Finite Group Databasebeta

← C2×#21422620 #28555815

Subgroups — C2×#21422620 #28555815

Order 15104.

1206 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837938C211yesmaximal, characteristic
2C2×#16837456C211yesmaximal, characteristic, complemented
3C2×#16837253C211yesmaximal, characteristic, complemented
4#21422620C218yesmaximal, direct product factor
5C2×#16837139C211yesmaximal, characteristic, complemented
6C2×#16837137C211yesmaximal, characteristic
7C2×#16837135C211yesmaximal, characteristic
8C2×C59×#328C211yesmaximal, characteristic, complemented
9#16837938—22no—
10#16837938—22no—
11#16837938C414yescomplemented
12#16837938C2214yes—
13C2×#14273722—21no—
14C2×#14273718C411yescharacteristic, complemented
15C2×#14273716C411yescharacteristic, complemented
16C2×#14273711—21no—
17C2×#14273701C2211yescharacteristic
18C2×Q8×Dic59C411yescharacteristic, complemented
19C2×#14273689C411yescharacteristic, complemented
20C2×(C22⋊D59)⋊C4—21no—
21C2×#14273685—21no—
22C22×Dic59⋊C4—21no—
23C22×C4×Dic59—21no—
24C22×C4×Dic59C2211yescharacteristic
25#16837456—22no—
26#16837456C2214yescomplemented
27C2×Dic59⋊C8—21no—
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273637—21no—
30#16837253—22no—
31#16837253C2214yescomplemented
32C2×C59⋊#95C2211yescharacteristic
33#16837139C2214yescomplemented
34#16837137C2214yes—
35#16837135C2214yes—
36C2×C22⋊Q8×C59C2211yescharacteristic
37C2×C22⋊C8×C59C2211yescharacteristic
38C2×#95×C59C2211yescharacteristic
39C59×#328C2214yescomplemented
40C22×D236C2×C411yescharacteristic
41C22×C22⋊D59—21no—
42C23×Dic59—21no—
43C23×Dic59D411yescharacteristic
44C23×Dic59D411yescharacteristic
45C22×C4×D59—21no—
46C22×C59⋊C8—21no—
47#14273722—24no—
48#14273722—24no—
49#14273718—22no—
50#14273718C2×C414yescomplemented
51#14273716—22no—
52#14273716C2×C414yescomplemented
53#14273711—24no—
54#14273711—24no—
55#14273701—21no—
56#14273701—21no—
57#14273701C2×C412yes—
58#14273701C2312yes—
59Q8×Dic59—22no—
60Q8×Dic59C2×C414yescomplemented
61#14273689—22no—
62#14273689C2×C414yescomplemented
63(C22⋊D59)⋊C4—24no—
64(C22⋊D59)⋊C4—24no—
65#14273685—24no—
66#14273685—24no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59C2×C411yescharacteristic
70C2×Dic59⋊C4—28no—
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—24no—
73C2×(C2×D59)⋊C4—21no—
74C2×(C2×D59)⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59—24no—
79C2×C4×Dic59—44no—
80C2×C4×Dic59C2×C411yescharacteristic
81C2×C4×Dic59C2×C412yes—
82C2×C4×Dic59C2×C411yescharacteristic
83C2×C4×Dic59D412yes—
84C2×C4×Dic59D411yescharacteristic
85C2×C4×Dic59D411yescharacteristic
86C2×C4×Dic59D412yes—
87C2×C4×Dic59C2312yes—
88Dic59⋊C8—24no—
89C22⋊(C59⋊C8)—21no—
90C22⋊(C59⋊C8)C2312yes—
91#14273637—24no—
92C59⋊#95—21no—
93C59⋊#95C2312yes—
94C22×Q8×C59C2×C411yescharacteristic
95C23×C4×C59—21no—
96C23×C4×C59C2311yescharacteristic
97C22×C8×C59—21no—
98C22⋊Q8×C59—21no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59C2×C412yes—
101C22⋊Q8×C59C2312yes—
102C2×C4⋊C4×C59—21no—
103C2×C4⋊C4×C59C2×C411yescharacteristic
104C2×C22⋊C4×C59—21no—
105C22⋊C8×C59—21no—
106C22⋊C8×C59C2312yes—
107#95×C59—21no—
108#95×C59C2312yes—
109C24×C59C2×D411yescharacteristic
110C23×D59—21no—
111C2×Q8×C59—21no—
