Finite Group Databasebeta

← #28749622

Subgroups — #28749622

Order 15104.

2858 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837878C211yesmaximal, characteristic
2C2×#16837878C211yesmaximal, characteristic
3C2×C4×Q8×D59C211yesmaximal, characteristic
4#21427015C213yesmaximal, complemented
5#21427014C211yesmaximal, characteristic, complemented
6#21427013C213yesmaximal, complemented
7Q8×C4⋊Dic59C211yesmaximal, characteristic, complemented
8C4⋊C4×Dic118C213yesmaximal, complemented
9#21427010C213yesmaximal, complemented
10Q8×Dic59⋊C4C211yesmaximal, characteristic, complemented
11#21426721C213yesmaximal
12#21426718C213yesmaximal
13#21426577C213yesmaximal
14#21426576C213yesmaximal
15C59×#2351C211yesmaximal, characteristic, complemented
16C22×Q8×D59C2211yescharacteristic
17#16837878—28no—
18#16837878C4116yescomplemented
19C2×Q8×Dic59C2211yescharacteristic
20C2×Q8×Dic59C2211yescharacteristic
21C2×Q8×Dic59C2211yescharacteristic
22C2×C4×Dic118C2213yes—
23C2×C4×Dic118C2213yes—
24C2×C4×Dic118C2213yes—
25C2×#14273692C2213yes—
26C2×#14273692C2213yes—
27C2×#14273692C2213yes—
28C4×Q8×D59—28no—
29C2×#14273685C2213yes—
30C2×#14273685C2213yes—
31C2×#14273684C2213yes—
32C2×#14273684C2213yes—
33C2×C4⋊C4×D59C2213yes—
34C2×C42×D59C2213yes—
35#16837301C2213yes—
36#16837301C2213yes—
37#16837300C2213yes—
38#16837300C2213yes—
39#16837299C2216yes—
40#16837299C2216yes—
41#16837299C2213yes—
42#16837299C2213yes—
43#16837298C2213yes—
44#16837298C2213yes—
45#16837297C2213yes—
46#16837297C2213yes—
47#16837276C2216yes—
48#16837276C2213yes—
49#16837276C2211yescharacteristic
50#16837273C2213yes—
51#16837273C2213yes—
52#16837272C2213yes—
53#16837271C2216yes—
54#16837271C2213yes—
55#16837267C2213yes—
56#16837266C2216yes—
57C4⋊C4×Dic59C2216yes—
58C4×Dic59⋊C4C2213yes—
59C4×Dic59⋊C4C2213yes—
60C4×C4⋊Dic59C2213yes—
61#16837253C2216yes—
62C2×C4×Q8×C59C2211yescharacteristic
63C2×C4×Q8×C59C2211yescharacteristic
64C2×C4×Q8×C59C2211yescharacteristic
65C59×#389C2213yes—
66C59×#389C2213yes—
67C59×#382C2213yes—
68C4×C4⋊C4×C59C2213yes—
69C22×Dic118C2313yes—
70C2×Q8×D59C2×C418yes—
71C2×Q8×D59D418yes—
72C2×Q8×D59Q818yes—
73C22×C4×D59C2311yescharacteristic
74C22×C4×D59C2313yes—
75C22×C4×D59C2313yes—
76Q8×Dic59—24no—
77Q8×Dic59—24no—
78Q8×Dic59C2×C418yescomplemented
79C4×Dic118—212no—
80C4×Dic118—212no—
81C4×Dic118C2×C4124yescomplemented
82#14273692—212no—
83#14273692—212no—
84#14273692C2×C4124yescomplemented
85#14273685—212no—
86#14273685C2×C4124yes—
87#14273684—212no—
88#14273684C2×C4124yes—
89C2×C4⋊Dic59C2311yescharacteristic
90C2×C4⋊Dic59C2313yes—
91C2×C4⋊Dic59C2313yes—
92C2×C4⋊Dic59C2313yes—
93C2×C4⋊Dic59C2316yes—
94C2×C4⋊Dic59C2313yes—
95C2×Dic59⋊C4C2313yes—
96C2×Dic59⋊C4C2311yescharacteristic
97C2×Dic59⋊C4C2313yes—
98C2×Dic59⋊C4C2316yes—
