Finite Group Databasebeta

← #28525036

Subgroups — #28525036

Order 15104.

1678 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837341C211yesmaximal, characteristic, complemented
2C2×#16837305C211yesmaximal, characteristic
3C2×C4⋊C4×Dic59C211yesmaximal, characteristic, complemented
4#21422363C211yesmaximal, characteristic, complemented
5#21422343C212yesmaximal, complemented
6#21422341C212yesmaximal, complemented
7#21422337C211yesmaximal, characteristic
8#21422333C212yesmaximal, complemented
9#21422333C212yesmaximal, complemented
10#21422322C211yesmaximal, characteristic, complemented
11C2×C59×#394C211yesmaximal, characteristic
12C22×C4⋊Dic59C2211yescharacteristic
13C22×C4⋊Dic59C2211yescharacteristic
14C22×C22⋊Dic59C2212yes—
15C22×C22⋊Dic59C2212yes—
16C22×C22⋊Dic59C2212yes—
17C22×C22⋊Dic59C2212yes—
18C22×C4×Dic59C2211yescharacteristic
19C22×C4×Dic59C2211yescharacteristic
20C22×C4×Dic59C2211yescharacteristic
21#16837341—24no—
22#16837305—24no—
23C4⋊C4×Dic59—24no—
24C2×C59⋊#95C2211yescharacteristic
25C2×C59⋊#95C2211yescharacteristic
26C2×C59⋊#95C2211yescharacteristic
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2212yes—
29C2×C59⋊#95C2212yes—
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2212yes—
32C2×C59⋊#95C2212yes—
33C23×D4×C59C2211yescharacteristic
34C22×C4⋊C4×C59C2211yescharacteristic
35C22×C22⋊C4×C59C2212yes—
36C22×C22⋊C4×C59C2212yes—
37C59×#394—24no—
38C2×#95×C59C2211yescharacteristic
39C23×Dic59C2312yes—
40C23×Dic59C2312yes—
41C23×Dic59C2311yescharacteristic
42C23×Dic59C2311yescharacteristic
43C23×Dic59C2311yescharacteristic
44C23×Dic59C2311yescharacteristic
45C2×C4⋊Dic59—22no—
46C2×C4⋊Dic59—22no—
47C2×C4⋊Dic59—22no—
48C2×C4⋊Dic59—22no—
49C2×C4⋊Dic59—22no—
50C2×C4⋊Dic59—22no—
51C2×C22⋊Dic59—24no—
52C2×C22⋊Dic59—24no—
53C2×C22⋊Dic59—24no—
54C2×C22⋊Dic59—24no—
55C2×C22⋊Dic59—24no—
56C2×C22⋊Dic59—24no—
57C2×C22⋊Dic59—24no—
58C2×C22⋊Dic59—42no—
59C2×C22⋊Dic59—42no—
60C2×C22⋊Dic59—42no—
61C2×C22⋊Dic59—42no—
62C2×C22⋊Dic59—42no—
63C2×C4×Dic59—22no—
64C2×C4×Dic59—22no—
65C2×C4×Dic59—22no—
66C2×C4×Dic59—22no—
67C2×C4×Dic59—42no—
68C2×C4×Dic59—42no—
69C2×C4×Dic59D414yes—
70C59⋊#95—22no—
71C59⋊#95—22no—
72C59⋊#95—24no—
73C59⋊#95—24no—
74C59⋊#95—22no—
75C59⋊#95—24no—
76C59⋊#95—42no—
77C59⋊#95—42no—
78C59⋊#95—42no—
79C25×C59C2312yes—
80C22×D4×C59—24no—
81C22×D4×C59—22no—
82C22×D4×C59—22no—
83C22×D4×C59—22no—
84C22×D4×C59—42no—
85C23×C4×C59C2312yes—
86C23×C4×C59C2311yescharacteristic
87C23×C4×C59C2311yescharacteristic
88C23×C4×C59C2311yescharacteristic
89C2×C4⋊C4×C59—22no—
90C2×C4⋊C4×C59—22no—
91C2×C4⋊C4×C59D414yes—
92C2×C22⋊C4×C59—24no—
93C2×C22⋊C4×C59—24no—
94C2×C22⋊C4×C59—24no—
95C2×C22⋊C4×C59—24no—
96C2×C22⋊C4×C59—24no—
97C2×C22⋊C4×C59—42no—
98#95×C59—22no—
99C24×C59—22no—
100C24×C59—22no—
101C24×C59—22no—
102C24×C59—22no—
103C24×C59—22no—
104C24×C59—22no—
