Finite Group Databasebeta

← #15907785

Subgroups — #15907785

Order 3328.

842 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13119754C211yesmaximal, characteristic
2#13117736C211yesmaximal, characteristic
3#13116271C212yesmaximal, complemented
4#13116031C212yesmaximal, complemented
5#13115629C211yesmaximal, characteristic
6#13115600C211yesmaximal, characteristic, complemented
7#13115140C212yesmaximal, complemented
8#13114963C211yesmaximal, characteristic, complemented
9#13114921C212yesmaximal, complemented
10C4×#88075C211yesmaximal, characteristic, complemented
11C13×#1803C211yesmaximal, characteristic, complemented
12C2×C4×Q8×C13C2211yescharacteristic
13C2×#88200C2212yes—
14C2×#88147—21no—
15C2×C4⋊C4×D13C2212yes—
16C2×C4×D52C2211yescharacteristic
17C2×C13×C4⋊C8C2211yescharacteristic
18C2×C13×#103C2212yes—
19C13×#384C2211yescharacteristic
20C2×#88075C2211yescharacteristic
21C2×#88075C2211yescharacteristic
22C2×#88060C2212yes—
23C2×#88060C2212yes—
24C2×#88057C2212yes—
25#11718655—22no—
26#11718650—22no—
27#11718648C2212yes—
28#11718648C2212yes—
29#11718634—21no—
30#11718600—21no—
31#11718596C2211yescharacteristic
32C4×C4⋊Dic13C2211yescharacteristic
33#11718589—21no—
34C2×C4×#1055C2211yescharacteristic
35C13×#340C2212yes—
36#11718143C2212yes—
37#11718121C2212yes—
38#11718121C2211yescharacteristic
39#11718121C2211yescharacteristic
40#11718121C2212yes—
41C22×Q8×C13C2311yescharacteristic
42C22×D52D411yescharacteristic
43C22×C4×D13—22no—
44C22×C4×D13C2312yes—
45C22×C8×C13C2312yes—
46C4×Q8×C13—22no—
47C4×Q8×C13—22no—
48C2×C13×C4⋊C4—21no—
49C2×C13×C4⋊C4—21no—
50C2×C13×C4⋊C4—21no—
51C2×C13×C4⋊C4C2312yes—
52C2×C13×C4⋊C4C2311yescharacteristic
53C2×C42×C13—21no—
54C2×C42×C13C2311yescharacteristic
55#88200—24no—
56#88200—42no—
57C2×#6103—22no—
58C2×#6103—22no—
59C2×#6103—22no—
60C2×#6103D412yes—
61C2×C4⋊Dic13—21no—
62C2×C4⋊Dic13—21no—
63C2×C4⋊Dic13D411yescharacteristic
64C2×C4⋊Dic13C2312yes—
65C2×C4⋊Dic13C2311yescharacteristic
66C2×C4⋊Dic13C2311yescharacteristic
67C2×#6101—22no—
68C2×#6101—22no—
69C2×C4×Dic13—21no—
70C22×#1055C2311yescharacteristic
71C22×#1055C2311yescharacteristic
72#88147—44no—
73C4⋊C4×D13—24no—
74C4⋊C4×D13—42no—
75C4×D52—22no—
76C4×D52—22no—
77C13×C4⋊C8—22no—
78C13×#103—42no—
79#88075—22no—
80#88075—22no—
81#88071—21no—
82#88071—22no—
83#88071—22no—
84#88060—42no—
85#88060—42no—
86#88057—42no—
87C4×#1055—22no—
88#24969—131nomaximal, Sylow
89C23×D13C2×D412yes—
90C2×Q8×C13—21no—
91C2×Q8×C13—21no—
92C2×Q8×C13—21no—
93C2×Q8×C13—21no—
94C2×Q8×C13—21no—
95C2×Q8×C13#4112yes—
96C22×C4×C13—21no—
97C22×C4×C13—21no—
98C22×C4×C13—22no—
99C22×C4×C13C2×D411yescharacteristic
100C22×C4×C13C2×D411yescharacteristic
