Finite Group Databasebeta

← #28520664

Subgroups — #28520664

Order 15104.

1982 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837321C211yesmaximal, characteristic, complemented
2C2×#16837297C212yesmaximal, complemented
3C2×#16837275C211yesmaximal, characteristic
4C2×C4×C22⋊Dic59C211yesmaximal, characteristic
5#21422367C211yesmaximal, characteristic, complemented
6#21422359C211yesmaximal, characteristic, complemented
7#21422346C211yesmaximal, characteristic, complemented
8#21422339C212yesmaximal
9#21422339C212yesmaximal
10#21422328C212yesmaximal, complemented
11C22×C22⋊Q8×C59C211yesmaximal, characteristic
12C22×Dic59⋊C4C2212yes—
13C22×Dic59⋊C4C2211yescharacteristic
14C22×Dic59⋊C4C2211yescharacteristic
15C22×Dic59⋊C4C2212yes—
16C22×C22⋊Dic59C2211yescharacteristic
17C22×C22⋊Dic59C2212yes—
18C22×C22⋊Dic59C2211yescharacteristic
19C22×C4×Dic59C2212yes—
20#16837321—24no—
21#16837297—28no—
22#16837275—24no—
23C4×C22⋊Dic59—24no—
24C2×C59⋊#95C2212yes—
25C2×C59⋊#95C2212yes—
26C2×C59⋊#95C2212yes—
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2212yes—
29C2×C59⋊#95C2211yescharacteristic
30C2×C59⋊#95C2211yescharacteristic
31C2×C59⋊#95C2211yescharacteristic
32C2×C59⋊#95C2212yes—
33C2×C59⋊#95C2211yescharacteristic
34C23×Q8×C59C2211yescharacteristic
35C24×C4×C59C2211yescharacteristic
36C2×C22⋊Q8×C59—24no—
37C2×C22⋊Q8×C59—28no—
38C22×C4⋊C4×C59C2211yescharacteristic
39C22×C4⋊C4×C59C2212yes—
40C22×C22⋊C4×C59C2212yes—
41C23×Dic59C2311yescharacteristic
42C23×Dic59C2311yescharacteristic
43C23×Dic59C2312yes—
44C23×Dic59C2311yescharacteristic
45C23×Dic59C2311yescharacteristic
46C23×Dic59C2312yes—
47C2×Dic59⋊C4—24no—
48C2×Dic59⋊C4—24no—
49C2×Dic59⋊C4—24no—
50C2×Dic59⋊C4—22no—
51C2×Dic59⋊C4—22no—
52C2×Dic59⋊C4—24no—
53C2×Dic59⋊C4—24no—
54C2×Dic59⋊C4—22no—
55C2×Dic59⋊C4—22no—
56C2×Dic59⋊C4—24no—
57C2×Dic59⋊C4—42no—
58C2×C22⋊Dic59—22no—
59C2×C22⋊Dic59—24no—
60C2×C22⋊Dic59—22no—
61C2×C22⋊Dic59—22no—
62C2×C22⋊Dic59—24no—
63C2×C22⋊Dic59—22no—
64C2×C22⋊Dic59—42no—
65C2×C22⋊Dic59Q818yescomplemented
66C2×C4×Dic59—24no—
67C2×C4×Dic59—24no—
68C2×C4×Dic59—42no—
69C59⋊#95—22no—
70C59⋊#95—22no—
71C59⋊#95—22no—
72C59⋊#95—24no—
73C59⋊#95—24no—
74C59⋊#95—24no—
75C59⋊#95—24no—
76C59⋊#95—24no—
77C59⋊#95—42no—
78C59⋊#95D414yes—
79C25×C59C2311yescharacteristic
80C22×Q8×C59—24no—
81C22×Q8×C59—22no—
82C22×Q8×C59—22no—
83C22×Q8×C59—22no—
84C22×Q8×C59—42no—
85C23×C4×C59—22no—
86C23×C4×C59—24no—
87C23×C4×C59—22no—
88C23×C4×C59—42no—
89C23×C4×C59D414yes—
90C23×C4×C59C2312yes—
91C23×C4×C59C2312yes—
92C23×C4×C59C2311yescharacteristic
93C23×C4×C59C2311yescharacteristic
94C22⋊Q8×C59—232no—
95C2×C4⋊C4×C59—24no—
96C2×C4⋊C4×C59—22no—
97C2×C4⋊C4×C59—24no—
98C2×C4⋊C4×C59—24no—
99C2×C4⋊C4×C59D418yes—
100C2×C22⋊C4×C59—24no—
101C2×C22⋊C4×C59—24no—
102C2×C22⋊C4×C59—24no—
103C24×C59—22no—
104C24×C59—21no—
