Finite Group Databasebeta

← #28506194

Subgroups — #28506194

Order 15104.

724 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#16837376C211yesmaximal, characteristic
2C2×#16837192C212yesmaximal, complemented
3#21424024C211yesmaximal, characteristic
4#21422214C212yesmaximal, complemented
5C59×#2406C211yesmaximal, characteristic
6C2×#14273686—21no—
7C2×#14273646—21no—
8C2×#14273644C2212yes—
9#16837376—24no—
10C2×#14273642—21no—
11C2×#14273642C2212yes—
12#16837366—21no—
13#16837192—24no—
14C22×#41×C59C2211yescharacteristic
15C2×#123×C59C411yescharacteristic, complemented
16C2×#123×C59C2211yescharacteristic
17C22×C4⋊C4×C59C411yescharacteristic, complemented
18C59×#395—21no—
19C59×#394—21no—
20C59×#386C411yescharacteristic, complemented
21C59×#386C2211yescharacteristic
22C59×#385—21no—
23C59×#385—21no—
24C59×#381C411yescharacteristic, complemented
25C2×#11745664—22no—
26#14273686—24no—
27#14273686—42no—
28C2×C4⋊Dic59—41no—
29C2×C22⋊Dic59—22no—
30C2×C4×Dic59—41no—
31#14273646—42no—
32#14273644—22no—
33#14273644—22no—
34#14273642—24no—
35#14273642—22no—
36#14273642D414yes—
37C22⋊(C59⋊C8)—41no—
38C22⋊(C59⋊C8)—41no—
39C2×#41×C59—22no—
40C2×#41×C59—22no—
41C2×#41×C59—41no—
42C2×#41×C59—41no—
43C2×#41×C59—42no—
44C22×Q8×C59D411yescharacteristic
45C22×D4×C59C2×C411yescharacteristic
46C22×D4×C59D411yescharacteristic
47C22×D4×C59C2311yescharacteristic
48C23×C4×C59C2×C411yescharacteristic
49C23×C4×C59C2×C411yescharacteristic
50C23×C4×C59C2×C411yescharacteristic
51C23×C4×C59D411yescharacteristic
52C23×C4×C59D411yescharacteristic
53#123×C59—22no—
54#123×C59—22no—
55#123×C59—42no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—41no—
61C2×C4⋊C4×C59—41no—
62C2×C22⋊C4×C59—21no—
63C2×C22⋊C4×C59—21no—
64C2×C22⋊C4×C59—21no—
65C2×C22⋊C4×C59—21no—
66C2×C22⋊C4×C59C2×C411yescharacteristic
67C2×C22⋊C4×C59C2×C411yescharacteristic
68#95×C59—21no—
69#95×C59—21no—
70#95×C59—21no—
71#95×C59—21no—
72C24×C59C22⋊C411yescharacteristic
73C24×C59C22×C411yescharacteristic
74C24×C59C2×D411yescharacteristic
75#41×C59—44no—
76#41×C59—44no—
77#41×C59—44no—
78#41×C59—44no—
79C2×Q8×C59—22no—
80C2×Q8×C59—22no—
81C2×Q8×C59—41no—
82C2×D4×C59—22no—
83C2×D4×C59—21no—
84C2×D4×C59—21no—
85C2×D4×C59—22no—
86C2×D4×C59—41no—
87C2×D4×C59—41no—
88C2×D4×C59—41no—
89C2×D4×C59—41no—
90C2×D4×C59—41no—
91C2×D4×C59C2×D412yes—
92C2×D4×C59#4112yes—
93C22×C4×C59—21no—
94C22×C4×C59—21no—
95C22×C4×C59—21no—
96C22×C4×C59—21no—
97C22×C4×C59—22no—
98C22×C4×C59—21no—
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—21no—
106C22×C4×C59—21no—
107C22×C4×C59—21no—
108C22×C4×C59—22no—
109C22×C4×C59—41no—
110C22×C4×C59—41no—
111C22×C4×C59—42no—
112C22×C4×C59—41no—
113C22×C4×C59—41no—
114C22×C4×C59—41no—
115C22×C4×C59C22⋊C411yescharacteristic
116C22×C4×C59C22⋊C411yescharacteristic
117C22×C4×C59C22⋊C411yescharacteristic
118C22×C4×C59C2×D411yescharacteristic
119C22×Dic59—42no—
120C4⋊C4×C59—42no—
121C4⋊C4×C59—41no—
122C4⋊C4×C59—41no—
123C4⋊C4×C59—42no—
124C4⋊C4×C59—42no—
125C22⋊C4×C59—22no—
126C22⋊C4×C59—42no—
127C22⋊C4×C59—42no—
128C22⋊C4×C59—42no—
129C22⋊C4×C59—41no—
130C22⋊C4×C59—41no—
131C22⋊C4×C59—41no—
132C22⋊Dic59—24no—
