Finite Group Databasebeta

← #28638629

Subgroups — #28638629

Order 15104.

870 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837756C211yesmaximal, characteristic, complemented
2C2×#16837614C211yesmaximal, characteristic
3#21429296C211yesmaximal, characteristic, complemented
4#21425509C211yesmaximal, characteristic, complemented
5#21425504C211yesmaximal, characteristic, complemented
6#21425411C211yesmaximal, characteristic, complemented
7#21425406C211yesmaximal, characteristic
8#21425053C211yesmaximal, characteristic, complemented
9#21424815C211yesmaximal, characteristic, complemented
10#21424451C211yesmaximal, characteristic, complemented
11#21424444C211yesmaximal, characteristic
12#21424111C211yesmaximal, characteristic, complemented
13#21424096C211yesmaximal, characteristic, complemented
14C4×#14273647C211yesmaximal, characteristic, complemented
15C59×#2367C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C22×#11745669C2211yescharacteristic
18C2×#14273719C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20#16837756—24no—
21#16837614—24no—
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×#14273648C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837324—21no—
35#16837324—21no—
36#16837304C2211yescharacteristic
37#16837304C2211yescharacteristic
38#16837276—21no—
39#16837276—21no—
40#16837272C2211yescharacteristic
41#16837268C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#124×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×C4×D4×C59C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#388—21no—
56C59×#388—21no—
57C59×#385—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C4×C22⋊C4×C59—21no—
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C2×#11745671—22no—
64C2×#11745671—22no—
65C2×#11745671—41no—
66C2×#11745669—22no—
67C2×#11745669—22no—
68C2×#11745669—41no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273719—22no—
74#14273719—22no—
75C4×D236—22no—
76C4×D236—22no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×(C2×D59)⋊C4—21no—
88C2×(C2×D59)⋊C4—21no—
89C4⋊(C59⋊C8)—22no—
90C4⋊(C59⋊C8)—41no—
91C4⋊(C59⋊C8)—41no—
92#14273651—22no—
93#14273651—22no—
94#14273651—22no—
95#14273648—22no—
96#14273648—22no—
97#14273648—22no—
98#14273647—22no—
99#14273647—22no—
100C4×C59⋊C8—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59C2311yescharacteristic
104C22×D4×C59C2311yescharacteristic
105C23×C4×C59—21no—
106#124×C59—22no—
107#124×C59D414yes—
108C4×Q8×C59—22no—
109C4×Q8×C59—41no—
110C4×D4×C59—22no—
111C4×D4×C59—41no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59D411yescharacteristic
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C42×C59—21no—
123C2×C42×C59—21no—
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126C2×C42×C59C2311yescharacteristic
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59—21no—
133C23×D59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59C2×D412yes—
140C2×D4×C59—21no—
141C2×D4×C59—21no—
142C2×D4×C59—21no—
143C2×D4×C59—21no—
144C2×D4×C59—21no—
145C2×D4×C59C2×D412yes—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—42no—
150C22×C4×C59—42no—
151C22×C4×C59—81no—
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2411yescharacteristic
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C2×D236—21no—