105C24×C59—22no—
106C24×C59—22no—
107C24×C59—22no—
108C24×C59—22no—
109C24×C59—22no—
110C24×C59—22no—
111C24×C59—22no—
112C24×C59—22no—
113C24×C59#4114yes—
114C24×C59C2411yescharacteristic
115C2×D4×C59—28no—
116C2×D4×C59—48no—
117C2×D4×C59—48no—
118C2×D4×C59—44no—
119C2×D4×C59—44no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—22no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—22no—
126C22×C4×C59—21no—
127C22×C4×C59—22no—
128C22×C4×C59—21no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—22no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59C2×D412yes—
138C22×C4×C59C2×D412yes—
139C22×C4×C59C2×D412yes—
140C22×C4×C59C2×D412yes—
141C22×C4×C59#4112yes—
142C22×C4×C59#4112yes—
143C22×C4×C59#4112yes—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—22no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—22no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—22no—
155C22×Dic59—22no—
156C22×Dic59—22no—
157C22×Dic59—22no—
158C22×Dic59—22no—
159C22×Dic59—22no—
160C22×Dic59—22no—
161C22×Dic59—22no—
162C22×Dic59—22no—
163C22×Dic59—22no—
164C22×Dic59—22no—
165C22×Dic59—22no—
166C22×Dic59—22no—
167C22×Dic59—22no—
168C22×Dic59—22no—
169C22×Dic59—22no—
170C22×Dic59—22no—
171C22×Dic59—22no—
172C22×Dic59—22no—
173C22×Dic59C2×D412yes—
174C22×Dic59C2×D412yes—
175C22×Dic59#4112yes—
176C22×Dic59#4112yes—
177C22×Dic59#4114yes—
178C4⋊C4×C59—28no—
179C22⋊C4×C59—48no—
180C22⋊C4×C59—48no—
181C22⋊Dic59—48no—
182C22⋊Dic59—48no—
183C22⋊Dic59—48no—
184C22⋊Dic59—48no—
185C4⋊Dic59—28no—
186C4⋊Dic59—44no—
187C4×Dic59—28no—
188C4×Dic59—44no—
189C4×Dic59—44no—
190C23×C59—24no—
191C23×C59—24no—
192C23×C59—24no—
193C23×C59—24no—
194C23×C59—24no—
195C23×C59—24no—
196C23×C59—24no—
197C23×C59—42no—
198C23×C59—42no—
199C23×C59—42no—
200C23×C59—42no—
201C23×C59—42no—
202C23×C59—42no—
203C23×C59—42no—
204C23×C59—42no—
205C23×C59—42no—
206C23×C59—42no—
207C23×C59—42no—
208C23×C59—42no—
209C23×C59—42no—
210C23×C59—42no—
211C23×C59—42no—
212C23×C59—42no—
213C23×C59—42no—
214C23×C59—42no—
215C23×C59—42no—
216C23×C59—42no—
217C23×C59—42no—
218C23×C59—42no—
219C23×C59—42no—
220C23×C59—42no—
221C23×C59—42no—
222C23×C59—42no—
223C23×C59—42no—
224C23×C59—42no—
225C23×C59C22×D411yescharacteristic
226C23×C59C22×D411yescharacteristic
227C23×C59C2×#4111yescharacteristic
228C23×C59C2×#4111yescharacteristic
229C23×C59C2×#4112yes—
230C23×C59C2×#4111yescharacteristic
231C23×C59C2×#4111yescharacteristic
232C23×C59C2×#4111yescharacteristic
233C23×C59C2×#4112yes—
234C23×C59H(5,𝔽2)12yes—
235C23×C59H(5,𝔽2)12yes—
236D4×C59—416no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—22no—
240C2×C4×C59—22no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—22no—
244C2×C4×C59—22no—
245C2×C4×C59—24no—
246C2×C4×C59—24no—
247C2×C4×C59—22no—
248C2×C4×C59—24no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—42no—
