Finite Group Databasebeta

← #15830648

Subgroups — #15830648

Order 3328.

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#13128701C211yesmaximal, characteristic
2#13123917C211yesmaximal, characteristic, complemented
3#13113411C214yesmaximal, complemented
4#13113399C211yesmaximal, characteristic, complemented
5#13113395C211yesmaximal, characteristic
6#13113334C211yesmaximal, characteristic, complemented
7#13113330C211yesmaximal, characteristic
8#13113320C214yesmaximal, complemented
9C13×#2356C211yesmaximal, characteristic, complemented
10C2×C4×Q8×C13C2211yescharacteristic
11#11719521—21no—
12#11719290C412yescomplemented
13#11719289C412yescomplemented
14#11719286—22no—
15#11719284C412yescomplemented
16#11719273—22no—
17#11719272C412yescomplemented
18#11719220—22no—
19#11719209—21no—
20#11719184C2214yes—
21#11719182C2211yescharacteristic
22#11719175C2211yescharacteristic
23C13×#391—21no—
24C13×#387—22no—
25C13×#383C2212yes—
26C13×#382C2211yescharacteristic
27C4×C13×C4⋊C4—21no—
28C4×C13×C4⋊C4C2211yescharacteristic
29C13×#376C2212yes—
30Q8×#1055—22no—
31Q8×#1055—22no—
32#11718586C2211yescharacteristic
33#11718586C2211yescharacteristic
34#11718586C2214yes—
35#11718044C2214yes—
36#11718044C2212yes—
37#11718044C2212yes—
38#11718043C2214yes—
39#11718043C2212yes—
40#11718043C2212yes—
41C22×Q8×C13D411yescharacteristic
42C2×#6127—21no—
43C4×Q8×C13C2×C412yes—
44C4×Q8×C13C2×C412yes—
45C4×Q8×C13D412yes—
46C4×Q8×C13D412yes—
47C2×C13×C4⋊C4—21no—
48C2×C13×C4⋊C4—21no—
49C2×C13×C4⋊C4D411yescharacteristic
50C2×C13×C4⋊C4D412yes—
51C2×C42×C13—21no—
52C2×C42×C13—22no—
53C2×C42×C13C2311yescharacteristic
54C2×C42×C13C2312yes—
55#88201—22no—
56#88200—22no—
57#88200—22no—
58#88198—22no—
59#88184—21no—
60#88184C2312yes—
61#88183—24no—
62C4×C22⋊D13—24no—
63#88180—21no—
64#88178—21no—
65#88178C2312yes—
66#88152—24no—
67#88150—24no—
68#88148—24no—
69#88147—22no—
70#88146—22no—
71C4⋊C4×D13—22no—
72#88143—24no—
73#88141—22no—
74#88130—24no—
75#88129—21no—
76#88129C2×C414yes—
77#88128C2×C411yescharacteristic
78C4×D52—21no—
79C4×D52C2×C412yes—
80C4×D52C2×C414yes—
81#88126—24no—
82#88124—21no—
83#88124C2×C414yes—
84#88123C2×C411yescharacteristic
85#88123C2×C411yescharacteristic
86#88123C2×C411yescharacteristic
87C4×Dic26—21no—
88C4×Dic26C2×C412yes—
89C4×Dic26C2×C414yes—
90C13×#95—22no—
91C13×#95—21no—
92C13×#95—22no—
93C13×#95—22no—
94C13×#95C2311yescharacteristic
95C13×#95C2311yescharacteristic
96C13×#95C2312yes—
97C22⋊#1055C2312yes—
98C22⋊#1055C2312yes—
99C4⋊#1055—21no—
100C4⋊#1055—21no—
101C4⋊#1055—24no—
102C4×#1055—21no—
103C4×#1055—21no—
104C4×#1055—24no—
