Finite Group Databasebeta

← C2×#21425104 #28615485

Subgroups — C2×#21425104 #28615485

Order 15104.

2182 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11745672C211yesmaximal, characteristic, complemented
2C22×#14273706C211yesmaximal, characteristic
3C22×#14273651C211yesmaximal, characteristic, complemented
4C22×#14273650C211yesmaximal, characteristic, complemented
5C22×C22⋊(C59⋊C8)C211yesmaximal, characteristic
6#21425104C218yesmaximal, direct product factor
7C2×#16837164C211yesmaximal, characteristic, complemented
8C22×C22⋊Q8×C59C211yesmaximal, characteristic
9C23×Dic118C2211yescharacteristic
10C22×#11745672—24no—
11C22×#11745672—24no—
12C22×#11745672—24no—
13C22×#11745672C2214yescomplemented
14C23×C59⋊C8C2211yescharacteristic
15C23×C59⋊C8C2211yescharacteristic
16C2×#14273706—24no—
17C2×#14273706—24no—
18C2×#14273706—22no—
19C2×#14273706C2214yes—
20C22×C4⋊Dic59C2211yescharacteristic
21C22×Dic59⋊C4—21no—
22C22×C22⋊Dic59—21no—
23C2×#14273651—24no—
24C2×#14273651C2214yescomplemented
25C2×#14273650—24no—
26C2×#14273650C2214yescomplemented
27C2×C22⋊(C59⋊C8)—24no—
28C2×C22⋊(C59⋊C8)C2214yes—
29#16837164C2214yescomplemented
30C23×Q8×C59C2211yescharacteristic
31C24×C4×C59C2211yescharacteristic
32C2×C22⋊Q8×C59—24no—
33C2×C22⋊Q8×C59—22no—
34C2×C22⋊Q8×C59—24no—
35C2×C22⋊Q8×C59C2214yes—
36C22×C4⋊C4×C59—21no—
37C22×C4⋊C4×C59C2211yescharacteristic
38C22×C22⋊C4×C59—21no—
39C22×Dic118—22no—
40C22×Dic118—22no—
41C22×Dic118—22no—
42C22×Dic118—22no—
43C22×Dic118—22no—
44C22×Dic118—21no—
45C22×Dic118D414yes—
46C22×Dic118C2312yes—
47C23×Dic59—21no—
48C23×Dic59—21no—
49C2×#11745672—232no—
50C2×#11745672—28no—
51C2×#11745672—28no—
52C2×#11745672—28no—
53C22×C59⋊C8—22no—
54C22×C59⋊C8—22no—
55C22×C59⋊C8—22no—
56C22×C59⋊C8—22no—
57C22×C59⋊C8—22no—
58C22×C59⋊C8D414yes—
59C22×C59⋊C8C2312yes—
60C22×C59⋊C8C2312yes—
61#14273706—216no—
62#14273706—28no—
63#14273706—28no—
64C2×C4⋊Dic59—22no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59D414yes—
67C2×C4⋊Dic59C2312yes—
68C2×Dic59⋊C4—28no—
69C2×Dic59⋊C4—24no—
70C2×C22⋊Dic59—28no—
71C2×C22⋊Dic59—24no—
72#14273651—28no—
73#14273650—28no—
74C22⋊(C59⋊C8)—28no—
75C25×C59D411yescharacteristic
76C22×Q8×C59—22no—
77C22×Q8×C59—21no—
78C22×Q8×C59—22no—
79C22×Q8×C59—22no—
80C22×Q8×C59—22no—
81C22×Q8×C59—22no—
82C22×Q8×C59D414yes—
83C22×Q8×C59C2312yes—
84C23×C4×C59—21no—
85C23×C4×C59—21no—
86C23×C4×C59—22no—
87C23×C4×C59—22no—
88C23×C4×C59—22no—
89C23×C4×C59—22no—
90C23×C4×C59—22no—
91C23×C4×C59—22no—
92C23×C4×C59D412yes—
93C23×C4×C59D411yescharacteristic
94C23×C4×C59C2312yes—
95C23×C4×C59C2311yescharacteristic
96C22⋊Q8×C59—216no—
97C22⋊Q8×C59—28no—
98C22⋊Q8×C59—28no—
99C2×C4⋊C4×C59—28no—
100C2×C4⋊C4×C59—24no—
101C2×C4⋊C4×C59—21no—
102C2×C4⋊C4×C59—22no—
