Finite Group Databasebeta

← C2×#21424405 #28589892

Subgroups — C2×#21424405 #28589892

Order 15104.

1966 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273703C211yesmaximal, characteristic
2C22×#14273685C211yesmaximal, characteristic
3C22×#14273651C211yesmaximal, characteristic, complemented
4C22×(C2×D59)⋊C8C211yesmaximal, characteristic
5#21424405C218yesmaximal, direct product factor
6C2×#16837165C211yesmaximal, characteristic, complemented
7C2×#16837164C211yesmaximal, characteristic, complemented
8C22×#103×C59C211yesmaximal, characteristic, complemented
9C23×C4×D59C2211yescharacteristic
10C23×C59⋊C8C2211yescharacteristic
11C2×#14273703—24no—
12C2×#14273703—22no—
13C2×#14273703C418yescomplemented
14C2×#14273703C2214yes—
15C2×#14273685—22no—
16C2×#14273685—24no—
17C2×#14273685—24no—
18C2×#14273685C2214yes—
19C22×C4⋊Dic59C2211yescharacteristic
20C22×C4⋊Dic59C2211yescharacteristic
21C22×Dic59⋊C4—21no—
22C22×(C2×D59)⋊C4—21no—
23C22×(C2×D59)⋊C4—21no—
24C22×C4×Dic59—21no—
25C2×#14273651—24no—
26C2×#14273651C2214yescomplemented
27C2×(C2×D59)⋊C8—24no—
28C2×(C2×D59)⋊C8C2214yes—
29#16837165C2214yescomplemented
30#16837164C2214yescomplemented
31C23×Q8×C59C2211yescharacteristic
32C23×C8×C59C2211yescharacteristic
33C22×C4⋊C4×C59C2211yescharacteristic
34C2×#103×C59—24no—
35C2×#103×C59C2214yescomplemented
36C24×D59D411yescharacteristic
37C23×Dic59—21no—
38C23×Dic59—21no—
39C23×Dic59D411yescharacteristic
40C22×C4×D59—22no—
41C22×C4×D59—22no—
42C22×C4×D59—22no—
43C22×C4×D59—22no—
44C22×C4×D59C2×C414yes—
45C22×C4×D59D412yes—
46C22×C4×D59D414yes—
47C22×C4×D59C2312yes—
48C22×C59⋊C8—22no—
49C22×C59⋊C8—22no—
50C22×C59⋊C8—22no—
51C22×C59⋊C8C2312yes—
52#14273703—216no—
53#14273703—28no—
54#14273703C2×C4116yescomplemented
55#14273685—216no—
56#14273685—28no—
57#14273685—28no—
58C2×C4⋊Dic59—21no—
59C2×C4⋊Dic59—22no—
60C2×C4⋊Dic59—21no—
61C2×C4⋊Dic59—22no—
62C2×C4⋊Dic59C2×C414yes—
63C2×C4⋊Dic59D414yes—
64C2×C4⋊Dic59C2312yes—
65C2×C4⋊Dic59C2312yes—
66C2×Dic59⋊C4—28no—
67C2×Dic59⋊C4—24no—
68C2×(C2×D59)⋊C4—28no—
69C2×(C2×D59)⋊C4—28no—
70C2×(C2×D59)⋊C4—24no—
71C2×(C2×D59)⋊C4—24no—
72C2×C4×Dic59—28no—
73C2×C4×Dic59—24no—
74#14273651—28no—
75(C2×D59)⋊C8—28no—
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×C59C2×C414yes—
83C22×Q8×C59C2312yes—
84C23×C4×C59—21no—
85C23×C4×C59—21no—
86C23×C4×C59C2311yescharacteristic
87C22×C8×C59—22no—
88C22×C8×C59—22no—
89C22×C8×C59—22no—
90C22×C8×C59C2312yes—
91C2×C4⋊C4×C59—21no—
92C2×C4⋊C4×C59—22no—
93C2×C4⋊C4×C59D414yes—
94C2×C4⋊C4×C59C2312yes—
95#103×C59—28no—
96C24×C59C2×D411yescharacteristic
97C23×D59—22no—
98C23×D59—22no—
99C23×D59—22no—
100C23×D59—22no—
101C23×D59—22no—
102C23×D59—22no—
103C23×D59C22⋊C414yes—
104C23×D59C2×D412yes—
105C2×Q8×C59—24no—
106C2×Q8×C59—22no—
107C2×Q8×C59—28no—
108C2×Q8×C59—28no—
109C2×Q8×C59—22no—
110C2×Q8×C59—22no—
111C2×Q8×C59—22no—
112C2×Q8×C59—22no—
113C2×Q8×C59—42no—
114C2×Q8×C59—44no—
115C2×Q8×C59—44no—
116C2×Q8×C59—42no—
117C2×Q8×C59C22×C414yes—
118C22×C4×C59—24no—
119C22×C4×C59—24no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—42no—
123C22×C4×C59—42no—
124C22×C4×C59—42no—
125C22×C4×C59—42no—
