Finite Group Databasebeta

← #13118500

Subgroups — #13118500

Order 1664.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6103C211yesmaximal, characteristic
2C2×#88151C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D13C211yesmaximal, characteristic
4C2×#88132C211yesmaximal, characteristic, complemented
5C13×#385C211yesmaximal, characteristic, complemented
6#11718873C211yesmaximal, characteristic, complemented
7#11718655C211yesmaximal, characteristic, complemented
8#11718655C211yesmaximal, characteristic, complemented
9#11718648C211yesmaximal, characteristic, complemented
10#11718616C211yesmaximal, characteristic
11#11718611C211yesmaximal, characteristic, complemented
12#11718338C211yesmaximal, characteristic, complemented
13#11718334C211yesmaximal, characteristic, complemented
14#11718329C211yesmaximal, characteristic, complemented
15#11718329C211yesmaximal, characteristic, complemented
16C24×D13C2211yescharacteristic
17C23×C4×C13C2211yescharacteristic
18C23×Dic13C2211yescharacteristic
19C22×C4×D13C2211yescharacteristic
20C22×C4×D13C2211yescharacteristic
21C22×Dic26C2211yescharacteristic, complemented
22C2×C13×C4⋊C4C2211yescharacteristic, complemented
23C2×C13×C4⋊C4C2211yescharacteristic, complemented
24C2×C13×C22⋊C4C2211yescharacteristic
25C2×C13×C22⋊C4C2211yescharacteristic
26C2×C22⋊Dic13C2211yescharacteristic
27C2×C22⋊Dic13C2211yescharacteristic
28C2×#6103—24no—
29C2×#6103C2211yescharacteristic
30C2×#6103C2211yescharacteristic
31C2×#6103C2211yescharacteristic
32C2×#6103C2211yescharacteristic
33C2×#6103C2211yescharacteristic
34C2×#6103C2211yescharacteristic
35C2×#6103C2211yescharacteristic
36C2×#6103C2211yescharacteristic
37C2×#6103C2211yescharacteristic
38C2×#6103C2211yescharacteristic
39C2×C4⋊Dic13C2211yescharacteristic, complemented
40C2×C4⋊Dic13C2211yescharacteristic, complemented
41C2×C4⋊Dic13C2211yescharacteristic, complemented
42C2×C4⋊Dic13C2211yescharacteristic, complemented
43C2×C4⋊Dic13C2211yescharacteristic, complemented
44C2×#6101C2211yescharacteristic, complemented
45C2×#6101C2211yescharacteristic, complemented
46#88151—24no—
47C22⋊C4×D13—24no—
48#88132—24no—
49C13×#95C2211yescharacteristic, complemented
50C13×#95C2211yescharacteristic, complemented
51#88071C2211yescharacteristic, complemented
52#88071C2211yescharacteristic, complemented
53#88071C2211yescharacteristic, complemented
54#88071C2211yescharacteristic, complemented
55C24×C13C2311yescharacteristic
56C23×D13—22no—
57C23×D13—22no—
58C23×D13—22no—
59C23×D13—22no—
60C23×D13—41no—
61C23×D13—41no—
62C23×D13Q814yescomplemented
63C23×D13C2311yescharacteristic
64C23×D13C2311yescharacteristic
65C22×C4×C13—22no—
66C22×C4×C13—22no—
67C22×C4×C13—41no—
68C22×C4×C13C2311yescharacteristic
69C22×C4×C13C2311yescharacteristic
70C22×C4×C13C2311yescharacteristic
71C22×C4×C13C2311yescharacteristic
72C22×C4×C13C2311yescharacteristic
73C22×C4×C13C2311yescharacteristic
74C22×Dic13—22no—
75C22×Dic13—22no—
