Finite Group Databasebeta

← #28495613

Subgroups — #28495613

Order 15104.

1454 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273642C212yesmaximal
2C2×#16837262C211yesmaximal, characteristic, complemented
3C2×#16837169C212yesmaximal, complemented
4#21422319C211yesmaximal, characteristic, complemented
5C2×C59×#386C211yesmaximal, characteristic
6C22×C4⋊Dic59—21no—
7C22×C22⋊Dic59C2212yes—
8C22×C22⋊Dic59C2212yes—
9C2×#14273642—24no—
10C2×#14273642—24no—
11C2×#14273642—24no—
12#16837262—24no—
13C2×C59⋊#95—21no—
14C2×C59⋊#95—22no—
15#16837169—24no—
16C23×D4×C59C2211yescharacteristic
17C24×C4×C59C2211yescharacteristic
18C22×C22⋊C4×C59C412yescomplemented
19C22×C22⋊C4×C59C2211yescharacteristic
20C59×#386—21no—
21C59×#386—22no—
22C59×#386—21no—
23C2×#95×C59C412yescomplemented
24C23×Dic59—22no—
25C23×Dic59—22no—
26C2×C4⋊Dic59—24no—
27C2×C4⋊Dic59—44no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—22no—
33C2×C22⋊Dic59—22no—
34C2×C22⋊Dic59—22no—
35C2×C22⋊Dic59—22no—
36C2×C22⋊Dic59—22no—
37C2×C22⋊Dic59—22no—
38C2×C22⋊Dic59D414yes—
39C2×C22⋊Dic59D414yes—
40#14273642—216no—
41C59⋊#95—24no—
42C59⋊#95—44no—
43C25×C59C2×C411yescharacteristic
44C25×C59C2311yescharacteristic
45C22×D4×C59—21no—
46C22×D4×C59—21no—
47C22×D4×C59—22no—
48C22×D4×C59—22no—
49C22×D4×C59—21no—
50C22×D4×C59—21no—
51C22×D4×C59—22no—
52C22×D4×C59—22no—
53C22×D4×C59D412yes—
54C22×D4×C59Q812yes—
55C23×C4×C59—22no—
56C23×C4×C59—22no—
57C23×C4×C59—44no—
58C23×C4×C59C2×C411yescharacteristic
59C23×C4×C59C2×C412yes—
60C23×C4×C59C2×C412yes—
61C23×C4×C59D411yescharacteristic
62C23×C4×C59D412yes—
63C23×C4×C59D411yescharacteristic
64C2×C22⋊C4×C59—22no—
65C2×C22⋊C4×C59—21no—
66C2×C22⋊C4×C59—21no—
67C2×C22⋊C4×C59—22no—
68C2×C22⋊C4×C59—22no—
69C2×C22⋊C4×C59—22no—
70C2×C22⋊C4×C59—44no—
71#95×C59—22no—
72#95×C59—22no—
73C24×C59—21no—
74C24×C59—21no—
75C24×C59—22no—
76C24×C59—22no—
77C24×C59—42no—
78C24×C59—42no—
79C24×C59—42no—
80C24×C59—42no—
81C24×C59C4⋊C412yescomplemented
82C24×C59C22×C411yescharacteristic
83C24×C59C2×D411yescharacteristic
84C24×C59C2×D411yescharacteristic
85C24×C59C2×D411yescharacteristic
86C24×C59C2×D412yes—
87C24×C59C2×D412yes—
88C24×C59C2×Q811yescharacteristic
89C24×C59#4111yescharacteristic
90C24×C59#4111yescharacteristic
91C24×C59#4112yes—
92C24×C59#4112yes—
93C2×D4×C59—24no—
94C2×D4×C59—22no—
95C2×D4×C59—22no—
96C2×D4×C59—22no—
97C2×D4×C59—22no—
98C2×D4×C59—24no—
99C2×D4×C59—24no—
100C2×D4×C59—24no—
101C2×D4×C59—41no—
102C2×D4×C59—42no—
103C2×D4×C59—42no—
104C2×D4×C59—42no—
105C2×D4×C59—41no—
106C2×D4×C59—42no—
107C2×D4×C59—42no—
108C2×D4×C59—41no—
109C2×D4×C59—42no—
110C2×D4×C59—41no—
111C22×C4×C59—21no—
112C22×C4×C59—21no—
113C22×C4×C59—24no—
114C22×C4×C59—21no—
115C22×C4×C59—21no—
116C22×C4×C59—24no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—21no—
120C22×C4×C59—24no—
121C22×C4×C59—21no—
122C22×C4×C59—24no—
123C22×C4×C59—24no—
124C22×C4×C59—22no—
125C22×C4×C59—24no—
126C22×C4×C59—42no—
127C22×C4×C59—42no—
128C22×C4×C59—42no—
129C22×C4×C59—48no—
130C22×C4×C59—48no—
131C22×C4×C59—42no—
132C22×C4×C59—42no—
