Finite Group Databasebeta

← #28605333

Subgroups — #28605333

Order 15104.

1034 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838129C211yesmaximal, characteristic
2C2×#16837530C211yesmaximal, characteristic
3#21427008C211yesmaximal, characteristic
4#21425063C211yesmaximal, characteristic
5#21425055C211yesmaximal, characteristic
6#21425054C211yesmaximal, characteristic
7#21425052C211yesmaximal, characteristic
8#21425046C211yesmaximal, characteristic
9#21424100C211yesmaximal, characteristic
10#21424100C211yesmaximal, characteristic
11#21424098C211yesmaximal, characteristic
12#21423927C211yesmaximal, characteristic
13#21423925C211yesmaximal, characteristic
14#21423915C211yesmaximal, characteristic
15C59×#1992C211yesmaximal, characteristic
16#16838129—24no—
17#16838129C418yescomplemented
18C2×#14273726—21no—
19C2×#14273726C2211yescharacteristic
20C2×#14273718C2211yescharacteristic
21C2×#14273717—21no—
22C2×#14273717—21no—
23C2×C4×Dic118C2211yescharacteristic
24C2×C4×Dic118C2211yescharacteristic
25C2×#14273692—21no—
26C2×#14273692—21no—
27C2×#14273692C2211yescharacteristic
28C2×#14273692C2211yescharacteristic
29#16837530—24no—
30C2×C4⋊(C59⋊C8)C2211yescharacteristic
31C2×Dic59⋊C8C2211yescharacteristic
32C2×Dic59⋊C8C2211yescharacteristic
33C2×#14273652C2211yescharacteristic
34C2×#14273652C2211yescharacteristic
35C2×#14273650C2211yescharacteristic
36C2×#14273650C2211yescharacteristic
37C2×#14273637C2211yescharacteristic
38C2×C8×Dic59C2211yescharacteristic
39#16837298—21no—
40#16837297—21no—
41#16837297—21no—
42#16837271C2211yescharacteristic
43#16837271C2211yescharacteristic
44C4⋊C4×Dic59C2211yescharacteristic
45C4⋊C4×Dic59C2211yescharacteristic
46C4×C4⋊Dic59C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53#16837164C2211yescharacteristic
54#16837164C2211yescharacteristic
55C2×#125×C59C2211yescharacteristic
56C2×#97×C59C2211yescharacteristic
57C59×#384C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#340C2211yescharacteristic
62C22×Dic118—21no—
63C22×Dic118D411yescharacteristic
64C22×Dic118C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273726—28no—
68#14273726—22no—
69#14273726C2×C414yes—
70#14273718—22no—
71#14273718C2×C414yes—
72#14273717—28no—
73#14273717—28no—
74C4×Dic118—22no—
75C4×Dic118C2×C414yes—
76C4×Dic118C2×C414yes—
77C4×Dic118D414yes—
78#14273692—28no—
79#14273692—22no—
80#14273692—22no—
81#14273692—44no—
82#14273692C2×C414yes—
83#14273692C2×C414yes—
84C2×C4⋊Dic59—21no—
85C2×C4⋊Dic59D411yescharacteristic
86C2×C4⋊Dic59C2311yescharacteristic
87C2×C4⋊Dic59C2311yescharacteristic
88C2×C4⋊Dic59C2311yescharacteristic
89C2×Dic59⋊C4—21no—
90C2×Dic59⋊C4—21no—
91C2×Dic59⋊C4—21no—
92C2×Dic59⋊C4—21no—
93C2×Dic59⋊C4—21no—
94C2×Dic59⋊C4—21no—
95C2×Dic59⋊C4—21no—
96C2×Dic59⋊C4—21no—
97C2×Dic59⋊C4—21no—
98C2×Dic59⋊C4D411yescharacteristic
99C2×Dic59⋊C4D411yescharacteristic
100C2×C4×Dic59—21no—
101C2×C4×Dic59—21no—
102C2×C4×Dic59—21no—
103C2×C4×Dic59—21no—
104C2×C4×Dic59—21no—
105C2×C4×Dic59—21no—
106C2×C4×Dic59C2311yescharacteristic
107C2×C4×Dic59C2311yescharacteristic
108C4⋊(C59⋊C8)—22no—
