Finite Group Databasebeta

← #28598318

Subgroups — #28598318

Order 15104.

1102 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837930C211yesmaximal, characteristic
2C2×#16837458C211yesmaximal, characteristic, complemented
3C2×#16837424C211yesmaximal, characteristic, complemented
4#21425034C212yesmaximal, complemented
5#21424892C212yesmaximal, complemented
6#21424392C212yesmaximal, complemented
7#21424390C212yesmaximal, complemented
8#21424080C211yesmaximal, characteristic, complemented
9#21424077C211yesmaximal, characteristic, complemented
10#21424077C211yesmaximal, characteristic, complemented
11C59×#1798C211yesmaximal, characteristic, complemented
12C22×#11745659C2212yes—
13#16837930—24no—
14#16837930—24no—
15C2×#14273722C411yescharacteristic
16C2×#14273720C411yescharacteristic
17C2×#14273717C411yescharacteristic
18C2×#14273713C411yescharacteristic
19C2×#14273712C411yescharacteristic
20C2×#14273709C411yescharacteristic
21C2×#14273705C411yescharacteristic
22C2×#14273700C411yescharacteristic
23C2×C4×Dic118C2211yescharacteristic
24C2×C4×D236C2211yescharacteristic
25C2×C4×C22⋊D59C2211yescharacteristic
26C2×C4×C22⋊D59C2211yescharacteristic
27C2×#14273684C2211yescharacteristic
28C2×C42×D59C2211yescharacteristic
29#16837458—24no—
30#16837424—24no—
31C2×C4⋊(C59⋊C8)C2211yescharacteristic
32C2×C4⋊(C59⋊C8)C2211yescharacteristic
33C2×Dic59⋊C8C2212yes—
34C2×C22⋊(C59⋊C8)C2212yes—
35C2×(C2×D59)⋊C8C2212yes—
36C2×(C2×D59)⋊C8C2212yes—
37C2×(C2×D59)⋊C8C2212yes—
38C2×#14273637C2212yes—
39#16837157C2212yes—
40#16837156C2212yes—
41#16837156C2212yes—
42C2×#117×C59C2211yescharacteristic
43C2×C4⋊C8×C59C2211yescharacteristic
44C2×C4⋊C8×C59C2211yescharacteristic
45C2×C22⋊C8×C59C2212yes—
46C59×#336C2212yes—
47C22×Dic118C2×C411yescharacteristic
48C22×D236C2×C411yescharacteristic
49C22×C22⋊D59C2×C411yescharacteristic
50C22×C22⋊D59C2×C411yescharacteristic
51C22×C4×D59C2×C411yescharacteristic
52C22×C4×D59C2×C411yescharacteristic
53C22×C4×D59C2311yescharacteristic
54C22×C4×D59C2311yescharacteristic
55C2×#11745659—24no—
56C2×#11745659—42no—
57C2×#11745659—42no—
58C22×C59⋊C8C2312yes—
59C22×C59⋊C8C2312yes—
60#14273722—24no—
61#14273720—24no—
62#14273717—24no—
63#14273713—24no—
64#14273712—24no—
65#14273709—24no—
66#14273705—24no—
67#14273700—24no—
68C4×Dic118—22no—
69C4×Dic118—22no—
70C4×D236—22no—
71C4×D236—22no—
72C4×C22⋊D59—22no—
73C4×C22⋊D59—22no—
74C4×C22⋊D59—22no—
75C4×C22⋊D59—22no—
76#14273684—22no—
77#14273684—22no—
78C2×C4⋊Dic59C2×C411yescharacteristic
79C2×C4⋊Dic59C2×C411yescharacteristic
80C2×Dic59⋊C4C2×C411yescharacteristic
81C2×Dic59⋊C4C2×C411yescharacteristic
82C2×Dic59⋊C4C2×C411yescharacteristic
83C2×Dic59⋊C4C2×C411yescharacteristic
84C2×Dic59⋊C4C2×C411yescharacteristic
85C2×Dic59⋊C4C2×C411yescharacteristic
86C2×C22⋊Dic59C2×C411yescharacteristic
87C2×C22⋊Dic59C2×C411yescharacteristic
