Finite Group Databasebeta

← #28617383

Subgroups — #28617383

Order 15104.

1178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837444C211yesmaximal, characteristic, complemented
2C2×#16837404C211yesmaximal, characteristic
3C2×#16837404C211yesmaximal, characteristic
4#21426586C211yesmaximal, characteristic
5C2×#16837300C211yesmaximal, characteristic, complemented
6#21425490C212yesmaximal, complemented
7#21425483C212yesmaximal, complemented
8#21425476C212yesmaximal, complemented
9#21425469C212yesmaximal, complemented
10#21423816C211yesmaximal, characteristic, complemented
11C2×C59×#423C211yesmaximal, characteristic, complemented
12C22×#11745685C2211yescharacteristic
13C2×#14273686C2211yescharacteristic
14C22×C4⋊Dic59C2211yescharacteristic
15C22×C4⋊Dic59C2211yescharacteristic
16C22×Dic59⋊C4—21no—
17#16837444—24no—
18C2×#14273661C2212yes—
19C2×#14273659C2212yes—
20#16837404—24no—
21#16837404—24no—
22C2×#14273652C2212yes—
23C2×#14273652C2212yes—
24C2×#14273649C2212yes—
25C2×#14273649C2212yes—
26#16837308C2211yescharacteristic
27#16837308C2211yescharacteristic
28#16837304C2211yescharacteristic
29#16837304C2211yescharacteristic
30#16837300—21no—
31#16837300—21no—
32#16837300C2212yes—
33#16837300C2212yes—
34#16837281—21no—
35#16837281—21no—
36C4×(C2×D59)⋊C4—21no—
37C4×(C2×D59)⋊C4—21no—
38C2×C59⋊#95—21no—
39#16837165C2212yes—
40#16837165C2212yes—
41C22×M16×C59C2211yescharacteristic
42C22×M16×C59C2211yescharacteristic
43C22×C42×C59C2211yescharacteristic
44C59×#423—24no—
45C2×C4⋊C8×C59C2212yes—
46C2×C4⋊C8×C59C2212yes—
47C2×#11745685—22no—
48C2×#11745685—22no—
49C2×#11745685—22no—
50C2×#11745685—22no—
51C2×#11745685—22no—
52C2×#11745685Q814yes—
53C22×Dic118C2311yescharacteristic
54C22×D236C2311yescharacteristic
55C22×C22⋊D59—21no—
56C23×Dic59—21no—
57C23×Dic59—21no—
58C22×C4×D59—21no—
59#14273686—22no—
60#14273686—22no—
61C2×C4⋊Dic59—21no—
62C2×C4⋊Dic59—22no—
63C2×C4⋊Dic59—22no—
64C2×C4⋊Dic59—22no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59D414yes—
67C2×C4⋊Dic59C2312yes—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2312yes—
71C2×Dic59⋊C4—22no—
72C2×Dic59⋊C4—22no—
73C2×Dic59⋊C4—44no—
74C2×C22⋊Dic59—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×C4×Dic59—21no—
80#14273661—24no—
81#14273659—24no—
82#14273652—24no—
83#14273652—24no—
84#14273649—24no—
85#14273649—24no—
86C59⋊#95—22no—
87C59⋊#95—22no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C23×C4×C59D411yescharacteristic
93C23×C4×C59D411yescharacteristic
94C23×C4×C59C2311yescharacteristic
95C2×M16×C59—22no—
96C2×M16×C59—22no—
97C2×M16×C59—22no—
98C2×M16×C59—22no—
99C2×M16×C59—22no—
100C2×M16×C59D414yes—
101C22×C8×C59C2312yes—
102C22×C8×C59C2312yes—
103C2×C42×C59—22no—
104C2×C42×C59—22no—
105C2×C42×C59D411yescharacteristic
106C2×C42×C59D411yescharacteristic
107C2×C42×C59C2311yescharacteristic
108C2×C42×C59C2311yescharacteristic
109C4⋊C8×C59—24no—
110C4⋊C8×C59—24no—
111C24×C59C2×D411yescharacteristic
112C23×D59—21no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—41no—
118C22×C4×C59—41no—
119C22×C4×C59C2×D411yescharacteristic
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D412yes—
125C22×C4×C59C2×D412yes—
126C22×C4×C59C2×Q812yes—
127C22×C4×C59#4112yes—
128C22×C4×C59#4112yes—
129C22×C4×C59#4112yes—
130C22×C4×C59C2411yescharacteristic