163C2×D236—21no—
164C2×D236—21no—
165C2×D236—21no—
166C2×D236—21no—
167C2×D236#4112yes—
168C2×C4×D59—42no—
169C2×C4×D59—81no—
170C2×C4×D59—81no—
171C4⋊C4×C59—22no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—41no—
176C4⋊C4×C59—42no—
177C4⋊C4×C59#4112yes—
178C22⋊C4×C59—24no—
179C22⋊C4×C59—24no—
180C22⋊C4×C59—42no—
181C22⋊C4×C59—42no—
182C22⋊C4×C59—81no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—42no—
188C42×C59—81no—
189C42×C59C2×D412yes—
190C42×C59#4112yes—
191#11745671—44no—
192#11745669—44no—
193(C2×D59)⋊C4—42no—
194(C2×D59)⋊C4—42no—
195(C2×D59)⋊C4—42no—
196(C2×D59)⋊C4—81no—
197(C2×D59)⋊C4—81no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59—81no—
205C4⋊Dic59—81no—
206Dic59⋊C4—42no—
207Dic59⋊C4—42no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8—21no—
219C2×C59⋊C8—21no—
220C23×C59—22no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59C22×D411yescharacteristic
228C22×D59—22no—
229C22×D59—41no—
230C22×D59—41no—
231C22×D59—41no—
232C22×D59—41no—
233C22×D59—41no—
234C22×D59—41no—
235Q8×C59—22no—
236Q8×C59—22no—
237Q8×C59—41no—
238Q8×C59—41no—
239D4×C59—22no—
240D4×C59—22no—
241D4×C59—41no—
242D4×C59—41no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—22no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—22no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
256C2×C4×C59—21no—
257C2×C4×C59—21no—
258C2×C4×C59—21no—
259C2×C4×C59—21no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59—41no—
265C2×C4×C59—41no—
266C2×C4×C59—41no—
267C2×C4×C59—84no—
268C2×C4×C59#12112yes—
269C2×C4×C59#12112yes—
270C2×C4×C59#13512yes—
271C2×C4×C59C22×D411yescharacteristic
272C2×C4×C59C2×#4111yescharacteristic
273C2×C4×C59C2×#4111yescharacteristic
274C2×C4×C59C2×#4111yescharacteristic
275C2×C4×C59H(5,𝔽2)11yescharacteristic
276C2×C4×C592−1+411yescharacteristic
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C2×Dic59—41no—
285C2×Dic59—41no—
286D236—22no—
287D236—22no—
288D236—41no—
289D236—41no—
290C4×D59—84no—
291C59⋊C8—41no—
292C59⋊C8—41no—
293C59⋊C8—41no—
294C59⋊C8—41no—
295#25327—591nomaximal, Sylow
296C22×C59—42no—
297C22×C59—42no—
298C22×C59—42no—
299C22×C59—81no—
300C22×C59—81no—
301C22×C59—81no—
302C22×C59—81no—
303C22×C59C2×#12111yescharacteristic
304C22×C59#54611yescharacteristic
305C22×C59#54811yescharacteristic
306C22×C59C2×#13511yescharacteristic
307C22×C59#57511yescharacteristic
308C22×C59#57611yescharacteristic
309C22×C59#57711yescharacteristic
310C2×D59—42no—
311C2×D59—42no—
312C2×D59—42no—
313C2×D59—81no—
314C2×D59—81no—
315C2×D59—81no—
316C2×D59—81no—
317C4×C59—22no—
318C4×C59—22no—
319C4×C59—22no—
320C4×C59—42no—
321C4×C59—41no—
322C4×C59—42no—
323C4×C59—41no—
324C4×C59—41no—
325C4×C59—81no—
326C4×C59#52911yescharacteristic
327C4×C59#54211yescharacteristic
328C4×C59#54611yescharacteristic
329C4×C59#54711yescharacteristic
330Dic59—81no—
331Dic59—81no—
332Dic59—81no—
333C2×#488—591no—
334C2×#463—591no—
335#2566—591no—
336#2367D5911yescharacteristic, complemented
337#2073—591no—
338#2072—591no—
339#2055—591no—
340#2054—591no—
341#1989—591no—
342#1939—591no—
343#1880—591no—
344#1879—591no—
345#1805—591no—
346#1802—591no—
347C4×#102—591no—
348C2×C59—82no—
349C2×C59#242711yescharacteristic
350C2×C59#303511yescharacteristic
