Finite Group Databasebeta

← C2×#21428562 #28757745

Subgroups — C2×#21428562 #28757745

Order 15104.

4222 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273712C212yesmaximal, complemented
2C22×#14273709C212yesmaximal, complemented
3C22×#14273700C211yesmaximal, characteristic
4C22×C4⋊C4×D59C211yesmaximal, characteristic, complemented
5#21428562C2116yesmaximal, direct product factor
6C2×#16837339C211yesmaximal, characteristic, complemented
7C2×#16837338C211yesmaximal, characteristic, complemented
8C2×#16837337C211yesmaximal, characteristic, complemented
9C2×#16837302C212yesmaximal, complemented
10C2×#16837271C212yesmaximal, complemented
11C2×#16837249C211yesmaximal, characteristic, complemented
12C2×C59×#399C211yesmaximal, characteristic, complemented
13C23×D236C2211yescharacteristic
14C23×C22⋊D59C2211yescharacteristic
15C23×C22⋊D59C2211yescharacteristic
16C23×C4×D59C2212yes—
17C23×C4×D59C2211yescharacteristic
18C2×#14273712—216no—
19C2×#14273712C22116yescomplemented
20C2×#14273709—216no—
21C2×#14273709C22116yescomplemented
22C2×#14273700—28no—
23C2×#14273700C2218yes—
24C22×C4⋊Dic59C2212yes—
25C22×C4⋊Dic59C2211yescharacteristic
26C22×Dic59⋊C4C2211yescharacteristic
27C22×Dic59⋊C4C2212yes—
28C22×Dic59⋊C4C2212yes—
29C22×Dic59⋊C4C2212yes—
30C2×C4⋊C4×D59—28no—
31C2×C4⋊C4×D59C2218yescomplemented
32C22×C22⋊Dic59C2212yes—
33C22×(C2×D59)⋊C4C2212yes—
34C22×(C2×D59)⋊C4C2212yes—
35C22×(C2×D59)⋊C4C2211yescharacteristic
36C22×(C2×D59)⋊C4C2212yes—
37C22×C4×Dic59C2212yes—
38#16837339C2218yescomplemented
39#16837338C2218yescomplemented
40#16837337C2218yescomplemented
41#16837302C22116yescomplemented
42#16837271C22116yescomplemented
43#16837249C2218yescomplemented
44C2×C59⋊#95C2211yescharacteristic
45C22×C4⋊C4×C59C2212yes—
46C22×C4⋊C4×C59C2211yescharacteristic
47C22×C22⋊C4×C59C2212yes—
48C22×C22⋊C4×C59C2211yescharacteristic
49C59×#399C2218yescomplemented
50C2×#95×C59C2211yescharacteristic
51C24×D59C2311yescharacteristic
52C24×D59C2311yescharacteristic
53C22×D236—24no—
54C22×D236—28no—
55C22×D236C2314yes—
56C22×C22⋊D59—24no—
57C22×C22⋊D59—28no—
58C22×C22⋊D59—24no—
59C22×C22⋊D59—28no—
60C22×C22⋊D59C2314yes—
61C22×C22⋊D59C2314yes—
62C23×Dic59C2312yes—
63C23×Dic59C2311yescharacteristic
64C23×Dic59C2312yes—
65C23×Dic59C2311yescharacteristic
66C22×C4×D59—28no—
67C22×C4×D59—28no—
68C22×C4×D59—28no—
69C22×C4×D59—28no—
70C22×C4×D59D418yes—
71C22×C4×D59C2318yes—
72C22×C4×D59C2314yes—
73#14273712—264no—
74#14273709—264no—
75#14273700—232no—
76C2×C4⋊Dic59—28no—
77C2×C4⋊Dic59—24no—
78C2×C4⋊Dic59C2318yes—
79C2×C4⋊Dic59C2314yes—
80C2×Dic59⋊C4—24no—
81C2×Dic59⋊C4—28no—
82C2×Dic59⋊C4—28no—
83C2×Dic59⋊C4—28no—
84C2×Dic59⋊C4C2314yes—
85C2×Dic59⋊C4C2318yes—
86C2×Dic59⋊C4C2318yes—
87C2×Dic59⋊C4C2318yes—
88C4⋊C4×D59—232no—
89C2×C22⋊Dic59—28no—
90C2×C22⋊Dic59C2318yes—
91C2×(C2×D59)⋊C4—24no—
92C2×(C2×D59)⋊C4—28no—