103C2×C4⋊C4×C59D414yes—
104C2×C4⋊C4×C59C2312yes—
105C2×C22⋊C4×C59—28no—
106C2×C22⋊C4×C59—24no—
107C24×C59—22no—
108C24×C59—22no—
109C24×C59—22no—
110C24×C59—22no—
111C24×C59—22no—
112C24×C59—22no—
113C24×C59Q1614yescomplemented
114C24×C59C2×D412yes—
115C24×C59C2×D411yescharacteristic
116C2×Q8×C59—24no—
117C2×Q8×C59—22no—
118C2×Q8×C59—28no—
119C2×Q8×C59—28no—
120C2×Q8×C59—22no—
121C2×Q8×C59—22no—
122C2×Q8×C59—22no—
123C2×Q8×C59—22no—
124C2×Q8×C59—42no—
125C2×Q8×C59—42no—
126C2×Q8×C59—44no—
127C2×Q8×C59—44no—
128C2×Q8×C59C2×D414yes—
129C22×C4×C59—24no—
130C22×C4×C59—24no—
131C22×C4×C59—22no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—24no—
137C22×C4×C59—24no—
138C22×C4×C59—24no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—42no—
146C22×C4×C59—42no—
147C22×C4×C59—42no—
148C22×C4×C59—42no—
149C22×C4×C59—44no—
150C22×C4×C59—44no—
151C22×C4×C59—44no—
152C22×C4×C59—44no—
153C22×C4×C59C2×D412yes—
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D412yes—
156C22×C4×C59C2×D412yes—
157C22×C4×C59C2×D412yes—
158C22×C4×C59C2×D412yes—
159C22×C4×C59C2×D412yes—
160C22×C4×C59#4112yes—
161C22×C4×C59C2411yescharacteristic
162C22×Dic59—24no—
163C22×Dic59—24no—
164C22×Dic59—22no—
165C22×Dic59—22no—
166C22×Dic59—42no—
167C22×Dic59—42no—
168C22×Dic59—42no—
169C22×Dic59—42no—
170C2×Dic118—28no—
171C2×Dic118—28no—
172C2×Dic118—22no—
173C2×Dic118—22no—
174C2×Dic118—22no—
175C2×Dic118—22no—
176C2×Dic118—24no—
177C2×Dic118—22no—
178C2×Dic118—42no—
179C2×Dic118—44no—
180C2×Dic118—42no—
181C2×Dic118—44no—
182C2×Dic118C2×D414yes—
183C4⋊C4×C59—216no—
184C4⋊C4×C59—24no—
185C4⋊C4×C59—22no—
186C4⋊C4×C59C2×D414yes—
187C22⋊C4×C59—216no—
188C22⋊Dic59—216no—
189#11745672—232no—
190C4⋊Dic59—24no—
191C4⋊Dic59—22no—
192C4⋊Dic59C2×D414yes—
193Dic59⋊C4—216no—
194C2×C59⋊C8—28no—
195C2×C59⋊C8—22no—
196C2×C59⋊C8—22no—
197C2×C59⋊C8—22no—
198C2×C59⋊C8—22no—
199C2×C59⋊C8—22no—
200C2×C59⋊C8—44no—
201C2×C59⋊C8C2×D414yes—
202C23×C59—22no—
203C23×C59—22no—
204C23×C59—22no—
205C23×C59—22no—
206C23×C59—22no—
207C23×C59—22no—
208C23×C59—28no—
209C23×C59—28no—
210C23×C59—28no—
211C23×C59—44no—
212C23×C59—44no—
213C23×C59—44no—
214C23×C59—44no—
215C23×C59#12012yes—
216C23×C59#12012yes—
217C23×C59#12112yes—
218C23×C59C2×Q1612yes—
219C23×C59C2×Q1614yescomplemented
220C23×C59#13512yes—
221C23×C59#13712yes—
222C23×C59#13712yes—
223C23×C59C22×D411yescharacteristic
224Q8×C59—24no—
225Q8×C59—24no—
226Q8×C59—42no—
227Q8×C59—42no—
228Q8×C59—48no—
229C2×C4×C59—22no—
230C2×C4×C59—24no—
231C2×C4×C59—24no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—42no—
237C2×C4×C59—42no—
238C2×C4×C59—44no—
239C2×C4×C59—48no—
240C2×C4×C59—48no—
241C2×C4×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59—42no—
244C2×C4×C59—42no—
245C2×C4×C59—48no—
