Finite Group Databasebeta

← #28510961

Subgroups — #28510961

Order 15104.

1774 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837299C212yesmaximal
2C2×#16837297C212yesmaximal
3C2×#16837271C212yesmaximal
4C2×#16837271C211yesmaximal, characteristic
5C2×#16837253C211yesmaximal, characteristic
6C2×#16837253C211yesmaximal, characteristic
7#21422361C211yesmaximal, characteristic
8#21422360C211yesmaximal, characteristic
9#21422359C211yesmaximal, characteristic
10Dic59⋊#95C212yesmaximal
11C59×#1444C211yesmaximal, characteristic
12C22×C4⋊Dic59C2212yes—
13C22×C4⋊Dic59C2211yescharacteristic
14C22×Dic59⋊C4C2212yes—
15C22×Dic59⋊C4C2212yes—
16C22×Dic59⋊C4C2212yes—
17C22×Dic59⋊C4C2212yes—
18C22×Dic59⋊C4C2211yescharacteristic
19C22×Dic59⋊C4C2211yescharacteristic
20C22×Dic59⋊C4C2212yes—
21C22×C4×Dic59C2211yescharacteristic
22C22×C4×Dic59C2211yescharacteristic
23C22×C4×Dic59C2212yes—
24C22×C4×Dic59C2211yescharacteristic
25#16837299—28no—
26#16837297—28no—
27#16837271—28no—
28#16837271—24no—
29#16837253—24no—
30#16837253C418yescomplemented
31C2×C59⋊#95C2211yescharacteristic
32C2×C59⋊#95C2212yes—
33C2×C59⋊#95C2212yes—
34C2×C59⋊#95C2212yes—
35C2×C59⋊#95C2211yescharacteristic
36C22×C4⋊C4×C59C2212yes—
37C22×C4⋊C4×C59C2211yescharacteristic
38C2×#95×C59C2211yescharacteristic
39C2×#95×C59C2212yes—
40C2×#95×C59C2211yescharacteristic
41C23×Dic59C2312yes—
42C23×Dic59C2311yescharacteristic
43C23×Dic59C2311yescharacteristic
44C23×Dic59C2312yes—
45C23×Dic59C2311yescharacteristic
46C23×Dic59C2311yescharacteristic
47C2×C4⋊Dic59—22no—
48C2×C4⋊Dic59—24no—
49C2×C4⋊Dic59—24no—
50C2×C4⋊Dic59—42no—
51C2×C4⋊Dic59—42no—
52C2×Dic59⋊C4—22no—
53C2×Dic59⋊C4—22no—
54C2×Dic59⋊C4—22no—
55C2×Dic59⋊C4—22no—
56C2×Dic59⋊C4—24no—
57C2×Dic59⋊C4—24no—
58C2×Dic59⋊C4—24no—
59C2×Dic59⋊C4—24no—
60C2×Dic59⋊C4—24no—
61C2×Dic59⋊C4—24no—
62C2×Dic59⋊C4—24no—
63C2×Dic59⋊C4—24no—
64C2×Dic59⋊C4—24no—
65C2×Dic59⋊C4—24no—
66C2×Dic59⋊C4—42no—
67C2×Dic59⋊C4—42no—
68C2×Dic59⋊C4C2×C418yes—
69C2×Dic59⋊C4D414yes—
70C2×Dic59⋊C4D418yes—
71C2×Dic59⋊C4Q814yes—
72C2×Dic59⋊C4Q818yes—
73C2×C4×Dic59—24no—
74C2×C4×Dic59—24no—
75C2×C4×Dic59—22no—
76C2×C4×Dic59—22no—
77C2×C4×Dic59—24no—
78C2×C4×Dic59—24no—
79C2×C4×Dic59C2×C418yes—
80C2×C4×Dic59C2×C414yes—
81C2×C4×Dic59D414yes—
82C2×C4×Dic59Q814yes—
83C59⋊#95—22no—
84C59⋊#95—24no—
85C59⋊#95—24no—
86C59⋊#95—24no—
87C59⋊#95C2×C414yes—
88C23×C4×C59C2312yes—
89C23×C4×C59C2311yescharacteristic
90C23×C4×C59C2311yescharacteristic
91C23×C4×C59C2311yescharacteristic
92C23×C4×C59C2312yes—
93C2×C4⋊C4×C59—24no—
94C2×C4⋊C4×C59—22no—
95C2×C4⋊C4×C59—24no—