93C2×(C2×D59)⋊C4—28no—
94C2×(C2×D59)⋊C4D4116yescomplemented
95C2×(C2×D59)⋊C4C2314yes—
96C2×(C2×D59)⋊C4C2318yes—
97C2×(C2×D59)⋊C4C2318yes—
98C2×(C2×D59)⋊C4C2318yes—
99C2×C4×Dic59—28no—
100C2×C4×Dic59C2318yes—
101C59⋊#95C2314yes—
102C25×C59C2311yescharacteristic
103C23×C4×C59C2312yes—
104C23×C4×C59C2311yescharacteristic
105C23×C4×C59C2311yescharacteristic
106C23×C4×C59C2312yes—
107C2×C4⋊C4×C59—28no—
108C2×C4⋊C4×C59—24no—
109C2×C4⋊C4×C59C2318yes—
110C2×C4⋊C4×C59C2314yes—
111C2×C22⋊C4×C59—24no—
112C2×C22⋊C4×C59—28no—
113C2×C22⋊C4×C59C2314yes—
114C2×C22⋊C4×C59C2318yes—
115#95×C59C2314yes—
116C24×C59—24no—
117C24×C59—22no—
118C24×C59—24no—
119C24×C59—22no—
120C24×C59—22no—
121C24×C59C2412yes—
122C24×C59C2411yescharacteristic
123C23×D59—24no—
124C23×D59—22no—
125C23×D59—22no—
126C23×D59—24no—
127C23×D59—22no—
128C23×D59—24no—
129C23×D59—22no—
130C23×D59—22no—
131C23×D59C2×D418yes—
132C23×D59C2×D414yes—
133C23×D59C2412yes—
134C23×D59C2412yes—
135C22×C4×C59—24no—
136C22×C4×C59—24no—
137C22×C4×C59—24no—
138C22×C4×C59—24no—
139C22×C4×C59—22no—
140C22×C4×C59—24no—
141C22×C4×C59—24no—
142C22×C4×C59—24no—
143C22×C4×C59C2×D418yes—
144C22×C4×C59C2×D414yes—
145C22×C4×C59C2414yes—
146C22×C4×C59C2412yes—
147C22×C4×C59C2412yes—
148C22×C4×C59C2414yes—
149C2×C22⋊D59—28no—
150C2×C22⋊D59—216no—
151C2×C22⋊D59—28no—
152C2×C22⋊D59—216no—
153C2×C22⋊D59—416no—
154C2×C22⋊D59—416no—
155C22×Dic59—24no—
156C22×Dic59—24no—
157C22×Dic59—24no—
158C22×Dic59—22no—
159C22×Dic59—24no—
160C22×Dic59—24no—
161C22×Dic59—24no—
162C22×Dic59C2×D418yes—
163C22×Dic59C2×D414yes—
164C22×Dic59#4118yes—
165C22×Dic59C2414yes—
166C22×Dic59C2412yes—
167C22×Dic59C2414yes—
168C22×Dic59C2412yes—
169C2×D236—28no—
170C2×D236—216no—
171C2×D236—416no—
172C2×C4×D59—232no—
173C2×C4×D59—216no—
174C2×C4×D59—216no—
175C2×C4×D59—216no—
176C2×C4×D59—216no—
177C2×C4×D59—432no—
178C2×C4×D59C2×D4116yes—
179C4⋊C4×C59—28no—
180C4⋊C4×C59—216no—
181C22⋊C4×C59—28no—
182C22⋊C4×C59—216no—
183C22⋊Dic59—216no—
184(C2×D59)⋊C4—28no—
185(C2×D59)⋊C4—216no—
186(C2×D59)⋊C4—216no—
187(C2×D59)⋊C4C2×D4132yescomplemented
188C4⋊Dic59—28no—
189C4⋊Dic59—216no—
190Dic59⋊C4—216no—
191Dic59⋊C4—28no—
192Dic59⋊C4—216no—
193Dic59⋊C4—216no—
194C4×Dic59—216no—
195C23×C59—24no—
196C23×C59—22no—
197C23×C59—24no—
198C23×C59—22no—
199C23×C59—22no—
200C23×C59—44no—
201C23×C59—44no—
202C23×C59—44no—
203C23×C59—48no—
204C23×C59—48no—
205C23×C59C22×D412yes—
206C23×C59C22×D414yes—
207C23×C59C2×#4114yes—
208C23×C592−1+412yes—
209C23×C592−1+412yes—
210C23×C59C2511yescharacteristic
211C22×D59—28no—
212C22×D59—28no—
213C22×D59—216no—
214C22×D59—216no—
215C22×D59—24no—
216C22×D59—22no—
217C22×D59—22no—
218C22×D59—24no—
219C22×D59—22no—
220C22×D59—24no—
221C22×D59—22no—
222C22×D59—22no—
223C22×D59—44no—
224C22×D59—44no—
225C22×D59—44no—
226C22×D59—48no—
227C22×D59—48no—
228C22×D59#127116yes—
229C22×D59C22×D418yes—
