Finite Group Databasebeta

← #28647546

Subgroups — #28647546

Order 15104.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837811C211yesmaximal, characteristic, complemented
2C2×#16837444C211yesmaximal, characteristic
3#21427809C211yesmaximal, characteristic
4#21427798C211yesmaximal, characteristic
5#21425574C211yesmaximal, characteristic, complemented
6#21425569C211yesmaximal, characteristic, complemented
7#21425435C211yesmaximal, characteristic, complemented
8#21425430C211yesmaximal, characteristic, complemented
9#21424834C211yesmaximal, characteristic, complemented
10#21424829C211yesmaximal, characteristic, complemented
11#21424297C211yesmaximal, characteristic
12#21424291C211yesmaximal, characteristic
13#21424253C211yesmaximal, characteristic, complemented
14#21424129C211yesmaximal, characteristic
15C59×#1753C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×#14273704—21no—
19C2×#14273686C2211yescharacteristic
20C22×C4⋊Dic59C2211yescharacteristic
21#16837811—24no—
22#16837444—24no—
23C2×#14273661C2211yescharacteristic
24C2×#14273661C2211yescharacteristic
25C2×#14273661C2211yescharacteristic
26C2×#14273661C2211yescharacteristic
27C2×#14273659C2211yescharacteristic
28C2×#14273659C2211yescharacteristic
29C2×#14273659C2211yescharacteristic
30C2×#14273659C2211yescharacteristic
31C2×#14273659C2211yescharacteristic
32C2×#14273659C2211yescharacteristic
33#16837338—21no—
34#16837334—21no—
35#16837325—21no—
36#16837323—21no—
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837300C2211yescharacteristic
41#16837299—21no—
42#16837285—21no—
43#16837277—21no—
44#16837272C2211yescharacteristic
45#16837272C2211yescharacteristic
46#16837262C2211yescharacteristic
47#16837262C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53#16837165C2211yescharacteristic
54C22×M16×C59C2211yescharacteristic
55C2×C22⋊C8×C59C2211yescharacteristic
56C2×#97×C59C2211yescharacteristic
57C2×C4×C8×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C59×#336C2211yescharacteristic
61C22×Dic118—21no—
62C23×Dic59—21no—
63#14273727—22no—
64#14273727—41no—
65#14273719—22no—
66#14273719—41no—
67#14273704—44no—
68#14273686—22no—
69#14273686—41no—
70C2×C4⋊Dic59—22no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—41no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×C4⋊Dic59C2311yescharacteristic
85C2×C4⋊Dic59C2311yescharacteristic
86C2×Dic59⋊C4—21no—
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4—21no—
90C2×Dic59⋊C4—21no—
91C2×Dic59⋊C4—21no—
92C2×C22⋊Dic59—21no—
93C2×C22⋊Dic59—21no—
94C2×C22⋊Dic59—21no—
95C2×C4×Dic59—21no—
96#14273661—22no—
97#14273661—22no—
98#14273661—22no—
99#14273661D414yes—
100#14273659—22no—
101#14273659—22no—
102#14273659—22no—
103#14273659—22no—
104#14273659—41no—
105#14273659—41no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C23×C4×C59C2311yescharacteristic
113C2×M16×C59—22no—
114C2×M16×C59—22no—
115C2×M16×C59Q814yescomplemented
116C22×C8×C59C2311yescharacteristic
117C22×C8×C59C2311yescharacteristic
118C22×C8×C59C2311yescharacteristic
119C22×C8×C59C2311yescharacteristic
120C2×C22⋊C4×C59D411yescharacteristic
121C2×C22⋊C4×C59D411yescharacteristic
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124C22⋊C8×C59Q814yescomplemented
125#97×C59—22no—
126C4×C8×C59—22no—
127#95×C59D411yescharacteristic
128#95×C59D411yescharacteristic
129C24×C59C2×D411yescharacteristic
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×Q812yes—
140C22×C4×C59C2×Q812yes—
141C22×C4×C59C2411yescharacteristic
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—42no—
151C22×Dic59—42no—
152C22×Dic59—81no—
153C2×Dic118—42no—
154C2×Dic118—42no—
155C2×Dic118—81no—
156M16×C59—28no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59C2×D412yes—
164C2×C8×C59C2×Q812yes—
165C2×C8×C59C2×Q812yes—
166C2×C8×C59C2×Q812yes—
167C2×C8×C59C2×Q812yes—