105#17616—131nomaximal, Sylow
106C2×Q8×C13—22no—
107C2×Q8×C13—22no—
108C2×Q8×C13C22⋊C412yes—
109C2×Q8×C13C22⋊C412yes—
110C22×C4×C13—21no—
111C22×C4×C13—22no—
112C22×C4×C13—21no—
113C22×C4×C13C2×D412yes—
114C22×C4×C13C2×D411yescharacteristic
115C22×C4×C13C2×D412yes—
116C22×C4×C13C2×D411yescharacteristic
117C22×C4×C13C2411yescharacteristic
118C2×C22⋊D13—22no—
119#6127—42no—
120#6127—42no—
121C2×D52—21no—
122C2×D52—21no—
123C2×D52C22×C412yes—
124C2×C4×D13—22no—
125C2×C4×D13—24no—
126C2×Dic26—21no—
127C2×Dic26—21no—
128C2×Dic26C22×C412yes—
129C13×C4⋊C4—42no—
130C13×C4⋊C4—42no—
131C13×C4⋊C4C22⋊C412yes—
132C13×C4⋊C4C22⋊C414yes—
133C13×C4⋊C4C22⋊C412yes—
134C13×C4⋊C4C22⋊C414yes—
135C42×C13—44no—
136C42×C13—42no—
137C42×C13C22×C411yescharacteristic
138C42×C13C22×C414yes—
139C42×C13C22×C411yescharacteristic
140C42×C13C2×D411yescharacteristic
141C42×C13C2×D411yescharacteristic
142C42×C13C2×D414yes—
143C22⋊Dic13—22no—
144#6103—24no—
145#6103—22no—
146#6103—24no—
147#6103—24no—
148#6103—22no—
149C4⋊Dic13—21no—
150C4⋊Dic13—21no—
151C4⋊Dic13—21no—
152C4⋊Dic13—21no—
153C4⋊Dic13C22×C412yes—
154C4⋊Dic13C22×C412yes—
155#6101—22no—
156#6101—24no—
157#6101—24no—
158#6101—24no—
159#6101—22no—
160C4×Dic13—22no—
161C4×Dic13—24no—
162C2×#1055—22no—
163C2×#1055—22no—
164#3507—131no—
165#2982—131no—
166#2356D1311yescharacteristic, complemented
167#1521—134no—
168#1518—131no—
169#1517—131no—
170#1503—131no—
171#1502—131no—
172#1500—134no—
173C23×C13C22×D411yescharacteristic
174C22×D13—22no—
175Q8×C13—24no—
176Q8×C13—24no—
177C2×C4×C13—22no—
178C2×C4×C13—22no—
179C2×C4×C13—22no—
180C2×C4×C13—42no—
181C2×C4×C13—41no—
182C2×C4×C13—41no—
183C2×C4×C13—41no—
184C2×C4×C13—41no—
185C2×C4×C13—42no—
186C2×C4×C13—41no—
187C2×C4×C13—41no—
188C2×C4×C13—42no—
189C2×C4×C13—41no—
190C2×C4×C13—42no—
191C2×C4×C13C2×C22⋊C414yes—
192C2×C4×C13C2×C22⋊C411yescharacteristic
193C2×C4×C13C2×C22⋊C414yes—
194C2×C4×C13C2×C22⋊C411yescharacteristic
195C2×C4×C13C2×C22⋊C411yescharacteristic
196C2×C4×C13C2×C22⋊C411yescharacteristic
197C2×C4×C13#13512yes—
198C2×C4×C13#13512yes—
199C2×C4×C13C23×C411yescharacteristic
200C2×C4×C13C22×D411yescharacteristic
201C22⋊D13—42no—
202C22⋊D13—42no—
203C2×Dic13—22no—
204C2×Dic13—22no—
205C2×Dic13—22no—
206D52—42no—
207D52—42no—
208D52—81no—
209C4×D13—44no—
210C4×D13—84no—
211Dic26—42no—
212Dic26—42no—
213Dic26—81no—
214#1055—42no—
215#1055—42no—
216#559—132no—
217#557—132no—
218#556—132no—
219#556—262no—
220#554—132no—
221#552—262no—
222#539—262no—
223#536—261no—
224#533—131no—
225#532—131no—
226#529—134no—
227#520—261no—
228C2×C4×Q8C2×D1311yescharacteristic
229C8×Q8—262no—