109Dic59⋊C8—22no—
110Dic59⋊C8—22no—
111#14273652—22no—
112#14273652Q814yescomplemented
113#14273650—22no—
114#14273650—22no—
115#14273637—22no—
116C8×Dic59—22no—
117C59⋊#95—21no—
118C59⋊#95—21no—
119C59⋊#95—21no—
120C22×C8×C59C2311yescharacteristic
121C22×C8×C59C2311yescharacteristic
122#125×C59—22no—
123#125×C59C2×C414yes—
124C2×C4⋊C4×C59—21no—
125C2×C4⋊C4×C59—21no—
126C2×C4⋊C4×C59—21no—
127C2×C4⋊C4×C59—21no—
128C2×C4⋊C4×C59C2311yescharacteristic
129C2×C4⋊C4×C59C2311yescharacteristic
130C2×C4⋊C4×C59C2311yescharacteristic
131C2×C4⋊C4×C59C2311yescharacteristic
132C2×C42×C59C2311yescharacteristic
133#97×C59—22no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59C2×D411yescharacteristic
146C22×Dic59C2×D411yescharacteristic
147C22×Dic59C2×D411yescharacteristic
148C22×Dic59C2×D411yescharacteristic
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—22no—
153C2×Dic118—22no—
154C2×Dic118—24no—
155C2×Dic118—42no—
156C2×Dic118—42no—
157C2×Dic118C22⋊C414yes—
158C2×Dic118C22×C412yes—
159C2×Dic118C2×Q812yes—
160C2×Dic118#4112yes—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59C2×Q812yes—
167C4⋊C4×C59—24no—
168C4⋊C4×C59—24no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—42no—
176C4⋊C4×C59—81no—
177C4⋊C4×C59C22×C412yes—
178C4⋊C4×C59C22×C412yes—
179C42×C59C22×C412yes—
180C42×C59C2×D412yes—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—24no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59C22⋊C414yes—
187C4⋊Dic59C22×C412yes—
188C4⋊Dic59C2×Q812yes—
189Dic59⋊C4—24no—
190Dic59⋊C4—24no—
191Dic59⋊C4—24no—
192Dic59⋊C4—24no—
193Dic59⋊C4—24no—
194Dic59⋊C4—42no—
195Dic59⋊C4—42no—
196Dic59⋊C4—81no—
197Dic59⋊C4—81no—
198Dic59⋊C4C22⋊C414yes—
199Dic59⋊C4C22⋊C414yes—
200C4×Dic59—24no—
201C4×Dic59—24no—
202C4×Dic59—24no—
203C4×Dic59—42no—
204C4×Dic59—42no—
205C4×Dic59—42no—
206C4×Dic59C22×C412yes—
207C4×Dic59C22×C412yes—
208C4×Dic59C2×D412yes—
209C4×Dic59C2×D412yes—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C23×C59C22×D411yescharacteristic
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59C2×C22⋊C412yes—
234C2×C4×C59C2×C22⋊C412yes—
235C2×C4×C59C23×C411yescharacteristic
236C2×C4×C59C22×D411yescharacteristic
237C2×C4×C59C22×Q811yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C592−1+411yescharacteristic
240C2×C4×C592−1+411yescharacteristic
241C2×Dic59—22no—
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—21no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—22no—
251C2×Dic59—22no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59C2×C22⋊C412yes—
260C2×Dic59C2×C22⋊C412yes—
261C2×Dic59C2×C22⋊C412yes—
262C2×Dic59C2×C22⋊C412yes—
263C2×Dic59AGL(1,ℤ/8)12yes—
264C2×Dic59#13712yes—
265Dic118—22no—
266Dic118—22no—
267Dic118—22no—
268Dic118—28no—
269Dic118—48no—
270Dic118C4×Q814yescomplemented
271C8×C59—22no—
272C8×C59—41no—
273C59⋊C8—41no—
274C59⋊C8—41no—
275#22524—591nomaximal, Sylow
276C22×C59C22×C22⋊C411yescharacteristic
277C22×C59#54811yescharacteristic
278C22×C59D4×Q811yescharacteristic
279C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C59C2×#13711yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57811yescharacteristic
