Finite Group Databasebeta

← #28498982

Subgroups — #28498982

Order 15104.

448 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 115 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#21423829C211yesmaximal, characteristic
2#21422410C211yesmaximal, characteristic
3#21422120C214yesmaximal
4C59×#2628C211yesmaximal, characteristic
5C2×#14273686—21no—
6#16837455—21no—
7#16837169—21no—
8#16837169C2212yes—
9#16837128—21no—
10#16837126C2212yes—
11C2×C4⋊Q8×C59C411yescharacteristic, complemented
12C2×#117×C59C2211yescharacteristic
13C59×#400C412yescomplemented
14C59×#399C2212yes—
15C59×#395—21no—
16C59×#393—21no—
17C59×#389—21no—
18C59×#382C411yescharacteristic, complemented
19C59×#380—21no—
20#14273686—24no—
21#14273686—42no—
22C2×C4⋊Dic59—41no—
23C2×C22⋊Dic59—22no—
24C2×C4×Dic59—41no—
25C4⋊(C59⋊C8)—41no—
26C22⋊(C59⋊C8)—22no—
27C4×C59⋊C8—41no—
28C22×Q8×C59—21no—
29C23×C4×C59D411yescharacteristic
30C4⋊Q8×C59—41no—
31C4⋊Q8×C59—41no—
32#117×C59—41no—
33#117×C59—41no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—22no—
37C2×C4⋊C4×C59C2×C412yes—
38C2×C4⋊C4×C59C2×C411yescharacteristic
39C2×C4⋊C4×C59D411yescharacteristic
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59C2311yescharacteristic
43C2×C42×C59C2×C411yescharacteristic
44C2×C42×C59D411yescharacteristic
45#95×C59—22no—
46#95×C59—21no—
47#95×C59—22no—
48#95×C59C2×C412yes—
49C24×C59C2×D411yescharacteristic
50C2×Q8×C59—42no—
51C2×Q8×C59—82no—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59—22no—
55C22×C4×C59—22no—
56C22×C4×C59—22no—
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22⋊C411yescharacteristic
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22⋊C411yescharacteristic
61C22×C4×C59SD1612yes—
62C22×C4×C59Q1612yes—
63C22×C4×C59C22×C411yescharacteristic
64C22×C4×C59C2×D411yescharacteristic
65C22×Dic59—42no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—42no—
69C4⋊C4×C59—41no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—82no—
73C22⋊C4×C59—41no—
74C22⋊C4×C59—41no—
75C22⋊C4×C59—82no—
76C42×C59—21no—
77C42×C59—21no—
78C42×C59—21no—
79C42×C59—21no—
80C22⋊Dic59—24no—
81C22⋊Dic59—42no—
82C4⋊Dic59—44no—
83C4×Dic59—44no—
84C2×C59⋊C8—42no—
85C23×C59—21no—
86C23×C59—21no—
87C23×C59—41no—
88C23×C59—41no—
89C23×C59C2×C22⋊C411yescharacteristic
90C23×C59C2×SD1611yescharacteristic
91C23×C59C2×Q1611yescharacteristic
92Q8×C59—84no—
93C2×C4×C59—21no—
94C2×C4×C59—21no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—21no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59—42no—
112C2×C4×C59—42no—
113C2×C4×C59#9911yescharacteristic
114C2×C4×C59#9911yescharacteristic
115C2×C4×C59#10312yes—
116C2×C4×C59#10312yes—
117C2×C4×C59#10412yes—
118C2×Dic59—44no—
119C2×Dic59—82no—
120C2×Dic59—82no—
121C59⋊C8—82no—
122#11718—591nomaximal, Sylow
123C22×C59—21no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—41no—
128C22×C59—41no—
129C22×C59—41no—
130C22×C59—41no—
131C22×C59—81no—
132C22×C59C2×#9911yescharacteristic
133C22×C59C2×#10311yescharacteristic
134C22×C59C2×#10411yescharacteristic
135C4×C59—22no—
136C4×C59—41no—
137C4×C59—41no—
138C4×C59—41no—
139C4×C59—41no—
140C4×C59—82no—
141C4×C59—82no—
142C4×C59#32812yes—
143C4×C59#32812yes—
144Dic59—84no—
145#2628D5911yescharacteristic
146#1740—591no—
