Finite Group Databasebeta

← #28638109

Subgroups — #28638109

Order 15104.

794 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837764C211yesmaximal, characteristic
2C2×#16837684C211yesmaximal, characteristic, complemented
3#21429076C211yesmaximal, characteristic
4#21425474C211yesmaximal, characteristic, complemented
5#21425467C211yesmaximal, characteristic
6#21425342C211yesmaximal, characteristic, complemented
7#21425337C211yesmaximal, characteristic
8#21425028C211yesmaximal, characteristic
9#21425028C211yesmaximal, characteristic
10#21425019C211yesmaximal, characteristic, complemented
11#21424820C211yesmaximal, characteristic, complemented
12#21424730C211yesmaximal, characteristic, complemented
13#21424723C211yesmaximal, characteristic
14#21423929C211yesmaximal, characteristic, complemented
15C59×#2369C211yesmaximal, characteristic, complemented
16C2×#14273727C2211yescharacteristic
17C2×#14273715C2211yescharacteristic
18C2×#14273715C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20#16837764—24no—
21#16837684—24no—
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×#14273650C2211yescharacteristic
34C2×#14273650C2211yescharacteristic
35C2×#14273647C2211yescharacteristic
36C2×#14273647C2211yescharacteristic
37C2×#14273647C2211yescharacteristic
38C2×#14273647C2211yescharacteristic
39C2×#14273638C2211yescharacteristic
40#16837324—21no—
41#16837324—21no—
42#16837308C2211yescharacteristic
43#16837304C2211yescharacteristic
44#16837294—21no—
45#16837294—21no—
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C4⋊Q8×C59C2211yescharacteristic
51C59×#391—21no—
52C59×#391—21no—
53C59×#389—21no—
54C59×#389C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C4×C4⋊C4×C59—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C22×Dic118C2311yescharacteristic
63C22×D236C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273727—22no—
69#14273727—41no—
70#14273715—22no—
71#14273715—22no—
72#14273715—22no—
73#14273715—41no—
74C4×Dic118—22no—
75C4×Dic118—22no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×(C2×D59)⋊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—
86C2×C4×Dic59—21no—
87#14273658—22no—
88#14273658—22no—
89C4⋊(C59⋊C8)—22no—
90C4⋊(C59⋊C8)—41no—
91C4⋊(C59⋊C8)—41no—
92#14273650—22no—
93#14273650—22no—
94#14273650—22no—
95#14273650—22no—
96#14273650—22no—
97#14273650—22no—
98#14273650—22no—
99#14273650—41no—
100#14273647—22no—
101#14273647—22no—
102#14273647—22no—
103#14273647—41no—
104#14273638—22no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C22×Q8×C59—21no—
108C4⋊Q8×C59—22no—
109C4⋊Q8×C59D414yes—
110C2×C4⋊C4×C59—21no—
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×C4⋊C4×C59C2311yescharacteristic
119C2×C42×C59—21no—
120C2×C42×C59—21no—
121C2×C42×C59—21no—
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124C2×C42×C59C2311yescharacteristic
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C23×D59—21no—
133C2×Q8×C59—24no—
134C2×Q8×C59—81no—
135C2×Q8×C59—81no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59C2411yescharacteristic
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118#4112yes—
168C4⋊C4×C59—24no—
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×C59C2×D412yes—
179C4⋊C4×C59C2×D412yes—
180C4⋊C4×C59#4112yes—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—42no—
186C42×C59—81no—
187C42×C59—81no—
188C42×C59C2×D412yes—
189C42×C59#4112yes—
190(C2×D59)⋊C4—42no—
191(C2×D59)⋊C4—42no—
192(C2×D59)⋊C4—42no—
193(C2×D59)⋊C4—42no—
194(C2×D59)⋊C4—42no—
195(C2×D59)⋊C4—42no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—42no—
199C4⋊Dic59—42no—
200Dic59⋊C4—42no—
201C4×Dic59—42no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
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—
214C23×C59C22×D411yescharacteristic
