Finite Group Databasebeta

← C2×#21422637 #28555942

Subgroups — C2×#21422637 #28555942

Order 15104.

1510 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837913C211yesmaximal, characteristic
2C2×#16837366C211yesmaximal, characteristic, complemented
3#21422637C218yesmaximal, direct product factor
4C2×#16837138C212yesmaximal, complemented
5C2×#16837135C212yesmaximal, complemented
6C2×C59×#380C211yesmaximal, characteristic, complemented
7C22×#11745685C411yescharacteristic, complemented
8C22×#11745664—21no—
9#16837913—22no—
10#16837913—22no—
11#16837913C414yescomplemented
12#16837913C2214yes—
13C2×#14273710—21no—
14C2×#14273706C411yescharacteristic, complemented
15C2×#14273706C2211yescharacteristic
16C2×#14273701C411yescharacteristic, complemented
17C2×#14273701C2211yescharacteristic
18C2×#14273696—21no—
19C2×C4×C22⋊D59—21no—
20C2×C4×C22⋊D59—21no—
21C2×#14273686C411yescharacteristic, complemented
22#16837366—22no—
23#16837366C2214yescomplemented
24C2×C22⋊(C59⋊C8)—21no—
25C2×C22⋊(C59⋊C8)C2212yes—
26#16837138C2218yescomplemented
27#16837135C2218yescomplemented
28C24×C4×C59C2211yescharacteristic
29C22×C4⋊C4×C59—21no—
30C22×C22⋊C4×C59—21no—
31C59×#380—22no—
32C59×#380C2214yescomplemented
33C2×#95×C59C2212yes—
34C2×#11745685—22no—
35C2×#11745685—44no—
36C2×#11745685C2×C414yescomplemented
37C22×Dic118C2×C411yescharacteristic
38C22×D236C2×C411yescharacteristic
39C22×C22⋊D59—21no—
40C22×C22⋊D59—21no—
41C22×C22⋊D59—21no—
42C22×C4×D59—21no—
43C2×#11745664—24no—
44C2×#11745664—44no—
45C22×C59⋊C8—22no—
46#14273710—24no—
47#14273710—24no—
48#14273706—22no—
49#14273706—21no—
50#14273706—21no—
51#14273706C2×C412yes—
52#14273706C2×C414yescomplemented
53#14273706C2312yes—
54#14273701—22no—
55#14273701—21no—
56#14273701—21no—
57#14273701C2×C412yes—
58#14273701C2×C414yescomplemented
59#14273701C2312yes—
60#14273696—24no—
61#14273696—24no—
62C4×C22⋊D59—24no—
63C4×C22⋊D59—24no—
64C4×C22⋊D59—24no—
65C4×C22⋊D59—24no—
66#14273686—22no—
67#14273686C2×C414yescomplemented
68C2×C4⋊Dic59C2×C411yescharacteristic
69C2×C4⋊Dic59C2×C411yescharacteristic
70C2×Dic59⋊C4—21no—
71C2×Dic59⋊C4—21no—
72C2×C22⋊Dic59—21no—
73C2×C22⋊Dic59—21no—
74C2×C22⋊Dic59—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×C4×Dic59—21no—
78C22⋊(C59⋊C8)—24no—
79C22⋊(C59⋊C8)—22no—
80C22⋊(C59⋊C8)C2314yes—
81C25×C59D411yescharacteristic
82C23×C4×C59—22no—
83C23×C4×C59—44no—
84C23×C4×C59C2×C411yescharacteristic
85C23×C4×C59C2×C411yescharacteristic
86C23×C4×C59C2×C412yes—
87C23×C4×C59D411yescharacteristic
88C23×C4×C59D411yescharacteristic
89C23×C4×C59D412yes—
90C23×C4×C59D411yescharacteristic
91C23×C4×C59D412yes—
