Finite Group Databasebeta

← #28484202

Subgroups — #28484202

Order 15104.

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

normal subgroups only

This group has 538 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#21433137C211yesmaximal, characteristic
2#21426808C211yesmaximal, characteristic
3#21422490C212yesmaximal, complemented
4#21421924C212yesmaximal, complemented
5C59×#1460C211yesmaximal, characteristic
6C2×#14273714C411yescharacteristic
7C2×C4×Dic118C2211yescharacteristic
8#16837480—24no—
9#16837455C2212yes—
10#16837366—22no—
11#16837307C411yescharacteristic
12#16837305C411yescharacteristic
13#16837290—22no—
14#16837275—22no—
15#16837271C411yescharacteristic
16#16837248C2211yescharacteristic
17#16837132—24no—
18C2×C4×D4×C59C2211yescharacteristic
19C59×#431C2212yes—
20C59×#323—22no—
21C22×Dic118C2×C411yescharacteristic
22C2×#11745664—24no—
23#14273714—22no—
24#14273714—22no—
25C4×Dic118—22no—
26C4×Dic118—22no—
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—22no—
32C2×C4×Dic59—21no—
33C2×C4×Dic59C2×C411yescharacteristic
34C2×C4×Dic59C2×C411yescharacteristic
35C4⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—24no—
37C22⋊(C59⋊C8)—22no—
38C4×C59⋊C8D414yes—
39C59⋊#95—22no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C22×D4×C59C2×C411yescharacteristic
43C23×C4×C59D412yes—
44C4×D4×C59—22no—
45C4×D4×C59—22no—
46C2×C4⋊C4×C59C2×C411yescharacteristic
47C2×C22⋊C4×C59D412yes—
48C2×C42×C59C2311yescharacteristic
49C4⋊C8×C59—22no—
50C22⋊C8×C59—22no—
51C4×C8×C59—22no—
52C24×C59C22⋊C412yes—
53C2×D4×C59—22no—
54C2×D4×C59—21no—
55C2×D4×C59—21no—
56C2×D4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—82no—
60C22×C4×C59C22⋊C412yes—
61C22×C4×C59C22×C411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×Dic59—21no—
65C22×Dic59—21no—
66C22×Dic59—21no—
67C22×Dic59—21no—
68C2×Dic118—22no—
69C2×Dic118—22no—
70C2×Dic118—21no—
71C2×Dic118M1612yescomplemented
72C2×C8×C59—42no—
73C4⋊C4×C59—22no—
74C22⋊C4×C59—24no—
75C42×C59C2×D412yes—
76C42×C59#4112yes—
77C22⋊Dic59—28no—
78C22⋊Dic59—82no—
79C4⋊Dic59—22no—
80Dic59⋊C4—41no—
81Dic59⋊C4—41no—
82Dic59⋊C4—82no—
83C4×Dic59—21no—
84C4×Dic59—22no—
85C4×Dic59—42no—
86C4×Dic59M1612yescomplemented
87#11745664—84no—
88C2×C59⋊C8—24no—
89C23×C59—22no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59—42no—
93C23×C59—42no—
94C23×C59C2×C22⋊C411yescharacteristic
95D4×C59—42no—
96D4×C59—42no—
97C2×C4×C59—22no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—44no—
101C2×C4×C59—42no—
102C2×C4×C59—82no—
103C2×C4×C59C4×D412yes—
104C2×C4×C59#12011yescharacteristic
105C2×C4×C59#12112yes—
106C2×C4×C59#12311yescharacteristic
107C2×C4×C59C2×M1611yescharacteristic
108C2×C4×C59C2×M1611yescharacteristic
109C2×C4×C59#13112yes—
110C2×Dic59—22no—
111C2×Dic59—22no—
112C2×Dic59—41no—
113C2×Dic59—41no—
114C2×Dic59—41no—
115C2×Dic59—41no—
116C2×Dic59—42no—
117C2×Dic59—41no—
118C2×Dic59—41no—
119C2×Dic59—41no—
120C2×Dic59—41no—
121Dic118—24no—
122Dic118—42no—
123C8×C59—82no—
124C59⋊C8—44no—
125#9103—591nomaximal, Sylow
126C22×C59—21no—
127C22×C59—21no—
128C22×C59—44no—
129C22×C59—42no—
130C22×C59—42no—
131C22×C59—42no—
132C22×C59—82no—
133C22×C59#38611yescharacteristic
134C22×C59#40711yescharacteristic
135C22×C59#40811yescharacteristic
136C4×C59—22no—
137C4×C59—21no—
138C4×C59—21no—
139C4×C59—42no—
140C4×C59#43512yes—
141C4×C59#43612yes—
142Dic59—42no—
143Dic59—42no—