230C22×D59C22×D414yes—
231C2×C4×C59—28no—
232C2×C4×C59—216no—
233C2×C4×C59—24no—
234C2×C4×C59—24no—
235C2×C4×C59—24no—
236C2×C4×C59—24no—
237C2×C4×C59—22no—
238C2×C4×C59—24no—
239C2×C4×C59—24no—
240C2×C4×C59—24no—
241C2×C4×C59—48no—
242C2×C4×C59—44no—
243C2×C4×C59C22×D418yes—
244C2×C4×C59C22×D414yes—
245C22⋊D59—416no—
246C22⋊D59—416no—
247C2×Dic59—216no—
248C2×Dic59—24no—
249C2×Dic59—24no—
250C2×Dic59—24no—
251C2×Dic59—24no—
252C2×Dic59—24no—
253C2×Dic59—22no—
254C2×Dic59—24no—
255C2×Dic59—48no—
256C2×Dic59—44no—
257C2×Dic59#123116yescomplemented
258C2×Dic59C22×D418yes—
259C2×Dic59C22×D414yes—
260C2×Dic59C2×#4118yes—
261D236—416no—
262C4×D59—232no—
263C4×D59—432no—
264C2×#2501—591nomaximal, Sylow
265C22×C59—42no—
266C22×C59—42no—
267C22×C59—42no—
268C22×C59—44no—
269C22×C59—44no—
270C22×C59—88no—
271C22×C59C2×#12314yes—
272C22×C59C2×#12714yes—
273C22×C59#53614yes—
274C22×C59#54718yes—
275C22×C59#54718yes—
276C22×C59C23×D411yescharacteristic
277C22×C59C23×D412yes—
278C22×C59C22×#4112yes—
279C22×C59C2×2−1+411yescharacteristic
280C22×C59C2×2−1+411yescharacteristic
281C2×D59—24no—
282C2×D59—24no—
283C2×D59—28no—
284C2×D59—28no—
285C2×D59—232no—
286C2×D59—42no—
287C2×D59—42no—
288C2×D59—42no—
289C2×D59—44no—
290C2×D59—44no—
291C2×D59—88no—
292C2×D59C2×#12718yes—
293C4×C59—24no—
294C4×C59—28no—
295C4×C59—44no—
296C4×C59—42no—
297Dic59—28no—
298Dic59—44no—
299Dic59—42no—
300Dic59C2×#12318yescomplemented
301C22×#123—592no—
302C22×#123—592no—
303C22×#121—591no—
304C23×C4⋊C4—591no—
305#2501—5916no—
306C2×#399—591no—
307C2×#399—591no—
308C2×#399—591no—
309C2×#399D5911yescharacteristic, complemented
310C2×#390—592no—
311C2×#384—592no—
312C2×C4×C22⋊C4—591no—
313C2×C59—82no—
314C2×C59#250118yescomplemented
315C2×C59C22×#12311yescharacteristic
316C2×C59C22×#12711yescharacteristic
317C2×C59C2×#53611yescharacteristic
318C2×C59C2×#54712yes—
319C2×C59C2×#54712yes—
320D59—28no—
321D59—82no—
322C23×D4—591no—
323C23×D4—591no—
324C23×D4—591no—
325C24×C4—591no—
326C24×C4—592no—
327C2×#123—5916no—
328C2×#123—5916no—
329C2×#123—11816no—
330C2×#123—11816no—
331C2×#121—598no—
332C2×#121—1188no—
333C22×C4⋊C4—591no—
334C22×C4⋊C4—592no—
335C22×C4⋊C4—592no—
336C22×C4⋊C4—592no—
337C22×C4⋊C4—592no—
338C22×C4⋊C4—591no—
339C22×C4⋊C4—598no—
340C22×C4⋊C4—1188no—
341C22×C4⋊C4C2×D5911yescharacteristic
342C22×C4⋊C4C2×D5912yes—
343C22×C22⋊C4—592no—
344C22×C22⋊C4—592no—
345C22×C22⋊C4—591no—
346C22×C22⋊C4—592no—
347C22×C22⋊C4—592no—
348C22×C22⋊C4C2×D5911yescharacteristic
349C22×C22⋊C4C2×D5912yes—
350C22×C42—592no—
351#399—598no—
352#399—598no—
353#399—598no—
354#399C2×D5918yescomplemented
355#390—5916no—
356#384—5916no—
357C4×C22⋊C4—598no—
358C2×#95—591no—
359C2×#95C2×D5911yescharacteristic
360C59C2×#250111yescharacteristic, complemented, Sylow
361C25—591no—
362C25—591no—
363C25C22×D5911yescharacteristic
364C22×D4—594no—