230C8×Q8—262no—
231#431—134no—
232#431—131no—
233#431—131no—
234#391—21no—
235#387—22no—
236#383C2×D1312yes—
237#382C2×D1311yescharacteristic
238C4×C4⋊C4—21no—
239C4×C4⋊C4C2×D1311yescharacteristic
240#376C2×D1312yes—
241#328—132no—
242#328—134no—
243#328—132no—
244#327—132no—
245#327—134no—
246#327—132no—
247C22×C13—21no—
248C22×C13—21no—
249C22×C13C22×C22⋊C411yescharacteristic
250C22×C13C2×#13511yescharacteristic
251C22×C13C2×#13511yescharacteristic
252C2×D13—42no—
253C2×D13—42no—
254C2×D13—42no—
255C4×C13—21no—
256C4×C13—21no—
257C4×C13—24no—
258C4×C13—21no—
259C4×C13—24no—
260C4×C13—21no—
261C4×C13—42no—
262C4×C13—81no—
263C4×C13—82no—
264C4×C13#41612yes—
265C4×C13#41612yes—
266Dic13—42no—
267Dic13—42no—
268Dic13—42no—
269C2×#41—261no—
270C22×Q8C22⋊D1311yescharacteristic
271C4⋊Q8—131no—
272C4⋊Q8—131no—
273C4⋊Q8—131no—
274C4⋊Q8—262no—
275#127—131no—
276#126—264no—
277#126—264no—
278#125—134no—
279#125—264no—
280#125—261no—
281#124—134no—
282#124—262no—
283#124—261no—
284#123—264no—
285#123—264no—
286C22⋊Q8—132no—
287C22⋊Q8—262no—
288C22⋊Q8—262no—
289C22⋊Q8—261no—
290C22⋊Q8—264no—
291#121—132no—
292#121—261no—
293C4×Q8—134no—
294C4×Q8—132no—
295C4×Q8—262no—
296C4×Q8—261no—
297C4×Q8C4×D1312yes—
298C4×Q8C4×D1312yes—
299C4×Q8D5212yes—
300C4×Q8D5212yes—
301C4×D4—134no—
302C4×D4—132no—
303C4×D4—264no—
304C4×D4—262no—
305C4×D4—261no—
306#117—261no—
307#117—262no—
308#117—264no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—262no—
312C2×C4⋊C4C22⋊D1311yescharacteristic
313C2×C4⋊C4C22⋊D1312yes—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42C22×D1312yes—
317C2×C42C22×D1311yescharacteristic
318C4⋊C8—264no—
319C4⋊C8—261no—
320C4⋊C8—261no—
321C22⋊C8—132no—
322C22⋊C8—132no—
323C4×C8—264no—
324C4×C8—261no—
325C4×C8—261no—
326#95—22no—
327#95—22no—
328#95—21no—
329#95—22no—
330#95C22×D1311yescharacteristic
331#95C22×D1312yes—
332#95C22×D1311yescharacteristic
333C2×C13—21no—
334C2×C13—21no—
335C2×C13#288411yescharacteristic
336C2×C13C2×#41611yescharacteristic
337C2×C13#290111yescharacteristic
338D13—82no—
339#41—522no—
340#41—522no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—132no—
344C2×Q8—261no—
345C2×Q8—261no—
346C2×Q8#610312yes—
347C2×Q8#610312yes—
348C2×D4—132no—
349C2×D4—262no—
350C2×D4—261no—
351C2×D4—261no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—262no—
356C22×C4—264no—
357C22×C4C2×C22⋊D1312yes—
358C22×C4C2×C22⋊D1312yes—
359C22×C4C2×C22⋊D1311yescharacteristic
360C22×C4C2×C22⋊D1311yescharacteristic
361C22×C4C23×D1311yescharacteristic
362C2×C8—262no—
363C2×C8—262no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—132no—
