Finite Group Databasebeta

← #28506213

Subgroups — #28506213

Order 15104.

782 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427711C211yesmaximal, characteristic
2C2×#16837194C212yesmaximal, complemented
3#21424014C211yesmaximal, characteristic, complemented
4#21422220C212yesmaximal, complemented
5C2×C59×#410C211yesmaximal, characteristic, complemented
6C2×#14273723C2211yescharacteristic
7C2×#14273716C411yescharacteristic, complemented
8C2×#14273701C411yescharacteristic, complemented
9C2×#14273642—21no—
10C2×#14273642C2212yes—
11#16837310—21no—
12#16837310—21no—
13#16837309C411yescharacteristic, complemented
14#16837305C2211yescharacteristic
15#16837281—21no—
16#16837281—21no—
17#16837265—21no—
18C4×C22⋊Dic59C411yescharacteristic, complemented
19#16837194—24no—
20C22×#41×C59C2211yescharacteristic
21C2×#117×C59—21no—
22C59×#410—24no—
23C2×#99×C59—21no—
24C2×#99×C59C2212yes—
25C22×D236C2×C411yescharacteristic
26C22×C22⋊D59—21no—
27C22×C22⋊D59—21no—
28C22×C22⋊D59—21no—
29#14273723—22no—
30#14273723—22no—
31#14273716—42no—
32#14273701—41no—
33#14273701—41no—
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—22no—
37C2×C22⋊Dic59—21no—
38C2×C22⋊Dic59—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42C2×C4×Dic59C2×C411yescharacteristic
43C2×C4×Dic59C2×C411yescharacteristic
44#14273642—22no—
45#14273642—22no—
46#14273642—42no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—41no—
51C59⋊#95—41no—
52C2×#41×C59—22no—
53C2×#41×C59—22no—
54C2×#41×C59—41no—
55C2×#41×C59—41no—
56C2×#41×C59—42no—
57C22×Q8×C59C2×C411yescharacteristic
58C22×D4×C59D411yescharacteristic
59C22×D4×C59D411yescharacteristic
60C22×D4×C59C2311yescharacteristic
61C23×C4×C59C2×C411yescharacteristic
62C23×C4×C59D411yescharacteristic
63C23×C4×C59D411yescharacteristic
64#117×C59—24no—
65#117×C59—42no—
66C2×C4⋊C4×C59—41no—
67C2×C22⋊C4×C59—22no—
68C2×C42×C59—41no—
69#99×C59—24no—
70#99×C59—22no—
71#99×C59D414yescomplemented
72C24×C59C22⋊C411yescharacteristic
73C24×C59C2×D411yescharacteristic
74C24×C59C2×D411yescharacteristic
75C23×D59—21no—
76#41×C59—44no—
77#41×C59—44no—
78#41×C59—44no—
79#41×C59—44no—
80C2×Q8×C59—22no—
81C2×Q8×C59—22no—
82C2×Q8×C59—41no—
83C2×D4×C59—21no—
84C2×D4×C59—22no—
85C2×D4×C59—21no—
86C2×D4×C59—22no—
87C2×D4×C59—41no—
88C2×D4×C59—41no—
89C2×D4×C59—41no—
90C2×D4×C59—41no—
91C2×D4×C59—41no—
92C2×D4×C59C2×D412yes—
93C2×D4×C59#4112yes—
94C22×C4×C59—21no—
95C22×C4×C59—21no—
96C22×C4×C59—21no—
97C22×C4×C59—21no—
98C22×C4×C59—22no—
99C22×C4×C59—22no—
100C22×C4×C59—21no—
101C22×C4×C59—21no—
102C22×C4×C59—21no—
103C22×C4×C59—21no—
104C22×C4×C59—22no—
105C22×C4×C59—41no—
106C22×C4×C59—41no—
107C22×C4×C59—42no—
108C22×C4×C59C22⋊C411yescharacteristic
109C22×C4×C59C22⋊C411yescharacteristic
110C22×C4×C59C22⋊C411yescharacteristic
111C22×C4×C59C22×C411yescharacteristic
112C2×C22⋊D59—42no—
113C2×C22⋊D59—42no—
114C2×C22⋊D59—42no—
115C2×C22⋊D59—41no—
116C2×C22⋊D59—42no—
117C2×C22⋊D59—41no—
118C2×C22⋊D59—81no—
119C2×C22⋊D59—81no—
120C2×C22⋊D59—81no—
121C2×C22⋊D59—81no—
122C22×Dic59—21no—
123C22×Dic59—21no—
124C22×Dic59—21no—
125C22×Dic59—42no—
126C2×D236—22no—
127C2×D236—41no—
128C2×D236—41no—
129C4⋊C4×C59—44no—
130C22⋊C4×C59—24no—
131C22⋊C4×C59—42no—
132C42×C59—44no—
133C22⋊Dic59—42no—
134C22⋊Dic59—42no—
135C22⋊Dic59—42no—
136C22⋊Dic59—81no—
137C22⋊Dic59—81no—
138(C2×D59)⋊C4—42no—
139(C2×D59)⋊C4—41no—
140(C2×D59)⋊C4—41no—
141(C2×D59)⋊C4—42no—
142C4⋊Dic59—41no—
143C4×Dic59—22no—
144C4×Dic59—41no—
145C23×C59—21no—
