Finite Group Databasebeta

← #28611486

Subgroups — #28611486

Order 15104.

942 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837660C211yesmaximal, characteristic
2C2×#16837508C211yesmaximal, characteristic, complemented
3#21427123C211yesmaximal, characteristic
4#21427101C211yesmaximal, characteristic
5#21424955C211yesmaximal, characteristic, complemented
6#21424837C211yesmaximal, characteristic, complemented
7#21424736C211yesmaximal, characteristic, complemented
8#21424729C211yesmaximal, characteristic, complemented
9#21424408C211yesmaximal, characteristic
10#21424403C211yesmaximal, characteristic
11#21424361C211yesmaximal, characteristic, complemented
12#21424347C211yesmaximal, characteristic, complemented
13#106×Dic59C211yesmaximal, characteristic, complemented
14#21423877C211yesmaximal, characteristic
15C2×C59×#428C211yesmaximal, characteristic
16C22×Dic236C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×#14273692C2211yescharacteristic
20C2×#14273692C2211yescharacteristic
21#16837660C418yescomplemented
22#16837508—24no—
23C2×#14273659C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×C22⋊(C59⋊C8)C2211yescharacteristic
31C2×#14273637C2211yescharacteristic
32C2×C8×Dic59C2211yescharacteristic
33#16837307C2211yescharacteristic
34#16837307C2211yescharacteristic
35#16837298C2211yescharacteristic
36#16837298C2211yescharacteristic
37#16837297—21no—
38#16837296—21no—
39#16837290—21no—
40#16837286—21no—
41#16837283—21no—
42#16837275—21no—
43C4⋊C4×Dic59C2211yescharacteristic
44C4⋊C4×Dic59C2211yescharacteristic
45C4×Dic59⋊C4—21no—
46C4×C22⋊Dic59—21no—
47C4×C22⋊Dic59C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53#16837156C2211yescharacteristic
54C22×M16×C59C2211yescharacteristic
55C2×#117×C59C2211yescharacteristic
56C22×C4⋊C4×C59C2211yescharacteristic
57C59×#428—24no—
58C2×#107×C59C2211yescharacteristic
59C2×#107×C59C2211yescharacteristic
60C2×#106×C59C2211yescharacteristic
61C2×#106×C59C2211yescharacteristic
62C22×Dic118C2311yescharacteristic
63C22×Dic118C2311yescharacteristic
64C2×Dic236—22no—
65C2×Dic236—22no—
66C2×Dic236C2×C414yescomplemented
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273706—22no—
70#14273706—22no—
71#14273706C2×C414yes—
72#14273706C2×C414yes—
73#14273692—22no—
74#14273692—22no—
75#14273692—41no—
76#14273692—41no—
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×C22⋊Dic59—21no—
86C2×C22⋊Dic59—21no—
87C2×C22⋊Dic59D411yescharacteristic
88C2×C22⋊Dic59D411yescharacteristic
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59C2311yescharacteristic
94C2×C4×Dic59C2311yescharacteristic
95#14273659C2×C414yescomplemented
96#14273652C2×C414yescomplemented
97#14273652C2×C414yescomplemented
98#14273650—22no—
99#14273650—22no—
100#14273650—41no—
101#14273650—41no—
102C22⋊(C59⋊C8)—22no—
103#14273637—22no—
104C8×Dic59—22no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C59⋊#95D411yescharacteristic
114C59⋊#95D411yescharacteristic
115C23×C4×C59—21no—
116C23×C4×C59C2311yescharacteristic
117C2×M16×C59—22no—
118C2×M16×C59—22no—
119C2×M16×C59C2×C414yes—
120C22×C8×C59C2311yescharacteristic
121C22×C8×C59C2311yescharacteristic
122#117×C59—22no—
123#117×C59—41no—
124C2×C4⋊C4×C59—22no—
125C2×C4⋊C4×C59—21no—
126C2×C4⋊C4×C59—41no—
127C2×C4⋊C4×C59C2311yescharacteristic
