Finite Group Databasebeta

← C2×#21434046 #29456497

Subgroups — C2×#21434046 #29456497

Order 15104.

2326 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273707C211yesmaximal, characteristic
2C2×#117×D59C211yesmaximal, characteristic
3#21434046C2116yesmaximal, direct product factor
4C2×#16837739C212yesmaximal, complemented
5C2×#16837746C212yesmaximal, complemented
6C2×#16837745C212yesmaximal, complemented
7C2×#16837734C212yesmaximal, complemented
8C2×#16837442C211yesmaximal, characteristic, complemented
9C2×#16837412C211yesmaximal, characteristic, complemented
10C2×#16837410C211yesmaximal, characteristic, complemented
11C2×#16837365C211yesmaximal, characteristic
12C2×C59×#428C211yesmaximal, characteristic, complemented
13C23×Dic118C2211yescharacteristic
14C23×C4×D59C2211yescharacteristic
15C22×#11745664C2211yescharacteristic
16C2×#14273707—21no—
17C2×#14273707—24no—
18C2×#14273707—21no—
19C2×#14273707C2212yes—
20C2×#14273707C2212yes—
21C2×#14273707C2218yes—
22C2×#14273686C2211yescharacteristic
23C2×#14273685C2211yescharacteristic
24C2×#14273685C2211yescharacteristic
25C2×#14273684—21no—
26C2×#14273683—21no—
27#117×D59—24no—
28#117×D59C2218yes—
29C22×C4⋊Dic59—21no—
30C2×C4⋊C4×D59C2211yescharacteristic
31C2×C4⋊C4×D59C2211yescharacteristic
32C22×(C2×D59)⋊C4—21no—
33C2×C22⋊C4×D59—21no—
34C2×C42×D59—21no—
35#16837746C22116yescomplemented
36#16837745C22116yescomplemented
37#16837739C22116yescomplemented
38#16837734C22116yescomplemented
39#16837442C2218yescomplemented
40C2×#14273660C2212yescomplemented
41C2×#14273658C2212yescomplemented
42#16837412C2218yescomplemented
43#16837410C2218yescomplemented
44C2×#14273652C2212yescomplemented
45C2×#14273652C2212yescomplemented
46C2×#14273650C2212yescomplemented
47C2×#14273650C2212yescomplemented
48#16837365C2218yes—
49C2×(C2×D59)⋊C8C2212yes—
50C2×(C2×D59)⋊C8C2212yes—
51C22×M16×C59C2211yescharacteristic
52C2×#117×C59C2211yescharacteristic
53C22×C4⋊C4×C59C2211yescharacteristic
54C59×#428C2218yescomplemented
55C2×#107×C59C2212yescomplemented
56C2×#106×C59C2212yescomplemented
57C24×D59D411yescharacteristic
58C22×Dic118—24no—
59C22×Dic118—24no—
60C22×Dic118—21no—
61C22×Dic118—22no—
62C22×Dic118C2312yes—
63C22×Dic118C2314yes—
64C23×Dic59—21no—
65C23×Dic59D411yescharacteristic
66C22×C4×D59—24no—
67C22×C4×D59—24no—
68C22×C4×D59—21no—
69C22×C4×D59—21no—
70C22×C4×D59D411yescharacteristic
71C22×C4×D59D414yes—
72C22×C4×D59D411yescharacteristic
73C22×C4×D59C2311yescharacteristic
74C22×C4×D59C2314yes—
75C22×C4×D59C2311yescharacteristic
76C2×#11745664—24no—
77C2×#11745664C2314yes—
78C22×C59⋊C8C2312yes—
