Finite Group Databasebeta

← #28651670

Subgroups — #28651670

Order 15104.

896 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837702C211yesmaximal, characteristic, complemented
2C2×C4×#11745669C211yesmaximal, characteristic
3#21427968C211yesmaximal, characteristic, complemented
4#21425620C211yesmaximal, characteristic, complemented
5#21425578C211yesmaximal, characteristic, complemented
6#21425481C211yesmaximal, characteristic, complemented
7#21425434C211yesmaximal, characteristic, complemented
8#21424826C211yesmaximal, characteristic, complemented
9#21424655C211yesmaximal, characteristic, complemented
10#21424444C211yesmaximal, characteristic
11#21424444C211yesmaximal, characteristic
12#21424357C211yesmaximal, characteristic, complemented
13#21424356C211yesmaximal, characteristic, complemented
14#21424343C211yesmaximal, characteristic, complemented
15C59×#2349C211yesmaximal, characteristic, complemented
16C22×#11745669C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19#16837702—24no—
20C4×#11745669—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273648C2211yescharacteristic
29C2×#14273648C2211yescharacteristic
30C2×#14273647C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×C4×C59⋊C8C2211yescharacteristic
34#16837337—21no—
35#16837326—21no—
36#16837304C2211yescharacteristic
37#16837304C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837273—21no—
41#16837261—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4×D4×C59C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C2×C4×D4×C59C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#385—21no—
56C59×#385—21no—
57C59×#382C2211yescharacteristic
58C59×#380—21no—
59C4×C4⋊C4×C59C2211yescharacteristic
60C2×#95×C59—21no—
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C2×#11745669—22no—
64C2×#11745669—22no—
65C2×#11745669Q814yes—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70#14273719—22no—
71#14273719—41no—
72C4×D236—22no—
73C4×D236—41no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86#14273660—22no—
87#14273660—22no—
88#14273660—22no—
89#14273658—22no—
90#14273658—41no—
91C4⋊(C59⋊C8)—22no—
92#14273648—22no—
93#14273648—22no—
94#14273648D414yes—
95#14273647—22no—
96#14273647—22no—
97#14273647—41no—
98C4×C59⋊C8—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×D4×C59C2311yescharacteristic
102C23×C4×C59—21no—
103C23×C4×C59—21no—
104C23×C4×C59—21no—
105C4×D4×C59—22no—
106C4×D4×C59—22no—
107C4×D4×C59—22no—
108C4×D4×C59—22no—
109C4×D4×C59—41no—
110C4×D4×C59—41no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59D411yescharacteristic
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#95×C59—21no—
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59D411yescharacteristic
129#95×C59D411yescharacteristic
130C24×C59—21no—
131C23×D59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59C2×D412yes—
136C2×D4×C59C2×Q812yes—
137C2×D4×C59#4112yes—
138C22×C4×C59—21no—
139C22×C4×C59—24no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—42no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59—42no—
146C22×C4×C59—81no—
147C22×C4×C59—81no—
148C22×C4×C59—81no—
149C22×C4×C59—81no—
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2411yescharacteristic
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C2×D236—21no—
161C2×D236—21no—
162C2×D236—21no—
163C2×D236—21no—
164C2×D236—21no—
165C2×D236C2×Q812yes—
166C2×C4×D59—42no—
167C2×C4×D59—42no—
168C2×C4×D59—81no—
169C4⋊C4×C59—22no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—41no—
176C4⋊C4×C59—81no—
177C4⋊C4×C59—81no—
178C4⋊C4×C59#4112yes—
179C22⋊C4×C59—24no—
180C22⋊C4×C59—42no—
181C22⋊C4×C59—42no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188#11745669—28no—
189(C2×D59)⋊C4—42no—
190(C2×D59)⋊C4—42no—
191(C2×D59)⋊C4—81no—
192(C2×D59)⋊C4—81no—
193(C2×D59)⋊C4—81no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59C2×D412yes—
200Dic59⋊C4—42no—
201Dic59⋊C4—42no—
202Dic59⋊C4—81no—
203Dic59⋊C4—81no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
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⋊C8C2×D412yes—
215C2×C59⋊C8C2×Q812yes—
