Finite Group Databasebeta

← #28503663

Subgroups — #28503663

Order 15104.

826 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 151 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×#16837374C212yesmaximal
2#21426686C211yesmaximal, characteristic
3#21422221C211yesmaximal, characteristic, complemented
4#21422220C212yesmaximal, complemented
5C59×#1393C211yesmaximal, characteristic, complemented
6C22×#11745687C2211yescharacteristic
7C2×#14273684C411yescharacteristic, complemented
8C2×#14273683—22no—
9C2×C4⋊C4×D59C411yescharacteristic, complemented
10#16837374—24no—
11#16837374C418yescomplemented
12C2×#14273642C2212yes—
13C2×#14273641—22no—
14#16837308C411yescharacteristic, complemented
15#16837307C411yescharacteristic, complemented
16#16837305C2211yescharacteristic
17#16837278—22no—
18#16837260—22no—
19C2×#99×C59C2212yes—
20C59×#390C2211yescharacteristic
21C2×#11745687—42no—
22C2×#11745687—42no—
23C2×#11745687C2×C414yes—
24C2×#11745687D412yes—
25C2×#11745687Q812yes—
26C22×Dic118C2×C411yescharacteristic
27C22×C22⋊D59D412yes—
28C23×Dic59D412yes—
29C22×C4×D59—21no—
30C22×C4×D59C2×C411yescharacteristic
31#14273684—42no—
32#14273683—28no—
33#14273683—44no—
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×Dic59⋊C4—21no—
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4—42no—
38C4⋊C4×D59—42no—
39C2×C22⋊Dic59—22no—
40C2×C22⋊Dic59—22no—
41C2×(C2×D59)⋊C4—21no—
42C2×C4×Dic59—42no—
43C2×C4×Dic59C2×C411yescharacteristic
44#14273642—22no—
45#14273642C2×C414yescomplemented
46#14273641—28no—
47C59⋊#95—22no—
48C59⋊#95—22no—
49C22×D4×C59C2311yescharacteristic
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C22⋊C4×C59—22no—
52C2×C22⋊C4×C59—22no—
53C2×C42×C59C2×C411yescharacteristic
54#99×C59—22no—
55#99×C59C2×C414yescomplemented
56C24×C59C2×D412yes—
57C23×D59C22⋊C411yescharacteristic
58C2×D4×C59—41no—
59C2×D4×C59—41no—
60C2×D4×C59C22×C412yes—
61C2×D4×C59C2×D411yescharacteristic
62C2×D4×C59C2×Q811yescharacteristic
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—42no—
66C22×C4×C59C22×C411yescharacteristic
67C2×C22⋊D59—42no—
68C2×C22⋊D59—42no—
69C2×C22⋊D59C22⋊C414yes—
70C2×C22⋊D59C4⋊C414yes—
71C22×Dic59—22no—
72C22×Dic59—22no—
73C22×Dic59—24no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—42no—
77C22×Dic59C22⋊C412yes—
78C22×Dic59C22⋊C414yes—
79C22×Dic59C22⋊C411yescharacteristic
80C22×Dic59C4⋊C414yes—
81#11745687—44no—
82#11745687—44no—
83#11745687—44no—
84#11745687—44no—
85C2×C4×D59—22no—
86C2×C4×D59—22no—
87C2×C4×D59—42no—
88C2×C4×D59—82no—
89C2×C4×D59C4212yes—
90C2×C4×D59C4⋊C412yes—
91C2×Dic118—41no—
92C2×Dic118—41no—
93C2×Dic118C4212yes—
94C2×Dic118C4⋊C412yes—
95C4⋊C4×C59—41no—
96C22⋊C4×C59—24no—
97C22⋊C4×C59—42no—
98C22⋊C4×C59—82no—
99C42×C59—41no—
100C22⋊Dic59—24no—
101C22⋊Dic59—42no—
102C22⋊Dic59—82no—
103(C2×D59)⋊C4—42no—
104C4⋊Dic59—41no—
105Dic59⋊C4—48no—
106Dic59⋊C4—42no—
107Dic59⋊C4—42no—
108C4×Dic59—48no—
109C4×Dic59—41no—
110C23×C59—22no—
