Finite Group Databasebeta

← #28633982

Subgroups — #28633982

Order 15104.

716 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429454C211yesmaximal, characteristic
2Q8×C4⋊Dic59C211yesmaximal, characteristic, complemented
3#21425660C211yesmaximal, characteristic, complemented
4#21425655C211yesmaximal, characteristic, complemented
5#21425501C211yesmaximal, characteristic
6#21425501C211yesmaximal, characteristic
7#21424866C211yesmaximal, characteristic, complemented
8#21424724C211yesmaximal, characteristic
9#21424717C211yesmaximal, characteristic
10#21424689C211yesmaximal, characteristic, complemented
11#21424682C211yesmaximal, characteristic, complemented
12#21424230C211yesmaximal, characteristic, complemented
13#21424097C211yesmaximal, characteristic
14#21423922C211yesmaximal, characteristic
15C59×#1985C211yesmaximal, characteristic, complemented
16C2×Q8×Dic59—21no—
17C2×C4×D236C2211yescharacteristic
18C2×#14273661C2211yescharacteristic
19C2×#14273659C2211yescharacteristic
20C2×#14273649C2211yescharacteristic
21C2×#14273649C2211yescharacteristic
22C2×#14273649C2211yescharacteristic
23C2×#14273649C2211yescharacteristic
24#16837337—21no—
25#16837313—21no—
26#16837304C2211yescharacteristic
27#16837304C2211yescharacteristic
28#16837301—21no—
29#16837301—21no—
30#16837301—21no—
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837273—21no—
34#16837273—21no—
35#16837272C2211yescharacteristic
36#16837272C2211yescharacteristic
37#16837272C2211yescharacteristic
38#16837268C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52C2×C4×Q8×C59C2211yescharacteristic
53C2×C4⋊C8×C59C2211yescharacteristic
54C2×#97×C59C2211yescharacteristic
55C59×#336C2211yescharacteristic
56C59×#336C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C4×D59—21no—
61Q8×Dic59—44no—
62C4×D236—22no—
63C4×D236—22no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87#14273661—22no—
88#14273659—22no—
89#14273649—22no—
90#14273649—22no—
91#14273649—22no—
92#14273649—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C22×Q8×C59D411yescharacteristic
101C22×C8×C59C2311yescharacteristic
102C22×C8×C59C2311yescharacteristic
103C22×C8×C59C2311yescharacteristic
104C22×C8×C59C2311yescharacteristic
105C4×Q8×C59—22no—
106C4×Q8×C59—22no—
107C2×C4⋊C4×C59D411yescharacteristic
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C42×C59C2311yescharacteristic
111C2×C42×C59C2311yescharacteristic
112C2×C42×C59C2311yescharacteristic
113C4⋊C8×C59—22no—
114#97×C59—22no—
115C23×D59—21no—
116C2×Q8×C59—22no—
117C2×Q8×C59—22no—
118C2×Q8×C59—22no—
119C22×C4×C59C2×D411yescharacteristic
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2411yescharacteristic
126C22×Dic59—21no—
127C22×Dic59—21no—
128C22×Dic59—21no—
129C22×Dic59—21no—
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—81no—
141C2×C4×D59—81no—
142C2×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C4⋊C4×C59—22no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—22no—
157C42×C59—21no—
158C42×C59—21no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59#4112yes—
162C42×C59#4112yes—
163(C2×D59)⋊C4—42no—
164(C2×D59)⋊C4—42no—
165(C2×D59)⋊C4—42no—
166(C2×D59)⋊C4—81no—
167(C2×D59)⋊C4—81no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—42no—
182C4⋊Dic59—42no—
183C4⋊Dic59—42no—
184C4⋊Dic59—81no—
185C4⋊Dic59—81no—
186C4⋊Dic59—81no—
187C4⋊Dic59#4112yes—
188Dic59⋊C4—81no—
189Dic59⋊C4—81no—
190C4×Dic59—42no—
191C4×Dic59—42no—
192C4×Dic59—42no—
193C23×C59C22×D411yescharacteristic
194C22×D59—22no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C22×D59—41no—
200C22×D59—41no—
201Q8×C59—28no—
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—21no—
