Finite Group Databasebeta

← #28618736

Subgroups — #28618736

Order 15104.

752 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273706C211yesmaximal, characteristic
2#21425663C211yesmaximal, characteristic, complemented
3#21425647C211yesmaximal, characteristic, complemented
4#21425578C211yesmaximal, characteristic, complemented
5#21425533C211yesmaximal, characteristic
6#21425522C211yesmaximal, characteristic
7#21425515C211yesmaximal, characteristic, complemented
8#21425299C211yesmaximal, characteristic
9#21425292C211yesmaximal, characteristic
10C4×#14273660C211yesmaximal, characteristic, complemented
11C4×#14273658C211yesmaximal, characteristic, complemented
12C4×#14273650C211yesmaximal, characteristic, complemented
13C4×#14273650C211yesmaximal, characteristic, complemented
14C4×C22⋊(C59⋊C8)C211yesmaximal, characteristic
15C59×#2283C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273650C2211yescharacteristic
23C2×#14273650C2211yescharacteristic
24C2×#14273650C2211yescharacteristic
25C2×#14273650C2211yescharacteristic
26C2×C22⋊(C59⋊C8)C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2211yescharacteristic
28C2×C4×C59⋊C8C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837308C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837286—21no—
33#16837275—21no—
34#16837262C2211yescharacteristic
35C4×Dic59⋊C4—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C22⋊Dic59—21no—
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C22×C42×C59C2211yescharacteristic
49C59×#383—21no—
50C59×#382C2211yescharacteristic
51C59×#382C2211yescharacteristic
52C59×#381—21no—
53C59×#380C2211yescharacteristic
54C59×#380C2211yescharacteristic
55C4×C4⋊C4×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C22⋊C4×C59—21no—
58C59×#376—21no—
59C22×Dic118C2311yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273706—22no—
65#14273706—22no—
66C4×Dic118—22no—
67C4×Dic118—22no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×C22⋊Dic59—21no—
75C2×C22⋊Dic59—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59—21no—
78#14273660—22no—
79#14273660—22no—
80#14273658—22no—
81#14273658—22no—
82#14273650—22no—
83#14273650—22no—
84#14273650—22no—
85#14273650—22no—
86C22⋊(C59⋊C8)—22no—
87C22⋊(C59⋊C8)—22no—
88C4×C59⋊C8—22no—
89C4×C59⋊C8—22no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C23×C4×C59D411yescharacteristic
96C23×C4×C59D411yescharacteristic
97C23×C4×C59C2311yescharacteristic
98C2×C4⋊C4×C59C2311yescharacteristic
99C2×C4⋊C4×C59C2311yescharacteristic
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C22⋊C4×C59—21no—
103C2×C22⋊C4×C59—21no—
104C2×C42×C59—22no—
105C2×C42×C59—22no—
106C2×C42×C59—21no—
107C2×C42×C59—21no—
108C2×C42×C59D411yescharacteristic
109C2×C42×C59D411yescharacteristic
110C2×C42×C59C2311yescharacteristic
111C2×C42×C59C2311yescharacteristic
112#95×C59—21no—
113#95×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—21no—
118#95×C59—21no—
119#95×C59—21no—
120C24×C59C2×D411yescharacteristic
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—41no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118#4112yes—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C22⋊C4×C59—42no—
162C22⋊C4×C59—42no—
163C42×C59—22no—
164C42×C59—22no—
165C42×C59—21no—
166C42×C59—21no—
167C42×C59—42no—
168C42×C59—42no—
169C42×C59#4112yes—
170C42×C59#4112yes—
171C22⋊Dic59—24no—
172C22⋊Dic59—42no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59#4112yes—