79#14273707—28no—
80#14273707—24no—
81#14273707—28no—
82#14273707—24no—
83#14273707C2318yes—
84#14273707C2318yes—
85#14273686—22no—
86#14273686C2314yes—
87#14273685—22no—
88#14273685—22no—
89#14273685C2314yes—
90#14273685C2314yes—
91#14273684—28no—
92#14273683—28no—
93C2×C4⋊Dic59—22no—
94C2×C4⋊Dic59—22no—
95C2×C4⋊Dic59—28no—
96C2×C4⋊Dic59C2311yescharacteristic
97C2×C4⋊Dic59C2311yescharacteristic
98C2×Dic59⋊C4—21no—
99C2×Dic59⋊C4—21no—
100C4⋊C4×D59—22no—
101C4⋊C4×D59—22no—
102C4⋊C4×D59C2314yes—
103C4⋊C4×D59C2314yes—
104C2×C22⋊Dic59—21no—
105C2×(C2×D59)⋊C4—22no—
106C2×(C2×D59)⋊C4—28no—
107C2×(C2×D59)⋊C4—22no—
108C2×(C2×D59)⋊C4—21no—
109C2×(C2×D59)⋊C4—21no—
110C22⋊C4×D59—28no—
111C2×C4×Dic59—21no—
112C2×C4×Dic59—21no—
113C2×C4×Dic59—21no—
114C42×D59—28no—
115#14273660C2318yescomplemented
116#14273658C2318yescomplemented
117#14273652C2318yescomplemented
118#14273652C2318yescomplemented
119#14273650C2318yescomplemented
120#14273650C2318yescomplemented
121(C2×D59)⋊C8C2318yes—
122(C2×D59)⋊C8C2318yes—
123C23×C4×C59—21no—
124C23×C4×C59C2311yescharacteristic
125C2×M16×C59—24no—
126C2×M16×C59C2314yes—
127C22×C8×C59C2312yes—
128#117×C59—22no—
129#117×C59C2314yes—
130C2×C4⋊C4×C59—21no—
131C2×C4⋊C4×C59—22no—
132C2×C4⋊C4×C59C2312yes—
133C2×C4⋊C4×C59C2314yes—
134C2×C4⋊C4×C59C2311yescharacteristic
135C2×C4⋊C4×C59C2311yescharacteristic
136C2×C22⋊C4×C59—21no—
137C2×C42×C59—21no—
138#107×C59C2318yescomplemented
139#106×C59C2318yescomplemented
140C24×C59C2×D411yescharacteristic
141C23×D59—24no—
142C23×D59—24no—
143C23×D59—42no—
144C23×D59C2×D411yescharacteristic
145C23×D59C2×D414yes—
146C23×D59C2×D411yescharacteristic
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—24no—
150C22×C4×C59—22no—
151C22×C4×C59—44no—
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D412yes—
154C22×C4×C59#4114yes—
155C22×C4×C59#4114yes—
156C22×C4×C59C2411yescharacteristic
157C22×C4×C59C2412yes—
158C22×Dic59—24no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—24no—
162C22×Dic59—22no—
163C22×Dic59—44no—
164C22×Dic59C2×D411yescharacteristic
165C22×Dic59C2×D411yescharacteristic
166C22×Dic59C2×D414yes—
167C2×C4×D59—24no—
168C2×C4×D59—216no—
169C2×C4×D59—24no—
170C2×C4×D59—22no—
171C2×C4×D59—22no—
172C2×C4×D59—28no—
173C2×C4×D59—22no—
174C2×C4×D59—22no—
175C2×C4×D59—28no—
176C2×C4×D59—24no—
177C2×C4×D59—24no—
178C2×C4×D59—44no—
179C2×C4×D59—44no—
180C2×C4×D59C2×D414yes—
181C2×C4×D59C2×D412yes—
182C2×C4×D59C2×D414yes—
183C2×C4×D59C2×D412yes—
184C2×C4×D59C2412yes—
185C2×C4×D59C2412yes—
186C2×Dic118—24no—
187C2×Dic118—28no—
188C2×Dic118—24no—
189C2×Dic118—28no—
190C2×Dic118—24no—