365C22×D4—594no—
366C22×D4—594no—
367C22×D4—1184no—
368C22×D4—1188no—
369C22×D4—1184no—
370C22×D4—1188no—
371C22×D4—1184no—
372C22×D4—1188no—
373C23×C4—591no—
374C23×C4—592no—
375C23×C4—598no—
376C23×C4—594no—
377C23×C4—592no—
378C23×C4—591no—
379C23×C4—598no—
380C23×C4—1188no—
381C23×C4—1188no—
382C23×C4—1188no—
383C23×C4—1188no—
384C23×C4C22×D5912yes—
385C23×C4C22×D5911yescharacteristic
386C23×C4C22×D5912yes—
387C23×C4C22×D5911yescharacteristic
388#123—11864no—
389#123—11864no—
390#121—11832no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—28no—
393C2×C4⋊C4—598no—
394C2×C4⋊C4—594no—
395C2×C4⋊C4—594no—
396C2×C4⋊C4—598no—
397C2×C4⋊C4—598no—
398C2×C4⋊C4—598no—
399C2×C4⋊C4—11832no—
400C2×C4⋊C4—1188no—
401C2×C4⋊C4—1184no—
402C2×C4⋊C4—1188no—
403C2×C4⋊C4—1188no—
404C2×C4⋊C4—1184no—
405C2×C4⋊C4—1188no—
406C2×C4⋊C4C22×D5914yes—
407C2×C4⋊C4C22×D5918yes—
408C2×C22⋊C4—24no—
409C2×C22⋊C4—28no—
410C2×C22⋊C4—5916no—
411C2×C22⋊C4—598no—
412C2×C22⋊C4—598no—
413C2×C22⋊C4—594no—
414C2×C22⋊C4—598no—
415C2×C22⋊C4—598no—
416C2×C22⋊C4—1188no—
417C2×C22⋊C4—1188no—
418C2×C22⋊C4—1184no—
419C2×C22⋊C4—1188no—
420C2×C22⋊C4C22×D5914yes—
421C2×C22⋊C4C22×D5918yes—
422C2×C42—598no—
423C2×C42—1188no—
424#95—594no—
425#95C22×D5914yes—
426C24—24no—
427C24—22no—
428C24—24no—
429C24—22no—
430C24—22no—
431C24—598no—
432C24—594no—
433C24—592no—
434C24—592no—
435C24—1184no—
436C24—1182no—
437C24—1182no—
438C24—1184no—
439C24—1182no—
440C24—1184no—
441C24—1182no—
442C24—1182no—
443C24C23×D5912yes—
444C24C23×D5911yescharacteristic
445C2×D4—1188no—
446C2×D4—11816no—
447C2×D4—1188no—
448C2×D4—11816no—
449C2×D4—1188no—
450C2×D4—11816no—
451C2×D4—23616no—
452C2×D4—23616no—
453C2×D4—23616no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—594no—
463C22×C4—598no—
464C22×C4—592no—
465C22×C4—598no—
466C22×C4—594no—
467C22×C4—5916no—
468C22×C4—594no—
469C22×C4—592no—
470C22×C4—11832no—
471C22×C4—11816no—
472C22×C4—11816no—
473C22×C4—11816no—
474C22×C4—11816no—
475C22×C4—1184no—
476C22×C4—1184no—
477C22×C4—1184no—
478C22×C4—1184no—
479C22×C4—1182no—
480C22×C4—1184no—
481C22×C4—1184no—
482C22×C4—23632no—
483C22×C4D4×D5918yes—
484C22×C4D4×D5914yes—
485C22×C4C23×D5914yes—
486C22×C4C23×D5912yes—
487C22×C4C23×D5912yes—
488C22×C4C23×D5914yes—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—11816no—
492C4⋊C4—1188no—
493C4⋊C4—11816no—
494C4⋊C4—11816no—
495C4⋊C4—1188no—
496C4⋊C4—11816no—
497C22⋊C4—28no—
498C22⋊C4—216no—
499C22⋊C4—5932no—
500C22⋊C4—11816no—
501C22⋊C4—11816no—
502C22⋊C4—1188no—
503C22⋊C4—11816no—
504C42—11816no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—48no—
514C23—48no—
515C23—5916no—
516C23—598no—
517C23—594no—
518C23—1188no—
519C23—1184no—
520C23—1188no—
521C23—1182no—
522C23—1182no—
523C23—11816no—
524C23—11816no—
525C23—1184no—
526C23—1182no—
527C23—1184no—
528C23—1182no—
529C23—1182no—
530C23—2364no—