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—42no—
286C4×C59—42no—
287C4×C59—81no—
288C4×C59—81no—
289C4×C59C2×C4×Q811yescharacteristic
290C4×C59#51911yescharacteristic
291C4×C59#54811yescharacteristic
292C4×C59D4×Q811yescharacteristic
293Dic59—24no—
294Dic59—22no—
295Dic59—22no—
296Dic59—22no—
297Dic59—42no—
298Dic59—42no—
299Dic59—81no—
300Dic59#41714yes—
301C2×#557—591no—
302C2×#446—591no—
303Q8×C4⋊C4—591no—
304#1992—591no—
305#1992D5911yescharacteristic
306#1990—591no—
307#1989—591no—
308#1989—591no—
309#1988—591no—
310#1803—591no—
311#1803—591no—
312#1803—591no—
313#1763—591no—
314#1763—591no—
315C4×#103—591no—
316C2×C59Q8×C22⋊C411yescharacteristic
317C2×C59C2×#41711yescharacteristic
318C2×C59#289811yescharacteristic
319C2×C59#336811yescharacteristic
320C2×C59#337411yescharacteristic
321C2×C59#338611yescharacteristic
322C2×C59#338711yescharacteristic
323#557—598no—
324#557—1184no—
325C2×C4⋊Q8—591no—
326C2×C4⋊Q8—1181no—
327C2×#125—591no—
328C2×#125—1181no—
329C2×#125—1181no—
330C2×#125C2×D5911yescharacteristic
331C2×C4×Q8—591no—
332C2×C4×Q8—591no—
333C2×C4×Q8—591no—
334C2×C4×Q8—591no—
335C2×C4×Q8—1181no—
336C2×C4×Q8—1181no—
337#446—1184no—
338C2×C4⋊C8—591no—
339C2×C4⋊C8—591no—
340C2×C4⋊C8—591no—
341C2×#103—591no—
342C2×#103—591no—
343C2×#103—591no—
344C2×#103—591no—
345C2×#97—591no—
346C2×#97C2×D5911yescharacteristic
347C2×C4×C8—591no—
348#389—1181no—
349#389—1181no—
350#389—1181no—
351#384—591no—
352#384—591no—
353#384C2×D5911yescharacteristic
354C4×C4⋊C4—591no—
355C4×C4⋊C4—591no—
356C4×C4⋊C4—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340C2×D5911yescharacteristic
366#340C2×D5911yescharacteristic
367#340C2×D5911yescharacteristic
368#340C2×D5911yescharacteristic
369C59#2252411yescharacteristic, complemented, Sylow
370C22×Q8—591no—
371C22×Q8—591no—
372C22×Q8—1181no—
373C22×C8—591no—
374C22×C8—591no—
375C22×C8C22×D5911yescharacteristic
376C22×C8C22×D5911yescharacteristic
377C4⋊Q8—594no—
378C4⋊Q8—1182no—
379C4⋊Q8—1188no—
380#125—22no—
381#125—594no—
382#125—1182no—
383#125—1188no—
384#125—1188no—
385#125C4×D5914yes—
386C4×Q8—594no—
387C4×Q8—594no—
388C4×Q8—594no—
389C4×Q8—594no—
390C4×Q8—594no—
391C4×Q8—1182no—
392C4×Q8—1182no—
393C4×Q8—1188no—
394C4×Q8—1182no—
395C4×Q8—2364no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—591no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C4⋊C4C22×D5911yescharacteristic
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C42—591no—
421C2×C42—591no—
422C2×C42—1181no—
423C2×C42—1181no—
424C2×C42—1181no—
425C2×C42—1181no—
426C2×C42—1181no—
427C2×C42—1181no—
428C2×C42C22×D5911yescharacteristic
429C4⋊C8—1182no—
430C4⋊C8—1182no—
431C4⋊C8—1182no—
432#103—594no—
433#103—1182no—
434#103—1182no—
435#103—1182no—
436#97—22no—
437#97—1182no—
438C4×C8—1182no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442C2×Q8—594no—
443C2×Q8—592no—
444C2×Q8—592no—
445C2×Q8—592no—
446C2×Q8—1182no—
447C2×Q8—1182no—
448C2×Q8—1184no—
449C2×Q8—1181no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C2×Q8—2362no—
