Finite Group Databasebeta

← #28611345

Subgroups — #28611345

Order 15104.

970 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837641C211yesmaximal, characteristic
2C2×#16837506C211yesmaximal, characteristic, complemented
3C22×#14273658C211yesmaximal, characteristic
4#21427105C211yesmaximal, characteristic
5#21427101C211yesmaximal, characteristic
6#21424964C211yesmaximal, characteristic, complemented
7#21424833C211yesmaximal, characteristic, complemented
8#21424733C211yesmaximal, characteristic, complemented
9#21424728C211yesmaximal, characteristic, complemented
10#21424401C211yesmaximal, characteristic
11#21424365C211yesmaximal, characteristic, complemented
12#21424362C211yesmaximal, characteristic, complemented
13#21423992C211yesmaximal, characteristic, complemented
14#21423875C211yesmaximal, characteristic
15C59×#1872C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C23×C59⋊C8C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×Q8×Dic59C2211yescharacteristic
20C2×#14273692C2211yescharacteristic
21C22×C4⋊Dic59C2211yescharacteristic
22#16837641C418yescomplemented
23#16837506—24no—
24C2×#14273660C2211yescharacteristic
25C2×#14273658—24no—
26C2×#14273658C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273652C2211yescharacteristic
31C2×#14273652C2211yescharacteristic
32C2×#14273651C2211yescharacteristic
33C2×#14273650C2211yescharacteristic
34C2×#14273637C2211yescharacteristic
35C2×#14273637C2211yescharacteristic
36#16837309C2211yescharacteristic
37#16837307C2211yescharacteristic
38#16837301—21no—
39#16837301C2211yescharacteristic
40#16837298C2211yescharacteristic
41#16837297—21no—
42#16837288—21no—
43#16837283—21no—
44#16837277—21no—
45#16837275—21no—
46C4⋊C4×Dic59C2211yescharacteristic
47C4⋊C4×Dic59C2211yescharacteristic
48C22⋊C4×Dic59—21no—
49C4×C22⋊Dic59—21no—
50C4×C22⋊Dic59C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53#16837156C2211yescharacteristic
54#16837156C2211yescharacteristic
55C2×C22⋊Q8×C59C2211yescharacteristic
56C22×C4⋊C4×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C2×#103×C59C2211yescharacteristic
59C2×C22⋊C8×C59C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#340C2211yescharacteristic
62C22×Dic118C2311yescharacteristic
63C23×Dic59—21no—
64C2×#11745672—22no—
65C2×#11745672—22no—
66C2×#11745672C2×C414yescomplemented
67C22×C59⋊C8—22no—
68C22×C59⋊C8—22no—
69C22×C59⋊C8C2×C414yes—
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72#14273706—22no—
73#14273706C2×C414yes—
74Q8×Dic59—22no—
75Q8×Dic59—41no—
76#14273692—22no—
77#14273692—41no—
78C2×C4⋊Dic59—22no—
79C2×C4⋊Dic59—21no—
80C2×C4⋊Dic59—21no—
81C2×C4⋊Dic59—21no—
82C2×C4⋊Dic59—41no—
83C2×C4⋊Dic59C2311yescharacteristic
84C2×C4⋊Dic59C2311yescharacteristic
85C2×C4⋊Dic59C2311yescharacteristic
86C2×Dic59⋊C4—21no—
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×C22⋊Dic59—21no—
90C2×C22⋊Dic59D411yescharacteristic
91C2×C22⋊Dic59D411yescharacteristic
92C2×C4×Dic59—21no—
93C2×C4×Dic59—21no—
94C2×C4×Dic59—21no—
95C2×C4×Dic59—21no—
96C2×C4×Dic59C2311yescharacteristic
97C2×C4×Dic59C2311yescharacteristic
98#14273660C2×C414yescomplemented
99#14273658—22no—
100#14273658—22no—
101#14273658—22no—
102#14273658D414yes—
103#14273652—22no—
104#14273652—41no—
105#14273651C2×C414yescomplemented
106#14273650C2×C414yescomplemented
107#14273637—22no—
108#14273637—22no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C59⋊#95—21no—
114C59⋊#95—21no—
115C59⋊#95—21no—
116C59⋊#95—21no—
117C59⋊#95D411yescharacteristic
118C59⋊#95D411yescharacteristic
119C22×Q8×C59C2311yescharacteristic