531C23—2364no—
532C23—2364no—
533C23—2368no—
534C23—2368no—
535C23C2×D4×D5912yes—
536C23C2×D4×D5914yes—
537C23#41×D5914yes—
538C23#1427377412yes—
539C23#1427377512yes—
540C23C24×D5911yescharacteristic
541D4—23616no—
542D4—23616no—
543D4—23616no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—216no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—5916no—
557C2×C4—594no—
558C2×C4—598no—
559C2×C4—598no—
560C2×C4—11832no—
561C2×C4—1184no—
562C2×C4—1184no—
563C2×C4—11816no—
564C2×C4—1184no—
565C2×C4—1184no—
566C2×C4—1182no—
567C2×C4—1184no—
568C2×C4—1184no—
569C2×C4—23632no—
570C2×C4—2368no—
571C2×C4—2364no—
572C2×C4C2×D4×D5918yes—
573C2×C4C2×D4×D5914yes—
574C22—44no—
575C22—44no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—88no—
580C22—598no—
581C22—11832no—
582C22—1188no—
583C22—1188no—
584C22—1184no—
585C22—1184no—
586C22—4728no—
587C22—2364no—
588C22—2364no—
589C22—2362no—
590C22—2362no—
591C22—2362no—
592C22#123×D5914yes—
593C22#127×D5914yes—
594C22#1683797514yes—
595C22#1683805018yes—
596C22#1683805418yes—
597C22C22×D4×D5912yes—
598C22C22×D4×D5911yescharacteristic
599C22C2×#41×D5912yes—
600C22C2×#1427377411yescharacteristic
601C22C2×#1427377511yescharacteristic
602C4—28no—
603C4—24no—
604C4—44no—
605C4—42no—
606C4—598no—
607C4—1188no—
608C4—2364no—
609C4—2362no—
610C2—82no—
611C2—1188no—
612C2—4722no—
613C2#2142856218yesdirect product factor
614C2C2×#123×D5911yescharacteristic
615C2C2×#127×D5911yescharacteristic
616C2C2×#1683797511yescharacteristic
617C2C2×#1683805012yes—
618C2C2×#1683805412yes—
subgroupG/H
centerC24C23×D59
commutator, cosocleC23×C59C25
FrattiniC23C24×D59
FittingC2×C59×#399C2
socleC24×C59C24
prime radicalC22×#14273700C2
hypercenterC2×#399D59
hypocenterC59C2×#2501
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C22×#14273712C2semidirect
2. C22×#14273709C2semidirect
4. C22×C4⋊C4×D59C2semidirect
5. #21428562C2direct (N × H)
6. C2×#16837339C2semidirect
7. C2×#16837338C2semidirect
8. C2×#16837337C2semidirect
9. C2×#16837302C2semidirect
10. C2×#16837271C2semidirect
11. C2×#16837249C2semidirect
12. C2×C59×#399C2semidirect
19. C2×#14273712C22semidirect
21. C2×#14273709C22semidirect
31. C2×C4⋊C4×D59C22semidirect
38. #16837339C22semidirect
39. #16837338C22semidirect
40. #16837337C22semidirect
41. #16837302C22semidirect
42. #16837271C22semidirect
43. #16837249C22semidirect
49. C59×#399C22semidirect
94. C2×(C2×D59)⋊C4D4semidirect
187. (C2×D59)⋊C4C2×D4semidirect
257. C2×Dic59#123semidirect
300. Dic59C2×#123semidirect
309. C2×#399D59semidirect
314. C2×C59#2501semidirect
354. #399C2×D59semidirect
360. C59C2×#2501semidirect
613. C2#21428562direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#142737122
C22×#142737092
C22×#142737002
C22×C4⋊C4×D592
#214285622
C2×#168373392
C2×#168373382
C2×#168373372
C2×#168373022
C2×#168372712
C2×#168372492
C2×C59×#3992
C2×#250159