Finite Group Databasebeta

← C2×#21424402 #28589882

Subgroups — C2×#21424402 #28589882

Order 15104.

2270 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273707C211yesmaximal, characteristic
2C22×C4⋊C4×D59C211yesmaximal, characteristic
3C22×#14273652C211yesmaximal, characteristic, complemented
4C22×#14273650C211yesmaximal, characteristic, complemented
5C22×(C2×D59)⋊C8C211yesmaximal, characteristic
6#21424402C218yesmaximal, direct product factor
7C2×#16837164C211yesmaximal, characteristic, complemented
8C2×C59×#340C211yesmaximal, characteristic, complemented
9C23×Dic118C2211yescharacteristic
10C23×C4×D59—21no—
11C23×C4×D59C2211yescharacteristic
12C23×C59⋊C8C2211yescharacteristic
13C2×#14273707—24no—
14C2×#14273707—22no—
15C2×#14273707C418yescomplemented
16C2×#14273707C2214yes—
17C22×C4⋊Dic59—21no—
18C22×C4⋊Dic59C2211yescharacteristic
19C22×Dic59⋊C4—21no—
20C2×C4⋊C4×D59—22no—
21C2×C4⋊C4×D59—24no—
22C2×C4⋊C4×D59—24no—
23C2×C4⋊C4×D59C2214yes—
24C22×(C2×D59)⋊C4—21no—
25C2×#14273652—24no—
26C2×#14273652C2214yescomplemented
27C2×#14273650—24no—
28C2×#14273650C2214yescomplemented
29C2×(C2×D59)⋊C8—24no—
30C2×(C2×D59)⋊C8C2214yes—
31#16837164C2214yescomplemented
32C23×C8×C59C2211yescharacteristic
33C22×C4⋊C4×C59C2211yescharacteristic
34C22×C4⋊C4×C59C2211yescharacteristic
35C59×#340C2214yescomplemented
36C24×D59D411yescharacteristic
37C22×Dic118—22no—
38C22×Dic118—22no—
39C22×Dic118—22no—
40C22×Dic118—22no—
41C22×Dic118—21no—
42C22×Dic118—22no—
43C22×Dic118C2×C414yescomplemented
44C22×Dic118C2312yes—
45C23×Dic59—21no—
46C23×Dic59—21no—
47C23×Dic59D411yescharacteristic
48C22×C4×D59—22no—
49C22×C4×D59—22no—
50C22×C4×D59—22no—
51C22×C4×D59—22no—
52C22×C4×D59—28no—
53C22×C4×D59—24no—
54C22×C4×D59—44no—
55C22×C4×D59—44no—
56C22×C4×D59C2×C414yes—
57C22×C4×D59D412yes—
58C22×C4×D59D414yes—
59C22×C4×D59C2312yes—
60C22×C59⋊C8—22no—
61C22×C59⋊C8—22no—
62C22×C59⋊C8—22no—
63C22×C59⋊C8C2312yes—
64#14273707—216no—
65#14273707—28no—
66#14273707C2×C4116yescomplemented
67C2×C4⋊Dic59—22no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—28no—
70C2×C4⋊Dic59—24no—
71C2×C4⋊Dic59D414yes—
72C2×C4⋊Dic59C2312yes—
73C2×Dic59⋊C4—28no—
74C2×Dic59⋊C4—24no—
75C4⋊C4×D59—216no—
76C4⋊C4×D59—28no—
77C4⋊C4×D59—28no—
78C2×(C2×D59)⋊C4—28no—
79C2×(C2×D59)⋊C4—24no—
80#14273652—28no—
81#14273650—28no—
82(C2×D59)⋊C8—28no—
83C23×C4×C59—21no—
84C23×C4×C59—21no—
85C23×C4×C59C2311yescharacteristic
86C22×C8×C59—22no—
87C22×C8×C59—22no—
88C22×C8×C59—22no—
89C22×C8×C59C2312yes—
90C2×C4⋊C4×C59—21no—
91C2×C4⋊C4×C59—22no—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59—22no—
94C2×C4⋊C4×C59C2×C414yescomplemented
95C2×C4⋊C4×C59D414yes—
96C2×C4⋊C4×C59C2312yes—
97C2×C4⋊C4×C59C2312yes—
98C24×C59C2×D411yescharacteristic
99C23×D59—22no—
100C23×D59—22no—
101C23×D59—22no—
102C23×D59—22no—