112C2×Q8×C59—42no—
113C2×Q8×C59C22×C412yes—
114C22×C4×C59—24no—
115C22×C4×C59—21no—
116C22×C4×C59—21no—
117C22×C4×C59—22no—
118C22×C4×C59—42no—
119C22×C4×C59—42no—
120C22×C4×C59—42no—
121C22×C4×C59C22×C411yescharacteristic
122C22×C4×C59C22×C411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2411yescharacteristic
127C2×C22⋊D59—22no—
128C2×C22⋊D59—22no—
129C2×C22⋊D59—42no—
130C2×C22⋊D59—42no—
131C22×Dic59—22no—
132C22×Dic59—22no—
133C22×Dic59—24no—
134C22×Dic59—24no—
135C22×Dic59—21no—
136C22×Dic59—22no—
137C22×Dic59—42no—
138C22×Dic59—42no—
139C22×Dic59C22⋊C411yescharacteristic
140C22×Dic59C22⋊C411yescharacteristic
141C22×Dic59C22⋊C411yescharacteristic
142C22×Dic59C22⋊C412yes—
143C22×Dic59C22⋊C412yes—
144C22×Dic59C22⋊C411yescharacteristic
145C22×Dic59C2×D412yes—
146C22×Dic59C2×D412yes—
147C2×D236—21no—
148C2×D236—42no—
149C2×D236C22×C412yes—
150C2×C4×D59—24no—
151C2×C4×D59—44no—
152C2×C8×C59—22no—
153C2×C8×C59—42no—
154C4⋊C4×C59—22no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59C22×C412yes—
158C22⋊C4×C59—22no—
159C22⋊C4×C59—22no—
160(C2×D59)⋊C4—22no—
161(C2×D59)⋊C4—22no—
162(C2×D59)⋊C4—24no—
163(C2×D59)⋊C4—24no—
164(C2×D59)⋊C4—24no—
165C4⋊Dic59—21no—
166C4⋊Dic59—24no—
167C4⋊Dic59—24no—
168C4⋊Dic59C22×C412yes—
169Dic59⋊C4—216no—
170Dic59⋊C4—24no—
171C4×Dic59—24no—
172C4×Dic59—48no—
173C4×Dic59C22⋊C412yes—
174C4×Dic59C22⋊C412yes—
175C4×Dic59C22⋊C412yes—
176C4×Dic59C22⋊C412yes—
177C4×Dic59C22×C412yes—
178C4×Dic59C22×C412yes—
179C4×Dic59C2×D412yes—
180C4×Dic59C2×D412yes—
181C2×C59⋊C8—22no—
182C2×C59⋊C8—42no—
183C23×C59—22no—
184C23×C59—22no—
185C23×C59C2×C22⋊C411yescharacteristic
186C23×C59C2×C22⋊C411yescharacteristic
187C23×C59AGL(1,ℤ/8)12yes—
188C23×C59#13712yes—
189C23×C59C22×D411yescharacteristic
190C22×D59—22no—
191C22×D59—42no—
192C22×D59—42no—
193C22×D59—42no—
194Q8×C59—42no—
195Q8×C59—42no—
196C2×C4×C59—22no—
197C2×C4×C59—22no—
198C2×C4×C59—24no—
199C2×C4×C59—22no—
200C2×C4×C59—22no—
201C2×C4×C59—41no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—41no—
205C2×C4×C59—42no—
206C2×C4×C59—48no—
207C2×C4×C59—42no—
208C2×C4×C59—42no—
209C2×C4×C59—42no—
210C2×C4×C59C2×C22⋊C411yescharacteristic
211C2×C4×C59C2×C22⋊C411yescharacteristic
212C2×C4×C59C23×C411yescharacteristic
213C2×C4×C59C22×D411yescharacteristic
214C22⋊D59—42no—
215C22⋊D59—42no—
216C22⋊D59—44no—
217C2×Dic59—22no—
218C2×Dic59—22no—
219C2×Dic59—24no—
220C2×Dic59—22no—
221C2×Dic59—22no—
222C2×Dic59—22no—
223C2×Dic59—24no—
224C2×Dic59—24no—
225C2×Dic59—24no—
226C2×Dic59—22no—
227C2×Dic59—42no—
228C2×Dic59—48no—
229C2×Dic59—42no—
230C2×Dic59—42no—
231C2×Dic59#10414yes—
232C2×Dic59C2×C22⋊C412yes—
233C2×Dic59C2×C22⋊C412yes—
234C2×Dic59C2×C22⋊C412yes—
235C2×Dic59C2×C22⋊C412yes—
236D236—42no—
237D236—42no—
238C4×D59—48no—
239C8×C59—42no—
240C59⋊C8—42no—
241C2×#1509—591nomaximal, Sylow
242C22×C59—21no—
243C22×C59—21no—
244C22×C59—21no—
245C22×C59—21no—
246C22×C59—24no—
247C22×C59—24no—
248C22×C59#41012yes—
249C22×C59#41712yes—
250C22×C59#41712yes—
251C22×C59C2×#10412yes—
252C22×C59C22×C22⋊C411yescharacteristic
253C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C59C2×#13711yescharacteristic
255C2×D59—42no—
256C2×D59—42no—
257C2×D59—42no—
258C2×D59—84no—
259C4×C59—21no—
