Finite Group Databasebeta

← #28619108

Subgroups — #28619108

Order 15104.

712 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426844C211yesmaximal, characteristic, complemented
2#21425664C211yesmaximal, characteristic, complemented
3#21425617C211yesmaximal, characteristic
4#21425597C211yesmaximal, characteristic, complemented
5#21425592C211yesmaximal, characteristic, complemented
6#21425484C211yesmaximal, characteristic
7#21425484C211yesmaximal, characteristic
8#21425424C211yesmaximal, characteristic, complemented
9#21425232C211yesmaximal, characteristic
10#21423994C211yesmaximal, characteristic, complemented
11C4×#14273660C211yesmaximal, characteristic, complemented
12#21423953C211yesmaximal, characteristic, complemented
13#21423921C211yesmaximal, characteristic
14C4×#14273648C211yesmaximal, characteristic
15C59×#2319C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17C2×#14273660C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273638C2211yescharacteristic
28C2×C4×C59⋊C8C2211yescharacteristic
29#16837306—21no—
30#16837306—21no—
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837285—21no—
34#16837285—21no—
35#16837268C2211yescharacteristic
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4×D4×C59C2211yescharacteristic
49C59×#390C2211yescharacteristic
50C59×#390C2211yescharacteristic
51C59×#389C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#388—21no—
54C59×#388—21no—
55C59×#381—21no—
56C59×#380—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#376C2211yescharacteristic
59C22×Dic118—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273727—22no—
65#14273727—22no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×C4×Dic59—21no—
74C2×C4×Dic59—21no—
75#14273660—22no—
76#14273660—22no—
77#14273660—22no—
78#14273658—22no—
79C4⋊(C59⋊C8)—22no—
80C4⋊(C59⋊C8)—22no—
81#14273648—22no—
82#14273648—22no—
83#14273648—22no—
84#14273648—22no—
85#14273638—22no—
86C4×C59⋊C8—22no—
87C59⋊#95—21no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C22×D4×C59C2311yescharacteristic
94C23×C4×C59—21no—
95C4×D4×C59—22no—
96C4×D4×C59Q814yescomplemented
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59C2311yescharacteristic
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C22⋊C4×C59—21no—
103C2×C22⋊C4×C59—21no—
104C2×C22⋊C4×C59—21no—
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59—21no—
107C2×C42×C59—21no—
108C2×C42×C59C2311yescharacteristic
109C2×C42×C59C2311yescharacteristic
110C2×C42×C59C2311yescharacteristic
111#95×C59—21no—
112#95×C59—21no—
113#95×C59—21no—
114#95×C59D411yescharacteristic
115#95×C59D411yescharacteristic
116#95×C59D411yescharacteristic
117#95×C59D411yescharacteristic
118C24×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59C2×Q812yes—
125C22×C4×C59—21no—
126C22×C4×C59—24no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—81no—
130C22×C4×C59—81no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C2×Dic118—24no—
143C2×Dic118—81no—
144C2×Dic118—81no—
145C4⋊C4×C59—21no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—81no—
151C4⋊C4×C59—81no—
152C4⋊C4×C59C2×Q812yes—
153C22⋊C4×C59—24no—
154C22⋊C4×C59—81no—
155C22⋊C4×C59—81no—
156C22⋊C4×C59—81no—
157C22⋊C4×C59—81no—
158C42×C59—21no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59—81no—
163C42×C59C2×Q812yes—
164C42×C59#4112yes—
165C4⋊Dic59—24no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59#4112yes—
173C4⋊Dic59#4112yes—
174Dic59⋊C4—81no—
175Dic59⋊C4—81no—