126C22×C4×C59C22×C412yes—
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D412yes—
129C22×C4×C59C2×D412yes—
130C22×C4×C59C2×D412yes—
131C22×C4×C59#4112yes—
132C22×C4×C59#4112yes—
133C22×C4×C59C2411yescharacteristic
134C22×Dic59—22no—
135C22×Dic59—22no—
136C22×Dic59—24no—
137C22×Dic59—24no—
138C22×Dic59—22no—
139C22×Dic59—22no—
140C22×Dic59—42no—
141C22×Dic59—42no—
142C22×Dic59—42no—
143C22×Dic59—42no—
144C22×Dic59C22⋊C414yes—
145C22×Dic59C2×D412yes—
146C2×C4×D59—28no—
147C2×C4×D59—28no—
148C2×C4×D59—24no—
149C2×C4×D59—28no—
150C2×C4×D59—28no—
151C2×C4×D59—24no—
152C2×C4×D59—22no—
153C2×C4×D59—22no—
154C2×C4×D59—22no—
155C2×C4×D59—22no—
156C2×C4×D59C22⋊C418yes—
157C2×C4×D59C22×C414yes—
158C2×C4×D59C2×D414yes—
159C2×C8×C59—22no—
160C2×C8×C59—22no—
161C2×C8×C59—22no—
162C2×C8×C59—44no—
163C4⋊C4×C59—24no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59C2×D414yes—
166(C2×D59)⋊C4—216no—
167(C2×D59)⋊C4—216no—
168C4⋊Dic59—24no—
169C4⋊Dic59—24no—
170C4⋊Dic59—22no—
171C4⋊Dic59—22no—
172C4⋊Dic59C22×C414yes—
173C4⋊Dic59C2×D414yes—
174Dic59⋊C4—216no—
175C4×Dic59—216no—
176C2×C59⋊C8—22no—
177C2×C59⋊C8—22no—
178C2×C59⋊C8—22no—
179C2×C59⋊C8—44no—
180C23×C59C2×C22⋊C412yes—
181C23×C59#12012yes—
182C23×C59#12312yes—
183C23×C59#13512yes—
184C23×C59#13512yes—
185C23×C59#13712yes—
186C23×C59#13712yes—
187C23×C59C22×D411yescharacteristic
188C22×D59—22no—
189C22×D59—22no—
190C22×D59—22no—
191C22×D59—22no—
192C22×D59—22no—
193C22×D59—22no—
194C22×D59—28no—
195C22×D59—28no—
196C22×D59—28no—
197C22×D59—44no—
198C22×D59—44no—
199C22×D59—44no—
200C22×D59—44no—
201C22×D59C2×C22⋊C414yes—
202Q8×C59—24no—
203Q8×C59—24no—
204Q8×C59—42no—
205Q8×C59—42no—
206Q8×C59—48no—
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—
245C8×C59—42no—
246C59⋊C8—42no—
247C2×#1873—591nomaximal, Sylow
248C22×C59#38614yes—
249C22×C59#41614yes—
250C22×C59#41914yes—
251C22×C59#45214yes—
252C22×C59#45214yes—
253C22×C59#48514yes—
254C22×C59#48514yes—
255C22×C59C22×C22⋊C411yescharacteristic
256C22×C59C2×#12011yescharacteristic
257C22×C59C2×#12311yescharacteristic
258C22×C59C2×#13511yescharacteristic
259C22×C59C2×#13511yescharacteristic
260C22×C59C2×#13711yescharacteristic
261C22×C59C2×#13711yescharacteristic
262C2×D59—24no—
263C2×D59—24no—
264C2×D59—24no—
265C2×D59—42no—
266C2×D59—42no—
267C2×D59—416no—
268C2×D59—42no—
269C2×D59—42no—
270C4×C59—44no—
271C4×C59—44no—
272C4×C59#38614yes—
273C4×C59C2×C4×D411yescharacteristic
274C4×C59C2×C4×D411yescharacteristic
275C4×C59C2×#12111yescharacteristic
276C4×C59C2×#12111yescharacteristic
277Dic59—24no—
278Dic59—44no—
279Dic59—44no—
280C22×C22⋊Q8—591no—
281C22×#117—591no—
282C22×#103—591no—
283C22×#103D5911yescharacteristic, complemented
284C22×C22⋊C8—591no—
285#1873—598no—
286C2×#340—591no—
287C2×#340—591no—
288C2×C59#187318yescomplemented
289C2×C59C2×#38611yescharacteristic
290C2×C59C2×#41611yescharacteristic
291C2×C59C2×#41911yescharacteristic
292C2×C59C2×#45211yescharacteristic
293C2×C59C2×#45211yescharacteristic
294C2×C59C2×#48511yescharacteristic
295C2×C59C2×#48511yescharacteristic
296D59—44no—
297C23×Q8C2×D5911yescharacteristic
298C24×C4—591no—
299C23×C8—591no—
