Finite Group Databasebeta

← #28510678

Subgroups — #28510678

Order 15104.

2222 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D59C211yesmaximal, characteristic
2C2×#16837267C211yesmaximal, characteristic, complemented
3C2×#16837259C211yesmaximal, characteristic
4C2×C4×(C2×D59)⋊C4C211yesmaximal, characteristic
5#21422354C214yesmaximal, complemented
6#21422326C211yesmaximal, characteristic, complemented
7#21422325C211yesmaximal, characteristic, complemented
8(C2×D59)⋊#95C214yesmaximal
9C2×C59×#382C211yesmaximal, characteristic, complemented
10C23×C4×D59C2212yes—
11C23×C4×D59C2211yescharacteristic
12C22×C4⋊Dic59C2211yescharacteristic
13C22×C4⋊Dic59C2211yescharacteristic
14C22×Dic59⋊C4C2212yes—
15C2×C4⋊C4×D59—24no—
16C2×C4⋊C4×D59—24no—
17C2×C4⋊C4×D59—24no—
18C22×(C2×D59)⋊C4C2212yes—
19C22×(C2×D59)⋊C4C2212yes—
20C22×C4×Dic59C2211yescharacteristic
21#16837267—24no—
22#16837259—24no—
23C4×(C2×D59)⋊C4C418yescomplemented
24C2×C59⋊#95C2212yes—
25C2×C59⋊#95C2214yes—
26C2×C59⋊#95C2214yes—
27C2×C59⋊#95C2214yes—
28C2×C59⋊#95C2212yes—
29C22×C4⋊C4×C59C2211yescharacteristic
30C22×C4⋊C4×C59C2211yescharacteristic
31C22×C42×C59C2211yescharacteristic
32C59×#382—24no—
33C2×#95×C59C2214yes—
34C24×D59C2311yescharacteristic
35C23×Dic59C2312yes—
36C23×Dic59C2312yes—
37C23×Dic59C2312yes—
38C23×Dic59C2311yescharacteristic
39C22×C4×D59—22no—
40C22×C4×D59—22no—
41C22×C4×D59—22no—
42C22×C4×D59—22no—
43C22×C4×D59—24no—
44C22×C4×D59—24no—
45C22×C4×D59—24no—
46C22×C4×D59—28no—
47C22×C4×D59—44no—
48C22×C4×D59C2×C414yes—
49C22×C4×D59D414yes—
50C22×C4×D59Q814yes—
51C2×C4⋊Dic59—22no—
52C2×C4⋊Dic59—22no—
53C2×C4⋊Dic59—22no—
54C2×C4⋊Dic59—22no—
55C2×C4⋊Dic59—22no—
56C2×C4⋊Dic59D414yes—
57C2×Dic59⋊C4—24no—
58C2×Dic59⋊C4—24no—
59C2×Dic59⋊C4—24no—
60C4⋊C4×D59—232no—
61C2×(C2×D59)⋊C4—24no—
62C2×(C2×D59)⋊C4—24no—
63C2×(C2×D59)⋊C4—24no—
64C2×(C2×D59)⋊C4—28no—
65C2×(C2×D59)⋊C4C2×C418yes—
66C2×C4×Dic59—22no—
67C2×C4×Dic59—22no—
68C2×C4×Dic59C2×C414yescomplemented
69C59⋊#95—24no—
70C59⋊#95—28no—
71C59⋊#95—44no—
72C59⋊#95—44no—
73C59⋊#95C2×C418yescomplemented
74C23×C4×C59C2312yes—
75C23×C4×C59C2312yes—
76C23×C4×C59C2312yes—
77C23×C4×C59C2311yescharacteristic
78C2×C4⋊C4×C59—22no—
79C2×C4⋊C4×C59—22no—
80C2×C4⋊C4×C59—22no—
81C2×C4⋊C4×C59—22no—
82C2×C4⋊C4×C59—22no—
83C2×C4⋊C4×C59D414yes—
84C2×C42×C59—22no—
85C2×C42×C59—22no—
86C2×C42×C59C2×C414yescomplemented
87#95×C59—28no—
88C24×C59C2411yescharacteristic
89C23×D59—21no—
90C23×D59—21no—
91C23×D59—21no—
92C23×D59—21no—
93C23×D59—22no—
94C23×D59—22no—
95C23×D59C22×C412yes—
96C23×D59C2×D412yes—
97C23×D59C2×Q812yes—
98C23×D59#4114yes—
99C23×D59#4114yes—
100C22×C4×C59—22no—
101C22×C4×C59—22no—