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#12312yes—
217C2×C4×C59#12312yes—
218C2×C4×C59AGL(1,ℤ/8)12yes—
219C2×C4×C59#13712yes—
220C2×C4×C59C2×#4111yescharacteristic
221C2×C4×C59C2×#4111yescharacteristic
222C2×C4×C59C2×#4111yescharacteristic
223C2×C4×C59C2×#4111yescharacteristic
224C2×C4×C592−1+411yescharacteristic
225C2×C4×C592−1+411yescharacteristic
226C2×Dic59—22no—
227C2×Dic59—41no—
228C2×Dic59—41no—
229C2×Dic59—41no—
230C2×Dic59—41no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—41no—
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—
245C2×Dic59—41no—
246C2×Dic59—41no—
247D236—22no—
248D236—22no—
249D236—41no—
250D236—41no—
251C4×D59—84no—
252C8×C59—41no—
253C8×C59—41no—
254C8×C59—41no—
255C8×C59—41no—
256#25002—591nomaximal, Sylow
257C22×C59C2×#12311yescharacteristic
258C22×C59#54811yescharacteristic
259C22×C59#54811yescharacteristic
260C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C59C2×#13711yescharacteristic
262C22×C59#57611yescharacteristic
263C22×C59#57711yescharacteristic
264C2×D59—42no—
265C2×D59—42no—
266C2×D59—42no—
267C2×D59—81no—
268C2×D59—81no—
269C2×D59—81no—
270C2×D59—81no—
271C4×C59—22no—
272C4×C59—22no—
273C4×C59—22no—
274C4×C59—22no—
275C4×C59—41no—
276C4×C59—41no—
277C4×C59#52911yescharacteristic
278C4×C59#54111yescharacteristic
279C4×C59#55211yescharacteristic
280C4×C59#55611yescharacteristic
281Dic59—42no—
282Dic59—81no—
283Dic59—81no—
284Dic59—81no—
285Dic59—81no—
286Dic59—81no—
287Dic59—81no—
288#2580—591no—
289Q8×C4⋊C4—591no—
290#2100—591no—
291#2099—591no—
292#2071—591no—
293#2071—591no—
294#1985D5911yescharacteristic, complemented
295#1949—591no—
296#1926—591no—
297#1925—591no—
298#1921—591no—
299#1920—591no—
300#1836—591no—
301#1802—591no—
302#1762—591no—
303C2×C59#242111yescharacteristic
304C2×C59#309211yescharacteristic
305C2×C59#309611yescharacteristic
306C2×C59#336611yescharacteristic
307C2×C59#337211yescharacteristic
308C2×C59#337311yescharacteristic
309C2×C59#337511yescharacteristic
310D59—82no—
311C2×C4×Q8—1181no—
312C2×C4×Q8C2×D5911yescharacteristic
313C2×C4×D4—591no—
314C2×#107—591no—
315C2×#106—591no—
316C2×C4⋊C8C2×D5911yescharacteristic
317C2×#102—591no—
318C2×#102—591no—
319C2×#102—591no—
320C2×#102—591no—
321C2×#97C2×D5911yescharacteristic
322#399—1181no—
323#393—1181no—
324#390—591no—
325#390—591no—
326#389—591no—
327#389—591no—
328#389—1181no—
329#389—1181no—
330#389—1181no—
331#385—1181no—
332#385—1181no—
333#384—591no—
334#384—591no—
335#384—591no—
336#382—591no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4—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#340—591no—
349#340—591no—
350#336C2×D5911yescharacteristic
351#336C2×D5911yescharacteristic
352#336C2×D5911yescharacteristic
353#336C2×D5911yescharacteristic
354C59#2500211yescharacteristic, complemented, Sylow
355C22×Q8D23611yescharacteristic
356C22×D4—591no—
357C23×C4—1181no—
358C22×C8C22×D5911yescharacteristic
359C22×C8C22×D5911yescharacteristic
360C22×C8C22×D5911yescharacteristic
361C22×C8C22×D5911yescharacteristic
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—2364no—
365C4×D4—1182no—
366C4×D4—1182no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4D23611yescharacteristic
383C2×C4⋊C4D23611yescharacteristic
384C2×C4⋊C4D23611yescharacteristic
385C2×C22⋊C4—1181no—
386C2×C22⋊C4—1181no—
387C2×C22⋊C4—1181no—
388C2×C22⋊C4—1181no—
389C2×C22⋊C4—1181no—
390C2×C42—1181no—
391C2×C42—1181no—
392C2×C42—1181no—
393C2×C42C22×D5911yescharacteristic
394C2×C42C22×D5911yescharacteristic
395C2×C42C22×D5911yescharacteristic
396#107—1182no—
397#106—1182no—
398C4⋊C8—22no—
399#102—1182no—
400#102—1182no—
401#102—1182no—
402#102—1182no—
