Finite Group Databasebeta

← #28480133

Subgroups — #28480133

Order 15104.

1276 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273706C211yesmaximal, characteristic
2#21424006C211yesmaximal, characteristic
3C2×#16837138C214yesmaximal, complemented
4C59×C22⋊#95C211yesmaximal, characteristic
5C23×Dic118C411yescharacteristic, complemented
6C2×#14273706—28no—
7C2×#14273706—21no—
8C2×#14273706—21no—
9C2×#14273706C412yescomplemented
10C2×#14273706C2212yes—
11C22×C4⋊Dic59C411yescharacteristic, complemented
12C22×Dic59⋊C4—21no—
13C22×C22⋊Dic59—21no—
14C2×C22⋊(C59⋊C8)—21no—
15C2×C22⋊(C59⋊C8)—21no—
16C2×C22⋊(C59⋊C8)C2212yes—
17#16837138—28no—
18C24×C4×C59C2211yescharacteristic
19C22×C22⋊C4×C59—21no—
20C2×#95×C59—21no—
21C2×#95×C59C2212yes—
22C22×Dic118—21no—
23C22×Dic118—24no—
24C22×Dic118—42no—
25C22×Dic118—42no—
26C22×Dic118C2×C412yes—
27C23×Dic59—21no—
28C23×Dic59—21no—
29C22×C59⋊C8—22no—
30#14273706—28no—
31#14273706—28no—
32#14273706—24no—
33#14273706—24no—
34#14273706—24no—
35#14273706D418yes—
36C2×C4⋊Dic59—21no—
37C2×C4⋊Dic59—24no—
38C2×C4⋊Dic59C2×C412yes—
39C2×Dic59⋊C4—28no—
40C2×Dic59⋊C4—22no—
41C2×Dic59⋊C4—22no—
42C2×C22⋊Dic59—28no—
43C2×C22⋊Dic59—22no—
44C2×C22⋊Dic59—22no—
45C22⋊(C59⋊C8)—24no—
46C22⋊(C59⋊C8)—42no—
47C22⋊(C59⋊C8)—42no—
48C25×C59D411yescharacteristic
49C23×C4×C59—24no—
50C23×C4×C59—22no—
51C23×C4×C59—42no—
52C23×C4×C59—42no—
53C23×C4×C59C2×C411yescharacteristic
54C23×C4×C59C2×C411yescharacteristic
55C23×C4×C59D411yescharacteristic
56C23×C4×C59D411yescharacteristic
57C23×C4×C59D411yescharacteristic
58C23×C4×C59C2311yescharacteristic
59C2×C22⋊C4×C59—24no—
60C2×C22⋊C4×C59—42no—
61C2×C22⋊C4×C59—42no—
62#95×C59—24no—
63#95×C59—42no—
64C24×C59—22no—
65C24×C59—22no—
66C24×C59—22no—
67C24×C59—22no—
68C24×C59—41no—
69C24×C59—41no—
70C24×C59C22⋊C411yescharacteristic
71C24×C59C22⋊C411yescharacteristic
72C24×C59SD1612yes—
73C24×C59Q1612yes—
74C24×C59C2×D411yescharacteristic
75C22×C4×C59—22no—
76C22×C4×C59—22no—
77C22×C4×C59—22no—
78C22×C4×C59—22no—
79C22×C4×C59—22no—
80C22×C4×C59—22no—
81C22×C4×C59—22no—
82C22×C4×C59—22no—
83C22×C4×C59—22no—
84C22×C4×C59—22no—
85C22×C4×C59—24no—
86C22×C4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59—41no—
89C22×C4×C59—41no—
90C22×C4×C59—41no—
91C22×C4×C59—41no—
92C22×C4×C59—41no—
93C22×C4×C59—416no—
94C22×C4×C59—42no—
95C22×C4×C59—42no—
96C22×C4×C59—42no—
97C22×C4×C59—42no—
98C22×C4×C59—42no—
99C22×C4×C59—42no—
100C22×C4×C59C22⋊C411yescharacteristic
101C22×C4×C59C22⋊C411yescharacteristic
102C22×C4×C59C22×C411yescharacteristic
103C22×C4×C59C2×D411yescharacteristic
104C22×C4×C59C2×D414yes—
105C22×Dic59—24no—
106C22×Dic59—24no—
107C22×Dic59—22no—
108C22×Dic59—22no—
109C22×Dic59—42no—
110C22×Dic59—42no—
111C22×Dic59—42no—
112C22×Dic59—42no—
113C2×Dic118—24no—
114C2×Dic118—22no—
115C2×Dic118—42no—
116C2×Dic118—416no—
117C2×Dic118—42no—
118C2×Dic118—42no—
119C2×Dic118—42no—
120C2×Dic118C22⋊C414yescomplemented
121C22⋊C4×C59—48no—
122C22⋊Dic59—28no—
123C22⋊Dic59—28no—
124C4⋊Dic59—24no—
125C4⋊Dic59—22no—
126C4⋊Dic59C22⋊C414yescomplemented
127Dic59⋊C4—28no—
128Dic59⋊C4—28no—
129C2×C59⋊C8—42no—
130C2×C59⋊C8—42no—
131C2×C59⋊C8—42no—
132C23×C59—21no—
133C23×C59—21no—
134C23×C59—21no—
135C23×C59—21no—
136C23×C59—21no—
137C23×C59—21no—
138C23×C59—21no—
139C23×C59—21no—
140C23×C59—24no—
141C23×C59—24no—
142C23×C59—21no—
143C23×C59—21no—
144C23×C59—21no—
145C23×C59—21no—
146C23×C59—21no—
147C23×C59—21no—
148C23×C59—24no—
