Finite Group Databasebeta

← #28646301

Subgroups — #28646301

Order 15104.

702 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427787C211yesmaximal, characteristic
2#21425664C211yesmaximal, characteristic, complemented
3#21425597C211yesmaximal, characteristic, complemented
4#21425592C211yesmaximal, characteristic, complemented
5#21425424C211yesmaximal, characteristic, complemented
6#21425048C211yesmaximal, characteristic, complemented
7#21424864C211yesmaximal, characteristic, complemented
8#21424688C211yesmaximal, characteristic, complemented
9#21424667C211yesmaximal, characteristic, complemented
10#21424293C211yesmaximal, characteristic
11#21424276C211yesmaximal, characteristic
12#21424191C211yesmaximal, characteristic
13#21424123C211yesmaximal, characteristic
14#21423812C211yesmaximal, characteristic
15C59×#2347C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C2×#14273727C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C22⋊(C59⋊C8)C2211yescharacteristic
26#16837325—21no—
27#16837314—21no—
28#16837306—21no—
29#16837300C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837289—21no—
32#16837268C2211yescharacteristic
33#16837262C2211yescharacteristic
34#16837164C2211yescharacteristic
35#16837164C2211yescharacteristic
36#16837164C2211yescharacteristic
37#16837164C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#117×C59C2211yescharacteristic
49C59×#389C2211yescharacteristic
50C59×#389C2211yescharacteristic
51C59×#388—21no—
52C59×#388—21no—
53C59×#383—21no—
54C59×#380C2211yescharacteristic
55C4×C4⋊C4×C59—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C22×Dic118—21no—
60C22×C59⋊C8—22no—
61C22×C59⋊C8—22no—
62C22×C59⋊C8—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67#14273727—22no—
68#14273727—41no—
69#14273686—22no—
70#14273686—41no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C4×Dic59—21no—
81#14273660—22no—
82#14273660—22no—
83#14273660—41no—
84#14273658—41no—
85C4⋊(C59⋊C8)—22no—
86C4⋊(C59⋊C8)—41no—
87C22⋊(C59⋊C8)—22no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C23×C4×C59C2311yescharacteristic
94#117×C59—22no—
95#117×C59Q814yescomplemented
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59C2311yescharacteristic
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×C59D411yescharacteristic
105C2×C22⋊C4×C59D411yescharacteristic
106C2×C42×C59—21no—
107C2×C42×C59—21no—
108C2×C42×C59—21no—
109C2×C42×C59C2311yescharacteristic
110C2×C42×C59C2311yescharacteristic
111#95×C59—21no—
112#95×C59—21no—
113#95×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59D411yescharacteristic
117#95×C59D411yescharacteristic
118C24×C59C2×D411yescharacteristic
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×Q812yes—
133C22×C4×C59#4112yes—
134C22×C4×C59C2411yescharacteristic
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C2×Dic118—42no—
140C2×Dic118—42no—
141C2×Dic118—81no—
142C4⋊C4×C59—21no—
143C4⋊C4×C59—21no—
144C4⋊C4×C59—21no—
145C4⋊C4×C59—21no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—42no—
149C4⋊C4×C59—42no—
150C4⋊C4×C59C2×Q812yes—
151C4⋊C4×C59C2×Q812yes—
152C22⋊C4×C59—24no—
153C22⋊C4×C59—22no—
154C22⋊C4×C59—41no—
155C22⋊C4×C59—81no—
156C42×C59—24no—
157C42×C59—21no—
158C42×C59—21no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59—81no—
162C42×C59—81no—
163C22⋊Dic59—42no—
164C22⋊Dic59—81no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—42no—
173C4⋊Dic59#4112yes—
174Dic59⋊C4—42no—
175Dic59⋊C4—81no—
176C4×Dic59—42no—
177C2×C59⋊C8—21no—
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—44no—
189C2×C59⋊C8#4112yes—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59C22⋊Q812yes—
