Finite Group Databasebeta

← #28605482

Subgroups — #28605482

Order 15104.

1074 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838121C211yesmaximal, characteristic
2C2×C8×Dic118C211yesmaximal, characteristic, complemented
3#21427048C211yesmaximal, characteristic, complemented
4#21425029C211yesmaximal, characteristic
5#21425027C211yesmaximal, characteristic
6#21425017C211yesmaximal, characteristic, complemented
7#21424906C211yesmaximal, characteristic
8#21424896C211yesmaximal, characteristic, complemented
9#21424109C211yesmaximal, characteristic, complemented
10#21424108C211yesmaximal, characteristic, complemented
11#21424094C211yesmaximal, characteristic
12#103×Dic59C211yesmaximal, characteristic, complemented
13C4×#14273652C211yesmaximal, characteristic, complemented
14#102×Dic59C211yesmaximal, characteristic
15C59×#1958C211yesmaximal, characteristic
16#16838121—24no—
17#16838121C418yescomplemented
18C2×#14273725C2211yescharacteristic
19C2×#14273720—21no—
20C2×#14273720—21no—
21C2×#14273714C2211yescharacteristic
22C2×#14273704—21no—
23C2×Q8×Dic59C2211yescharacteristic
24C2×C4×Dic118C2211yescharacteristic
25C2×C4×Dic118C2211yescharacteristic
26C2×D4×Dic59C2211yescharacteristic
27C2×#14273683—21no—
28C8×Dic118—24no—
29C2×C4⋊(C59⋊C8)C2211yescharacteristic
30C2×Dic59⋊C8C2211yescharacteristic
31C2×Dic59⋊C8C2211yescharacteristic
32C2×#14273652C2211yescharacteristic
33C2×#14273652C2211yescharacteristic
34C2×#14273651C2211yescharacteristic
35C2×#14273651C2211yescharacteristic
36C2×#14273650C2211yescharacteristic
37C2×#14273650C2211yescharacteristic
38C2×#14273648C2211yescharacteristic
39C2×#14273648C2211yescharacteristic
40C2×C8×Dic59C2211yescharacteristic
41C2×C8×Dic59C2211yescharacteristic
42#16837306—21no—
43#16837287—21no—
44#16837284—21no—
45#16837271C2211yescharacteristic
46#16837267C2211yescharacteristic
47C4⋊C4×Dic59C2211yescharacteristic
48C4⋊C4×Dic59C2211yescharacteristic
49C4×C4⋊Dic59C2211yescharacteristic
50#16837253—21no—
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53#16837164C2211yescharacteristic
54#16837164C2211yescharacteristic
55C2×#124×C59C2211yescharacteristic
56C2×#103×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C2×#102×C59C2211yescharacteristic
59C2×#102×C59C2211yescharacteristic
60C2×C4×C8×C59C2211yescharacteristic
61C59×#382C2211yescharacteristic
62C22×Dic118D411yescharacteristic
63C22×Dic118C2311yescharacteristic
64C23×Dic59—21no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273725—22no—
68#14273725C2×C414yes—
69#14273720—28no—
70#14273720—28no—
71#14273714—22no—
72#14273714C2×C414yes—
73#14273704—28no—
74Q8×Dic59—22no—
75Q8×Dic59C2×C414yes—
76C4×Dic118—22no—
77C4×Dic118C2×C414yes—
78C4×Dic118C2×C414yes—
79C4×Dic118D414yes—
80D4×Dic59—22no—
81D4×Dic59C2×C414yes—
82#14273683—28no—
83C2×C4⋊Dic59—21no—
84C2×C4⋊Dic59D411yescharacteristic
85C2×C4⋊Dic59C2311yescharacteristic
86C2×C4⋊Dic59C2311yescharacteristic
87C2×C4⋊Dic59C2311yescharacteristic
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4—21no—