252C2×C4×C59—42no—
253C2×C4×C59—42no—
254C2×C4×C59—42no—
255C2×C4×C59—42no—
256C2×C4×C59—42no—
257C2×C4×C59—42no—
258C2×C4×C59—42no—
259C2×C4×C59#12114yes—
260C2×C4×C59#12414yes—
261C2×Dic59—22no—
262C2×Dic59—22no—
263C2×Dic59—24no—
264C2×Dic59—24no—
265C2×Dic59—22no—
266C2×Dic59—22no—
267C2×Dic59—22no—
268C2×Dic59—22no—
269C2×Dic59—22no—
270C2×Dic59—24no—
271C2×Dic59—24no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—42no—
277C2×Dic59—42no—
278C2×Dic59—42no—
279C2×Dic59—42no—
280C2×Dic59—42no—
281C2×Dic59—42no—
282C2×Dic59—42no—
283C2×Dic59—42no—
284C2×Dic59—42no—
285C2×Dic59—42no—
286C2×Dic59—42no—
287C2×Dic59—42no—
288C2×Dic59—42no—
289C2×Dic59—42no—
290C2×Dic59—42no—
291C2×Dic59—42no—
292C2×Dic59—42no—
293C2×Dic59—42no—
294C2×Dic59#12414yes—
295#15107—591nomaximal, Sylow
296C22×C59—44no—
297C22×C59—44no—
298C22×C59—44no—
299C22×C59—44no—
300C22×C59—44no—
301C22×C59—44no—
302C22×C59—44no—
303C22×C59—82no—
304C22×C59—82no—
305C22×C59—82no—
306C22×C59—82no—
307C22×C59—82no—
308C22×C59—82no—
309C22×C59—82no—
310C22×C59—82no—
311C22×C59C2×#12111yescharacteristic
312C22×C59#52511yescharacteristic
313C22×C59#52511yescharacteristic
314C22×C59#52511yescharacteristic
315C22×C59C2×#12411yescharacteristic
316C22×C59C2×#12411yescharacteristic
317C22×C59#52912yes—
318C22×C59#52911yescharacteristic
319C22×C59#52911yescharacteristic
320C22×C59#52912yes—
321C22×C59#53211yescharacteristic
322C22×C59#53411yescharacteristic
323C22×C59#53511yescharacteristic
324C22×C59#53812yes—
325C22×C59#54012yes—
326C22×C59#54612yes—
327C22×C59#55012yes—
328C22×C59#55112yes—
329C22×C59#55112yes—
330C22×C59#55312yes—
331C22×C59#55312yes—
332C22×C59#55312yes—
333C22×C59#55912yes—
334C4×C59—24no—
335C4×C59—24no—
336C4×C59—42no—
337C4×C59—82no—
338Dic59—24no—
339Dic59—42no—
340Dic59—42no—
341Dic59—42no—
342Dic59—82no—
343Dic59—82no—
344C2×#400—591no—
345C2×#394D5911yescharacteristic
346C2×#390—591no—
347C2×C4×C4⋊C4—591no—
348#1445—591no—
349#1440—592no—
350#1439—592no—
351#1438—591no—
352#1437—592no—
353#1437—592no—
354#1434—591no—
355C2×C59—84no—
356C2×C59#239811yescharacteristic
357C2×C59#249012yes—
358C2×C59#251612yes—
359C2×C59#253411yescharacteristic
360C2×C59#254811yescharacteristic
361C2×C59#255412yes—
362C2×C59#259111yescharacteristic
363C2×C59#259512yes—
364C2×C59#259811yescharacteristic
365C2×C59#263511yescharacteristic
366C2×C59#285011yescharacteristic
367C23×D4C2×D5911yescharacteristic
368C22×C4⋊C4—591no—
369C22×C4⋊C4—591no—
370C22×C4⋊C4C2×D5911yescharacteristic
371C22×C22⋊C4—592no—
372C22×C22⋊C4—592no—
373C22×C22⋊C4—592no—
374C22×C22⋊C4—592no—
375C22×C22⋊C4C2×D5912yes—
376C22×C22⋊C4C2×D5912yes—
377C22×C42—591no—
378C22×C42—591no—
379C22×C42—591no—
380#400—1184no—
381#394—24no—
382#390—1184no—
383C4×C4⋊C4—1184no—
384C2×#95—591no—
385C2×#95—592no—
386C2×#95—592no—