128C2×C4⋊C4×C59C2311yescharacteristic
129C2×C4⋊C4×C59C2311yescharacteristic
130C2×C4⋊C4×C59C2311yescharacteristic
131C2×C22⋊C4×C59—21no—
132C2×C42×C59—21no—
133#107×C59—22no—
134#107×C59—22no—
135#106×C59—22no—
136#106×C59D414yes—
137C24×C59C2×D411yescharacteristic
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—42no—
147C22×C4×C59—42no—
148C22×C4×C59—81no—
149C22×C4×C59C22×C412yes—
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59#4112yes—
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59C2×D411yescharacteristic
158C22×Dic59C2×D411yescharacteristic
159C22×Dic59C2×D411yescharacteristic
160C22×Dic59C2×D411yescharacteristic
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118C22×C412yes—
172C2×Dic118C22×C412yes—
173M16×C59—28no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59C22×C412yes—
177C2×C8×C59C22×C412yes—
178C2×C8×C59C2×D412yes—
179C2×C8×C59#4112yes—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—42no—
187C4⋊C4×C59C2×D412yes—
188C4⋊C4×C59C2×D412yes—
189C22⋊C4×C59—42no—
190C42×C59—42no—
191C22⋊Dic59—24no—
192C22⋊Dic59—24no—
193C22⋊Dic59—22no—
194C22⋊Dic59—41no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59C22×C412yes—
198C4⋊Dic59C22×C412yes—
199Dic59⋊C4—24no—
200Dic59⋊C4—24no—
201Dic59⋊C4—42no—
202Dic59⋊C4—42no—
203Dic59⋊C4—42no—
204Dic59⋊C4—81no—
205C4×Dic59—21no—
206C4×Dic59—21no—
207C4×Dic59—21no—
208C4×Dic59—21no—
209C4×Dic59—42no—
210C4×Dic59—42no—
211C4×Dic59—42no—
212C4×Dic59—81no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8—21no—
219Dic236—28no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59C4×D412yes—
227C23×C59C22×D411yescharacteristic
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—84no—
247C2×C4×C59C4×D412yes—
248C2×C4×C59C23×C411yescharacteristic
249C2×C4×C59C22×D411yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59H(5,𝔽2)11yescharacteristic
253C2×C4×C592−1+411yescharacteristic
254C2×Dic59—21no—
255C2×Dic59—21no—
256C2×Dic59—22no—
257C2×Dic59—22no—
258C2×Dic59—22no—
259C2×Dic59—22no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—21no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—21no—
267C2×Dic59—21no—
268C2×Dic59—21no—
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#13712yes—
278Dic118—22no—
279Dic118—22no—
280Dic118—22no—
281Dic118—22no—
282Dic118—41no—
283Dic118—41no—
284Dic118—41no—
285Dic118—41no—
286C8×C59—22no—
287C8×C59C4×D414yes—
288C59⋊C8—41no—
289C59⋊C8—41no—
290#23038—591nomaximal, Sylow
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59—41no—
298C22×C59C2×C4×D411yescharacteristic
299C22×C59D4211yescharacteristic
300C22×C59#54711yescharacteristic
301C22×C59C2×#13711yescharacteristic
302C22×C59#57511yescharacteristic
303C22×C59#57711yescharacteristic
304C22×C59#57811yescharacteristic
305C4×C59—22no—
306C4×C59—42no—
307C4×C59—42no—
308C4×C59—81no—
309C4×C59—81no—
310C4×C59C2×C4×D411yescharacteristic
311C4×C59#52011yescharacteristic
312C4×C59#54811yescharacteristic
313C4×C59#55311yescharacteristic
314Dic59—22no—
315Dic59—41no—
316Dic59—42no—
317Dic59—42no—
318Dic59—42no—
319Dic59—42no—
320Dic59—41no—
321Dic59—41no—
322C2×#470—591no—
323C2×#441—591no—
324C2×#428D5911yescharacteristic
325#2362—591no—
326#2360—591no—
327#1969—591no—
328#1944—591no—
329#1928—591no—
330#1927—591no—