191C2×Dic118—22no—
192C2×Dic118—48no—
193C2×Dic118—44no—
194C2×Dic118C2414yes—
195M16×C59—28no—
196C2×C8×C59—24no—
197C2×C8×C59C2414yes—
198C4⋊C4×C59—24no—
199C4⋊C4×C59—22no—
200C4⋊C4×C59—21no—
201C4⋊C4×C59—21no—
202C4⋊C4×C59C2412yes—
203C4⋊C4×C59C2414yes—
204C4⋊C4×C59C2412yes—
205C22⋊C4×C59—24no—
206C42×C59—24no—
207C22⋊Dic59—24no—
208(C2×D59)⋊C4—24no—
209(C2×D59)⋊C4—24no—
210(C2×D59)⋊C4—28no—
211(C2×D59)⋊C4—28no—
212C4⋊Dic59—21no—
213C4⋊Dic59—21no—
214C4⋊Dic59—28no—
215C4⋊Dic59—28no—
216C4⋊Dic59C2412yes—
217C4⋊Dic59C2412yes—
218Dic59⋊C4—24no—
219Dic59⋊C4—24no—
220C4×Dic59—24no—
221C4×Dic59—24no—
222C4×Dic59—24no—
223#11745664—28no—
224C2×C59⋊C8—24no—
225C2×C59⋊C8C2414yes—
226C23×C59—22no—
227C23×C59—22no—
228C23×C59#12314yes—
229C23×C59C22×D412yes—
230C23×C59C22×D411yescharacteristic
231C22×D59—22no—
232C22×D59—22no—
233C22×D59—28no—
234C22×D59—22no—
235C22×D59—22no—
236C22×D59—22no—
237C22×D59—28no—
238C22×D59—44no—
239C22×D59—416no—
240C22×D59#13714yes—
241C22×D59C22×D412yes—
242C22×D59C22×D412yes—
243C2×C4×C59—22no—
244C2×C4×C59—22no—
245C2×C4×C59—24no—
246C2×C4×C59—48no—
247C2×C4×C59—42no—
248C2×C4×C59—42no—
249C2×C4×C59—44no—
250C2×C4×C59#12318yes—
251C2×C4×C59#12314yes—
252C2×C4×C59C22×D412yes—
253C2×C4×C59C22×D411yescharacteristic
254C2×C4×C59C2×#4112yes—
255C2×C4×C59C2×#4114yes—
256C2×C4×C59C2×#4112yes—
257C2×C4×C59C2×#4114yes—
258C2×C4×C59C2511yescharacteristic
259C2×Dic59—28no—
260C2×Dic59—22no—
261C2×Dic59—22no—
262C2×Dic59—22no—
263C2×Dic59—22no—
264C2×Dic59—24no—
265C2×Dic59—48no—
266C2×Dic59—44no—
267C2×Dic59—42no—
268C2×Dic59—42no—
269C2×Dic59C22×D412yes—
270C2×Dic59C22×D412yes—
271C4×D59—28no—
272C4×D59—24no—
273C4×D59—28no—
274C4×D59—24no—
275C4×D59—22no—
276C4×D59—22no—
277C4×D59—22no—
278C4×D59—22no—
279C4×D59—48no—
280C4×D59—48no—
281Dic118—24no—
282Dic118—24no—
283Dic118—48no—
284Dic118—44no—
285C8×C59—24no—
286C59⋊C8—24no—
287C2×#3095—591nomaximal, Sylow
288C22×C59—28no—
289C22×C59—22no—
290C22×C59—22no—
291C22×C59C2×#12314yes—
292C22×C59C2×#12312yes—
293C22×C59C2×#13712yes—
294C22×C59#57512yes—
295C22×C59C23×D411yescharacteristic
296C2×D59—28no—
297C2×D59—22no—
298C2×D59—22no—
299C2×D59—22no—
300C2×D59—22no—
301C2×D59—22no—
302C2×D59—416no—
303C2×D59—44no—
304C2×D59C2×#13714yes—
305C4×C59—42no—
306C4×C59—42no—
307C4×C59—44no—
308C4×C59C2×#12314yes—
309C4×C59C2×#12312yes—
310C4×C59C22×#4111yescharacteristic
311C4×C59C22×#4111yescharacteristic
312Dic59—22no—
313Dic59—22no—
314Dic59—44no—
315Dic59—42no—
316Dic59—42no—
317C22×C22⋊Q8—591no—
318C22×#117—591no—
319#3095—5916no—
320C2×#484—592no—
321C2×#485—592no—
