Finite Group Databasebeta

← #28498878

Subgroups — #28498878

Order 15104.

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

normal subgroups only

This group has 468 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#21422096C214yesmaximal
4C59×#2621C211yesmaximal, characteristic
5C2×#14273686—21no—
6#16837455—21no—
7#16837169—21no—
8#16837169C2212yes—
9#16837128—21no—
10#16837126C2212yes—
11C2×C4⋊Q8×C59C411yescharacteristic, complemented
12C2×C22⋊Q8×C59—21no—
13C2×#117×C59C2211yescharacteristic
14C59×#398—21no—
15C59×#398C412yescomplemented
16C59×#397—21no—
17C59×#397C2212yes—
18C59×#391—21no—
19C59×#382C411yescharacteristic, complemented
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—
32C22⋊Q8×C59—42no—
33C22⋊Q8×C59—42no—
34#117×C59—41no—
35#117×C59—41no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—22no—
39C2×C4⋊C4×C59C2×C412yes—
40C2×C4⋊C4×C59C2×C411yescharacteristic
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59C2311yescharacteristic
45C2×C42×C59C2×C411yescharacteristic
46C2×C42×C59D411yescharacteristic
47#95×C59—21no—
48#95×C59—22no—
49#95×C59—22no—
50#95×C59C2×C412yes—
51C24×C59C2×D411yescharacteristic
52C2×Q8×C59—41no—
53C2×Q8×C59—41no—
54C2×Q8×C59—42no—
55C2×Q8×C59—42no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59C22⋊C411yescharacteristic
62C22×C4×C59C22⋊C411yescharacteristic
63C22×C4×C59C22⋊C411yescharacteristic
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59D812yes—
66C22×C4×C59SD1612yes—
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×Dic59—42no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59—41no—
72C4⋊C4×C59—42no—
73C4⋊C4×C59—42no—
74C4⋊C4×C59—42no—
75C4⋊C4×C59—44no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—41no—
78C22⋊C4×C59—42no—
79C22⋊C4×C59—42no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C42×C59—21no—
84C22⋊Dic59—24no—
85C22⋊Dic59—42no—
86C4⋊Dic59—44no—
87C4×Dic59—44no—
88C2×C59⋊C8—42no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—41no—
92C23×C59—41no—
93C23×C59C2×C22⋊C411yescharacteristic
94C23×C59C2×D811yescharacteristic
95C23×C59C2×SD1611yescharacteristic
96Q8×C59—82no—
97Q8×C59—82no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—22no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—42no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59#9911yescharacteristic
119C2×C4×C59#9911yescharacteristic
120C2×C4×C59#10212yes—
121C2×C4×C59#10212yes—
122C2×C4×C59#10412yes—
123C2×Dic59—44no—
124C2×Dic59—82no—
125C2×Dic59—82no—
126C59⋊C8—82no—
127#11694—591nomaximal, Sylow
128C22×C59—21no—
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—21no—
132C22×C59—41no—
133C22×C59—41no—
134C22×C59—41no—
135C22×C59—41no—
136C22×C59—81no—
137C22×C59C2×#9911yescharacteristic
138C22×C59C2×#10211yescharacteristic
139C22×C59C2×#10411yescharacteristic
140C4×C59—22no—
141C4×C59—41no—
142C4×C59—41no—
143C4×C59—41no—
144C4×C59—41no—
145C4×C59—82no—
146C4×C59—82no—
147C4×C59#32712yes—
148C4×C59#32712yes—
149Dic59—84no—
150#2621D5911yescharacteristic
151#1740—591no—
152#1458—591no—
153#1344—594no—
154C2×C59—21no—
155C2×C59—21no—
156C2×C59—41no—
157C2×C59—41no—
158C2×C59C2×#32711yescharacteristic
159C2×C59#212311yescharacteristic
160C2×C59#212911yescharacteristic
161C2×C4⋊Q8Dic5911yescharacteristic, complemented
162C2×C22⋊Q8—21no—
163C2×#117—1181no—
164C2×#117C2×D5911yescharacteristic
165#431—1181no—
166#398—21no—
167#398Dic5912yescomplemented
168#397—21no—
169#397C2×D5912yes—
170#391—21no—
171#382Dic5911yescharacteristic, complemented
172#342—592no—
173#342—1181no—
174#324—1181no—
175#323—592no—
176C59#1169411yescharacteristic, complemented, Sylow
177C22×Q8—21no—
178C23×C4C22⋊D5911yescharacteristic
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183#117—41no—
184#117—41no—
185#117—1184no—
186#117—2362no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—2361no—
191C2×C4⋊C4C2×Dic5912yes—
192C2×C4⋊C4C2×Dic5911yescharacteristic
193C2×C4⋊C4C22⋊D5911yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—1182no—
196C2×C22⋊C4C22⋊D5911yescharacteristic
197C2×C22⋊C4C22×D5911yescharacteristic
198C2×C42—2361no—
199C2×C42C2×Dic5911yescharacteristic
200C2×C42C22⋊D5911yescharacteristic
201C4⋊C8—2361no—
202C22⋊C8—1182no—
203C4×C8—2361no—
204#95—21no—
205#95—22no—
206#95—22no—
207#95C2×Dic5912yes—
208C24C2×C22⋊D5911yescharacteristic
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—2362no—
219C22×C4#1174566912yes—
220C22×C4#1174567012yes—
221C22×C4C22⋊Dic5911yescharacteristic
222C22×C4C22⋊Dic5911yescharacteristic
223C22×C4C22⋊Dic5911yescharacteristic
224C22×C4C22⋊Dic5911yescharacteristic
225C22×C4C22×Dic5911yescharacteristic
226C22×C4C2×C22⋊D5911yescharacteristic
227C2×C8—2362no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—2364no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—1184no—
240C22⋊C4—2362no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—2364no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23C2×C22⋊Dic5911yescharacteristic
251C23C2×#1174566911yescharacteristic
252C23C2×#1174567011yescharacteristic
253Q8—82no—
254Q8—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—4722no—
276C2×C4—4722no—
277C2×C4—2364no—
278C2×C4#1427364211yescharacteristic
279C2×C4#1427364211yescharacteristic
280C2×C4#1427364812yes—
281C2×C4#1427364812yes—
282C2×C4#1427365412yes—
283C8—4722no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22C2×#1427364211yescharacteristic
294C22C2×#1427364811yescharacteristic
295C22C2×#1427365411yescharacteristic
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—82no—
302C4—82no—
303C4—4724no—
304C4#1683713612yes—
305C4#1683713612yes—
306C2—21no—
307C2—21no—
308C2—41no—
309C2—41no—
310C2C2×#1683713611yescharacteristic
311C2#2142575011yescharacteristic
312C2#2142577111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2621C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#2621D59
hypocenterC59#11694
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C59C4semidirect
15. C59×#398C4semidirect
19. C59×#382C4semidirect
161. C2×C4⋊Q8Dic59semidirect
167. #398Dic59semidirect
171. #382Dic59semidirect
176. C59#11694semidirect

Maximal subgroups

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

subgroupindex
#214238292
#214224102
#214220962
C59×#26212
#1169459