216C23×C59—22no—
217C23×C59—22no—
218C23×C59—22no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59—41no—
222C23×C59—41no—
223C23×C59C22×D411yescharacteristic
224C22×D59—22no—
225C22×D59—41no—
226C22×D59—41no—
227C22×D59—41no—
228C22×D59—41no—
229C22×D59—41no—
230C22×D59—41no—
231D4×C59—22no—
232D4×C59—22no—
233D4×C59—22no—
234D4×C59C22⋊Q814yes—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—22no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
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—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—84no—
262C2×C4×C59—84no—
263C2×C4×C59—84no—
264C2×C4×C59#13512yes—
265C2×C4×C59C22×D411yescharacteristic
266C2×C4×C59C22×Q811yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×C4×C59H(5,𝔽2)11yescharacteristic
270C2×C4×C592−1+411yescharacteristic
271C2×Dic59—22no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280D236—22no—
281D236—22no—
282D236—41no—
283D236—41no—
284C4×D59—84no—
285C59⋊C8—22no—
286C59⋊C8—22no—
287C59⋊C8—41no—
288C59⋊C8—41no—
289#26370—591nomaximal, Sylow
290C22×C59—24no—
291C22×C59—42no—
292C22×C59—42no—
293C22×C59—42no—
294C22×C59—42no—
295C22×C59—42no—
296C22×C59—42no—
297C22×C59#52511yescharacteristic
298C22×C59D4211yescharacteristic
299C22×C59D4×Q811yescharacteristic
300C22×C59C2×#13511yescharacteristic
301C22×C59#57511yescharacteristic
302C22×C59#57611yescharacteristic
303C22×C59#57611yescharacteristic
304C2×D59—42no—
305C2×D59—42no—
306C2×D59—42no—
307C2×D59—81no—
308C2×D59—81no—
309C2×D59—81no—
310C2×D59—81no—
311C4×C59—22no—
312C4×C59—41no—
313C4×C59—41no—
314C4×C59—42no—
315C4×C59—41no—
316C4×C59—41no—
317C4×C59—41no—
318C4×C59—81no—
319C4×C59—81no—
320C4×C59C2×C22⋊Q811yescharacteristic
321C4×C59#53911yescharacteristic
322C4×C59#54711yescharacteristic
323C4×C59#55411yescharacteristic
324Dic59—42no—
325Dic59—81no—
326Dic59—81no—
327C2×#478—591no—
328C2×C4×D8—591no—
329#2454—591no—
330#2349D5911yescharacteristic, complemented
331#2092—591no—
332#2085—591no—
333#2067—591no—
334#2059—591no—
335#1942—591no—
336#1916—591no—
337#1879—591no—
338#1879—591no—
339#1864—591no—
340#1864—591no—
341#1862—591no—
342C2×C59—44no—
343C2×C59#244911yescharacteristic
344C2×C59#310211yescharacteristic
345C2×C59#312211yescharacteristic
346C2×C59#326811yescharacteristic
347C2×C59#328011yescharacteristic
348C2×C59#338011yescharacteristic
349C2×C59#338911yescharacteristic
350D59—82no—
351C22×D8—591no—
352C2×#125—591no—
353C2×C4×D4—591no—
354C2×C4×D4C2×D5911yescharacteristic
355C2×C4×D4C2×D5911yescharacteristic
356C2×C4×D4C2×D5911yescharacteristic
357#478—1184no—
358C4×D8—1184no—
359C2×#107—591no—
360C2×#107—591no—
361C2×#107—591no—
362C2×#106—591no—
363C2×#106—591no—
364C2×C4⋊C8—591no—
365C2×#102—591no—
366C2×#102—591no—
367C2×#102—591no—
368C2×#102—591no—
369C2×#102—591no—
370C2×#102—591no—
371C2×C4×C8—591no—
372#399—1181no—
373#396—1181no—
374#390—591no—
375#390—591no—
376#389—591no—
377#389—591no—
378#389C2×D5911yescharacteristic
379#389C2×D5911yescharacteristic
380#385—21no—
381#385—21no—
382#385—1181no—
383#382C2×D5911yescharacteristic
384#380—21no—
385#380—1181no—
386C4×C4⋊C4C2×D5911yescharacteristic
387C2×#95—21no—
388#340—591no—
389#340—591no—
390#340—591no—
391#340—591no—
392#336—591no—
393#336—591no—
394#336—591no—
395#336—591no—
396C59#2637011yescharacteristic, complemented, Sylow
397C22×D4—591no—
398C22×D4C22×D5911yescharacteristic
399C23×C4—21no—
400C23×C4—21no—
401C23×C4—21no—
402C23×C4—1181no—
403C2×D8—594no—
404C2×D8—1182no—
405C2×D8—1182no—
406C22×C8—591no—
407C22×C8—591no—
408C22×C8—591no—
409C22×C8—591no—
410#125—1182no—
411#125—2361no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—41no—
417C4×D4—41no—
418C4×D4—1182no—
419C4×D4—2361no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4C22⋊D5911yescharacteristic
430C2×C4⋊C4C22⋊D5911yescharacteristic
431C2×C4⋊C4C22×D5911yescharacteristic
432C2×C4⋊C4C22×D5911yescharacteristic
433C2×C4⋊C4C22×D5911yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1181no—
438C2×C22⋊C4—1181no—
439C2×C22⋊C4—1181no—
440C2×C22⋊C4—1181no—
441C2×C22⋊C4—1181no—