90C2×Dic59⋊C4—21no—
91C2×Dic59⋊C4—21no—
92C2×Dic59⋊C4—21no—
93C2×Dic59⋊C4D411yescharacteristic
94C2×Dic59⋊C4D411yescharacteristic
95C2×C22⋊Dic59—21no—
96C2×C22⋊Dic59—21no—
97C2×C22⋊Dic59—21no—
98C2×C4×Dic59—21no—
99C2×C4×Dic59—21no—
100C2×C4×Dic59—21no—
101C2×C4×Dic59—21no—
102C2×C4×Dic59—21no—
103C2×C4×Dic59C2311yescharacteristic
104C2×C4×Dic59C2311yescharacteristic
105C4⋊(C59⋊C8)—22no—
106Dic59⋊C8—22no—
107Dic59⋊C8—22no—
108#14273652—22no—
109#14273652—22no—
110#14273651—22no—
111#14273651—41no—
112#14273650—41no—
113#14273650—41no—
114#14273648—22no—
115#14273648—41no—
116C8×Dic59—22no—
117C8×Dic59—22no—
118C59⋊#95—21no—
119C59⋊#95—21no—
120C59⋊#95—21no—
121C59⋊#95—21no—
122C59⋊#95—21no—
123C22×Q8×C59C2311yescharacteristic
124C22×D4×C59C2311yescharacteristic
125C22×C8×C59C2311yescharacteristic
126C22×C8×C59C2311yescharacteristic
127#124×C59—22no—
128#124×C59C2×C414yes—
129C2×C4⋊C4×C59C2311yescharacteristic
130C2×C4⋊C4×C59C2311yescharacteristic
131C2×C22⋊C4×C59—21no—
132C2×C22⋊C4×C59—21no—
133C2×C42×C59C2311yescharacteristic
134#103×C59—22no—
135#103×C59—41no—
136#102×C59—22no—
137#102×C59—41no—
138C4×C8×C59—22no—
139#95×C59—21no—
140#95×C59—21no—
141C24×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59—21no—
144C2×Q8×C59—21no—
145C2×Q8×C59—21no—
146C2×Q8×C59—21no—
147C2×Q8×C59C22×C412yes—
148C2×D4×C59—21no—
149C2×D4×C59—21no—
150C2×D4×C59—21no—
151C2×D4×C59—21no—
152C2×D4×C59—21no—
153C2×D4×C59C22×C412yes—
154C22×C4×C59—21no—
155C22×C4×C59—21no—
156C22×C4×C59—21no—
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2411yescharacteristic
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—24no—
164C22×Dic59—21no—
165C22×Dic59—42no—
166C22×Dic59—42no—
167C22×Dic59C2×D411yescharacteristic
168C22×Dic59C2×D411yescharacteristic
169C22×Dic59C2×D411yescharacteristic
170C22×Dic59C2×D411yescharacteristic
171C2×Dic118—22no—
172C2×Dic118—22no—
173C2×Dic118—21no—
174C2×Dic118—21no—
175C2×Dic118—21no—
176C2×Dic118—21no—
177C2×Dic118—21no—
178C2×Dic118C22⋊C414yes—
179C2×Dic118C22×C412yes—
180C2×C8×C59—21no—
181C2×C8×C59—21no—
182C2×C8×C59—21no—
183C2×C8×C59—21no—
184C2×C8×C59—21no—
185C2×C8×C59#4112yes—
186C4⋊C4×C59—21no—
187C4⋊C4×C59—21no—
188C4⋊C4×C59—21no—
189C4⋊C4×C59—21no—
190C22⋊C4×C59—24no—
191C22⋊C4×C59—24no—
192C42×C59C22×C412yes—
193C42×C59C2×D412yes—
194C22⋊Dic59—24no—
195C22⋊Dic59—24no—
196C22⋊Dic59—24no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—24no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59C22⋊C414yes—
203C4⋊Dic59C22×C412yes—
204C4⋊Dic59#4112yes—
205Dic59⋊C4—24no—
206Dic59⋊C4—24no—
207Dic59⋊C4—24no—
208Dic59⋊C4—42no—
209Dic59⋊C4—81no—
210Dic59⋊C4C22⋊C414yes—
211Dic59⋊C4C22⋊C414yes—
212C4×Dic59—24no—
213C4×Dic59—24no—
214C4×Dic59—42no—
215C4×Dic59—81no—
216C4×Dic59—81no—
217C4×Dic59C22×C412yes—
218C4×Dic59C22×C412yes—
219C4×Dic59C2×D412yes—
220C4×Dic59C2×D412yes—
221C2×C59⋊C8—21no—
222C2×C59⋊C8—21no—
223C2×C59⋊C8—21no—
224C2×C59⋊C8—21no—
225C2×C59⋊C8—21no—