246C2×C4×C59#12014yes—
247C2×C4×C59#12012yes—
248C2×C4×C59#12012yes—
249C2×C4×C59#12114yes—
250C2×C4×C59#12114yes—
251C2×C4×C59#12112yes—
252C2×C4×C59#12714yes—
253C2×C4×C59C22×D411yescharacteristic
254C2×C4×C59C22×D411yescharacteristic
255C2×C4×C59C22×D411yescharacteristic
256C2×C4×C59C22×D411yescharacteristic
257C2×C4×C59C22×D411yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×Dic59—24no—
260C2×Dic59—24no—
261C2×Dic59—42no—
262C2×Dic59—42no—
263C2×Dic59—42no—
264C2×Dic59—42no—
265C2×Dic59—48no—
266C2×Dic59—48no—
267Dic118—24no—
268Dic118—24no—
269Dic118—42no—
270Dic118—42no—
271Dic118—48no—
272C59⋊C8—24no—
273C59⋊C8—42no—
274C2×#2001—591nomaximal, Sylow
275C22×C59—24no—
276C22×C59—24no—
277C22×C59—24no—
278C22×C59—42no—
279C22×C59—42no—
280C22×C59—42no—
281C22×C59—416no—
282C22×C59—42no—
283C22×C59#39214yes—
284C22×C59#45114yes—
285C22×C59#45114yes—
286C22×C59#45214yes—
287C22×C59#45214yes—
288C22×C59#46714yes—
289C22×C59#47014yes—
290C22×C59C2×#12011yescharacteristic
291C22×C59C2×#12011yescharacteristic
292C22×C59C2×#12111yescharacteristic
293C22×C59C22×Q1611yescharacteristic
294C22×C59C2×#13511yescharacteristic
295C22×C59C2×#13711yescharacteristic
296C22×C59C2×#13711yescharacteristic
297C4×C59—42no—
298C4×C59—44no—
299C4×C59—44no—
300C4×C59#39214yes—
301C4×C59C2×#12011yescharacteristic
302C4×C59C2×#12111yescharacteristic
303C4×C59C2×#12111yescharacteristic
304C4×C59C2×#12711yescharacteristic
305Dic59—44no—
306Dic59—44no—
307C23×Q16—591no—
308C22×C22⋊Q8—591no—
309C22×C22⋊Q8D5911yescharacteristic
310C22×#103—591no—
311C22×#103—591no—
312C22×C22⋊C8—591no—
313#2001—598no—
314C2×#340—591no—
315C2×C59—44no—
316C2×C59#200118yescomplemented
317C2×C59C2×#39211yescharacteristic
318C2×C59C2×#45111yescharacteristic
319C2×C59C2×#45111yescharacteristic
320C2×C59C2×#45211yescharacteristic
321C2×C59C2×#45211yescharacteristic
322C2×C59C2×#46711yescharacteristic
323C2×C59C2×#47011yescharacteristic
324C23×Q8—591no—
325C23×Q8C2×D5911yescharacteristic
326C24×C4C2×D5911yescharacteristic
327C22×Q16—594no—
328C22×Q16—1184no—
329C22×Q16—1184no—
330C22×Q16—1184no—
331C23×C8—591no—
332C23×C8—591no—
333C2×C22⋊Q8—24no—
334C2×C22⋊Q8—22no—
335C2×C22⋊Q8—24no—
336C2×C22⋊Q8—594no—
337C2×C22⋊Q8—1184no—
338C2×C22⋊Q8—1184no—
339C2×C22⋊Q8—1182no—
340C2×C22⋊Q8C2×D5914yes—
341C22×C4⋊C4—21no—
342C22×C4⋊C4—591no—
343C22×C4⋊C4—1181no—
344C22×C4⋊C4C2×D5911yescharacteristic
345C22×C22⋊C4—21no—
346C22×C22⋊C4—1181no—
347C2×#103—594no—
348C2×#103—594no—
349C2×#103—1184no—
350C2×#103—1184no—
351C2×C22⋊C8—594no—
352C2×C22⋊C8—1184no—
353#340—594no—
354C59C2×#200111yescharacteristic, complemented, Sylow
355C25C22⋊D5911yescharacteristic
356C22×Q8—22no—
357C22×Q8—21no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—594no—