300C23×C8C2×D5911yescharacteristic
301C2×C22⋊Q8—598no—
302C2×C22⋊Q8—594no—
303C2×C22⋊Q8—1182no—
304C2×C22⋊Q8—1184no—
305C2×#117—594no—
306C2×#117—1184no—
307C2×#117—1182no—
308C2×#117—1184no—
309C22×C4⋊C4—591no—
310C22×C4⋊C4—591no—
311C22×C4⋊C4—1181no—
312C22×C4⋊C4C2×D5911yescharacteristic
313C22×C22⋊C4—1181no—
314C22×C22⋊C4—1181no—
315C22×C42—1181no—
316C2×#103—24no—
317C2×#103—594no—
318C2×#103—1184no—
319C2×#103C2×D5914yescomplemented
320C2×C22⋊C8—594no—
321C2×C22⋊C8—1184no—
322#340—594no—
323#340—594no—
324C59C2×#187311yescharacteristic, complemented, Sylow
325C25—591no—
326C22×Q8—22no—
327C22×Q8—21no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8—22no—
331C22×Q8—22no—
332C22×Q8C4×D5914yes—
333C22×Q8C22×D5912yes—
334C23×C4—21no—
335C23×C4—21no—
336C23×C4—591no—
337C23×C4—594no—
338C23×C4—592no—
339C23×C4—594no—
340C23×C4—592no—
341C23×C4—1182no—
342C23×C4—1182no—
343C23×C4—1182no—
344C23×C4—1182no—
345C23×C4—1181no—
346C23×C4—1181no—
347C23×C4C22×D5911yescharacteristic
348C22×C8—22no—
349C22×C8—22no—
350C22×C8—22no—
351C22×C8—592no—
352C22×C8—1182no—
353C22×C8—1182no—
354C22×C8—1182no—
355C22×C8C22×D5912yes—
356C22⋊Q8—5916no—
357C22⋊Q8—11816no—
358C22⋊Q8—1188no—
359#117—1188no—
360#117—11816no—
361#117—1188no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—594no—
365C2×C4⋊C4—594no—
366C2×C4⋊C4—592no—
367C2×C4⋊C4—592no—
368C2×C4⋊C4—1188no—
369C2×C4⋊C4—1184no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1182no—
373C2×C4⋊C4—1182no—
374C2×C4⋊C4C22⋊D5914yes—
375C2×C4⋊C4C22×D5912yes—
376C2×C22⋊C4—1184no—
377C2×C22⋊C4—1188no—
378C2×C22⋊C4—1184no—
379C2×C22⋊C4—1188no—
380C2×C42—1184no—
381C2×C42—1188no—
382#103—28no—
383#103—1188no—
384C22⋊C8—1188no—
385C24—594no—
386C24—592no—
387C24—1182no—
388C24—1182no—
389C24—1182no—
390C24—1182no—
391C24—1182no—
392C24—1182no—
393C24D4×D5911yescharacteristic
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—28no—
397C2×Q8—28no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8C2×C4×D5914yes—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—594no—
416C22×C4—592no—
417C22×C4—598no—
418C22×C4—594no—
419C22×C4—594no—
420C22×C4—1182no—
421C22×C4—1182no—
422C22×C4—1188no—
423C22×C4—1188no—
424C22×C4—1184no—
425C22×C4—1188no—
426C22×C4—1188no—
427C22×C4—1184no—
428C22×C4—1182no—
429C22×C4—1182no—
430C22×C4—1182no—
431C22×C4—1182no—
432C22×C4—1184no—
433C22×C4—1184no—
434C22×C4—1182no—
435C22×C4—1182no—
436C22×C4—2362no—
437C22×C4—2362no—
438C22×C4—2362no—
439C22×C4—2362no—
440C22×C4C2×C4×D5912yes—
441C22×C4#1174568512yes—
442C22×C4D4×D5911yescharacteristic
443C22×C4D4×D5912yes—
444C22×C4D4×D5912yes—
445C22×C4#1174568712yes—
446C22×C4C2×C22⋊D5912yes—
447C22×C4C23×D5911yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—44no—
452C2×C8—1182no—
453C2×C8—1182no—
454C2×C8—1182no—
455C2×C8—2364no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—594no—
459C4⋊C4—594no—
460C4⋊C4—11816no—
461C4⋊C4—1184no—
462C4⋊C4—1184no—
463C4⋊C4—1182no—
464C4⋊C4—1182no—
465C4⋊C4C2×C22⋊D5914yes—
466C22⋊C4—11816no—
467C22⋊C4—11816no—
468C42—11816no—
469C23—594no—
470C23—1182no—
471C23—1188no—
472C23—1188no—