387C2×#95—592no—
388C2×#95—591no—
389C2×#95—591no—
390C2×#95—592no—
391C2×#95—592no—
392C2×#95—592no—
393C2×#95C2×D5911yescharacteristic
394C59#1510711yescharacteristic, complemented, Sylow
395C25C22×D5912yes—
396C22×D4—24no—
397C22×D4—22no—
398C22×D4—22no—
399C22×D4—22no—
400C22×D4—42no—
401C23×C4—591no—
402C23×C4—591no—
403C23×C4—591no—
404C23×C4—591no—
405C23×C4—592no—
406C23×C4—592no—
407C23×C4C22×D5912yes—
408C23×C4C22×D5911yescharacteristic
409C23×C4C22×D5911yescharacteristic
410C23×C4C22×D5911yescharacteristic
411C2×C4⋊C4—22no—
412C2×C4⋊C4—22no—
413C2×C4⋊C4—1182no—
414C2×C4⋊C4—1182no—
415C2×C4⋊C4—1182no—
416C2×C4⋊C4—1182no—
417C2×C4⋊C4—1182no—
418C2×C4⋊C4—1182no—
419C2×C4⋊C4C22⋊D5914yes—
420C2×C22⋊C4—24no—
421C2×C22⋊C4—24no—
422C2×C22⋊C4—24no—
423C2×C22⋊C4—24no—
424C2×C22⋊C4—24no—
425C2×C22⋊C4—42no—
426C2×C22⋊C4—1184no—
427C2×C22⋊C4—1184no—
428C2×C22⋊C4—1184no—
429C2×C22⋊C4—1184no—
430C2×C22⋊C4—1184no—
431C2×C22⋊C4—1184no—
432C2×C22⋊C4—1184no—
433C2×C22⋊C4—2362no—
434C2×C22⋊C4—2362no—
435C2×C22⋊C4—2362no—
436C2×C22⋊C4—2362no—
437C2×C22⋊C4—2362no—
438C2×C42—594no—
439C2×C42—1182no—
440C2×C42—1182no—
441C2×C42—1182no—
442C2×C42—1182no—
443C2×C42—2362no—
444C2×C42—2362no—
445#95—22no—
446#95—1184no—
447#95—1182no—
448#95—1182no—
449#95—1182no—
450#95—1184no—
451#95—1184no—
452#95—2362no—
453#95—2362no—
454#95—2362no—
455C24—22no—
456C24—22no—
457C24—22no—
458C24—22no—
459C24—22no—
460C24—22no—
461C24—22no—
462C24—22no—
463C24—22no—
464C24—22no—
465C24—22no—
466C24—22no—
467C24—22no—
468C24—22no—
469C24#1174568714yes—
470C24C23×D5911yescharacteristic
471C2×D4—28no—
472C2×D4—48no—
473C2×D4—48no—
474C2×D4—44no—
475C2×D4—44no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—592no—
494C22×C4—594no—
495C22×C4—592no—
496C22×C4—592no—
497C22×C4—592no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4—1182no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—1181no—
504C22×C4—1181no—
505C22×C4—1182no—
506C22×C4—1181no—
507C22×C4—1181no—
508C22×C4—1182no—
509C22×C4—1182no—
510C22×C4—1182no—
511C22×C4—1182no—
512C22×C4—1182no—
513C22×C4—1182no—
514C22×C4—1182no—
515C22×C4—1182no—
516C22×C4—1182no—
517C22×C4—1182no—
518C22×C4—1182no—
519C22×C4—1182no—
520C22×C4—1182no—
521C22×C4—1182no—
522C22×C4—1182no—
523C22×C4—1182no—
524C22×C4—1182no—
525C22×C4—1182no—
526C22×C4—1182no—
527C22×C4D4×D5912yes—
528C22×C4#1174568712yes—
529C22×C4#1174568712yes—
530C22×C4#1174568712yes—
531C22×C4C2×C22⋊D5912yes—
532C22×C4C2×C22⋊D5912yes—
533C22×C4C2×C22⋊D5912yes—
534C4⋊C4—28no—
535C4⋊C4—1188no—
536C4⋊C4—2364no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—2368no—
540C22⋊C4—2368no—
541C22⋊C4—2368no—
542C22⋊C4—2368no—
543C42—1188no—
544C42—2364no—
545C42—2364no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23C2×D4×D5911yescharacteristic
582C23C22×C22⋊D5911yescharacteristic