120C23×C4×C59—21no—
121C23×C4×C59C2311yescharacteristic
122C22×C8×C59C2311yescharacteristic
123C22×C8×C59C2311yescharacteristic
124C22⋊Q8×C59—22no—
125C22⋊Q8×C59C2×C414yes—
126C2×C4⋊C4×C59—21no—
127C2×C4⋊C4×C59—22no—
128C2×C4⋊C4×C59—21no—
129C2×C4⋊C4×C59—41no—
130C2×C4⋊C4×C59C2311yescharacteristic
131C2×C4⋊C4×C59C2311yescharacteristic
132C2×C4⋊C4×C59C2311yescharacteristic
133C2×C22⋊C4×C59—21no—
134#103×C59—22no—
135#103×C59—41no—
136C22⋊C8×C59—22no—
137C24×C59C2×D411yescharacteristic
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59C22×C412yes—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—42no—
153C22×C4×C59—42no—
154C22×C4×C59—81no—
155C22×C4×C59C22×C412yes—
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59#4112yes—
158C22×C4×C59C2411yescharacteristic
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—42no—
164C22×Dic59—42no—
165C22×Dic59—81no—
166C22×Dic59C2×D411yescharacteristic
167C22×Dic59C2×D411yescharacteristic
168C22×Dic59C2×D411yescharacteristic
169C22×Dic59C2×D411yescharacteristic
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118—21no—
173C2×Dic118—21no—
174C2×Dic118—21no—
175C2×Dic118C22×C412yes—
176C2×C8×C59—21no—
177C2×C8×C59—21no—
178C2×C8×C59—21no—
179C2×C8×C59—21no—
180C2×C8×C59—21no—
181C2×C8×C59—21no—
182C4⋊C4×C59—24no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—21no—
187C4⋊C4×C59—42no—
188C4⋊C4×C59C22×C412yes—
189C4⋊C4×C59C2×D412yes—
190C22⋊C4×C59—24no—
191C22⋊Dic59—24no—
192C22⋊Dic59—22no—
193C22⋊Dic59—22no—
194#11745672—28no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—42no—
200C4⋊Dic59—42no—
201C4⋊Dic59—81no—
202C4⋊Dic59C22×C412yes—
203C4⋊Dic59C2×D412yes—
204Dic59⋊C4—24no—
205Dic59⋊C4—42no—
206Dic59⋊C4—81no—
207C4×Dic59—21no—
208C4×Dic59—21no—
209C4×Dic59—21no—
210C4×Dic59—21no—
211C4×Dic59—42no—
212C4×Dic59—42no—
213C4×Dic59—81no—
214C4×Dic59—81no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—28no—
218C2×C59⋊C8C22×C412yes—
219C2×C59⋊C8C22×C412yes—
220C2×C59⋊C8C2×D412yes—
221C2×C59⋊C8#4112yes—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59C4×D412yes—
227C23×C59#13712yes—
228C23×C59#13712yes—
229C23×C59C22×D411yescharacteristic
230Q8×C59—22no—
231Q8×C59—22no—
232Q8×C59—41no—
233Q8×C59—41no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—22no—
240C2×C4×C59—22no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
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—41no—
251C2×C4×C59—41no—
252C2×C4×C59—84no—
253C2×C4×C59C4×D412yes—
254C2×C4×C59C23×C411yescharacteristic
255C2×C4×C59C22×D411yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59H(5,𝔽2)11yescharacteristic
259C2×C4×C592−1+411yescharacteristic
260C2×Dic59—22no—
261C2×Dic59—22no—
262C2×Dic59—21no—
263C2×Dic59—21no—
264C2×Dic59—22no—
265C2×Dic59—21no—
266C2×Dic59—21no—
267C2×Dic59—21no—
268C2×Dic59—21no—
269C2×Dic59—21no—
270C2×Dic59—21no—
271C2×Dic59—21no—
272C2×Dic59—21no—
273C2×Dic59—21no—
274C2×Dic59—21no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C2×Dic59—84no—
285Dic118—22no—
286Dic118—22no—
287Dic118—41no—
288Dic118—41no—
289C8×C59—41no—
290C8×C59—41no—
291C59⋊C8—22no—
292C59⋊C8C4×D414yes—
293#23028—591nomaximal, Sylow
294C22×C59—22no—
295C22×C59—22no—
296C22×C59—22no—
297C22×C59—22no—
298C22×C59—22no—
299C22×C59—22no—
300C22×C59#44114yescomplemented
301C22×C59C2×C4×D411yescharacteristic
302C22×C59D4211yescharacteristic
303C22×C59#54711yescharacteristic
304C22×C59C2×#13711yescharacteristic
305C22×C59C2×#13711yescharacteristic
306C22×C59#57711yescharacteristic
307C22×C59#57711yescharacteristic
308C4×C59—22no—