322C2×#485—592no—
323C2×#483—592no—
324C2×#428—591no—
325C2×#428D5911yescharacteristic, complemented
326C2×#419—591no—
327C2×#419—591no—
328C2×#407—591no—
329C2×C59—24no—
330C2×C59#309514yescomplemented
331C2×C59C22×#12311yescharacteristic
332C2×C59C22×#13711yescharacteristic
333C2×C59C2×#57511yescharacteristic
334D59—24no—
335D59—42no—
336C23×Q8—591no—
337C24×C4—591no—
338C22×M16—591no—
339C22×M16C2×D5911yescharacteristic
340C2×C22⋊Q8—592no—
341C2×C22⋊Q8—592no—
342C2×C22⋊Q8—598no—
343C2×C22⋊Q8—1184no—
344C2×C22⋊Q8—1181no—
345C2×C22⋊Q8—1181no—
346C2×#117—591no—
347C2×#117—598no—
348C2×#117—591no—
349C2×#117—591no—
350C2×#117—1181no—
351C2×#117—1181no—
352C2×#117—1184no—
353C2×#117C2×D5911yescharacteristic
354C22×C4⋊C4—591no—
355C22×C4⋊C4—591no—
356C22×C4⋊C4—1181no—
357C22×C4⋊C4C2×D5911yescharacteristic
358C22×C22⋊C4—1181no—
359C22×C22⋊C4—1181no—
360C22×C42—1181no—
361#485—5916no—
362#485—5916no—
363#484—5916no—
364#483—5916no—
365#428—598no—
366#428C2×D5918yescomplemented
367C2×#107—592no—
368C2×#107C2×D5912yescomplemented
369C2×#106—592no—
370C2×#106C2×D5912yescomplemented
371#419—598no—
372#419—598no—
373C2×#103—592no—
374C2×#103—592no—
375C2×#103—592no—
376C2×#103—592no—
377#407—598no—
378C2×C22⋊C8—592no—
379C2×C22⋊C8—592no—
380C59C2×#309511yescharacteristic, complemented, Sylow
381C25—591no—
382C22×Q8—592no—
383C22×Q8—594no—
384C22×Q8—1182no—
385C22×Q8—1181no—
386C22×Q8—1184no—
387C22×Q8—1184no—
388C23×C4—21no—
389C23×C4—591no—
390C23×C4—594no—
391C23×C4—591no—
392C23×C4—591no—
393C23×C4—591no—
394C23×C4—591no—
395C23×C4—594no—
396C23×C4—1181no—
397C23×C4—1181no—
398C23×C4—1184no—
399C23×C4—1184no—
400C23×C4—1181no—
401C23×C4C22×D5911yescharacteristic
402C2×M16—24no—
403C2×M16—594no—
404C2×M16—1184no—
405C2×M16C22×D5914yes—
406C22×C8—592no—
407C22×C8C22×D5912yes—
408C22⋊Q8—598no—
409C22⋊Q8—598no—
410C22⋊Q8—1188no—
411C22⋊Q8—1184no—
412C22⋊Q8—1184no—
413C22⋊Q8—1188no—
414#117—22no—
415#117—594no—
416#117—594no—
417#117—594no—
418#117—1188no—
419#117—1188no—
420#117—1182no—
421#117—1182no—
422#117—1182no—
423#117C22×D5914yes—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—591no—
427C2×C4⋊C4—591no—
428C2×C4⋊C4—594no—
429C2×C4⋊C4—594no—
430C2×C4⋊C4—1182no—
431C2×C4⋊C4—1182no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4—1188no—
435C2×C4⋊C4—1182no—
436C2×C4⋊C4—1182no—
437C2×C4⋊C4C22×D5911yescharacteristic
438C2×C4⋊C4C22×D5911yescharacteristic
439C2×C4⋊C4C22×D5912yes—
440C2×C4⋊C4C22×D5914yes—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1188no—
443C2×C22⋊C4—1181no—
444C2×C22⋊C4—1181no—
445C2×C22⋊C4—1188no—