131C2×C22⋊D59—24no—
132C2×C22⋊D59—42no—
133C2×C22⋊D59—42no—
134C22×Dic59—24no—
135C22×Dic59—21no—
136C22×Dic59—22no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—22no—
140C22×Dic59—42no—
141C22×Dic59—42no—
142C22×Dic59—42no—
143C22×Dic59—42no—
144C22×Dic59—42no—
145#11745685—28no—
146#11745685—28no—
147#11745685—44no—
148#11745685—44no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236C2×Q812yes—
155C2×C4×D59—24no—
156C2×C4×D59—42no—
157C2×C4×D59—42no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118C2×Q812yes—
164M16×C59—28no—
165M16×C59—44no—
166C2×C8×C59—22no—
167C2×C8×C59—22no—
168C2×C8×C59—22no—
169C2×C8×C59—22no—
170C2×C8×C59—22no—
171C2×C8×C59C2×D414yes—
172C42×C59—22no—
173C42×C59—22no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C22⋊Dic59—42no—
179(C2×D59)⋊C4—42no—
180(C2×D59)⋊C4—42no—
181(C2×D59)⋊C4—42no—
182(C2×D59)⋊C4—42no—
183C4⋊Dic59—24no—
184C4⋊Dic59—22no—
185C4⋊Dic59—22no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—22no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—42no—
192C4⋊Dic59C2×D414yes—
193Dic59⋊C4—44no—
194Dic59⋊C4—44no—
195C4×Dic59—42no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59#12112yes—
201C23×C59#12112yes—
202C23×C59C22⋊Q812yes—
203C23×C59C22×D411yescharacteristic
204C22×D59—22no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—41no—
210C22×D59—41no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—44no—
222C2×C4×C59—44no—
223C2×C4×C59#12114yes—
224C2×C4×C59#12112yes—
225C2×C4×C59#12112yes—
226C2×C4×C59C22⋊Q814yes—
227C2×C4×C59C22⋊Q814yes—
228C2×C4×C59C22⋊Q812yes—
229C2×C4×C59#12614yes—
230C2×C4×C59AGL(1,ℤ/8)12yes—
231C2×C4×C59#13712yes—
232C2×C4×C59C22×D411yescharacteristic
233C2×C4×C59C22×D411yescharacteristic
234C2×C4×C59C22×Q811yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C22⋊D59—48no—
239C2×Dic59—24no—
240C2×Dic59—22no—
241C2×Dic59—48no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—42no—
245C2×Dic59—42no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—42no—
249C2×Dic59—42no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—42no—
255C2×Dic59—84no—
256D236—22no—
257D236—22no—
258D236—41no—
259D236—41no—
260C4×D59—48no—
261Dic118—22no—
262Dic118—22no—
263Dic118—41no—
264Dic118—41no—
265C8×C59—24no—
266C8×C59—42no—
267#23617—591nomaximal, Sylow
268C22×C59—22no—
269C22×C59—22no—
270C22×C59—22no—
271C22×C59—22no—
272C22×C59—22no—
273C22×C59—22no—
274C22×C59#39614yes—
275C22×C59C2×#12111yescharacteristic
276C22×C59C2×#12111yescharacteristic
277C22×C59C2×C22⋊Q811yescharacteristic
278C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C59C2×#13711yescharacteristic
280C22×C59#57511yescharacteristic
281C22×C59#57511yescharacteristic
282C2×D59—42no—
283C2×D59—42no—
284C2×D59—42no—
285C2×D59—81no—
286C2×D59—81no—
287C2×D59—81no—
288C2×D59—81no—
289C4×C59—22no—
290C4×C59—22no—
291C4×C59—41no—
292C4×C59—41no—
293C4×C59#39614yes—
294C4×C59C2×#12111yescharacteristic
295C4×C59C2×C22⋊Q811yescharacteristic
296C4×C59C2×C22⋊Q811yescharacteristic
297C4×C59C2×#12611yescharacteristic
298Dic59—42no—
299Dic59—44no—
300Dic59—81no—
301Dic59—81no—
302Dic59—82no—
303C2×#428—591no—
304C2×#423D5911yescharacteristic, complemented
305C2×#417—591no—
306C2×#417—591no—
307#2297—591no—
308C2×#389—591no—
309#2068—592no—