92C23×C4×C59C2311yescharacteristic
93C23×C4×C59C2312yes—
94C2×C4⋊C4×C59—24no—
95C2×C4⋊C4×C59—44no—
96C2×C22⋊C4×C59—24no—
97C2×C22⋊C4×C59—44no—
98#95×C59—22no—
99#95×C59C2314yes—
100C24×C59—22no—
101C24×C59—22no—
102C24×C59—42no—
103C24×C59—42no—
104C24×C59C22⋊C411yescharacteristic
105C24×C59C22⋊C411yescharacteristic
106C24×C59C22⋊C412yes—
107C24×C59C2×D411yescharacteristic
108C24×C59C2×D412yes—
109C23×D59—21no—
110C22×C4×C59—24no—
111C22×C4×C59—24no—
112C22×C4×C59—24no—
113C22×C4×C59—24no—
114C22×C4×C59—44no—
115C22×C4×C59—44no—
116C22×C4×C59—44no—
117C22×C4×C59—44no—
118C22×C4×C59—42no—
119C22×C4×C59—42no—
120C22×C4×C59—42no—
121C22×C4×C59—44no—
122C22×C4×C59—44no—
123C22×C4×C59—44no—
124C22×C4×C59C22⋊C411yescharacteristic
125C22×C4×C59C22⋊C412yes—
126C22×C4×C59C22⋊C412yes—
127C22×C4×C59C22⋊C412yes—
128C22×C4×C59C22⋊C412yes—
129C22×C4×C59C22⋊C412yes—
130C22×C4×C59C22⋊C411yescharacteristic
131C22×C4×C59C22×C411yescharacteristic
132C22×C4×C59C22×C412yes—
133C22×C4×C59C22×C411yescharacteristic
134C22×C4×C59C22×C412yes—
135C22×C4×C59C2×D412yes—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D412yes—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D412yes—
141C22×C4×C59C2411yescharacteristic
142C2×C22⋊D59—22no—
143C2×C22⋊D59—24no—
144C2×C22⋊D59—24no—
145C2×C22⋊D59—22no—
146C2×C22⋊D59—44no—
147C2×C22⋊D59—44no—
148C2×C22⋊D59—42no—
149C2×C22⋊D59—42no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153#11745685—48no—
154#11745685—48no—
155C2×D236—21no—
156C2×D236—42no—
157C2×D236C22×C412yes—
158C2×C4×D59—24no—
159C2×C4×D59—44no—
160C2×Dic118—21no—
161C2×Dic118—42no—
162C2×Dic118C22×C412yes—
163C4⋊C4×C59—48no—
164C22⋊C4×C59—48no—
165C22⋊Dic59—24no—
166C22⋊Dic59—22no—
167C22⋊Dic59—24no—
168C22⋊Dic59—22no—
169(C2×D59)⋊C4—22no—
170(C2×D59)⋊C4—24no—
171(C2×D59)⋊C4—22no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59C22×C412yes—
175C4⋊Dic59C22×C412yes—
176Dic59⋊C4—22no—
177Dic59⋊C4—24no—
178Dic59⋊C4—22no—
179C4×Dic59—24no—
180#11745664—48no—
181C2×C59⋊C8—24no—
182C2×C59⋊C8—44no—
183C23×C59—22no—
184C23×C59—22no—
185C23×C59—22no—
186C23×C59—22no—
187C23×C59—22no—
188C23×C59—22no—
189C23×C59—24no—
190C23×C59—22no—
191C23×C59—22no—
192C23×C59—22no—
193C23×C59—22no—
194C23×C59—42no—
195C23×C59—42no—
196C23×C59—42no—
197C23×C59—42no—
198C23×C59—48no—
199C23×C59—48no—
200C23×C59C2×C22⋊C412yes—
201C23×C59C2×C22⋊C412yes—
202C23×C59C2×C22⋊C411yescharacteristic
203C23×C59C2×C22⋊C411yescharacteristic
204C23×C59AGL(1,ℤ/8)12yes—
205C23×C59#13712yes—
206C23×C59C22×D411yescharacteristic
207C22×D59—22no—
208C22×D59—42no—
209C22×D59—42no—
210C22×D59—42no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—24no—