105C24×C59—21no—
106C24×C59—22no—
107C24×C59—21no—
108C24×C59—21no—
109C24×C59—22no—
110C24×C59—21no—
111C24×C59—21no—
112C24×C59C2×D412yes—
113C24×C59C2×Q814yes—
114C24×C59C2411yescharacteristic
115C2×Q8×C59—28no—
116C2×Q8×C59—48no—
117C2×Q8×C59—48no—
118C2×Q8×C59—44no—
119C2×Q8×C59—44no—
120C22×C4×C59—22no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59—28no—
138C22×C4×C59—28no—
139C22×C4×C59—44no—
140C22×C4×C59—48no—
141C22×C4×C59—48no—
142C22×C4×C59C2×D414yes—
143C22×C4×C59C2×D414yes—
144C22×C4×C59C2×D412yes—
145C22×C4×C59C2×D414yes—
146C22×C4×C59C2×D412yes—
147C22×C4×C59C2×D412yes—
148C22×C4×C59#4112yes—
149C22×Dic59—22no—
150C22×Dic59—22no—
151C22×Dic59—22no—
152C22×Dic59—22no—
153C22×Dic59—22no—
154C22×Dic59—21no—
155C22×Dic59—22no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—22no—
159C22×Dic59—22no—
160C22×Dic59—22no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—22no—
164C22×Dic59—21no—
165C22×Dic59—22no—
166C22×Dic59—21no—
167C22×Dic59—22no—
168C22×Dic59—21no—
169C22×Dic59—21no—
170C22×Dic59—22no—
171C22×Dic59—21no—
172C22×Dic59—21no—
173C22×Dic59—22no—
174C22×Dic59—21no—
175C22×Dic59—22no—
176C22×Dic59—22no—
177C22×Dic59—21no—
178C22×Dic59—21no—
179C22×Dic59C2×D414yes—
180C22×Dic59C2×Q814yes—
181C22×Dic59C2×Q814yes—
182C22×Dic59#4112yes—
183C22×Dic59#4112yes—
184C22×Dic59#4114yes—
185C4⋊C4×C59—28no—
186C4⋊C4×C59—216no—
187C22⋊C4×C59—216no—
188C22⋊Dic59—28no—
189C22⋊Dic59—216no—
190C22⋊Dic59—44no—
191Dic59⋊C4—216no—
192Dic59⋊C4—48no—
193Dic59⋊C4—44no—
194Dic59⋊C4—44no—
195C4×Dic59—48no—
196C23×C59—22no—
197C23×C59—22no—
198C23×C59—24no—
199C23×C59—22no—
200C23×C59—22no—
201C23×C59—24no—
202C23×C59—22no—
203C23×C59—22no—
204C23×C59—24no—
205C23×C59—24no—
206C23×C59—24no—
207C23×C59—24no—
208C23×C59—42no—
209C23×C59—42no—
210C23×C59—42no—
211C23×C59—42no—
212C23×C59—42no—
213C23×C59—42no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59C4⋊Q814yes—
219C23×C59C22×D411yescharacteristic
220C23×C59C22×D411yescharacteristic
221C23×C59C22×D412yes—
222C23×C59C22×Q812yes—
223C23×C59C2×#4112yes—
224C23×C59C2×#4111yescharacteristic
225C23×C59C2×#4111yescharacteristic
226C23×C59C2×#4111yescharacteristic
227C23×C59H(5,𝔽2)12yes—
228C23×C592−1+411yescharacteristic
229C23×C592−1+411yescharacteristic
230Q8×C59—416no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—24no—
236C2×C4×C59—24no—
237C2×C4×C59—24no—
238C2×C4×C59—24no—
239C2×C4×C59—24no—
240C2×C4×C59—24no—
241C2×C4×C59—24no—
242C2×C4×C59—24no—
243C2×C4×C59—24no—
244C2×C4×C59—24no—
245C2×C4×C59—416no—
246C2×C4×C59—42no—
247C2×C4×C59—42no—
248C2×C4×C59—42no—
249C2×C4×C59—42no—
250C2×C4×C59—42no—
251C2×C4×C59—42no—
252C2×C4×C59—42no—
253C2×C4×C59—42no—
254C2×C4×C59#12014yes—
255C2×C4×C59#12114yes—
256C2×Dic59—24no—
257C2×Dic59—24no—
258C2×Dic59—24no—
259C2×Dic59—24no—
260C2×Dic59—24no—