101C22×C4×C13C2411yescharacteristic
102C22×Dic13—21no—
103C22×Dic13—21no—
104C22×Dic13—22no—
105C22×Dic13C2×D412yes—
106C2×D52—22no—
107C2×D52—22no—
108C2×D52SD1614yes—
109C2×C4×D13—22no—
110C2×C4×D13—22no—
111C2×C4×D13—22no—
112C2×C4×D13—22no—
113C2×C4×D13—22no—
114C2×C4×D13—22no—
115C2×C4×D13—44no—
116C2×C4×D13—44no—
117C2×C4×D13—82no—
118C2×C8×C13—22no—
119C2×C8×C13—22no—
120C2×C8×C13—22no—
121C13×C4⋊C4—22no—
122C13×C4⋊C4—21no—
123C13×C4⋊C4—21no—
124C13×C4⋊C4—22no—
125C13×C4⋊C4—42no—
126C13×C4⋊C4—42no—
127C13×C4⋊C4—81no—
128C42×C13—42no—
129C42×C13#4112yes—
130C42×C13#4112yes—
131#6103—24no—
132#6103—44no—
133#6103—44no—
134#6103—82no—
135C4⋊Dic13—22no—
136C4⋊Dic13—21no—
137C4⋊Dic13—21no—
138C4⋊Dic13—22no—
139C4⋊Dic13—22no—
140C4⋊Dic13—21no—
141C4⋊Dic13—42no—
142C4⋊Dic13—81no—
143C4⋊Dic13#4112yes—
144#6101—44no—
145#6101—44no—
146C4×Dic13—42no—
147C2×#1055—21no—
148C2×#1055—21no—
149C2×#1055—21no—
150C2×#1055—21no—
151C2×#1055—21no—
152C2×#1055—21no—
153#2538—131no—
154D4×C4⋊C4—131no—
155#2077—132no—
156#2035—132no—
157#1954—131no—
158#1948—131no—
159#1872—132no—
160#1834—131no—
161#1823—132no—
162#1803D1311yescharacteristic, complemented
163C4×#103—131no—
164C23×C13C22×D411yescharacteristic
165C22×D13—22no—
166C22×D13—22no—
167C22×D13—22no—
168C22×D13—22no—
169C22×D13—22no—
170C22×D13—22no—
171C22×D13C2×SD1614yes—
172Q8×C13—22no—
173Q8×C13—22no—
174Q8×C13—41no—
175Q8×C13—41no—
176C2×C4×C13—21no—
177C2×C4×C13—21no—
178C2×C4×C13—22no—
179C2×C4×C13—41no—
180C2×C4×C13—41no—
181C2×C4×C13—41no—
182C2×C4×C13—42no—
183C2×C4×C13—42no—
184C2×C4×C13—41no—
185C2×C4×C13—42no—
186C2×C4×C13—41no—
187C2×C4×C13#12312yes—
188C2×C4×C13#12312yes—
189C2×C4×C13C2×SD1612yes—
190C2×C4×C13#13712yes—
191C2×C4×C13C2×#4111yescharacteristic
192C2×C4×C13C2×#4111yescharacteristic
193C2×C4×C13C2×#4111yescharacteristic
194C2×C4×C13C2×#4111yescharacteristic
195C2×C4×C13H(5,𝔽2)11yescharacteristic
196C2×C4×C13H(5,𝔽2)11yescharacteristic
197C2×Dic13—22no—
198C2×Dic13—22no—
199C2×Dic13—22no—
200C2×Dic13—22no—
201C2×Dic13—41no—
202C2×Dic13—41no—
203C2×Dic13—42no—
204C2×Dic13—41no—
205C2×Dic13—41no—
206C2×Dic13—42no—
207C2×Dic13—42no—
208C2×Dic13—41no—
209D52—28no—
210C4×D13—42no—
211C4×D13—42no—
212C4×D13—42no—
213C4×D13—42no—
214C4×D13—88no—
215C8×C13—42no—
216#1055—41no—
217#1055—41no—
218C2×C22⋊Q8—132no—
219C2×C4×Q8C2×D1311yescharacteristic
220C2×C4×D4—131no—
221C2×C4×D4—261no—
222C22×C4⋊C4—132no—
223C2×#106—132no—
224C2×C4⋊C8C2×D1311yescharacteristic
225C2×#103—131no—
226C2×#103—131no—
227C2×#103C2×D1312yes—
228C2×C22⋊C8—132no—
229C2×C22⋊C8—132no—
230C2×C4×C8—131no—
231#396—262no—
232#390—261no—
233#389—261no—
234#389—261no—
235#385—262no—
236#384—131no—