88C2×(C2×D59)⋊C4C2×C411yescharacteristic
89C2×(C2×D59)⋊C4C2×C411yescharacteristic
90C2×(C2×D59)⋊C4C2×C411yescharacteristic
91C2×(C2×D59)⋊C4C2×C411yescharacteristic
92C2×(C2×D59)⋊C4C2×C411yescharacteristic
93C2×(C2×D59)⋊C4C2×C411yescharacteristic
94C2×C4×Dic59C2×C411yescharacteristic
95C2×C4×Dic59C2×C411yescharacteristic
96C2×C4×Dic59C2311yescharacteristic
97C2×C4×Dic59C2311yescharacteristic
98C42×D59D414yes—
99C42×D59D414yes—
100C4⋊(C59⋊C8)—22no—
101C4⋊(C59⋊C8)—22no—
102Dic59⋊C8—24no—
103C22⋊(C59⋊C8)—42no—
104(C2×D59)⋊C8—24no—
105(C2×D59)⋊C8—42no—
106(C2×D59)⋊C8—42no—
107#14273637—24no—
108C23×C4×C59C2311yescharacteristic
109C22×C8×C59C2312yes—
110C22×C8×C59C2312yes—
111#117×C59—22no—
112#117×C59—22no—
113C2×C4⋊C4×C59C2×C411yescharacteristic
114C2×C4⋊C4×C59C2×C411yescharacteristic
115C2×C22⋊C4×C59C2×C411yescharacteristic
116C2×C22⋊C4×C59C2×C411yescharacteristic
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C4⋊C8×C59—22no—
120C4⋊C8×C59—22no—
121C22⋊C8×C59—42no—
122C24×C59C22×C411yescharacteristic
123C23×D59C22×C411yescharacteristic
124C23×D59C22×C411yescharacteristic
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59C22×C411yescharacteristic
132C22×C4×C59C22×C411yescharacteristic
133C22×C4×C59C22×C411yescharacteristic
134C22×C4×C59C22×C411yescharacteristic
135C22×C4×C59C22×C411yescharacteristic
136C22×C4×C59C2411yescharacteristic
137C2×C22⋊D59—22no—
138C2×C22⋊D59—22no—
139C2×C22⋊D59—42no—
140C2×C22⋊D59—42no—
141C22×Dic59C22×C411yescharacteristic
142C22×Dic59C22×C411yescharacteristic
143C22×Dic59C22×C411yescharacteristic
144C22×Dic59C22×C411yescharacteristic
145C22×Dic59C22×C411yescharacteristic
146C22×Dic59C22×C411yescharacteristic
147C2×D236—22no—
148C2×D236—42no—
149C2×C4×D59—21no—
150C2×C4×D59—21no—
151C2×C4×D59—21no—
152C2×C4×D59—21no—
153C2×C4×D59—21no—
154C2×C4×D59—21no—
155C2×C4×D59—21no—
156C2×C4×D59—21no—
157C2×C4×D59—21no—
158C2×C4×D59—21no—
159C2×C4×D59—42no—
160C2×C4×D59—42no—
161C2×C4×D59C22⋊C414yes—
162C2×C4×D59C22⋊C414yes—
163C2×C4×D59C2×D412yes—
164C2×C4×D59C2×D412yes—
165C2×Dic118—22no—
166C2×Dic118—42no—
167C2×C8×C59—22no—
168C2×C8×C59—22no—
169C2×C8×C59—22no—
170C2×C8×C59—22no—
171C2×C8×C59—22no—
172C2×C8×C59—22no—
173C4⋊C4×C59—22no—
174C4⋊C4×C59—22no—
175C22⋊C4×C59—22no—
176C22⋊C4×C59—22no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59C2×D412yes—
180C42×C59C2×D412yes—
181C22⋊Dic59—22no—
182C22⋊Dic59—22no—
183(C2×D59)⋊C4—22no—
184(C2×D59)⋊C4—22no—
185(C2×D59)⋊C4—22no—
186(C2×D59)⋊C4—22no—
187(C2×D59)⋊C4—22no—
188(C2×D59)⋊C4—22no—
189C4⋊Dic59—22no—
190C4⋊Dic59—22no—
191Dic59⋊C4—22no—
192Dic59⋊C4—22no—
193Dic59⋊C4—22no—
194Dic59⋊C4—22no—
195Dic59⋊C4—22no—
196Dic59⋊C4—22no—
197C4×Dic59—21no—
198C4×Dic59—21no—
199C4×Dic59C22⋊C414yes—
200C4×Dic59C22⋊C414yes—
201C4×Dic59C2×D412yes—
202C4×Dic59C2×D412yes—
203C2×C59⋊C8—22no—
204C2×C59⋊C8—22no—
205C2×C59⋊C8—22no—
206C2×C59⋊C8—22no—
