Finite Group Databasebeta

← #28479681

Subgroups — #28479681

Order 15104.

510 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 121 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#21432786C211yesmaximal, characteristic
2#21423875C212yesmaximal, complemented
3#21422588C212yesmaximal
4#21421981C211yesmaximal, characteristic, complemented
5C59×#1298C211yesmaximal, characteristic, complemented
6C2×#14273735C411yescharacteristic, complemented
7C23×C59⋊C8C2211yescharacteristic
8#16837913C2211yescharacteristic
9#16837470—22no—
10(C22⋊D59)⋊C8—22no—
11#16837430C411yescharacteristic, complemented
12#16837370C411yescharacteristic, complemented
13#16837360C411yescharacteristic, complemented
14C2×#14273637—22no—
15C4×C22⋊Dic59C2212yes—
16#16837156—22no—
17#16837140C414yescomplemented
18#16837137C414yescomplemented
19C59×#407C2211yescharacteristic
20C2×C22⋊C8×C59C2212yes—
21C2×#11745685C2×C411yescharacteristic
22#14273735—42no—
23C2×#11745659—21no—
24C22×C59⋊C8—42no—
25C22×C59⋊C8C2×C411yescharacteristic
26C22×C59⋊C8C2×C411yescharacteristic
27C22×C59⋊C8D412yes—
28C22×C59⋊C8D412yes—
29C2×C8×D59—21no—
30#14273710—21no—
31#14273706C2×C412yes—
32#14273701C2×C412yes—
33#14273696—21no—
34C4×C22⋊D59—22no—
35#14273686C2×C411yescharacteristic
36C2×C22⋊Dic59—22no—
37C2×C4×Dic59—22no—
38Dic59⋊C8—22no—
39(C2×D59)⋊C8—22no—
40#14273637—21no—
41#14273637—44no—
42C8×Dic59—21no—
43C59⋊#95C2×C414yes—
44C23×C4×C59C2311yescharacteristic
45C2×M16×C59C2×C411yescharacteristic
46C2×M16×C59C2×C411yescharacteristic
47C22×C8×C59—22no—
48C22⋊C8×C59—22no—
49C22⋊C8×C59—22no—
50C22⋊C8×C59C2×C414yes—
51C24×C59#4111yescharacteristic
52C22×C4×C59—41no—
53C22×C4×C59—41no—
54C22×C4×C59C22×C411yescharacteristic
55C22×C4×C59C22×C412yes—
56C22×C4×C59C2×D412yes—
57C22×C4×C59#4111yescharacteristic
58C2×C22⋊D59—21no—
59C2×C22⋊D59—22no—
60C22×Dic59—24no—
61#11745685—41no—
62#11745685—41no—
63C2×D236C4211yescharacteristic
64C2×C4×D59—21no—
65C2×Dic118C4211yescharacteristic
66M16×C59—41no—
67M16×C59—41no—
68C2×C8×C59—21no—
69C2×C8×C59—21no—
70C2×C8×C59—24no—
71C2×C8×C59—42no—
72C2×C8×C59—42no—
73C22⋊Dic59—21no—
74C22⋊Dic59—22no—
75C22⋊Dic59—44no—
76(C2×D59)⋊C4—22no—
77C4⋊Dic59C4212yes—
78Dic59⋊C4—22no—
79C4×Dic59—21no—
80C4×Dic59—42no—
81C4×Dic59—42no—
82C2×C59⋊C8—24no—
83C2×C59⋊C8—22no—
84C2×C59⋊C8—41no—
85C2×C59⋊C8—42no—
86C2×C59⋊C8C22⋊C411yescharacteristic
87C2×C59⋊C8C22⋊C412yes—
88C2×C59⋊C8C22⋊C411yescharacteristic
89#11745659—42no—
90C8×D59—42no—
91C23×C59—24no—
92C23×C59—41no—
93C23×C59#11711yescharacteristic
94C23×C59C4×D412yes—
95C22×D59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—22no—
99C2×C4×C59—21no—
100C2×C4×C59—42no—
101C2×C4×C59—41no—
102C2×C4×C59C2×C4211yescharacteristic
103C2×C4×C59C2×C22⋊C411yescharacteristic
104C2×C4×C59C4×D412yes—
105C22⋊D59—42no—
106C22⋊D59—44no—
107C2×Dic59—22no—
108C2×Dic59—21no—
109C2×Dic59—44no—
110C2×Dic59—44no—
111C2×Dic59—44no—
112D236—41no—
113C4×D59—41no—
114C4×D59—41no—
115Dic118—41no—
116C8×C59—44no—
117C8×C59—41no—
118C8×C59—41no—
119C59⋊C8—21no—
120C59⋊C8—22no—
121C59⋊C8#9912yes—
122#8239—591nomaximal, Sylow
123C22×C59—22no—
124C22×C59—22no—
125C22×C59—21no—
126C22×C59—22no—
127C22×C59—44no—
128C22×C59C4×C22⋊C411yescharacteristic
129C22×C59#44414yes—
130C2×D59—41no—
131C2×D59—42no—
132C4×C59—22no—
133C4×C59—21no—
134C4×C59C2×#9911yescharacteristic
135C4×C59#41011yescharacteristic
136Dic59—41no—
137Dic59—42no—
138Dic59—84no—
139#2929—591no—
140#1752—592no—
141#1501—592no—