147#1458—591no—
148#1352—594no—
149C2×C59—21no—
150C2×C59—21no—
151C2×C59—41no—
152C2×C59—41no—
153C2×C59C2×#32811yescharacteristic
154C2×C59#212311yescharacteristic
155C2×C59#212811yescharacteristic
156C2×C4⋊Q8Dic5911yescharacteristic, complemented
157C2×#117—1181no—
158C2×#117C2×D5911yescharacteristic
159#431—1181no—
160#400Dic5912yescomplemented
161#399C2×D5912yes—
162#395—21no—
163#393—21no—
164#389—21no—
165#382Dic5911yescharacteristic, complemented
166#380—21no—
167#342—592no—
168#342—1181no—
169#324—1181no—
170#323—592no—
171C59#1171811yescharacteristic, complemented, Sylow
172C22×Q8—21no—
173C23×C4C22⋊D5911yescharacteristic
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176#117—41no—
177#117—41no—
178#117—1184no—
179#117—2362no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—2361no—
184C2×C4⋊C4C2×Dic5912yes—
185C2×C4⋊C4C2×Dic5911yescharacteristic
186C2×C4⋊C4C22⋊D5911yescharacteristic
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1182no—
189C2×C22⋊C4C22⋊D5911yescharacteristic
190C2×C22⋊C4C22×D5911yescharacteristic
191C2×C42—2361no—
192C2×C42C2×Dic5911yescharacteristic
193C2×C42C22⋊D5911yescharacteristic
194C4⋊C8—2361no—
195C22⋊C8—1182no—
196C4×C8—2361no—
197#95—22no—
198#95—21no—
199#95—22no—
200#95C2×Dic5912yes—
201C24C2×C22⋊D5911yescharacteristic
202C2×Q8—42no—
203C2×Q8—82no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—2362no—
210C22×C4#1174567112yes—
211C22×C4#1174567212yes—
212C22×C4C22⋊Dic5911yescharacteristic
213C22×C4C22⋊Dic5911yescharacteristic
214C22×C4C22⋊Dic5911yescharacteristic
215C22×C4C22⋊Dic5911yescharacteristic
216C22×C4C22×Dic5911yescharacteristic
217C22×C4C2×C22⋊D5911yescharacteristic
218C2×C8—2362no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—82no—
226C4⋊C4—2364no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—82no—
230C22⋊C4—1184no—
231C22⋊C4—2362no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—2364no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23C2×C22⋊Dic5911yescharacteristic
242C23C2×#1174567111yescharacteristic
243C23C2×#1174567211yescharacteristic
244Q8—84no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—4722no—
266C2×C4—4722no—
267C2×C4—2364no—
268C2×C4#1427364211yescharacteristic
269C2×C4#1427364211yescharacteristic
270C2×C4#1427365112yes—
271C2×C4#1427365112yes—
272C2×C4#1427365412yes—
273C8—4722no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#1427364211yescharacteristic
284C22C2×#1427365111yescharacteristic
285C22C2×#1427365411yescharacteristic
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4—82no—
293C4—4724no—
294C4#1683713912yes—
295C4#1683713912yes—
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#1683713911yescharacteristic
301C2#2142575011yescharacteristic
302C2#2142576811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2628C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2628D59
hypocenterC59#11718
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C59C4semidirect
13. C59×#400C4semidirect
18. C59×#382C4semidirect
156. C2×C4⋊Q8Dic59semidirect
160. #400Dic59semidirect
165. #382Dic59semidirect
171. C59#11718semidirect

Maximal subgroups

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

subgroupindex
#214238292
#214224102
#214221202
C59×#26282
#1171859