8C22×D5311yescharacteristic
261C4⋊Q8C22×D5311yescharacteristic
262#125—21no—
263#125—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×D4—532no—
273C4×D4—532no—
274C4×D4—1061no—
275C4×D4—1061no—
276#117—21no—
277#117—21no—
278C2×C4⋊C4—531no—
279C2×C22⋊C4—1061no—
280C2×C42C22×D5311yescharacteristic
281#107—22no—
282#107—1062no—
283#106—22no—
284#106—1062no—
285C4⋊C8—532no—
286C4⋊C8—532no—
287C4⋊C8—532no—
288C4⋊C8C22⋊D5312yes—
289#103—22no—
290#103—22no—
291#103—1062no—
292#103—1062no—
293#102—1062no—
294#102—1062no—
295C22⋊C8—1061no—
296C22⋊C8—1061no—
297C22⋊C8—1061no—
298C22⋊C8—1062no—
299C22⋊C8—1062no—
300C22⋊C8C22⋊D5312yes—
301#97—1061no—
302C4×C8—1061no—
303C4×C8—1061no—
304C4×C8C22×D5311yescharacteristic
305C4×C8C22×D5311yescharacteristic
306C24—1061no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×D4—532no—
312C2×D4—1064no—
313C2×D4—1061no—
314C22×C4—1061no—
315C22×C4—1062no—
316C22×C4—1062no—
317C22×C4—4242no—
318C22×C4D4×D5311yescharacteristic
319C22×C4#1171994711yescharacteristic
320C22×C4C2×C22⋊D5311yescharacteristic
321SD16—2124no—
322SD16—2124no—
323C2×C8—1062no—
324C2×C8—1062no—
325C2×C8—1062no—
326C2×C8—4242no—
327C2×C8—4242no—
328C2×C8C2×C22⋊D5312yes—
329C2×C8C2×C22⋊D5312yes—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—532no—
343C4⋊C4—1061no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—1062no—
347C22⋊C4—1062no—
348C42—41no—
349C42—41no—
350C42D4×D5311yescharacteristic
351C42#1171994711yescharacteristic
352C42C2×C22⋊D5311yescharacteristic
353C42C23×D5311yescharacteristic
354C23—1064no—
355C23—1062no—
356C23—4241no—
357C23—4241no—
358C23#1313318811yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—81no—
363D4—534no—
364D4—1064no—
365D4—1062no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—1062no—
374C2×C4—4242no—
375C2×C4—4241no—
376C2×C4—2122no—
377C2×C4—2122no—
378C2×C4#1313318811yescharacteristic
379C2×C4#1313318811yescharacteristic
380C2×C4#1313318811yescharacteristic
381C2×C4C2×D4×D5311yescharacteristic
382C2×C4C22×C22⋊D5311yescharacteristic
383C2×C4C2×#1171994711yescharacteristic
384C8—22no—
385C8—22no—
386C8—2122no—
387C8—2122no—
388C8—2122no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—1064no—
393C22—4244no—
394C22—2122no—
395C22—2122no—
396C22C2×#1313318811yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—2122no—
406C4#135×D5312yes—
407C4#1681143112yes—
408C2—41no—
409C2—2124no—
410C2#2028422311yescharacteristic
411C2#2028429211yescharacteristic
412C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3231C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#3231D53
hypocenterC53#19395
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3231

Complements

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

NH ≅ G/Ntype
4. #20274163C2semidirect
5. #20274135C2semidirect
6. #20274101C2semidirect
7. #20274087C2semidirect
8. #20273495C2semidirect
9. #20273425C2semidirect
10. #20273380C2semidirect
11. C53×#3231C2semidirect
30. #16810376C22semidirect
31. #16810374C22semidirect
32. #16810372C22semidirect
33. #16810369C22semidirect
34. #16810369C22semidirect
35. #16810369C22semidirect
38. C53×#500C22semidirect
39. C53×#498C22semidirect
40. C53×#487C22semidirect
41. C53×#487C22semidirect
202. #3231D53semidirect
221. #500C2×D53semidirect
222. #498C2×D53semidirect
223. #487C2×D53semidirect
224. #487C2×D53semidirect
251. C53#19395semidirect

Maximal subgroups

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

subgroupindex
#202848102
#202847192
#202837382
#202741632
#202741352
#202741012
#202740872
#202734952
#202734252
#202733802
C53×#32312
#1939553