133C22×C4×C59C22⋊C412yes—
134C22×C4×C59C22⋊C412yes—
135C22×C4×C59C4⋊C412yescomplemented
136C22×Dic59—24no—
137C22×Dic59—24no—
138C22×Dic59—44no—
139C22×Dic59—44no—
140C22×Dic59—44no—
141C22×Dic59—44no—
142C22×Dic59—42no—
143C22×Dic59—42no—
144C22×Dic59—42no—
145C22×Dic59—42no—
146C22⋊C4×C59—44no—
147C22⋊C4×C59—42no—
148C22⋊C4×C59—41no—
149C22⋊C4×C59—41no—
150C22⋊C4×C59—44no—
151C22⋊Dic59—24no—
152C22⋊Dic59—24no—
153C22⋊Dic59—24no—
154C22⋊Dic59—24no—
155C22⋊Dic59—42no—
156C22⋊Dic59—42no—
157C22⋊Dic59—42no—
158C22⋊Dic59—42no—
159C4⋊Dic59—48no—
160C23×C59—22no—
161C23×C59—22no—
162C23×C59—22no—
163C23×C59—22no—
164C23×C59—22no—
165C23×C59—22no—
166C23×C59—22no—
167C23×C59—24no—
168C23×C59—42no—
169C23×C59—42no—
170C23×C59—42no—
171C23×C59—42no—
172C23×C59—44no—
173C23×C59—42no—
174C23×C59—42no—
175C23×C59—42no—
176C23×C59—42no—
177C23×C59—44no—
178C23×C59—44no—
179C23×C59—44no—
180C23×C59—41no—
181C23×C59—42no—
182C23×C59—41no—
183C23×C59—42no—
184C23×C59—42no—
185C23×C59—82no—
186C23×C59—82no—
187C23×C59—84no—
188C23×C59C2×C22⋊C411yescharacteristic
189C23×C59C2×C4⋊C411yescharacteristic
190C23×C59#11711yescharacteristic
191C23×C59C4×D412yes—
192C23×C59C4×D412yes—
193C23×C59#12011yescharacteristic
194C23×C59#12012yes—
195C23×C59#12112yes—
196C23×C59#12112yes—
197C23×C59#12112yes—
198C23×C59#12111yescharacteristic
199C23×C59#12111yescharacteristic
200C23×C59C22⋊Q812yes—
201C23×C59C22⋊Q812yes—
202C23×C59C22⋊Q811yescharacteristic
203C23×C59C22⋊Q811yescharacteristic
204C23×C59#12311yescharacteristic
205C23×C59#12312yes—
206C23×C59#12312yes—
207C23×C59#12412yes—
208C23×C59#12411yescharacteristic
209C23×C59#12612yes—
210C23×C59#12711yescharacteristic
211D4×C59—42no—
212D4×C59—42no—
213D4×C59—44no—
214D4×C59—44no—
215D4×C59—42no—
216D4×C59—42no—
217C2×C4×C59—24no—
218C2×C4×C59—24no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—24no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—48no—
227C2×C4×C59—48no—
228C2×C4×C59—42no—
229C2×C4×C59—42no—
230C2×C4×C59—41no—
231C2×C4×C59—42no—
232C2×C4×C59—41no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—84no—
237C2×C4×C59#9914yescomplemented
238C2×Dic59—44no—
239C2×Dic59—44no—
240C2×Dic59—48no—
241C2×Dic59—44no—
242C2×Dic59—44no—
243C2×Dic59—84no—
244C2×Dic59—84no—
245C2×Dic59—82no—
246C2×Dic59—82no—
247C2×Dic59—84no—
248#11043—591nomaximal, Sylow
249C22×C59—22no—
250C22×C59—22no—
251C22×C59—22no—
252C22×C59—22no—
253C22×C59—22no—
254C22×C59—22no—
255C22×C59—21no—
256C22×C59—22no—
257C22×C59—21no—
258C22×C59—22no—
259C22×C59—21no—
260C22×C59—22no—
261C22×C59—22no—
262C22×C59—21no—
263C22×C59—24no—
264C22×C59—22no—
265C22×C59—24no—
266C22×C59—22no—
267C22×C59—42no—
268C22×C59—42no—
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—42no—
273C22×C59—42no—
274C22×C59—42no—
275C22×C59—44no—
276C22×C59—42no—
277C22×C59—84no—
278C22×C59—84no—
279C22×C59—81no—
280C22×C59—81no—
281C22×C59—82no—
282C22×C59#38011yescharacteristic
283C22×C59#38512yes—
284C22×C59#38611yescharacteristic
285C22×C59#38812yes—
286C22×C59#39011yescharacteristic