146C23×C59—21no—
147C23×C59—21no—
148C23×C59—21no—
149C23×C59—21no—
150C23×C59—21no—
151C23×C59—21no—
152C23×C59—41no—
153C23×C59—41no—
154C23×C59—41no—
155C23×C59—41no—
156C23×C59—41no—
157C23×C59—41no—
158C23×C59C2×C22⋊C411yescharacteristic
159C23×C59#12011yescharacteristic
160C23×C59#12112yes—
161C23×C59#12311yescharacteristic
162C22×D59—41no—
163C22×D59—41no—
164C22×D59—41no—
165C22×D59—41no—
166C22×D59—41no—
167C22×D59—41no—
168C22×D59—41no—
169Q8×C59—44no—
170D4×C59—42no—
171D4×C59—42no—
172D4×C59—44no—
173D4×C59—42no—
174D4×C59—42no—
175C2×C4×C59—22no—
176C2×C4×C59—22no—
177C2×C4×C59—22no—
178C2×C4×C59—22no—
179C2×C4×C59—21no—
180C2×C4×C59—21no—
181C2×C4×C59—21no—
182C2×C4×C59—21no—
183C2×C4×C59—21no—
184C2×C4×C59—21no—
185C2×C4×C59—21no—
186C2×C4×C59—21no—
187C2×C4×C59—21no—
188C2×C4×C59—42no—
189C2×C4×C59—42no—
190C2×C4×C59—42no—
191C2×C4×C59—42no—
192C2×C4×C59—44no—
193C2×C4×C59—82no—
194C2×C4×C59—82no—
195C2×C4×C59#9911yescharacteristic
196C2×C4×C59#9911yescharacteristic
197C2×C4×C59#9912yes—
198C2×C4×C59C4×D412yes—
199C22⋊D59—82no—
200C22⋊D59—84no—
201C22⋊D59—82no—
202C22⋊D59—84no—
203C2×Dic59—22no—
204C2×Dic59—41no—
205C2×Dic59—41no—
206C2×Dic59—41no—
207C2×Dic59—41no—
208C2×Dic59—41no—
209C2×Dic59—41no—
210C2×Dic59—41no—
211C2×Dic59—41no—
212C2×Dic59—82no—
213C2×Dic59—82no—
214C2×Dic59—82no—
215D236—82no—
216#13089—591nomaximal, Sylow
217C22×C59—21no—
218C22×C59—21no—
219C22×C59—22no—
220C22×C59—21no—
221C22×C59—21no—
222C22×C59—41no—
223C22×C59—42no—
224C22×C59—41no—
225C22×C59—41no—
226C22×C59—41no—
227C22×C59—41no—
228C22×C59—41no—
229C22×C59—41no—
230C22×C59—41no—
231C22×C59—41no—
232C22×C59—81no—
233C22×C59—81no—
234C22×C59—81no—
235C22×C59—81no—
236C22×C59#38611yescharacteristic
237C22×C59C2×#9911yescharacteristic
238C22×C59C2×#9911yescharacteristic
239C2×D59—81no—
240C2×D59—81no—
241C2×D59—81no—
242C2×D59—81no—
243C2×D59—81no—
244C2×D59—81no—
245C2×D59—81no—
246C4×C59—22no—
247C4×C59—21no—
248C4×C59—21no—
249C4×C59—22no—
250C4×C59—41no—
251C4×C59—84no—
252Dic59—42no—
253Dic59—81no—
254Dic59—81no—
255Dic59—162no—
256C2×#410D5911yescharacteristic, complemented
257#2431—591no—
258C2×#353—592no—
259#1782—591no—
260#1393—592no—
261C2×C59—21no—
262C2×C59—21no—
263C2×C59—42no—
264C2×C59—41no—
265C2×C59—41no—
266C2×C59—81no—
267C2×C59#189811yescharacteristic
268C2×C59#212611yescharacteristic
269C2×C59#212811yescharacteristic
270D59—161no—
271C22×#41C2×D5911yescharacteristic
272C2×#127—591no—
273C2×#124—591no—
274C2×#121—591no—
275C2×#117—21no—
276#410—24no—
277C2×#99—21no—
278C2×#99—592no—
279C2×#99—1181no—
280C2×#99C2×D5912yes—
281#392—1181no—
282#392—1181no—
283#391—591no—
284#390—591no—
285#386—1181no—
286#386—1181no—
287#381—1181no—
288C4×C22⋊C4—591no—
289#353—1184no—
290C59#1308911yescharacteristic, complemented, Sylow
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—41no—
294C2×#41—41no—
295C2×#41—42no—
296C22×Q8C4×D5911yescharacteristic
297C22×D4—591no—
298C22×D4—1181no—
299C22×D4—1181no—
300C22×D4—1181no—
301C22×D4D23611yescharacteristic
302C22×D4C22⋊D5911yescharacteristic
303C22×D4C22×D5911yescharacteristic
304C23×C4C4×D5911yescharacteristic
305C23×C4D23611yescharacteristic
306C23×C4C22⋊D5911yescharacteristic
307#127—1182no—
308#127—1182no—
309#124—2362no—
310#121—2361no—
311#121—2361no—
312#117—24no—
313#117—42no—
314C2×C4⋊C4—41no—
315C2×C4⋊C4—591no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—1181no—
318C2×C22⋊C4—1181no—
319C2×C22⋊C4—1181no—