197C23×C59C22×D411yescharacteristic
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—22no—
203C2×C4×C59—22no—
204C2×C4×C59—22no—
205C2×C4×C59—22no—
206C2×C4×C59—22no—
207C2×C4×C59—22no—
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—41no—
218C2×C4×C59—41no—
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×C59C22⋊Q812yes—
228C2×C4×C59#13512yes—
229C2×C4×C59C22×Q811yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59H(5,𝔽2)11yescharacteristic
234C2×C4×C592−1+411yescharacteristic
235C2×Dic59—22no—
236C2×Dic59—41no—
237C2×Dic59—41no—
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—
247Dic118—84no—
248C59⋊C8—22no—
249C59⋊C8—41no—
250C59⋊C8—41no—
251C59⋊C8—41no—
252#25989—591nomaximal, Sylow
253C22×C59—22no—
254C22×C59—22no—
255C22×C59—22no—
256C22×C59—41no—
257C22×C59—41no—
258C22×C59—41no—
259C22×C59—41no—
260C22×C59C2×C22⋊Q811yescharacteristic
261C22×C59#54611yescharacteristic
262C22×C59#54711yescharacteristic
263C22×C59C2×#13511yescharacteristic
264C22×C59#57511yescharacteristic
265C22×C59#57611yescharacteristic
266C22×C59#57811yescharacteristic
267C4×C59—22no—
268C4×C59—22no—
269C4×C59—42no—
270C4×C59—42no—
271C4×C59—41no—
272C4×C59—41no—
273C4×C59—41no—
274C4×C59—81no—
275C4×C59—81no—
276C4×C59#53311yescharacteristic
277C4×C59#54411yescharacteristic
278C4×C59#55111yescharacteristic
279C4×C59#55211yescharacteristic
280Dic59—42no—
281Dic59—81no—
282Dic59—81no—
283Dic59—81no—
284#2439—591no—
285#2347D5911yescharacteristic
286#2101—591no—
287#2088—591no—
288#2087—591no—
289#2057—591no—
290#1988—591no—
291#1949—591no—
292#1921—591no—
293#1918—591no—
294#1854—591no—
295#1850—591no—
296#1825—591no—
297#1808—591no—
298#1735—591no—
299C2×C59—42no—
300C2×C59#243811yescharacteristic
301C2×C59#307611yescharacteristic
302C2×C59#308311yescharacteristic
303C2×C59#329711yescharacteristic
304C2×C59#331311yescharacteristic
305C2×C59#334211yescharacteristic
306C2×C59#334711yescharacteristic
307C23×C8—591no—
308C2×C4⋊Q8—591no—
309C2×#117—591no—
310C2×#117C2×D5911yescharacteristic
311C2×#107—591no—
312C2×#107—591no—
313C2×#107—591no—
314C2×#106—591no—
315C2×C4⋊C8—591no—
316C2×C4⋊C8—591no—
317C2×C22⋊C8—591no—
318#396—1181no—
319#393—1181no—
320#391—1181no—
321#389—591no—
322#389—591no—
323#389C2×D5911yescharacteristic
324#389C2×D5911yescharacteristic
325#388—21no—
326#388—21no—
327#387—1181no—
328#383—21no—
329#382—591no—
330#380—591no—
331#380C2×D5911yescharacteristic
332C4×C4⋊C4—21no—
333C4×C4⋊C4C2×D5911yescharacteristic
334C4×C4⋊C4C2×D5911yescharacteristic
335C4×C22⋊C4C2×D5911yescharacteristic
336#340—591no—
337#340—591no—
338#340—591no—
339#340—591no—
340#340—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#336—591no—
349#336—591no—
350C59#2598911yescharacteristic, complemented, Sylow
351C22×Q8—1181no—
352C23×C4C22×D5911yescharacteristic
353C22×C8—591no—
354C22×C8—591no—
355C22×C8—591no—
356C22×C8—591no—
357C22×C8—1182no—
358C22×C8—1182no—
359C22×C8—2361no—
360C4⋊Q8—1182no—
361C4⋊Q8—2361no—
362#117—22no—
363#117—1182no—
364#117—2361no—
365#117Dic11814yescomplemented
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4C22×D5911yescharacteristic
376C2×C4⋊C4C22×D5911yescharacteristic
377C2×C4⋊C4C22×D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1181no—
382C2×C22⋊C4—1181no—
383C2×C22⋊C4C22⋊D5911yescharacteristic
384C2×C22⋊C4C22⋊D5911yescharacteristic
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—1181no—
389C2×C42C22×D5911yescharacteristic
390C2×C42C22×D5911yescharacteristic
391#107—1182no—
392#107—1182no—
393#107—2361no—
394#106—2361no—
395C4⋊C8—1182no—
396C4⋊C8—2361no—