226C2×C59⋊C8—21no—
227C23×C59—22no—
228C23×C59—41no—
229C23×C59—41no—
230C23×C59—41no—
231C23×C59—41no—
232C23×C59—41no—
233C23×C59—41no—
234C23×C59C22×D411yescharacteristic
235Q8×C59—22no—
236Q8×C59—22no—
237Q8×C59—41no—
238Q8×C59—41no—
239D4×C59—22no—
240D4×C59—22no—
241D4×C59—41no—
242D4×C59—41no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—22no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59C2×C22⋊C412yes—
257C2×C4×C59C2×C22⋊C412yes—
258C2×C4×C59C23×C411yescharacteristic
259C2×C4×C59C22×D411yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C2×C4×C59H(5,𝔽2)11yescharacteristic
263C2×C4×C592−1+411yescharacteristic
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—21no—
267C2×Dic59—21no—
268C2×Dic59—21no—
269C2×Dic59—21no—
270C2×Dic59—22no—
271C2×Dic59—22no—
272C2×Dic59—22no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—48no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59C2×C22⋊C412yes—
284C2×Dic59C2×C22⋊C412yes—
285C2×Dic59C2×C22⋊C412yes—
286C2×Dic59C2×C22⋊C412yes—
287C2×Dic59#13512yes—
288C2×Dic59#13512yes—
289Dic118—28no—
290Dic118—22no—
291Dic118—22no—
292Dic118—41no—
293Dic118—41no—
294C8×C59—22no—
295C8×C59—41no—
296C59⋊C8—41no—
297C59⋊C8—41no—
298#22541—591nomaximal, Sylow
299C22×C59—42no—
300C22×C59—42no—
301C22×C59—42no—
302C22×C59—81no—
303C22×C59—81no—
304C22×C59—81no—
305C22×C59—81no—
306C22×C59C22×C22⋊C411yescharacteristic
307C22×C59#54811yescharacteristic
308C22×C59#55011yescharacteristic
309C22×C59C2×#13511yescharacteristic
310C22×C59C2×#13511yescharacteristic
311C22×C59#57711yescharacteristic
312C22×C59#57811yescharacteristic
313C4×C59—22no—
314C4×C59—22no—
315C4×C59—42no—
316C4×C59—81no—
317C4×C59—81no—
318C4×C59C4×#4111yescharacteristic
319C4×C59#52011yescharacteristic
320C4×C59#54811yescharacteristic
321C4×C59#55011yescharacteristic
322Dic59—22no—
323Dic59—22no—
324Dic59—22no—
325Dic59—42no—
326Dic59—42no—
327Dic59—42no—
328Dic59—81no—
329Dic59#41614yes—
330C2×#556—591no—
331C2×C8×Q8—591no—
332#2355—591no—
333#1983—591no—
334#1983—591no—
335#1981—591no—
336#1958—591no—
337#1958D5911yescharacteristic
338#1956—591no—
339#1805—591no—
340#1805—591no—
341#1802—591no—
342C4×#103—591no—
343C4×#103—591no—
344C4×#102—591no—
345C2×C59—82no—
346C2×C59#234111yescharacteristic
347C2×C59C2×#41611yescharacteristic
348C2×C59#289811yescharacteristic
349C2×C59#336011yescharacteristic
350C2×C59#336411yescharacteristic
351C2×C59#339311yescharacteristic
352C2×C59#339411yescharacteristic
353#556—598no—
354#556—1184no—
355C2×C4⋊Q8—591no—
356C2×#126—1181no—
357C2×#126—1181no—
358C2×#124—591no—
359C2×#124C2×D5911yescharacteristic
360C2×C22⋊Q8—1181no—
361C2×C4×Q8—591no—
362C2×C4×Q8—591no—
363C2×C4×Q8—591no—
364C2×C4×D4—591no—
365C2×#117—1181no—
366C8×Q8—1184no—
367C2×C4⋊C8—591no—
368C2×C4⋊C8—591no—
369C2×C4⋊C8—591no—
370C2×#103—591no—
371C2×#103—591no—
372C2×#103—591no—
373C2×#103—591no—
374C2×#103—591no—
375C2×#103—591no—
376C2×#103C2×D5911yescharacteristic
377C2×#103C2×D5911yescharacteristic