310#2067—592no—
311#2066—592no—
312#2065—592no—
313#1736—591no—
314C2×C59—24no—
315C2×C59C2×#39611yescharacteristic
316C2×C59#303611yescharacteristic
317C2×C59#303711yescharacteristic
318C2×C59#305511yescharacteristic
319C2×C59#305611yescharacteristic
320C2×C59#307911yescharacteristic
321C2×C59#308011yescharacteristic
322D59—82no—
323C22×#41—591no—
324C22×M16C2×D5911yescharacteristic
325C22×M16C2×D5911yescharacteristic
326C2×#117—591no—
327C22×C4⋊C4—591no—
328C22×C4⋊C4—591no—
329C22×C4⋊C4—1181no—
330C22×C42C2×D5911yescharacteristic
331#428—1184no—
332C2×#107—592no—
333C2×#106—592no—
334#423—24no—
335C2×C4⋊C8C2×D5912yes—
336C2×C4⋊C8C2×D5912yes—
337#417—1184no—
338#417—1184no—
339C2×#103—592no—
340C2×#103—592no—
341C2×#102—592no—
342C2×#102—592no—
343#391—591no—
344#391—591no—
345#390—591no—
346#390—591no—
347#389—592no—
348#389—592no—
349#389—1181no—
350#389—1181no—
351#386—1181no—
352#386—1181no—
353C4×C22⋊C4—1181no—
354C4×C22⋊C4—1181no—
355C2×#95—1181no—
356#340—592no—
357#340—592no—
358C59#2361711yescharacteristic, complemented, Sylow
359C2×#41—594no—
360C2×#41—1182no—
361C2×#41—1182no—
362C2×#41—1182no—
363C2×#41—1182no—
364C2×#41—1182no—
365C22×Q8—591no—
366C22×D4—591no—
367C22×D4—1181no—
368C23×C4—1181no—
369C23×C4—1181no—
370C23×C4—1181no—
371C23×C4D23611yescharacteristic
372C23×C4D23611yescharacteristic
373C23×C4C22×D5911yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—22no—
377C2×M16—22no—
378C2×M16—22no—
379C2×M16C22⋊D5914yes—
380C22×C8C22×D5912yes—
381C22×C8C22×D5912yes—
382#117—1182no—
383#117—1182no—
384C2×C4⋊C4—594no—
385C2×C4⋊C4—592no—
386C2×C4⋊C4—592no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—1182no—
392C2×C4⋊C4—1182no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1182no—
395C2×C4⋊C4—1182no—
396C2×C4⋊C4—2364no—
397C2×C22⋊C4—1181no—
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—1181no—
405C2×C42D23611yescharacteristic
406C2×C42D23611yescharacteristic
407C2×C42C22×D5911yescharacteristic
408C2×C42C22×D5911yescharacteristic
409#107—1184no—
410#106—1184no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413#103—1184no—
414#103—1184no—
415#102—1184no—
416#102—1184no—
417#95—1182no—
418#95—1182no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423C24—1181no—
424C24C2×D23611yescharacteristic
425#41—1188no—
426#41—1188no—
427#41—2364no—
428#41—2364no—
429C2×Q8—592no—
430C2×Q8—1181no—
431C2×Q8—1181no—
432C2×Q8—1181no—
433C2×Q8—1181no—
434C2×Q8—1181no—
435C2×D4—592no—
436C2×D4—1184no—
437C2×D4—1181no—
438C2×D4—1181no—
439C2×D4—1181no—
440C2×D4—1181no—
441C2×D4—1181no—
442C2×D4—2362no—
443C2×D4—2362no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—1184no—
451C22×C4—1181no—
452C22×C4—1182no—
453C22×C4—1184no—
454C22×C4—1181no—
455C22×C4—1181no—
456C22×C4—1182no—
457C22×C4—2362no—
458C22×C4—2362no—
459C22×C4—2362no—
460C22×C4—2362no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4—2362no—
464C22×C4C2×D23611yescharacteristic
465C22×C4C2×D23611yescharacteristic
466C22×C4C2×D23611yescharacteristic
467C22×C4C2×D23611yescharacteristic
468C22×C4C2×D23611yescharacteristic
469C22×C4#1174568512yes—
470C22×C4D4×D5912yes—
471C22×C4#1174568712yes—
472C22×C4Q8×D5912yes—
473C22×C4#1174568912yes—
474C22×C4C2×C22⋊D5912yes—
475C22×C4C23×D5911yescharacteristic
476M16—28no—
