Finite Group Databasebeta

← #28564958

Subgroups — #28564958

Order 15104.

1068 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 179 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#21434553C211yesmaximal, characteristic, complemented
2#21434187C212yesmaximal, complemented
3#21423241C214yesmaximal, complemented
4#21423179C214yesmaximal, complemented
5#21422822C211yesmaximal, characteristic, complemented
6#21422539C212yesmaximal, complemented
7C2×C59×#555C211yesmaximal, characteristic, complemented
8C2×#14273758—22no—
9#16837952C2211yescharacteristic
10#16837794—22no—
11#16837781C2212yescomplemented
12#16837776C2212yescomplemented
13#16837720—22no—
14#16837692C2214yescomplemented
15#16837597C2214yescomplemented
16#16837536—22no—
17#16837512—24no—
18#16837459C2212yes—
19C4×#11745664C2211yescharacteristic
20#16837151C2214yescomplemented
21D4⋊(C59⋊C8)C2218yescomplemented
22C59×#555—22no—
23C59×#555—24no—
24C59×#555C2214yes—
25C2×C4⋊Q8×C59C2211yescharacteristic
26C2×#124×C59—22no—
27C2×#121×C59—22no—
28C2×C4×D4×C59C2212yes—
29C2×#117×C59—22no—
30C2×#11745685—21no—
31#14273758—48no—
32C2×#11745672—24no—
33C2×#11745671—24no—
34C2×#11745669—24no—
35C2×#11745664—21no—
36#14273727C2311yescharacteristic
37#14273724C2311yescharacteristic
38#14273715—21no—
39#14273701—22no—
40C4×D236—22no—
41#14273658—24no—
42C4⋊(C59⋊C8)C2314yes—
43#14273648—24no—
44#14273647—24no—
45C22⋊(C59⋊C8)—22no—
46#14273638—22no—
47#14273638—21no—
48C4×C59⋊C8C2312yes—
49C22×Q8×C59D412yes—
50C22×D4×C59—22no—
51C22×D4×C59D412yes—
52C23×C4×C59—22no—
53C4⋊Q8×C59—22no—
54C4⋊Q8×C59D412yes—
55C4⋊Q8×C59C2312yes—
56#124×C59—28no—
57#124×C59—28no—
58#121×C59—28no—
59#121×C59—28no—
60C4×D4×C59—22no—
61C4×D4×C59—22no—
62C4×D4×C59D414yes—
63C4×D4×C59C2314yes—
64#117×C59—28no—
65#117×C59—28no—
66C2×C4⋊C4×C59—22no—
67C2×C4⋊C4×C59D412yes—
68C2×C22⋊C4×C59—24no—
69C2×C22⋊C4×C59—22no—
70C2×C42×C59—22no—
71C2×C42×C59C2311yescharacteristic
72C24×C59—22no—
73C2×Q8×C59—22no—
74C2×Q8×C59—44no—
75C2×Q8×C59C2×D414yes—
76C2×D4×C59—28no—
77C2×D4×C59—22no—
78C2×D4×C59—44no—
79C2×D4×C59—84no—
80C2×D4×C59C2×D414yes—
81C22×C4×C59—22no—
82C22×C4×C59—22no—
83C22×C4×C59—24no—
84C22×C4×C59—24no—
85C22×C4×C59—48no—
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D412yes—
88C2×C22⋊D59—41no—
89#11745685—44no—
90C2×D236—22no—
91C2×D236—21no—
92C2×C4×D59—41no—
93C2×Dic118—21no—
94C4⋊C4×C59—24no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59—24no—
97C4⋊C4×C59C2×D414yes—
98C22⋊C4×C59—216no—
99C22⋊C4×C59—24no—
100C22⋊C4×C59—24no—
101C42×C59—28no—
102C42×C59C2×D411yescharacteristic
103C42×C59C2×D412yes—
104C42×C59C2411yescharacteristic
105#11745672—48no—
106#11745671—48no—
107#11745669—48no—
108(C2×D59)⋊C4—42no—
109C4⋊Dic59—22no—
110#11745664—24no—
111C2×C59⋊C8—22no—
112C2×C59⋊C8—24no—
113C23×C59—24no—
114C23×C59—44no—
115C23×C59—44no—
116C23×C59—44no—
117C23×C59#12011yescharacteristic
118C22×D59—41no—
119Q8×C59—44no—
120Q8×C59—82no—
121D4×C59—24no—
122D4×C59—24no—
123D4×C59—44no—
124D4×C59—88no—
125C2×C4×C59—22no—
126C2×C4×C59—24no—
127C2×C4×C59—24no—
128C2×C4×C59—24no—
129C2×C4×C59—48no—
130C2×C4×C59—44no—
131C2×C4×C59—44no—
132C2×C4×C59—48no—
133C2×C4×C59#12012yes—
134C2×C4×C59#12011yescharacteristic
135C2×C4×C59C22×D411yescharacteristic
136C2×C4×C59C22×D412yes—
137C22⋊D59—82no—
138C2×Dic59—41no—
139D236—42no—
140D236—44no—
141C4×D59—82no—
142Dic118—42no—
143C59⋊C8—24no—
144C59⋊C8—44no—
145#18547—591nomaximal, Sylow
146C22×C59—22no—
147C22×C59—44no—
148C22×C59—44no—
149C22×C59—44no—
150C22×C59—88no—
151C22×C59#45312yes—
152C22×C59C2×#12011yescharacteristic
153C2×D59—82no—
154C2×D59—81no—
155C4×C59—22no—
156C4×C59—21no—
157C4×C59—21no—
158C4×C59—44no—
159C4×C59—44no—
160C4×C59#57514yes—
161Dic59—81no—
162C2×#555D5911yescharacteristic, complemented
