Finite Group Databasebeta

← #28486329

Subgroups — #28486329

Order 15104.

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

normal subgroups only

This group has 712 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
1#21432541C211yesmaximal, characteristic
2#21424014C211yesmaximal, characteristic
3#21422629C212yesmaximal, complemented
4#21421949C212yesmaximal, complemented
5C59×#2432C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×#14273642—21no—
8C2×#14273642—21no—
9#16837369—24no—
10#16837366—21no—
11#16837366C2212yes—
12#16837265C2212yes—
13#16837139—24no—
14C22×#41×C59C2211yescharacteristic
15C2×C4⋊Q8×C59C411yescharacteristic, complemented
16C2×#124×C59C411yescharacteristic, complemented
17C2×C22⋊Q8×C59C2211yescharacteristic
18C59×#394—21no—
19C59×#394—21no—
20C59×#391C411yescharacteristic, complemented
21C59×#390C411yescharacteristic, complemented
22C59×#385—21no—
23C59×#385—21no—
24C4×C22⋊C4×C59C2211yescharacteristic
25C2×#11745664—22no—
26C2×#11745664—22no—
27C2×#11745664—42no—
28C2×#11745664—42no—
29C22×C59⋊C8—22no—
30C2×C22⋊Dic59—22no—
31#14273642—42no—
32#14273642—42no—
33C22⋊(C59⋊C8)—24no—
34C22⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)D414yes—
36C59⋊#95—22no—
37C59⋊#95—22no—
38C2×#41×C59—22no—
39C2×#41×C59—22no—
40C2×#41×C59—42no—
41C2×#41×C59—42no—
42C22×Q8×C59C2×C411yescharacteristic
43C22×Q8×C59C2×C411yescharacteristic
44C22×D4×C59C2×C411yescharacteristic
45C22×D4×C59D411yescharacteristic
46C22×D4×C59D411yescharacteristic
47C23×C4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C23×C4×C59C2311yescharacteristic
50C4⋊Q8×C59—42no—
51#124×C59—42no—
52C22⋊Q8×C59—22no—
53C22⋊Q8×C59—22no—
54C2×C4⋊C4×C59—21no—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59C2×C411yescharacteristic
58C2×C22⋊C4×C59—21no—
59C2×C22⋊C4×C59—21no—
60C2×C22⋊C4×C59—21no—
61C2×C22⋊C4×C59—21no—
62C2×C22⋊C4×C59—21no—
63C2×C22⋊C4×C59—21no—
64C2×C42×C59C2×C411yescharacteristic
65C2×C42×C59C2×C411yescharacteristic
66#95×C59—21no—
67#95×C59—21no—
68#95×C59—21no—
69C24×C59C22⋊C411yescharacteristic
70C24×C59C22⋊C411yescharacteristic
71C24×C59C2×D411yescharacteristic
72#41×C59—44no—
73#41×C59—44no—
74#41×C59—48no—
75C2×Q8×C59—22no—
76C2×Q8×C59—22no—
77C2×Q8×C59—22no—
78C2×Q8×C59—41no—
79C2×Q8×C59—41no—
80C2×Q8×C59—41no—
81C2×D4×C59—22no—
82C2×D4×C59—22no—
83C2×D4×C59—22no—
84C2×D4×C59—22no—
85C2×D4×C59—22no—
86C2×D4×C59—22no—
87C2×D4×C59—41no—
88C2×D4×C59—41no—
89C2×D4×C59—41no—
90C22×C4×C59—22no—
91C22×C4×C59—22no—
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—21no—
95C22×C4×C59—21no—
96C22×C4×C59—21no—
97C22×C4×C59—21no—
98C22×C4×C59—21no—
99C22×C4×C59—21no—
100C22×C4×C59—41no—
101C22×C4×C59—41no—
102C22×C4×C59—41no—
103C22×C4×C59—41no—
104C22×C4×C59C22⋊C411yescharacteristic
105C22×C4×C59C22⋊C411yescharacteristic
106C22×C4×C59C22×C411yescharacteristic
107C22×C4×C59C2×D411yescharacteristic
108C22×C4×C59C2×D412yes—
109C22×C4×C59#4112yes—
110C22×Dic59—42no—