96C2×C4⋊C4×C59—42no—
97C2×C4⋊C4×C59—42no—
98#95×C59—22no—
99#95×C59—24no—
100#95×C59C2×C414yes—
101C24×C59C2411yescharacteristic
102C22×C4×C59—22no—
103C22×C4×C59—22no—
104C22×C4×C59—22no—
105C22×C4×C59—22no—
106C22×C4×C59—22no—
107C22×C4×C59—22no—
108C22×C4×C59—22no—
109C22×C4×C59—21no—
110C22×C4×C59—21no—
111C22×C4×C59—22no—
112C22×C4×C59—21no—
113C22×C4×C59—21no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59C22×C412yes—
124C22×C4×C59C22×C414yes—
125C22×C4×C59C2×D412yes—
126C22×C4×C59C2×D412yes—
127C22×C4×C59C2×D414yes—
128C22×C4×C59C2×Q812yes—
129C22×C4×C59C2×Q812yes—
130C22×C4×C59C2×Q814yes—
131C22×Dic59—22no—
132C22×Dic59—22no—
133C22×Dic59—22no—
134C22×Dic59—22no—
135C22×Dic59—22no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—22no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—22no—
142C22×Dic59—22no—
143C22×Dic59—22no—
144C22×Dic59—22no—
145C22×Dic59—22no—
146C22×Dic59—22no—
147C22×Dic59—22no—
148C22×Dic59—22no—
149C22×Dic59—22no—
150C22×Dic59C22×C414yes—
151C22×Dic59C22×C412yes—
152C22×Dic59C22×C412yes—
153C22×Dic59C2×D412yes—
154C22×Dic59C2×D414yes—
155C22×Dic59C2×D412yes—
156C22×Dic59C2×D412yes—
157C22×Dic59C2×D414yes—
158C22×Dic59C2×D412yes—
159C22×Dic59C2×Q812yes—
160C22×Dic59C2×Q814yes—
161C22×Dic59C2×Q812yes—
162C22×Dic59C2×Q812yes—
163C22×Dic59C2×Q814yes—
164C22×Dic59C2×Q812yes—
165C22×Dic59#4112yes—
166C22×Dic59#4112yes—
167C4⋊C4×C59—48no—
168C4⋊C4×C59—44no—
169C4⋊Dic59—48no—
170C4⋊Dic59—44no—
171Dic59⋊C4—28no—
172Dic59⋊C4—28no—
173Dic59⋊C4—48no—
174Dic59⋊C4—48no—
175Dic59⋊C4—48no—
176Dic59⋊C4—48no—
177Dic59⋊C4C4⋊C4132yescomplemented
178C4×Dic59—216no—
179C4×Dic59—44no—
180C4×Dic59—44no—
181C4×Dic59C4⋊C4116yescomplemented
182C23×C59C23×C411yescharacteristic
183C23×C59C22×D411yescharacteristic
184C23×C59C22×D412yes—
185C23×C59C22×D411yescharacteristic
186C23×C59C22×Q811yescharacteristic
187C23×C59C22×Q812yes—
188C23×C59C22×Q811yescharacteristic
189C23×C59C2×#4111yescharacteristic
190C23×C59C2×#4111yescharacteristic
191C23×C59H(5,𝔽2)12yes—
192C23×C592−1+412yes—
193C2×C4×C59—24no—
194C2×C4×C59—24no—
195C2×C4×C59—24no—
196C2×C4×C59—22no—
197C2×C4×C59—22no—
198C2×C4×C59—24no—
199C2×C4×C59—24no—
200C2×C4×C59—24no—
201C2×C4×C59—24no—
202C2×C4×C59—24no—
203C2×C4×C59—22no—
204C2×C4×C59—22no—
205C2×C4×C59—24no—
206C2×C4×C59—24no—
207C2×C4×C59—41no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—41no—
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—42no—
219C2×C4×C59—42no—
220C2×C4×C59C2×C4⋊C414yes—
221C2×C4×C59C2×C4⋊C418yes—
222C2×Dic59—24no—
223C2×Dic59—24no—
224C2×Dic59—24no—