260C4×C59—21no—
261C4×C59—21no—
262C4×C59—21no—
263C4×C59—44no—
264C4×C59—42no—
265C4×C59—42no—
266Dic59—22no—
267Dic59—22no—
268Dic59—22no—
269Dic59—44no—
270Dic59—42no—
271Dic59C2×#10414yes—
272C2×#529—591no—
273C2×#432—591no—
274C2×C4×C4⋊C4—591no—
275#1509—598no—
276C2×#328—591no—
277C2×#328D5911yescharacteristic, complemented
278C2×#327—591no—
279C2×#327—591no—
280C2×C59—21no—
281C2×C59—21no—
282C2×C59—22no—
283C2×C59#150914yescomplemented
284C2×C59C2×#41011yescharacteristic
285C2×C59C2×#41711yescharacteristic
286C2×C59C22×#10411yescharacteristic
287D59—82no—
288#529—594no—
289#529—594no—
290#529—1182no—
291#529—1182no—
292C2×#126—1181no—
293C2×#125—591no—
294C2×#124—591no—
295C2×#123—1181no—
296C2×C22⋊Q8C2×D5911yescharacteristic
297C2×#121—591no—
298C2×C4×Q8—591no—
299C2×C4×D4—591no—
300C2×C4×D4—1181no—
301C2×#117—1181no—
302C22×C4⋊C4—1181no—
303C22×C42—591no—
304C22×C42—1181no—
305#432—594no—
306#432—1182no—
307C2×C4⋊C8—1181no—
308C2×C22⋊C8—591no—
309C2×C22⋊C8C2×D5911yescharacteristic
310C2×#97—1181no—
311C4×C4⋊C4—594no—
312C4×C4⋊C4—1182no—
313C2×#95—591no—
314C2×#95C2×D5911yescharacteristic
315#328—594no—
316#328C2×D5914yescomplemented
317#327—594no—
318#327—594no—
319C59C2×#150911yescharacteristic, complemented, Sylow
320C22×Q8C4×D5911yescharacteristic
321C22×D4—591no—
322C22×D4—1181no—
323C23×C4—21no—
324C23×C4—591no—
325C23×C4—591no—
326C23×C4—1181no—
327C23×C4—1181no—
328C23×C4C22×D5911yescharacteristic
329C22×C8—21no—
330C22×C8—1181no—
331#126—1184no—
332#126—1184no—
333#125—594no—
334#125—1182no—
335#124—594no—
336#124—1182no—
337#123—1184no—
338#123—1184no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8C4×D5912yes—
342C22⋊Q8C22×D5912yes—
343#121—592no—
344#121—592no—
345#121—1181no—
346#121—1181no—
347C4×Q8—594no—
348C4×Q8—1182no—
349C4×D4—594no—
350C4×D4—1182no—
351C4×D4—1184no—
352C4×D4—1184no—
353#117—1184no—
354#117—1184no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—591no—
357C2×C4⋊C4—1184no—
358C2×C4⋊C4—1181no—
359C2×C4⋊C4—1181no—
360C2×C4⋊C4—1181no—
361C2×C4⋊C4—1188no—
362C2×C4⋊C4C4×D5911yescharacteristic
363C2×C22⋊C4—21no—
364C2×C22⋊C4—1181no—
365C2×C22⋊C4—1181no—
366C2×C22⋊C4—1181no—
367C2×C22⋊C4—1181no—
368C2×C42—591no—
369C2×C42—591no—
370C2×C42—592no—
371C2×C42—591no—
372C2×C42—591no—
373C2×C42—592no—
374C2×C42—592no—
375C2×C42—592no—
376C2×C42—1181no—
377C2×C42—1184no—
378C2×C42—2364no—
379C4⋊C8—1184no—
380C22⋊C8—21no—
381C22⋊C8—592no—
382C22⋊C8—1181no—
383C22⋊C8C22×D5912yes—
384#97—1184no—
385#95—21no—
386#95—592no—
387#95—1181no—
388#95C22×D5912yes—
389C24—1181no—
390C24D4×D5911yescharacteristic
391C2×Q8—21no—
392C2×Q8—42no—
393C2×Q8C2×C4×D5912yes—
394C2×D4—592no—
395C2×D4—1182no—
396C2×D4—1182no—
397C2×D4—1181no—
398C2×D4—2362no—
399C2×D4—2362no—
400C2×D4—2362no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—591no—
409C22×C4—592no—
410C22×C4—592no—
411C22×C4—591no—
412C22×C4—591no—
413C22×C4—591no—
414C22×C4—592no—
415C22×C4—592no—
416C22×C4—1184no—
417C22×C4—1184no—
418C22×C4—1182no—
419C22×C4—1182no—
420C22×C4—1184no—
421C22×C4—1181no—
422C22×C4—1182no—
423C22×C4—2364no—
424C22×C4—2362no—
425C22×C4—2362no—
426C22×C4C2×C4×D5911yescharacteristic
427C22×C4C2×C4×D5911yescharacteristic