237#384C2×D1311yescharacteristic
238#380—132no—
239#380—132no—
240C4×C4⋊C4—131no—
241#340—132no—
242#340—131no—
243#340—132no—
244#340—131no—
245#340—132no—
246#340C2×D1312yes—
247C22×C13C2×#12311yescharacteristic
248C22×C13#54611yescharacteristic
249C22×C13#54611yescharacteristic
250C22×C13C22×SD1611yescharacteristic
251C22×C13C2×#13711yescharacteristic
252C22×C13#57611yescharacteristic
253C22×C13#57811yescharacteristic
254C2×D13—24no—
255C2×D13—24no—
256C2×D13—24no—
257C2×D13—42no—
258C2×D13—42no—
259C2×D13—42no—
260C2×D13—42no—
261C4×C13—22no—
262C4×C13—42no—
263C4×C13—81no—
264C4×C13—82no—
265C4×C13#48314yes—
266C4×C13#52911yescharacteristic
267C4×C13#54011yescharacteristic
268C4×C13#55311yescharacteristic
269C4×C13#55511yescharacteristic
270Dic13—42no—
271Dic13—42no—
272Dic13—82no—
273Dic13—81no—
274C22×Q8C22×D1311yescharacteristic
275C22×D4—131no—
276C23×C4—132no—
277C23×C4—262no—
278C22×C8—131no—
279C22×C8—131no—
280C22×C8C22×D1312yes—
281C22⋊Q8—264no—
282C22⋊Q8—522no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×D4—262no—
286C4×D4—262no—
287C4×D4—524no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—131no—
292C2×C4⋊C4—131no—
293C2×C4⋊C4—132no—
294C2×C4⋊C4—131no—
295C2×C4⋊C4—264no—
296C2×C4⋊C4—262no—
297C2×C4⋊C4—262no—
298C2×C4⋊C4—261no—
299C2×C4⋊C4—261no—
300C2×C4⋊C4—522no—
301C2×C4⋊C4C22×D1312yes—
302C2×C4⋊C4C22×D1311yescharacteristic
303C2×C22⋊C4—132no—
304C2×C22⋊C4—262no—
305C2×C22⋊C4—262no—
306C2×C22⋊C4—262no—
307C2×C42—21no—
308C2×C42—261no—
309C2×C42C22×D1311yescharacteristic
310#106—522no—
311C4⋊C8—22no—
312#103—42no—
313#103—262no—
314#103—262no—
315C22⋊C8—522no—
316C22⋊C8—522no—
317C4×C8—262no—
318#95—262no—
319#95—262no—
320#95—261no—
321C2×C13#242011yescharacteristic
322C2×C13C2×#48311yescharacteristic
323C2×C13#309811yescharacteristic
324C2×C13#329511yescharacteristic
325C2×C13#329811yescharacteristic
326C2×C13#332811yescharacteristic
327C2×C13#333311yescharacteristic
328D13—44no—
329C24—132no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8#613112yes—
336C2×D4—134no—
337C2×D4—262no—
338C2×D4—262no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—132no—
343C22×C4—261no—
344C22×C4—262no—
345C22×C4—262no—
346C22×C4—262no—
347C22×C4—262no—
348C22×C4—262no—
349C22×C4—262no—
350C22×C4—261no—
351C22×C4—262no—
352C22×C4—524no—
353C22×C4—524no—
354C22×C4—1042no—
355C22×C4D4×D1311yescharacteristic
356C22×C4D4×D1311yescharacteristic
357C22×C4C23×D1311yescharacteristic
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—261no—
362C2×C8—261no—
363C2×C8—261no—
364C2×C8—261no—
365C2×C8—261no—
366C2×C8—261no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—81no—
374C4⋊C4—132no—
375C4⋊C4—262no—
376C4⋊C4—261no—