168C2×C8×C59#4112yes—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—22no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C22⋊Dic59—42no—
176C22⋊Dic59—42no—
177C22⋊Dic59—81no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—42no—
193C4⋊Dic59—42no—
194C4⋊Dic59—42no—
195C4⋊Dic59—42no—
196C4⋊Dic59—81no—
197C4⋊Dic59—81no—
198C4⋊Dic59C2×D412yes—
199C4⋊Dic59C2×D412yes—
200Dic59⋊C4—42no—
201Dic59⋊C4—42no—
202Dic59⋊C4—42no—
203Dic59⋊C4—42no—
204Dic59⋊C4—42no—
205Dic59⋊C4—81no—
206C4×Dic59—42no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59C4⋊Q812yes—
214C23×C59C22×D411yescharacteristic
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59C4⋊Q812yes—
232C2×C4×C59AGL(1,ℤ/8)12yes—
233C2×C4×C59#13712yes—
234C2×C4×C59C22×D411yescharacteristic
235C2×C4×C59C22×Q811yescharacteristic
236C2×C4×C59C22×Q811yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59H(5,𝔽2)11yescharacteristic
239C2×C4×C592−1+411yescharacteristic
240C2×Dic59—22no—
241C2×Dic59—22no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—84no—
265Dic118—84no—
266C8×C59—22no—
267C8×C59—22no—
268C8×C59—22no—
269C8×C59C22⋊Q814yescomplemented
270#26078—591nomaximal, Sylow
271C22×C59—22no—
272C22×C59—22no—
273C22×C59—22no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59C2×C4⋊Q811yescharacteristic
279C22×C59D4211yescharacteristic
280C22×C59#54711yescharacteristic
281C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C59C2×#13711yescharacteristic
283C22×C59#57711yescharacteristic
284C22×C59#57711yescharacteristic
285C4×C59—22no—
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59C2×C22⋊Q811yescharacteristic
291C4×C59#54411yescharacteristic
292C4×C59D4×Q811yescharacteristic
293C4×C59#55411yescharacteristic
294Dic59—42no—
295Dic59—42no—
296Dic59—81no—
297Dic59—81no—
298Dic59—81no—
299Dic59—81no—
300Dic59—81no—
301Dic59—81no—
302C2×#501—591no—
303C2×#428—591no—
304#2441—591no—
305#2440—591no—
306#2084—591no—
307#2083—591no—
308#2059—591no—
309#2058—591no—
310#1943—591no—
311#1942—591no—
312#1854—591no—
313#1853—591no—
314#1844—591no—
315#1809—591no—
316#1753D5911yescharacteristic
317C2×C59—42no—
318C2×C59#244011yescharacteristic
319C2×C59#316311yescharacteristic
320C2×C59#316511yescharacteristic
321C2×C59#327211yescharacteristic
322C2×C59#327811yescharacteristic
323C2×C59#334411yescharacteristic
324C2×C59#334511yescharacteristic
325C22×M16C2×D5911yescharacteristic
326C2×C4⋊Q8—591no—
327C2×#125—591no—
328C2×C22⋊Q8—1181no—
329C2×#117—591no—
330C22×C4⋊C4—591no—
331#501—1184no—
332#428—1184no—
333C2×#107—591no—
334C2×#107—591no—
335C2×#107—591no—
336C2×#107—591no—
337C2×#106—591no—
338C2×#106—591no—
339C2×#106—591no—
340C2×#106—591no—
341C2×#106—591no—
342C2×#106—591no—
343C2×C22⋊C8C2×D5911yescharacteristic
344C2×#97C2×D5911yescharacteristic
345C2×C4×C8C2×D5911yescharacteristic
346#399—1181no—
347#398—1181no—
348#396—1181no—
349#395—1181no—
350#389—591no—
351#389—591no—
352#389—591no—
353#389—591no—
354#389—1181no—
355#387—1181no—
356#385—1181no—
357#384—591no—
358#384—591no—
359#380—591no—
360#380—591no—
361C4×C22⋊C4C2×D5911yescharacteristic
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#336C2×D5911yescharacteristic
369#336C2×D5911yescharacteristic
370C59#2607811yescharacteristic, complemented, Sylow
371C22×Q8—1181no—
372C23×C4—1181no—
373C23×C4C22×D5911yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16Dic11814yescomplemented
377C22×C8C22×D5911yescharacteristic
378C22×C8C22×D5911yescharacteristic
379C22×C8C22×D5911yescharacteristic
380C22×C8C22×D5911yescharacteristic
381C4⋊Q8—1182no—
382C4⋊Q8—2361no—
383#125—1182no—
384#125—2361no—
385C22⋊Q8—2364no—
386#117—1182no—
387#117—2361no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—1182no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—2361no—
410C2×C22⋊C4—1181no—
411C2×C22⋊C4—1181no—
412C2×C22⋊C4—1181no—
413C2×C22⋊C4D23611yescharacteristic