163#3152—591no—
164#3109—592no—
165#1634—594no—
166#1624—594no—
167#1558—591no—
168#1490—592no—
169C2×C59—22no—
170C2×C59—84no—
171C2×C59#300512yes—
172C2×C59C2×#45311yescharacteristic
173D59—161no—
174C2×#137—1182no—
175#555—22no—
176#555—24no—
177#555C2×D5914yes—
178#532—591no—
179C2×C4⋊Q8C2×D5911yescharacteristic
180C2×#124—22no—
181C2×#121—22no—
182C2×C4×D4C2×D5912yes—
183C2×#117—22no—
184#497—1182no—
185#494—592no—
186#492—592no—
187#481—1182no—
188#476—594no—
189#459—594no—
190#447—1182no—
191#442—1184no—
192#432—592no—
193C4×M16—591no—
194#333—594no—
195D4⋊C8—598no—
196C59#1854711yescharacteristic, complemented, Sylow
197C2×#41—1181no—
198C22×Q8C22⋊D5912yes—
199C22×D4—22no—
200C22×D4C22⋊D5912yes—
201C23×C4—22no—
202#137—2368no—
203C2×Q16—1184no—
204C2×SD16—1184no—
205C2×D8—1184no—
206C2×M16—1181no—
207C4⋊Q8—22no—
208C4⋊Q8—591no—
209C4⋊Q8C22⋊D5912yes—
210C4⋊Q8C22×D5912yes—
211#127—591no—
212#124—28no—
213#124—28no—
214#124—1181no—
215#121—28no—
216#121—28no—
217#121—1182no—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—1182no—
221C4×D4C22⋊D5914yes—
222C4×D4C22×D5914yes—
223#117—28no—
224#117—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C22⋊D5912yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C42—22no—
230C2×C42C22×D5911yescharacteristic
231#106—1184no—
232C4⋊C8—594no—
233#102—1184no—
234#102—1184no—
235C22⋊C8—1182no—
236#97—1181no—
237#97—1182no—
238C4×C8—592no—
239C24—22no—
240#41—2364no—
241C2×Q8—22no—
242C2×Q8—44no—
243C2×Q8—1181no—
244C2×Q8C2×C22⋊D5914yes—
245C2×D4—28no—
246C2×D4—22no—
247C2×D4—44no—
248C2×D4—84no—
249C2×D4—1182no—
250C2×D4—1181no—
251C2×D4—2361no—
252C2×D4C2×C22⋊D5914yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—48no—
258C22×C4—2361no—
259C22×C4C2×C22⋊D5912yes—
260C22×C4C2×C22⋊D5911yescharacteristic
261Q16—2368no—
262SD16—2368no—
263D8—2368no—
264M16—1184no—
265C2×C8—1182no—
266C2×C8—1184no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—1182no—
271C4⋊C4C2×C22⋊D5914yes—
272C22⋊C4—216no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—2362no—
276C42—28no—
277C42C2×C22⋊D5911yescharacteristic
278C42C2×C22⋊D5912yes—
279C42C23×D5911yescharacteristic
280C23—24no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—2361no—
285C23#1427369611yescharacteristic
286Q8—44no—
287Q8—82no—
288Q8—2362no—
289D4—24no—
290D4—24no—
291D4—44no—
292D4—88no—
293D4—4722no—
294D4—2362no—
295D4—2364no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—48no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—4722no—
305C2×C4—2361no—
306C2×C4#1427369611yescharacteristic
307C2×C4#1427369612yes—
308C2×C4C22×C22⋊D5911yescharacteristic
309C2×C4C22×C22⋊D5912yes—
310C8—1184no—
311C8—2364no—
312C22—22no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—88no—
317C22—4721no—
318C22—4722no—
319C22#1683757012yes—
320C22C2×#1427369611yescharacteristic
321C4—21no—
322C4—21no—
323C4—22no—
324C4—44no—
325C4—44no—
326C4—4721no—
327C4#1683820014yes—
328C2—22no—
329C2—84no—
330C2—9441no—
331C2#2143333812yes—
332C2C2×#1683757011yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×C59×#555C2
socleC22×C59C2×#120
hypercenterC2×#555D59
hypocenterC59#18547
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #21434553C2semidirect
2. #21434187C2semidirect
3. #21423241C2semidirect
4. #21423179C2semidirect
5. #21422822C2semidirect
6. #21422539C2semidirect
7. C2×C59×#555C2semidirect
11. #16837781C22semidirect
12. #16837776C22semidirect
14. #16837692C22semidirect
15. #16837597C22semidirect
20. #16837151C22semidirect
21. D4⋊(C59⋊C8)C22semidirect
162. C2×#555D59semidirect
196. C59#18547semidirect

Maximal subgroups

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

subgroupindex
#214345532
#214341872
#214232412
#214231792
#214228222
#214225392
C2×C59×#5552
#1854759