111C4⋊C4×C59—22no—
112C4⋊C4×C59—42no—
113C4⋊C4×C59—42no—
114C4⋊C4×C59—41no—
115C4⋊C4×C59—41no—
116C22⋊C4×C59—42no—
117C22⋊C4×C59—42no—
118C22⋊C4×C59—42no—
119C22⋊C4×C59—42no—
120C22⋊C4×C59—42no—
121C22⋊C4×C59—42no—
122C42×C59—41no—
123C42×C59—41no—
124C22⋊Dic59—42no—
125C22⋊Dic59—42no—
126#11745664—44no—
127#11745664—42no—
128#11745664—42no—
129C2×C59⋊C8—24no—
130C2×C59⋊C8—44no—
131C23×C59—21no—
132C23×C59—21no—
133C23×C59—21no—
134C23×C59—21no—
135C23×C59—21no—
136C23×C59—21no—
137C23×C59—21no—
138C23×C59—41no—
139C23×C59—41no—
140C23×C59—41no—
141C23×C59—41no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59#9912yes—
145C23×C59C2×C22⋊C411yescharacteristic
146C23×C59#12112yes—
147Q8×C59—44no—
148Q8×C59—82no—
149D4×C59—44no—
150D4×C59—44no—
151D4×C59—44no—
152C2×C4×C59—22no—
153C2×C4×C59—22no—
154C2×C4×C59—24no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—22no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
164C2×C4×C59—21no—
165C2×C4×C59—42no—
166C2×C4×C59—42no—
167C2×C4×C59—42no—
168C2×C4×C59—42no—
169C2×C4×C59—41no—
170C2×C4×C59—41no—
171C2×C4×C59—41no—
172C2×C4×C59—41no—
173C2×C4×C59—41no—
174C2×C4×C59—41no—
175C2×C4×C59—41no—
176C2×C4×C59—41no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59#9912yes—
181C2×C4×C59C4×D412yes—
182C2×C4×C59#12011yescharacteristic
183C2×C4×C59#12311yescharacteristic
184C2×Dic59—82no—
185C2×Dic59—82no—
186C2×Dic59—82no—
187C59⋊C8—44no—
188#9443—591nomaximal, Sylow
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—22no—
192C22×C59—22no—
193C22×C59—22no—
194C22×C59—22no—
195C22×C59—42no—
196C22×C59—42no—
197C22×C59—41no—
198C22×C59—41no—
199C22×C59—41no—
200C22×C59—41no—
201C22×C59—41no—
202C22×C59—81no—
203C22×C59—81no—
204C22×C59—81no—
205C22×C59—81no—
206C22×C59#38611yescharacteristic
207C22×C59C2×#9911yescharacteristic
208C22×C59C2×#9911yescharacteristic
209C4×C59—21no—
210C4×C59—21no—
211C4×C59—22no—
212C4×C59—22no—
213C4×C59—42no—
214C4×C59—41no—
215C4×C59—81no—
216C4×C59—81no—
217C4×C59—81no—
218Dic59—162no—
219#2879—591no—
220#2432D5911yescharacteristic
221#1782—591no—
222#1511—592no—
223#1284—592no—
224C2×C59—21no—
225C2×C59—21no—
226C2×C59—42no—
227C2×C59—42no—
228C2×C59—81no—
229C2×C59#188411yescharacteristic
230C2×C59#189811yescharacteristic
231C2×C59#190911yescharacteristic
232C22×#41C2×D5911yescharacteristic
233C22×M16—1181no—
234C2×C4⋊Q8Dic5911yescharacteristic, complemented
235C2×#124Dic5911yescharacteristic, complemented
236C2×C22⋊Q8C2×D5911yescharacteristic
237C2×#99—1181no—
238C2×#99—1181no—
239#408—1184no—
240#407—592no—
241#407—1181no—
242#394—21no—
243#394—21no—
244#391Dic5911yescharacteristic, complemented
245#390Dic5911yescharacteristic, complemented
246#385—21no—
247#385—21no—
248#381—592no—
249C4×C22⋊C4C2×D5911yescharacteristic
250#328—1184no—
251C59#944311yescharacteristic, complemented, Sylow
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—42no—
255C2×#41—42no—
256C22×Q8C2×Dic5911yescharacteristic