103C23×D59C22⋊C414yes—
104C23×D59SD1614yes—
105C23×D59Q1614yes—
106C23×D59C2×D412yes—
107C22×C4×C59—24no—
108C22×C4×C59—24no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—42no—
112C22×C4×C59—42no—
113C22×C4×C59—42no—
114C22×C4×C59—42no—
115C22×C4×C59C22×C412yes—
116C22×C4×C59C2×D411yescharacteristic
117C22×C4×C59C2×D412yes—
118C22×C4×C59C2×D412yes—
119C22×C4×C59C2×D412yes—
120C22×C4×C59#4112yes—
121C22×C4×C59#4112yes—
122C22×C4×C59C2411yescharacteristic
123C22×Dic59—22no—
124C22×Dic59—22no—
125C22×Dic59—24no—
126C22×Dic59—24no—
127C22×Dic59—22no—
128C22×Dic59—22no—
129C22×Dic59—42no—
130C22×Dic59—42no—
131C22×Dic59—42no—
132C22×Dic59—42no—
133C22×Dic59C22⋊C414yes—
134C22×Dic59C2×D412yes—
135C2×C4×D59—28no—
136C2×C4×D59—28no—
137C2×C4×D59—24no—
138C2×C4×D59—28no—
139C2×C4×D59—28no—
140C2×C4×D59—24no—
141C2×C4×D59—22no—
142C2×C4×D59—22no—
143C2×C4×D59—22no—
144C2×C4×D59—22no—
145C2×C4×D59—216no—
146C2×C4×D59—48no—
147C2×C4×D59—432no—
148C2×C4×D59—48no—
149C2×C4×D59C22⋊C418yes—
150C2×C4×D59C22×C414yes—
151C2×C4×D59C2×D414yes—
152C2×Dic118—28no—
153C2×Dic118—28no—
154C2×Dic118—22no—
155C2×Dic118—22no—
156C2×Dic118—22no—
157C2×Dic118—22no—
158C2×Dic118—24no—
159C2×Dic118—22no—
160C2×Dic118—42no—
161C2×Dic118—44no—
162C2×Dic118—44no—
163C2×Dic118—42no—
164C2×Dic118C22×C414yescomplemented
165C2×C8×C59—22no—
166C2×C8×C59—22no—
167C2×C8×C59—22no—
168C2×C8×C59—44no—
169C4⋊C4×C59—24no—
170C4⋊C4×C59—24no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59C22×C414yescomplemented
174C4⋊C4×C59C2×D414yes—
175(C2×D59)⋊C4—216no—
176C4⋊Dic59—24no—
177C4⋊Dic59—22no—
178C4⋊Dic59—216no—
179C4⋊Dic59C2×D414yes—
180Dic59⋊C4—216no—
181C2×C59⋊C8—22no—
182C2×C59⋊C8—22no—
183C2×C59⋊C8—22no—
184C2×C59⋊C8—44no—
185C23×C59C2×C22⋊C412yes—
186C23×C59#12012yes—
187C23×C59#12312yes—
188C23×C59C2×SD1612yes—
189C23×C59C2×Q1612yes—
190C23×C59AGL(1,ℤ/8)12yes—
191C23×C59#13712yes—
192C23×C59C22×D411yescharacteristic
193C22×D59—22no—
194C22×D59—22no—
195C22×D59—22no—
196C22×D59—22no—
197C22×D59—28no—
198C22×D59—28no—
199C22×D59—28no—
200C22×D59—28no—
201C22×D59—28no—
202C22×D59—28no—
203C22×D59#103116yescomplemented
204C22×D59C2×C22⋊C414yes—
205C22×D59C2×SD1614yes—
206C22×D59C2×Q1614yes—
207C2×C4×C59—24no—
208C2×C4×C59—24no—
209C2×C4×C59—42no—
210C2×C4×C59—42no—
211C2×C4×C59—42no—
212C2×C4×C59—42no—
213C2×C4×C59—48no—
214C2×C4×C59—48no—
215C2×C4×C59C2×C22⋊C412yes—
216C2×C4×C59C4×D414yes—
217C2×C4×C59C4×D414yes—
218C2×C4×C59#12012yes—
219C2×C4×C59#12114yes—
220C2×C4×C59#12114yes—
221C2×C4×C59#12312yes—
222C2×C4×C59C23×C411yescharacteristic
223C2×C4×C59C22×D411yescharacteristic
224C2×C4×C59C22×D411yescharacteristic
225C2×C4×C59C22×D411yescharacteristic
226C2×C4×C59C2×#4111yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×Dic59—22no—
229C2×Dic59—22no—
230C2×Dic59—28no—
231C2×Dic59—24no—