133C22⋊Dic59—42no—
134C4⋊Dic59—44no—
135C4×Dic59—44no—
136#11745664—42no—
137#11745664—42no—
138C2×C59⋊C8—42no—
139C23×C59—21no—
140C23×C59—21no—
141C23×C59—21no—
142C23×C59—21no—
143C23×C59—21no—
144C23×C59—21no—
145C23×C59—21no—
146C23×C59—41no—
147C23×C59—41no—
148C23×C59—41no—
149C23×C59—41no—
150C23×C59—41no—
151C23×C59—41no—
152C23×C59C2×C22⋊C411yescharacteristic
153C23×C59C4×D412yes—
154C23×C59#12011yescharacteristic
155C23×C59#12311yescharacteristic
156Q8×C59—44no—
157D4×C59—42no—
158D4×C59—42no—
159D4×C59—44no—
160D4×C59—42no—
161D4×C59—42no—
162C2×C4×C59—22no—
163C2×C4×C59—22no—
164C2×C4×C59—22no—
165C2×C4×C59—22no—
166C2×C4×C59—21no—
167C2×C4×C59—21no—
168C2×C4×C59—21no—
169C2×C4×C59—21no—
170C2×C4×C59—21no—
171C2×C4×C59—21no—
172C2×C4×C59—22no—
173C2×C4×C59—21no—
174C2×C4×C59—21no—
175C2×C4×C59—21no—
176C2×C4×C59—42no—
177C2×C4×C59—42no—
178C2×C4×C59—42no—
179C2×C4×C59—42no—
180C2×C4×C59—41no—
181C2×C4×C59—42no—
182C2×C4×C59—41no—
183C2×C4×C59—42no—
184C2×C4×C59—41no—
185C2×C4×C59—41no—
186C2×C4×C59—41no—
187C2×C4×C59—41no—
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59—41no—
191C2×C4×C59—41no—
192C2×C4×C59—41no—
193C2×C4×C59—82no—
194C2×C4×C59#9911yescharacteristic
195C2×C4×C59#9911yescharacteristic
196C2×C4×C59#9912yes—
197C2×C4×C59#12112yes—
198C2×Dic59—44no—
199C2×Dic59—82no—
200C2×Dic59—82no—
201C59⋊C8—82no—
202#13085—591nomaximal, Sylow
203C22×C59—21no—
204C22×C59—21no—
205C22×C59—21no—
206C22×C59—21no—
207C22×C59—22no—
208C22×C59—42no—
209C22×C59—41no—
210C22×C59—41no—
211C22×C59—42no—
212C22×C59—42no—
213C22×C59—41no—
214C22×C59—41no—
215C22×C59—41no—
216C22×C59—41no—
217C22×C59—41no—
218C22×C59—41no—
219C22×C59—41no—
220C22×C59—81no—
221C22×C59—81no—
222C22×C59#38611yescharacteristic
223C22×C59C2×#9911yescharacteristic
224C22×C59C2×#9911yescharacteristic
225C4×C59—22no—
226C4×C59—21no—
227C4×C59—21no—
228C4×C59—22no—
229C4×C59—42no—
230C4×C59—41no—
231C4×C59—81no—
232C4×C59—81no—
233C4×C59—81no—
234Dic59—84no—
235C2×#410—591no—
236#2406D5911yescharacteristic
237C2×#352—592no—
238#1785—591no—
239#1390—592no—
240C2×C59—21no—
241C2×C59—21no—
242C2×C59—41no—
243C2×C59—42no—
244C2×C59—41no—
245C2×C59—81no—
246C2×C59#189811yescharacteristic
247C2×C59#212011yescharacteristic
248C2×C59#212311yescharacteristic
249C22×#41C2×D5911yescharacteristic
250C2×#123Dic5911yescharacteristic, complemented
251C2×#123C2×D5911yescharacteristic
252C2×#117—1181no—
253C22×C4⋊C4Dic5911yescharacteristic, complemented
254C2×#101—1181no—
255C2×#100—592no—
256#410—1184no—
257C2×#99—592no—
258C2×#99—1181no—
259#407—1181no—
260#395—21no—
261#394—21no—
262#386Dic5911yescharacteristic, complemented
263#386C2×D5911yescharacteristic
264#385—21no—
265#385—21no—
266#381Dic5911yescharacteristic, complemented
267#352—1184no—
268C59#1308511yescharacteristic, complemented, Sylow
269C2×#41—22no—
270C2×#41—22no—
271C2×#41—41no—
272C2×#41—41no—
273C2×#41—42no—
274C22×Q8C22⋊D5911yescharacteristic
275C22×D4C2×Dic5911yescharacteristic
276C22×D4C22⋊D5911yescharacteristic
277C22×D4C22×D5911yescharacteristic
278C23×C4C2×Dic5911yescharacteristic
279C23×C4C2×Dic5911yescharacteristic