215C2×C4×C59—24no—
216C2×C4×C59—24no—
217C2×C4×C59—24no—
218C2×C4×C59—24no—
219C2×C4×C59—24no—
220C2×C4×C59—42no—
221C2×C4×C59—42no—
222C2×C4×C59—41no—
223C2×C4×C59—42no—
224C2×C4×C59—41no—
225C2×C4×C59—42no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—44no—
229C2×C4×C59—44no—
230C2×C4×C59—44no—
231C2×C4×C59—88no—
232C2×C4×C59#9914yes—
233C2×C4×C59C2×C22⋊C411yescharacteristic
234C2×C4×C59C2×C22⋊C412yes—
235C2×C4×C59C2×C22⋊C411yescharacteristic
236C2×C4×C59C2×C22⋊C412yes—
237C2×C4×C59C23×C411yescharacteristic
238C2×C4×C59C22×D411yescharacteristic
239C22⋊D59—48no—
240C22⋊D59—48no—
241C22⋊D59—42no—
242C22⋊D59—42no—
243C22⋊D59—44no—
244C2×Dic59—22no—
245C2×Dic59—22no—
246C2×Dic59—22no—
247C2×Dic59—42no—
248C2×Dic59—42no—
249C2×Dic59—42no—
250D236—42no—
251D236—42no—
252C4×D59—48no—
253Dic118—42no—
254Dic118—42no—
255C59⋊C8—44no—
256C2×#1513—591nomaximal, Sylow
257C22×C59—21no—
258C22×C59—21no—
259C22×C59—21no—
260C22×C59—21no—
261C22×C59—24no—
262C22×C59—24no—
263C22×C59—22no—
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—22no—
269C22×C59—44no—
270C22×C59—44no—
271C22×C59—48no—
272C22×C59—44no—
273C22×C59—44no—
274C22×C59C2×#9912yes—
275C22×C59#41712yes—
276C22×C59#41712yes—
277C22×C59#42112yes—
278C22×C59C22×C22⋊C411yescharacteristic
279C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C59C2×#13711yescharacteristic
281C2×D59—42no—
282C2×D59—42no—
283C2×D59—42no—
284C2×D59—84no—
285C4×C59—21no—
286C4×C59—21no—
287C4×C59—21no—
288C4×C59—21no—
289C4×C59—22no—
290C4×C59—84no—
291C4×C59C2×#9914yes—
292Dic59—42no—
293Dic59—42no—
294Dic59—42no—
295C2×#525—591no—
296C2×#407—591no—
297C2×#380D5911yescharacteristic, complemented
298#1513—598no—
299C2×#328—592no—
300C2×#327—592no—
301C2×C59—21no—
302C2×C59—21no—
303C2×C59—22no—
304C2×C59—42no—
305C2×C59—42no—
306C2×C59#151314yescomplemented
307C2×C59C22×#9911yescharacteristic
308C2×C59C2×#41711yescharacteristic
309C2×C59C2×#42111yescharacteristic
310D59—82no—
311C22×#41—591no—
312C24×C4C2×D5911yescharacteristic
313C22×M16—1181no—
314#525—594no—
315#525—594no—
316#525—1182no—
317#525—1182no—
318C2×#123—1181no—
319C2×C22⋊Q8—591no—
320C2×C22⋊Q8—591no—
321C2×#121—591no—
322C2×#121—591no—
323C2×#120—1181no—
324C2×C4×D4—1181no—
325C2×C4×D4—1181no—
326C2×#117—591no—
327C22×C4⋊C4—21no—
328C22×C22⋊C4—21no—
329#407—594no—
330#407—1182no—
331C2×C22⋊C8—592no—
332C2×C22⋊C8—1181no—
333#380—22no—
334#380C2×D5914yescomplemented
335C2×#95C2×D5912yes—
336#328—598no—
337#327—598no—
338C59C2×#151311yescharacteristic, complemented, Sylow
339C25C22⋊D5911yescharacteristic
340C2×#41—594no—
341C2×#41—1182no—
342C2×#41—2364no—