225C2×Dic59—24no—
226C2×Dic59—24no—
227C2×Dic59—24no—
228C2×Dic59—22no—
229C2×Dic59—22no—
230C2×Dic59—24no—
231C2×Dic59—24no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—42no—
237C2×Dic59—42no—
238C2×Dic59—42no—
239C2×Dic59—42no—
240C2×Dic59—42no—
241C2×Dic59—42no—
242C2×Dic59—42no—
243C2×Dic59—42no—
244C2×Dic59—42no—
245C2×Dic59C2×C4⋊C418yes—
246C2×Dic59C2×C4⋊C414yes—
247C2×Dic59C2×C4⋊C418yes—
248C2×Dic59C2×C4⋊C414yes—
249C2×Dic59C4×D414yes—
250C2×Dic59C4×Q814yes—
251C2×Dic59#12114yes—
252C2×Dic59C22⋊Q814yes—
253C2×Dic59#12514yes—
254C2×Dic59#12714yes—
255C2×Dic59C4⋊Q814yes—
256C2×Dic59C4⋊Q818yes—
257C2×Dic59C4⋊Q814yes—
258#13910—591nomaximal, Sylow
259C22×C59C22×C4⋊C411yescharacteristic
260C22×C59C22×C4⋊C412yes—
261C22×C59C2×C4×D411yescharacteristic
262C22×C59C2×C4×Q811yescharacteristic
263C22×C59#52012yes—
264C22×C59C2×#12111yescharacteristic
265C22×C59C2×C22⋊Q811yescharacteristic
266C22×C59C2×#12511yescharacteristic
267C22×C59C2×#12711yescharacteristic
268C22×C59C2×C4⋊Q811yescharacteristic
269C22×C59C2×C4⋊Q812yes—
270C22×C59C2×C4⋊Q811yescharacteristic
271C22×C59#53511yescharacteristic
272C22×C59#53611yescharacteristic
273C22×C59#54412yes—
274C22×C59D4212yes—
275C22×C59#54712yes—
276C22×C59D4×Q812yes—
277C22×C59D4×Q812yes—
278C22×C59#55012yes—
279C22×C59#55412yes—
280C22×C59#55712yes—
281C22×C59Q8212yes—
282C4×C59—24no—
283C4×C59—28no—
284C4×C59—42no—
285C4×C59—42no—
286C4×C59—82no—
287Dic59—28no—
288Dic59—42no—
289Dic59—42no—
290Dic59—82no—
291Dic59#38418yescomplemented
292Dic59#38918yescomplemented
293C2×#389—592no—
294C2×#389—592no—
295C2×#384—592no—
296C2×#384—591no—
297C2×C4×C4⋊C4—591no—
298C2×C4×C4⋊C4—591no—
299#1444—591no—
300#1444—591no—
301#1444—591no—
302#1444D5911yescharacteristic
303C4⋊#95—592no—
304C2×C59C2×#38411yescharacteristic
305C2×C59C2×#38911yescharacteristic
306C2×C59#231711yescharacteristic
307C2×C59D4×C4⋊C412yes—
308C2×C59Q8×C4⋊C412yes—
309C2×C59#254111yescharacteristic
310C2×C59#254211yescharacteristic
311C2×C59#254611yescharacteristic
312C2×C59#255112yes—
313C2×C59#255212yes—
314C2×C59#256711yescharacteristic
315C22×C4⋊C4—592no—
316C22×C4⋊C4—591no—
317C22×C4⋊C4—591no—
318C22×C4⋊C4—592no—
319C22×C4⋊C4—592no—
320C22×C4⋊C4—592no—
321C22×C4⋊C4—592no—
322C22×C4⋊C4—592no—
323C22×C4⋊C4—591no—
324C22×C4⋊C4C2×D5912yes—
325C22×C4⋊C4C2×D5911yescharacteristic
326C22×C42—591no—
327C22×C42—592no—
328C22×C42—591no—
329C22×C42—591no—
330#389—1188no—
331#389—1188no—
332#384—1184no—
333#384—1188no—
334C4×C4⋊C4—598no—
335C4×C4⋊C4—1184no—
336C2×#95—591no—
337C2×#95—592no—
338C2×#95—592no—
339C2×#95—592no—