149C23×C59—24no—
150C23×C59—42no—
151C23×C59—42no—
152C23×C59—44no—
153C23×C59—44no—
154C23×C59—44no—
155C23×C59—44no—
156C23×C59#9912yes—
157C23×C59#10312yes—
158C23×C59#10312yes—
159C23×C59#10412yes—
160C23×C59C2×C22⋊C411yescharacteristic
161C23×C59#12012yes—
162C23×C59C2×SD1611yescharacteristic
163C23×C59C2×Q1611yescharacteristic
164C23×C59#13712yes—
165C2×C4×C59—28no—
166C2×C4×C59—21no—
167C2×C4×C59—21no—
168C2×C4×C59—21no—
169C2×C4×C59—21no—
170C2×C4×C59—21no—
171C2×C4×C59—21no—
172C2×C4×C59—21no—
173C2×C4×C59—21no—
174C2×C4×C59—21no—
175C2×C4×C59—44no—
176C2×C4×C59—44no—
177C2×C4×C59—44no—
178C2×C4×C59—44no—
179C2×C4×C59—44no—
180C2×C4×C59—44no—
181C2×C4×C59—44no—
182C2×C4×C59—44no—
183C2×C4×C59—82no—
184C2×C4×C59—82no—
185C2×C4×C59—82no—
186C2×C4×C59—82no—
187C2×C4×C59#9912yes—
188C2×C4×C59C2×C22⋊C412yes—
189C2×C4×C59#12012yes—
190C2×Dic59—24no—
191C2×Dic59—24no—
192C2×Dic59—48no—
193C2×Dic59—48no—
194C2×Dic59—42no—
195C2×Dic59—42no—
196C2×Dic59—42no—
197C2×Dic59—42no—
198Dic118—48no—
199Dic118—44no—
200Dic118—44no—
201C59⋊C8—82no—
202#8329—591nomaximal, Sylow
203C22×C59—21no—
204C22×C59—21no—
205C22×C59—22no—
206C22×C59—22no—
207C22×C59—22no—
208C22×C59—22no—
209C22×C59—21no—
210C22×C59—22no—
211C22×C59—21no—
212C22×C59—22no—
213C22×C59—22no—
214C22×C59—22no—
215C22×C59—22no—
216C22×C59—22no—
217C22×C59—22no—
218C22×C59—22no—
219C22×C59—22no—
220C22×C59—48no—
221C22×C59—48no—
222C22×C59—41no—
223C22×C59—41no—
224C22×C59—41no—
225C22×C59—41no—
226C22×C59#32814yescomplemented
227C22×C59#37911yescharacteristic
228C22×C59C2×#9911yescharacteristic
229C22×C59C2×#9911yescharacteristic
230C22×C59C2×#10311yescharacteristic
231C22×C59#41911yescharacteristic
232C22×C59C2×#10411yescharacteristic
233C22×C59#42111yescharacteristic
234C22×C59#44812yes—
235C22×C59#45112yes—
236C4×C59—22no—
237C4×C59—22no—
238C4×C59—22no—
239C4×C59—41no—
240C4×C59—84no—
241Dic59—44no—
242Dic59—44no—
243C22×C22⋊Q8—591no—
244#1779—591no—
245C2×#328—594no—
246C22⋊#95D5911yescharacteristic
247C2×C59—22no—
248C2×C59—24no—
249C2×C59—22no—
250C2×C59—42no—
251C2×C59C2×#32811yescharacteristic
252C2×C59#150811yescharacteristic
253C2×C59#151111yescharacteristic
254C2×C59#152011yescharacteristic
255C2×C59#178111yescharacteristic
256C2×C59#178711yescharacteristic
257C2×C59#178911yescharacteristic
258C23×Q8—591no—
259C24×C4C2×D5911yescharacteristic
260C2×C22⋊Q8—592no—
261C2×C22⋊Q8—592no—
262C2×C22⋊Q8—1188no—
263C2×C22⋊Q8—1181no—
264C2×C22⋊Q8—1181no—
265C22×C4⋊C4—591no—
266C22×C4⋊C4—1181no—
267C22×C22⋊C4—21no—
268C22×C22⋊C4—1181no—
269C2×C22⋊C8—592no—
270C2×C22⋊C8—1181no—
271C2×C22⋊C8—1181no—
272C2×#95—21no—
273C2×#95C2×D5912yes—
274#328—1188no—
275C59#832911yescharacteristic, complemented, Sylow
276C25C22⋊D5911yescharacteristic
277C22×Q8—592no—
278C22×Q8—1184no—
279C22×Q8—1181no—
280C22×Q8—2362no—
281C22×Q8—2362no—
282C23×C4—24no—
283C23×C4—22no—
284C23×C4—42no—
285C23×C4—42no—
286C23×C4—1181no—
287C23×C4—1181no—
288C23×C4C4×D5911yescharacteristic
289C23×C4C4×D5911yescharacteristic
290C23×C4D23611yescharacteristic
291C23×C4D23611yescharacteristic
292C23×C4C22⋊D5911yescharacteristic
293C23×C4C22×D5911yescharacteristic
294C22×C8—1182no—
295C22⋊Q8—598no—
296C22⋊Q8—1184no—
297C22⋊Q8—1184no—
298C22⋊Q8—1188no—
299C22⋊Q8—1188no—
300C22⋊Q8—1184no—
301C2×C4⋊C4—592no—
302C2×C4⋊C4—1184no—
303C2×C4⋊C4—1188no—
304C2×C4⋊C4—1181no—
305C2×C4⋊C4—1182no—
306C2×C4⋊C4—1182no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—1188no—