343C22×Q8—591no—
344C22×D4—591no—
345C22×D4—1181no—
346C22×D4—1181no—
347C22×D4—1181no—
348C23×C4—22no—
349C23×C4—44no—
350C23×C4—1181no—
351C23×C4C4×D5911yescharacteristic
352C23×C4C4×D5911yescharacteristic
353C23×C4C4×D5912yes—
354C23×C4D23611yescharacteristic
355C23×C4D23611yescharacteristic
356C23×C4D23612yes—
357C23×C4C22⋊D5911yescharacteristic
358C23×C4C22⋊D5912yes—
359C23×C4C22×D5911yescharacteristic
360C23×C4C22×D5912yes—
361C2×M16—1184no—
362C2×M16—2364no—
363C22×C8—1182no—
364#123—1184no—
365#123—1184no—
366C22⋊Q8—592no—
367C22⋊Q8—594no—
368C22⋊Q8—592no—
369C22⋊Q8—1182no—
370C22⋊Q8—1181no—
371C22⋊Q8—1181no—
372#121—592no—
373#121—594no—
374#121—592no—
375#121—1182no—
376#121—1181no—
377#121—1181no—
378#120—1184no—
379#120—1184no—
380C4×D4—1184no—
381C4×D4—1184no—
382C4×D4—1184no—
383C4×D4—1184no—
384#117—594no—
385#117—1182no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—44no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—44no—
394C2×C22⋊C4—1181no—
395C2×C22⋊C4—1181no—
396C2×C22⋊C4—1181no—
397C2×C22⋊C4—1181no—
398C2×C22⋊C4—1181no—
399C2×C42—1181no—
400C22⋊C8—594no—
401C22⋊C8—1182no—
402C22⋊C8—1184no—
403#95—22no—
404#95C22×D5914yes—
405C24—22no—
406C24—22no—
407C24—42no—
408C24—42no—
409C24—1181no—
410C24(C2×D59)⋊C411yescharacteristic
411C24(C2×D59)⋊C411yescharacteristic
412C24(C2×D59)⋊C412yes—
413C24C2×C22⋊D5911yescharacteristic
414C24C2×C22⋊D5912yes—
415#41—2368no—
416#41—2368no—
417C2×Q8—592no—
418C2×Q8—1181no—
419C2×Q8—2362no—
420C2×D4—592no—
421C2×D4—1184no—
422C2×D4—1182no—
423C2×D4—1184no—
424C2×D4—1182no—
425C2×D4—1181no—
426C2×D4—2362no—
427C2×D4—2364no—
428C2×D4—2364no—
429C2×D4—2362no—
430C2×D4—2362no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1184no—
449C22×C4—2364no—
450C22×C4(C2×D59)⋊C412yes—
451C22×C4(C2×D59)⋊C412yes—
452C22×C4(C2×D59)⋊C412yes—
453C22×C4(C2×D59)⋊C411yescharacteristic
454C22×C4(C2×D59)⋊C412yes—
455C22×C4(C2×D59)⋊C412yes—
456C22×C4(C2×D59)⋊C411yescharacteristic
457C22×C4C2×C4×D5912yes—
458C22×C4C2×C4×D5911yescharacteristic
459C22×C4C2×C4×D5912yes—
460C22×C4C2×C4×D5911yescharacteristic
461C22×C4C2×D23612yes—
462C22×C4C2×D23611yescharacteristic
463C22×C4C2×D23611yescharacteristic
464C22×C4C2×D23612yes—
465C22×C4C2×C22⋊D5912yes—
466C22×C4C2×C22⋊D5911yescharacteristic
467C22×C4C23×D5911yescharacteristic
468M16—2368no—
469C2×C8—1184no—
470C2×C8—2364no—
471C4⋊C4—48no—
472C4⋊C4—592no—
473C4⋊C4—592no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1184no—
477C4⋊C4—1182no—
478C4⋊C4—1182no—
479C22⋊C4—48no—
480C22⋊C4—1184no—
481C22⋊C4—1184no—
482C22⋊C4—1182no—
483C22⋊C4—1182no—
484C22⋊C4—1184no—
485C22⋊C4—1182no—
