Finite Group Databasebeta

← C2×#21423144 #28560939

Subgroups — C2×#21423144 #28560939

Order 15104.

1200 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837735C211yesmaximal, characteristic, complemented
2C2×#16837718C211yesmaximal, characteristic, complemented
3C2×#16837338C211yesmaximal, characteristic, complemented
4#21423144C218yesmaximal, direct product factor
5C2×#16837139C211yesmaximal, characteristic, complemented
6C2×#16837136C211yesmaximal, characteristic, complemented
7C2×#16837126C211yesmaximal, characteristic, complemented
8C2×C59×#538C211yesmaximal, characteristic
9C22×C4⋊Dic59C2211yescharacteristic
10C22×Dic59⋊C4—21no—
11C22×C22⋊Dic59—21no—
12#16837735—22no—
13#16837735C2214yescomplemented
14#16837718—22no—
15#16837718C2214yescomplemented
16C2×#14273660—21no—
17C2×#14273658—21no—
18C2×#14273651—21no—
19C2×#14273648—21no—
20C2×C22⋊(C59⋊C8)C2211yescharacteristic
21C2×C22⋊(C59⋊C8)C2211yescharacteristic
22#16837338—22no—
23#16837338C2214yescomplemented
24C2×C59⋊#95C2211yescharacteristic
25#16837139C2214yescomplemented
26#16837136C2214yescomplemented
27#16837126C2214yescomplemented
28C59×#538—22no—
29C59×#538—22no—
30C59×#538—22no—
31C59×#538C2214yes—
32C2×#126×C59—21no—
33C2×#124×C59—21no—
34C2×#123×C59—21no—
35C2×C22⋊Q8×C59C2211yescharacteristic
36C2×#121×C59—21no—
37C2×#121×C59C2211yescharacteristic
38C2×#120×C59—21no—
39C2×C4×D4×C59—21no—
40C22×C22⋊C4×C59C2211yescharacteristic
41C23×Dic59—21no—
42C23×Dic59—21no—
43C22×C59⋊C8—21no—
44C22×C59⋊C8—21no—
45C2×C4⋊Dic59—21no—
46C2×C4⋊Dic59—21no—
47C2×C4⋊Dic59—21no—
48C2×C4⋊Dic59—21no—
49C2×C4⋊Dic59—21no—
50C2×C4⋊Dic59C2312yes—
51C2×Dic59⋊C4—24no—
52C2×Dic59⋊C4—44no—
53C2×C22⋊Dic59—24no—
54C2×C22⋊Dic59—44no—
55#14273660—24no—
56#14273658—24no—
57#14273651—24no—
58#14273648—24no—
59C22⋊(C59⋊C8)—21no—
60C22⋊(C59⋊C8)—21no—
61C22⋊(C59⋊C8)C2312yes—
62C22⋊(C59⋊C8)C2312yes—
63C59⋊#95—21no—
64C59⋊#95C2312yes—
65C25×C59D411yescharacteristic
66C22×Q8×C59D411yescharacteristic
67C22×D4×C59—21no—
68C22×D4×C59—21no—
69C22×D4×C59—21no—
70C22×D4×C59D411yescharacteristic
71C23×C4×C59—21no—
72C23×C4×C59D411yescharacteristic
73C23×C4×C59C2311yescharacteristic
74#126×C59—24no—
75#126×C59—24no—
76#124×C59—24no—
77#124×C59—24no—
78#123×C59—24no—
79#123×C59—24no—
80C22⋊Q8×C59—21no—
81C22⋊Q8×C59—21no—
82C22⋊Q8×C59D412yes—
83C22⋊Q8×C59C2312yes—
84#121×C59—24no—
85#121×C59—24no—
86#121×C59—21no—
87#121×C59—21no—
88#121×C59D412yes—
89#121×C59C2312yes—
90#120×C59—24no—
91#120×C59—24no—
92C4×D4×C59—24no—
93C4×D4×C59—24no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59D411yescharacteristic
97C2×C4⋊C4×C59D411yescharacteristic
98C2×C22⋊C4×C59—21no—
99C2×C22⋊C4×C59—21no—
100C2×C22⋊C4×C59—21no—
101C2×C22⋊C4×C59—21no—
102C2×C22⋊C4×C59—21no—
103C2×C22⋊C4×C59—21no—
104C2×C22⋊C4×C59—21no—
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59—21no—
107C2×C22⋊C4×C59D412yes—
108C2×C22⋊C4×C59C2312yes—
109C2×C42×C59—21no—