102C22×C4×C59—22no—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—24no—
107C22×C4×C59—22no—
108C22×C4×C59—24no—
109C22×C4×C59—22no—
110C22×C4×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—24no—
114C22×C4×C59—24no—
115C22×C4×C59C22×C414yes—
116C22×C4×C59C22×C412yes—
117C22×C4×C59C2×D414yes—
118C22×C4×C59C2×D412yes—
119C22×C4×C59C2×D412yes—
120C22×C4×C59C2×D412yes—
121C22×C4×C59C2×Q812yes—
122C22×C4×C59#4112yes—
123C22×C4×C59#4112yes—
124C22×Dic59—21no—
125C22×Dic59—21no—
126C22×Dic59—21no—
127C22×Dic59—21no—
128C22×Dic59—22no—
129C22×Dic59—22no—
130C22×Dic59—22no—
131C22×Dic59—22no—
132C22×Dic59—22no—
133C22×Dic59—22no—
134C22×Dic59—24no—
135C22×Dic59—22no—
136C22×Dic59—24no—
137C22×Dic59—22no—
138C22×Dic59—22no—
139C22×Dic59—24no—
140C22×Dic59—22no—
141C22×Dic59—22no—
142C22×Dic59—24no—
143C22×Dic59C22×C414yes—
144C22×Dic59C22×C412yes—
145C22×Dic59C2×D414yes—
146C22×Dic59C2×D412yes—
147C22×Dic59C2×Q812yes—
148C2×C4×D59—28no—
149C2×C4×D59—28no—
150C2×C4×D59—28no—
151C2×C4×D59—28no—
152C2×C4×D59—28no—
153C2×C4×D59—28no—
154C2×C4×D59—216no—
155C2×C4×D59—416no—
156C2×C4×D59—416no—
157C2×C4×D59—48no—
158C2×C4×D59—48no—
159C2×C4×D59C4⋊C4116yes—
160C4⋊C4×C59—28no—
161C4⋊C4×C59—44no—
162C42×C59—28no—
163(C2×D59)⋊C4—216no—
164(C2×D59)⋊C4—48no—
165C4⋊Dic59—28no—
166C4⋊Dic59—44no—
167Dic59⋊C4—216no—
168C4×Dic59—28no—
169C23×C59C23×C411yescharacteristic
170C23×C59C22×D411yescharacteristic
171C23×C59C22×D411yescharacteristic
172C23×C59C22×D411yescharacteristic
173C23×C59C22×Q811yescharacteristic
174C23×C59C2×#4112yes—
175C23×C59C2×#4112yes—
176C23×C59C2×#4111yescharacteristic
177C23×C59C2×#4111yescharacteristic
178C23×C59H(5,𝔽2)12yes—
179C23×C592−1+412yes—
180C22×D59—22no—
181C22×D59—22no—
182C22×D59—24no—
183C22×D59—22no—
184C22×D59—22no—
185C22×D59—22no—
186C22×D59—22no—
187C22×D59—22no—
188C22×D59—22no—
189C22×D59—24no—
190C22×D59—24no—
191C22×D59—24no—
192C22×D59—24no—
193C22×D59—24no—
194C22×D59—24no—
195C22×D59—24no—
196C22×D59—24no—
197C22×D59—24no—
198C22×D59C2×C4⋊C414yes—
199C22×D59#11718yes—
200C22×D59#12414yes—
201C22×D59#12414yes—
202C22×D59#12518yes—
203C2×C4×C59—24no—
204C2×C4×C59—24no—
205C2×C4×C59—24no—
206C2×C4×C59—24no—
207C2×C4×C59—28no—
208C2×C4×C59—42no—
209C2×C4×C59—42no—
210C2×C4×C59—44no—
211C2×C4×C59—42no—
212C2×C4×C59—42no—
213C2×C4×C59—42no—
214C2×C4×C59—42no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59—44no—
219C2×C4×C59—44no—
220C2×C4×C59C2×C4⋊C414yes—
221C2×C4×C59C4×D414yes—
222C2×C4×C59C4×D414yes—
223C2×C4×C59#12014yes—
224C2×C4×C59C22⋊Q814yes—
225C2×C4×C59C22⋊Q814yes—
226C2×C4×C59#12314yes—
227C2×Dic59—22no—
228C2×Dic59—22no—
229C2×Dic59—22no—
230C2×Dic59—22no—
231C2×Dic59—22no—
232C2×Dic59—22no—
