Finite Group Databasebeta

← #28564996

Subgroups — #28564996

Order 15104.

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

normal subgroups only

This group has 982 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#21434476C211yesmaximal, characteristic, complemented
2#21434195C212yesmaximal, complemented
3#21423324C218yesmaximal, complemented
4#21422822C211yesmaximal, characteristic, complemented
5#21422539C212yesmaximal, complemented
6C2×C59×#555C211yesmaximal, characteristic
7#16837959C2211yescharacteristic
8#16837770—22no—
9#16837752C2214yescomplemented
10#16837737—22no—
11#16837682C2214yescomplemented
12#16837607C2214yescomplemented
13#16837553—22no—
14#16837502—24no—
15#16837459C2212yes—
16#16837411—22no—
17C4×#11745664C2211yescharacteristic
18#16837151C2214yescomplemented
19D4⋊(C59⋊C8)C2218yescomplemented
20C59×#555—22no—
21C59×#555—24no—
22C59×#555C2214yes—
23C2×C4⋊Q8×C59C2211yescharacteristic
24C2×#124×C59—22no—
25C2×#121×C59—22no—
26C2×C4×D4×C59C2212yes—
27C2×#117×C59—22no—
28C2×#11745670—24no—
29C2×#11745664—21no—
30#14273727C2312yes—
31#14273719—21no—
32#14273706—22no—
33C4×Dic118—22no—
34#14273686—21no—
35#14273660—24no—
36C4⋊(C59⋊C8)C2314yes—
37#14273651—28no—
38#14273650—24no—
39#14273648—24no—
40C22⋊(C59⋊C8)—22no—
41#14273638—22no—
42#14273638—21no—
43C4×C59⋊C8C2312yes—
44C22×Q8×C59D412yes—
45C22×D4×C59—22no—
46C22×D4×C59D412yes—
47C23×C4×C59—22no—
48C4⋊Q8×C59—22no—
49C4⋊Q8×C59D412yes—
50C4⋊Q8×C59C2312yes—
51#124×C59—28no—
52#124×C59—28no—
53#121×C59—28no—
54#121×C59—28no—
55C4×D4×C59—22no—
56C4×D4×C59—22no—
57C4×D4×C59D414yes—
58C4×D4×C59C2314yes—
59#117×C59—28no—
60#117×C59—28no—
61C2×C4⋊C4×C59—22no—
62C2×C4⋊C4×C59D412yes—
63C2×C22⋊C4×C59—24no—
64C2×C22⋊C4×C59—22no—
65C2×C42×C59—22no—
66C2×C42×C59C2311yescharacteristic
67C24×C59—22no—
68C2×Q8×C59—22no—
69C2×Q8×C59—44no—
70C2×Q8×C59C2×D414yes—
71C2×D4×C59—28no—
72C2×D4×C59—22no—
73C2×D4×C59—44no—
74C2×D4×C59—84no—
75C2×D4×C59C2×D414yes—
76C22×C4×C59—22no—
77C22×C4×C59—22no—
78C22×C4×C59—24no—
79C22×C4×C59—24no—
80C22×C4×C59—48no—
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2×D412yes—
83C2×Dic118—22no—
84C4⋊C4×C59—24no—
85C4⋊C4×C59—22no—
86C4⋊C4×C59—24no—
87C4⋊C4×C59C2×D414yes—
88C22⋊C4×C59—216no—
89C22⋊C4×C59—24no—
90C22⋊C4×C59—24no—
91C42×C59—28no—
92C42×C59C2×D411yescharacteristic
93C42×C59C2×D412yes—
94C42×C59C2411yescharacteristic
95C22⋊Dic59—41no—
96#11745670—48no—
97C4⋊Dic59—22no—
98C4⋊Dic59—22no—
99Dic59⋊C4—42no—
100C4×Dic59—41no—
101#11745664—42no—
102C2×C59⋊C8—22no—
103C2×C59⋊C8—24no—
104C23×C59—24no—
105C23×C59—44no—
106C23×C59—44no—
107C23×C59—44no—
108C23×C59#12011yescharacteristic
109Q8×C59—44no—
110Q8×C59—82no—
111D4×C59—24no—
112D4×C59—24no—
113D4×C59—44no—
114D4×C59—88no—
115C2×C4×C59—22no—
116C2×C4×C59—24no—
117C2×C4×C59—24no—
118C2×C4×C59—24no—
119C2×C4×C59—48no—
120C2×C4×C59—44no—
121C2×C4×C59—44no—
122C2×C4×C59—48no—
123C2×C4×C59#12011yescharacteristic
124C2×C4×C59#12012yes—
125C2×C4×C59C22×D411yescharacteristic
126C2×C4×C59C22×D412yes—
127C2×Dic59—42no—
128Dic118—44no—
129C59⋊C8—42no—
130C59⋊C8—44no—
131#18552—591nomaximal, Sylow
132C22×C59—22no—
133C22×C59—44no—
134C22×C59—44no—
135C22×C59—44no—
136C22×C59—88no—
137C22×C59#45512yes—
138C22×C59C2×#12011yescharacteristic
139C4×C59—22no—
140C4×C59—21no—
141C4×C59—21no—
142C4×C59—44no—
143C4×C59—44no—
144C4×C59#57514yes—
145Dic59—82no—
