Finite Group Databasebeta

← #28638853

Subgroups — #28638853

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837710C211yesmaximal, characteristic, complemented
2#21429262C211yesmaximal, characteristic
3#21429005C211yesmaximal, characteristic
4#21425529C211yesmaximal, characteristic, complemented
5#21425519C211yesmaximal, characteristic, complemented
6#21425300C211yesmaximal, characteristic, complemented
7#21425293C211yesmaximal, characteristic, complemented
8#21425061C211yesmaximal, characteristic, complemented
9#21424866C211yesmaximal, characteristic, complemented
10#21424519C211yesmaximal, characteristic, complemented
11#21424519C211yesmaximal, characteristic, complemented
12#21424125C211yesmaximal, characteristic, complemented
13#21424121C211yesmaximal, characteristic, complemented
14C4×#14273652C211yesmaximal, characteristic, complemented
15C2×C8×D4×C59C211yesmaximal, characteristic
16C2×#14273719C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20#16837710—24no—
21C2×#14273661C2211yescharacteristic
22C2×#14273661C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273652C2211yescharacteristic
28C2×#14273652C2211yescharacteristic
29#16837340—21no—
30#16837333—21no—
31#16837316—21no—
32#16837316—21no—
33#16837296—21no—
34#16837296—21no—
35#16837290—21no—
36#16837290—21no—
37#16837268C2211yescharacteristic
38#16837262C2211yescharacteristic
39#16837262C2211yescharacteristic
40#16837262C2211yescharacteristic
41#16837262C2211yescharacteristic
42#16837262C2211yescharacteristic
43#16837262C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C23×C8×C59C2211yescharacteristic
54C23×C8×C59C2211yescharacteristic
55C2×C4×D4×C59C2211yescharacteristic
56C8×D4×C59—24no—
57C2×C4⋊C8×C59C2211yescharacteristic
58C2×C22⋊C8×C59C2211yescharacteristic
59C2×C22⋊C8×C59C2211yescharacteristic
60C2×C4×C8×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62#14273719—22no—
63#14273719—22no—
64#14273706—22no—
65#14273706—22no—
66#14273706—41no—
67#14273706—41no—
68C4×Dic118—22no—
69C4×Dic118—22no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C22⋊Dic59—21no—
85C2×C22⋊Dic59—21no—
86C2×C22⋊Dic59—21no—
87C2×C22⋊Dic59—21no—
88C2×C22⋊Dic59—21no—
89C2×C22⋊Dic59—21no—
90C2×C4×Dic59—21no—
91C2×C4×Dic59—21no—
92#14273661—22no—
93#14273661—41no—
94#14273659—22no—
95#14273659—41no—
96#14273652—22no—
97#14273652—22no—
98#14273652—22no—
99#14273652—22no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C22×D4×C59D411yescharacteristic
110C23×C4×C59C2311yescharacteristic
111C23×C4×C59C2311yescharacteristic
112C22×C8×C59—22no—
113C22×C8×C59—22no—
114C22×C8×C59—22no—
115C22×C8×C59—22no—
116C22×C8×C59—41no—
117C22×C8×C59—41no—
118C22×C8×C59C2311yescharacteristic
119C22×C8×C59C2311yescharacteristic
120C22×C8×C59C2311yescharacteristic
121C22×C8×C59C2311yescharacteristic
122C4×D4×C59—22no—
123C4×D4×C59—22no—
124C2×C4⋊C4×C59D411yescharacteristic
125C2×C22⋊C4×C59D411yescharacteristic
126C2×C22⋊C4×C59D411yescharacteristic
127C2×C42×C59C2311yescharacteristic
128C4⋊C8×C59D414yes—
129C22⋊C8×C59—22no—
130C22⋊C8×C59—22no—
131C4×C8×C59—22no—
132C24×C59C2×D411yescharacteristic
133C24×C59C2×D411yescharacteristic
134C2×D4×C59—22no—
135C2×D4×C59—22no—
136C2×D4×C59—22no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2411yescharacteristic
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118#4112yes—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59—21no—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C2×C8×C59—44no—
178C2×C8×C59—44no—
179C2×C8×C59C2×D412yes—
180C2×C8×C59C2×D412yes—
181C2×C8×C59#4112yes—
182C4⋊C4×C59—22no—
183C22⋊C4×C59—22no—
184C22⋊C4×C59—22no—
185C42×C59C2×D412yes—
186C42×C59#4112yes—
187C22⋊Dic59—42no—
188C22⋊Dic59—42no—
189C22⋊Dic59—42no—
190C22⋊Dic59—42no—
191C22⋊Dic59—42no—
192C22⋊Dic59—42no—
193C22⋊Dic59—81no—
194C22⋊Dic59—81no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59—21no—
205C4⋊Dic59—21no—