309C4×C59—42no—
310C4×C59—42no—
311C4×C59—42no—
312C4×C59—81no—
313C4×C59C2×C4×D411yescharacteristic
314C4×C59#52011yescharacteristic
315C4×C59#54811yescharacteristic
316C4×C59#55311yescharacteristic
317Dic59—41no—
318Dic59—41no—
319Dic59—42no—
320Dic59—42no—
321Dic59—41no—
322Dic59—41no—
323Dic59—42no—
324Dic59—81no—
325C2×#467—591no—
326C2×#441—591no—
327C22×#106—591no—
328#2360—591no—
329#2360—591no—
330#1971—591no—
331#1943—591no—
332#1927—591no—
333#1927—591no—
334#1872—591no—
335#1872D5911yescharacteristic
336#1865—591no—
337#1865—591no—
338#1776—591no—
339#1752—591no—
340C2×C59—24no—
341C2×C59#236111yescharacteristic
342C2×C59C2×#44111yescharacteristic
343C2×C59#295211yescharacteristic
344C2×C59#326611yescharacteristic
345C2×C59#326611yescharacteristic
346C2×C59#334411yescharacteristic
347C2×C59#334411yescharacteristic
348C22×Q16—591no—
349C23×C8—591no—
350C2×C22⋊Q8—591no—
351C2×C22⋊Q8C2×D5911yescharacteristic
352C2×C4×Q8—591no—
353C2×C4×Q8—591no—
354C22×C4⋊C4—591no—
355C22×C4⋊C4C2×D5911yescharacteristic
356#467—598no—
357#441—1184no—
358C2×#107—591no—
359C2×#106—591no—
360C2×#106—591no—
361C2×#106—591no—
362C2×#106—591no—
363C2×#106—1184no—
364C2×#103—591no—
365C2×#103—591no—
366C2×#103—591no—
367C2×#103—591no—
368C2×#103C2×D5911yescharacteristic
369C2×#103C2×D5911yescharacteristic
370C2×C22⋊C8C2×D5911yescharacteristic
371C2×#97—591no—
372C2×#97—591no—
373#391—591no—
374#391—591no—
375#389—591no—
376#389—591no—
377#389—1181no—
378#389—1181no—
379#387—1181no—
380#387—1181no—
381#385—1181no—
382#385—1181no—
383C4×C4⋊C4—591no—
384C4×C4⋊C4—591no—
385C4×C22⋊C4—591no—
386C4×C22⋊C4—1181no—
387C4×C22⋊C4—1181no—
388#340—591no—
389#340—591no—
390#340C2×D5911yescharacteristic
391#340C2×D5911yescharacteristic
392#336—591no—
393#336—591no—
394C59#2302811yescharacteristic, complemented, Sylow
395C22×Q8—591no—
396C22×Q8C22×D5911yescharacteristic
397C23×C4—21no—
398C23×C4—1181no—
399C23×C4C22×D5911yescharacteristic
400C2×Q16—594no—
401C2×Q16—1182no—
402C2×Q16—1182no—
403C22×C8—591no—
404C22×C8—594no—
405C22×C8—591no—
406C22×C8—1182no—
407C22×C8—1182no—
408C22×C8C22×D5911yescharacteristic
409C22×C8C22×D5911yescharacteristic
410C22⋊Q8—22no—
411C22⋊Q8—594no—
412C22⋊Q8—1182no—
413C22⋊Q8C4×D5914yes—
414C4×Q8—1182no—
415C4×Q8—1182no—
416C4×Q8—2361no—
417C4×Q8—2361no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—41no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—1182no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—2361no—
433C2×C4⋊C4C22×D5911yescharacteristic
434C2×C4⋊C4C22×D5911yescharacteristic
435C2×C4⋊C4C22×D5911yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—591no—
438C2×C22⋊C4—591no—
439C2×C22⋊C4—1181no—
440C2×C42—591no—
441C2×C42—591no—
442C2×C42—1181no—
443C2×C42—1181no—
444C2×C42—1181no—
445C2×C42—1181no—
446#107—594no—
447#106—594no—
448#106—1182no—
449#106—1182no—
450#106—1182no—
451#103—22no—
452#103—41no—
453#103—594no—
454#103—594no—
455#103—1182no—
456#103—2361no—
457C22⋊C8—22no—
458#97—1182no—
459#97—1182no—
460#95—591no—
461#95—591no—
462#95—1181no—
463#95—1181no—
464#95—1181no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468#95—1181no—
469#95—1181no—
470C24D4×D5911yescharacteristic
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—592no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×Q8—1181no—
481C2×Q8—1181no—
482C2×Q8C2×C4×D5912yes—
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—42no—
492C22×C4—42no—
493C22×C4—81no—
494C22×C4—591no—
495C22×C4—591no—
496C22×C4—591no—