233C2×Dic59—24no—
234C2×Dic59—24no—
235C2×Dic59—24no—
236C2×Dic59—24no—
237C2×Dic59—28no—
238C2×Dic59—42no—
239C2×Dic59—42no—
240C2×Dic59—44no—
241C2×Dic59—42no—
242C2×Dic59—42no—
243C2×Dic59—42no—
244C2×Dic59—42no—
245C2×Dic59—42no—
246C2×Dic59—42no—
247C2×Dic59—42no—
248C2×Dic59—44no—
249C2×Dic59—44no—
250C2×Dic59C2×C4⋊C414yes—
251C4×D59—232no—
252C4×D59—432no—
253#13881—591nomaximal, Sylow
254C22×C59C22×C4⋊C411yescharacteristic
255C22×C59C2×#11712yes—
256C22×C59C2×C4×D411yescharacteristic
257C22×C59C2×C4×D411yescharacteristic
258C22×C59#52012yes—
259C22×C59C2×#12011yescharacteristic
260C22×C59C2×C22⋊Q811yescharacteristic
261C22×C59C2×C22⋊Q811yescharacteristic
262C22×C59C2×#12311yescharacteristic
263C22×C59C2×#12411yescharacteristic
264C22×C59C2×#12411yescharacteristic
265C22×C59C2×#12512yes—
266C22×C59#53511yescharacteristic
267C22×C59#53611yescharacteristic
268C22×C59#54412yes—
269C22×C59#54612yes—
270C22×C59#54612yes—
271C22×C59#54812yes—
272C22×C59#54812yes—
273C22×C59#55212yes—
274C22×C59#55212yes—
275C22×C59#55312yes—
276C22×C59#55312yes—
277C2×D59—24no—
278C2×D59—28no—
279C2×D59—28no—
280C2×D59—24no—
281C2×D59—24no—
282C2×D59—28no—
283C2×D59—28no—
284C2×D59—24no—
285C2×D59—24no—
286C2×D59—24no—
287C2×D59#38218yescomplemented
288C4×C59—44no—
289C4×C59—44no—
290C4×C59—82no—
291C4×C59#38518yes—
292Dic59—24no—
293Dic59—44no—
294Dic59—44no—
295Dic59—82no—
296C23×C4⋊C4—591no—
297C2×#382—591no—
298C2×#382D5911yescharacteristic, complemented
299C2×#380—591no—
300C2×C4×C22⋊C4—591no—
301#1442—594no—
302C4⋊#95—591no—
303C4⋊#95—591no—
304C22⋊#95—594no—
305C2×C59C2×#38211yescharacteristic
306C2×C59C2×#38511yescharacteristic
307C2×C59#231711yescharacteristic
308C2×C59#235212yes—
309C2×C59#235212yes—
310C2×C59#243511yescharacteristic
311C2×C59#243611yescharacteristic
312C2×C59#244212yes—
313C2×C59#244212yes—
314C2×C59#250311yescharacteristic
315C2×C59#250411yescharacteristic
316D59—28no—
317C24×C4—591no—
318C24×C4—592no—
319C22×C4⋊C4—591no—
320C22×C4⋊C4—592no—
321C22×C4⋊C4—591no—
322C22×C4⋊C4—1184no—
323C22×C4⋊C4—1184no—
324C22×C4⋊C4—1184no—
325C22×C4⋊C4C2×D5911yescharacteristic
326C22×C4⋊C4C2×D5911yescharacteristic
327C22×C22⋊C4—592no—
328C22×C22⋊C4—592no—
329C22×C42—591no—
330C22×C42C2×D5911yescharacteristic
331#382—24no—
332#382—1184no—
333#380—1184no—
334C4×C22⋊C4—598no—
335C2×#95—592no—
336C2×#95—594no—
337C2×#95—592no—
338C2×#95—594no—
339C2×#95—594no—
340C2×#95C2×D5914yes—
341C59#1388111yescharacteristic, complemented, Sylow
342C25—591no—
343C23×C4—591no—
344C23×C4—594no—
345C23×C4—594no—
346C23×C4—594no—
347C23×C4—592no—
348C23×C4—592no—
349C23×C4—592no—
350C23×C4—1184no—
351C23×C4—1184no—
352C23×C4—1188no—
353C23×C4—1182no—