110C24×C59—21no—
111C24×C59—21no—
112C24×C59—22no—
113C24×C59—21no—
114C24×C59—22no—
115C24×C59—42no—
116C24×C59—42no—
117C24×C59C2×D411yescharacteristic
118C24×C59C2×D412yes—
119C2×Q8×C59—21no—
120C2×Q8×C59—42no—
121C2×Q8×C59C2×D412yes—
122C2×D4×C59—22no—
123C2×D4×C59—22no—
124C2×D4×C59—24no—
125C2×D4×C59—24no—
126C2×D4×C59—21no—
127C2×D4×C59—42no—
128C2×D4×C59—42no—
129C2×D4×C59—42no—
130C2×D4×C59—82no—
131C2×D4×C59—82no—
132C2×D4×C59C2×D412yes—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—24no—
138C22×C4×C59—21no—
139C22×C4×C59—42no—
140C22×C4×C59—42no—
141C22×C4×C59—82no—
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D412yes—
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2411yescharacteristic
149C22×Dic59—24no—
150C22×Dic59—22no—
151C22×Dic59—22no—
152C22×Dic59—42no—
153C22×Dic59—42no—
154C22×Dic59—42no—
155C22×Dic59—42no—
156C22×Dic59—41no—
157C22×Dic59—41no—
158C4⋊C4×C59—24no—
159C4⋊C4×C59—22no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59C2×D412yes—
164C4⋊C4×C59C2×D412yes—
165C22⋊C4×C59—24no—
166C22⋊C4×C59—24no—
167C22⋊C4×C59—24no—
168C22⋊C4×C59—22no—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—22no—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—22no—
176C42×C59—24no—
177C22⋊Dic59—48no—
178C4⋊Dic59—22no—
179C4⋊Dic59—22no—
180C4⋊Dic59—41no—
181C4⋊Dic59—41no—
182Dic59⋊C4—48no—
183C2×C59⋊C8—22no—
184C2×C59⋊C8—22no—
185C2×C59⋊C8—42no—
186C2×C59⋊C8—42no—
187C23×C59—22no—
188C23×C59—22no—
189C23×C59—22no—
190C23×C59—22no—
191C23×C59—22no—
192C23×C59—22no—
193C23×C59—42no—
194C23×C59—42no—
195C23×C59—48no—
196C23×C59—42no—
197C23×C59—42no—
198C23×C59—42no—
199C23×C59—42no—
200C23×C59—42no—
201C23×C59—42no—
202C23×C59—42no—
203C23×C59—81no—
204C23×C59—81no—
205C23×C59—84no—
206C23×C59#12011yescharacteristic
207C23×C59#12011yescharacteristic
208C23×C59#13512yes—
209C23×C59#13712yes—
210C23×C59C22×D411yescharacteristic
211Q8×C59—42no—
212Q8×C59—81no—
213D4×C59—42no—
214D4×C59—42no—
215D4×C59—42no—
216D4×C59—44no—
217D4×C59—81no—
218D4×C59—84no—
219D4×C59—84no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—81no—
236C2×C4×C59—84no—
237C2×C4×C59#12011yescharacteristic
238C2×C4×C59#12011yescharacteristic
239C2×C4×C59C22×D411yescharacteristic
240C2×C4×C59C22×D411yescharacteristic
241C2×Dic59—24no—
242C2×Dic59—48no—
243C2×Dic59—42no—
244C2×Dic59—42no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248C2×Dic59—82no—
249C2×Dic59—82no—
250C59⋊C8—42no—
251C59⋊C8—42no—
252C2×#1618—591nomaximal, Sylow
253C22×C59—24no—
254C22×C59—24no—
255C22×C59—21no—
256C22×C59—21no—
257C22×C59—21no—
258C22×C59—21no—
259C22×C59—44no—
260C22×C59—42no—
261C22×C59—42no—
262C22×C59—42no—
263C22×C59—42no—
264C22×C59—42no—
265C22×C59—42no—
266C22×C59—82no—
267C22×C59—84no—
268C22×C59—84no—
269C22×C59—84no—
270C22×C59#45212yes—
271C22×C59#45212yes—
272C22×C59#45312yes—
273C22×C59#45812yes—
274C22×C59C2×#12011yescharacteristic