207C2×C59⋊C8—22no—
208C2×C59⋊C8—22no—
209#11745659—48no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—22no—
214C23×C59—22no—
215C23×C59C23×C411yescharacteristic
216C22×D59—21no—
217C22×D59—21no—
218C22×D59—22no—
219C22×D59—22no—
220C22×D59—21no—
221C22×D59—21no—
222C22×D59—21no—
223C22×D59—22no—
224C22×D59—22no—
225C22×D59C2×C22⋊C412yes—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59C2×C22⋊C412yes—
240C2×C4×C59C2×C22⋊C412yes—
241C2×C4×C59C22×D411yescharacteristic
242C2×C4×C59C22×D411yescharacteristic
243C2×C4×C59H(5,𝔽2)11yescharacteristic
244C2×C4×C59H(5,𝔽2)11yescharacteristic
245C2×C4×C592−1+411yescharacteristic
246C2×C4×C592−1+411yescharacteristic
247C22⋊D59—44no—
248C22⋊D59—44no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—22no—
252C2×Dic59—22no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—21no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—22no—
259C2×Dic59—22no—
260C2×Dic59C2×C22⋊C412yes—
261C2×Dic59C2×C22⋊C412yes—
262C2×Dic59C2×C22⋊C412yes—
263D236—44no—
264C4×D59—22no—
265C4×D59—22no—
266C4×D59—22no—
267C4×D59—22no—
268C4×D59—44no—
269C4×D59—44no—
270C4×D59—41no—
271C4×D59—41no—
272C4×D59—41no—
273C4×D59—41no—
274Dic118—44no—
275C8×C59—42no—
276C8×C59—42no—
277C59⋊C8—42no—
278C59⋊C8—42no—
279#21854—591nomaximal, Sylow
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—42no—
284C22×C59—42no—
285C22×C59C22×C22⋊C411yescharacteristic
286C22×C59#52011yescharacteristic
287C22×C59#52011yescharacteristic
288C22×C59#56812yes—
289C22×C59#56812yes—
290C2×D59—22no—
291C2×D59—22no—
292C2×D59—22no—
293C2×D59—42no—
294C2×D59—42no—
295C2×D59—41no—
296C2×D59—41no—
297C2×D59—41no—
298C2×D59—42no—
299C2×D59—42no—
300C4×C59—22no—
301C4×C59—22no—
302C4×C59—41no—
303C4×C59—41no—
304C4×C59—41no—
305C4×C59#53511yescharacteristic
306C4×C59#53611yescharacteristic
307C4×C59#53711yescharacteristic
308C4×C59#53711yescharacteristic
309Dic59—22no—
310Dic59—22no—
311Dic59—22no—
312Dic59—41no—
313Dic59—41no—
314Dic59—41no—
315C2×#529—591no—
316C2×#432—591no—
317C2×#423—591no—
318#1985—592no—
319#1955—592no—
320#1870—592no—
321#1870—592no—
322#1798—591no—
323#1798—591no—
324#1798—591no—
325#1798D5911yescharacteristic, complemented
326C2×C59—81no—
327C2×C59#231111yescharacteristic
328C2×C59#287911yescharacteristic
329C2×C59#287911yescharacteristic
330C2×C59#299512yes—
331C2×C59#299512yes—
332D59—42no—
333D59—81no—
334C22×M16—592no—
335#529—1184no—
336#529—1184no—
337C2×#126—591no—
338C2×#126—591no—
339C2×#125—591no—
340C2×#124—591no—
341C2×#123—591no—
342C2×#123—591no—
343C2×C22⋊Q8—591no—
344C2×#121—591no—
345C2×C4×Q8—591no—
346C2×C4×D4—591no—
347C2×C4×D4—591no—
348C2×C4×D4—591no—
349C2×#117—591no—
350C2×#117C2×D5911yescharacteristic
351C22×C42—591no—
352#432—1184no—
353#423—1184no—
354C2×C4⋊C8—592no—
355C2×C4⋊C8—591no—
356C2×C4⋊C8—591no—
357C2×C4⋊C8C2×D5911yescharacteristic