354C23×C4—1182no—
355C23×C4—1182no—
356C23×C4—1182no—
357C23×C4—1184no—
358C23×C4—2364no—
359C23×C4C22×D5912yes—
360C23×C4C22×D5912yes—
361C23×C4C22×D5912yes—
362C23×C4C22×D5911yescharacteristic
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—594no—
369C2×C4⋊C4—11832no—
370C2×C4⋊C4—1184no—
371C2×C4⋊C4—1184no—
372C2×C4⋊C4—1182no—
373C2×C4⋊C4—1184no—
374C2×C4⋊C4—1182no—
375C2×C4⋊C4—1182no—
376C2×C4⋊C4—1182no—
377C2×C4⋊C4—1182no—
378C2×C4⋊C4C22⋊D5914yes—
379C2×C22⋊C4—598no—
380C2×C22⋊C4—1184no—
381C2×C22⋊C4—1184no—
382C2×C22⋊C4—1184no—
383C2×C22⋊C4—1188no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—594no—
387C2×C42—1182no—
388C2×C42—1182no—
389C2×C42C4×D5914yescomplemented
390#95—28no—
391#95—598no—
392#95—1184no—
393#95—1188no—
394#95—2364no—
395#95—2364no—
396C24—592no—
397C24—592no—
398C24—592no—
399C24—594no—
400C24—594no—
401C24—1181no—
402C24—1181no—
403C24—1181no—
404C24—1181no—
405C24—1182no—
406C24—1182no—
407C24C23×D5911yescharacteristic
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—592no—
424C22×C4—592no—
425C22×C4—592no—
426C22×C4—5916no—
427C22×C4—594no—
428C22×C4—594no—
429C22×C4—11816no—
430C22×C4—1181no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1188no—
435C22×C4—1188no—
436C22×C4—1188no—
437C22×C4—1188no—
438C22×C4—1188no—
439C22×C4—1188no—
440C22×C4—1182no—
441C22×C4—1182no—
442C22×C4—1182no—
443C22×C4—1182no—
444C22×C4—1182no—
445C22×C4—1184no—
446C22×C4—1182no—
447C22×C4—1182no—
448C22×C4—1184no—
449C22×C4—1182no—
450C22×C4—1182no—
451C22×C4—1184no—
452C22×C4—1182no—
453C22×C4—1182no—
454C22×C4—1184no—
455C22×C4—23616no—
456C22×C4—23616no—
457C22×C4—2368no—
458C22×C4—2368no—
459C22×C4C2×C4×D5914yes—
460C22×C4C2×C4×D5912yes—
461C22×C4#1174568512yes—
462C22×C4D4×D5912yes—
463C22×C4D4×D5912yes—
464C22×C4Q8×D5912yes—
465C22×C4#1174568912yes—
466C22×C4C2×C22⋊D5914yes—
467C22×C4C2×C22⋊D5912yes—
468C4⋊C4—28no—
469C4⋊C4—44no—
470C4⋊C4—11816no—
471C4⋊C4—1188no—
472C4⋊C4—2364no—
473C22⋊C4—11816no—
474C22⋊C4—2368no—
475C42—28no—
476C42—1188no—
477C23—594no—
478C23—594no—
479C23—598no—
480C23—594no—
481C23—598no—
482C23—1182no—
483C23—1182no—
484C23—1182no—
485C23—1182no—
486C23—1182no—
487C23—1184no—
488C23—1184no—
489C23—1184no—
490C23—1184no—
491C23—1184no—
492C23—1184no—
493C23—1184no—
494C23—1184no—
495C23—1184no—
496C23—1182no—
497C23—1182no—
498C23—1184no—
499C23—1182no—
500C23C22×C4×D5911yescharacteristic
501C23C2×D4×D5911yescharacteristic
502C23C2×D4×D5911yescharacteristic
503C23C22×C22⋊D5911yescharacteristic
504C23C2×Q8×D5911yescharacteristic
505C23#41×D5912yes—
506C23#41×D5912yes—
507C23C2×#1174568511yescharacteristic
508C23C2×#1174568911yescharacteristic
509C23#1427377312yes—