331#1873—591no—
332#1872—591no—
333#1865—591no—
334#1863—591no—
335C4×#106—591no—
336#1753—591no—
337C2×C59—42no—
338C2×C59#236111yescharacteristic
339C2×C59C4×#13711yescharacteristic
340C2×C59#294711yescharacteristic
341C2×C59#327511yescharacteristic
342C2×C59#327611yescharacteristic
343C2×C59#334411yescharacteristic
344C2×C59#334911yescharacteristic
345C22×Q16—591no—
346C22×M16C2×D5911yescharacteristic
347C2×C22⋊Q8—591no—
348C2×C22⋊Q8—591no—
349C2×C4×Q8—591no—
350C2×C4×Q8—591no—
351C2×#117C2×D5911yescharacteristic
352C22×C4⋊C4C2×D5911yescharacteristic
353#470—598no—
354#441—1184no—
355#428—24no—
356C2×#107C2×D5911yescharacteristic
357C2×#107C2×D5911yescharacteristic
358C2×#106—591no—
359C2×#106C2×D5911yescharacteristic
360C2×#106C2×D5911yescharacteristic
361C2×#103—591no—
362C2×#103—591no—
363C2×#103—591no—
364C2×#103—591no—
365C2×#103—591no—
366C2×#103—591no—
367C2×C22⋊C8—591no—
368C2×#97—591no—
369C2×C4×C8—591no—
370#391—591no—
371#391—591no—
372#389—591no—
373#389—591no—
374#389—1181no—
375#388—1181no—
376#388—1181no—
377#387—1181no—
378#387—1181no—
379#385—1181no—
380C4×C4⋊C4—591no—
381C4×C4⋊C4—591no—
382C4×C4⋊C4—1181no—
383C4×C22⋊C4—591no—
384C4×C22⋊C4—1181no—
385#340—591no—
386#340—591no—
387#340—591no—
388#340—591no—
389#336—591no—
390#336—591no—
391C59#2303811yescharacteristic, complemented, Sylow
392C22×Q8—591no—
393C22×Q8—591no—
394C23×C4—21no—
395C23×C4C22×D5911yescharacteristic
396C2×Q16—594no—
397C2×Q16—1182no—
398C2×Q16—1182no—
399C2×M16—22no—
400C2×M16—22no—
401C2×M16C4×D5914yes—
402C22×C8—591no—
403C22×C8—591no—
404C22×C8C22×D5911yescharacteristic
405C22×C8C22×D5911yescharacteristic
406C22⋊Q8—594no—
407C22⋊Q8—594no—
408C22⋊Q8—1182no—
409C22⋊Q8—1182no—
410C4×Q8—1182no—
411C4×Q8—1182no—
412C4×Q8—2361no—
413C4×Q8—2361no—
414#117—22no—
415#117—41no—
416C2×C4⋊C4—22no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—41no—
419C2×C4⋊C4—591no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4C22×D5911yescharacteristic
428C2×C4⋊C4C22×D5911yescharacteristic
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C4⋊C4C22×D5911yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—591no—
433C2×C22⋊C4—591no—
434C2×C22⋊C4—1181no—
435C2×C22⋊C4—1181no—
436C2×C42—21no—
437C2×C42—591no—
438C2×C42—591no—
439C2×C42—1181no—
440C2×C42—1181no—
441C2×C42—1181no—
442C2×C42—1181no—
443#107—22no—
444#107—22no—
445#106—22no—
446#106—594no—
447#106C22⋊D5914yes—
448#103—594no—
449#103—594no—
450#103—1182no—
451#103—1182no—
452#103—2361no—
453#103—2361no—
454C22⋊C8—1182no—
455#97—1182no—
456C4×C8—1182no—
457#95—591no—
458#95—591no—
459#95—1181no—
460#95—1181no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464#95—1181no—
465#95—1181no—
466#95—1181no—
467C24D4×D5911yescharacteristic
468C2×Q8—592no—
469C2×Q8—592no—
470C2×Q8—1181no—
471C2×Q8—1181no—
472C2×Q8—1181no—
473C2×Q8—1181no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×Q8—1181no—
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—42no—
489C22×C4—42no—
490C22×C4—81no—
491C22×C4—591no—
492C22×C4—591no—
493C22×C4—591no—
494C22×C4—591no—
495C22×C4—1181no—
496C22×C4—1181no—
497C22×C4—1181no—
498C22×C4—1181no—
499C22×C4C2×C4×D5912yes—
500C22×C4D4×D5911yescharacteristic
501C22×C4#1174568912yes—
502C22×C4C23×D5911yescharacteristic