215C22×D59—41no—
216C22×D59—41no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222Q8×C59—84no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—22no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59#12112yes—
251C2×C4×C59#12112yes—
252C2×C4×C59#13712yes—
253C2×C4×C59C22×D411yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59H(5,𝔽2)11yescharacteristic
258C2×C4×C592−1+411yescharacteristic
259C2×Dic59—22no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268D236—41no—
269D236—41no—
270D236—41no—
271D236—41no—
272Dic118—22no—
273Dic118—22no—
274Dic118—41no—
275Dic118—41no—
276Dic118—41no—
277Dic118—41no—
278Dic118—41no—
279Dic118—41no—
280C59⋊C8—41no—
281C59⋊C8—41no—
282C59⋊C8—41no—
283C59⋊C8—41no—
284#25287—591nomaximal, Sylow
285C22×C59C2×#12111yescharacteristic
286C22×C59#54611yescharacteristic
287C22×C59#54811yescharacteristic
288C22×C59C2×#13711yescharacteristic
289C22×C59#57511yescharacteristic
290C22×C59#57711yescharacteristic
291C22×C59#57711yescharacteristic
292C2×D59—81no—
293C2×D59—81no—
294C2×D59—81no—
295C2×D59—81no—
296C2×D59—81no—
297C2×D59—81no—
298C2×D59—81no—
299C4×C59—22no—
300C4×C59—22no—
301C4×C59—22no—
302C4×C59—42no—
303C4×C59—42no—
304C4×C59—42no—
305C4×C59—41no—
306C4×C59—41no—
307C4×C59—41no—
308C4×C59—81no—
309C4×C59#52911yescharacteristic
310C4×C59#54211yescharacteristic
311C4×C59#54811yescharacteristic
312C4×C59#55011yescharacteristic
313Dic59—42no—
314Dic59—81no—
315Dic59—81no—
316C2×#489—591no—
317C2×#475—591no—
318#2545—591no—
319#2369D5911yescharacteristic, complemented
320#2066—591no—
321#2065—591no—
322#2043—591no—
323#2042—591no—
324#1983—591no—
325#1983—591no—
326#1981—591no—
327#1940—591no—
328#1927—591no—
329#1926—591no—
330#1763—591no—
331C2×C59#242611yescharacteristic
332C2×C59#303711yescharacteristic
333C2×C59#304211yescharacteristic
334C2×C59#331611yescharacteristic
335C2×C59#332111yescharacteristic
336C2×C59#336911yescharacteristic
337C2×C59#337611yescharacteristic
338D59—161no—
339C2×C4⋊Q8—591no—
340C2×C4⋊Q8C2×D5911yescharacteristic
341C2×#124—591no—
342C2×#124—591no—
343C2×C4×Q8—591no—
344#489—1184no—
345#475—1184no—
346C2×#106—591no—
347C2×#106—591no—
348C2×C4⋊C8—591no—
349C2×C4⋊C8—591no—
350C2×C4⋊C8—591no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#103—591no—
358C2×#103—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×#102—591no—
362C2×#102—591no—
363C2×#97—591no—
364#396—1181no—
365#396—1181no—
366#391—21no—
367#391—21no—
368#391—591no—
369#390—591no—
370#389—21no—
371#389C2×D5911yescharacteristic
372#389C2×D5911yescharacteristic
373#388—1181no—
374#388—1181no—
375C4×C4⋊C4—21no—
376C4×C4⋊C4C2×D5911yescharacteristic
377C4×C4⋊C4C2×D5911yescharacteristic
378C4×C4⋊C4C2×D5911yescharacteristic
379C4×C4⋊C4C2×D5911yescharacteristic
380#340—591no—
381#340—591no—
382#336—591no—
383#336—591no—
384C59#2528711yescharacteristic, complemented, Sylow
385C22×Q8—21no—
386C22×Q8—591no—
387C22×Q8—591no—
388C22×D4—591no—
389C22×C8—591no—
390C22×C8—591no—
391C22×C8—591no—
392C22×C8—591no—
393C4⋊Q8—22no—
394C4⋊Q8—1182no—
395C4⋊Q8—2361no—
396C4⋊Q8D23614yes—
397#124—1182no—
398#124—1182no—
399#124—1182no—
400#124—2361no—
401C4×Q8—1182no—
402C4×Q8—1182no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4C22⋊D5911yescharacteristic
411C2×C4⋊C4C22⋊D5911yescharacteristic
412C2×C4⋊C4C22×D5911yescharacteristic
413C2×C4⋊C4C22×D5911yescharacteristic
414C2×C4⋊C4C22×D5911yescharacteristic
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C22⋊C4—1181no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C22⋊C4—1181no—
422C2×C42—21no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—1181no—
426C2×C42C22×D5911yescharacteristic
427C2×C42C22×D5911yescharacteristic
428C2×C42C22×D5911yescharacteristic