378C2×#102—591no—
379C2×#102—591no—
380C2×#102C2×D5911yescharacteristic
381C2×#102C2×D5911yescharacteristic
382C2×C4×C8—591no—
383C2×C4×C8—591no—
384C2×C4×C8C2×D5911yescharacteristic
385#391—1181no—
386#387—1181no—
387#387—1181no—
388#384—591no—
389#382—591no—
390#382C2×D5911yescharacteristic
391C4×C4⋊C4—591no—
392C4×C4⋊C4—591no—
393C4×C4⋊C4—591no—
394C4×C4⋊C4—1181no—
395#340—591no—
396#340—591no—
397#340—591no—
398#340—591no—
399C59#2254111yescharacteristic, complemented, Sylow
400C22×Q8—591no—
401C22×Q8—591no—
402C22×Q8C22×D5911yescharacteristic
403C22×D4C22×D5911yescharacteristic
404C23×C4—1181no—
405C22×C8—591no—
406C22×C8—591no—
407C22×C8C22×D5911yescharacteristic
408C22×C8C22×D5911yescharacteristic
409C4⋊Q8—594no—
410C4⋊Q8—1182no—
411#126—1188no—
412#126—1188no—
413#124—22no—
414#124—594no—
415#124—1182no—
416#124C4×D5914yes—
417C22⋊Q8—1188no—
418C4×Q8—594no—
419C4×Q8—594no—
420C4×Q8—594no—
421C4×Q8—594no—
422C4×Q8—1182no—
423C4×Q8—1182no—
424C4×D4—594no—
425C4×D4—1182no—
426#117—1188no—
427C2×C4⋊C4—591no—
428C2×C4⋊C4—591no—
429C2×C4⋊C4—591no—
430C2×C4⋊C4—591no—
431C2×C4⋊C4—591no—
432C2×C4⋊C4—591no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4—1181no—
435C2×C4⋊C4—1181no—
436C2×C4⋊C4—1181no—
437C2×C4⋊C4—1181no—
438C2×C4⋊C4—1181no—
439C2×C4⋊C4C22×D5911yescharacteristic
440C2×C4⋊C4C22×D5911yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1181no—
444C2×C22⋊C4—1181no—
445C2×C22⋊C4—1181no—
446C2×C42—591no—
447C2×C42—591no—
448C2×C42—1181no—
449C2×C42—1181no—
450C2×C42—1181no—
451C2×C42—1181no—
452C2×C42—1181no—
453C2×C42C22×D5911yescharacteristic
454C4⋊C8—1182no—
455C4⋊C8—1182no—
456C4⋊C8—1182no—
457#103—22no—
458#103—41no—
459#103—1182no—
460#103—1182no—
461#103—1182no—
462#103—2361no—
463#103—2361no—
464#103—2361no—
465#102—22no—
466#102—41no—
467#102—1182no—
468#102—2361no—
469C4×C8—22no—
470C4×C8—1182no—
471C4×C8—1182no—
472#95—21no—
473#95—21no—
474#95—1181no—
475#95—1181no—
476#95—1181no—
477#95—1181no—
478#95—1181no—
479C24—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—594no—
486C2×Q8—592no—
487C2×Q8—1182no—
488C2×Q8—1182no—
489C2×Q8—1181no—
490C2×Q8—1181no—
491C2×Q8—1181no—
492C2×Q8—1181no—
493C2×Q8—1181no—
494C2×Q8C2×C4×D5912yes—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4C2×C4×D5912yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—591no—
505C22×C4—591no—
506C22×C4—591no—
507C22×C4—591no—
508C22×C4—1184no—
509C22×C4—1181no—
510C22×C4—1181no—
511C22×C4—1181no—
512C22×C4—1181no—
513C22×C4—2362no—
514C22×C4—2362no—
515C22×C4D4×D5911yescharacteristic
516C22×C4D4×D5911yescharacteristic
517C22×C4C23×D5911yescharacteristic
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—1181no—
524C2×C8—1181no—
525C2×C8—1181no—
526C2×C8—1181no—
527C2×C8—1181no—
528C2×C8—1181no—
529C2×C8#1174568712yes—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—594no—
535C4⋊C4—594no—
536C4⋊C4—594no—
537C4⋊C4—592no—
538C4⋊C4—592no—
539C4⋊C4—1184no—
540C4⋊C4—1184no—
541C4⋊C4—1184no—
542C4⋊C4—1184no—
543C4⋊C4—1181no—