232C2×Dic59—24no—
233C2×Dic59—42no—
234C2×Dic59—42no—
235C2×Dic59—42no—
236C2×Dic59—42no—
237C2×Dic59—48no—
238C2×Dic59—48no—
239C2×Dic59C2×C22⋊C414yes—
240C4×D59—24no—
241C4×D59—24no—
242C4×D59—24no—
243C4×D59—24no—
244C4×D59—216no—
245C4×D59—432no—
246Dic118—24no—
247Dic118—24no—
248Dic118—42no—
249Dic118—42no—
250Dic118—48no—
251C8×C59—42no—
252C59⋊C8—42no—
253C2×#1872—591nomaximal, Sylow
254C22×C59#38614yes—
255C22×C59C2×#10314yes—
256C22×C59#41714yes—
257C22×C59#45014yes—
258C22×C59#45114yes—
259C22×C59#48314yes—
260C22×C59#48414yes—
261C22×C59C22×C22⋊C411yescharacteristic
262C22×C59C2×#12011yescharacteristic
263C22×C59C2×#12311yescharacteristic
264C22×C59C22×SD1611yescharacteristic
265C22×C59C22×Q1611yescharacteristic
266C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C59C2×#13711yescharacteristic
268C2×D59—24no—
269C2×D59—24no—
270C2×D59—24no—
271C2×D59—232no—
272C2×D59—24no—
273C2×D59—24no—
274C2×D59—24no—
275C2×D59C2×#10318yescomplemented
276C4×C59—44no—
277C4×C59—44no—
278C4×C59#38614yes—
279C4×C59C2×C4×D411yescharacteristic
280C4×C59C2×C4×D411yescharacteristic
281C4×C59C2×#12111yescharacteristic
282C4×C59C2×#12111yescharacteristic
283Dic59—24no—
284Dic59—44no—
285Dic59—44no—
286C22×C22⋊Q8—591no—
287C23×C4⋊C4—591no—
288C22×#103—591no—
289C22×#103—591no—
290C22×C22⋊C8—591no—
291#1872—598no—
292C2×#340—591no—
293C2×#340D5911yescharacteristic, complemented
294C2×C59#187218yescomplemented
295C2×C59C2×#38611yescharacteristic
296C2×C59C22×#10311yescharacteristic
297C2×C59C2×#41711yescharacteristic
298C2×C59C2×#45011yescharacteristic
299C2×C59C2×#45111yescharacteristic
300C2×C59C2×#48311yescharacteristic
301C2×C59C2×#48411yescharacteristic
302D59—28no—
303C23×Q8—591no—
304C24×C4—591no—
305C24×C4—1181no—
306C23×C8—591no—
307C23×C8C2×D5911yescharacteristic
308C2×C22⋊Q8—598no—
309C2×C22⋊Q8—594no—
310C2×C22⋊Q8—1182no—
311C2×C22⋊Q8—1184no—
312C22×C4⋊C4—591no—
313C22×C4⋊C4—594no—
314C22×C4⋊C4—1184no—
315C22×C4⋊C4—1182no—
316C22×C4⋊C4—1181no—
317C22×C4⋊C4—1181no—
318C22×C4⋊C4—1184no—
319C22×C4⋊C4C2×D5911yescharacteristic
320C22×C4⋊C4C2×D5911yescharacteristic
321C22×C22⋊C4—1181no—
322C2×#103—594no—
323C2×#103—594no—
324C2×#103—1184no—
325C2×#103—1184no—
326C2×C22⋊C8—594no—
327C2×C22⋊C8—1184no—
328#340—594no—
329#340C2×D5914yescomplemented
330C59C2×#187211yescharacteristic, complemented, Sylow
331C25—591no—
332C22×Q8—594no—
333C22×Q8—592no—
334C22×Q8—1182no—
335C22×Q8—1181no—
336C22×Q8—1182no—
337C22×Q8—1182no—
338C22×Q8—1182no—
339C22×Q8—1182no—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—591no—
343C23×C4—594no—
344C23×C4—592no—
345C23×C4—594no—
346C23×C4—592no—
347C23×C4—1184no—
348C23×C4—1188no—
349C23×C4—1182no—
350C23×C4—1182no—
351C23×C4—1182no—
352C23×C4—1182no—
353C23×C4—1181no—
354C23×C4—1181no—
355C23×C4—2364no—