280C23×C4C2×Dic5911yescharacteristic
281C23×C4C22⋊D5911yescharacteristic
282C23×C4C22⋊D5911yescharacteristic
283C2×M16—1182no—
284#123—22no—
285#123—22no—
286#123—42no—
287#117—1184no—
288#117—2362no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—41no—
295C2×C4⋊C4—2361no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—1182no—
301C2×C22⋊C4C2×Dic5911yescharacteristic
302C2×C22⋊C4C2×Dic5911yescharacteristic
303C2×C42—2361no—
304#101—2362no—
305#100—1182no—
306#100—1182no—
307#99—594no—
308#99—1184no—
309#99—1182no—
310C22⋊C8—2361no—
311C22⋊C8—2361no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316C24C22⋊Dic5911yescharacteristic
317C24C22×Dic5911yescharacteristic
318C24C2×C22⋊D5911yescharacteristic
319#41—44no—
320#41—44no—
321#41—44no—
322#41—44no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—22no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4D4×D5912yes—
336C2×D4#1174568712yes—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—2362no—
360C22×C4C22⋊Dic5911yescharacteristic
361C22×C4C22⋊Dic5911yescharacteristic
362C22×C4C22⋊Dic5911yescharacteristic
363C22×C4C2×C22⋊D5911yescharacteristic
364M16—2362no—
365M16—2362no—
366C2×C8—2362no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—2364no—
373C22⋊C4—22no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—1184no—
381C22⋊C4—2362no—
382C42—2364no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23C2×C22⋊Dic5911yescharacteristic
397C23D4×Dic5912yes—
398C23#1427369511yescharacteristic
399C23#1427370811yescharacteristic
400Q8—44no—
401D4—42no—
402D4—42no—
403D4—44no—
404D4—42no—
405D4—42no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—82no—
438C2×C4—4722no—
439C2×C4—4722no—
440C2×C4—2364no—
441C2×C4#1427364212yes—
442C2×C4#1427364211yescharacteristic
443C2×C4#1427364211yescharacteristic
444C2×C4#1427369912yes—
445C8—4722no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—42no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—81no—
464C22—81no—
465C22#1683728211yescharacteristic
466C22C2×#1427364211yescharacteristic
467C22C2×#1427364211yescharacteristic
468C4—22no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—42no—
473C4—41no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—4724no—
478C2—21no—
479C2—21no—
480C2—41no—
481C2—42no—
482C2—41no—
483C2—81no—
484C2#2142455011yescharacteristic
485C2#2142573711yescharacteristic
486C2#2142575011yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#2406C2
socleC22×C59C2×#99
prime radicalC22×#41×C59C22
cosocleC22×D4×C59C23
hypercenter#2406D59
hypocenterC59#13085
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2406

Complements

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

NH ≅ G/Ntype
2. C2×#16837192C2semidirect
4. #21422214C2semidirect
15. C2×#123×C59C4semidirect
17. C22×C4⋊C4×C59C4semidirect
20. C59×#386C4semidirect
24. C59×#381C4semidirect
250. C2×#123Dic59semidirect
253. C22×C4⋊C4Dic59semidirect
262. #386Dic59semidirect
266. #381Dic59semidirect
268. C59#13085semidirect

Maximal subgroups

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

subgroupindex
C2×#168373762
C2×#168371922
#214240242
#214222142
C59×#24062
#1308559