261C2×Dic59—24no—
262C2×Dic59—24no—
263C2×Dic59—24no—
264C2×Dic59—22no—
265C2×Dic59—22no—
266C2×Dic59—22no—
267C2×Dic59—24no—
268C2×Dic59—22no—
269C2×Dic59—24no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—42no—
274C2×Dic59—42no—
275C2×Dic59—41no—
276C2×Dic59—42no—
277C2×Dic59—42no—
278C2×Dic59—41no—
279C2×Dic59—42no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C2×Dic59—41no—
285C2×Dic59—41no—
286C2×Dic59—41no—
287C2×Dic59—42no—
288C2×Dic59—42no—
289C2×Dic59—42no—
290C2×Dic59—42no—
291C2×Dic59C22⋊Q818yescomplemented
292#14763—591nomaximal, Sylow
293C22×C59—24no—
294C22×C59—24no—
295C22×C59—24no—
296C22×C59—42no—
297C22×C59—42no—
298C22×C59—42no—
299C22×C59—42no—
300C22×C59—44no—
301C22×C59—44no—
302C22×C59—44no—
303C22×C59—44no—
304C22×C59C2×#12011yescharacteristic
305C22×C59C2×#12111yescharacteristic
306C22×C59C2×C22⋊Q812yes—
307C22×C59#52511yescharacteristic
308C22×C59#52512yes—
309C22×C59C2×C4⋊Q811yescharacteristic
310C22×C59#53211yescharacteristic
311C22×C59#53212yes—
312C22×C59#53312yes—
313C22×C59#53411yescharacteristic
314C22×C59#53611yescharacteristic
315C22×C59#53912yes—
316C22×C59#54212yes—
317C22×C59D4212yes—
318C22×C59#54612yes—
319C22×C59#54711yescharacteristic
320C22×C59#54811yescharacteristic
321C22×C59D4×Q812yes—
322C22×C59D4×Q812yes—
323C22×C59#55211yescharacteristic
324C22×C59#55412yes—
325C22×C59#55412yes—
326C22×C59#55611yescharacteristic
327C4×C59—24no—
328C4×C59—24no—
329C4×C59—44no—
330C4×C59—44no—
331Dic59—28no—
332Dic59—42no—
333Dic59—42no—
334Dic59—44no—
335Dic59—81no—
336Dic59—81no—
337C22×C22⋊Q8D5911yescharacteristic
338C2×#395—591no—
339C2×#389—592no—
340C2×#385—591no—
341C2×C4×C22⋊C4—591no—
342#1446—591no—
343#1444—591no—
344#1441—591no—
345#1439—592no—
346#1439—592no—
347#1436—592no—
348C2×C59—44no—
349C2×C59#238811yescharacteristic
350C2×C59#241011yescharacteristic
351C2×C59#241511yescharacteristic
352C2×C59#243411yescharacteristic
353C2×C59#244912yes—
354C2×C59#245212yes—
355C2×C59#246512yes—
356C2×C59#255612yes—
357C2×C59#258311yescharacteristic
358C2×C59#266311yescharacteristic
359C2×C59#285711yescharacteristic
360C23×Q8C2×D5911yescharacteristic
361C24×C4C2×D5911yescharacteristic
362C2×C22⋊Q8—24no—
363C2×C22⋊Q8—28no—
364C22×C4⋊C4—591no—
365C22×C4⋊C4—592no—
366C22×C4⋊C4—591no—
367C22×C4⋊C4—592no—
368C22×C4⋊C4C2×D5911yescharacteristic
369C22×C4⋊C4C2×D5912yes—
370C22×C22⋊C4—592no—
371C22×C22⋊C4—591no—
372C22×C22⋊C4—591no—
373C22×C22⋊C4C2×D5912yes—
374C22×C42—592no—
375#395—1184no—
376#389—1188no—
377#385—1184no—
378C4×C22⋊C4—1184no—
379C2×#95—591no—
380C2×#95—592no—
381C2×#95—591no—
382C2×#95—592no—
383C2×#95—592no—
384C2×#95—592no—
385C2×#95—592no—
386C2×#95—591no—
387C2×#95—591no—
388C2×#95—592no—
389C59#1476311yescharacteristic, complemented, Sylow
390C25C22×D5911yescharacteristic
391C22×Q8—22no—
392C22×Q8—22no—