356C23×C4—2364no—
357C23×C4C22×D5911yescharacteristic
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—592no—
362C22×C8—1182no—
363C22×C8—1182no—
364C22×C8—1182no—
365C22×C8C22×D5912yes—
366C22⋊Q8—5916no—
367C22⋊Q8—11816no—
368C22⋊Q8—1188no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—594no—
374C2×C4⋊C4—592no—
375C2×C4⋊C4—1188no—
376C2×C4⋊C4—11816no—
377C2×C4⋊C4—1184no—
378C2×C4⋊C4—1188no—
379C2×C4⋊C4—1184no—
380C2×C4⋊C4—1188no—
381C2×C4⋊C4—1182no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1188no—
384C2×C4⋊C4C4×D5914yescomplemented
385C2×C4⋊C4D23614yes—
386C2×C4⋊C4C22×D5912yes—
387C2×C4⋊C4C22×D5912yes—
388C2×C22⋊C4—1188no—
389C2×C22⋊C4—1184no—
390#103—1188no—
391#103—1188no—
392C22⋊C8—1188no—
393C24—594no—
394C24—594no—
395C24—594no—
396C24—592no—
397C24—1182no—
398C24—1182no—
399C24—1182no—
400C24—1182no—
401C24D4×D5911yescharacteristic
402C2×Q8—594no—
403C2×Q8—1184no—
404C2×Q8—1182no—
405C2×Q8—1188no—
406C2×Q8—1188no—
407C2×Q8—1182no—
408C2×Q8—1182no—
409C2×Q8—1182no—
410C2×Q8—1182no—
411C2×Q8—2362no—
412C2×Q8—2364no—
413C2×Q8—2364no—
414C2×Q8—2362no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—594no—
424C22×C4—592no—
425C22×C4—598no—
426C22×C4—594no—
427C22×C4—594no—
428C22×C4—11816no—
429C22×C4—1182no—
430C22×C4—1182no—
431C22×C4—1188no—
432C22×C4—1188no—
433C22×C4—1184no—
434C22×C4—1188no—
435C22×C4—1188no—
436C22×C4—1184no—
437C22×C4—1182no—
438C22×C4—1182no—
439C22×C4—1182no—
440C22×C4—1182no—
441C22×C4—1184no—
442C22×C4—1184no—
443C22×C4—1182no—
444C22×C4—1182no—
445C22×C4—2368no—
446C22×C4—2368no—
447C22×C4—23632no—
448C22×C4—2362no—
449C22×C4—2362no—
450C22×C4—2362no—
451C22×C4—2362no—
452C22×C4C2×C4×D5912yes—
453C22×C4C2×D23612yes—
454C22×C4#1174568512yes—
455C22×C4D4×D5911yescharacteristic
456C22×C4D4×D5912yes—
457C22×C4D4×D5912yes—
458C22×C4#1174568912yes—
459C22×C4C23×D5911yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C2×C8—1182no—
465C2×C8—1182no—
466C2×C8—1182no—
467C2×C8—2364no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—594no—
473C4⋊C4—11816no—
474C4⋊C4—11816no—
475C4⋊C4—1184no—
476C4⋊C4—1182no—
477C4⋊C4C2×C4×D5914yescomplemented
478C4⋊C4C2×D23614yes—
479C22⋊C4—11816no—
480C23—5916no—
481C23—594no—
482C23—594no—
483C23—594no—
484C23—1182no—
485C23—1182no—
486C23—1188no—
487C23—1182no—
488C23—1182no—
489C23—1188no—
490C23—1188no—
491C23—1188no—
492C23—1188no—
493C23—1188no—
494C23C22⋊C4×D5912yes—
495C23#1427369712yes—
496C23#1427371212yes—
497C23SD16×D5912yes—
498C23Q16×D5912yes—
499C23#1427375112yes—
500C23#1427375612yes—
501C23C2×D4×D5911yescharacteristic
502Q8—1184no—
503Q8—1184no—
504Q8—2362no—
505Q8—2362no—
506Q8—2368no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—48no—
515C2×C4—594no—
516C2×C4—1182no—
517C2×C4—1182no—
518C2×C4—1188no—
519C2×C4—1184no—
520C2×C4—1184no—
