Finite Group Databasebeta

← #28634053

Subgroups — #28634053

Order 15104.

700 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429014C211yesmaximal, characteristic
2#21425663C211yesmaximal, characteristic, complemented
3#21425658C211yesmaximal, characteristic, complemented
4#21425510C211yesmaximal, characteristic
5#21425500C211yesmaximal, characteristic
6#21425037C211yesmaximal, characteristic, complemented
7#21424854C211yesmaximal, characteristic, complemented
8#21424716C211yesmaximal, characteristic
9#21424716C211yesmaximal, characteristic
10#21424688C211yesmaximal, characteristic, complemented
11#21424667C211yesmaximal, characteristic, complemented
12#21424229C211yesmaximal, characteristic, complemented
13#21424103C211yesmaximal, characteristic
14#21423924C211yesmaximal, characteristic
15C59×#2355C211yesmaximal, characteristic, complemented
16C2×#14273703—21no—
17C2×C4×D236C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273647C2211yescharacteristic
22C2×#14273647C2211yescharacteristic
23C2×#14273647C2211yescharacteristic
24C2×#14273647C2211yescharacteristic
25C2×#14273638C2211yescharacteristic
26#16837324—21no—
27#16837324—21no—
28#16837304C2211yescharacteristic
29#16837304C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837272C2211yescharacteristic
33#16837259—21no—
34#16837164C2211yescharacteristic
35#16837164C2211yescharacteristic
36#16837164C2211yescharacteristic
37#16837164C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4×Q8×C59C2211yescharacteristic
49C59×#391—21no—
50C59×#387—21no—
51C59×#387—21no—
52C59×#384C2211yescharacteristic
53C59×#382C2211yescharacteristic
54C59×#382C2211yescharacteristic
55C4×C4⋊C4×C59—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C4×D59—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273703—44no—
66C4×D236—22no—
67C4×D236—22no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80#14273660—22no—
81#14273660—22no—
82C4⋊(C59⋊C8)—22no—
83#14273647—22no—
84#14273647—22no—
85#14273647—22no—
86#14273647—22no—
87#14273638—22no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C22×Q8×C59D411yescharacteristic
91C4×Q8×C59—22no—
92C4×Q8×C59—22no—
93C2×C4⋊C4×C59—21no—
94C2×C4⋊C4×C59—21no—
95C2×C4⋊C4×C59D411yescharacteristic
96C2×C4⋊C4×C59D411yescharacteristic
97C2×C4⋊C4×C59D411yescharacteristic
98C2×C4⋊C4×C59C2311yescharacteristic
99C2×C4⋊C4×C59C2311yescharacteristic
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C42×C59—21no—
103C2×C42×C59—21no—
104C2×C42×C59—21no—
105C2×C42×C59C2311yescharacteristic
106C2×C42×C59C2311yescharacteristic
107C2×C42×C59C2311yescharacteristic
108#95×C59—21no—
109#95×C59—21no—
110#95×C59—21no—
111#95×C59—21no—
112#95×C59—21no—
113#95×C59—21no—
114#95×C59—21no—
115C23×D59—21no—
116C2×Q8×C59—22no—
117C2×Q8×C59—22no—
118C2×Q8×C59—41no—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2411yescharacteristic
130C22×Dic59—21no—
131C22×Dic59—21no—
132C22×Dic59—21no—
133C2×D236—21no—
134C2×D236—21no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236#4112yes—
139C2×C4×D59—42no—
140C2×C4×D59—42no—
141C2×C4×D59—42no—
142C4⋊C4×C59—22no—
143C4⋊C4×C59—22no—
144C4⋊C4×C59—22no—
145C4⋊C4×C59—21no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—42no—
153C4⋊C4×C59—81no—
154C4⋊C4×C59#4112yes—
155C42×C59—21no—
156C42×C59—21no—
157C42×C59—21no—
158C42×C59—21no—
159C42×C59—42no—
160C42×C59—81no—
161C42×C59—81no—
162C42×C59#4112yes—
163C42×C59#4112yes—
164(C2×D59)⋊C4—42no—
165(C2×D59)⋊C4—42no—
166(C2×D59)⋊C4—42no—
167(C2×D59)⋊C4—81no—
168(C2×D59)⋊C4—81no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—42no—