287C22×C59#39211yescharacteristic
288C22×C59#39311yescharacteristic
289C22×C59#39412yes—
290C22×C59#39612yes—
291C22×C59#39711yescharacteristic
292C22×C59#39911yescharacteristic
293C22×C59C2×#9912yes—
294C22×C59#41012yes—
295C4×C59—24no—
296C4×C59—42no—
297C4×C59—42no—
298C4×C59—82no—
299Dic59—84no—
300Dic59—84no—
301C22×#99—592no—
302C2×#386D5911yescharacteristic
303C2×#380—591no—
304C2×#342—592no—
305C22⋊#95—591no—
306C2×C59—21no—
307C2×C59—22no—
308C2×C59—22no—
309C2×C59—22no—
310C2×C59—21no—
311C2×C59—22no—
312C2×C59—21no—
313C2×C59—21no—
314C2×C59—82no—
315C2×C59#143911yescharacteristic
316C2×C59#179912yes—
317C2×C59#189811yescharacteristic
318C2×C59#189911yescharacteristic
319C2×C59#196412yes—
320C23×D4C2×D5911yescharacteristic
321C24×C4C2×D5911yescharacteristic
322C22×C4⋊C4—1181no—
323C22×C22⋊C4—592no—
324C22×C22⋊C4—592no—
325C22×C22⋊C4Dic5912yescomplemented
326C22×C22⋊C4C2×D5911yescharacteristic
327C2×#99—1184no—
328C2×#99—1184no—
329C2×#99—1184no—
330#386—21no—
331#386—22no—
332#386—21no—
333#380—1184no—
334C2×#95—1181no—
335C2×#95—1182no—
336C2×#95Dic5912yescomplemented
337#342—1184no—
338C59#1104311yescharacteristic, complemented, Sylow
339C25C2×Dic5911yescharacteristic
340C25C22×D5911yescharacteristic
341C22×D4—21no—
342C22×D4—21no—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—21no—
348C22×D4—21no—
349C22×D4Dic11812yes—
350C22×D4D23612yes—
351C23×C4—22no—
352C23×C4—22no—
353C23×C4—44no—
354C23×C4—1182no—
355C23×C4—1182no—
356C23×C4C2×Dic5911yescharacteristic
357C23×C4C2×Dic5912yes—
358C23×C4C2×Dic5912yes—
359C23×C4C22⋊D5911yescharacteristic
360C23×C4C22⋊D5912yes—
361C23×C4C22⋊D5911yescharacteristic
362C2×C4⋊C4—1184no—
363C2×C4⋊C4—2364no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4—594no—
372C2×C22⋊C4—594no—
373C2×C22⋊C4—1182no—
374C2×C22⋊C4—1182no—
375C2×C22⋊C4—1182no—
376C2×C22⋊C4—1182no—
377C2×C22⋊C4—1182no—
378C2×C22⋊C4—1182no—
379C2×C22⋊C4—1182no—
380C2×C22⋊C4—1182no—
381C2×C22⋊C4—1182no—
382C2×C22⋊C4—1182no—
383#99—11816no—
384#95—22no—
385#95—22no—
386#95—1184no—
387#95—2364no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C4⋊Dic5912yescomplemented
397C24C2×Dic11811yescharacteristic
398C24C2×D23611yescharacteristic
399C24#1174568511yescharacteristic
400C24#1174568511yescharacteristic
401C24D4×D5912yes—
402C24D4×D5912yes—
403C24#1174568712yes—
404C24#1174568712yes—
405C24C22×Dic5911yescharacteristic
406C24C2×C22⋊D5911yescharacteristic
407C24C2×C22⋊D5911yescharacteristic
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—41no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—41no—
424C2×D4—42no—
425C2×D4—41no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—1184no—
449C22×C4—1184no—
450C22×C4—2364no—
451C22×C4—2364no—
452C22×C4—2364no—
453C22×C4—2364no—
454C22×C4—2362no—
455C22×C4—2362no—
456C22×C4—2362no—
457C22×C4—2362no—
458C22×C4C4⋊Dic5912yescomplemented
459C22×C4C22⋊Dic5912yes—
460C22×C4C22⋊Dic5912yes—
461C4⋊C4—2368no—
462C22⋊C4—44no—
463C22⋊C4—42no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—44no—
467C22⋊C4—1184no—
468C22⋊C4—1184no—
