Finite Group Databasebeta

← #28634340

Subgroups — #28634340

Order 15104.

834 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837738C211yesmaximal, characteristic, complemented
2C2×#16837744C211yesmaximal, characteristic, complemented
3#21429018C211yesmaximal, characteristic
4#21427025C211yesmaximal, characteristic
5#21425529C211yesmaximal, characteristic, complemented
6#21425519C211yesmaximal, characteristic, complemented
7#21425300C211yesmaximal, characteristic, complemented
8#21425293C211yesmaximal, characteristic, complemented
9#21424855C211yesmaximal, characteristic, complemented
10#21424297C211yesmaximal, characteristic, complemented
11#21424286C211yesmaximal, characteristic, complemented
12#21424264C211yesmaximal, characteristic, complemented
13#21424112C211yesmaximal, characteristic, complemented
14#21423927C211yesmaximal, characteristic, complemented
15C59×#1955C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20C22×C4⋊Dic59C2211yescharacteristic
21#16837744—24no—
22#16837738—24no—
23C2×#14273661C2211yescharacteristic
24C2×#14273661C2211yescharacteristic
25C2×#14273661C2211yescharacteristic
26C2×#14273661C2211yescharacteristic
27C2×#14273659C2211yescharacteristic
28C2×#14273659C2211yescharacteristic
29C2×#14273652C2211yescharacteristic
30C2×#14273652C2211yescharacteristic
31C2×#14273652C2211yescharacteristic
32C2×#14273652C2211yescharacteristic
33#16837325—21no—
34#16837325—21no—
35#16837305—21no—
36#16837298—21no—
37#16837295—21no—
38#16837288—21no—
39#16837282—21no—
40#16837272C2211yescharacteristic
41#16837268C2211yescharacteristic
42#16837262C2211yescharacteristic
43#16837262C2211yescharacteristic
44#16837262C2211yescharacteristic
45#16837262C2211yescharacteristic
46C4×C4⋊Dic59C2211yescharacteristic
47C22⋊C4×Dic59—21no—
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53#16837165C2211yescharacteristic
54C2×C4×D4×C59C2211yescharacteristic
55C2×C4⋊C8×C59C2211yescharacteristic
56C2×C22⋊C8×C59C2211yescharacteristic
57C2×C22⋊C8×C59C2211yescharacteristic
58C2×C22⋊C8×C59C2211yescharacteristic
59C2×C22⋊C8×C59C2211yescharacteristic
60C2×#97×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C23×Dic59—21no—
63#14273706—22no—
64#14273706—22no—
65#14273706—41no—
66#14273706—41no—
67C4×Dic118—22no—
68C4×Dic118—22no—
69#14273686—22no—
70#14273686—41no—
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—41no—
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×C22⋊Dic59—21no—
87C2×C22⋊Dic59—21no—
88C2×C22⋊Dic59—21no—
89C2×C22⋊Dic59—21no—
90C2×C22⋊Dic59—21no—
91C2×C22⋊Dic59—21no—
92C2×C22⋊Dic59—21no—
93C2×C4×Dic59—21no—
94C2×C4×Dic59—21no—
95C2×C4×Dic59—21no—
96#14273661—22no—
97#14273661—22no—
98#14273661—22no—
99#14273661—41no—
100#14273659—22no—
101#14273659—41no—
102#14273652—22no—
103#14273652—22no—
104#14273652—22no—
105#14273652D414yescomplemented
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C22×D4×C59D411yescharacteristic
114C23×C4×C59C2311yescharacteristic
115C23×C4×C59C2311yescharacteristic
116C22×C8×C59C2311yescharacteristic
117C22×C8×C59C2311yescharacteristic
118C22×C8×C59C2311yescharacteristic
119C22×C8×C59C2311yescharacteristic
120C4×D4×C59—22no—
121C4×D4×C59—22no—
122C2×C4⋊C4×C59D411yescharacteristic
123C2×C22⋊C4×C59D411yescharacteristic
124C2×C22⋊C4×C59D411yescharacteristic
125C2×C42×C59C2311yescharacteristic
126C4⋊C8×C59—22no—
127C22⋊C8×C59—22no—
128C22⋊C8×C59—22no—
129C22⋊C8×C59—22no—
130C22⋊C8×C59—22no—
131#97×C59—22no—
132C24×C59C2×D411yescharacteristic
133C24×C59C2×D411yescharacteristic
134C2×D4×C59—22no—
135C2×D4×C59—22no—
136C2×D4×C59—41no—
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—
162C22×Dic59—42no—
163C22×Dic59—42no—
164C22×Dic59—81no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118C2×D412yes—
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—21no—
178C2×C8×C59—21no—
179C2×C8×C59—21no—
180C2×C8×C59—21no—
181C2×C8×C59C2×D412yes—
182C2×C8×C59#4112yes—
183C4⋊C4×C59—22no—
184C22⋊C4×C59—22no—
185C22⋊C4×C59—22no—