503Q16—1188no—
504M16—28no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8C2×C4×D5912yes—
514C2×C8C2×C4×D5912yes—
515C2×C8#1174568512yes—
516C2×C8C2×C22⋊D5912yes—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—42no—
524C4⋊C4—592no—
525C4⋊C4—592no—
526C4⋊C4—1184no—
527C4⋊C4—1184no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—4721no—
531C4⋊C4—2362no—
532C4⋊C4—2362no—
533C4⋊C4—2362no—
534C4⋊C4C2×C22⋊D5912yes—
535C4⋊C4C2×C22⋊D5912yes—
536C22⋊C4—42no—
537C22⋊C4—1182no—
538C22⋊C4—1184no—
539C22⋊C4—1184no—
540C22⋊C4—2361no—
541C42—42no—
542C42—1181no—
543C42—1181no—
544C42—1181no—
545C42—1181no—
546C42—4721no—
547C42—2362no—
548C42—2362no—
549C42—2362no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23#1427368912yes—
557C23C2×D4×D5911yescharacteristic
558Q8—1182no—
559Q8—1182no—
560Q8—1182no—
561Q8—1182no—
562Q8—2361no—
563Q8—2361no—
564Q8—2361no—
565Q8—2361no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—592no—
586C2×C4—1181no—
587C2×C4—1181no—
588C2×C4—1181no—
589C2×C4—1181no—
590C2×C4—1181no—
591C2×C4—1181no—
592C2×C4—1181no—
593C2×C4—1181no—
594C2×C4—1181no—
595C2×C4—1181no—
596C2×C4—1181no—
597C2×C4—1182no—
598C2×C4—1182no—
599C2×C4—1182no—
600C2×C4—1182no—
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#1427368912yes—
610C2×C4C22×C4×D5911yescharacteristic
611C2×C4C22×C22⋊D5911yescharacteristic
612C2×C4C2×#1174568511yescharacteristic
613C2×C4C2×#1174568911yescharacteristic
614C2×C4#1427377211yescharacteristic
615C2×C4#1427377511yescharacteristic
616C8—22no—
617C8—2361no—
618C8—2361no—
619C8C4×C22⋊D5914yes—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22C2×#1427368911yescharacteristic
628C22D4×C22⋊D5911yescharacteristic
629C22#1683805311yescharacteristic
630C22#137×D5911yescharacteristic
631C22#1683819811yescharacteristic
632C22#1683821211yescharacteristic
633C22#1683821811yescharacteristic
634C4—22no—
635C4—42no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—1182no—
640C4—2361no—
641C4—2362no—
642C4—2361no—
643C4—2362no—
644C4—2362no—
645C4—2362no—
646C4—2361no—
647C4C2×C4×C22⋊D5911yescharacteristic
648C4#1683788311yescharacteristic
649C4#1683806311yescharacteristic
650C4#1683810011yescharacteristic
651C2—42no—
652C2#2142711211yescharacteristic
653C2#2143291411yescharacteristic
654C2#2143293711yescharacteristic
655C2#2143572711yescharacteristic
656C2#2143573611yescharacteristic
657C2#2143655211yescharacteristic
658C2#2143662711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#428C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenterC2×#428D59
hypocenterC59#23038
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C2×#16837508C2semidirect
5. #21424955C2semidirect
6. #21424837C2semidirect
7. #21424736C2semidirect
8. #21424729C2semidirect
11. #21424361C2semidirect
12. #21424347C2semidirect
13. #106×Dic59C2semidirect
21. #16837660C4semidirect
66. C2×Dic236C2×C4semidirect
95. #14273659C2×C4semidirect
96. #14273652C2×C4semidirect
97. #14273652C2×C4semidirect
391. C59#23038semidirect

Maximal subgroups

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

subgroupindex
C2×#168376602
C2×#168375082
#214271232
#214271012
#214249552
#214248372
#214247362
#214247292
#214244082
#214244032
#214243612
#214243472
#106×Dic592
#214238772
C2×C59×#4282
#2303859