363C22×Q8—592no—
364C22×Q8—1182no—
365C22×Q8—1181no—
366C22×Q8—1182no—
367C22×Q8—1182no—
368C22×Q8—1182no—
369C22×Q8—1182no—
370C22×Q8C22⋊D5914yes—
371C22×Q8C22×D5912yes—
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—22no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—1181no—
381C23×C4—1181no—
382C23×C4C22⋊D5912yes—
383C23×C4C22⋊D5911yescharacteristic
384C23×C4C22×D5912yes—
385C23×C4C22×D5911yescharacteristic
386C2×Q16—1188no—
387C2×Q16—1188no—
388C2×Q16—11832no—
389C2×Q16—1188no—
390C22×C8—594no—
391C22×C8—592no—
392C22×C8—592no—
393C22×C8—1182no—
394C22×C8—1182no—
395C22×C8—1182no—
396C22×C8—1182no—
397C22×C8—1182no—
398C22⋊Q8—216no—
399C22⋊Q8—28no—
400C22⋊Q8—28no—
401C22⋊Q8—1188no—
402C22⋊Q8—1188no—
403C22⋊Q8—11816no—
404C2×C4⋊C4—28no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—594no—
409C2×C4⋊C4—592no—
410C2×C4⋊C4—1188no—
411C2×C4⋊C4—1182no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1184no—
414C2×C4⋊C4C22⋊D5914yes—
415C2×C4⋊C4C22×D5912yes—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—24no—
418C2×C22⋊C4—1188no—
419C2×C22⋊C4—1184no—
420#103—1188no—
421#103—1188no—
422C22⋊C8—1188no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24#1174567214yescomplemented
430C24C2×C22⋊D5912yes—
431C24C2×C22⋊D5911yescharacteristic
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—28no—
435C2×Q8—28no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—594no—
445C2×Q8—1184no—
446C2×Q8—1182no—
447C2×Q8—1188no—
448C2×Q8—1188no—
449C2×Q8—1182no—
450C2×Q8—1182no—
451C2×Q8—1182no—
452C2×Q8—1182no—
453C2×Q8—2362no—
454C2×Q8—2362no—
455C2×Q8—2364no—
456C2×Q8—2364no—
457C2×Q8C2×C22⋊D5914yes—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—1184no—
483C22×C4—1184no—
484C22×C4—1182no—
485C22×C4—1182no—
486C22×C4—2362no—
487C22×C4—2362no—
488C22×C4—2362no—
489C22×C4—2362no—
490C22×C4D4×D5912yes—
491C22×C4D4×D5912yes—
492C22×C4D4×D5912yes—
493C22×C4#1174568712yes—
494C22×C4C2×C22⋊D5912yes—
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C2×C22⋊D5912yes—
497C22×C4C2×C22⋊D5912yes—
498C22×C4C23×D5911yescharacteristic
499Q16—11832no—
500C2×C8—594no—
501C2×C8—1188no—
502C2×C8—1182no—
503C2×C8—1182no—
504C2×C8—1182no—
505C2×C8—1182no—
506C2×C8—1182no—
507C2×C8—2364no—
508C4⋊C4—216no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—594no—
512C4⋊C4—11816no—
513C4⋊C4—1184no—
514C4⋊C4—1182no—
515C4⋊C4C2×C22⋊D5914yes—
516C22⋊C4—216no—
517C22⋊C4—11816no—
518C23—22no—
519C23—22no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—28no—
525C23—28no—
526C23—22no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23#1427369512yes—
532C23#1427369612yes—
533C23#1427369912yes—
534C23C2×#1174567212yes—
535C23C2×#1174567214yescomplemented
536C23#1427374712yes—