510C23#1427377412yes—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—594no—
529C2×C4—1182no—
530C2×C4—1182no—
531C2×C4—1182no—
532C2×C4—1182no—
533C2×C4—1182no—
534C2×C4—1182no—
535C2×C4—11832no—
536C2×C4—1188no—
537C2×C4—1184no—
538C2×C4—1184no—
539C2×C4—1184no—
540C2×C4—1184no—
541C2×C4—23632no—
542C2×C4—2362no—
543C2×C4—2362no—
544C2×C4—2362no—
545C2×C4—2364no—
546C2×C4—2364no—
547C2×C4—2362no—
548C2×C4—2362no—
549C2×C4—2364no—
550C2×C4—2362no—
551C2×C4—2362no—
552C2×C4—2362no—
553C2×C4—2362no—
554C2×C4C4⋊C4×D5914yes—
555C2×C4C4×C22⋊D5914yes—
556C2×C4#1427368914yes—
557C2×C4#1427369514yes—
558C2×C4#1427370314yes—
559C2×C4#1427370514yes—
560C2×C4#1427371214yes—
561C22—598no—
562C22—1184no—
563C22—1184no—
564C22—1184no—
565C22—1184no—
566C22—1188no—
567C22—1188no—
568C22—1184no—
569C22—1188no—
570C22—1184no—
571C22—1188no—
572C22C2×C4⋊C4×D5911yescharacteristic
573C22#117×D5912yes—
574C22C2×C4×C22⋊D5911yescharacteristic
575C22C2×#1427368911yescharacteristic
576C22#1683788212yes—
577C22C2×#1427369511yescharacteristic
578C22C2×#1427370311yescharacteristic
579C22C2×#1427370511yescharacteristic
580C22C2×#1427371211yescharacteristic
581C22#124×D5911yescharacteristic
582C22#124×D5911yescharacteristic
583C22#125×D5912yes—
584C22#1683797011yescharacteristic
585C22#1683797211yescharacteristic
586C22#1683803212yes—
587C22#1683804012yes—
588C22#1683804512yes—
589C22#1683805912yes—
590C22#1683806012yes—
591C22#1683809012yes—
592C22#1683809112yes—
593C22#1683809612yes—
594C22#1683810512yes—
595C4—44no—
596C4—44no—
597C4—82no—
598C4—1184no—
599C4—4722no—
600C4—2364no—
601C4—2364no—
602C4#1683727318yes—
603C2—1188no—
604C2#382×D5911yescharacteristic
605C2C2×#1683727311yescharacteristic
606C2#2142672111yescharacteristic
607C2#2142702212yes—
608C2#2142702612yes—
609C2#2142774411yescharacteristic
610C2#2142775111yescharacteristic
611C2#2142781612yes—
612C2#2142782312yes—
613C2#2142857811yescharacteristic
614C2#2142858511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC2×C59×#382C2
socle, cosocleC24×C59C24
prime radicalC24×D59C23
hypercenterC2×#382D59
hypocenterC59#13881
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×#16837267C2semidirect
5. #21422354C2semidirect
6. #21422326C2semidirect
7. #21422325C2semidirect
9. C2×C59×#382C2semidirect
23. C4×(C2×D59)⋊C4C4semidirect
68. C2×C4×Dic59C2×C4semidirect
73. C59⋊#95C2×C4semidirect
86. C2×C42×C59C2×C4semidirect
287. C2×D59#382semidirect
298. C2×#382D59semidirect
341. C59#13881semidirect
389. C2×C42C4×D59semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D592
C2×#168372672
C2×#168372592
C2×C4×(C2×D59)⋊C42
#214223542
#214223262
#214223252
(C2×D59)⋊#952
C2×C59×#3822
#1388159