Finite Group Databasebeta

← C4×#16837540 #28594278

Subgroups — C4×#16837540 #28594278

Order 15104.

1532 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic118C211yesmaximal, characteristic, complemented
2C2×#16837540C211yesmaximal, characteristic
3C2×C4×#11745670C212yesmaximal, complemented
4#21424454C212yesmaximal, complemented
5#21424414C211yesmaximal, characteristic
6#21424363C212yesmaximal, complemented
7#21424288C211yesmaximal, characteristic, complemented
8#21424030C211yesmaximal, characteristic, complemented
9C4×#14273650C212yesmaximal, complemented
10C4×C22⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
11C4×#121×C59C211yesmaximal, characteristic
12C23×Dic118C2211yescharacteristic
13C22×#11745670C2212yes—
14C2×C4×Dic118—28no—
15C2×C4×Dic118—21no—
16C2×C4×Dic118—21no—
17C2×C4×Dic118C2212yes—
18C2×C4×Dic118C2212yes—
19C22×C4⋊Dic59C2211yescharacteristic
20C22×Dic59⋊C4—21no—
21C22×C4×Dic59—21no—
22#16837540C418yesdirect product factor
23C4×#11745670—28no—
24C2×#14273658C2212yes—
25C2×#14273650C2212yes—
26C2×#14273650C2212yes—
27C2×#14273650C2212yes—
28C2×#14273648C2212yes—
29C2×C22⋊(C59⋊C8)C2211yescharacteristic
30C2×C22⋊(C59⋊C8)C2211yescharacteristic
31C2×C4×C59⋊C8C2212yes—
32#16837164C2212yes—
33#16837164C2212yes—
34#16837157C2212yes—
35C2×#121×C59C2211yescharacteristic
36C2×C4×D4×C59—21no—
37C2×C4×D4×C59C2211yescharacteristic
38C22×C42×C59C2211yescharacteristic
39C59×#390C2211yescharacteristic
40C59×#389C2211yescharacteristic
41C59×#388—21no—
42C59×#386—21no—
43C59×#380C2211yescharacteristic
44C4×C4⋊C4×C59C2211yescharacteristic
45C4×C22⋊C4×C59—21no—
46C22×Dic118—21no—
47C22×Dic118—22no—
48C22×Dic118—24no—
49C22×Dic118—24no—
50C22×Dic118C2×C414yescomplemented
51C22×Dic118C2312yes—
52C23×Dic59—21no—
53C23×Dic59—21no—
54C2×#11745670—24no—
55C2×#11745670—24no—
56C2×#11745670C2×C418yescomplemented
57C22×C59⋊C8C2312yes—
58C22×C59⋊C8C2312yes—
59C4×Dic118—28no—
60C4×Dic118—28no—
61C4×Dic118—24no—
62C4×Dic118—24no—
63C4×Dic118—24no—
64C4×Dic118D418yescomplemented
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—24no—
67C2×C4⋊Dic59C2312yes—
68C2×Dic59⋊C4—28no—
69C2×Dic59⋊C4—22no—
70C2×Dic59⋊C4—22no—
71C2×C4×Dic59—28no—
72C2×C4×Dic59—22no—
73C2×C4×Dic59—22no—
74#14273658—24no—
75#14273650—24no—
76#14273650—24no—
77#14273650C2×C418yescomplemented
78#14273648—24no—
79C22⋊(C59⋊C8)—22no—
80C22⋊(C59⋊C8)C2×C414yescomplemented
81C4×C59⋊C8—24no—
82C22×D4×C59—21no—
83C22×D4×C59C2311yescharacteristic
84C23×C4×C59—21no—
85C23×C4×C59D411yescharacteristic
86C23×C4×C59D411yescharacteristic
87C23×C4×C59C2311yescharacteristic
88#121×C59—22no—
89#121×C59C2×C414yes—
90C4×D4×C59—24no—
91C4×D4×C59—44no—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59C2311yescharacteristic
94C2×C4⋊C4×C59C2311yescharacteristic
95C2×C4⋊C4×C59C2311yescharacteristic
96C2×C22⋊C4×C59—21no—
97C2×C22⋊C4×C59—21no—
98C2×C22⋊C4×C59—21no—
99C2×C22⋊C4×C59—21no—
100C2×C22⋊C4×C59—21no—
101C2×C42×C59—24no—
102C2×C42×C59—21no—
103C2×C42×C59D411yescharacteristic
104C2×C42×C59D411yescharacteristic
105C2×C42×C59C2311yescharacteristic
106C2×C42×C59C2311yescharacteristic
107#95×C59—21no—
108#95×C59—21no—
109#95×C59—21no—
110C24×C59—21no—
111C24×C59C2×D411yescharacteristic