111C23×C59—22no—
112C23×C59—42no—
113C23×C59—42no—
114C23×C59C2×C22⋊C412yes—
115C23×C59C2×C22⋊C411yescharacteristic
116C23×C59C2×C4⋊C412yes—
117C22×D59—21no—
118C22×D59—21no—
119C22×D59—41no—
120C22×D59—41no—
121C22×D59#9512yes—
122D4×C59—42no—
123D4×C59—41no—
124D4×C59—41no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—44no—
128C2×C4×C59—41no—
129C2×C4×C59—42no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—82no—
134C2×C4×C59—82no—
135C2×C4×C59C2×C4211yescharacteristic
136C2×C4×C59C2×C4⋊C411yescharacteristic
137C22⋊D59—44no—
138C22⋊D59—44no—
139C2×Dic59—22no—
140C2×Dic59—22no—
141C2×Dic59—24no—
142C2×Dic59—24no—
143C2×Dic59—24no—
144C2×Dic59—24no—
145C2×Dic59—44no—
146C2×Dic59—41no—
147C2×Dic59—42no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—82no—
152C2×Dic59—82no—
153C2×Dic59#9514yes—
154C2×Dic59#9512yes—
155C2×Dic59#9912yes—
156C2×Dic59#9912yes—
157C4×D59—24no—
158C4×D59—24no—
159C4×D59—84no—
160Dic118—42no—
161Dic118—42no—
162#12583—591nomaximal, Sylow
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—22no—
166C22×C59—22no—
167C22×C59—42no—
168C22×C59—42no—
169C22×C59—42no—
170C22×C59—42no—
171C22×C59—82no—
172C22×C59C2×#9511yescharacteristic
173C22×C59C2×#9911yescharacteristic
174C22×C59C2×#9911yescharacteristic
175C2×D59—22no—
176C2×D59—42no—
177C2×D59—41no—
178C2×D59—41no—
179C2×D59—42no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—81no—
183C4×C59—84no—
184C4×C59—81no—
185Dic59—24no—
186Dic59—81no—
187Dic59—84no—
188Dic59—81no—
189Dic59#34214yes—
190C2×#410—592no—
191#2311—591no—
192#1393—592no—
193#1393—591no—
194#1393D5911yescharacteristic, complemented
195C2×C59—21no—
196C2×C59—21no—
197C2×C59—42no—
198C2×C59—42no—
199C2×C59C2×#34211yescharacteristic
200C2×C59#212811yescharacteristic
201C2×C59#212911yescharacteristic
202D59—42no—
203C22×#41—591no—
204C2×#117—591no—
205C2×#117—1182no—
206C22×C4⋊C4—591no—
207#410—598no—
208#410—1184no—
209C2×#99—592no—
210C2×#99—1182no—
211C2×#99C2×D5912yes—
212#391—591no—
213#391—591no—
214#390—591no—
215#390C2×D5911yescharacteristic
216#386—1182no—
217#380—1182no—
218C59#1258311yescharacteristic, complemented, Sylow
219C2×#41—594no—
220C2×#41—592no—
221C2×#41—592no—
222C2×#41—2362no—
223C2×#41—2362no—
224C22×Q8—591no—
225C22×D4—592no—
226C22×D4C22×D5911yescharacteristic
227C23×C4—592no—
228C23×C4—591no—
229C23×C4—1181no—
230#117—1188no—
231#117—2362no—
232#117—2364no—
233C2×C4⋊C4—591no—
234C2×C4⋊C4—1181no—
235C2×C4⋊C4—1181no—
236C2×C4⋊C4—2362no—
237C2×C4⋊C4—2362no—
238C2×C4⋊C4C4×D5911yescharacteristic
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—1182no—
242C2×C22⋊C4—1182no—
243C2×C22⋊C4—1181no—
244C2×C42—591no—
245C2×C42—2362no—
246C2×C42C4×D5911yescharacteristic
247#99—22no—
248#99—594no—
249#99—1188no—
250#99—1182no—
251#99C4×D5914yescomplemented
252#95—1182no—
253#95—1182no—
254C24—591no—
255C24D4×D5912yes—
256#41—2364no—
257#41—2364no—
258#41—2364no—
259#41—2364no—