367C4⋊C4—132no—
368C4⋊C4—261no—
369C4⋊C4—264no—
370C4⋊C4—261no—
371C4⋊C4—264no—
372C4⋊C4—261no—
373C4⋊C4—262no—
374C4⋊C4—264no—
375C4⋊C4—261no—
376C4⋊C4—262no—
377C4⋊C4#610312yes—
378C4⋊C4#610312yes—
379C4⋊C4#610314yes—
380C4⋊C4#610314yes—
381C22⋊C4—262no—
382C22⋊C4—264no—
383C22⋊C4—264no—
384C22⋊C4—262no—
385C22⋊C4—264no—
386C22⋊C4—262no—
387C42—44no—
388C42—42no—
389C42—264no—
390C42—262no—
391C42C2×C4×D1311yescharacteristic
392C42C2×C4×D1314yes—
393C42C2×C4×D1311yescharacteristic
394C42C2×D5214yes—
395C42C2×D5211yescharacteristic
396C42C2×D5211yescharacteristic
397C13#1761611yescharacteristic, complemented, Sylow
398C23—262no—
399C23C22×C22⋊D1311yescharacteristic
400Q8—24no—
401Q8—24no—
402Q8—522no—
403Q8—522no—
404Q8—1041no—
405D4—522no—
406D4—522no—
407D4—522no—
408D4—522no—
409D4—1041no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—262no—
425C2×C4—262no—
426C2×C4—262no—
427C2×C4—524no—
428C2×C4—1044no—
429C2×C4C2×#610314yes—
430C2×C4C2×#610314yes—
431C2×C4C2×#610311yescharacteristic
432C2×C4C2×#610311yescharacteristic
433C2×C4C2×#610311yescharacteristic
434C2×C4C2×#610311yescharacteristic
435C2×C4#8820412yes—
436C2×C4#8820412yes—
437C2×C4C22×C4×D1311yescharacteristic
438C2×C4C22×D5211yescharacteristic
439C8—522no—
440C8—522no—
441C22—21no—
442C22—21no—
443C22—522no—
444C22—522no—
445C22—522no—
446C22C22×#610311yescharacteristic
447C22C2×#8820411yescharacteristic
448C22C2×#8820411yescharacteristic
449C4—24no—
450C4—24no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—82no—
458C4—522no—
459C4—522no—
460C4—522no—
461C4#1171866312yes—
462C4#1171866312yes—
463C2—21no—
464C2—21no—
465C2—1042no—
466C2#1312329911yescharacteristic
467C2C2×#1171866311yescharacteristic
468C2#1312338411yescharacteristic
subgroupG/H
centerC22C22×#6103
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2356C2
socleC22×C13C22×C22⋊C4
prime radical#11719182C22
cosocleC22×C4×C13C24
hypercenter#2356D13
hypocenterC13#17616
LCSG ▷ C2×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2356

Complements

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

NH ≅ G/Ntype
2. #13123917C2semidirect
3. #13113411C2semidirect
4. #13113399C2semidirect
6. #13113334C2semidirect
8. #13113320C2semidirect
9. C13×#2356C2semidirect
12. #11719290C4semidirect
13. #11719289C4semidirect
15. #11719284C4semidirect
17. #11719272C4semidirect
166. #2356D13semidirect
397. C13#17616semidirect

Maximal subgroups

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

subgroupindex
#131287012
#131239172
#131134112
#131133992
#131133952
#131133342
#131133302
#131133202
C13×#23562
#1761613