275C22×C59C2×#13511yescharacteristic
276C22×C59C2×#13711yescharacteristic
277C4×C59—21no—
278C4×C59—21no—
279C4×C59—21no—
280C4×C59—21no—
281C4×C59—42no—
282C4×C59—42no—
283C4×C59—42no—
284C4×C59—42no—
285Dic59—44no—
286Dic59—82no—
287C2×#538D5911yescharacteristic
288C2×#484—591no—
289C2×#481—591no—
290C2×#399—591no—
291#1618—598no—
292C2×#328—591no—
293C2×#327—591no—
294C2×#323—591no—
295C2×C59—21no—
296C2×C59—21no—
297C2×C59—22no—
298C2×C59—82no—
299C2×C59—82no—
300C2×C59#161814yescomplemented
301C2×C59C2×#45211yescharacteristic
302C2×C59C2×#45311yescharacteristic
303C2×C59C2×#45811yescharacteristic
304#538—22no—
305#538—22no—
306#538—22no—
307#538C2×D5914yes—
308C2×#126—21no—
309C2×#124—21no—
310C2×#123—21no—
311C2×C22⋊Q8C2×D5911yescharacteristic
312C2×#121—21no—
313C2×#121C2×D5911yescharacteristic
314C2×#120—21no—
315C2×C4×D4—21no—
316C22×C4⋊C4—591no—
317C22×C4⋊C4—1181no—
318C22×C22⋊C4—1181no—
319C22×C22⋊C4C2×D5911yescharacteristic
320#484—594no—
321#484—1182no—
322#481—594no—
323#481—1182no—
324C2×#107—1181no—
325C2×#106—1181no—
326C2×#103—1181no—
327C2×#102—1181no—
328C2×C22⋊C8—591no—
329C2×C22⋊C8—591no—
330#399—594no—
331#399—1182no—
332C2×#95—591no—
333#328—594no—
334#327—594no—
335#323—594no—
336C59C2×#161811yescharacteristic, complemented, Sylow
337C25C22⋊D5911yescharacteristic
338C22×Q8C22⋊D5911yescharacteristic
339C22×D4—21no—
340C22×D4—21no—
341C22×D4—21no—
342C22×D4C22⋊D5911yescharacteristic
343C23×C4—21no—
344C23×C4—1181no—
345C23×C4—1181no—
346C23×C4C22⋊D5911yescharacteristic
347C23×C4C22×D5911yescharacteristic
348C22×C8—1181no—
349C22×C8—1181no—
350#126—24no—
351#126—24no—
352#124—24no—
353#124—24no—
354#123—24no—
355#123—24no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358C22⋊Q8C22⋊D5912yes—
359C22⋊Q8C22×D5912yes—
360#121—24no—
361#121—24no—
362#121—21no—
363#121—21no—
364#121C22⋊D5912yes—
365#121C22×D5912yes—
366#120—24no—
367#120—24no—
368C4×D4—24no—
369C4×D4—24no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—592no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1184no—
379C2×C4⋊C4—2364no—
380C2×C4⋊C4C22⋊D5911yescharacteristic
381C2×C4⋊C4C22⋊D5911yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—1184no—
392C2×C22⋊C4—2364no—
393C2×C22⋊C4C22⋊D5912yes—
394C2×C22⋊C4C22×D5912yes—
395C2×C42—21no—
396#107—1184no—
397#106—1184no—
398#103—1184no—
399#102—1184no—
400C22⋊C8—592no—
401C22⋊C8—592no—
402C22⋊C8—1181no—
403C22⋊C8—1181no—
404#95—592no—
405#95—1181no—
406C24—21no—
407C24—21no—
408C24—22no—
409C24—21no—
410C24—22no—
411C24—42no—
412C24—42no—
413C24C2×C22⋊D5911yescharacteristic
414C24C2×C22⋊D5912yes—
415C2×Q8—21no—
416C2×Q8—42no—
417C2×Q8C2×C22⋊D5912yes—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4C2×C22⋊D5912yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—82no—
438C22×C4—1184no—
439C22×C4—1182no—
440C22×C4—1182no—
441C22×C4—2362no—
442C22×C4—2362no—
443C22×C4—2361no—
444C22×C4—2361no—
445C22×C4—2362no—