428C22×C4D4×D5911yescharacteristic
429C22×C4D4×D5911yescharacteristic
430C22×C4D4×D5911yescharacteristic
431C22×C4C23×D5911yescharacteristic
432C2×C8—22no—
433C2×C8—42no—
434C2×C8—1182no—
435C2×C8—2362no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—592no—
440C4⋊C4—1184no—
441C4⋊C4—1184no—
442C4⋊C4—1184no—
443C4⋊C4—11816no—
444C4⋊C4—1181no—
445C4⋊C4C2×C4×D5912yes—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—1184no—
449C22⋊C4—1184no—
450C22⋊C4—1184no—
451C22⋊C4—1182no—
452C22⋊C4—1182no—
453C42—592no—
454C42—592no—
455C42—592no—
456C42—592no—
457C42—592no—
458C42—592no—
459C42—592no—
460C42—592no—
461C42—1184no—
462C42—2368no—
463C23—22no—
464C23—22no—
465C23—1182no—
466C23—2362no—
467C23—2362no—
468C23—2362no—
469C23C22⋊C4×D5911yescharacteristic
470C23C22⋊C4×D5911yescharacteristic
471C23#1427375412yes—
472C23#1427375912yes—
473C23C2×D4×D5911yescharacteristic
474Q8—42no—
475Q8—42no—
476D4—2362no—
477D4—2362no—
478D4—2364no—
479D4—2362no—
480D4—2362no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—594no—
496C2×C4—592no—
497C2×C4—592no—
498C2×C4—592no—
499C2×C4—592no—
500C2×C4—1184no—
501C2×C4—1184no—
502C2×C4—1184no—
503C2×C4—1182no—
504C2×C4—1182no—
505C2×C4—1182no—
506C2×C4—1182no—
507C2×C4—1182no—
508C2×C4—1184no—
509C2×C4—1182no—
510C2×C4—2368no—
511C2×C4—2368no—
512C2×C4—2362no—
513C2×C4—2362no—
514C2×C4—2362no—
515C2×C4C22⋊C4×D5911yescharacteristic
516C2×C4C22⋊C4×D5911yescharacteristic
517C2×C4C22×C4×D5911yescharacteristic
518C2×C4C2×D4×D5911yescharacteristic
519C8—42no—
520C8—2362no—
521C22—24no—
522C22—24no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—4724no—
528C22—2362no—
529C22—2362no—
530C22—2362no—
531C22#1683737412yes—
532C22#1683740212yes—
533C22#1683740212yes—
534C22#104×D5912yes—
535C22C2×C22⋊C4×D5911yescharacteristic
536C22C2×#1427375411yescharacteristic
537C22C2×#1427375911yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—44no—
543C4—42no—
544C4—42no—
545C4—594no—
546C4—1182no—
547C4—1182no—
548C4—1182no—
549C4—2364no—
550C4—2362no—
551C2—21no—
552C2—22no—
553C2—21no—
554C2—4722no—
555C2#2142262014yesdirect product factor
556C2C2×#1683737411yescharacteristic
557C2C2×#1683740211yescharacteristic
558C2C2×#104×D5911yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#328C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenterC2×#328D59
hypocenterC59C2×#1509
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. C2×#16837456C2semidirect
3. C2×#16837253C2semidirect
4. #21422620C2direct (N × H)
5. C2×#16837139C2semidirect
8. C2×C59×#328C2semidirect
11. #16837938C4semidirect
14. C2×#14273718C4semidirect
15. C2×#14273716C4semidirect
18. C2×Q8×Dic59C4semidirect
19. C2×#14273689C4semidirect
26. #16837456C22semidirect
31. #16837253C22semidirect
33. #16837139C22semidirect
39. C59×#328C22semidirect
50. #14273718C2×C4semidirect
52. #14273716C2×C4semidirect
60. Q8×Dic59C2×C4semidirect
62. #14273689C2×C4semidirect
277. C2×#328D59semidirect
283. C2×C59#1509semidirect
316. #328C2×D59semidirect
319. C59C2×#1509semidirect
555. C2#21422620direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168379382
C2×#168374562
C2×#168372532
#214226202
C2×#168371392
C2×#168371372
C2×#168371352
C2×C59×#3282
C2×#150959