Finite Group Databasebeta

← C4×#16837702 #28596295

Subgroups — C4×#16837702 #28596295

Order 15104.

948 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837702C211yesmaximal, characteristic
2C4×#14273719C211yesmaximal, characteristic, complemented
3#21425053C211yesmaximal, characteristic, complemented
4#21424864C211yesmaximal, characteristic, complemented
5#21424688C211yesmaximal, characteristic, complemented
6#21424674C211yesmaximal, characteristic, complemented
7#21424357C211yesmaximal, characteristic
8#21424343C211yesmaximal, characteristic
9#21424106C211yesmaximal, characteristic
10C4×#14273660C211yesmaximal, characteristic, complemented
11C4×#14273658C211yesmaximal, characteristic, complemented
12C4×C4⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
13C4×#14273648C211yesmaximal, characteristic
14C4×#14273648C211yesmaximal, characteristic
15C42×D4×C59C211yesmaximal, characteristic
16C2×#14273719C2211yescharacteristic
17#16837702C418yesdirect product factor
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×C4×C59⋊C8C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837286—21no—
33#16837286—21no—
34#16837268C2211yescharacteristic
35C4×Dic59⋊C4—21no—
36C4×Dic59⋊C4—21no—
37C4×C4⋊Dic59C2211yescharacteristic
38C4×C4⋊Dic59C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C22×C42×C59—21no—
53C4×C4⋊C4×C59C2211yescharacteristic
54C4×C4⋊C4×C59C2211yescharacteristic
55C4×C4⋊C4×C59C2211yescharacteristic
56C4×C22⋊C4×C59—21no—
57C4×C22⋊C4×C59—21no—
58C4×C22⋊C4×C59—21no—
59C43×C59C2211yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273719—22no—
65#14273719C2×C414yescomplemented
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×Dic59⋊C4—21no—
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76#14273660—22no—
77#14273660C2×C414yescomplemented
78#14273658—22no—
79#14273658C2×C414yescomplemented
80C4⋊(C59⋊C8)—22no—
81C4⋊(C59⋊C8)C2×C414yescomplemented
82#14273648—22no—
83#14273648—22no—
84#14273648C2×C414yes—
85#14273648C2×C414yes—
86C4×C59⋊C8—22no—
87C4×C59⋊C8—22no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C22×D4×C59C2311yescharacteristic
95C23×C4×C59—21no—
96C23×C4×C59—21no—
97C23×C4×C59—21no—
98C4×D4×C59—22no—
99C4×D4×C59—22no—
100C4×D4×C59—22no—
101C4×D4×C59—22no—
102C4×D4×C59C2×C414yes—
103C4×D4×C59Q814yes—
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C4⋊C4×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C22⋊C4×C59—21no—
108C2×C22⋊C4×C59—21no—
109C2×C22⋊C4×C59—21no—
110C2×C42×C59—21no—
111C2×C42×C59—21no—
112C2×C42×C59—21no—
113C2×C42×C59—44no—
114C2×C42×C59D411yescharacteristic
115C2×C42×C59D411yescharacteristic
116C2×C42×C59D411yescharacteristic
117C2×C42×C59D411yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C2×C42×C59C2311yescharacteristic
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124C24×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59C22×C412yes—
129C2×D4×C59C2×Q812yes—
130C2×D4×C59#4112yes—
131C22×C4×C59—24no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—24no—
136C22×C4×C59—24no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59—42no—
140C22×C4×C59—42no—
141C22×C4×C59—42no—
142C22×C4×C59—42no—
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2411yescharacteristic
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59C22×C412yes—
158C4⋊C4×C59C2×Q812yes—
159C4⋊C4×C59#4112yes—
160C22⋊C4×C59—24no—
161C22⋊C4×C59—24no—
162C22⋊C4×C59—24no—
163C42×C59—22no—
164C42×C59—22no—
165C42×C59—22no—
166C42×C59—21no—
167C42×C59—21no—
168C42×C59—41no—
169C42×C59—42no—
170C42×C59—42no—
171C42×C59—42no—
172C42×C59C22×C412yes—
173C42×C59C2×Q812yes—
174C42×C59#4112yes—
175C42×C59#4112yes—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59C22×C412yes—
183C4⋊Dic59C22×C412yes—
184Dic59⋊C4—24no—