446C22×C4—2362no—
447C22×C4C2×C22⋊D5911yescharacteristic
448C22×C4C2×C22⋊D5911yescharacteristic
449C22×C4C2×C22⋊D5912yes—
450C22×C4C2×C22⋊D5911yescharacteristic
451C22×C4C2×C22⋊D5911yescharacteristic
452C22×C4C2×C22⋊D5911yescharacteristic
453C22×C4C23×D5911yescharacteristic
454C2×C8—1182no—
455C2×C8—1182no—
456C2×C8—2362no—
457C2×C8—2362no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—1182no—
464C4⋊C4—1182no—
465C4⋊C4—2368no—
466C4⋊C4—2361no—
467C4⋊C4—2361no—
468C4⋊C4C2×C22⋊D5912yes—
469C4⋊C4C2×C22⋊D5912yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—2368no—
482C42—24no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—48no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—81no—
500C23—81no—
501C23—84no—
502C23#1427369611yescharacteristic
503C23#1427369611yescharacteristic
504C23#1427374712yes—
505C23#1427375712yes—
506C23C22×C22⋊D5911yescharacteristic
507Q8—42no—
508Q8—81no—
509D4—42no—
510D4—42no—
511D4—42no—
512D4—44no—
513D4—81no—
514D4—84no—
515D4—84no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—81no—
532C2×C4—84no—
533C2×C4—1184no—
534C2×C4—4722no—
535C2×C4—4722no—
536C2×C4—2362no—
537C2×C4—2362no—
538C2×C4—2362no—
539C2×C4—2368no—
540C2×C4—2362no—
541C2×C4—2362no—
542C2×C4#1427369611yescharacteristic
543C2×C4#1427369611yescharacteristic
544C2×C4C22×C22⋊D5911yescharacteristic
545C2×C4C22×C22⋊D5911yescharacteristic
546C8—2362no—
547C8—2362no—
548C22—24no—
549C22—24no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—82no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22#1683756412yes—
566C22#1683756412yes—
567C22#1683757012yes—
568C22#1683759212yes—
569C22C2×#1427369611yescharacteristic
570C22C2×#1427374711yescharacteristic
571C22C2×#1427375711yescharacteristic
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—4722no—
581C4—2364no—
582C2—21no—
583C2—21no—
584C2—22no—
585C2—82no—
586C2—82no—
587C2#2142314414yesdirect product factor
588C2C2×#1683756411yescharacteristic
589C2C2×#1683757011yescharacteristic
590C2C2×#1683759211yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
Fitting, prime radicalC2×C59×#538C2
socleC23×C59#120
hypercenterC2×#538D59
hypocenterC59C2×#1618
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#538

Complements

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

NH ≅ G/Ntype
1. C2×#16837735C2semidirect
2. C2×#16837718C2semidirect
3. C2×#16837338C2semidirect
4. #21423144C2direct (N × H)
5. C2×#16837139C2semidirect
6. C2×#16837136C2semidirect
7. C2×#16837126C2semidirect
13. #16837735C22semidirect
15. #16837718C22semidirect
23. #16837338C22semidirect
25. #16837139C22semidirect
26. #16837136C22semidirect
27. #16837126C22semidirect
300. C2×C59#1618semidirect
336. C59C2×#1618semidirect
587. C2#21423144direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168377352
C2×#168377182
C2×#168373382
#214231442
C2×#168371392
C2×#168371362
C2×#168371262
C2×C59×#5382
C2×#161859