340C2×#95—591no—
341C2×#95C2×D5912yes—
342C2×#95C2×D5911yescharacteristic
343C2×#95C2×D5911yescharacteristic
344C59#1391011yescharacteristic, complemented, Sylow
345C23×C4—591no—
346C23×C4—591no—
347C23×C4—592no—
348C23×C4—591no—
349C23×C4—591no—
350C23×C4—592no—
351C23×C4C22×D5912yes—
352C23×C4C22×D5911yescharacteristic
353C23×C4C22×D5911yescharacteristic
354C23×C4C22×D5912yes—
355C23×C4C22×D5911yescharacteristic
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—42no—
360C2×C4⋊C4—42no—
361C2×C4⋊C4—598no—
362C2×C4⋊C4—598no—
363C2×C4⋊C4—598no—
364C2×C4⋊C4—594no—
365C2×C4⋊C4—594no—
366C2×C4⋊C4—1182no—
367C2×C4⋊C4—1182no—
368C2×C4⋊C4—1182no—
369C2×C4⋊C4—1182no—
370C2×C4⋊C4—1184no—
371C2×C4⋊C4—1184no—
372C2×C4⋊C4—1184no—
373C2×C4⋊C4—1184no—
374C2×C4⋊C4—1184no—
375C2×C4⋊C4—1184no—
376C2×C4⋊C4—1184no—
377C2×C4⋊C4—1184no—
378C2×C4⋊C4—1184no—
379C2×C4⋊C4—1184no—
380C2×C4⋊C4—1184no—
381C2×C4⋊C4—1182no—
382C2×C4⋊C4—1184no—
383C2×C4⋊C4—2362no—
384C2×C4⋊C4—2362no—
385C2×C4⋊C4—2362no—
386C2×C4⋊C4—2362no—
387C2×C42—594no—
388C2×C42—594no—
389C2×C42—594no—
390C2×C42—598no—
391C2×C42—1184no—
392C2×C42—1184no—
393C2×C42—1182no—
394C2×C42—1182no—
395C2×C42—1184no—
396C2×C42—1184no—
397#95—22no—
398#95—24no—
399#95—594no—
400#95—1184no—
401#95—1184no—
402#95—1182no—
403#95—1184no—
404#95C4×D5914yes—
405C24C23×D5911yescharacteristic
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—592no—
428C22×C4—592no—
429C22×C4—592no—
430C22×C4—592no—
431C22×C4—592no—
432C22×C4—594no—
433C22×C4—592no—
434C22×C4—592no—
435C22×C4—592no—
436C22×C4—594no—
437C22×C4—592no—
438C22×C4—592no—
439C22×C4—594no—
440C22×C4—594no—
441C22×C4—594no—
442C22×C4—592no—
443C22×C4—592no—
444C22×C4—1182no—
445C22×C4—1182no—
446C22×C4—1182no—
447C22×C4—1182no—
448C22×C4—1182no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1182no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—1182no—
455C22×C4—1182no—
456C22×C4—1182no—
457C22×C4—1182no—
458C22×C4—1182no—
459C22×C4—1182no—
460C22×C4—1182no—
461C22×C4—1182no—
462C22×C4—1182no—
463C22×C4C2×C4×D5912yes—
464C22×C4C2×C4×D5914yes—
465C22×C4D4×D5912yes—
466C22×C4D4×D5912yes—
467C22×C4D4×D5914yes—
468C22×C4Q8×D5912yes—
469C22×C4Q8×D5912yes—
470C22×C4Q8×D5914yes—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—5932no—
474C4⋊C4—1188no—
475C4⋊C4—1188no—
476C4⋊C4—2368no—
477C4⋊C4—2368no—
478C4⋊C4—2364no—
479C4⋊C4—2368no—
480C4⋊C4—2368no—
481C4⋊C4—2368no—
482C42—5916no—
483C42—11816no—
484C42—2364no—
485C42—2364no—
486C23C22×C4×D5911yescharacteristic
487C23C2×D4×D5911yescharacteristic
488C23C2×D4×D5911yescharacteristic