320C2×C22⋊C4—1181no—
321C2×C22⋊C4—1181no—
322C2×C22⋊C4—1181no—
323C2×C22⋊C4—1182no—
324C2×C42—41no—
325C2×C42—591no—
326C2×C42—591no—
327#99—24no—
328#99—22no—
329#99—1182no—
330#99—1182no—
331#99—2362no—
332#99C22⋊D5914yescomplemented
333#95—1181no—
334#95—1181no—
335#95—1181no—
336#95—2361no—
337#95—2361no—
338C24—1181no—
339C24(C2×D59)⋊C411yescharacteristic
340C24C2×D23611yescharacteristic
341C24C2×C22⋊D5911yescharacteristic
342#41—44no—
343#41—44no—
344#41—44no—
345#41—44no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×D4—21no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—1182no—
359C2×D4—4721no—
360C2×D4—4721no—
361C2×D4—4721no—
362C2×D4—4721no—
363C2×D4—2361no—
364C2×D4—2361no—
365C2×D4—2362no—
366C2×D4—2362no—
367C2×D4—2362no—
368C2×D4—2361no—
369C2×D4—2362no—
370C2×D4—2361no—
371C2×D4#1174568512yes—
372C2×D4C2×C22⋊D5912yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—1181no—
388C22×C4—1181no—
389C22×C4—1181no—
390C22×C4—2362no—
391C22×C4(C2×D59)⋊C411yescharacteristic
392C22×C4(C2×D59)⋊C411yescharacteristic
393C22×C4(C2×D59)⋊C411yescharacteristic
394C22×C4C2×C4×D5911yescharacteristic
395C4⋊C4—44no—
396C4⋊C4—2361no—
397C22⋊C4—24no—
398C22⋊C4—42no—
399C22⋊C4—4721no—
400C22⋊C4—4721no—
401C22⋊C4—2362no—
402C22⋊C4—2362no—
403C22⋊C4—2362no—
404C22⋊C4—2362no—
405C22⋊C4—2361no—
406C22⋊C4—2361no—
407C22⋊C4—2362no—
408C42—44no—
409C42—1182no—
410C42—2361no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—2361no—
425C23—2361no—
426C23—2361no—
427C23—2361no—
428C23—2361no—
429C23—2361no—
430C23—2361no—
431C23C2×(C2×D59)⋊C411yescharacteristic
432C23#1427369611yescharacteristic
433C23#1427370112yes—
434C23#1427371011yescharacteristic
435Q8—44no—
436D4—42no—
437D4—42no—
438D4—44no—
439D4—42no—
440D4—42no—
441D4—4722no—
442D4—4724no—
443D4—4722no—
444D4—4724no—
445D4—4722no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—1182no—
467C2×C4—4722no—
468C2×C4—4722no—
469C2×C4—4722no—
470C2×C4—2361no—
471C2×C4—2361no—
472C2×C4—2361no—
473C2×C4—2361no—
474C2×C4—2361no—
475C2×C4—2361no—
476C2×C4—2361no—
477C2×C4—2361no—
478C2×C4#1427364112yes—
479C2×C4#1427364111yescharacteristic
480C2×C4#1427364111yescharacteristic
481C2×C4C4×C22⋊D5912yes—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—22no—
487C22—41no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—4721no—
502C22—4721no—
503C22—4721no—
504C22—4721no—
505C22—4721no—
506C22—4721no—
507C22—4721no—
508C22#1683728111yescharacteristic
509C22C2×#1427364111yescharacteristic
510C22C2×#1427364111yescharacteristic
511C4—21no—
512C4—21no—
513C4—22no—
514C4—22no—
515C4—41no—
516C4—84no—
517C4—9442no—
518C4—4721no—
519C4—4721no—
520C4—2362no—
521C2—21no—
522C2—21no—
523C2—41no—
524C2—42no—
525C2—41no—
526C2—81no—
527C2—9441no—
528C2#2142454911yescharacteristic
529C2#2142575911yescharacteristic
530C2#2142576711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC2×C59×#410C2
socleC22×C59C2×#99
prime radical#21427711C2
cosocleC22×D4×C59C23
hypercenterC2×#410D59
hypocenterC59#13089
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#16837194C2semidirect
3. #21424014C2semidirect
4. #21422220C2semidirect
5. C2×C59×#410C2semidirect
7. C2×#14273716C4semidirect
8. C2×#14273701C4semidirect
13. #16837309C4semidirect
18. C4×C22⋊Dic59C4semidirect
71. #99×C59D4semidirect
256. C2×#410D59semidirect
290. C59#13089semidirect
332. #99C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214277112
C2×#168371942
#214240142
#214222202
C2×C59×#4102
#1308959