358C2×C4⋊C8C2×D5911yescharacteristic
359C2×C22⋊C8—592no—
360C2×C22⋊C8—592no—
361C2×C22⋊C8—592no—
362C2×C22⋊C8—592no—
363C2×C22⋊C8C2×D5912yes—
364C2×#97—592no—
365#336—592no—
366#336—592no—
367#336—592no—
368#336C2×D5912yes—
369C59#2185411yescharacteristic, complemented, Sylow
370C22×Q8—591no—
371C22×D4—591no—
372C22×D4—591no—
373C22×D4—591no—
374C23×C4—591no—
375C23×C4—591no—
376C23×C4—591no—
377C23×C4—591no—
378C23×C4C22×D5911yescharacteristic
379C2×M16—1184no—
380C2×M16—2362no—
381C2×M16—2362no—
382C22×C8—592no—
383C22×C8—592no—
384C22×C8C22×D5912yes—
385C22×C8C22×D5912yes—
386#126—1184no—
387#126—1184no—
388#125—1184no—
389#124—1184no—
390#123—1184no—
391#123—1184no—
392C22⋊Q8—1184no—
393#121—1184no—
394C4×Q8—1182no—
395C4×Q8—1182no—
396C4×D4—1182no—
397C4×D4—1182no—
398C4×D4—1182no—
399C4×D4—1182no—
400C4×D4—1182no—
401C4×D4—1182no—
402#117—22no—
403#117—22no—
404#117—1182no—
405#117—1182no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4C4×D5911yescharacteristic
415C2×C4⋊C4C4×D5911yescharacteristic
416C2×C22⋊C4—591no—
417C2×C22⋊C4—591no—
418C2×C22⋊C4—591no—
419C2×C22⋊C4—591no—
420C2×C22⋊C4—591no—
421C2×C22⋊C4—591no—
422C2×C22⋊C4—591no—
423C2×C22⋊C4—591no—
424C2×C22⋊C4C4×D5911yescharacteristic
425C2×C22⋊C4C4×D5911yescharacteristic
426C2×C42—591no—
427C2×C42—591no—
428C2×C42—591no—
429C2×C42—594no—
430C2×C42—594no—
431C2×C42—591no—
432C2×C42C22×D5911yescharacteristic
433C2×C42C22×D5911yescharacteristic
434C4⋊C8—22no—
435C4⋊C8—22no—
436C4⋊C8—1184no—
437C4⋊C8—1182no—
438C4⋊C8—1182no—
439C22⋊C8—42no—
440C22⋊C8—1184no—
441C22⋊C8—2362no—
442C22⋊C8—2362no—
443C22⋊C8—2362no—
444#97—1184no—
445C24—591no—
446C24—591no—
447C24C2×C4×D5911yescharacteristic
448C2×Q8—1182no—
449C2×Q8—2362no—
450C2×D4—1182no—
451C2×D4—1182no—
452C2×D4—1182no—
453C2×D4—2362no—
454C2×D4—2362no—
455C2×D4—2362no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—591no—
463C22×C4—591no—
464C22×C4—591no—
465C22×C4—592no—
466C22×C4—592no—
467C22×C4—594no—
468C22×C4—594no—
469C22×C4—591no—
470C22×C4—591no—
471C22×C4—591no—
472C22×C4—1181no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4—1181no—
482C22×C4—2362no—
483C22×C4—2362no—
484C22×C4C2×C4×D5911yescharacteristic
485C22×C4C2×C4×D5911yescharacteristic
486C22×C4C2×C4×D5911yescharacteristic
487C22×C4C2×C4×D5911yescharacteristic
488C22×C4C2×C4×D5911yescharacteristic
489C22×C4C23×D5911yescharacteristic
490M16—2368no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—1182no—
498C2×C8—1182no—
499C2×C8—1182no—
500C2×C8—1182no—
501C2×C8—1182no—
502C2×C8—1182no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—1182no—
506C4⋊C4—1182no—
507C4⋊C4—1182no—
508C4⋊C4—1182no—
509C4⋊C4—1182no—
510C4⋊C4—1182no—
511C4⋊C4—1182no—
512C4⋊C4—1182no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—1182no—
516C22⋊C4—1182no—
517C22⋊C4—1182no—
518C22⋊C4—1182no—
519C22⋊C4—1182no—