99C2×Dic59⋊C4C2313yes—
100C2×Dic59⋊C4C2316yes—
101C2×Dic59⋊C4C2313yes—
102C2×Dic59⋊C4C2313yes—
103C2×Dic59⋊C4C2313yes—
104C2×Dic59⋊C4C2313yes—
105C4⋊C4×D59—212no—
106C2×(C2×D59)⋊C4C2311yescharacteristic
107C2×(C2×D59)⋊C4C2313yes—
108C2×(C2×D59)⋊C4C2313yes—
109C2×(C2×D59)⋊C4C2311yescharacteristic
110C2×C4×Dic59C2313yes—
111C2×C4×Dic59C2316yes—
112C2×C4×Dic59C2313yes—
113C2×C4×Dic59C2313yes—
114C2×C4×Dic59C2313yes—
115C2×C4×Dic59C2316yes—
116C2×C4×Dic59C2313yes—
117C42×D59—212no—
118C59⋊#95C2313yes—
119C59⋊#95C2313yes—
120C59⋊#95C2311yescharacteristic
121C59⋊#95C2313yes—
122C59⋊#95C2313yes—
123C59⋊#95C2313yes—
124C59⋊#95C2316yes—
125C22×Q8×C59C2311yescharacteristic
126C4×Q8×C59—24no—
127C4×Q8×C59—24no—
128C4×Q8×C59C2×C418yescomplemented
129C2×C4⋊C4×C59C2313yes—
130C2×C4⋊C4×C59C2313yes—
131C2×C4⋊C4×C59C2313yes—
132C2×C4⋊C4×C59C2313yes—
133C2×C4⋊C4×C59C2313yes—
134C2×C42×C59C2313yes—
135C2×C42×C59C2313yes—
136C2×C42×C59C2313yes—
137#95×C59C2311yescharacteristic
138#95×C59C2313yes—
139#95×C59C2316yes—
140C23×D59C2411yescharacteristic
141C2×Q8×C59C22×C414yes—
142C2×Q8×C59C2×D414yes—
143C2×Q8×C59C2×Q814yes—
144C22×C4×C59C2413yes—
145C22×C4×C59C2413yes—
146C22×C4×C59C2411yescharacteristic
147C22×C4×C59C2411yescharacteristic
148C22×C4×C59C2413yes—
149C22×C4×C59C2411yescharacteristic
150C22×C4×C59C2413yes—
151C22×Dic59C2411yescharacteristic
152C22×Dic59C2413yes—
153C22×Dic59C2413yes—
154C22×Dic59C2411yescharacteristic
155C22×Dic59C2413yes—
156C22×Dic59C2411yescharacteristic
157C22×Dic59C2413yes—
158Q8×D59C4⋊C4164yescomplemented
159C2×C4×D59—22no—
160C2×C4×D59—22no—
161C2×C4×D59—22no—
162C2×C4×D59—26no—
163C2×C4×D59—43no—
164C2×C4×D59—43no—
165C2×C4×D59C22×C4112yes—
166C2×C4×D59C2×D4112yes—
167C2×C4×D59C2×Q8112yes—
168C2×Dic118C22×C4112yes—
169C2×Dic118C2×D4112yes—
170C2×Dic118C2×Q8112yes—
171C4⋊C4×C59—26no—
172C4⋊C4×C59—26no—
173C4⋊C4×C59—26no—
174C4⋊C4×C59—43no—
175C4⋊C4×C59C22×C4112yes—
176C42×C59—26no—
177C42×C59—26no—
178C42×C59C22×C4112yes—
179(C2×D59)⋊C4—22no—
180(C2×D59)⋊C4—26no—
181(C2×D59)⋊C4C22×C414yes—
182(C2×D59)⋊C4C22×C4112yes—
183C4⋊Dic59—26no—
184C4⋊Dic59—26no—
185C4⋊Dic59—22no—
186C4⋊Dic59—43no—
187C4⋊Dic59—46no—
188C4⋊Dic59C22×C4112yes—
189Dic59⋊C4—22no—
190Dic59⋊C4—212no—
191Dic59⋊C4—26no—
192Dic59⋊C4—26no—
193Dic59⋊C4—26no—
194Dic59⋊C4—46no—
195Dic59⋊C4—43no—
196Dic59⋊C4—43no—
197Dic59⋊C4C22×C4112yes—
198Dic59⋊C4C22×C4112yes—
199C4×Dic59—26no—
200C4×Dic59—26no—
201C4×Dic59—26no—
202C4×Dic59—26no—
203C4×Dic59—212no—
204C4×Dic59C22×C4112yes—
205C4×Dic59C22×C4124yes—
206C23×C59C2511yescharacteristic
207C22×D59—21no—
208C22×D59—21no—
209C22×D59—21no—
210C22×D59—21no—
211C22×D59C23×C412yes—
212C22×D59C22×D412yes—
213C22×D59C22×Q812yes—
214Q8×C59C2×C4⋊C4116yes—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—23no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—23no—