477M16—44no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8C2×C22⋊D5914yes—
484C4⋊C4—594no—
485C4⋊C4—1184no—
486C4⋊C4—1182no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1182no—
490C4⋊C4—1182no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—2364no—
494C4⋊C4—2362no—
495C4⋊C4—2364no—
496C22⋊C4—2362no—
497C22⋊C4—2362no—
498C22⋊C4—2362no—
499C22⋊C4—2362no—
500C22⋊C4—2362no—
501C42—22no—
502C42—22no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—2362no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—1182no—
513C23—2361no—
514C23—2361no—
515C23—2361no—
516C23—2361no—
517C23—2361no—
518C23—2361no—
519C23#1427370112yes—
520C23#1427370212yes—
521C23#1427370712yes—
522C23C22×D23611yescharacteristic
523Q8—1182no—
524Q8—1182no—
525Q8—2361no—
526Q8—2361no—
527D4—1182no—
528D4—1182no—
529D4—2368no—
530D4—2361no—
531D4—2361no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—1184no—
545C2×C4—1182no—
546C2×C4—4724no—
547C2×C4—2362no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2362no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2362no—
556C2×C4—2368no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2362no—
560C2×C4—2362no—
561C2×C4—2368no—
562C2×C4#1427369914yes—
563C2×C4#1427370112yes—
564C2×C4#1427370212yes—
565C2×C4#1427370314yes—
566C2×C4#1427370514yes—
567C2×C4#1427370712yes—
568C2×C4#1427372214yes—
569C2×C4#1427375512yes—
570C2×C4#1427376012yes—
571C2×C4C2×D4×D5911yescharacteristic
572C2×C4C22×C22⋊D5911yescharacteristic
573C2×C4C2×Q8×D5911yescharacteristic
574C2×C4C2×#1174568711yescharacteristic
575C2×C4C2×#1174568511yescharacteristic
576C2×C4C2×#1174568911yescharacteristic
577C8—24no—
578C8—42no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—4721no—
586C22—4721no—
587C22—4721no—
588C22—4721no—
589C22—2362no—
590C22—2362no—
591C22—2362no—
592C22#1683732414yes—
593C22C2×#1427370111yescharacteristic
594C22C2×#1427370211yescharacteristic
595C22C2×#1427370711yescharacteristic
596C22C2×#1427375511yescharacteristic
597C22C2×#1427376011yescharacteristic
598C22#1683820211yescharacteristic
599C22#1683820211yescharacteristic
600C4—22no—
601C4—22no—
602C4—41no—
603C4—41no—
604C4—4722no—
605C4—4721no—
606C4—4721no—
607C4—2362no—
608C4—2364no—
609C4#1683732414yes—
610C4C2×#1427369911yescharacteristic
611C4C2×#1427370311yescharacteristic
612C4C2×#1427370511yescharacteristic
613C4C2×#1427372211yescharacteristic
614C2—24no—
615C2—4722no—
616C2C2×#1683732411yescharacteristic
617C2#2143356211yescharacteristic
618C2#2143357011yescharacteristic
619C2#2143371511yescharacteristic
620C2#2143372611yescharacteristic
621C2#2143391211yescharacteristic
622C2#2143391911yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#423C2
socleC23×C59C22×D4
prime radicalC22×#11745685C22
hypercenterC2×#423D59
hypocenterC59#23617
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
1. C2×#16837444C2semidirect
5. C2×#16837300C2semidirect
6. #21425490C2semidirect
7. #21425483C2semidirect
8. #21425476C2semidirect
9. #21425469C2semidirect
10. #21423816C2semidirect
11. C2×C59×#423C2semidirect
304. C2×#423D59semidirect
358. C59#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#168374442
C2×#168374042
#214265862
C2×#168373002
#214254902
#214254832
#214254762
#214254692
#214238162
C2×C59×#4232
#2361759