537C23#1427375712yes—
538C23#1427375812yes—
539C23C22×C22⋊D5911yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—42no—
543Q8—42no—
544Q8—48no—
545Q8—1184no—
546Q8—1184no—
547Q8—2362no—
548Q8—2362no—
549Q8—2368no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—1184no—
568C2×C4—1184no—
569C2×C4—2362no—
570C2×C4—2362no—
571C2×C4—2362no—
572C2×C4—2362no—
573C2×C4—2368no—
574C2×C4—2368no—
575C2×C4#1427369512yes—
576C2×C4#1427369514yes—
577C2×C4#1427369612yes—
578C2×C4#1427369814yes—
579C2×C4#1427369914yes—
580C2×C4#1427369912yes—
581C2×C4#1427372314yes—
582C2×C4C2×D4×D5911yescharacteristic
583C2×C4C2×D4×D5911yescharacteristic
584C2×C4C2×D4×D5911yescharacteristic
585C2×C4C22×C22⋊D5911yescharacteristic
586C2×C4C22×C22⋊D5911yescharacteristic
587C2×C4C2×#1174568711yescharacteristic
588C8—1184no—
589C8—2362no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—416no—
596C22—42no—
597C22—42no—
598C22#1683731014yes—
599C22#1683755714yes—
600C22#1683755814yes—
601C22#1683756214yes—
602C22#1683756414yes—
603C22#1683764114yes—
604C22#1683765814yes—
605C22C2×#1427369511yescharacteristic
606C22C2×#1427369611yescharacteristic
607C22C2×#1427369911yescharacteristic
608C22C22×#1174567211yescharacteristic
609C22C2×#1427374711yescharacteristic
610C22C2×#1427375711yescharacteristic
611C22C2×#1427375811yescharacteristic
612C4—42no—
613C4—44no—
614C4—44no—
615C4—2364no—
616C4—2364no—
617C4#1683731014yes—
618C4C2×#1427369511yescharacteristic
619C4C2×#1427369811yescharacteristic
620C4C2×#1427369911yescharacteristic
621C4C2×#1427372311yescharacteristic
622C2—44no—
623C2#2142510418yesdirect product factor
624C2C2×#1683731011yescharacteristic
625C2C2×#1683755711yescharacteristic
626C2C2×#1683755811yescharacteristic
627C2C2×#1683756211yescharacteristic
628C2C2×#1683756411yescharacteristic
629C2C2×#1683764111yescharacteristic
630C2C2×#1683765811yescharacteristic
subgroupG/H
centerC24C2×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC22×C22⋊Q8×C59C2
socleC24×C59C2×D4
prime radicalC25×C59D4
hypercenterC22×C22⋊Q8D59
hypocenterC59C2×#2001
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C23×#11745672C2semidirect
3. C22×#14273651C2semidirect
4. C22×#14273650C2semidirect
6. #21425104C2direct (N × H)
7. C2×#16837164C2semidirect
13. C22×#11745672C22semidirect
24. C2×#14273651C22semidirect
26. C2×#14273650C22semidirect
29. #16837164C22semidirect
113. C24×C59Q16semidirect
219. C23×C59C2×Q16semidirect
316. C2×C59#2001semidirect
354. C59C2×#2001semidirect
429. C24#11745672semidirect
535. C23C2×#11745672semidirect
623. C2#21425104direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117456722
C22×#142737062
C22×#142736512
C22×#142736502
C22×C22⋊(C59⋊C8)2
#214251042
C2×#168371642
C22×C22⋊Q8×C592
C2×#200159