403#97—22no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411C24—1181no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×D4—592no—
416C2×D4—1181no—
417C2×D4—1181no—
418C2×D4—1181no—
419C2×D4—1181no—
420C2×D4—1181no—
421C22×C4—1181no—
422C22×C4—1181no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4—1181no—
427C22×C4—1181no—
428C22×C4—4721no—
429C22×C4—4721no—
430C22×C4—2362no—
431C22×C4C2×D23611yescharacteristic
432C22×C4C2×D23611yescharacteristic
433C22×C4C2×D23611yescharacteristic
434C22×C4C2×D23611yescharacteristic
435C22×C4C2×D23611yescharacteristic
436C22×C4C2×D23611yescharacteristic
437C22×C4C23×D5911yescharacteristic
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—592no—
454C4⋊C4—1181no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—4721no—
468C4⋊C4—4721no—
469C4⋊C4—4721no—
470C4⋊C4—4721no—
471C4⋊C4—4721no—
472C4⋊C4—2362no—
473C4⋊C4—2362no—
474C4⋊C4—2362no—
475C22⋊C4—4721no—
476C22⋊C4—4721no—
477C22⋊C4—2362no—
478C22⋊C4—2362no—
479C22⋊C4—2362no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—2362no—
485C42—2362no—
486C42—2362no—
487C42#1174568712yes—
488C42#1174568712yes—
489C23—1182no—
490C23—2361no—
491C23—2361no—
492C23—2361no—
493C23—2361no—
494C23—2361no—
495C23—2361no—
496C23C22×D23611yescharacteristic
497Q8—28no—
498D4—1182no—
499D4—1182no—
500D4—2361no—
501D4—2361no—
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—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—1182no—
517C2×C4—4724no—
518C2×C4—2361no—
519C2×C4—2361no—
520C2×C4—2361no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4—2361no—
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—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4#1427371112yes—
539C2×C4#1427371112yes—
540C2×C4#1427375512yes—
541C2×C4#1427376012yes—
542C2×C4#41×D5911yescharacteristic
543C2×C4#41×D5911yescharacteristic
544C2×C4C2×#1174568711yescharacteristic
545C2×C4C2×#1174568711yescharacteristic
546C2×C4#1427377511yescharacteristic
547C2×C4#1427377511yescharacteristic
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C22—4721no—
553C22—4721no—
554C22—4721no—
555C22—4721no—
556C22—2362no—
557C22—2362no—
558C22—2362no—
559C22C2×#1427371111yescharacteristic
560C22#1683806411yescharacteristic
561C22#1683806411yescharacteristic
562C22C2×#1427375511yescharacteristic
563C22C2×#1427376011yescharacteristic
564C22#1683820711yescharacteristic
565C22#1683821411yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—41no—
571C4—41no—
572C4—4721no—
573C4—4721no—
574C4—4721no—
575C4—4721no—
576C4—4721no—
577C4—4721no—
578C4—2362no—
579C4#1683793111yescharacteristic
580C4#1683801311yescharacteristic
581C4#1683808611yescharacteristic
582C4#1683812311yescharacteristic
583C2—4722no—
584C2#2142759711yescharacteristic
585C2#2143402111yescharacteristic
586C2#2143405511yescharacteristic
587C2#2143680511yescharacteristic
588C2#2143687911yescharacteristic
589C2#2143689011yescharacteristic
590C2#2143691211yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1985C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#1985D59
hypocenterC59#25002
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊Dic59C2semidirect
3. #21425660C2semidirect
4. #21425655C2semidirect
7. #21424866C2semidirect
10. #21424689C2semidirect
11. #21424682C2semidirect
12. #21424230C2semidirect
15. C59×#1985C2semidirect
294. #1985D59semidirect
354. C59#25002semidirect

Maximal subgroups

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

subgroupindex
#214294542
Q8×C4⋊Dic592
#214256602
#214256552
#214255012
#214248662
#214247242
#214247172
#214246892
#214246822
#214242302
#214240972
#214239222
C59×#19852
#2500259