497C22×C4—591no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4—4721no—
503C22×C4—2362no—
504C22×C4—2362no—
505C22×C4C2×C4×D5912yes—
506C22×C4D4×D5911yescharacteristic
507C22×C4#1174568712yes—
508C22×C4C23×D5911yescharacteristic
509Q16—1188no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—592no—
517C2×C8—592no—
518C2×C8—592no—
519C2×C8—592no—
520C2×C8—1188no—
521C2×C8—1181no—
522C2×C8—1181no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—592no—
530C4⋊C4—592no—
531C4⋊C4—1184no—
532C4⋊C4—1181no—
533C4⋊C4—1181no—
534C4⋊C4—1181no—
535C4⋊C4—1181no—
536C4⋊C4—4721no—
537C4⋊C4—4721no—
538C4⋊C4—2362no—
539C4⋊C4—2362no—
540C4⋊C4—2362no—
541C4⋊C4C2×C4×D5912yes—
542C4⋊C4D4×D5912yes—
543C22⋊C4—24no—
544C22⋊C4—1184no—
545C22⋊C4—1182no—
546C22⋊C4—1182no—
547C42—1181no—
548C42—1181no—
549C42—1181no—
550C42—1181no—
551C42—4721no—
552C42—4721no—
553C42—2362no—
554C42—2362no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23(C22⋊D59)⋊C412yes—
560C23#1427375612yes—
561C23#1427375912yes—
562C23C2×D4×D5911yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—1182no—
568Q8—1182no—
569Q8—2361no—
570Q8—2361no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—1181no—
591C2×C4—1181no—
592C2×C4—1181no—
593C2×C4—1181no—
594C2×C4—1181no—
595C2×C4—1181no—
596C2×C4—1181no—
597C2×C4—1181no—
598C2×C4—1181no—
599C2×C4—1181no—
600C2×C4—1182no—
601C2×C4—1181no—
602C2×C4—1181no—
603C2×C4—1182no—
604C2×C4—1182no—
605C2×C4—4724no—
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(C22⋊D59)⋊C412yes—
616C2×C4C22×C4×D5911yescharacteristic
617C2×C4C2×D4×D5911yescharacteristic
618C2×C4#41×D5911yescharacteristic
619C2×C4C2×#1174568711yescharacteristic
620C2×C4#1427377211yescharacteristic
621C2×C4#1427377411yescharacteristic
622C8—41no—
623C8—41no—
624C8—594no—
625C8—1182no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22#1683750614yescomplemented
633C22C2×(C22⋊D59)⋊C411yescharacteristic
634C22#1683803411yescharacteristic
635C22#1683805011yescharacteristic
636C22C2×#1427375611yescharacteristic
637C22C2×#1427375911yescharacteristic
638C22#1683821111yescharacteristic
639C22#1683821211yescharacteristic
640C4—22no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—81no—
645C4—4721no—
646C4—2361no—
647C4—2361no—
648C4—2362no—
649C4—2362no—
650C4—2361no—
651C4—2362no—
652C4—2361no—
653C4C4×D4×D5911yescharacteristic
654C4#1683788111yescharacteristic
655C4#1683805811yescharacteristic
656C4#1683809511yescharacteristic
657C2—24no—
658C2#2142710711yescharacteristic
659C2C2×#1683750611yescharacteristic
660C2#2143297511yescharacteristic
661C2#2143559911yescharacteristic
662C2#2143561011yescharacteristic
663C2#2143654911yescharacteristic
664C2#2143655511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1872C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1872D59
hypocenterC59#23028
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#16837506C2semidirect
6. #21424964C2semidirect
7. #21424833C2semidirect
8. #21424733C2semidirect
9. #21424728C2semidirect
11. #21424365C2semidirect
12. #21424362C2semidirect
13. #21423992C2semidirect
22. #16837641C4semidirect
66. C2×#11745672C2×C4semidirect
98. #14273660C2×C4semidirect
105. #14273651C2×C4semidirect
106. #14273650C2×C4semidirect
300. C22×C59#441semidirect
394. C59#23028semidirect
632. C22#16837506semidirect

Maximal subgroups

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

subgroupindex
C2×#168376412
C2×#168375062
C22×#142736582
#214271052
#214271012
#214249642
#214248332
#214247332
#214247282
#214244012
#214243652
#214243622
#214239922
#214238752
C59×#18722
#2302859