142#1298D5911yescharacteristic, complemented
143#1297—591no—
144C2×C59—24no—
145C2×C59—41no—
146C2×C59C4×#9911yescharacteristic
147C2×C59#176512yes—
148D59—81no—
149C2×#131—591no—
150C23×C8—591no—
151#525—591no—
152#435—1182no—
153C8×D4—1182no—
154#424—591no—
155#408—591no—
156#407C2×D5911yescharacteristic
157C2×C22⋊C8C2×D5912yes—
158#405—591no—
159C2×#97—1182no—
160C4×C22⋊C4—592no—
161#336—1182no—
162#328—594no—
163#327—594no—
164C59#823911yescharacteristic, complemented, Sylow
165C2×#41—591no—
166C23×C4C22×D5911yescharacteristic
167#131—2362no—
168C2×M16—1181no—
169C2×M16C4×D5911yescharacteristic
170C2×M16C4×D5911yescharacteristic
171C22×C8—22no—
172C22×C8—591no—
173C22×C8—592no—
174C22×C8—591no—
175C22×C8—592no—
176C22×C8—1181no—
177C22×C8—2362no—
178#123—1181no—
179C22⋊Q8—592no—
180#121—592no—
181#120—1181no—
182C4×D4—1182no—
183#117—591no—
184C2×C22⋊C4—1182no—
185C2×C42—1182no—
186C4⋊C8—1182no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—1182no—
190C22⋊C8C4×D5914yes—
191#97—1181no—
192#97—2364no—
193C4×C8—1181no—
194#95—594no—
195C24#1174568911yescharacteristic
196#41—2361no—
197#41—2361no—
198C2×Q8—591no—
199C2×D4—591no—
200C2×D4—1181no—
201C2×D4—1182no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—1181no—
205C22×C4—1184no—
206C22×C4C2×C4×D5911yescharacteristic
207C22×C4C2×C4×D5912yes—
208C22×C4D4×D5912yes—
209C22×C4#1174568711yescharacteristic
210M16—41no—
211M16—41no—
212M16—2362no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—24no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—591no—
219C2×C8—591no—
220C2×C8—592no—
221C2×C8—1184no—
222C2×C8—1182no—
223C2×C8—2362no—
224C2×C8—2361no—
225C2×C8—2362no—
226C4⋊C4—592no—
227C4⋊C4—1182no—
228C22⋊C4—1181no—
229C22⋊C4—1182no—
230C22⋊C4—1182no—
231C22⋊C4—2364no—
232C42—1181no—
233C42—2362no—
234C42—2362no—
235C23—24no—
236C23—41no—
237C23—1181no—
238C23#1427368511yescharacteristic
239C23#1427368912yes—
240Q8—2361no—
241D4—2361no—
242D4—2362no—
243D4—2364no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—1182no—
251C2×C4—1181no—
252C2×C4—2364no—
253C2×C4—2364no—
254C2×C4—2364no—
255C2×C4—2361no—
256C2×C4—2361no—
257C2×C4C42×D5911yescharacteristic
258C2×C4C22⋊C4×D5911yescharacteristic
259C2×C4(C22⋊D59)⋊C412yes—
260C8—44no—
261C8—41no—
262C8—41no—
263C8—592no—
264C8—1182no—
265C8—1181no—
266C22—22no—
267C22—22no—
268C22—21no—
269C22—22no—
270C22—44no—
271C22—2361no—
272C22—2362no—
273C22#1683724911yescharacteristic
274C22#1683752314yes—
275C4—22no—
276C4—21no—
277C4—4724no—
278C4—2361no—
279C4—2362no—
280C4#99×D5911yescharacteristic
281C4#1683737411yescharacteristic
282C2—24no—
283C2—41no—
284C2—4721no—
285C2#2142388311yescharacteristic
286C2#2142393912yes—
subgroupG/H
centerC2×C4C22⋊C4×D59
commutatorC2×C4×C59C2×C42
FrattiniC23×C4C22×D59
FittingC59×#1298C2
socleC22×C59C4×C22⋊C4
prime radical#16837913C22
cosocleC23×C4×C59C23
hypercenter#1298D59
hypocenterC59#8239
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1298

Complements

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

NH ≅ G/Ntype
2. #21423875C2semidirect
4. #21421981C2semidirect
5. C59×#1298C2semidirect
6. C2×#14273735C4semidirect
11. #16837430C4semidirect
12. #16837370C4semidirect
13. #16837360C4semidirect
17. #16837140C4semidirect
18. #16837137C4semidirect
142. #1298D59semidirect
164. C59#8239semidirect

Maximal subgroups

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

subgroupindex
#214327862
#214238752
#214225882
#214219812
C59×#12982
#823959