144Dic59—81no—
145Dic59—81no—
146#2975—591no—
147#2326—591no—
148#1479—592no—
149#1460D5911yescharacteristic
150#1274—592no—
151C2×C59—21no—
152C2×C59—21no—
153C2×C59—82no—
154C2×C59#187011yescharacteristic
155C2×C59#190911yescharacteristic
156C2×C59#191111yescharacteristic
157C2×#124—591no—
158C2×C4×Q8—591no—
159C2×C4×D4C2×D5911yescharacteristic
160#436—1184no—
161#431—592no—
162#431C2×D5912yes—
163#407—1182no—
164#391—591no—
165#390—591no—
166#388—1182no—
167#385—1182no—
168#384—591no—
169#376—591no—
170Q8⋊C8—1184no—
171#323—22no—
172C59#910311yescharacteristic, complemented, Sylow
173C22×Q8—591no—
174C22×D4C4×D5911yescharacteristic
175C23×C4D23612yes—
176C2×M16—1184no—
177#124—1182no—
178#124—1182no—
179C4×Q8—1182no—
180C4×Q8—1182no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—591no—
184C2×C4⋊C4—1182no—
185C2×C4⋊C4—1181no—
186C2×C4⋊C4C4×D5911yescharacteristic
187C2×C22⋊C4—1182no—
188C2×C22⋊C4—1182no—
189C2×C22⋊C4C22⋊D5912yes—
190C2×C42—591no—
191C2×C42—591no—
192C2×C42—1181no—
193C2×C42C22×D5911yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—1182no—
196C22⋊C8—22no—
197C22⋊C8—1184no—
198C22⋊C8—1182no—
199C4×C8—22no—
200C4×C8—594no—
201#95—1182no—
202#95—1181no—
203#95—1181no—
204C24(C2×D59)⋊C412yes—
205C2×Q8—592no—
206C2×Q8—1181no—
207C2×Q8—1182no—
208C2×Q8—1182no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—82no—
216C22×C4—1181no—
217C22×C4—1181no—
218C22×C4—1181no—
219C22×C4—1181no—
220C22×C4(C2×D59)⋊C412yes—
221C22×C4C2×C4×D5911yescharacteristic
222C22×C4C2×D23611yescharacteristic
223C22×C4C2×C22⋊D5911yescharacteristic
224M16—4724no—
225C2×C8—42no—
226C2×C8—1184no—
227C4⋊C4—22no—
228C4⋊C4—1182no—
229C4⋊C4—4722no—
230C4⋊C4—2361no—
231C4⋊C4—2361no—
232C22⋊C4—24no—
233C22⋊C4—1188no—
234C22⋊C4—4722no—
235C42—592no—
236C42—1181no—
237C42—1182no—
238C42—2362no—
239C42D4×D5912yes—
240C42#1174568712yes—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23C2×(C2×D59)⋊C411yescharacteristic
247Q8—1184no—
248Q8—2362no—
249D4—42no—
250D4—42no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—1182no—
258C2×C4—1182no—
259C2×C4—2361no—
260C2×C4—2362no—
261C2×C4—2361no—
262C2×C4—2361no—
263C2×C4—2361no—
264C2×C4—2361no—
265C2×C4—2361no—
266C2×C4—2361no—
267C2×C4—2361no—
268C2×C4(C22⋊D59)⋊C412yes—
269C2×C4#1427369711yescharacteristic
270C2×C4#1427369912yes—
271C2×C4#1427371111yescharacteristic
272C2×C4M16×D5911yescharacteristic
273C2×C4M16×D5911yescharacteristic
274C2×C4#1427373312yes—
275C8—82no—
276C8—2364no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—82no—
284C22#1683727911yescharacteristic
285C22#1683736511yescharacteristic
286C22#1683736811yescharacteristic
287C4—22no—
288C4—21no—
289C4—21no—
290C4—42no—
291C4—4721no—
292C4—4721no—
293C4—2362no—
294C4—2362no—
295C4#1683747612yes—
296C4#1683747712yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2142439111yescharacteristic
301C2#2142461611yescharacteristic
302C2#2142462611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1460C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#1460D59
hypocenterC59#9103
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #21422490C2semidirect
4. #21421924C2semidirect
71. C2×Dic118M16semidirect
86. C4×Dic59M16semidirect
172. C59#9103semidirect

Maximal subgroups

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

subgroupindex
#214331372
#214268082
#214224902
#214219242
C59×#14602
#910359