206C4⋊Dic59—21no—
207C4⋊Dic59—21no—
208C4⋊Dic59—21no—
209Dic59⋊C4—42no—
210Dic59⋊C4—42no—
211Dic59⋊C4—42no—
212Dic59⋊C4—42no—
213Dic59⋊C4—42no—
214C4×Dic59—42no—
215C4×Dic59—81no—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59—21no—
227C23×C59—21no—
228C23×C59#13512yes—
229C23×C59#13512yes—
230C23×C59C22×D411yescharacteristic
231D4×C59—28no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59#12112yes—
251C2×C4×C59#12112yes—
252C2×C4×C59C22×D411yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59H(5,𝔽2)11yescharacteristic
257C2×C4×C592−1+411yescharacteristic
258C2×Dic59—22no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282Dic118—22no—
283Dic118—22no—
284Dic118—41no—
285Dic118—41no—
286C8×C59—22no—
287C8×C59—22no—
288C8×C59—22no—
289C8×C59—41no—
290#25343—591nomaximal, Sylow
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—22no—
295C22×C59—22no—
296C22×C59—22no—
297C22×C59—41no—
298C22×C59—41no—
299C22×C59—41no—
300C22×C59—41no—
301C22×C59—41no—
302C22×C59—41no—
303C22×C59—41no—
304C22×C59—41no—
305C22×C59C2×#12111yescharacteristic
306C22×C59#54611yescharacteristic
307C22×C59#54711yescharacteristic
308C22×C59C2×#13511yescharacteristic
309C22×C59C2×#13511yescharacteristic
310C22×C59#57711yescharacteristic
311C22×C59#57811yescharacteristic
312C4×C59—22no—
313C4×C59—22no—
314C4×C59—41no—
315C4×C59—41no—
316C4×C59#52911yescharacteristic
317C4×C59#54211yescharacteristic
318C4×C59#54611yescharacteristic
319C4×C59#54811yescharacteristic
320Dic59—42no—
321Dic59—81no—
322Dic59—81no—
323Dic59—81no—
324Dic59—81no—
325Dic59—81no—
326Dic59—81no—
327Dic59—81no—
328C2×#479—591no—
329C2×C8×D4D5911yescharacteristic
330#2562—591no—
331#2536—591no—
332#2077—591no—
333#2075—591no—
334#2036—591no—
335#2035—591no—
336#1991—591no—
337#1949—591no—
338#1892—591no—
339#1892—591no—
340#1808—591no—
341#1807—591no—
342C4×#103—591no—
343C2×C59—42no—
344C2×C59—42no—
345C2×C59#242711yescharacteristic
346C2×C59#305011yescharacteristic
347C2×C59#306311yescharacteristic
348C2×C59#330011yescharacteristic
349C2×C59#330111yescharacteristic
350C2×C59#333611yescharacteristic
351C2×C59#333711yescharacteristic
352C23×C8C2×D5911yescharacteristic
353C23×C8C2×D5911yescharacteristic
354C2×#125—591no—
355C2×C22⋊Q8—591no—
356C2×C22⋊Q8—591no—
357C2×C4×Q8—591no—
358C2×C4×D4C2×D5911yescharacteristic
359#479—1184no—
360C8×D4—24no—
361C2×#107—591no—
362C2×#107—591no—
363C2×#106—591no—
364C2×#106—591no—
365C2×C4⋊C8C2×D5911yescharacteristic
366C2×#103—591no—
367C2×#103—591no—
368C2×#103—591no—
369C2×#103—591no—
370C2×C22⋊C8C2×D5911yescharacteristic
371C2×C22⋊C8C2×D5911yescharacteristic
372C2×C4×C8C2×D5911yescharacteristic
373#400—1181no—
374#398—1181no—
375#394—1181no—
376#394—1181no—
377#388—1181no—
378#388—1181no—
379#388—1181no—
380#388—1181no—
381#382—591no—
382#380—591no—
383#380—591no—
384#380—591no—
385#380—591no—
386#380—591no—
387#380—591no—
388C4×C4⋊C4—591no—
389#340—591no—
390#340—591no—
391#340—591no—
392#340—591no—
393#340—591no—
394#340—591no—
395#340—591no—
396#340—591no—
397C59#2534311yescharacteristic, complemented, Sylow
398C22×Q8—591no—
399C22×D4D23611yescharacteristic
400C23×C4C22×D5911yescharacteristic
401C23×C4C22×D5911yescharacteristic
402C22×C8—22no—
403C22×C8—22no—
404C22×C8—22no—
405C22×C8—22no—
406C22×C8—41no—
407C22×C8—41no—
408C22×C8C22×D5911yescharacteristic
409C22×C8C22×D5911yescharacteristic
410C22×C8C22×D5911yescharacteristic
411C22×C8C22×D5911yescharacteristic
412#125—1182no—
413#125—1182no—
414C22⋊Q8—1182no—
415C22⋊Q8—1182no—
416C22⋊Q8—2361no—
417C22⋊Q8—2361no—
418C4×Q8—1182no—
419C4×Q8—1182no—
420C4×D4—22no—
421C4×D4—22no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—591no—
426C2×C4⋊C4—591no—
427C2×C4⋊C4—591no—
428C2×C4⋊C4—591no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4D23611yescharacteristic
435C2×C22⋊C4—1181no—
436C2×C22⋊C4—1181no—
437C2×C22⋊C4—1181no—
438C2×C22⋊C4—1181no—
439C2×C22⋊C4—1181no—