186C42×C59#4112yes—
187C42×C59#4112yes—
188C22⋊Dic59—42no—
189C22⋊Dic59—42no—
190C22⋊Dic59—42no—
191C22⋊Dic59—42no—
192C22⋊Dic59—81no—
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—42no—
209C4⋊Dic59—42no—
210C4⋊Dic59—81no—
211C4⋊Dic59C2×D412yes—
212Dic59⋊C4—42no—
213Dic59⋊C4—42no—
214Dic59⋊C4—42no—
215C4×Dic59—42no—
216C4×Dic59—42no—
217C4×Dic59—81no—
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—21no—
229C23×C59—21no—
230C23×C59—21no—
231C23×C59#13712yes—
232C23×C59C22×D411yescharacteristic
233D4×C59—44no—
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—21no—
243C2×C4×C59—21no—
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#12312yes—
253C2×C4×C59#12312yes—
254C2×C4×C59C22×D411yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
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—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—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C2×Dic59—84no—
285Dic118—22no—
286Dic118—22no—
287Dic118—41no—
288Dic118—41no—
289C8×C59—22no—
290C8×C59—22no—
291C8×C59—41no—
292C8×C59—41no—
293#25028—591nomaximal, Sylow
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×C59—41no—
306C22×C59—41no—
307C22×C59—41no—
308C22×C59C2×#12311yescharacteristic
309C22×C59D4211yescharacteristic
310C22×C59#54711yescharacteristic
311C22×C59C2×#13711yescharacteristic
312C22×C59#57511yescharacteristic
313C22×C59#57611yescharacteristic
314C22×C59#57811yescharacteristic
315C4×C59—22no—
316C4×C59—22no—
317C4×C59—41no—
318C4×C59—41no—
319C4×C59#53211yescharacteristic
320C4×C59#54211yescharacteristic
321C4×C59#54711yescharacteristic
322C4×C59#55311yescharacteristic
323Dic59—42no—
324Dic59—42no—
325Dic59—81no—
326Dic59—81no—
327Dic59—81no—
328Dic59—81no—
329Dic59—81no—
330Dic59—81no—
331C2×#484—591no—
332C2×#485—591no—
333#2538—591no—
334#2352—591no—
335#2077—591no—
336#2075—591no—
337#2036—591no—
338#2035—591no—
339#1955D5911yescharacteristic
340#1947—591no—
341#1854—591no—
342#1852—591no—
343#1847—591no—
344#1805—591no—
345#1763—591no—
346C2×C59—42no—
347C2×C59—81no—
348C2×C59#242211yescharacteristic
349C2×C59#309511yescharacteristic
350C2×C59#309811yescharacteristic
351C2×C59#326811yescharacteristic
352C2×C59#327011yescharacteristic
353C2×C59#334611yescharacteristic
354C2×C59#334911yescharacteristic
355C2×C22⋊Q8—591no—
356C2×C22⋊Q8—591no—
357C2×C4×Q8—591no—
358C2×C4×D4C2×D5911yescharacteristic
359C2×#117—591no—
360C22×C4⋊C4—591no—
361#485—1184no—
362#484—1184no—
363C2×#107—591no—
364C2×#107—591no—
365C2×#107—591no—
366C2×#107—591no—
367C2×#106—591no—
368C2×#106—591no—
369C2×C4⋊C8C2×D5911yescharacteristic
370C2×#103—591no—
371C2×#103—591no—
372C2×#103—591no—
373C2×#103—591no—
374C2×C22⋊C8C2×D5911yescharacteristic
375C2×C22⋊C8C2×D5911yescharacteristic
376C2×C22⋊C8C2×D5911yescharacteristic
377C2×C22⋊C8C2×D5911yescharacteristic
378C2×#97C2×D5911yescharacteristic
379#396—1181no—
380#396—1181no—
381#390—1181no—
382#389—1181no—
383#388—1181no—
384#387—1181no—
385#386—1181no—
386#384—591no—
387#382—591no—
388#380—591no—
389#380—591no—
390#380—591no—
391#380—591no—
392C4×C4⋊C4—591no—
393C4×C22⋊C4—1181no—
394#340—591no—
395#340—591no—
396#340—591no—
397#340—591no—
398#340—591no—
399#340—591no—
400C59#2502811yescharacteristic, complemented, Sylow
401C22×Q8—591no—
402C22×D4D23611yescharacteristic
403C23×C4—1181no—
404C23×C4C22×D5911yescharacteristic
405C23×C4C22×D5911yescharacteristic
406C22×C8C22×D5911yescharacteristic
407C22×C8C22×D5911yescharacteristic
408C22×C8C22×D5911yescharacteristic
409C22×C8C22×D5911yescharacteristic
410C22⋊Q8—1182no—
411C22⋊Q8—1182no—
412C22⋊Q8—2361no—
413C22⋊Q8—2361no—
414C4×Q8—1182no—
415C4×Q8—1182no—
416C4×D4—22no—
417C4×D4—22no—
418#117—1182no—
419#117—2361no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—591no—
426C2×C4⋊C4—591no—
427C2×C4⋊C4—1182no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4—2361no—
435C2×C4⋊C4D23611yescharacteristic
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×C42—1181no—