397C22⋊C8—1182no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—1181no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95C22⋊D5911yescharacteristic
409#95C22⋊D5911yescharacteristic
410C24C2×C22⋊D5911yescharacteristic
411C2×Q8—4721no—
412C2×Q8—2362no—
413C2×Q8—2362no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—1181no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4C2×Dic11812yes—
427C22×C4#1174568512yes—
428C22×C4C2×C22⋊D5911yescharacteristic
429C22×C4C2×C22⋊D5911yescharacteristic
430C22×C4C2×C22⋊D5911yescharacteristic
431C22×C4C2×C22⋊D5911yescharacteristic
432C22×C4C2×C22⋊D5911yescharacteristic
433C22×C4C23×D5911yescharacteristic
434C2×C8—592no—
435C2×C8—1181no—
436C2×C8—1181no—
437C2×C8—1181no—
438C2×C8—1181no—
439C2×C8—1181no—
440C2×C8—1181no—
441C2×C8—1181no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—2364no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—592no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—4721no—
464C4⋊C4—2362no—
465C4⋊C4—2362no—
466C4⋊C4C2×Dic11812yes—
467C4⋊C4C2×Dic11812yes—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—41no—
471C22⋊C4—81no—
472C22⋊C4—4721no—
473C22⋊C4—2362no—
474C42—24no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—81no—
480C42—81no—
481C42—2362no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23#1427370612yes—
489C23C22×C22⋊D5911yescharacteristic
490Q8—4724no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
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—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—1182no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4—2361no—
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#1427370612yes—
533C2×C4#1427374712yes—
534C2×C4C22×Dic11811yescharacteristic
535C2×C4#41×D5911yescharacteristic
536C2×C4#41×D5911yescharacteristic
537C2×C4C2×#1174568511yescharacteristic
538C2×C4#1427377311yescharacteristic
539C2×C4#1427377411yescharacteristic
540C8—1182no—
541C8—2361no—
542C8—2361no—
543C8—2361no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#1427370611yescharacteristic
552C22#1683804011yescharacteristic
553C22#1683805111yescharacteristic
554C22C2×#1427374711yescharacteristic
555C22#1683820011yescharacteristic
556C22#1683820511yescharacteristic
557C22#1683821711yescharacteristic
558C4—22no—
559C4—22no—
560C4—42no—
561C4—42no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—4721no—
568C4—4721no—
569C4—4721no—
570C4—2362no—
571C4#1683795511yescharacteristic
572C4#1683803011yescharacteristic
573C4#1683807611yescharacteristic
574C4#1683808511yescharacteristic
575C2—42no—
576C2#2142776611yescharacteristic
577C2#2143388411yescharacteristic
578C2#2143394711yescharacteristic
579C2#2143594011yescharacteristic
580C2#2143615611yescharacteristic
581C2#2143652211yescharacteristic
582C2#2143659011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2347C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2347D59
hypocenterC59#25989
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2347

Complements

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

NH ≅ G/Ntype
2. #21425664C2semidirect
3. #21425597C2semidirect
4. #21425592C2semidirect
5. #21425424C2semidirect
6. #21425048C2semidirect
7. #21424864C2semidirect
8. #21424688C2semidirect
9. #21424667C2semidirect
95. #117×C59Q8semidirect
350. C59#25989semidirect
365. #117Dic118semidirect

Maximal subgroups

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

subgroupindex
#214277872
#214256642
#214255972
#214255922
#214254242
#214250482
#214248642
#214246882
#214246672
#214242932
#214242762
#214241912
#214241232
#214238122
C59×#23472
#2598959