446C2×C22⋊C4—1182no—
447C2×C22⋊C4—1182no—
448C2×C22⋊C4—1181no—
449C2×C42—21no—
450C2×C42—1188no—
451C2×C42—1181no—
452C2×C42—1181no—
453C2×C42—1181no—
454#107—598no—
455#107C22×D5918yescomplemented
456#106—598no—
457#106C22×D5918yescomplemented
458#103—598no—
459#103—598no—
460#103—598no—
461#103—598no—
462C22⋊C8—598no—
463C22⋊C8—598no—
464C24—591no—
465C24—591no—
466C24—594no—
467C24—1184no—
468C24—1184no—
469C24—2362no—
470C24D4×D5911yescharacteristic
471C2×Q8—594no—
472C2×Q8—1184no—
473C2×Q8—1182no—
474C2×Q8—1184no—
475C2×Q8—1188no—
476C2×Q8—1184no—
477C2×Q8—1188no—
478C2×Q8—2368no—
479C2×Q8—2364no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—44no—
485C22×C4—594no—
486C22×C4—591no—
487C22×C4—591no—
488C22×C4—594no—
489C22×C4—592no—
490C22×C4—592no—
491C22×C4—594no—
492C22×C4—592no—
493C22×C4—592no—
494C22×C4—1184no—
495C22×C4—1184no—
496C22×C4—1184no—
497C22×C4—1184no—
498C22×C4—11816no—
499C22×C4—1184no—
500C22×C4—1182no—
501C22×C4—1182no—
502C22×C4—1188no—
503C22×C4—1182no—
504C22×C4—1182no—
505C22×C4—1188no—
506C22×C4—1184no—
507C22×C4—1181no—
508C22×C4—1181no—
509C22×C4—1182no—
510C22×C4—2364no—
511C22×C4—2364no—
512C22×C4—2364no—
513C22×C4#1174568514yes—
514C22×C4D4×D5912yes—
515C22×C4D4×D5911yescharacteristic
516C22×C4#1174568914yes—
517C22×C4C23×D5912yes—
518C22×C4C23×D5911yescharacteristic
519M16—28no—
520M16—1188no—
521C2×C8—24no—
522C2×C8—594no—
523C2×C8—1184no—
524C2×C8C23×D5914yes—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—592no—
530C4⋊C4—592no—
531C4⋊C4—1184no—
532C4⋊C4—1184no—
533C4⋊C4—1188no—
534C4⋊C4—1188no—
535C4⋊C4—1181no—
536C4⋊C4—1181no—
537C4⋊C4C23×D5912yes—
538C4⋊C4C23×D5912yes—
539C4⋊C4C23×D5914yes—
540C22⋊C4—24no—
541C22⋊C4—1184no—
542C22⋊C4—1184no—
543C22⋊C4—1188no—
544C22⋊C4—1188no—
545C22⋊C4—1184no—
546C42—24no—
547C42—1184no—
548C42—1184no—
549C42—1184no—
550C23—22no—
551C23—22no—
552C23—594no—
553C23—592no—
554C23—592no—
555C23—1188no—
556C23—1182no—
557C23—1182no—
558C23—1188no—
559C23—1182no—
560C23—1182no—
561C23—1182no—
562C23—2364no—
563C23—23616no—
564C23#1427371214yes—
565C23C2×D4×D5912yes—
566C23C2×D4×D5911yescharacteristic
567Q8—1184no—
568Q8—1184no—
569Q8—2368no—
570Q8—2364no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—592no—
579C2×C4—592no—
580C2×C4—1188no—
581C2×C4—1182no—
582C2×C4—1182no—
583C2×C4—1188no—
584C2×C4—1184no—
585C2×C4—1184no—
586C2×C4—1188no—
587C2×C4—1182no—
588C2×C4—1182no—
589C2×C4—1182no—
590C2×C4—1182no—
591C2×C4—1182no—
592C2×C4—1182no—
593C2×C4—1184no—
594C2×C4—2368no—
595C2×C4—2368no—
596C2×C4—2368no—
597C2×C4—2364no—
598C2×C4—2362no—
599C2×C4—2362no—