414C2×C22⋊C4D23611yescharacteristic
415C2×C42—1181no—
416C2×C42C22×D5911yescharacteristic
417C2×C42C22×D5911yescharacteristic
418#107—594no—
419#107—1182no—
420#107—1182no—
421#107—1182no—
422#106—1182no—
423#106—1182no—
424#106—1182no—
425#106—1182no—
426#106—2361no—
427#106—2361no—
428C22⋊C8Dic11814yescomplemented
429#97—22no—
430C4×C8—22no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95D23611yescharacteristic
438#95D23611yescharacteristic
439C24C2×D23611yescharacteristic
440C2×Q8—4721no—
441C2×Q8—2362no—
442C2×Q8—2362no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—4721no—
456C22×C4—2362no—
457C22×C4—2362no—
458C22×C4C2×Dic11812yes—
459C22×C4C2×Dic11812yes—
460C22×C4C2×D23611yescharacteristic
461C22×C4C2×D23611yescharacteristic
462C22×C4C2×D23611yescharacteristic
463C22×C4C2×D23611yescharacteristic
464C22×C4C2×D23611yescharacteristic
465C22×C4C23×D5911yescharacteristic
466M16—28no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8C2×Dic11812yes—
474C2×C8C2×Dic11812yes—
475C2×C8C2×Dic11812yes—
476C2×C8C2×Dic11812yes—
477C2×C8D4×D5912yes—
478C2×C8#1174568712yes—
479C4⋊C4—592no—
480C4⋊C4—592no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—4721no—
496C4⋊C4—4721no—
497C4⋊C4—4721no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4—2362no—
501C4⋊C4—2362no—
502C4⋊C4—2362no—
503C4⋊C4—2362no—
504C4⋊C4—2362no—
505C4⋊C4—2362no—
506C4⋊C4—2362no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—4721no—
510C22⋊C4—2362no—
511C22⋊C4—2362no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—2362no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1427372712yes—
524C23C22×D23611yescharacteristic
525Q8—4724no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—1182no—
543C2×C4—1182no—
544C2×C4—4724no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4#1427372712yes—
568C2×C4#1427375512yes—
569C2×C4#1427376012yes—
570C2×C4C2×D4×D5911yescharacteristic
571C2×C4C22×Dic11811yescharacteristic
572C2×C4C22×Dic11811yescharacteristic
573C2×C4C2×#1174568711yescharacteristic
574C2×C4#1427377311yescharacteristic
575C2×C4#1427377411yescharacteristic
576C8—22no—
577C8—22no—
578C8—22no—
579C8#1427370414yescomplemented
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1427372711yescharacteristic
588C22D4×D23611yescharacteristic
589C22#1683805611yescharacteristic
590C22C2×#1427375511yescharacteristic
591C22C2×#1427376011yescharacteristic
592C22#1683821411yescharacteristic
593C22#1683821411yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—4721no—
600C4—4721no—
601C4—4721no—
602C4—4721no—
603C4—4721no—
604C4—4721no—
605C4—2362no—
606C4—2362no—
607C4C2×#1427370411yescharacteristic
608C4#1683803011yescharacteristic
609C4D4×Dic11811yescharacteristic
610C4#1683810911yescharacteristic
611C2—42no—
612C2#2142779811yescharacteristic
613C2#2143462711yescharacteristic
614C2#2143464211yescharacteristic
615C2#2143569811yescharacteristic
616C2#2143576411yescharacteristic
617C2#2143655311yescharacteristic
618C2#2143656811yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1753C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1753D59
hypocenterC59#26078
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#16837811C2semidirect
5. #21425574C2semidirect
6. #21425569C2semidirect
7. #21425435C2semidirect
8. #21425430C2semidirect
9. #21424834C2semidirect
10. #21424829C2semidirect
13. #21424253C2semidirect
115. C2×M16×C59Q8semidirect
124. C22⋊C8×C59Q8semidirect
269. C8×C59C22⋊Q8semidirect
370. C59#26078semidirect
376. C2×M16Dic118semidirect
428. C22⋊C8Dic118semidirect
579. C8#14273704semidirect

Maximal subgroups

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

subgroupindex
C2×#168378112
C2×#168374442
#214278092
#214277982
#214255742
#214255692
#214254352
#214254302
#214248342
#214248292
#214242972
#214242912
#214242532
#214241292
C59×#17532
#2607859