76C22×Dic13—41no—
77C22×Dic13C2311yescharacteristic
78C22×Dic13C2311yescharacteristic
79C22×Dic13C2311yescharacteristic
80C22×Dic13C2311yescharacteristic
81C22×Dic13C2311yescharacteristic
82C22×Dic13C2311yescharacteristic
83C2×C4×D13—22no—
84C2×C4×D13—22no—
85C2×C4×D13—22no—
86C2×C4×D13—22no—
87C2×C4×D13—41no—
88C2×C4×D13—41no—
89C2×Dic26—22no—
90C2×Dic26—22no—
91C2×Dic26—41no—
92C13×C4⋊C4—22no—
93C13×C4⋊C4—41no—
94C13×C22⋊C4—22no—
95C13×C22⋊C4D414yes—
96C22⋊Dic13—22no—
97C22⋊Dic13—22no—
98#6103—22no—
99#6103—22no—
100#6103—22no—
101#6103—22no—
102#6103—22no—
103#6103—22no—
104#6103—22no—
105#6103—41no—
106#6103—41no—
107#6103D414yes—
108C4⋊Dic13—22no—
109C4⋊Dic13—22no—
110C4⋊Dic13—22no—
111C4⋊Dic13—41no—
112C4⋊Dic13—41no—
113#6101—22no—
114#6101—22no—
115#2437—131nomaximal, Sylow
116C23×C13—21no—
117C23×C13—21no—
118C23×C13—21no—
119C23×C13—21no—
120C23×C13C2×D412yes—
121C23×C13C2×Q812yes—
122C23×C13#4112yes—
123C23×C13C2411yescharacteristic
124C22×D13—21no—
125C22×D13—21no—
126C22×D13—21no—
127C22×D13—21no—
128C22×D13—28no—
129C22×D13—21no—
130C22×D13—21no—
131C22×D13—21no—
132C22×D13—21no—
133C22×D13—21no—
134C22×D13—21no—
135C22×D13—28no—
136C22×D13—44no—
137C22×D13—44no—
138C22×D13—44no—
139C22×D13—44no—
140C22×D13—44no—
141C22×D13C2×D412yes—
142C22×D13C2×Q812yes—
143C22×D13C2×Q812yes—
144C22×D13#4112yes—
145C2×C4×C13—21no—
146C2×C4×C13—21no—
147C2×C4×C13—21no—
148C2×C4×C13—21no—
149C2×C4×C13—21no—
150C2×C4×C13—21no—
151C2×C4×C13—21no—
152C2×C4×C13—21no—
153C2×C4×C13—21no—
154C2×C4×C13—21no—
155C2×C4×C13—21no—
156C2×C4×C13—21no—
157C2×C4×C13—21no—
158C2×C4×C13—21no—
159C2×C4×C13—21no—
160C2×C4×C13—44no—
161C2×C4×C13C2×D412yes—
162C2×C4×C13C2×D412yes—
163C2×C4×C13C2×D412yes—
164C2×Dic13—21no—
165C2×Dic13—21no—
166C2×Dic13—21no—
167C2×Dic13—21no—
168C2×Dic13—21no—
169C2×Dic13—21no—
170C2×Dic13—21no—
171C2×Dic13—21no—
172C2×Dic13—21no—
173C2×Dic13—21no—
174C2×Dic13—21no—
175C2×Dic13—21no—
176C2×Dic13—21no—
177C2×Dic13—21no—
178C2×Dic13—21no—
179C2×Dic13—21no—
180C2×Dic13—21no—
181C2×Dic13—44no—
182C2×Dic13C2×D412yes—
183C4×D13—44no—
184C4×D13—44no—
185Dic26—44no—
186C2×C22⋊Q8—131no—
187C2×C22⋊Q8—131no—
188C22×C22⋊C4—131no—
189C22×C22⋊C4—131no—
190#399—131no—
191#396—131no—
192#396—131no—
193#393—131no—
194#385—131no—
195#385—131no—
196#385—131no—
197#385D1311yescharacteristic, complemented
198#380—131no—
199#380—131no—
200#379—131no—
201C22×C13—22no—
202C22×C13—22no—
203C22×C13—22no—
204C22×C13—22no—
205C22×C13—22no—
206C22×C13—22no—
207C22×C13C22⋊Q814yescomplemented
208C22×C13C22×D411yescharacteristic
209C22×C13C22×D411yescharacteristic
210C22×C13C22×Q811yescharacteristic
211C22×C13C2×#4111yescharacteristic
212C22×C13C2×#4111yescharacteristic
213C22×C13H(5,𝔽2)11yescharacteristic
214C22×C13H(5,𝔽2)11yescharacteristic
215C2×D13—22no—
216C2×D13—22no—
217C2×D13—22no—
218C2×D13—22no—
219C2×D13—22no—
220C2×D13—22no—
221C2×D13—22no—
222C2×D13—22no—
223C2×D13—22no—