544C4⋊C4—1181no—
545C4⋊C4—1181no—
546C4⋊C4—1181no—
547C4⋊C4—4721no—
548C4⋊C4—2362no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—1184no—
552C22⋊C4—1184no—
553C22⋊C4—1184no—
554C42—592no—
555C42—592no—
556C42—592no—
557C42—592no—
558C42—1184no—
559C42—1184no—
560C42—4721no—
561C42—4721no—
562C42—2362no—
563C42C2×C4×D5912yes—
564C42D4×D5912yes—
565C23—22no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23C2×D4×D5911yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—41no—
576Q8—41no—
577Q8—1188no—
578Q8—1182no—
579Q8—1182no—
580Q8—2361no—
581Q8—2361no—
582D4—22no—
583D4—22no—
584D4—41no—
585D4—41no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—592no—
600C2×C4—592no—
601C2×C4—592no—
602C2×C4—592no—
603C2×C4—592no—
604C2×C4—592no—
605C2×C4—1181no—
606C2×C4—1181no—
607C2×C4—1181no—
608C2×C4—1181no—
609C2×C4—1181no—
610C2×C4—1181no—
611C2×C4—1182no—
612C2×C4—1182no—
613C2×C4—1182no—
614C2×C4—2368no—
615C2×C4—2361no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4—2361no—
620C2×C4—2361no—
621C2×C4—2361no—
622C2×C4—2361no—
623C2×C4—2361no—
624C2×C4C22⋊C4×D5912yes—
625C2×C4C22⋊C4×D5912yes—
626C2×C4C22×C4×D5911yescharacteristic
627C2×C4C2×D4×D5911yescharacteristic
628C2×C4#41×D5911yescharacteristic
629C2×C4C2×#1174568711yescharacteristic
630C2×C4#1427377211yescharacteristic
631C2×C4#1427377411yescharacteristic
632C8—22no—
633C8—41no—
634C8—2361no—
635C8—2361no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22C2×C22⋊C4×D5911yescharacteristic
644C22#1683806011yescharacteristic
645C22#1683807211yescharacteristic
646C22#135×D5911yescharacteristic
647C22#135×D5911yescharacteristic
648C22#1683820911yescharacteristic
649C22#1683821611yescharacteristic
650C4—22no—
651C4—22no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—594no—
656C4—1182no—
657C4—1182no—
658C4—1182no—
659C4—4721no—
660C4—2362no—
661C4—2362no—
662C4—2362no—
663C4C4×#1174568711yescharacteristic
664C4#1683788111yescharacteristic
665C4#1683805811yescharacteristic
666C4#1683807011yescharacteristic
667C2—82no—
668C2#2142693511yescharacteristic
669C2#416×D5911yescharacteristic
670C2#2143262511yescharacteristic
671C2#2143674211yescharacteristic
672C2#2143677811yescharacteristic
673C2#2143708111yescharacteristic
674C2#2143708911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1958C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#1958D59
hypocenterC59#22541
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×C8×Dic118C2semidirect
3. #21427048C2semidirect
6. #21425017C2semidirect
8. #21424896C2semidirect
9. #21424109C2semidirect
10. #21424108C2semidirect
12. #103×Dic59C2semidirect
13. C4×#14273652C2semidirect
17. #16838121C4semidirect
399. C59#22541semidirect

Maximal subgroups

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

subgroupindex
C2×#168381212
C2×C8×Dic1182
#214270482
#214250292
#214250272
#214250172
#214249062
#214248962
#214241092
#214241082
#214240942
#103×Dic592
C4×#142736522
#102×Dic592
C59×#19582
#2254159