179Dic59⋊C4—24no—
180Dic59⋊C4—42no—
181Dic59⋊C4—42no—
182C4×Dic59—42no—
183C4×Dic59—42no—
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—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8#4112yes—
195C2×C59⋊C8#4112yes—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59—21no—
202C23×C59#13512yes—
203C23×C59C22×D411yescharacteristic
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—44no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—44no—
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—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59#12312yes—
234C2×C4×C59#12312yes—
235C2×C4×C59#13512yes—
236C2×C4×C59#13712yes—
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255Dic118—22no—
256Dic118—22no—
257Dic118—41no—
258Dic118—41no—
259C59⋊C8—22no—
260C59⋊C8—22no—
261C59⋊C8—41no—
262C59⋊C8—41no—
263#23759—591nomaximal, Sylow
264C22×C59—22no—
265C22×C59—22no—
266C22×C59—22no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59C2×#12311yescharacteristic
272C22×C59#52511yescharacteristic
273C22×C59#53211yescharacteristic
274C22×C59C2×#13511yescharacteristic
275C22×C59C2×#13511yescharacteristic
276C22×C59C2×#13711yescharacteristic
277C22×C59#57511yescharacteristic
278C4×C59—22no—
279C4×C59—41no—
280C4×C59—41no—
281C4×C59—41no—
282C4×C59—81no—
283C4×C59—81no—
284C4×C59—81no—
285C4×C59—81no—
286C4×C59#48514yes—
287C4×C59#52911yescharacteristic
288C4×C59#52911yescharacteristic
289C4×C59#52911yescharacteristic
290C4×C59#52911yescharacteristic
291Dic59—42no—
292Dic59—42no—
293Dic59—81no—
294Dic59—81no—
295C4×C22⋊Q8—591no—
296#2283D5911yescharacteristic
297#2101—591no—
298#2097—591no—
299#2085—591no—
300#2077—591no—
301#2075—591no—
302#2074—591no—
303#2036—591no—
304#2035—591no—
305C4×#107—591no—
306C4×#106—591no—
307C4×#103—591no—
308C4×#103—591no—
309C4×C22⋊C8—591no—
310C2×C59—42no—
311C2×C59#239511yescharacteristic
312C2×C59C2×#48511yescharacteristic
313C2×C59#309111yescharacteristic
314C2×C59#310211yescharacteristic
315C2×C59#310811yescharacteristic
316C2×C59#316811yescharacteristic
317C2×C59#317211yescharacteristic
318C2×C22⋊Q8—591no—
319C2×C4×Q8—591no—
320C22×C42C2×D5911yescharacteristic
321C2×#107—591no—
322C2×#107—591no—
323C2×#106—591no—
324C2×#106—591no—
325C2×#103—591no—
326C2×#103—591no—
327C2×#103—591no—
328C2×#103—591no—
329C2×C22⋊C8—591no—
330C2×C22⋊C8—591no—
331C2×C4×C8—591no—
332C2×C4×C8—591no—
333#391—591no—
334#389—591no—
335#387—1181no—
336#385—1181no—
337#383—21no—
338#382C2×D5911yescharacteristic
339#382C2×D5911yescharacteristic
340#381—21no—
341#380—591no—
342#380C2×D5911yescharacteristic
343#380C2×D5911yescharacteristic
344C4×C4⋊C4—591no—
345C4×C4⋊C4—1181no—
346C4×C4⋊C4C2×D5911yescharacteristic
347C4×C4⋊C4C2×D5911yescharacteristic
348C4×C22⋊C4—21no—
349C4×C22⋊C4—1181no—
350#376—21no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#336—591no—
360#336—591no—
361C59#2375911yescharacteristic, complemented, Sylow
362C22×Q8—591no—
363C23×C4C22⋊D5911yescharacteristic
364C23×C4C22⋊D5911yescharacteristic
365C23×C4C22×D5911yescharacteristic
366C22×C8—591no—
367C22×C8—591no—
368C22×C8—591no—
369C22×C8—591no—
370C22⋊Q8—1182no—
371C22⋊Q8—1182no—
372C4×Q8—1182no—
373C4×Q8—1182no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C4⋊C4C22×D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1181no—
387C2×C22⋊C4—1181no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—1181no—
393C2×C42—1181no—
394C2×C42C22⋊D5911yescharacteristic