351C2×C59#304511yescharacteristic
352C2×C59#329711yescharacteristic
353C2×C59#332611yescharacteristic
354C2×C59#336011yescharacteristic
355C2×C59#337011yescharacteristic
356D59—82no—
357C22×SD16—591no—
358C22×D8—591no—
359C2×#125—591no—
360C2×#124C2×D5911yescharacteristic
361C2×C4×Q8C2×D5911yescharacteristic
362C2×C4×D4—591no—
363C2×C4×D4C2×D5911yescharacteristic
364#488—1184no—
365#463—1184no—
366C2×C4⋊C8—591no—
367C2×C4⋊C8—591no—
368C2×C4⋊C8—591no—
369C2×#103—591no—
370C2×#103—591no—
371C2×#103—591no—
372C2×#102—591no—
373C2×#102—591no—
374C2×#102—591no—
375C2×#102—591no—
376C2×#102—591no—
377C2×C4×C8—591no—
378#396—1181no—
379#396—1181no—
380#391C2×D5911yescharacteristic
381#390—591no—
382#390—591no—
383#390C2×D5911yescharacteristic
384#388—21no—
385#388—21no—
386#385—21no—
387#385—1181no—
388#385—1181no—
389#384—591no—
390#382—591no—
391C4×C4⋊C4C2×D5911yescharacteristic
392C4×C4⋊C4C2×D5911yescharacteristic
393C4×C22⋊C4—21no—
394#340—591no—
395#340—591no—
396#340—591no—
397#340—591no—
398#340—591no—
399#340—591no—
400#336—591no—
401#336—591no—
402C59#2532711yescharacteristic, complemented, Sylow
403C22×Q8C22×D5911yescharacteristic
404C22×D4—591no—
405C22×D4C22×D5911yescharacteristic
406C23×C4—21no—
407C23×C4—1181no—
408C2×SD16—1182no—
409C2×SD16—1182no—
410C2×SD16—2361no—
411C2×D8—1182no—
412C2×D8—1182no—
413C2×D8—2361no—
414C22×C8—591no—
415C22×C8—591no—
416C22×C8—591no—
417C22×C8—591no—
418#125—1182no—
419#125—1182no—
420#124—22no—
421#124D23614yes—
422C4×Q8—22no—
423C4×Q8—41no—
424C4×D4—22no—
425C4×D4—41no—
426C4×D4—1182no—
427C4×D4—1182no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—591no—
430C2×C4⋊C4—591no—
431C2×C4⋊C4—591no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4—1181no—
435C2×C4⋊C4—1181no—
436C2×C4⋊C4C22⋊D5911yescharacteristic
437C2×C4⋊C4C22⋊D5911yescharacteristic
438C2×C4⋊C4C22×D5911yescharacteristic
439C2×C4⋊C4C22×D5911yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—1181no—
446C2×C22⋊C4—1181no—
447C2×C22⋊C4—1181no—
448C2×C22⋊C4—1181no—
449C2×C22⋊C4—1181no—
450C2×C42—21no—
451C2×C42—21no—
452C2×C42C22×D5911yescharacteristic
453C2×C42C22×D5911yescharacteristic
454C2×C42C22×D5911yescharacteristic
455C4⋊C8—1182no—
456C4⋊C8—2361no—
457C4⋊C8—2361no—
458#103—1182no—
459#103—1182no—
460#103—1182no—
461#102—1182no—
462#102—1182no—
463#102—1182no—
464#102—1182no—
465#102—1182no—
466C4×C8—1182no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—1181no—
471#95—1181no—
472#95C22⋊D5911yescharacteristic
473#95C22⋊D5911yescharacteristic
474C24—21no—
475C24—1181no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8C2×D23612yes—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—592no—
488C2×D4—1181no—
489C2×D4—1181no—
490C2×D4—1181no—
491C2×D4—1181no—
492C2×D4—1181no—
493C2×D4C2×D23612yes—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—4721no—
504C22×C4—4721no—
505C22×C4—2362no—
506C22×C4C2×C22⋊D5911yescharacteristic
507C22×C4C2×C22⋊D5911yescharacteristic
508C22×C4C2×C22⋊D5911yescharacteristic
509C22×C4C2×C22⋊D5911yescharacteristic
510C22×C4C2×C22⋊D5911yescharacteristic
511C22×C4C2×C22⋊D5911yescharacteristic
512C22×C4C23×D5911yescharacteristic
513SD16—2364no—
514D8—2364no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—1181no—