393C22×Q8—24no—
394C22×Q8—22no—
395C22×Q8—42no—
396C23×C4—22no—
397C23×C4—22no—
398C23×C4—24no—
399C23×C4—42no—
400C23×C4—592no—
401C23×C4—591no—
402C23×C4—591no—
403C23×C4—592no—
404C23×C4—591no—
405C23×C4—591no—
406C23×C4C22⋊D5914yes—
407C23×C4C22×D5912yes—
408C23×C4C22×D5912yes—
409C23×C4C22×D5911yescharacteristic
410C23×C4C22×D5911yescharacteristic
411C22⋊Q8—232no—
412C2×C4⋊C4—24no—
413C2×C4⋊C4—22no—
414C2×C4⋊C4—24no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4—1184no—
417C2×C4⋊C4—1184no—
418C2×C4⋊C4—1184no—
419C2×C4⋊C4—1184no—
420C2×C4⋊C4—1182no—
421C2×C4⋊C4—1182no—
422C2×C4⋊C4—1182no—
423C2×C4⋊C4—1182no—
424C2×C4⋊C4—1184no—
425C2×C4⋊C4—1184no—
426C2×C4⋊C4—2362no—
427C2×C4⋊C4C22⋊D5918yes—
428C2×C22⋊C4—24no—
429C2×C22⋊C4—24no—
430C2×C22⋊C4—24no—
431C2×C22⋊C4—598no—
432C2×C22⋊C4—1184no—
433C2×C22⋊C4—1184no—
434C2×C22⋊C4—1182no—
435C2×C22⋊C4—1182no—
436C2×C22⋊C4—1182no—
437C2×C22⋊C4—1182no—
438C2×C22⋊C4—2362no—
439C2×C42—1184no—
440C2×C42—1184no—
441C2×C42—2362no—
442#95—594no—
443#95—1184no—
444#95—1184no—
445#95—1184no—
446#95—1182no—
447#95—1182no—
448#95—1182no—
449#95—1184no—
450#95—1184no—
451#95—2362no—
452C24—21no—
453C24—22no—
454C24—22no—
455C24—21no—
456C24—21no—
457C24—21no—
458C24—22no—
459C24—21no—
460C24—21no—
461C24Q8×D5914yes—
462C24C2×C22⋊D5912yes—
463C24C23×D5911yescharacteristic
464C2×Q8—28no—
465C2×Q8—48no—
466C2×Q8—48no—
467C2×Q8—44no—
468C2×Q8—44no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—28no—
487C22×C4—28no—
488C22×C4—44no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—594no—
492C22×C4—594no—
493C22×C4—594no—
494C22×C4—594no—
495C22×C4—592no—
496C22×C4—592no—
497C22×C4—1182no—
498C22×C4—1182no—
499C22×C4—1182no—
500C22×C4—1182no—
501C22×C4—1181no—
502C22×C4—1182no—
503C22×C4—1182no—
504C22×C4—1181no—
505C22×C4—1181no—
506C22×C4—1182no—
507C22×C4—1182no—
508C22×C4—1182no—
509C22×C4—1181no—
510C22×C4—1181no—
511C22×C4—1182no—
512C22×C4—1181no—
513C22×C4—1182no—
514C22×C4—1181no—
515C22×C4—1182no—
516C22×C4—1181no—
517C22×C4—1181no—
518C22×C4—1181no—
519C22×C4—1182no—
520C22×C4—1181no—
521C22×C4—1182no—
522C22×C4—1181no—
523C22×C4—1182no—
524C22×C4—1182no—
525C22×C4—1181no—
526C22×C4—1181no—
527C22×C4D4×D5912yes—
528C22×C4#1174568712yes—
529C22×C4C2×C22⋊D5914yes—
530C22×C4C2×C22⋊D5914yes—
531C22×C4C2×C22⋊D5912yes—
532C22×C4C2×C22⋊D5914yes—
533C22×C4C2×C22⋊D5912yes—
534C4⋊C4—28no—
535C4⋊C4—216no—
536C4⋊C4—11816no—
537C4⋊C4—2368no—
538C4⋊C4—2364no—
539C4⋊C4—2364no—
540C22⋊C4—216no—
541C22⋊C4—11816no—
542C22⋊C4—1188no—
543C22⋊C4—2364no—
544C42—2368no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23#1427372514yes—
568C23C2×D4×D5911yescharacteristic
569C23C22×C22⋊D5911yescharacteristic
570C23C22×C22⋊D5912yes—
571C23C2×Q8×D5912yes—
572C23#41×D5912yes—
573C23#41×D5911yescharacteristic
574C23#41×D5911yescharacteristic