176C4×Dic59—81no—
177C4×Dic59—81no—
178C2×C59⋊C8—21no—
179C2×C59⋊C8—21no—
180C2×C59⋊C8—21no—
181C2×C59⋊C8—21no—
182C2×C59⋊C8—21no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8#4112yes—
189C2×C59⋊C8#4112yes—
190C23×C59—22no—
191C23×C59—41no—
192C23×C59—41no—
193C23×C59—41no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59C22×D411yescharacteristic
198D4×C59—22no—
199D4×C59—22no—
200D4×C59—41no—
201D4×C59—41no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—22no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—84no—
228C2×C4×C59C22⋊Q812yes—
229C2×C4×C59C22⋊Q812yes—
230C2×C4×C59C22×Q811yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×Dic59—22no—
237C2×Dic59—22no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248Dic118—84no—
249C59⋊C8—22no—
250C59⋊C8—22no—
251C59⋊C8—41no—
252C59⋊C8—41no—
253#23787—591nomaximal, Sylow
254C22×C59—42no—
255C22×C59—42no—
256C22×C59—42no—
257C22×C59—81no—
258C22×C59—81no—
259C22×C59—81no—
260C22×C59—81no—
261C22×C59C2×C22⋊Q811yescharacteristic
262C22×C59#52511yescharacteristic
263C22×C59#53211yescharacteristic
264C22×C59#57611yescharacteristic
265C22×C59#57611yescharacteristic
266C22×C59#57711yescharacteristic
267C22×C59#57811yescharacteristic
268C4×C59—22no—
269C4×C59—22no—
270C4×C59—42no—
271C4×C59—41no—
272C4×C59—41no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59—81no—
276C4×C59—81no—
277C4×C59#52911yescharacteristic
278C4×C59#52911yescharacteristic
279C4×C59#53311yescharacteristic
280C4×C59#53311yescharacteristic
281Dic59—42no—
282Dic59—42no—
283Dic59—81no—
284Dic59—81no—
285#2331—591no—
286#2319D5911yescharacteristic
287#2101—591no—
288#2092—591no—
289#2088—591no—
290#2087—591no—
291#2067—591no—
292#2067—591no—
293#2057—591no—
294#2023—591no—
295#1776—591no—
296C4×#107—591no—
297#1768—591no—
298#1762—591no—
299C4×#102—591no—
300C2×C59—82no—
301C2×C59#239611yescharacteristic
302C2×C59#308111yescharacteristic
303C2×C59#308311yescharacteristic
304C2×C59#312111yescharacteristic
305C2×C59#312211yescharacteristic
306C2×C59#317511yescharacteristic
307C2×C59#317911yescharacteristic
308C2×C4⋊Q8—591no—
309C2×C4×D4C2×D5911yescharacteristic
310C2×#107—591no—
311C2×#107—591no—
312C2×#107—591no—
313C2×#106—591no—
314C2×C4⋊C8—591no—
315C2×C4⋊C8—591no—
316C2×#102—591no—
317C2×#102—591no—
318C2×#102—591no—
319C2×#102—591no—
320C2×#97—591no—
321C2×C4×C8—591no—
322#391—1181no—
323#391—1181no—
324#390C2×D5911yescharacteristic
325#390C2×D5911yescharacteristic
326#389—591no—
327#389—591no—
328#389C2×D5911yescharacteristic
329#389C2×D5911yescharacteristic
330#388—21no—
331#388—21no—
332#387—1181no—
333#387—1181no—
334#382—591no—
335#381—21no—
336#380—21no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4C2×D5911yescharacteristic
340#376C2×D5911yescharacteristic
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#336—591no—
350#336—591no—
351C59#2378711yescharacteristic, complemented, Sylow
352C22×Q8—1181no—
353C22×D4C22×D5911yescharacteristic
354C23×C4—21no—
355C22×C8—591no—
356C22×C8—591no—
357C22×C8—591no—
358C22×C8—591no—
359C4⋊Q8—1182no—
360C4⋊Q8—1182no—
361C4×D4—22no—
362C4×D4Dic11814yescomplemented
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4C22×D5911yescharacteristic
373C2×C4⋊C4C22×D5911yescharacteristic
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C42—21no—
381C2×C42—1181no—
382C2×C42—1181no—
383C2×C42C22×D5911yescharacteristic
384C2×C42C22×D5911yescharacteristic
385C2×C42C22×D5911yescharacteristic