453C2×Q8—2362no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—591no—
459C22×C4—591no—
460C22×C4—591no—
461C22×C4—591no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—1181no—
466C22×C4D4×D5911yescharacteristic
467C22×C4D4×D5911yescharacteristic
468C22×C4C23×D5911yescharacteristic
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8Q8×D5912yes—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—594no—
492C4⋊C4—594no—
493C4⋊C4—594no—
494C4⋊C4—592no—
495C4⋊C4—592no—
496C4⋊C4—1184no—
497C4⋊C4—1184no—
498C4⋊C4—1184no—
499C4⋊C4—1184no—
500C4⋊C4—1184no—
501C4⋊C4—1184no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—4721no—
507C4⋊C4—4721no—
508C4⋊C4—2362no—
509C4⋊C4—2362no—
510C4⋊C4C2×C4×D5912yes—
511C4⋊C4C2×C4×D5912yes—
512C42—592no—
513C42—592no—
514C42—592no—
515C42—592no—
516C42—1184no—
517C42—1184no—
518C42—1184no—
519C42—2362no—
520C42—2362no—
521C42—2362no—
522C42C2×C4×D5912yes—
523C42D4×D5912yes—
524C23C2×D4×D5911yescharacteristic
525Q8—594no—
526Q8—1188no—
527Q8—1182no—
528Q8—1182no—
529Q8—1182no—
530Q8—2368no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—592no—
548C2×C4—592no—
549C2×C4—592no—
550C2×C4—592no—
551C2×C4—592no—
552C2×C4—592no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1181no—
559C2×C4—1182no—
560C2×C4—1182no—
561C2×C4—1182no—
562C2×C4—1182no—
563C2×C4—1182no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4C22⋊C4×D5912yes—
572C2×C4C22⋊C4×D5912yes—
573C2×C4C22×C4×D5911yescharacteristic
574C2×C4C2×D4×D5911yescharacteristic
575C2×C4C2×Q8×D5911yescharacteristic
576C2×C4#41×D5911yescharacteristic
577C2×C4#1427377411yescharacteristic
578C2×C4#1427377511yescharacteristic
579C8—22no—
580C8—41no—
581C8—2361no—
582C8—2361no—
583C22C2×C22⋊C4×D5911yescharacteristic
584C22#1683806011yescharacteristic
585C22#1683806711yescharacteristic
586C22AGL(1,ℤ/8)×D5911yescharacteristic
587C22#137×D5911yescharacteristic
588C22#1683821211yescharacteristic
589C22#1683821811yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—81no—
595C4—81no—
596C4—594no—
597C4—1182no—
598C4—1182no—
599C4—1182no—
600C4—1184no—
601C4—4721no—
602C4—2362no—
603C4—2362no—
604C4C4×Q8×D5911yescharacteristic
605C4#1683787811yescharacteristic
606C4#1683806211yescharacteristic
607C4#1683806511yescharacteristic
608C2#2142693011yescharacteristic
609C2#417×D5911yescharacteristic
610C2#2143262711yescharacteristic
611C2#2143682211yescharacteristic
612C2#2143689611yescharacteristic
613C2#2143701411yescharacteristic
614C2#2143702511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1992C2
socle, prime radicalC23×C59C22×D4
cosocleC22×C4×C59C24
hypercenter#1992D59
hypocenterC59#22524
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
17. #16838129C4semidirect
112. #14273652Q8semidirect
270. Dic118C4×Q8semidirect
369. C59#22524semidirect

Maximal subgroups

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

subgroupindex
C2×#168381292
C2×#168375302
#214270082
#214250632
#214250552
#214250542
#214250522
#214250462
#214241002
#214240982
#214239272
#214239252
#214239152
C59×#19922
#2252459