257C22×Q8C2×Dic5911yescharacteristic
258C22×D4C2×Dic5911yescharacteristic
259C22×D4C22⋊D5911yescharacteristic
260C22×D4C22⋊D5911yescharacteristic
261C23×C4C22⋊D5911yescharacteristic
262C23×C4C22⋊D5911yescharacteristic
263C23×C4C22×D5911yescharacteristic
264C2×M16—1182no—
265C2×M16—1182no—
266C2×M16—2362no—
267C2×M16—2362no—
268C22×C8—1182no—
269C4⋊Q8—42no—
270#124—42no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C2×Dic5911yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—1182no—
284C2×C42C2×Dic5911yescharacteristic
285C2×C42C2×Dic5911yescharacteristic
286#99—2362no—
287#99—2362no—
288C22⋊C8—594no—
289C22⋊C8—1184no—
290C22⋊C8—1182no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—1182no—
295#95—1182no—
296C24C22⋊Dic5911yescharacteristic
297C24C22⋊Dic5911yescharacteristic
298C24C2×C22⋊D5911yescharacteristic
299#41—44no—
300#41—44no—
301#41—48no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—2362no—
332C22×C4C22⋊Dic5911yescharacteristic
333C22×C4C22⋊Dic5911yescharacteristic
334C22×C4D4×D5912yes—
335C22×C4#1174568712yes—
336C22×C4C22×Dic5911yescharacteristic
337C22×C4C2×C22⋊D5911yescharacteristic
338M16—2364no—
339M16—2362no—
340M16—2362no—
341C2×C8—1184no—
342C2×C8—2364no—
343C4⋊C4—22no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—2362no—
355C22⋊C4—2362no—
356C42—41no—
357C42—41no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23#1427364212yes—
372C23C2×C22⋊Dic5911yescharacteristic
373C23#1427369912yes—
374Q8—44no—
375Q8—82no—
376D4—44no—
377D4—44no—
378D4—44no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—4722no—
408C2×C4—4722no—
409C2×C4—4722no—
410C2×C4#1427364212yes—
411C2×C4D4×Dic5912yes—
412C2×C4#1427369511yescharacteristic
413C2×C4#1427370811yescharacteristic
414C8—2364no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22#1683728211yescharacteristic
433C22C2×#1427364211yescharacteristic
434C22C2×#1427364211yescharacteristic
435C4—21no—
436C4—21no—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—41no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—9442no—
445C2—21no—
446C2—21no—
447C2—42no—
448C2—42no—
449C2—81no—
450C2#2142448011yescharacteristic
451C2#2142455011yescharacteristic
452C2#2142461911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2432C2
socleC22×C59#386
prime radicalC22×#41×C59C22
cosocleC23×C4×C59C23
hypercenter#2432D59
hypocenterC59#9443
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2432

Complements

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

NH ≅ G/Ntype
3. #21422629C2semidirect
4. #21421949C2semidirect
15. C2×C4⋊Q8×C59C4semidirect
16. C2×#124×C59C4semidirect
20. C59×#391C4semidirect
21. C59×#390C4semidirect
234. C2×C4⋊Q8Dic59semidirect
235. C2×#124Dic59semidirect
244. #391Dic59semidirect
245. #390Dic59semidirect
251. C59#9443semidirect

Maximal subgroups

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

subgroupindex
#214325412
#214240142
#214226292
#214219492
C59×#24322
#944359