112C2×D4×C59—24no—
113C2×D4×C59—22no—
114C2×D4×C59—22no—
115C2×D4×C59—21no—
116C2×D4×C59—42no—
117C2×D4×C59—42no—
118C2×D4×C59C22×C412yes—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—21no—
125C22×C4×C59—22no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—41no—
129C22×C4×C59—42no—
130C22×C4×C59—42no—
131C22×C4×C59—42no—
132C22×C4×C59SD1614yescomplemented
133C22×C4×C59C22×C412yes—
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59#4112yes—
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—24no—
142C22×Dic59—24no—
143C22×Dic59—24no—
144C22×Dic59—22no—
145C22×Dic59—24no—
146C22×Dic59—22no—
147C22×Dic59—42no—
148C22×Dic59—42no—
149C2×Dic118—24no—
150C2×Dic118—24no—
151C2×Dic118—216no—
152C2×Dic118—24no—
153C2×Dic118—22no—
154C2×Dic118—22no—
155C2×Dic118—22no—
156C2×Dic118—216no—
157C2×Dic118C22×C414yes—
158C2×Dic118C2×D414yes—
159C2×Dic118#4114yes—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—42no—
164C4⋊C4×C59C22×C412yes—
165C22⋊C4×C59—24no—
166C22⋊C4×C59—42no—
167C22⋊C4×C59—42no—
168C22⋊C4×C59—42no—
169C22⋊C4×C59—42no—
170C42×C59—22no—
171C42×C59—22no—
172C42×C59—22no—
173C42×C59—42no—
174C42×C59C2×D414yes—
175#11745670—216no—
176C4⋊Dic59—24no—
177C4⋊Dic59—22no—
178C4⋊Dic59C2×D414yes—
179Dic59⋊C4—28no—
180Dic59⋊C4—28no—
181C4×Dic59—28no—
182C4×Dic59—28no—
183C2×C59⋊C8—22no—
184C2×C59⋊C8—22no—
185C2×C59⋊C8—22no—
186C2×C59⋊C8—22no—
187C2×C59⋊C8—22no—
188C2×C59⋊C8C22×C414yes—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59—22no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59C4×D412yes—
201C23×C59C2×SD1612yes—
202C23×C59#13512yes—
203C23×C59C22×D411yescharacteristic
204D4×C59—24no—
205D4×C59—48no—
206D4×C59—42no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—24no—
214C2×C4×C59—24no—
215C2×C4×C59—28no—
216C2×C4×C59—22no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—44no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—84no—
230C2×C4×C59C4×D412yes—
231C2×C4×C59#12012yes—
232C2×C4×C59#12012yes—
233C2×C4×C59C2×SD1612yes—
234C2×C4×C59C2×SD1612yes—
235C2×C4×C59#13712yes—
236C2×C4×C59C23×C411yescharacteristic
237C2×C4×C59C22×D412yes—
238C2×C4×C59C2×#4112yes—
239C2×C4×C59C2×#4111yescharacteristic
240C2×Dic59—216no—
241C2×Dic59—24no—
242C2×Dic59—24no—
243C2×Dic59—24no—
244C2×Dic59—24no—
245C2×Dic59—48no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248Dic118—216no—
249Dic118—24no—
250Dic118—24no—
251Dic118—24no—
252Dic118C4×D418yescomplemented
253C59⋊C8—24no—
254C59⋊C8—42no—
255C4×#448—591nomaximal, Sylow
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—22no—
259C22×C59—22no—
260C22×C59—22no—
261C22×C59—22no—
262C22×C59—42no—
263C22×C59—42no—
264C22×C59—42no—
265C22×C59—81no—
266C22×C59—81no—
267C22×C59—81no—
268C22×C59—81no—
269C22×C59C4×SD1614yescomplemented
270C22×C59C2×C4×D411yescharacteristic
271C22×C59C2×#12011yescharacteristic
272C22×C59#52511yescharacteristic
273C22×C59C22×SD1611yescharacteristic
274C22×C59C2×#13511yescharacteristic
275C22×C59C2×#13711yescharacteristic
276C22×C59#57511yescharacteristic
277C4×C59—22no—
278C4×C59—22no—
279C4×C59—22no—
280C4×C59—42no—
281C4×C59—41no—
282C4×C59—81no—
283C4×C59—81no—
284C4×C59#44814yescomplemented
285C4×C59C2×C4×D412yes—
286C4×C59#52512yes—