386#107—1182no—
387#107—1182no—
388#107—1182no—
389#106—1182no—
390C4⋊C8—1182no—
391C4⋊C8—1182no—
392#102—1182no—
393#102—1182no—
394#102—1182no—
395#102—1182no—
396#97—1182no—
397C4×C8—1182no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—1181no—
402#95—1181no—
403#95—1181no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95C22⋊D5911yescharacteristic
408#95C22⋊D5911yescharacteristic
409#95C22⋊D5911yescharacteristic
410#95C22⋊D5911yescharacteristic
411C24—21no—
412C2×Q8—1184no—
413C2×Q8—4721no—
414C2×Q8—4721no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4C2×Dic11812yes—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—1181no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4C2×C22⋊D5911yescharacteristic
432C22×C4C2×C22⋊D5911yescharacteristic
433C22×C4C2×C22⋊D5911yescharacteristic
434C22×C4C2×C22⋊D5911yescharacteristic
435C22×C4C2×C22⋊D5911yescharacteristic
436C22×C4C2×C22⋊D5911yescharacteristic
437C22×C4C23×D5911yescharacteristic
438C2×C8—592no—
439C2×C8—592no—
440C2×C8—1181no—
441C2×C8—1181no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—592no—
458C4⋊C4—592no—
459C4⋊C4—1184no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—4721no—
467C4⋊C4—4721no—
468C4⋊C4C2×Dic11812yes—
469C22⋊C4—24no—
470C22⋊C4—81no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—81no—
479C42—4721no—
480C42—4721no—
481C42C2×Dic11812yes—
482C42#1174568512yes—
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C22×C22⋊D5911yescharacteristic
491Q8—4724no—
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—84no—
522C2×C4—1182no—
523C2×C4—1182no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4#1427370612yes—
535C2×C4#1427370612yes—
536C2×C4C22×Dic11811yescharacteristic
537C2×C4#41×D5911yescharacteristic
538C2×C4#41×D5911yescharacteristic
539C2×C4#41×D5911yescharacteristic
540C2×C4#41×D5911yescharacteristic
541C2×C4C2×#1174568511yescharacteristic
542C8—1182no—
543C8—1182no—
544C8—2361no—
545C8—2361no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#1427370611yescharacteristic
554C22#1683790911yescharacteristic
555C22#1683794811yescharacteristic
556C22#1683820511yescharacteristic
557C22#1683820511yescharacteristic
558C22#1683821311yescharacteristic
559C22#1683821711yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—4721no—
570C4—4721no—
571C4—2362no—
572C4—2362no—
573C4#1683793011yescharacteristic
574C4#1683793011yescharacteristic
575C4#1683795511yescharacteristic
576C4#1683795511yescharacteristic
577C2—82no—
578C2#2142736411yescharacteristic
579C2#2143392711yescharacteristic
580C2#2143394711yescharacteristic
581C2#2143431911yescharacteristic
582C2#2143433411yescharacteristic
583C2#2143472911yescharacteristic
584C2#2143478111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2319C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2319D59
hypocenterC59#23787
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
1. #21426844C2semidirect
2. #21425664C2semidirect
4. #21425597C2semidirect
5. #21425592C2semidirect
8. #21425424C2semidirect
10. #21423994C2semidirect
11. C4×#14273660C2semidirect
12. #21423953C2semidirect
96. C4×D4×C59Q8semidirect
351. C59#23787semidirect
362. C4×D4Dic118semidirect

Maximal subgroups

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

subgroupindex
#214268442
#214256642
#214256172
#214255972
#214255922
#214254842
#214254242
#214252322
#214239942
C4×#142736602
#214239532
#214239212
C4×#142736482
C59×#23192
#2378759