525C2×C8—1181no—
526C2×C8—1181no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—41no—
532C4⋊C4—42no—
533C4⋊C4—1181no—
534C4⋊C4—1181no—
535C4⋊C4—1181no—
536C4⋊C4—1181no—
537C4⋊C4—1181no—
538C4⋊C4—1181no—
539C4⋊C4—4721no—
540C4⋊C4—4721no—
541C4⋊C4—2362no—
542C4⋊C4—2362no—
543C4⋊C4#1174568712yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—81no—
549C22⋊C4—4721no—
550C22⋊C4—4721no—
551C22⋊C4—2362no—
552C22⋊C4—2362no—
553C22⋊C4—2362no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—42no—
559C42—81no—
560C42C2×D23612yes—
561C42#1174568512yes—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—1182no—
570C23—2361no—
571C23—2361no—
572C23—2361no—
573C23—2361no—
574C23—2361no—
575C23—2361no—
576C23C22×C22⋊D5911yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581D4—22no—
582D4—22no—
583D4—41no—
584D4—41no—
585D4—1182no—
586D4—1182no—
587D4—2361no—
588D4—2361no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—84no—
614C2×C4—4724no—
615C2×C4—2361no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4—2361no—
620C2×C4—2361no—
621C2×C4—2361no—
622C2×C4—2361no—
623C2×C4—2361no—
624C2×C4#1427370112yes—
625C2×C4#1427370112yes—
626C2×C4#1427374712yes—
627C2×C4C22×D23611yescharacteristic
628C2×C4#41×D5911yescharacteristic
629C2×C4C2×#1174568711yescharacteristic
630C2×C4C2×#1174568511yescharacteristic
631C2×C4#1427377211yescharacteristic
632C2×C4#1427377411yescharacteristic
633C8—2361no—
634C8—2361no—
635C8—2361no—
636C8—2361no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—4721no—
645C22—4721no—
646C22—4721no—
647C22—4721no—
648C22—2362no—
649C22—2362no—
650C22—2362no—
651C22C2×#1427370111yescharacteristic
652C22#1683804611yescharacteristic
653C22#1683805911yescharacteristic
654C22C2×#1427374711yescharacteristic
655C22#1683820011yescharacteristic
656C22#1683820511yescharacteristic
657C22#1683821011yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—41no—
662C4—42no—
663C4—41no—
664C4—41no—
665C4—42no—
666C4—81no—
667C4—4721no—
668C4—4721no—
669C4—4721no—
670C4#1683793411yescharacteristic
671C4#1683801711yescharacteristic
672C4#1683804711yescharacteristic
673C4#1683805611yescharacteristic
674C2—82no—
675C2—4722no—
676C2#2142766711yescharacteristic
677C2#2143355111yescharacteristic
678C2#2143363511yescharacteristic
679C2#2143594611yescharacteristic
680C2#2143634911yescharacteristic
681C2#2143674111yescharacteristic
682C2#2143684311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2367C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
hypercenter#2367D59
hypocenterC59#25327
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
1. C2×#16837756C2semidirect
3. #21429296C2semidirect
4. #21425509C2semidirect
5. #21425504C2semidirect
6. #21425411C2semidirect
8. #21425053C2semidirect
9. #21424815C2semidirect
10. #21424451C2semidirect
12. #21424111C2semidirect
13. #21424096C2semidirect
14. C4×#14273647C2semidirect
15. C59×#2367C2semidirect
336. #2367D59semidirect
402. C59#25327semidirect

Maximal subgroups

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

subgroupindex
C2×#168377562
C2×#168376142
#214292962
#214255092
#214255042
#214254112
#214254062
#214250532
#214248152
#214244512
#214244442
#214241112
#214240962
C4×#142736472
C59×#23672
#2532759