575C23C2×#1174568711yescharacteristic
576C23#1427377212yes—
577C23#1427377511yescharacteristic
578C23#1427377511yescharacteristic
579Q8—416no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—416no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—598no—
604C2×C4—1184no—
605C2×C4—1184no—
606C2×C4—1184no—
607C2×C4—1184no—
608C2×C4—1184no—
609C2×C4—1184no—
610C2×C4—1184no—
611C2×C4—1182no—
612C2×C4—1182no—
613C2×C4—1184no—
614C2×C4—1182no—
615C2×C4—1184no—
616C2×C4—1182no—
617C2×C4—1184no—
618C2×C4—2361no—
619C2×C4—2361no—
620C2×C4—2361no—
621C2×C4—2361no—
622C2×C4—2361no—
623C2×C4—2361no—
624C2×C4—2362no—
625C2×C4—2362no—
626C2×C4—2362no—
627C2×C4—2362no—
628C2×C4—2361no—
629C2×C4—2361no—
630C2×C4—2361no—
631C2×C4—2362no—
632C2×C4—2362no—
633C2×C4—2361no—
634C2×C4—2362no—
635C2×C4—2362no—
636C2×C4—2361no—
637C2×C4—2362no—
638C2×C4—2361no—
639C2×C4#1427369614yes—
640C2×C4#1427369914yes—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—44no—
652C22C2×#1427369611yescharacteristic
653C22C2×#1427369911yescharacteristic
654C22C22⋊Q8×D5912yes—
655C22#1683790911yescharacteristic
656C22#1683790912yes—
657C22C2×#1427372511yescharacteristic
658C22#1683794811yescharacteristic
659C22#1683794812yes—
660C22#1683795412yes—
661C22#1683796111yescharacteristic
662C22#1683797611yescharacteristic
663C22#1683799812yes—
664C22#1683802212yes—
665C22D4×C22⋊D5912yes—
666C22#1683804612yes—
667C22#1683805411yescharacteristic
668C22#1683806211yescharacteristic
669C22Q8×C22⋊D5912yes—
670C22Q8×C22⋊D5912yes—
671C22#1683808611yescharacteristic
672C22#1683810612yes—
673C22#1683810612yes—
674C22#1683812311yescharacteristic
675C4—24no—
676C4—24no—
677C4—44no—
678C4—44no—
679C4—1188no—
680C4—4721no—
681C4—4721no—
682C4—2362no—
683C4—2362no—
684C4—2364no—
685C2—44no—
686C2#2142730911yescharacteristic
687C2#2142747011yescharacteristic
688C2#2142752611yescharacteristic
689C2#2142773611yescharacteristic
690C2#2142790212yes—
691C2#2142794512yes—
692C2#2142811312yes—
693C2#2142917812yes—
694C2#2142947711yescharacteristic
695C2#2143024711yescharacteristic
696C2#2143244411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D59
commutator, socle, cosocleC24×C59C24
FittingC22×C22⋊Q8×C59C2
prime radicalC25×C59C23
hypercenterC22×C22⋊Q8D59
hypocenterC59#14763
LCSG ▷ C24×C59 ▷ C59
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16837321C2semidirect
2. C2×#16837297C2semidirect
5. #21422367C2semidirect
6. #21422359C2semidirect
7. #21422346C2semidirect
10. #21422328C2semidirect
65. C2×C22⋊Dic59Q8semidirect
291. C2×Dic59C22⋊Q8semidirect
389. C59#14763semidirect

Maximal subgroups

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

subgroupindex
C2×#168373212
C2×#168372972
C2×#168372752
C2×C4×C22⋊Dic592
#214223672
#214223592
#214223462
#214223392
#214223282
C22×C22⋊Q8×C592
#1476359