489C23C2×D4×D5912yes—
490C23C2×Q8×D5911yescharacteristic
491C23C2×Q8×D5911yescharacteristic
492C23C2×Q8×D5912yes—
493C23#41×D5911yescharacteristic
494C23#41×D5911yescharacteristic
495C23#1427377212yes—
496C23#1427377412yes—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—594no—
525C2×C4—594no—
526C2×C4—594no—
527C2×C4—594no—
528C2×C4—594no—
529C2×C4—598no—
530C2×C4—594no—
531C2×C4—594no—
532C2×C4—594no—
533C2×C4—598no—
534C2×C4—594no—
535C2×C4—594no—
536C2×C4—598no—
537C2×C4—1184no—
538C2×C4—1184no—
539C2×C4—1184no—
540C2×C4—1184no—
541C2×C4—1184no—
542C2×C4—1184no—
543C2×C4—1182no—
544C2×C4—1182no—
545C2×C4—1184no—
546C2×C4—1184no—
547C2×C4—2362no—
548C2×C4—2362no—
549C2×C4—2362no—
550C2×C4—2362no—
551C2×C4—2362no—
552C2×C4—2362no—
553C2×C4—2362no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2362no—
559C2×C4—2362no—
560C2×C4C4⋊C4×D5914yes—
561C2×C4C4⋊C4×D5918yes—
562C22C2×C4⋊C4×D5911yescharacteristic
563C22C2×C4⋊C4×D5912yes—
564C22C4×D4×D5911yescharacteristic
565C22C4×Q8×D5911yescharacteristic
566C22#1683788112yes—
567C22#121×D5911yescharacteristic
568C22C22⋊Q8×D5911yescharacteristic
569C22#125×D5911yescharacteristic
570C22#127×D5911yescharacteristic
571C22C4⋊Q8×D5911yescharacteristic
572C22C4⋊Q8×D5912yes—
573C22C4⋊Q8×D5911yescharacteristic
574C22#1683796711yescharacteristic
575C22#1683797411yescharacteristic
576C22#1683803112yes—
577C22#1683803412yes—
578C22#1683805012yes—
579C22#1683806512yes—
580C22#1683806512yes—
581C22#1683807012yes—
582C22#1683810612yes—
583C22#1683812912yes—
584C22#1683813212yes—
585C4—24no—
586C4—28no—
587C4—42no—
588C4—42no—
589C4—82no—
590C4—598no—
591C4—598no—
592C4—1188no—
593C4—4722no—
594C4—2362no—
595C4—2362no—
596C2#384×D5911yescharacteristic
597C2#389×D5911yescharacteristic
598C2#2142671511yescharacteristic
599C2#2142699412yes—
600C2#2142701012yes—
601C2#2142903711yescharacteristic
602C2#2142904411yescharacteristic
603C2#2142908011yescharacteristic
604C2#2142912112yes—
605C2#2142912612yes—
606C2#2142930011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×#1444C2
socle, prime radical, cosocleC24×C59C24
hypercenter#1444D59
hypocenterC59#13910
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
30. #16837253C4semidirect
177. Dic59⋊C4C4⋊C4semidirect
181. C4×Dic59C4⋊C4semidirect
291. Dic59#384semidirect
292. Dic59#389semidirect
344. C59#13910semidirect

Maximal subgroups

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

subgroupindex
C2×#168372992
C2×#168372972
C2×#168372712
C2×#168372532
#214223612
#214223602
#214223592
Dic59⋊#952
C59×#14442
#1391059