224C2×D13—416no—
225C2×D13—41no—
226C2×D13—41no—
227C2×D13—41no—
228C2×D13—41no—
229C2×D13C22⋊Q814yescomplemented
230C4×C13—22no—
231C4×C13—22no—
232C4×C13—22no—
233C4×C13—41no—
234C4×C13—41no—
235C4×C13—41no—
236Dic13—22no—
237Dic13—41no—
238Dic13—41no—
239Dic13—41no—
240Dic13—41no—
241Dic13—41no—
242C25—131no—
243C22×Q8—131no—
244C23×C4—131no—
245C23×C4—131no—
246C23×C4—131no—
247C23×C4C2×D1311yescharacteristic
248C22⋊Q8—264no—
249C22⋊Q8—264no—
250C2×C4⋊C4—131no—
251C2×C4⋊C4—131no—
252C2×C4⋊C4—131no—
253C2×C4⋊C4—131no—
254C2×C4⋊C4—131no—
255C2×C4⋊C4—131no—
256C2×C4⋊C4—131no—
257C2×C4⋊C4C2×D1311yescharacteristic, complemented
258C2×C4⋊C4C2×D1311yescharacteristic, complemented
259C2×C22⋊C4—131no—
260C2×C22⋊C4—131no—
261C2×C22⋊C4—131no—
262C2×C22⋊C4—131no—
263C2×C22⋊C4—131no—
264C2×C22⋊C4—131no—
265C2×C22⋊C4—131no—
266C2×C22⋊C4—131no—
267C2×C22⋊C4—131no—
268C2×C22⋊C4—131no—
269C2×C22⋊C4—131no—
270C2×C22⋊C4—131no—
271C2×C22⋊C4—264no—
272C2×C22⋊C4—264no—
273C2×C22⋊C4C2×D1311yescharacteristic
274C2×C22⋊C4C2×D1311yescharacteristic
275#95—131no—
276#95—131no—
277#95—131no—
278#95—131no—
279#95C2×D1311yescharacteristic, complemented
280#95C2×D1311yescharacteristic, complemented
281C2×C13—24no—
282C2×C13C2×C22⋊Q811yescharacteristic
283C2×C13C2×C22⋊Q811yescharacteristic
284C2×C13#54311yescharacteristic
285C2×C13D4211yescharacteristic
286C2×C13#54611yescharacteristic
287C2×C13#54611yescharacteristic
288C2×C13#55111yescharacteristic
289D13—24no—
290D13—42no—
291C24—131no—
292C24—131no—
293C24—134no—
294C24—262no—
295C24—262no—
296C24—262no—
297C24—262no—
298C24—521no—
299C24—521no—
300C24C22×D1311yescharacteristic
301C2×Q8—262no—
302C2×Q8—262no—
303C2×Q8—521no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—131no—
308C22×C4—131no—
309C22×C4—131no—
310C22×C4—131no—
311C22×C4—131no—
312C22×C4—131no—
313C22×C4—262no—
314C22×C4—262no—
315C22×C4—262no—
316C22×C4—262no—
317C22×C4—262no—
318C22×C4—262no—
319C22×C4—521no—
320C22×C4—521no—
321C22×C4—521no—
322C22×C4C22×D1311yescharacteristic
323C22×C4C22×D1311yescharacteristic
324C22×C4C22×D1311yescharacteristic
325C22×C4C22×D1311yescharacteristic
326C22×C4C22×D1311yescharacteristic
327C22×C4C22×D1311yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—41no—
330C4⋊C4—262no—
331C4⋊C4—262no—
332C4⋊C4—262no—
333C4⋊C4—262no—
334C4⋊C4—262no—
335C4⋊C4—521no—
336C4⋊C4—521no—
337C22⋊C4—22no—
338C22⋊C4—134no—
339C22⋊C4—262no—
340C22⋊C4—262no—
341C22⋊C4—262no—
342C22⋊C4—262no—
343C22⋊C4—262no—
344C22⋊C4—262no—
345C22⋊C4—262no—
346C22⋊C4—262no—
347C22⋊C4—262no—
348C22⋊C4—521no—
349C22⋊C4—521no—
350C22⋊C4D5214yes—
351C13#243711yescharacteristic, complemented, Sylow
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—132no—
357C23—132no—
358C23—132no—
359C23—132no—
360C23—268no—
361C23—261no—
362C23—261no—
363C23—261no—
364C23—261no—
365C23—268no—
366C23—261no—
367C23—261no—
368C23—261no—
369C23—261no—
370C23—261no—
371C23—261no—