221C2×C4×C59—23no—
222C2×C4×C59—23no—
223C2×C4×C59—21no—
224C2×C4×C59—23no—
225C2×C4×C59—23no—
226C2×C4×C59—21no—
227C2×C4×C59—23no—
228C2×C4×C59—23no—
229C2×C4×C59—21no—
230C2×C4×C59C23×C416yes—
231C2×C4×C59C23×C412yes—
232C2×C4×C59C23×C416yes—
233C2×C4×C59C22×D416yes—
234C2×C4×C59C22×Q816yes—
235C2×C4×C592−1+412yes—
236C2×Dic59—21no—
237C2×Dic59—23no—
238C2×Dic59—21no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—23no—
242C2×Dic59—21no—
243C2×Dic59—23no—
244C2×Dic59—23no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—23no—
248C2×Dic59—23no—
249C2×Dic59—23no—
250C2×Dic59—23no—
251C2×Dic59C23×C412yes—
252C2×Dic59C23×C416yes—
253C2×Dic59C23×C416yes—
254C2×Dic59C22×D416yes—
255C2×Dic59C22×Q816yes—
256C2×Dic592−1+412yes—
257C4×D59—28no—
258C4×D59—412no—
259C4×D59C2×C4⋊C4148yes—
260Dic118C2×C4⋊C4148yes—
261#34249—591nomaximal, Sylow
262C22×C59C24×C411yescharacteristic
263C22×C59C23×D411yescharacteristic
264C22×C59C23×Q811yescharacteristic
265C22×C59C2×2−1+411yescharacteristic
266C22×C59C2×2−1+411yescharacteristic
267C22×C59#58511yescharacteristic
268C22×C59#58511yescharacteristic
269C2×D59—22no—
270C2×D59—22no—
271C2×D59—22no—
272C2×D59—22no—
273C2×D59—22no—
274C2×D59—22no—
275C2×D59C22×C4⋊C414yes—
276C4×C59—22no—
277C4×C59—26no—
278C4×C59—22no—
279C4×C59—43no—
280C4×C59—43no—
281C4×C59—41no—
282C4×C59C22×C4⋊C4112yes—
283Dic59—26no—
284Dic59—22no—
285Dic59—22no—
286Dic59—41no—
287Dic59—43no—
288Dic59—43no—
289Dic59C22×C4⋊C4112yes—
290C2×#519—591no—
291C2×#519—591no—
292C22×C4×Q8—591no—
293#2351—593no—
294#2351—593no—
295#2351—591no—
296#2351D5911yescharacteristic, complemented
297Q8×C4⋊C4—593no—
298Q8×C4⋊C4—593no—
299Q8×C4⋊C4—591no—
300Q8×C4⋊C4—591no—
301#2317—593no—
302#2317—593no—
303#2296—593no—
304#2296—593no—
305C2×C59C23×C4⋊C411yescharacteristic
306C2×C59C4×2−1+411yescharacteristic
307C2×C59C4×2−1+411yescharacteristic
308C2×C59#348611yescharacteristic
309C2×C59#348711yescharacteristic
310C2×C59#350211yescharacteristic
311C2×C59#350211yescharacteristic
312D59—24no—
313C23×Q8—591no—
314#519—5916no—
315#519—1188no—
316C2×C4×Q8—591no—
317C2×C4×Q8—591no—
318C2×C4×Q8—591no—
319C2×C4×Q8—593no—
320C2×C4×Q8—593no—
321C2×C4×Q8—593no—
322C2×C4×Q8—593no—
323C2×C4×Q8—593no—
324C2×C4×Q8—593no—
325C2×C4×Q8—1188no—
326C2×C4×Q8C2×D5911yescharacteristic
327C2×C4×Q8C2×D5911yescharacteristic
328C2×C4×Q8C2×D5911yescharacteristic
329C2×#117—593no—
330C2×#117—593no—
331C2×#117—593no—
332C2×#117—593no—
333C22×C4⋊C4—593no—
334C22×C42—593no—
335#389—593no—
336#389—593no—
337#389—593no—
338#389—593no—
339#389—593no—
340#389—593no—
341#389—593no—
342#389—593no—
343#389—596no—
344#389—596no—
345#389—593no—
346#389—593no—
347#389C2×D5913yes—
348#389C2×D5913yes—