469C22⋊C4—1184no—
470C22⋊C4—1184no—
471C22⋊C4—2362no—
472C22⋊C4—2362no—
473C22⋊C4—2362no—
474C22⋊C4—2362no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—42no—
491C23—44no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—42no—
500C23—84no—
501C23—82no—
502C23—82no—
503C23C2×C22⋊Dic5911yescharacteristic
504C23C2×C4⋊Dic5911yescharacteristic
505C23#1427368611yescharacteristic
506C23D4×Dic5912yes—
507C23D4×Dic5912yes—
508C23#1427369511yescharacteristic
509C23#1427369712yes—
510C23#1427369812yes—
511C23#1427369912yes—
512C23#1427370012yes—
513C23#1427370111yescharacteristic
514C23#1427370111yescharacteristic
515C23#1427370412yes—
516C23#1427370412yes—
517C23#1427370611yescharacteristic
518C23#1427370611yescharacteristic
519C23#1427370811yescharacteristic
520C23#1427370912yes—
521C23#1427371112yes—
522C23#1427371312yes—
523C23#1427371411yescharacteristic
524C23#1427372012yes—
525C23#1427372311yescharacteristic
526D4—42no—
527D4—42no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—4724no—
553C2×C4—4722no—
554C2×C4—4722no—
555C2×C4—4724no—
556C2×C4—4724no—
557C2×C4—2364no—
558C2×C4—2368no—
559C2×C4—2364no—
560C2×C4—2364no—
561C2×C4—2364no—
562C2×C4#1427364214yescomplemented
563C22—24no—
564C22—22no—
565C22—24no—
566C22—22no—
567C22—21no—
568C22—21no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—21no—
573C22—22no—
574C22—21no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—44no—
591C22—84no—
592C22—84no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22#1683726211yescharacteristic
597C22#1683727712yes—
598C22#1683728211yescharacteristic
599C22#1683729512yes—
600C22#1683730511yescharacteristic
601C22#1683731111yescharacteristic
602C22#1683731411yescharacteristic
603C22#1683731812yes—
604C22#1683732512yes—
605C22#1683732711yescharacteristic
606C22#1683733811yescharacteristic
607C22C2×#1427364212yes—
608C22#1683737612yes—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—4724no—
614C4—4724no—
615C2—21no—
616C2—22no—
617C2—22no—
618C2—22no—
619C2—21no—
620C2—22no—
621C2—21no—
622C2—21no—
623C2—82no—
624C2#2142234111yescharacteristic
625C2#2142408412yes—
626C2#2142455011yescharacteristic
627C2#2142455511yescharacteristic
628C2#2142493312yes—
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC24×C59C22×C4
FrattiniC25C22×D59
FittingC2×C59×#386C2
socleC23×C59C2×C22⋊C4
prime radicalC23×D4×C59C22
cosocleC25×C59C23
hypercenterC2×#386D59
hypocenterC59#11043
LCSG ▷ C24×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
2. C2×#16837262C2semidirect
3. C2×#16837169C2semidirect
4. #21422319C2semidirect
18. C22×C22⋊C4×C59C4semidirect
23. C2×#95×C59C4semidirect
81. C24×C59C4⋊C4semidirect
135. C22×C4×C59C4⋊C4semidirect
237. C2×C4×C59#99semidirect
325. C22×C22⋊C4Dic59semidirect
336. C2×#95Dic59semidirect
338. C59#11043semidirect
396. C24C4⋊Dic59semidirect
458. C22×C4C4⋊Dic59semidirect
562. C2×C4#14273642semidirect

Maximal subgroups

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

subgroupindex
C22×#142736422
C2×#168372622
C2×#168371692
#214223192
C2×C59×#3862
#1104359