583C23#41×D5911yescharacteristic
584C23#41×D5911yescharacteristic
585C23#41×D5912yes—
586C23C2×#1174568711yescharacteristic
587C23C2×#1174568712yes—
588C23C2×#1174568711yescharacteristic
589C23C2×#1174568711yescharacteristic
590C23#1427377212yes—
591C23#1427377312yes—
592D4—416no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—594no—
616C2×C4—1182no—
617C2×C4—1182no—
618C2×C4—1184no—
619C2×C4—1184no—
620C2×C4—1182no—
621C2×C4—1182no—
622C2×C4—1182no—
623C2×C4—1182no—
624C2×C4—1182no—
625C2×C4—1184no—
626C2×C4—1184no—
627C2×C4—2362no—
628C2×C4—2362no—
629C2×C4—2362no—
630C2×C4—2362no—
631C2×C4—2362no—
632C2×C4—2362no—
633C2×C4—2362no—
634C2×C4—2362no—
635C2×C4—2362no—
636C2×C4—2362no—
637C2×C4—2362no—
638C2×C4—2362no—
639C2×C4—2362no—
640C2×C4—2362no—
641C2×C4—2361no—
642C2×C4—2361no—
643C2×C4—2361no—
644C2×C4—2361no—
645C2×C4—2362no—
646C2×C4—2362no—
647C2×C4—2362no—
648C2×C4—2362no—
649C2×C4#1427369914yes—
650C2×C4#1427371414yes—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22C2×#1427369911yescharacteristic
667C22#1683790811yescharacteristic
668C22#1683791211yescharacteristic
669C22#1683791211yescharacteristic
670C22#124×D5911yescharacteristic
671C22C2×#1427371411yescharacteristic
672C22#1683792912yes—
673C22#1683793311yescharacteristic
674C22#1683793311yescharacteristic
675C22#1683793312yes—
676C22#1683794811yescharacteristic
677C22#1683796211yescharacteristic
678C22#1683797011yescharacteristic
679C22#1683798812yes—
680C22#1683800512yes—
681C22#1683804612yes—
682C22#1683807112yes—
683C22#1683807512yes—
684C22#1683808312yes—
685C22#1683809712yes—
686C22#1683809912yes—
687C22#1683810412yes—
688C22#1683813412yes—
689C4—24no—
690C4—24no—
691C4—42no—
692C4—82no—
693C4—1184no—
694C4—4722no—
695C4—4722no—
696C4—2362no—
697C4—2362no—
698C4—2362no—
699C2—84no—
700C2#2142738411yescharacteristic
701C2#2142845012yes—
702C2#2142873112yes—
703C2#2142898311yescharacteristic
704C2#2142909611yescharacteristic
705C2#2142915212yes—
706C2#2142957011yescharacteristic
707C2#2142961812yes—
708C2#2142965711yescharacteristic
709C2#2142998811yescharacteristic
710C2#2143239511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D59
commutator, socle, cosocleC24×C59C24
FittingC2×C59×#394C2
prime radicalC23×D4×C59C22
hypercenterC2×#394D59
hypocenterC59#15107
LCSG ▷ C24×C59 ▷ C59
UCSC1 ◁ C24 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. C2×#16837341C2semidirect
3. C2×C4⋊C4×Dic59C2semidirect
4. #21422363C2semidirect
5. #21422343C2semidirect
6. #21422341C2semidirect
8. #21422333C2semidirect
9. #21422333C2semidirect
10. #21422322C2semidirect
394. C59#15107semidirect

Maximal subgroups

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

subgroupindex
C2×#168373412
C2×#168373052
C2×C4⋊C4×Dic592
#214223632
#214223432
#214223412
#214223372
#214223332
#214223222
C2×C59×#3942
#1510759