260C2×Q8—592no—
261C2×Q8—592no—
262C2×Q8—2361no—
263C2×Q8—2361no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—594no—
267C2×D4—594no—
268C2×D4—2362no—
269C2×D4—2362no—
270C2×D4C2×C4×D5912yes—
271C2×D4D4×D5911yescharacteristic
272C2×D4Q8×D5911yescharacteristic
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—591no—
277C22×C4—594no—
278C22×C4—592no—
279C22×C4—594no—
280C22×C4—592no—
281C22×C4—592no—
282C22×C4—1182no—
283C22×C4—1182no—
284C22×C4—1182no—
285C22×C4—1182no—
286C22×C4—1184no—
287C22×C4—1181no—
288C22×C4—1181no—
289C22×C4—4722no—
290C22×C4—2362no—
291C22×C4—2362no—
292C22×C4C2×C4×D5911yescharacteristic
293C4⋊C4—41no—
294C4⋊C4—2361no—
295C4⋊C4—2362no—
296C4⋊C4—2362no—
297C4⋊C4—2368no—
298C22⋊C4—24no—
299C22⋊C4—42no—
300C22⋊C4—82no—
301C22⋊C4—1184no—
302C22⋊C4—4722no—
303C22⋊C4—2362no—
304C22⋊C4—2362no—
305C42—41no—
306C42—2361no—
307C42—2368no—
308C23—22no—
309C23—22no—
310C23—42no—
311C23—42no—
312C23—592no—
313C23—1181no—
314C23—1181no—
315C23—2361no—
316C23—2361no—
317C23C22⋊C4×D5912yes—
318C23C22⋊C4×D5911yescharacteristic
319C23C4⋊C4×D5912yes—
320Q8—2362no—
321Q8—2362no—
322D4—42no—
323D4—41no—
324D4—41no—
325D4—2364no—
326D4—2364no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—592no—
338C2×C4—592no—
339C2×C4—592no—
340C2×C4—594no—
341C2×C4—1184no—
342C2×C4—1184no—
343C2×C4—1184no—
344C2×C4—1184no—
345C2×C4—1184no—
346C2×C4—1184no—
347C2×C4—1182no—
348C2×C4—1182no—
349C2×C4—4724no—
350C2×C4—4722no—
351C2×C4—4722no—
352C2×C4—2364no—
353C2×C4—2361no—
354C2×C4—2362no—
355C2×C4—2361no—
356C2×C4—2361no—
357C2×C4—2361no—
358C2×C4C42×D5911yescharacteristic
359C2×C4C4⋊C4×D5911yescharacteristic
360C22—21no—
361C22—21no—
362C22—22no—
363C22—22no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—82no—
369C22—1182no—
370C22—2362no—
371C22—2362no—
372C22—2361no—
373C22—2361no—
374C22#95×D5911yescharacteristic
375C22#99×D5911yescharacteristic
376C22#99×D5911yescharacteristic
377C4—21no—
378C4—21no—
379C4—81no—
380C4—84no—
381C4—81no—
382C4—594no—
383C4—1184no—
384C4—4721no—
385C4—4724no—
386C4—4721no—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—2362no—
392C2#342×D5911yescharacteristic
393C2#2142576611yescharacteristic
394C2#2142576911yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC2×C4×C59C2×C42
FrattiniC22×D4C22×D59
FittingC59×#1393C2
socleC22×C59C2×#99
prime radicalC22×#11745687C22
cosocleC22×D4×C59C23
hypercenter#1393D59
hypocenterC59#12583
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
3. #21422221C2semidirect
4. #21422220C2semidirect
5. C59×#1393C2semidirect
7. C2×#14273684C4semidirect
9. C2×C4⋊C4×D59C4semidirect
11. #16837374C4semidirect
14. #16837308C4semidirect
15. #16837307C4semidirect
45. #14273642C2×C4semidirect
55. #99×C59C2×C4semidirect
194. #1393D59semidirect
218. C59#12583semidirect
251. #99C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168373742
#214266862
#214222212
#214222202
C59×#13932
#1258359