372C23—524no—
373C23—524no—
374C23—524no—
375C23—524no—
376C23—524no—
377C23C2×D5212yes—
378C23#612912yes—
379C23Q8×D1312yes—
380C23C23×D1311yescharacteristic
381Q8—524no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—44no—
398C2×C4—132no—
399C2×C4—261no—
400C2×C4—261no—
401C2×C4—261no—
402C2×C4—261no—
403C2×C4—261no—
404C2×C4—261no—
405C2×C4—261no—
406C2×C4—261no—
407C2×C4—261no—
408C2×C4—261no—
409C2×C4—261no—
410C2×C4—261no—
411C2×C4—261no—
412C2×C4—261no—
413C2×C4—261no—
414C2×C4—261no—
415C2×C4—261no—
416C2×C4—524no—
417C2×C4—524no—
418C2×C4—524no—
419C2×C4C2×D5212yes—
420C2×C4C2×D5212yes—
421C2×C4D4×D1312yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—134no—
429C22—262no—
430C22—262no—
431C22—262no—
432C22—262no—
433C22—262no—
434C22—262no—
435C22—262no—
436C22—262no—
437C22—262no—
438C22—5216no—
439C22—521no—
440C22—521no—
441C22—521no—
442C22—521no—
443C22#8815114yescomplemented
444C22C22×D5211yescharacteristic
445C22C2×D4×D1311yescharacteristic
446C22C2×#612911yescharacteristic
447C22#8825111yescharacteristic
448C22C2×Q8×D1311yescharacteristic
449C22#41×D1311yescharacteristic
450C22#8825611yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—262no—
458C4—521no—
459C4—521no—
460C4—521no—
461C4—521no—
462C4—521no—
463C2—24no—
464C2—264no—
465C2—522no—
466C2C2×#8815111yescharacteristic
467C2D4×D5211yescharacteristic
468C2#1171925911yescharacteristic
469C2#1171929611yescharacteristic
470C2D13×C22⋊Q811yescharacteristic
471C2#1171933911yescharacteristic
472C2#1171936311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#385C2
prime radicalC24×D13C22
hypercenter#385D13
hypocenterC13#2437
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#88151C2semidirect
4. C2×#88132C2semidirect
5. C13×#385C2semidirect
6. #11718873C2semidirect
7. #11718655C2semidirect
8. #11718655C2semidirect
9. #11718648C2semidirect
11. #11718611C2semidirect
12. #11718338C2semidirect
13. #11718334C2semidirect
14. #11718329C2semidirect
15. #11718329C2semidirect
21. C22×Dic26C22semidirect
22. C2×C13×C4⋊C4C22semidirect
23. C2×C13×C4⋊C4C22semidirect
39. C2×C4⋊Dic13C22semidirect
40. C2×C4⋊Dic13C22semidirect
41. C2×C4⋊Dic13C22semidirect
42. C2×C4⋊Dic13C22semidirect
43. C2×C4⋊Dic13C22semidirect
44. C2×#6101C22semidirect
45. C2×#6101C22semidirect
49. C13×#95C22semidirect
50. C13×#95C22semidirect
51. #88071C22semidirect
52. #88071C22semidirect
53. #88071C22semidirect
54. #88071C22semidirect
62. C23×D13Q8semidirect
197. #385D13semidirect
207. C22×C13C22⋊Q8semidirect
229. C2×D13C22⋊Q8semidirect
257. C2×C4⋊C4C2×D13semidirect
258. C2×C4⋊C4C2×D13semidirect
279. #95C2×D13semidirect
280. #95C2×D13semidirect
351. C13#2437semidirect
443. C22#88151semidirect

Maximal subgroups

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

subgroupindex
C22×#61032
C2×#881512
C2×C22⋊C4×D132
C2×#881322
C13×#3852
#117188732
#117186552
#117186482
#117186162
#117186112
#117183382
#117183342
#117183292
#243713