448C2×C42C22×D5911yescharacteristic
449#107—1182no—
450#107—1182no—
451#107—1182no—
452#107—2361no—
453#106—1182no—
454#106—2361no—
455C4⋊C8—22no—
456#103—594no—
457#103—1182no—
458#103—1182no—
459#103—1182no—
460C22⋊C8—22no—
461C22⋊C8—22no—
462C22⋊C8—22no—
463C22⋊C8—22no—
464#97—22no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468#95—1181no—
469#95—1181no—
470#95—1181no—
471#95—1181no—
472C24C2×D23611yescharacteristic
473C24C2×D23611yescharacteristic
474C2×Q8—592no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—41no—
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—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—1181no—
496C22×C4—1181no—
497C22×C4—1181no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—4721no—
504C22×C4—2362no—
505C22×C4—2362no—
506C22×C4C2×D23611yescharacteristic
507C22×C4C2×D23611yescharacteristic
508C22×C4C2×D23611yescharacteristic
509C22×C4C2×D23611yescharacteristic
510C22×C4C23×D5911yescharacteristic
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8D4×D5912yes—
522C2×C8#1174568712yes—
523C4⋊C4—22no—
524C4⋊C4—592no—
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—1181no—
535C4⋊C4—1181no—
536C4⋊C4—1181no—
537C4⋊C4—1181no—
538C4⋊C4—4721no—
539C4⋊C4—2362no—
540C4⋊C4—2362no—
541C4⋊C4—2362no—
542C4⋊C4—2362no—
543C4⋊C4—2362no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—4721no—
547C22⋊C4—4721no—
548C22⋊C4—4721no—
549C22⋊C4—2362no—
550C22⋊C4—2362no—
551C22⋊C4—2362no—
552C22⋊C4—2362no—
553C42—4721no—
554C42—2362no—
555C42—2362no—
556C42#1174568712yes—
557C42#1174568712yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23#1427376012yes—
572C23C22×D23611yescharacteristic
573Q8—1182no—
574Q8—1182no—
575Q8—2361no—
576Q8—2361no—
577D4—44no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—1182no—
597C2×C4—1182no—
598C2×C4—4724no—
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—2361no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4—2361no—
620C2×C4—2361no—
621C2×C4#1427371112yes—
622C2×C4#1427371112yes—
623C2×C4C2×D4×D5911yescharacteristic
624C2×C4C2×#1174568711yescharacteristic
625C2×C4C2×#1174568711yescharacteristic
626C2×C4C2×#1174568711yescharacteristic
627C2×C4#1427377311yescharacteristic
628C2×C4#1427377411yescharacteristic
629C8—22no—
630C8—22no—
631C8—41no—
632C8—41no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22C2×#1427371111yescharacteristic
648C22D4×D23611yescharacteristic
649C22#1683805611yescharacteristic
650C22C2×#1427376011yescharacteristic
651C22#1683820211yescharacteristic
652C22#1683820711yescharacteristic
653C22#1683821911yescharacteristic
654C4—22no—
655C4—22no—
656C4—41no—
657C4—41no—
658C4—4721no—
659C4—4721no—
660C4—4721no—
661C4—4721no—
662C4—4721no—
663C4—4721no—
664C4—2362no—
665C4—2362no—
666C4#1683795011yescharacteristic
667C4#1683802111yescharacteristic
668C4#1683805211yescharacteristic
669C4#1683810411yescharacteristic
670C2—42no—
671C2—81no—
672C2#2142760811yescharacteristic
673C2#2143404511yescharacteristic
674C2#2143407611yescharacteristic
675C2#2143564111yescharacteristic
676C2#2143567111yescharacteristic
677C2#2143658311yescharacteristic
678C2#2143662811yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1955C2
socleC23×C59C22×D4
prime radicalC22×D4×C59D4
hypercenter#1955D59
hypocenterC59#25028
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. C2×#16837738C2semidirect
2. C2×#16837744C2semidirect
5. #21425529C2semidirect
6. #21425519C2semidirect
7. #21425300C2semidirect
8. #21425293C2semidirect
9. #21424855C2semidirect
10. #21424297C2semidirect
11. #21424286C2semidirect
12. #21424264C2semidirect
13. #21424112C2semidirect
14. #21423927C2semidirect
105. #14273652D4semidirect
400. C59#25028semidirect

Maximal subgroups

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

subgroupindex
C2×#168377382
C2×#168377442
#214290182
#214270252
#214255292
#214255192
#214253002
#214252932
#214248552
#214242972
#214242862
#214242642
#214241122
#214239272
C59×#19552
#2502859