349#385—593no—
350#385—593no—
351#385—596no—
352#385—593no—
353#385—591no—
354#384—596no—
355#384—593no—
356#384—593no—
357#382—593no—
358#382—596no—
359#382C2×D5913yes—
360C4×C4⋊C4—596no—
361C4×C4⋊C4—593no—
362C4×C4⋊C4—593no—
363C4×C4⋊C4—593no—
364C4×C4⋊C4—596no—
365C4×C4⋊C4C2×D5913yes—
366C59#3424911yescharacteristic, complemented, Sylow
367C22×Q8—598no—
368C22×Q8—598no—
369C22×Q8—598no—
370C22×Q8—593no—
371C22×Q8C22×D5911yescharacteristic
372C23×C4—593no—
373C23×C4—591no—
374C23×C4—593no—
375C4×Q8—24no—
376C4×Q8—24no—
377C4×Q8—598no—
378C4×Q8—5924no—
379C4×Q8—5924no—
380C4×Q8—11812no—
381C4×Q8—11812no—
382C4×Q8—11812no—
383C4×Q8—11812no—
384C4×Q8—1184no—
385C4×Q8—1184no—
386C4×Q8C4×D5918yescomplemented
387#117—5924no—
388#117—5924no—
389#117—11812no—
390#117—11812no—
391C2×C4⋊C4—593no—
392C2×C4⋊C4—593no—
393C2×C4⋊C4—593no—
394C2×C4⋊C4—596no—
395C2×C4⋊C4—593no—
396C2×C4⋊C4—593no—
397C2×C4⋊C4—593no—
398C2×C4⋊C4—593no—
399C2×C4⋊C4—593no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—593no—
403C2×C4⋊C4—593no—
404C2×C4⋊C4—596no—
405C2×C4⋊C4—596no—
406C2×C4⋊C4—593no—
407C2×C4⋊C4—11812no—
408C2×C4⋊C4C22×D5913yes—
409C2×C4⋊C4C22×D5913yes—
410C2×C4⋊C4C22×D5913yes—
411C2×C4⋊C4C22×D5913yes—
412C2×C4⋊C4C22×D5913yes—
413C2×C22⋊C4—593no—
414C2×C22⋊C4—591no—
415C2×C22⋊C4—591no—
416C2×C22⋊C4—593no—
417C2×C42—596no—
418C2×C42—593no—
419C2×C42—596no—
420C2×C42—593no—
421C2×C42—593no—
422C2×C42—593no—
423C2×C42—593no—
424C2×C42—11812no—
425C2×C42C22×D5913yes—
426C2×C42C22×D5913yes—
427C2×C42C22×D5913yes—
428#95—591no—
429#95—593no—
430#95—593no—
431#95—593no—
432#95—593no—
433#95—596no—
434#95—593no—
435#95C22×D5916yes—
436#95C22×D5911yescharacteristic
437#95C22×D5913yes—
438C24—591no—
439C2×Q8—5964no—
440C2×Q8—5912no—
441C2×Q8—5912no—
442C2×Q8—5912no—
443C2×Q8C2×C4×D5914yes—
444C2×Q8D4×D5914yes—
445C2×Q8Q8×D5914yes—
446C22×C4—593no—
447C22×C4—5912no—
448C22×C4—5912no—
449C22×C4—5912no—
450C22×C4—591no—
451C22×C4—593no—
452C22×C4—591no—
453C22×C4—591no—
454C22×C4—593no—
455C22×C4—593no—
456C22×C4—1182no—
457C22×C4—1182no—
458C22×C4—1186no—
459C22×C4—1182no—
460C22×C4—2363no—
461C22×C4—2363no—
462C22×C4C23×D5913yes—
463C22×C4C23×D5913yes—
464C22×C4C23×D5911yescharacteristic
465C22×C4C23×D5913yes—
466C22×C4C23×D5911yescharacteristic
467C22×C4C23×D5911yescharacteristic
468C22×C4C23×D5913yes—
469C4⋊C4—26no—
470C4⋊C4—26no—
471C4⋊C4—26no—
472C4⋊C4—43no—
473C4⋊C4—5912no—
474C4⋊C4—5912no—
475C4⋊C4—5912no—
476C4⋊C4—11812no—
477C4⋊C4—1186no—
478C4⋊C4—1186no—
479C4⋊C4—1186no—
480C4⋊C4—1182no—
481C4⋊C4—1186no—
482C4⋊C4—1186no—
483C4⋊C4—1182no—
484C4⋊C4—2363no—
485C4⋊C4—2363no—
486C4⋊C4—2366no—
487C4⋊C4—2366no—
488C4⋊C4—2363no—
489C4⋊C4C2×C4×D59112yes—
490C22⋊C4—5912no—
491C22⋊C4—594no—
492C22⋊C4—1182no—
493C22⋊C4—1186no—
494C42—26no—