473C23—1182no—
474C23—1182no—
475C23—1182no—
476C23—1182no—
477C23—1188no—
478C23—1182no—
479C23—2364no—
480C23—2364no—
481C23—2364no—
482C23—2364no—
483C23C22⋊C4×D5912yes—
484C23#1427369512yes—
485C23#1427370912yes—
486C23#1427374812yes—
487C23#1427375012yes—
488C23#1427375612yes—
489C23#1427375912yes—
490C23C2×D4×D5911yescharacteristic
491Q8—24no—
492Q8—24no—
493Q8—42no—
494Q8—42no—
495Q8—48no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—594no—
505C2×C4—1182no—
506C2×C4—1188no—
507C2×C4—1182no—
508C2×C4—1184no—
509C2×C4—1184no—
510C2×C4—11816no—
511C2×C4—1184no—
512C2×C4—1184no—
513C2×C4—1184no—
514C2×C4—1184no—
515C2×C4—2368no—
516C2×C4—2362no—
517C2×C4—2362no—
518C2×C4—2362no—
519C2×C4—2362no—
520C2×C4—2368no—
521C2×C4C22⋊C4×D5912yes—
522C2×C4(C22⋊D59)⋊C414yes—
523C2×C4C4×C22⋊D5914yes—
524C2×C4#1427369512yes—
525C2×C4#1427369814yes—
526C2×C4#1427370014yes—
527C2×C4#1427370912yes—
528C2×C4C22×C4×D5911yescharacteristic
529C2×C4C2×D4×D5911yescharacteristic
530C2×C4C2×D4×D5911yescharacteristic
531C2×C4C22×C22⋊D5911yescharacteristic
532C2×C4C2×#1174568711yescharacteristic
533C2×C4C2×#1174568511yescharacteristic
534C8—42no—
535C8—2362no—
536C22—1184no—
537C22—1184no—
538C22—1184no—
539C22—23616no—
540C22—2362no—
541C22—2362no—
542C22—2362no—
543C22—2362no—
544C22#1683727814yes—
545C22#1683739714yes—
546C22#1683740914yes—
547C22#1683756114yes—
548C22#1683756314yes—
549C22#1683774114yes—
550C22#1683774214yes—
551C22C2×C22⋊C4×D5911yescharacteristic
552C22C2×#1427369511yescharacteristic
553C22C2×#1427370911yescharacteristic
554C22C2×#1427374811yescharacteristic
555C22C2×#1427375011yescharacteristic
556C22C2×#1427375611yescharacteristic
557C22C2×#1427375911yescharacteristic
558C4—44no—
559C4—44no—
560C4—1184no—
561C4—2364no—
562C4—2364no—
563C4#1683727814yes—
564C4C2×(C22⋊D59)⋊C411yescharacteristic
565C4C2×C4×C22⋊D5911yescharacteristic
566C4C2×#1427369811yescharacteristic
567C4C2×#1427370011yescharacteristic
568C2—2364no—
569C2#2142440518yesdirect product factor
570C2C2×#1683727811yescharacteristic
571C2C2×#1683739711yescharacteristic
572C2C2×#1683740911yescharacteristic
573C2C2×#1683756111yescharacteristic
574C2C2×#1683756311yescharacteristic
575C2C2×#1683774111yescharacteristic
576C2C2×#1683774211yescharacteristic
subgroupG/H
centerC24D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC22×#103×C59C2
socleC24×C59C2×D4
prime radicalC24×D59D4
cosocleC22×C4×C59C24
hypercenterC22×#103D59
hypocenterC59C2×#1873
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C22×#14273651C2semidirect
5. #21424405C2direct (N × H)
6. C2×#16837165C2semidirect
7. C2×#16837164C2semidirect
8. C22×#103×C59C2semidirect
13. C2×#14273703C4semidirect
26. C2×#14273651C22semidirect
29. #16837165C22semidirect
30. #16837164C22semidirect
35. C2×#103×C59C22semidirect
54. #14273703C2×C4semidirect
283. C22×#103D59semidirect
288. C2×C59#1873semidirect
319. C2×#103C2×D59semidirect
324. C59C2×#1873semidirect
569. C2#21424405direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142737032
C22×#142736852
C22×#142736512
C22×(C2×D59)⋊C82
#214244052
C2×#168371652
C2×#168371642
C22×#103×C592
C2×#187359