185Dic59⋊C4—24no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188C4×Dic59—42no—
189C4×Dic59—42no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8C22×C412yes—
201C2×C59⋊C8C22×C412yes—
202C23×C59—22no—
203C23×C59—22no—
204C23×C59—22no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59C22×D411yescharacteristic
210D4×C59—22no—
211D4×C59—22no—
212D4×C59—22no—
213D4×C59C4×Q814yes—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—48no—
229C2×C4×C59—48no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—48no—
237C2×C4×C59C4×D412yes—
238C2×C4×C59C4×D412yes—
239C2×C4×C59C22⋊Q812yes—
240C2×C4×C59C22⋊Q812yes—
241C2×C4×C59#13512yes—
242C2×C4×C59#13512yes—
243C2×C4×C59AGL(1,ℤ/8)12yes—
244C2×C4×C59C23×C411yescharacteristic
245C2×C4×C59C22×Q811yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×Dic59—22no—
251C2×Dic59—22no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C59⋊C8—22no—
263C59⋊C8—22no—
264C59⋊C8—41no—
265C59⋊C8—41no—
266C4×#478—591nomaximal, Sylow
267C22×C59—24no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59C2×C4×D411yescharacteristic
275C22×C59C2×C22⋊Q811yescharacteristic
276C22×C59#52511yescharacteristic
277C22×C59C2×#13511yescharacteristic
278C22×C59C2×#13511yescharacteristic
279C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C59#57511yescharacteristic
281C4×C59—22no—
282C4×C59—22no—
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—42no—
286C4×C59—41no—
287C4×C59—42no—
288C4×C59—42no—
289C4×C59#47814yescomplemented
290C4×C59C2×C4×Q811yescharacteristic
291C4×C59C4×#4111yescharacteristic
292C4×C59#52911yescharacteristic
293C4×C59#53311yescharacteristic
294Dic59—42no—
295Dic59—42no—
296Dic59—81no—
297Dic59—81no—
298C2×#478—591no—
299C4×#125—591no—
300C42×D4D5911yescharacteristic
301#1989—591no—
302#1949—591no—
303#1921—591no—
304#1919—591no—
305#1864—591no—
306#1862—591no—
307#1804—591no—
308C4×#107—591no—
309C4×#106—591no—
310C4×C4⋊C8—591no—
311C4×#102—591no—
312C4×#102—591no—
313C2×C59—44no—
314C2×C59C4×C22⋊Q811yescharacteristic
315C2×C59C4×#13511yescharacteristic
316C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
317C2×C59C2×#47811yescharacteristic
318C2×C59#307611yescharacteristic
319C2×C59#310211yescharacteristic
320C2×C59#310511yescharacteristic
321C2×#125—591no—
322C2×C4×D4C2×D5911yescharacteristic
323C2×C4×D4C2×D5911yescharacteristic
324C2×C4×D4C2×D5911yescharacteristic
325C22×C42—21no—
326#478—598no—
327C2×#107—591no—
328C2×#107—591no—
329C2×#106—591no—
330C2×#106—591no—
331C2×C4⋊C8—591no—
332C2×C4⋊C8—591no—
333C2×#102—591no—
334C2×#102—591no—
335C2×#102—591no—
336C2×#102—591no—
337C2×C4×C8—591no—
338C2×C4×C8—591no—
339#389—591no—
340#389—591no—
341#387—1181no—
342#387—1181no—
343#382—591no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4—591no—
346C4×C4⋊C4—1181no—
347C4×C4⋊C4—1181no—
348C4×C4⋊C4C2×D5911yescharacteristic
349C4×C4⋊C4C2×D5911yescharacteristic
350C4×C4⋊C4C2×D5911yescharacteristic
351C4×C22⋊C4—21no—
352C4×C22⋊C4—21no—
353C4×C22⋊C4—21no—
354C43C2×D5911yescharacteristic
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#336—591no—
364#336—591no—
365C59C4×#47811yescharacteristic, complemented, Sylow
366C22×D4C22×D5911yescharacteristic
367C23×C4—21no—
368C23×C4—21no—
369C23×C4—21no—
370C22×C8—591no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8—591no—
374#125—594no—
375#125—1182no—
376C4×D4—22no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—22no—
380C4×D4Dic11814yes—
381C4×D4C4×D5914yes—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—1181no—
387C2×C4⋊C4—1181no—
388C2×C4⋊C4—1181no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4C22×D5911yescharacteristic
391C2×C4⋊C4C22×D5911yescharacteristic
392C2×C4⋊C4C22×D5911yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—44no—
400C2×C42—1181no—
401C2×C42—1181no—
402C2×C42C22⋊D5911yescharacteristic