287Dic59—28no—
288Dic59—44no—
289C22×C4×Q8—591no—
290C2×#448—591no—
291C2×C4×SD16—592no—
292C4×#121D5911yescharacteristic
293#1880—592no—
294#1874—591no—
295#1865—592no—
296#1853—591no—
297#1787—591no—
298C4×#103—592no—
299C4×C22⋊C8—591no—
300C2×C59—24no—
301C2×C59—82no—
302C2×C59C4×#12011yescharacteristic
303C2×C59C2×C4×SD1611yescharacteristic
304C2×C59C4×#13711yescharacteristic
305C2×C59C2×#44811yescharacteristic
306C2×C59#300211yescharacteristic
307C2×C59#310311yescharacteristic
308C2×C59#310811yescharacteristic
309C23×Q8—591no—
310C22×SD16—592no—
311C2×#121C2×D5911yescharacteristic
312C2×C4×Q8—592no—
313C2×C4×Q8—592no—
314C2×C4×Q8—1188no—
315C2×C4×Q8—1181no—
316C2×C4×Q8—1181no—
317C2×C4×D4—21no—
318C2×C4×D4C2×D5911yescharacteristic
319C22×C4⋊C4—591no—
320C22×C4⋊C4—1181no—
321C22×C42—1181no—
322C22×C42C2×D5911yescharacteristic
323#448—598no—
324C4×SD16—1188no—
325C2×#106—592no—
326C2×#103—592no—
327C2×#103—592no—
328C2×#103—592no—
329C2×#102—592no—
330C2×C22⋊C8—591no—
331C2×C22⋊C8—591no—
332C2×C4×C8—592no—
333#390C2×D5911yescharacteristic
334#389C2×D5911yescharacteristic
335#388—21no—
336#386—21no—
337#380C2×D5911yescharacteristic
338C4×C4⋊C4C2×D5911yescharacteristic
339C4×C22⋊C4—21no—
340#340—592no—
341#340—592no—
342#336—592no—
343C59C4×#44811yescharacteristic, complemented, Sylow
344C22×Q8—594no—
345C22×Q8—592no—
346C22×Q8—1184no—
347C22×Q8—1184no—
348C22×Q8—1182no—
349C22×Q8—1181no—
350C22×D4—21no—
351C22×D4C22×D5911yescharacteristic
352C23×C4—21no—
353C23×C4—1181no—
354C23×C4—1181no—
355C23×C4C22⋊D5911yescharacteristic
356C23×C4C22⋊D5911yescharacteristic
357C23×C4C22×D5911yescharacteristic
358C2×SD16—598no—
359C2×SD16—1184no—
360C2×SD16—1184no—
361C22×C8—592no—
362C22×C8—592no—
363#121—22no—
364#121C4×D5914yes—
365C4×Q8—598no—
366C4×Q8—1184no—
367C4×Q8—1184no—
368C4×Q8—1188no—
369C4×Q8—1188no—
370C4×Q8—1184no—
371C4×D4—24no—
372C4×D4—44no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—592no—
375C2×C4⋊C4—1184no—
376C2×C4⋊C4—1188no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1182no—
379C2×C4⋊C4—1182no—
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C4⋊C4C22×D5911yescharacteristic
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C42—24no—
389C2×C42—21no—
390C2×C42—1188no—
391C2×C42—1182no—
392C2×C42—1182no—
393C2×C42C22⋊D5911yescharacteristic
394C2×C42C22⋊D5911yescharacteristic
395C2×C42C22×D5911yescharacteristic
396C2×C42C22×D5911yescharacteristic
397#106—1184no—
398#103—598no—
399#103—1184no—
400#103—1184no—
401#102—1184no—
402C22⋊C8—594no—
403C22⋊C8—1182no—
404C4×C8—1184no—
405#95—21no—
406#95—21no—
407#95—21no—
408C24—21no—
409C24C2×C22⋊D5911yescharacteristic
410C2×Q8—594no—
411C2×Q8—594no—
412C2×Q8—594no—
413C2×Q8—11816no—
414C2×Q8—1184no—
415C2×Q8—1184no—
416C2×Q8—11816no—
417C2×Q8—1184no—
418C2×Q8—1182no—
419C2×Q8—1182no—
420C2×Q8—1182no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×C4×D5912yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—1184no—
442C22×C4—1184no—
443C22×C4—1184no—
444C22×C4—1182no—
445C22×C4—1184no—
446C22×C4—1182no—
447C22×C4—2362no—
448C22×C4—2362no—
449C22×C4#1174567014yescomplemented
450C22×C4C2×C4×D5912yes—
451C22×C4#1174568512yes—
452C22×C4C2×C22⋊D5911yescharacteristic