395C2×C42C22⋊D5911yescharacteristic
396C2×C42C22×D5911yescharacteristic
397C2×C42C22×D5911yescharacteristic
398#107—1182no—
399#107—1182no—
400#106—1182no—
401#106—1182no—
402#103—1182no—
403#103—1182no—
404#103—1182no—
405#103—1182no—
406C22⋊C8—1182no—
407C22⋊C8—1182no—
408C4×C8—1182no—
409C4×C8—1182no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423C24C2×C22⋊D5911yescharacteristic
424C2×Q8—592no—
425C2×Q8—1181no—
426C2×Q8—1181no—
427C2×Q8—1181no—
428C2×Q8—1181no—
429C2×Q8—1181no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—41no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4C2×C22⋊D5911yescharacteristic
451C22×C4C2×C22⋊D5911yescharacteristic
452C22×C4C2×C22⋊D5911yescharacteristic
453C22×C4C2×C22⋊D5911yescharacteristic
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C23×D5911yescharacteristic
456C2×C8—592no—
457C2×C8—592no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—592no—
477C4⋊C4—1184no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—2362no—
484C4⋊C4—2362no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—1184no—
488C22⋊C4—2362no—
489C42—22no—
490C42—22no—
491C42—21no—
492C42—21no—
493C42—42no—
494C42—42no—
495C42—2362no—
496C42—2362no—
497C42#1174568512yes—
498C42#1174568512yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#1427374712yes—
506C23C22×C22⋊D5911yescharacteristic
507Q8—1182no—
508Q8—1182no—
509Q8—2361no—
510Q8—2361no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—1182no—
541C2×C4—1182no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4#1427371012yes—
553C2×C4#1427371012yes—
554C2×C4#1427374712yes—
555C2×C4#1427375712yes—
556C2×C4#41×D5911yescharacteristic
557C2×C4#41×D5911yescharacteristic
558C2×C4#41×D5911yescharacteristic
559C2×C4#41×D5911yescharacteristic
560C2×C4C2×#1174568511yescharacteristic
561C2×C4C2×#1174568511yescharacteristic
562C8—1182no—
563C8—1182no—
564C8—2361no—
565C8—2361no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22C2×#1427371011yescharacteristic
574C22#1683790911yescharacteristic
575C22#1683794811yescharacteristic
576C22C2×#1427374711yescharacteristic
577C22C2×#1427374711yescharacteristic
578C22C2×#1427375711yescharacteristic
579C22#1683820011yescharacteristic
580C4—22no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—4721no—
589C4—4721no—
590C4—2362no—
591C4—2362no—
592C4#1683774314yes—
593C4#1683793011yescharacteristic
594C4#1683793011yescharacteristic
595C4#1683793011yescharacteristic
596C4#1683793011yescharacteristic
597C2—42no—
598C2#2142735311yescharacteristic
599C2C2×#1683774311yescharacteristic
600C2#2143400911yescharacteristic
601C2#2143410811yescharacteristic
602C2#2143417211yescharacteristic
603C2#2143466011yescharacteristic
604C2#2143470311yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2283C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2283D59
hypocenterC59#23759
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
2. #21425663C2semidirect
3. #21425647C2semidirect
4. #21425578C2semidirect
7. #21425515C2semidirect
10. C4×#14273660C2semidirect
11. C4×#14273658C2semidirect
12. C4×#14273650C2semidirect
13. C4×#14273650C2semidirect
361. C59#23759semidirect

Maximal subgroups

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

subgroupindex
C4×#142737062
#214256632
#214256472
#214255782
#214255332
#214255222
#214255152
#214252992
#214252922
C4×#142736602
C4×#142736582
C4×#142736502
C4×C22⋊(C59⋊C8)2
C59×#22832
#2375959