600C2×C4#1427371218yes—
601C2×C4#1427371214yes—
602C2×C4C2×D4×D5912yes—
603C2×C4C2×D4×D5911yescharacteristic
604C2×C4C2×#1174568512yes—
605C2×C4C2×#1174568514yes—
606C2×C4C2×#1174568912yes—
607C2×C4C2×#1174568914yes—
608C2×C4C24×D5911yescharacteristic
609C8—24no—
610C8—1184no—
611C22—22no—
612C22—22no—
613C22—28no—
614C22—594no—
615C22—1182no—
616C22—1182no—
617C22—1188no—
618C22—1182no—
619C22—1182no—
620C22—1182no—
621C22—23616no—
622C22—2364no—
623C22C2×#1427371214yes—
624C22C2×#1427371212yes—
625C22#137×D5912yes—
626C22#1683819812yes—
627C22C22×D4×D5911yescharacteristic
628C4—42no—
629C4—44no—
630C4—42no—
631C4—1182no—
632C4—1182no—
633C4—2362no—
634C4—2362no—
635C4—2364no—
636C4C2×#1427371214yes—
637C4C2×#1427371212yes—
638C4C22×#1174568511yescharacteristic
639C4C22×#1174568911yescharacteristic
640C2—24no—
641C2—1184no—
642C2—2362no—
643C2#2143404614yesdirect product factor
644C2C22×#1427371211yescharacteristic
645C2C2×#137×D5911yescharacteristic
646C2C2×#1683819811yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC2×C59×#428C2
socleC23×C59C22×D4
prime radicalC24×D59D4
hypercenterC2×#428D59
hypocenterC59C2×#3095
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #21434046C2direct (N × H)
4. C2×#16837739C2semidirect
5. C2×#16837746C2semidirect
6. C2×#16837745C2semidirect
7. C2×#16837734C2semidirect
8. C2×#16837442C2semidirect
9. C2×#16837412C2semidirect
10. C2×#16837410C2semidirect
12. C2×C59×#428C2semidirect
35. #16837746C22semidirect
36. #16837745C22semidirect
37. #16837739C22semidirect
38. #16837734C22semidirect
39. #16837442C22semidirect
40. C2×#14273660C22semidirect
41. C2×#14273658C22semidirect
42. #16837412C22semidirect
43. #16837410C22semidirect
44. C2×#14273652C22semidirect
45. C2×#14273652C22semidirect
46. C2×#14273650C22semidirect
47. C2×#14273650C22semidirect
54. C59×#428C22semidirect
55. C2×#107×C59C22semidirect
56. C2×#106×C59C22semidirect
115. #14273660C23semidirect
116. #14273658C23semidirect
117. #14273652C23semidirect
118. #14273652C23semidirect
119. #14273650C23semidirect
120. #14273650C23semidirect
138. #107×C59C23semidirect
139. #106×C59C23semidirect
325. C2×#428D59semidirect
330. C2×C59#3095semidirect
366. #428C2×D59semidirect
368. C2×#107C2×D59semidirect
370. C2×#106C2×D59semidirect
380. C59C2×#3095semidirect
455. #107C22×D59semidirect
457. #106C22×D59semidirect
643. C2#21434046direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142737072
C2×#117×D592
#214340462
C2×#168377392
C2×#168377462
C2×#168377452
C2×#168377342
C2×#168374422
C2×#168374122
C2×#168374102
C2×#168373652
C2×C59×#4282
C2×#309559