146C2×#555D5911yescharacteristic
147#3139—591no—
148#3110—592no—
149#1646—598no—
150#1558—591no—
151#1490—592no—
152C2×C59—22no—
153C2×C59—84no—
154C2×C59#300512yes—
155C2×C59C2×#45511yescharacteristic
156#555—22no—
157#555—24no—
158#555C2×D5914yes—
159#533—591no—
160C2×C4⋊Q8C2×D5911yescharacteristic
161C2×#124—22no—
162C2×#121—22no—
163C2×C4×D4C2×D5912yes—
164C2×#117—22no—
165#491—1182no—
166#487—594no—
167#484—1182no—
168#474—594no—
169#461—594no—
170#450—1182no—
171#440—1184no—
172#432—592no—
173#419—1182no—
174C4×M16—591no—
175#333—594no—
176D4⋊C8—598no—
177C59#1855211yescharacteristic, complemented, Sylow
178C22×Q8C22⋊D5912yes—
179C22×D4—22no—
180C22×D4C22⋊D5912yes—
181C23×C4—22no—
182C2×SD16—1184no—
183C2×M16—1181no—
184C4⋊Q8—22no—
185C4⋊Q8—592no—
186C4⋊Q8C22⋊D5912yes—
187C4⋊Q8C22×D5912yes—
188#125—1181no—
189#124—28no—
190#124—28no—
191C22⋊Q8—1182no—
192#121—28no—
193#121—28no—
194C4×Q8—1182no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4C22⋊D5914yes—
198C4×D4C22×D5914yes—
199#117—28no—
200#117—28no—
201#117—1181no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4C22⋊D5912yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C42—22no—
207C2×C42C22×D5911yescharacteristic
208#107—1184no—
209C4⋊C8—594no—
210#103—1188no—
211#103—1184no—
212#102—1184no—
213C22⋊C8—1182no—
214#97—1181no—
215#97—1182no—
216C4×C8—592no—
217C24—22no—
218C2×Q8—22no—
219C2×Q8—44no—
220C2×Q8—1182no—
221C2×Q8C2×C22⋊D5914yes—
222C2×D4—28no—
223C2×D4—22no—
224C2×D4—44no—
225C2×D4—84no—
226C2×D4C2×C22⋊D5914yes—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—48no—
232C22×C4C2×C22⋊D5912yes—
233C22×C4C2×C22⋊D5911yescharacteristic
234SD16—2368no—
235M16—2362no—
236C2×C8—1182no—
237C2×C8—1184no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—1182no—
242C4⋊C4—1182no—
243C4⋊C4—2362no—
244C4⋊C4C2×C22⋊D5914yes—
245C22⋊C4—216no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—2361no—
249C42—28no—
250C42—2361no—
251C42C2×C22⋊D5911yescharacteristic
252C42C2×C22⋊D5912yes—
253C42C23×D5911yescharacteristic
254C23—24no—
255C23—44no—
256C23—44no—
257C23—44no—
258C23#1427369611yescharacteristic
259Q8—44no—
260Q8—82no—
261Q8—2364no—
262D4—24no—
263D4—24no—
264D4—44no—
265D4—88no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—48no—
274C2×C4—2362no—
275C2×C4#1427369611yescharacteristic
276C2×C4#1427369612yes—
277C2×C4C22×C22⋊D5911yescharacteristic
278C2×C4C22×C22⋊D5912yes—
279C8—2362no—
280C8—2364no—
281C22—22no—
282C22—44no—
283C22—44no—
284C22—44no—
285C22—88no—
286C22#1683758012yes—
287C22C2×#1427369611yescharacteristic
288C4—21no—
289C4—21no—
290C4—22no—
291C4—44no—
292C4—44no—
293C4—4722no—
294C4#1683820014yes—
295C2—22no—
296C2—84no—
297C2#2143333812yes—
298C2C2×#1683758011yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
Fitting, prime radicalC2×C59×#555C2
socleC22×C59C2×#120
hypercenterC2×#555D59
hypocenterC59#18552
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. #21434476C2semidirect
2. #21434195C2semidirect
3. #21423324C2semidirect
4. #21422822C2semidirect
5. #21422539C2semidirect
9. #16837752C22semidirect
11. #16837682C22semidirect
12. #16837607C22semidirect
18. #16837151C22semidirect
19. D4⋊(C59⋊C8)C22semidirect
177. C59#18552semidirect

Maximal subgroups

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

subgroupindex
#214344762
#214341952
#214233242
#214228222
#214225392
C2×C59×#5552
#1855259