377C4⋊C4—261no—
378C4⋊C4—262no—
379C4⋊C4—261no—
380C4⋊C4—262no—
381C4⋊C4—524no—
382C4⋊C4—524no—
383C4⋊C4—522no—
384C4⋊C4—1041no—
385C22⋊C4—264no—
386C22⋊C4—524no—
387C22⋊C4—524no—
388C22⋊C4—1042no—
389C42—42no—
390C42—522no—
391C42#612712yes—
392C42#612912yes—
393C13#2496911yescharacteristic, complemented, Sylow
394C23—134no—
395C23—262no—
396C23—262no—
397C23—262no—
398C23—262no—
399C23—262no—
400C23—262no—
401C23C2×D4×D1311yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—41no—
405Q8—41no—
406D4—268no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—262no—
419C2×C4—262no—
420C2×C4—262no—
421C2×C4—262no—
422C2×C4—521no—
423C2×C4—521no—
424C2×C4—522no—
425C2×C4—522no—
426C2×C4—522no—
427C2×C4—522no—
428C2×C4—521no—
429C2×C4—522no—
430C2×C4—522no—
431C2×C4—521no—
432C2×C4—522no—
433C2×C4—521no—
434C2×C4—1048no—
435C2×C4#8813712yes—
436C2×C4#8813712yes—
437C2×C4SD16×D1312yes—
438C2×C4#8817212yes—
439C2×C4C2×#612711yescharacteristic
440C2×C4C2×#612911yescharacteristic
441C2×C4#8825111yescharacteristic
442C2×C4C2×#613111yescharacteristic
443C2×C4#41×D1311yescharacteristic
444C2×C4#8825611yescharacteristic
445C8—42no—
446C8—521no—
447C8—521no—
448C22—264no—
449C22—264no—
450C22—264no—
451C22—522no—
452C22—522no—
453C22—522no—
454C22—522no—
455C22C2×#8813711yescharacteristic
456C22#1171931711yescharacteristic
457C22#1171933511yescharacteristic
458C22C2×SD16×D1311yescharacteristic
459C22C2×#8817211yescharacteristic
460C22#1171947911yescharacteristic
461C22#1171948411yescharacteristic
462C4—22no—
463C4—42no—
464C4—82no—
465C4—81no—
466C4—522no—
467C4—522no—
468C4—1042no—
469C4—1041no—
470C4#1171846614yes—
471C4#1171921911yescharacteristic
472C4#1171923711yescharacteristic
473C4#1171932411yescharacteristic
474C4#1171932911yescharacteristic
475C2—524no—
476C2#1311833011yescharacteristic
477C2C2×#1171846611yescharacteristic
478C2#1312481311yescharacteristic
479C2#1312666311yescharacteristic
480C2#1312669811yescharacteristic
481C2#1312712211yescharacteristic
482C2#1312717111yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#1803C2
socleC23×C13C22×D4
prime radicalC22×D52D4
hypercenter#1803D13
hypocenterC13#24969
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
3. #13116271C2semidirect
4. #13116031C2semidirect
6. #13115600C2semidirect
7. #13115140C2semidirect
8. #13114963C2semidirect
9. #13114921C2semidirect
10. C4×#88075C2semidirect
11. C13×#1803C2semidirect
162. #1803D13semidirect
393. C13#24969semidirect

Maximal subgroups

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

subgroupindex
#131197542
#131177362
#131162712
#131160312
#131156292
#131156002
#131151402
#131149632
#131149212
C4×#880752
C13×#18032
#2496913