442C2×C42C22×D5911yescharacteristic
443C2×C42C22×D5911yescharacteristic
444C2×C42C22×D5911yescharacteristic
445#107—1182no—
446#107—1182no—
447#107—1182no—
448#106—1182no—
449#106—2361no—
450C4⋊C8—1182no—
451#102—594no—
452#102—1182no—
453#102—1182no—
454#102—1182no—
455#102—1182no—
456#102—2361no—
457C4×C8—1182no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—21no—
462#95—1181no—
463#95—1181no—
464#95C22⋊D5911yescharacteristic
465#95C22⋊D5911yescharacteristic
466C24—21no—
467C24—1181no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—592no—
472C2×D4—1181no—
473C2×D4—1181no—
474C2×D4—1181no—
475C2×D4—1181no—
476C2×D4—1181no—
477C2×D4#1174568512yes—
478C2×D4D4×D5912yes—
479C2×D4Q8×D5912yes—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—1181no—
493C22×C4—1181no—
494C22×C4—1181no—
495C22×C4—4721no—
496C22×C4—2362no—
497C22×C4—2362no—
498C22×C4C2×C22⋊D5911yescharacteristic
499C22×C4C2×C22⋊D5911yescharacteristic
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C2×C22⋊D5911yescharacteristic
502C22×C4C2×C22⋊D5911yescharacteristic
503C22×C4C2×C22⋊D5911yescharacteristic
504C22×C4C23×D5911yescharacteristic
505D8—1188no—
506C2×C8—592no—
507C2×C8—592no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—41no—
525C4⋊C4—81no—
526C4⋊C4—81no—
527C4⋊C4—592no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—4721no—
534C4⋊C4—4721no—
535C4⋊C4—2362no—
536C4⋊C4—2362no—
537C4⋊C4#1174568512yes—
538C22⋊C4—24no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—4721no—
542C22⋊C4—4721no—
543C22⋊C4—4721no—
544C22⋊C4—2362no—
545C22⋊C4—2362no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—1182no—
560C23—2361no—
561C23—2361no—
562C23—2361no—
563C23—2361no—
564C23—2361no—
565C23—2361no—
566C23C22×C22⋊D5911yescharacteristic
567D4—22no—
568D4—22no—
569D4—22no—
570D4—1182no—
571D4—1182no—
572D4—2361no—
573D4—2361no—
574D4#1427370514yes—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—1182no—
605C2×C4—4724no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4#1427374712yes—
615C2×C4C2×D4×D5911yescharacteristic
616C2×C4C2×Q8×D5911yescharacteristic
617C2×C4C2×#1174568511yescharacteristic
618C2×C4C2×#1174568511yescharacteristic
619C2×C4#1427377211yescharacteristic
620C2×C4#1427377511yescharacteristic
621C8—1182no—
622C8—1182no—
623C8—2361no—
624C8—2361no—
625C22—24no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—4721no—
633C22—4721no—
634C22—4721no—
635C22—4721no—
636C22—2362no—
637C22—2362no—
638C22—2362no—
639C22#1683791311yescharacteristic
640C22D4×C22⋊D5911yescharacteristic
641C22Q8×C22⋊D5911yescharacteristic
642C22C2×#1427374711yescharacteristic
643C22#1683820011yescharacteristic
644C22#1683820511yescharacteristic
645C22#1683820511yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—42no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—81no—
655C4—4721no—
656C4—4721no—
657C4—2362no—
658C4C2×#1427370511yescharacteristic
659C4#1683799911yescharacteristic
660C4#1683805311yescharacteristic
661C4#1683810711yescharacteristic
662C2—44no—
663C2—4722no—
664C2#2142790511yescharacteristic
665C2#2143411411yescharacteristic
666C2#2143433911yescharacteristic
667C2#2143563211yescharacteristic
668C2#2143577711yescharacteristic
669C2#2143695911yescharacteristic
670C2#2143705011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2349C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
hypercenter#2349D59
hypocenterC59#26370
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
1. C2×#16837702C2semidirect
3. #21427968C2semidirect
4. #21425620C2semidirect
5. #21425578C2semidirect
6. #21425481C2semidirect
7. #21425434C2semidirect
8. #21424826C2semidirect
9. #21424655C2semidirect
12. #21424357C2semidirect
13. #21424356C2semidirect
14. #21424343C2semidirect
15. C59×#2349C2semidirect
330. #2349D59semidirect
396. C59#26370semidirect

Maximal subgroups

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

subgroupindex
C2×#168377022
C2×C4×#117456692
#214279682
#214256202
#214255782
#214254812
#214254342
#214248262
#214246552
#214244442
#214243572
#214243562
#214243432
C59×#23492
#2637059