486C22⋊C4—1182no—
487C42—1184no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—48no—
504C23—48no—
505C23—1182no—
506C23—2362no—
507C23—2362no—
508C23—2362no—
509C23C2×(C2×D59)⋊C412yes—
510C23C2×(C2×D59)⋊C412yes—
511C23C2×(C2×D59)⋊C411yescharacteristic
512C23C2×(C2×D59)⋊C411yescharacteristic
513C23#1427375312yes—
514C23#1427375812yes—
515C23C22×C22⋊D5911yescharacteristic
516Q8—2362no—
517Q8—2362no—
518D4—2362no—
519D4—2362no—
520D4—2368no—
521D4—2368no—
522D4—2364no—
523D4—2362no—
524D4—2362no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—88no—
546C2×C4—1182no—
547C2×C4—1182no—
548C2×C4—1182no—
549C2×C4—2362no—
550C2×C4—2362no—
551C2×C4—2362no—
552C2×C4—2368no—
553C2×C4#1427364114yes—
554C2×C4C2×(C2×D59)⋊C411yescharacteristic
555C2×C4C2×(C2×D59)⋊C412yes—
556C2×C4C2×(C2×D59)⋊C411yescharacteristic
557C2×C4C2×(C2×D59)⋊C412yes—
558C2×C4C22×C4×D5911yescharacteristic
559C2×C4C22×D23611yescharacteristic
560C8—2364no—
561C22—24no—
562C22—24no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—4724no—
579C22—2362no—
580C22—2362no—
581C22—2362no—
582C22C2×#1427364112yes—
583C22#1683740112yes—
584C22#1683740112yes—
585C22#1683742012yes—
586C22C22×(C2×D59)⋊C411yescharacteristic
587C22C2×#1427375311yescharacteristic
588C22C2×#1427375811yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—22no—
593C4—21no—
594C4—84no—
595C4—2362no—
596C4—2362no—
597C4—2362no—
598C4C2×#1427364114yes—
599C2—21no—
600C2—22no—
601C2—21no—
602C2—42no—
603C2—42no—
604C2—4722no—
605C2#2142263714yesdirect product factor
606C2C22×#1427364111yescharacteristic
607C2C2×#1683740111yescharacteristic
608C2C2×#1683742011yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#380C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#16837913C2
cosocleC22×C4×C59C24
hypercenterC2×#380D59
hypocenterC59C2×#1513
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#16837366C2semidirect
3. #21422637C2direct (N × H)
4. C2×#16837138C2semidirect
5. C2×#16837135C2semidirect
6. C2×C59×#380C2semidirect
7. C22×#11745685C4semidirect
11. #16837913C4semidirect
14. C2×#14273706C4semidirect
16. C2×#14273701C4semidirect
21. C2×#14273686C4semidirect
23. #16837366C22semidirect
26. #16837138C22semidirect
27. #16837135C22semidirect
32. C59×#380C22semidirect
36. C2×#11745685C2×C4semidirect
52. #14273706C2×C4semidirect
58. #14273701C2×C4semidirect
67. #14273686C2×C4semidirect
297. C2×#380D59semidirect
306. C2×C59#1513semidirect
334. #380C2×D59semidirect
338. C59C2×#1513semidirect
605. C2#21422637direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168379132
C2×#168373662
#214226372
C2×#168371382
C2×#168371352
C2×C59×#3802
C2×#151359