176C4⋊Dic59—81no—
177Dic59⋊C4—42no—
178Dic59⋊C4—42no—
179C2×C59⋊C8—21no—
180C2×C59⋊C8—21no—
181C2×C59⋊C8—21no—
182C2×C59⋊C8—21no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C23×C59C22×D411yescharacteristic
192C22×D59—22no—
193C22×D59—41no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199Q8×C59—44no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—22no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59#12312yes—
228C2×C4×C59#12312yes—
229C2×C4×C59AGL(1,ℤ/8)12yes—
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C592−1+411yescharacteristic
235C2×C4×C592−1+411yescharacteristic
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245D236—22no—
246D236—22no—
247D236—41no—
248D236—41no—
249C4×D59—84no—
250C59⋊C8—41no—
251C59⋊C8—41no—
252C59⋊C8—41no—
253C59⋊C8—41no—
254#25007—591nomaximal, Sylow
255C22×C59C2×#12311yescharacteristic
256C22×C59#54811yescharacteristic
257C22×C59#54811yescharacteristic
258C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C59#57511yescharacteristic
260C22×C59#57611yescharacteristic
261C22×C59#57811yescharacteristic
262C2×D59—42no—
263C2×D59—42no—
264C2×D59—42no—
265C2×D59—81no—
266C2×D59—81no—
267C2×D59—81no—
268C2×D59—81no—
269C4×C59—22no—
270C4×C59—22no—
271C4×C59—22no—
272C4×C59—41no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59—42no—
276C4×C59—81no—
277C4×C59—81no—
278C4×C59—81no—
279C4×C59#52911yescharacteristic
280C4×C59#54111yescharacteristic
281C4×C59#55211yescharacteristic
282C4×C59#55611yescharacteristic
283Dic59—81no—
284Dic59—81no—
285Dic59—81no—
286#2538—591no—
287#2355D5911yescharacteristic, complemented
288#2101—591no—
289#2100—591no—
290#2073—591no—
291#2071—591no—
292#1985—591no—
293#1947—591no—
294#1925—591no—
295#1925—591no—
296#1921—591no—
297#1918—591no—
298#1836—591no—
299#1804—591no—
300#1762—591no—
301C2×C59#242111yescharacteristic
302C2×C59#309411yescharacteristic
303C2×C59#309811yescharacteristic
304C2×C59#336611yescharacteristic
305C2×C59#336711yescharacteristic
306C2×C59#337011yescharacteristic
307C2×C59#337711yescharacteristic
308D59—82no—
309C2×C22⋊Q8—1181no—
310C2×C4×Q8C2×D5911yescharacteristic
311C2×C4×D4—591no—
312C2×#107—591no—
313C2×#107—591no—
314C2×C4⋊C8—591no—
315C2×#102—591no—
316C2×#102—591no—
317C2×#102—591no—
318C2×#102—591no—
319C2×#97—591no—
320#396—1181no—
321#396—1181no—
322#391—21no—
323#390—591no—
324#390—591no—
325#389—591no—
326#389—591no—
327#387—21no—
328#387—21no—
329#384—591no—
330#384C2×D5911yescharacteristic
331#382C2×D5911yescharacteristic
332#382C2×D5911yescharacteristic
333#380—1181no—
334C4×C4⋊C4—21no—
335C4×C4⋊C4C2×D5911yescharacteristic
336C4×C4⋊C4C2×D5911yescharacteristic
337C4×C4⋊C4C2×D5911yescharacteristic
338#340—591no—
339#340—591no—
340#340—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#336—591no—
349#336—591no—
350#336—591no—
351#336—591no—
352C59#2500711yescharacteristic, complemented, Sylow
353C22×Q8C22⋊D5911yescharacteristic
354C22×D4—591no—
355C23×C4—1181no—
356C22×C8—591no—
357C22×C8—591no—
358C22×C8—591no—
359C22×C8—591no—
360C22⋊Q8—2364no—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×D4—1182no—
364C4×D4—1182no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4C22⋊D5911yescharacteristic
375C2×C4⋊C4C22⋊D5911yescharacteristic
376C2×C4⋊C4C22⋊D5911yescharacteristic
377C2×C4⋊C4C22×D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C22⋊C4—1181no—
382C2×C22⋊C4—1181no—
383C2×C22⋊C4—1181no—
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1181no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42C22×D5911yescharacteristic