520C22⋊C4—1182no—
521C22⋊C4—1182no—
522C22⋊C4—1182no—
523C42—21no—
524C42—21no—
525C42—594no—
526C42—594no—
527C42—592no—
528C42—592no—
529C42—1181no—
530C42—1181no—
531C42D4×D5912yes—
532C42D4×D5912yes—
533C23—22no—
534C23—22no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—592no—
539C23—1182no—
540C23—1182no—
541C23—1181no—
542C23—1181no—
543C23—1182no—
544C23—1182no—
545C23—1181no—
546C23—1181no—
547C23—1181no—
548C23C22×C4×D5911yescharacteristic
549Q8—2364no—
550D4—2364no—
551D4—2364no—
552D4—2364no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—592no—
567C2×C4—592no—
568C2×C4—592no—
569C2×C4—1182no—
570C2×C4—1182no—
571C2×C4—1181no—
572C2×C4—1181no—
573C2×C4—1182no—
574C2×C4—1182no—
575C2×C4—1182no—
576C2×C4—1182no—
577C2×C4—1182no—
578C2×C4—1182no—
579C2×C4—1181no—
580C2×C4—1181no—
581C2×C4—1181no—
582C2×C4—1181no—
583C2×C4—1181no—
584C2×C4—2364no—
585C2×C4—2364no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4C22⋊C4×D5912yes—
591C2×C4C22⋊C4×D5912yes—
592C2×C4C2×D4×D5911yescharacteristic
593C2×C4C2×D4×D5911yescharacteristic
594C2×C4#1427377211yescharacteristic
595C2×C4#1427377311yescharacteristic
596C2×C4#1427377411yescharacteristic
597C2×C4#1427377511yescharacteristic
598C8—42no—
599C8—42no—
600C8—2362no—
601C8—2362no—
602C22—42no—
603C22—42no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—1182no—
608C22—1182no—
609C22—1182no—
610C22—2362no—
611C22—2362no—
612C22—2361no—
613C22—2361no—
614C22—2361no—
615C22—2362no—
616C22—2362no—
617C22C2×C22⋊C4×D5911yescharacteristic
618C22#1683788211yescharacteristic
619C22#1683788311yescharacteristic
620C22#1683816712yes—
621C22#1683816712yes—
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—1182no—
628C4—1182no—
629C4—1182no—
630C4—2361no—
631C4—2361no—
632C4—2361no—
633C4#1683796811yescharacteristic
634C4#1683797511yescharacteristic
635C4#1683798111yescharacteristic
636C4#1683798211yescharacteristic
637C2—81no—
638C2—4721no—
639C2—2362no—
640C2#2142668511yescharacteristic
641C2#2143253811yescharacteristic
642C2#2143253811yescharacteristic
643C2#2143326112yes—
644C2#2143326412yes—
subgroupG/H
center, FrattiniC22×C4C23×D59
commutator, socleC23×C59C23×C4
FittingC59×#1798C2
prime radicalC2×#14273700C4
cosocleC22×C4×C59C24
hypercenter#1798D59
hypocenterC59#21854
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. C2×#16837458C2semidirect
3. C2×#16837424C2semidirect
4. #21425034C2semidirect
5. #21424892C2semidirect
6. #21424392C2semidirect
7. #21424390C2semidirect
8. #21424080C2semidirect
9. #21424077C2semidirect
10. #21424077C2semidirect
11. C59×#1798C2semidirect
325. #1798D59semidirect
369. C59#21854semidirect

Maximal subgroups

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

subgroupindex
C2×#168379302
C2×#168374582
C2×#168374242
#214250342
#214248922
#214243922
#214243902
#214240802
#214240772
C59×#17982
#2185459