521C2×C4—11816no—
522C2×C4—1184no—
523C2×C4—1184no—
524C2×C4—1184no—
525C2×C4—1184no—
526C2×C4—23632no—
527C2×C4—2362no—
528C2×C4—2362no—
529C2×C4—2362no—
530C2×C4—2362no—
531C2×C4—2368no—
532C2×C4—2368no—
533C2×C4C22⋊C4×D5912yes—
534C2×C4#1427368914yes—
535C2×C4C4×D23614yes—
536C2×C4#1427369712yes—
537C2×C4#1427370014yes—
538C2×C4#1427370214yes—
539C2×C4#1427371212yes—
540C2×C4C22×C4×D5911yescharacteristic
541C2×C4C2×D4×D5911yescharacteristic
542C2×C4C2×D4×D5911yescharacteristic
543C2×C4C22×D23611yescharacteristic
544C2×C4C2×#1174568511yescharacteristic
545C2×C4C2×#1174568911yescharacteristic
546C8—42no—
547C8—2362no—
548C22—598no—
549C22—11832no—
550C22—1184no—
551C22—1184no—
552C22—1184no—
553C22—1184no—
554C22—1184no—
555C22—1184no—
556C22#1683728014yes—
557C22#103×D5914yes—
558C22#1683740014yes—
559C22#1683755414yes—
560C22#1683755914yes—
561C22#1683773414yes—
562C22#1683773914yes—
563C22C2×C22⋊C4×D5911yescharacteristic
564C22C2×#1427369711yescharacteristic
565C22C2×#1427371211yescharacteristic
566C22C2×SD16×D5911yescharacteristic
567C22C2×Q16×D5911yescharacteristic
568C22C2×#1427375111yescharacteristic
569C22C2×#1427375611yescharacteristic
570C4—44no—
571C4—44no—
572C4—1184no—
573C4—2364no—
574C4—2364no—
575C4#1683728014yes—
576C4C2×#1427368911yescharacteristic
577C4C2×C4×D23611yescharacteristic
578C4C2×#1427370011yescharacteristic
579C4C2×#1427370211yescharacteristic
580C2—1188no—
581C2#2142440218yesdirect product factor
582C2C2×#1683728011yescharacteristic
583C2C2×#103×D5911yescharacteristic
584C2C2×#1683740011yescharacteristic
585C2C2×#1683755411yescharacteristic
586C2C2×#1683755911yescharacteristic
587C2C2×#1683773411yescharacteristic
588C2C2×#1683773911yescharacteristic
subgroupG/H
centerC24D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#340C2
socleC24×C59C2×D4
prime radicalC24×D59D4
cosocleC22×C4×C59C24
hypercenterC2×#340D59
hypocenterC59C2×#1872
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C22×#14273652C2semidirect
4. C22×#14273650C2semidirect
6. #21424402C2direct (N × H)
7. C2×#16837164C2semidirect
8. C2×C59×#340C2semidirect
15. C2×#14273707C4semidirect
26. C2×#14273652C22semidirect
28. C2×#14273650C22semidirect
31. #16837164C22semidirect
35. C59×#340C22semidirect
43. C22×Dic118C2×C4semidirect
66. #14273707C2×C4semidirect
94. C2×C4⋊C4×C59C2×C4semidirect
164. C2×Dic118C22×C4semidirect
173. C4⋊C4×C59C22×C4semidirect
203. C22×D59#103semidirect
275. C2×D59C2×#103semidirect
293. C2×#340D59semidirect
294. C2×C59#1872semidirect
329. #340C2×D59semidirect
330. C59C2×#1872semidirect
384. C2×C4⋊C4C4×D59semidirect
477. C4⋊C4C2×C4×D59semidirect
581. C2#21424402direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142737072
C22×C4⋊C4×D592
C22×#142736522
C22×#142736502
C22×(C2×D59)⋊C82
#214244022
C2×#168371642
C2×C59×#3402
C2×#187259