390C2×C42C22×D5911yescharacteristic
391C2×C42C22×D5911yescharacteristic
392#107—1182no—
393#107—1182no—
394C4⋊C8—1182no—
395#102—1182no—
396#102—1182no—
397#102—1182no—
398#102—1182no—
399#97—1182no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—1181no—
408#95—1181no—
409C24—1181no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—41no—
413C2×D4—592no—
414C2×D4—1181no—
415C2×D4—1181no—
416C2×D4—1181no—
417C2×D4—1181no—
418C2×D4—1181no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4—2362no—
427C22×C4—2362no—
428C22×C4—2362no—
429C22×C4C2×C22⋊D5911yescharacteristic
430C22×C4C2×C22⋊D5911yescharacteristic
431C22×C4C2×C22⋊D5911yescharacteristic
432C22×C4C2×C22⋊D5911yescharacteristic
433C22×C4C2×C22⋊D5911yescharacteristic
434C22×C4C2×C22⋊D5911yescharacteristic
435C22×C4C23×D5911yescharacteristic
436C2×C8—1181no—
437C2×C8—1181no—
438C2×C8—1181no—
439C2×C8—1181no—
440C2×C8—1181no—
441C2×C8—1181no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—4721no—
467C4⋊C4—2362no—
468C4⋊C4—2362no—
469C4⋊C4—2362no—
470C4⋊C4#1174568912yes—
471C22⋊C4—4721no—
472C22⋊C4—4721no—
473C22⋊C4—2362no—
474C22⋊C4—2362no—
475C22⋊C4—2362no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—42no—
481C42—81no—
482C42—81no—
483C42#1174568512yes—
484C42#1174568512yes—
485C23—1182no—
486C23—2361no—
487C23—2361no—
488C23—2361no—
489C23—2361no—
490C23—2361no—
491C23—2361no—
492C23C22×C22⋊D5911yescharacteristic
493Q8—44no—
494D4—1182no—
495D4—1182no—
496D4—2361no—
497D4—2361no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—4724no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4#1427371012yes—
536C2×C4#1427371012yes—
537C2×C4#1427375212yes—
538C2×C4#41×D5911yescharacteristic
539C2×C4C2×#1174568511yescharacteristic
540C2×C4C2×#1174568511yescharacteristic
541C2×C4C2×#1174568911yescharacteristic
542C2×C4#1427377411yescharacteristic
543C2×C4#1427377511yescharacteristic
544C8—2361no—
545C8—2361no—
546C8—2361no—
547C8—2361no—
548C22—4721no—
549C22—4721no—
550C22—4721no—
551C22—4721no—
552C22—2362no—
553C22—2362no—
554C22—2362no—
555C22C2×#1427371011yescharacteristic
556C22#1683805911yescharacteristic
557C22#1683806311yescharacteristic
558C22C2×#1427375211yescharacteristic
559C22#1683820011yescharacteristic
560C22#1683820511yescharacteristic
561C22#1683821711yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—4721no—
573C4—4721no—
574C4—4721no—
575C4#1683793911yescharacteristic
576C4#1683801011yescharacteristic
577C4#1683809311yescharacteristic
578C4#1683812511yescharacteristic
579C2—4722no—
580C2#2142759311yescharacteristic
581C2#2143403611yescharacteristic
582C2#2143407411yescharacteristic
583C2#2143680311yescharacteristic
584C2#2143681011yescharacteristic
585C2#2143684311yescharacteristic
586C2#2143693811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2355C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2355D59
hypocenterC59#25007
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2355

Complements

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

NH ≅ G/Ntype
2. #21425663C2semidirect
3. #21425658C2semidirect
6. #21425037C2semidirect
7. #21424854C2semidirect
10. #21424688C2semidirect
11. #21424667C2semidirect
12. #21424229C2semidirect
15. C59×#2355C2semidirect
287. #2355D59semidirect
352. C59#25007semidirect

Maximal subgroups

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

subgroupindex
#214290142
#214256632
#214256582
#214255102
#214255002
#214250372
#214248542
#214247162
#214246882
#214246672
#214242292
#214241032
#214239242
C59×#23552
#2500759