403C2×C42C22⋊D5911yescharacteristic
404C2×C42C22⋊D5911yescharacteristic
405C2×C42C22⋊D5911yescharacteristic
406C2×C42C22×D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408C2×C42C22×D5911yescharacteristic
409#107—594no—
410#107—1182no—
411#106—594no—
412#106—1182no—
413C4⋊C8—594no—
414C4⋊C8—1182no—
415#102—594no—
416#102—594no—
417#102—1182no—
418#102—1182no—
419C4×C8—1182no—
420C4×C8—1182no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×Dic11812yes—
435C2×D4C2×C4×D5912yes—
436C2×D4#1174568512yes—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4C2×C22⋊D5911yescharacteristic
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C2×C22⋊D5911yescharacteristic
456C22×C4C2×C22⋊D5911yescharacteristic
457C22×C4C2×C22⋊D5911yescharacteristic
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460C2×C8—592no—
461C2×C8—592no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—592no—
476C4⋊C4—592no—
477C4⋊C4—1184no—
478C4⋊C4—1184no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—2362no—
486C4⋊C4—2362no—
487C4⋊C4C2×Dic11812yes—
488C4⋊C4C2×C4×D5912yes—
489C4⋊C4#1174568512yes—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—21no—
497C42—21no—
498C42—41no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—2362no—
503C42—2362no—
504C42C2×Dic11812yes—
505C42C2×C4×D5912yes—
506C42#1174568512yes—
507C42#1174568512yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C22×C22⋊D5911yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4C4×Dic11814yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—48no—
543C2×C4—1182no—
544C2×C4—1182no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4C4×C22⋊D5912yes—
556C2×C4C4×C22⋊D5912yes—
557C2×C4#1427370612yes—
558C2×C4#1427370612yes—
559C2×C4#1427374712yes—
560C2×C4#1427374712yes—
561C2×C4#1427375312yes—
562C2×C4C22×C4×D5911yescharacteristic
563C2×C4C22×Dic11811yescharacteristic
564C2×C4C2×#1174568511yescharacteristic
565C2×C4C2×#1174568511yescharacteristic
566C2×C4C2×#1174568511yescharacteristic
567C2×C4C2×#1174568511yescharacteristic
568C8—1182no—
569C8—1182no—
570C8—2361no—
571C8—2361no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×C4×C22⋊D5911yescharacteristic
580C22C2×#1427370611yescharacteristic
581C22#1683791311yescharacteristic
582C22C2×#1427374711yescharacteristic
583C22C2×#1427374711yescharacteristic
584C22C2×#1427375311yescharacteristic
585C22#1683820011yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—4721no—
595C4—4721no—
596C4—2362no—
597C4—2362no—
598C4#1683770214yesdirect product factor
599C4C2×C4×Dic11811yescharacteristic
600C4C4×#1174568511yescharacteristic
601C4#1683793611yescharacteristic
602C4#1683795911yescharacteristic
603C2—44no—
604C2C4×#1427370611yescharacteristic
605C2C4×#1427374711yescharacteristic
606C2C4×#1427375311yescharacteristic
607C2C2×#1683770211yescharacteristic
608C2#2143388411yescharacteristic
609C2#2143411411yescharacteristic
610C2#2143414311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC42×D4×C59C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenterC42×D4D59
hypocenterC59C4×#478
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. C4×#14273719C2semidirect
3. #21425053C2semidirect
4. #21424864C2semidirect
5. #21424688C2semidirect
6. #21424674C2semidirect
10. C4×#14273660C2semidirect
11. C4×#14273658C2semidirect
12. C4×C4⋊(C59⋊C8)C2semidirect
17. #16837702C4direct (N × H)
65. #14273719C2×C4semidirect
77. #14273660C2×C4semidirect
79. #14273658C2×C4semidirect
81. C4⋊(C59⋊C8)C2×C4semidirect
289. C4×C59#478semidirect
365. C59C4×#478semidirect
598. C4#16837702direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168377022
C4×#142737192
#214250532
#214248642
#214246882
#214246742
#214243572
#214243432
#214241062
C4×#142736602
C4×#142736582
C4×C4⋊(C59⋊C8)2
C4×#142736482
C42×D4×C592
C4×#47859