453C22×C4C2×C22⋊D5911yescharacteristic
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C2×C22⋊D5911yescharacteristic
456C22×C4C2×C22⋊D5911yescharacteristic
457C22×C4C23×D5911yescharacteristic
458SD16—11816no—
459C2×C8—594no—
460C2×C8—1182no—
461C2×C8—1182no—
462C2×C8—1182no—
463C2×C8—1182no—
464C2×C8—1182no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—594no—
470C4⋊C4—1188no—
471C4⋊C4—1184no—
472C4⋊C4—1188no—
473C4⋊C4—1182no—
474C4⋊C4C2×C4×D5912yes—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—42no—
484C42—1188no—
485C42—1188no—
486C42D4×D5914yes—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23C4×C22⋊D5912yes—
499C23C2×#1174567012yes—
500C23#1427374712yes—
501C23C22×C22⋊D5911yescharacteristic
502Q8—598no—
503Q8—11816no—
504Q8—1184no—
505Q8—1184no—
506Q8—1184no—
507D4—24no—
508D4—48no—
509D4—42no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—84no—
533C2×C4—11816no—
534C2×C4—1184no—
535C2×C4—1184no—
536C2×C4—1184no—
537C2×C4—1184no—
538C2×C4—2368no—
539C2×C4—2362no—
540C2×C4—2362no—
541C2×C4C4×C22⋊D5912yes—
542C2×C4#1427369512yes—
543C2×C4#1427369512yes—
544C2×C4C2×#1174567012yes—
545C2×C4C2×#1174567012yes—
546C2×C4#1427375712yes—
547C2×C4C22×C4×D5911yescharacteristic
548C2×C4C2×D4×D5912yes—
549C2×C4#41×D5912yes—
550C2×C4C2×#1174568511yescharacteristic
551C8—1184no—
552C8—2362no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22C4×#1174567014yescomplemented
567C22C2×C4×C22⋊D5911yescharacteristic
568C22C2×#1427369511yescharacteristic
569C22#1683790911yescharacteristic
570C22C22×#1174567011yescharacteristic
571C22C2×#1427374711yescharacteristic
572C22C2×#1427375711yescharacteristic
573C22#1683820011yescharacteristic
574C4—22no—
575C4—22no—
576C4—22no—
577C4—42no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—1188no—
582C4—2364no—
583C4#1683754014yesdirect product factor
584C4C4×D4×D5912yes—
585C4#1683791112yes—
586C2—24no—
587C2—82no—
588C2C4×#1427369511yescharacteristic
589C2C2×C4×#1174567011yescharacteristic
590C2C4×#1427375711yescharacteristic
591C2C2×#1683754011yescharacteristic
592C2#2143331211yescharacteristic
593C2#2143412311yescharacteristic
594C2#2143417211yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#121×C59C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
cosocleC22×C4×C59C24
hypercenterC4×#121D59
hypocenterC59C4×#448
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic118C2semidirect
3. C2×C4×#11745670C2semidirect
4. #21424454C2semidirect
6. #21424363C2semidirect
7. #21424288C2semidirect
8. #21424030C2semidirect
9. C4×#14273650C2semidirect
10. C4×C22⋊(C59⋊C8)C2semidirect
22. #16837540C4direct (N × H)
50. C22×Dic118C2×C4semidirect
56. C2×#11745670C2×C4semidirect
64. C4×Dic118D4semidirect
77. #14273650C2×C4semidirect
80. C22⋊(C59⋊C8)C2×C4semidirect
132. C22×C4×C59SD16semidirect
252. Dic118C4×D4semidirect
269. C22×C59C4×SD16semidirect
284. C4×C59#448semidirect
343. C59C4×#448semidirect
449. C22×C4#11745670semidirect
566. C22C4×#11745670semidirect
583. C4#16837540direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Dic1182
C2×#168375402
C2×C4×#117456702
#214244542
#214244142
#214243632
#214242882
#214240302
C4×#142736502
C4×C22⋊(C59⋊C8)2
C4×#121×C592
C4×#44859