495C42—26no—
496C42—5924no—
497C42—5912no—
498C42—1186no—
499C42—1186no—
500C42—11812no—
501C42—1186no—
502C42—1186no—
503C42C2×C4×D59112yes—
504C23—592no—
505C23—592no—
506C23—592no—
507C23—1181no—
508C23—1181no—
509C23—1181no—
510C23—1181no—
511C23C24×D5911yescharacteristic
512Q8—5948no—
513Q8C4⋊C4×D59116yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—23no—
518C2×C4—23no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—23no—
524C2×C4—23no—
525C2×C4—21no—
526C2×C4—23no—
527C2×C4—21no—
528C2×C4—23no—
529C2×C4—596no—
530C2×C4—596no—
531C2×C4—596no—
532C2×C4—5948no—
533C2×C4—596no—
534C2×C4—592no—
535C2×C4—592no—
536C2×C4—1188no—
537C2×C4—1181no—
538C2×C4—1183no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1183no—
543C2×C4—1181no—
544C2×C4—1183no—
545C2×C4—1183no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—1183no—
549C2×C4—1183no—
550C2×C4—1183no—
551C2×C4—1183no—
552C2×C4—23612no—
553C2×C4C22×C4×D5916yes—
554C2×C4C22×C4×D5912yes—
555C2×C4C22×C4×D5916yes—
556C2×C4C2×D4×D5916yes—
557C2×C4C2×Q8×D5916yes—
558C2×C4#1427377412yes—
559C22—594no—
560C22—1182no—
561C22—1182no—
562C22—1182no—
563C22—1182no—
564C22—1182no—
565C22—1182no—
566C22C23×C4×D5911yescharacteristic
567C22C22×D4×D5911yescharacteristic
568C22C22×Q8×D5911yescharacteristic
569C222−1+4×D5911yescharacteristic
570C22C2×#1427377411yescharacteristic
571C22#1683823811yescharacteristic
572C22#1683823911yescharacteristic
573C4—22no—
574C4—22no—
575C4—26no—
576C4—43no—
577C4—43no—
578C4—41no—
579C4—5912no—
580C4—1182no—
581C4—1186no—
582C4—1182no—
583C4—2361no—
584C4—2363no—
585C4—2363no—
586C4C2×C4⋊C4×D59112yes—
587C2—1184no—
588C2C22×C4⋊C4×D5911yescharacteristic
589C2#2143730511yescharacteristic
590C2C4×#1427377411yescharacteristic
591C2#2143774211yescharacteristic
592C2#2143775311yescharacteristic
593C2#2143790011yescharacteristic
594C2#2143790311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D59
commutatorC22×C59C24×C4
FittingC59×#2351C2
socle, cosocleC23×C59C25
prime radicalC23×D59C24
hypercenter#2351D59
hypocenterC59#34249
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #21427015C2semidirect
5. #21427014C2semidirect
6. #21427013C2semidirect
7. Q8×C4⋊Dic59C2semidirect
8. C4⋊C4×Dic118C2semidirect
9. #21427010C2semidirect
10. Q8×Dic59⋊C4C2semidirect
15. C59×#2351C2semidirect
18. #16837878C4semidirect
78. Q8×Dic59C2×C4semidirect
81. C4×Dic118C2×C4semidirect
84. #14273692C2×C4semidirect
128. C4×Q8×C59C2×C4semidirect
158. Q8×D59C4⋊C4semidirect
296. #2351D59semidirect
366. C59#34249semidirect
386. C4×Q8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168378782
C2×C4×Q8×D592
#214270152
#214270142
#214270132
Q8×C4⋊Dic592
C4⋊C4×Dic1182
#214270102
Q8×Dic59⋊C42
#214267212
#214267182
#214265772
#214265762
C59×#23512
#3424959