440C2×C22⋊C4—1181no—
441C2×C22⋊C4—1181no—
442C2×C22⋊C4—1181no—
443C2×C22⋊C4D23611yescharacteristic
444C2×C22⋊C4D23611yescharacteristic
445C2×C42—1181no—
446C2×C42—1181no—
447C2×C42C22×D5911yescharacteristic
448#107—1182no—
449#107—2361no—
450#106—1182no—
451#106—2361no—
452C4⋊C8C22⋊D5914yes—
453#103—1182no—
454#103—1182no—
455#103—1182no—
456#103—1182no—
457C22⋊C8—22no—
458C22⋊C8—22no—
459C4×C8—22no—
460#95—1181no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464#95—1181no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468#95—1181no—
469C24C2×D23611yescharacteristic
470C24C2×D23611yescharacteristic
471C2×Q8—592no—
472C2×Q8—1181no—
473C2×Q8—1181no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—1181no—
493C22×C4—1181no—
494C22×C4—1181no—
495C22×C4—1181no—
496C22×C4—1181no—
497C22×C4—1181no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4C2×D23611yescharacteristic
501C22×C4C2×D23611yescharacteristic
502C22×C4C2×D23611yescharacteristic
503C22×C4C2×D23611yescharacteristic
504C22×C4C23×D5911yescharacteristic
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—44no—
515C2×C8—44no—
516C2×C8#1174568712yes—
517C2×C8C2×C22⋊D5912yes—
518C2×C8C2×C22⋊D5912yes—
519C4⋊C4—22no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—1181no—
534C4⋊C4—2362no—
535C4⋊C4—2362no—
536C4⋊C4—2362no—
537C4⋊C4—2362no—
538C4⋊C4—2362no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—4721no—
542C22⋊C4—4721no—
543C22⋊C4—2362no—
544C22⋊C4—2362no—
545C22⋊C4—2362no—
546C22⋊C4—2362no—
547C22⋊C4—2362no—
548C22⋊C4—2362no—
549C42—4721no—
550C42—2362no—
551C42#1174568512yes—
552C42C2×C22⋊D5912yes—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23#1427374912yes—
566C23#1427374912yes—
567C23C22×D23611yescharacteristic
568Q8—1182no—
569Q8—1182no—
570Q8—2361no—
571Q8—2361no—
572D4—28no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—1182no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4—2361no—
615C2×C4#1427370112yes—
616C2×C4#1427370112yes—
617C2×C4C22×C22⋊D5911yescharacteristic
618C2×C4#41×D5911yescharacteristic
619C2×C4C2×#1174568711yescharacteristic
620C2×C4C2×#1174568511yescharacteristic
621C2×C4#1427377211yescharacteristic
622C2×C4#1427377411yescharacteristic
623C8—22no—
624C8—22no—
625C8—22no—
626C8—41no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22C2×#1427370111yescharacteristic
642C22#1683804711yescharacteristic
643C22#1683805611yescharacteristic
644C22C2×#1427374911yescharacteristic
645C22C2×#1427374911yescharacteristic
646C22#1683821411yescharacteristic
647C22#1683821911yescharacteristic
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—4721no—
653C4—4721no—
654C4—4721no—
655C4—4721no—
656C4—4721no—
657C4—4721no—
658C4—4721no—
659C4—2362no—
660C4#1683793411yescharacteristic
661C4#1683801711yescharacteristic
662C4#1683804611yescharacteristic
663C4#1683805911yescharacteristic
664C2—42no—
665C2—42no—
666C2#2142766711yescharacteristic
667C2#2143367711yescharacteristic
668C2#2143378711yescharacteristic
669C2#2143598511yescharacteristic
670C2#2143600011yescharacteristic
671C2#2143646511yescharacteristic
672C2#2143647711yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C8×D4×C59C2
socleC23×C59C22×D4
prime radicalC22×D4×C59D4
hypercenterC2×C8×D4D59
hypocenterC59#25343
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16837710C2semidirect
4. #21425529C2semidirect
5. #21425519C2semidirect
6. #21425300C2semidirect
7. #21425293C2semidirect
8. #21425061C2semidirect
9. #21424866C2semidirect
10. #21424519C2semidirect
11. #21424519C2semidirect
12. #21424125C2semidirect
13. #21424121C2semidirect
14. C4×#14273652C2semidirect
397. C59#25343semidirect

Maximal subgroups

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

subgroupindex
C2×#168377102
#214292622
#214290052
#214255292
#214255192
#214253002
#214252932
#214250612
#214248662
#214245192
#214241252
#214241212
C4×#142736522
C2×C8×D4×C592
#2534359