311C2×C22⋊C4—1182no—
312C2×C22⋊C4—1182no—
313C22⋊C8—1184no—
314C22⋊C8—2362no—
315C22⋊C8—2362no—
316#95—24no—
317#95—42no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—41no—
323C24—41no—
324C24(C2×D59)⋊C411yescharacteristic
325C24(C2×D59)⋊C411yescharacteristic
326C24#1174567012yes—
327C24#1174567212yes—
328C24C2×C22⋊D5911yescharacteristic
329C2×Q8—594no—
330C2×Q8—1184no—
331C2×Q8—1182no—
332C2×Q8—23616no—
333C2×Q8—2362no—
334C2×Q8—2362no—
335C2×Q8—2362no—
336C2×Q8—2362no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—416no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—1184no—
363C22×C4—1184no—
364C22×C4—1182no—
365C22×C4—1182no—
366C22×C4—2362no—
367C22×C4—2362no—
368C22×C4—2362no—
369C22×C4—2362no—
370C22×C4(C2×D59)⋊C411yescharacteristic
371C22×C4(C2×D59)⋊C411yescharacteristic
372C22×C4C2×C4×D5911yescharacteristic
373C22×C4C2×D23611yescharacteristic
374C22×C4D4×D5914yes—
375C2×C8—2362no—
376C2×C8—2362no—
377C2×C8—2362no—
378C4⋊C4—594no—
379C4⋊C4—1188no—
380C4⋊C4—1184no—
381C4⋊C4—1182no—
382C4⋊C4—1188no—
383C22⋊C4—48no—
384C22⋊C4—1188no—
385C22⋊C4—1188no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—24no—
395C23—24no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—24no—
401C23—24no—
402C23—21no—
403C23—21no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23#1427364112yes—
411C23#1427365012yes—
412C23#1427365012yes—
413C23#1427365512yes—
414C23C2×(C2×D59)⋊C411yescharacteristic
415C23#1427369512yes—
416C23C2×#1174567011yescharacteristic
417C23C2×#1174567211yescharacteristic
418C23#1427375712yes—
419Q8—2364no—
420Q8—2368no—
421Q8—2364no—
422C2×C4—28no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—1184no—
445C2×C4—1184no—
446C2×C4—2368no—
447C2×C4—2368no—
448C2×C4—2362no—
449C2×C4—2362no—
450C2×C4—2362no—
451C2×C4—2362no—
452C2×C4#1427364112yes—
453C2×C4C22⋊C4×D5912yes—
454C2×C4#1427369712yes—
455C8—4722no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—21no—
461C22—22no—
462C22—21no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—48no—
474C22—48no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22#1683713814yescomplemented
480C22#1683725711yescharacteristic
481C22C2×#1427364111yescharacteristic
482C22C2×#1427364111yescharacteristic
483C22C2×#1427365011yescharacteristic
484C22#1683741011yescharacteristic
485C22C2×#1427365511yescharacteristic
486C22#1683742011yescharacteristic
487C22#1683754012yes—
488C22#1683755712yes—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—41no—
493C4—84no—
494C4—2364no—
495C4—2364no—
496C2—22no—
497C2—24no—
498C2—22no—
499C2—42no—
500C2C2×#1683713811yescharacteristic
501C2#2142261511yescharacteristic
502C2#2142262811yescharacteristic
503C2#2142266711yescharacteristic
504C2#2142401011yescharacteristic
505C2#2142403011yescharacteristic
506C2#2142404111yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×C22⋊#95C2
socleC23×C59C2×C22⋊C4
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenterC22⋊#95D59
hypocenterC59#8329
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#16837138C2semidirect
5. C23×Dic118C4semidirect
9. C2×#14273706C4semidirect
11. C22×C4⋊Dic59C4semidirect
120. C2×Dic118C22⋊C4semidirect
126. C4⋊Dic59C22⋊C4semidirect
226. C22×C59#328semidirect
275. C59#8329semidirect
479. C22#16837138semidirect

Maximal subgroups

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

subgroupindex
C22×#142737062
#214240062
C2×#168371382
C59×C22⋊#952
#832959