429#106—1182no—
430#106—1182no—
431C4⋊C8—1182no—
432C4⋊C8—2361no—
433C4⋊C8—2361no—
434#103—1182no—
435#103—1182no—
436#103—1182no—
437#103—1182no—
438#103—1182no—
439#103—1182no—
440#103—1182no—
441#103—2361no—
442#102—1182no—
443#102—1182no—
444#102—1182no—
445#102—2361no—
446#97—1182no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—1181no—
453#95—1181no—
454#95C22⋊D5911yescharacteristic
455#95C22⋊D5911yescharacteristic
456C24—1181no—
457C2×Q8—24no—
458C2×Q8—81no—
459C2×Q8—81no—
460C2×Q8—592no—
461C2×Q8—1181no—
462C2×Q8—1181no—
463C2×Q8—1181no—
464C2×Q8—1181no—
465C2×Q8—1181no—
466C2×Q8—1181no—
467C2×Q8—1181no—
468C2×Q8—1181no—
469C2×Q8—1181no—
470C2×Q8—1181no—
471C2×Q8—1181no—
472C2×D4—1181no—
473C2×D4—1181no—
474C2×D4—1181no—
475C2×D4—1181no—
476C2×D4—1181no—
477C2×D4—1181no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C23×D5911yescharacteristic
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—2362no—
517C4⋊C4—2362no—
518C4⋊C4—2362no—
519C4⋊C4C2×D23612yes—
520C4⋊C4C2×D23612yes—
521C4⋊C4#1174568912yes—
522C22⋊C4—2362no—
523C22⋊C4—2362no—
524C22⋊C4—2362no—
525C22⋊C4—2362no—
526C22⋊C4—2362no—
527C22⋊C4—2362no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—42no—
533C42—81no—
534C42—81no—
535C42—2362no—
536C42C2×D23612yes—
537C42#1174568512yes—
538C23—2361no—
539C23—2361no—
540C23—2361no—
541C23—2361no—
542C23—2361no—
543C23—2361no—
544C23—2361no—
545C23C22×C22⋊D5911yescharacteristic
546Q8—84no—
547Q8—1182no—
548Q8—1182no—
549Q8—2361no—
550Q8—2361no—
551Q8—2361no—
552Q8—2361no—
553Q8—2361no—
554Q8—2361no—
555D4—2361no—
556D4—2361no—
557D4—2361no—
558D4—2361no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—1182no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4#1427370112yes—
596C2×C4#1427370112yes—
597C2×C4#1427375812yes—
598C2×C4C22×D23611yescharacteristic
599C2×C4#41×D5911yescharacteristic
600C2×C4C2×#1174568511yescharacteristic
601C2×C4C2×#1174568911yescharacteristic
602C2×C4#1427377311yescharacteristic
603C2×C4#1427377511yescharacteristic
604C8—2361no—
605C8—2361no—
606C8—2361no—
607C8—2361no—
608C22—4721no—
609C22—4721no—
610C22—4721no—
611C22—4721no—
612C22—4721no—
613C22—4721no—
614C22—4721no—
615C22C2×#1427370111yescharacteristic
616C22#1683804011yescharacteristic
617C22#1683806311yescharacteristic
618C22C2×#1427375811yescharacteristic
619C22#1683820011yescharacteristic
620C22#1683821311yescharacteristic
621C22#1683821311yescharacteristic
622C4—22no—
623C4—22no—
624C4—22no—
625C4—42no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—81no—
632C4—4721no—
633C4—4721no—
634C4—2362no—
635C4#1683793911yescharacteristic
636C4#1683802411yescharacteristic
637C4#1683806411yescharacteristic
638C4#1683807411yescharacteristic
639C2—9441no—
640C2#2142765611yescharacteristic
641C2#2143356811yescharacteristic
642C2#2143360811yescharacteristic
643C2#2143620111yescharacteristic
644C2#2143627611yescharacteristic
645C2#2143683611yescharacteristic
646C2#2143692311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2369C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2369D59
hypocenterC59#25287
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2369

Complements

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

NH ≅ G/Ntype
2. C2×#16837684C2semidirect
4. #21425474C2semidirect
6. #21425342C2semidirect
10. #21425019C2semidirect
11. #21424820C2semidirect
12. #21424730C2semidirect
14. #21423929C2semidirect
15. C59×#2369C2semidirect
319. #2369D59semidirect
384. C59#25287semidirect

Maximal subgroups

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

subgroupindex
C2×#168377642
C2×#168376842
#214290762
#214254742
#214254672
#214253422
#214253372
#214250282
#214250192
#214248202
#214247302
#214247232
#214239292
C59×#23692
#2528759